diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzneag" "b/data_all_eng_slimpj/shuffled/split2/finalzzneag" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzneag" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\n\\label{intro}\n\nTexts are hierarchic constructs which consist of several autonomous\nlevels \\cite{hutchins,valgina,hasan}: letters, words, phrases, clauses,\nsentences, paragraphs. If a text is looked at statistically, i.e.\nwithout understanding its meaning (e.g. because it is written in an\nunknown system), how can it be efficiently distinguished from a\nmeaningless collection of words \\cite{baa,orlov,arapov}? Several such\ndistinctions are well-known, e.g. {\\it (i)} meaningful texts have a\nlarge number of words that appear in a text only few times, in\nparticular once (rare words or {\\it hapax legomena}) \\cite{baa}. {\\it\n(ii)} Ranked frequencies of words obey the Zipf's law\n\\cite{estoup,condon,zipf}. {\\it (iii)} Letters and words of a text\ndemonstrate long-range (power-law) correlations\n\\cite{lrc_schenkel,lrc_shnerb,lrc_ebeling,lrc_eckmann,lrc_manin,lrc_altmann}. \n\nHowever, these characteristics can be reproduced by a sufficiently\nsimple stochastic models putting in doubt their direct relation to the\nmeaningfulness of a text. {\\it (i)} Simple stochastic models can\nrecover quite precisely the detailed structure of the hapax legomena\n\\cite{pre}. {\\it (ii)} Zipf's law can be deduced from statistical\napproaches\n\\cite{shrejder,li,simon,zane,kanter,hill,pre,liu,baek,vakarin,dover,latham,mandelbrot,mandel,manin}. The\nfirst model of this type was a random text, where words are generated\nthrough random combinations of letters, i.e. the most primitive\nstochastic process \\cite{mandelbrot,li}. Its drawbacks\n\\cite{howes,seb,cancho} (e.g. many words having the same frequency) are\navoided by more refined models \\cite{simon,zane,kanter,hill,pre}. \nMore generally, it was recently understood that Zipf's law is a statistical regularity\nthat emerges in samples which are informative about the underlying generative process \\cite{cubero}. \n{\\it (iii)} Physics and mathematics of stochastic processes offer a plethora\nof models and approaches for generating long-range correlations\n\\cite{buck} \\footnote{E.g. Ref.~\\cite{lrc_schenkel} points out that\nlong-range correlations are found also in a dictionary, where the\nmeaning of text (as opposed to the meaning of words and phrases) is\nabsent. Ref.~\\cite{lrc_manin} also argued against a direct relation between\nlong-range correlations and semantic structures.}. \n\nHere we contribute to resolving the above question by recalling that\nmeaningful texts evolve sequentially (linearly) from beginning to end.\nThis was taken as one of basic features of language \\cite{sure},\nwhich|together with other design features|allows to distinguish human\nlanguage from other communication systems \\cite{hockett}. Thus we divide\ntexts into two halves, each one containing the same amount of words.\nThereby we neutralize confound variables that are involved in a complex\ntext-producing process (style, genre, subject, the author's motives and\nvocabulary {\\it etc}), because they are the same in both halves. Hence\nby comparing the two halves with each other we hope to see statistical\nregularities that are normally shielded by above variables. Statistical\nregularities hold for the majority of texts, such that it is highly\nunlikely to get this majority for random reasons (as checked by the\n$3\\sigma$ rule). The significance of results will be checked by\nwell-accepted statistical tests; for our purpose this is the W-test\n(Wilcoxon's test) \\cite{wilcoxon}. \n\nIn two sets of several hundred texts we noted the following statistical\nregularities. {\\it (1)} The first half has a larger number of different\nwords, i.e. a larger vocabulary. {\\it (2)} It also has a larger number\nof rare words, i.e. words that appear once or twice. {\\it (3)} The\nfirst half is less compressible than the second half. The\ncompressibility was studied via several different standard approaches,\ne.g. the Lempel-Ziv complexity and the zip algorithm. Lesser\ncompressibility relates to more information in the sense of Shannon\n\\cite{cover}. {\\it (4)} Common words of both halves tend to have a\nlarger overall frequency in the second half. These four features\nsignificantly correlate with each other as quantified by Pearson's\ncorrelation coefficient. {\\it (5)} The words in the first half are\ndistribued less homogeneously, since they have a larger difference\nbetween the frequency and (inverse) spatial period. \n\nOne possible explanation of these result is that the first part of the\ntext normally contains the exposition (which sometimes can be up to 20\n\\% of the text), where the background information about events,\nsettings, and characters is introduced to readers. The first part also\nplots the main conflict (open issue), whose denouement (solution) comes\nin the second half \\footnote{\\label{foo1}Scientific texts contain\nclosely related aspects: introduction, critique of existing approaches,\nstatement of the problem, resolution of the problem, implications of the\nresolution {\\it etc}. The discussion on differences between the halves\napplies also here.}. Hence {\\it (1)-(4)} can be hypothetically explained\nby the fact that the exposition|hence the first half|needs more\ndifferent words {\\it (1)}, more rare words {\\it (2)}, is more\ninformative (in the sense of Shannon) {\\it (3)}, and introduces words that\nare employed in the second half {\\it (4, 5)} (i.e. the second half\nprocesses information introduced in the first half). We emphasize that\nthis explanation is hypothetical, its direct validity is yet to be\nchecked via more refined methods to be developed in future. \n\n{\\it (6)} Many other features|in particular those related to\nhigher-order hierarchic structures of the text\n\\cite{hutchins,valgina,hasan}|do not show any significant difference\nbetween the first and second half: number of sentences, paragraphs,\nrepetitiveness of words (as quantified by Yule's constant), number of\npunctuation signs {\\it etc}. Among such features we especially mention\nthe overall number of letters, since there is a weak statistical evidence\n(the W-test is not always passed) that this quantity is still larger in\nthe first half. \n\nChecking these features does not require any understanding of the text,\ni.e. it is not required that the meaning of words is understood, or\neven their writing system is known. We show that (expectedly) neither\nof them survives if the words of the text are randomly permuted, and only\nafter that the resulting ``text\" is divided into two parts. Hence these\nfeatures are specific for meaningful texts and they can be employed for\ndistinguishing meaningful texts from a random collection of words. \n\n\\comment{Moreover, they do no demand any text parsing, beyond\nconservation of its natural (linear) order. Not even the parsing into\nwords is necessary, since this can be done by detecting the space as the\nmost frequent symbol in the text. }\n\nThis paper is organized as follows. The next section reviews our data\ncollection method and recalls the W-test. Sections \\ref{I}--\\ref{spato}\npresent results from the above points {\\it (1)-(5)}.\nSection \\ref{nego} reviews negative results from point {\\it (6)}. \nThe last section contains the outlook of this research\nand relates it with existing literatures. Our main results\nare briefly summarized in Table~\\ref{tab0}. All other tables are given in\nAppendix \\ref{tablo}. \n\n\n\\begin{table}\n\\centering\n\\begin{tabular}{l|c|c} \\hline\\hline\n& First half & Second half \\\\\n\\hline\nNumber of different words; cf.~(\\ref{udosh}) & + & -- \\\\\n\\hline\nNumber of rare words (absolute and relative); cf.~(\\ref{cov}, \\ref{boris}) & + & -- \\\\\n\\hline\nCompressibility of the size; cf.~(\\ref{compo}, \\ref{sss}) & -- & + \\\\\n\\hline\nThe overall frequency of common words; cf.~(\\ref{cc}) & -- & + \\\\\n\\hline\nDifference between frequency and inverse spatial period; cf.~(\\ref{ort}) & + & -- \\\\\n\\hline\\hline\nNumber of letters; cf.~(\\ref{guppi}) & + & -- \\\\\n\\hline\\hline\nRepetitiveness of words (Yule's constant); see (\\ref{yule}) & $\\emptyset $ & $\\emptyset $ \\\\\n\\hline\nNumber of punctuation signs & $\\emptyset $ & $\\emptyset $ \\\\\n\\hline\nAverage length of words & $\\emptyset $ & $\\emptyset $ \\\\\n\\hline\nNumber of sentences & $\\emptyset $ & $\\emptyset $ \\\\\n\\hline\nAverage length of sentence & $\\emptyset $ & $\\emptyset $ \\\\\n\\hline\nNumber of paragraphs & $\\emptyset $ & $\\emptyset $ \\\\\n\\hline\nSize in bytes & $\\emptyset $ & $\\emptyset $ \\\\\n\\hline\n\\hline\n\\end{tabular}\n\\caption{\\label{tab0}Qualitative comparison of\nvarious features of texts between first and second halves: + (--) means\nthat the feature is larger (smaller) in the corresponding half.\n$\\emptyset $ means that the sought difference does not show up. \nFeatures are divided into two\ngroups by double-lines. The first four features correlate with each other.\nThe number of letters is separated, \nsince there is a weak evidence toward its validity (the \nvalues of test quantities are close to their threshold values). }\n\\end{table}\n\n\n\n\n\\section{Data collection and testing}\n\n\\subsection{Studied texts}\n\nWe selected English texts with a single narrative that are written on relatively few tightly\nconnected subjects, and are sufficiently short for not containing ``texts\ninside of texts'' \\footnote{To ilustrate this point, consider ``War and Peace'' by\nLeo Tolstoy. This big novel is written in two different languages\n(Russian and French), and contains a big amount of heroes and\ncircumstances. It does have several parallel narratives, and describes\nsituations in the course of twenty years. Clearly, this is a text of\ntexts, and it would not be meaningful to focus on dividing it over two\nhalves. But we can divide over two halves one of its (long) chapters. We\nhave not done this so far. }.\nWe divided studied texts into two halves (each half \ncontains equal number of words) \\footnote{We got a\npreliminary evidence that, as expected, dividing texts into more than two\nparts will obscure the text difference effects shown in Table I. Note that\nRef.~\\cite{zano} studies text division into several parts, but that was\ndone for a different purposes. For long texts containing\n$155000-220000$ words, Ref.~\\cite{zano} noted that such texts can be divided into several\nsub-texts of the size of $1000-3000$ words. The criterion of separation\nis qualitatively close to the above concept of ``books inside\nof a book\", because it looked at the spatial clustering of key-words.\nI.e. a group of key-words appearing mostly in one part of the text and\nnot in others, will effectively define a sub-text.} along the flow of the narrative, i.e. from\nthe beginning to end. Several aspects of texts are left unchanged: each half\nis sufficiently large for statistics to apply, they\nhave the same overall number of words, the same author, genre {\\it etc}.\nThe halves are semantically different, since the first half can be understood\nwithout the second half, but the second half generally cannot be\nunderstood alone. Also, the structure of narrative is different: the\nfirst half normally contains the exposition, where actors, situations\nand conflicts are set and defined, while the second half normally\ncontains the denouement; cf.~Footnote~\\ref{foo1}. \n\nWe have chosen to work with two datasets \\cite{online}. The first dataset was taken\nfrom the Gutenberg project at \\cite{gutenberg}. It\nconsists of 156 fiction novels; for each novel the overall number of\nwords is in the range $ [10000, 50000]$, which is sufficiently large for\nstatistics to apply, but sufficiently short to ensure that they do not\namount to ``books inside of a book''. This range of the overall number\nof words $ [10000, 50000]$ is motivated from our own experience as\nreaders. \n\nThe texts within the first dataset are thematically close, since they\nare all fiction novels. The second dataset consists of 350 thematically\ndiverse texts taken from various online sources and collected at\n\\cite{ting}. When collecting those texts we tried to ensure that they\ndo not contain texts that are meaningless to divide into halves, i.e.\nthat they do not contain effectively independent narratives. Hence we\ndid not include in this dataset biographies, poems, collections of short\nstories or essays (in particular, folk stories), lectures, proceedings,\nletters. \n\n\\subsection{Testing the difference between the halves}\n\nAfter processing, the typical form of our data are pairs of numbers for\neach text: $\\{x_{1,i}, x_{2,i}\\}_{i=1}^M$, where $x_{1,i}$ and $x_{2,i}$\nare certain features of (resp.) the first and second half of a text $i$,\nwith $M$ being the overall number of texts in the dataset. E.g. $x_{1,i}$ and $x_{2,i}$\nare the number of different words for (resp.) first and second halves of a text; see below.\n\nTo inquire on\nwhether this data indicate on a difference between two halves, we\nformulate two natural hypotheses: ${\\cal H}_1$ (${\\cal H}_0$) means that\nthe difference $x_{1,i}-x_{2,i}$ does (does not) follow a symmetric\ndistrbution around the zero. Now some understanding on excluding ${\\cal H}_0$ can be gained\nby looking at the percentage of cases, where $x_{1,i}30$ suffices, which always holds in our cases) the law of large numbers works and\n$W$ is a Gaussian random variable, since it is a weigted sum of a large number of uncorrelated random variables. \nIts average is zero, since ${\\rm sgn}[x_{2,i}-x_{i,i}]$ assume values $\\pm 1$ with equal probability (once \n${\\cal H}_0$ is assumed to hold). Its dispersion is calculated directly from (\\ref{gnu2}) \\cite{wilcoxon}:\n\\begin{eqnarray}\n\\label{sigma}\n\\sigma_W^2(M)=\\langle W^2\\rangle=\\sum_{k=1}^M k^2=\\frac{M(M+1)(2M+1)}{6}.\n\\end{eqnarray}\nHence ${\\cal H}_0$ can be excluded via the $3\\sigma$ rule, if \n\\begin{eqnarray}\n\\label{3sigma}\n|W|>3\\sigma_W (M).\n\\end{eqnarray}\nWe accept the $3\\sigma$ rule (\\ref{3sigma}) as the minimal threshold for claiming the statistical\nsignificance of our results. However, we emphasize that the absolute majority of\nour results hold the much stronger $5\\sigma$ rule; see tables in Appendix \\ref{tablo}. \n\n\\section{Words: different, rare, common}\n\\label{I}\n\n\\subsection{Different words (vocabulary)}\n\nThe basic hierarchic level of text is that of words. Neglecting phenomena\nof synonymy and homonymy (which are rare in English, but not at all rare\ne.g. in Chinese \\cite{epjb}), we can say that every word has several\nclosely related meanings (polysemy). Neglecting also the difference\nbetween polysemic meanings, the number of independent meanings in a text\ncan be estimated via the number of different words $n$. Tables\n\\ref{n_156} and \\ref{n_350} show that the first half of a meaningful\ntext has statistically more different words than the second half:\n\\begin{eqnarray}\n\\label{udosh}\nn_1>n_2\n\\end{eqnarray}\nAs expected, this result disappears after random shuffling (random\npermutation of words) of texts that destroys its linear structure; see\nTables \\ref{shuffle_156} and \\ref{shuffle_350}. \n\n\\subsection{Rare words (hapax legomena)}\n\nIn any meaningful text, a sizable number of words appear only very\nfew times ({\\it hapax legomena}). These rare words amount to a finite\nfraction of $n$ (i.e. the number of different words). The existence and\nthe (large) number of rare events is not peculiar for texts, since there\nare statistical distributions that can generate samples with a large\nnumber of rare events \\cite{baa,pre}. One reason\nwhy many rare words should appear in a meaningful text is that a typical\nsentence contains functional words (which come from a small pool), but\nit also has to contain some rare words, which then necessarily have to\ncome from a large pool \\cite{latham} \\footnote{\\label{lato}E.g. this sentence\ncontains rare words {\\it typical} and {\\it pool} that in the present text\nare met only 3 and 2 times, respectively. It also contains frequent\nwords {\\it words}, {\\it since}, {\\it large}.}. \n\n\\comment{\\begin{eqnarray}\n\\label{kusho}\n\\sum_{m=1}^k V_m^{[1]} \\geq \\sum_{m=1}^k V_m^{[2]}, \\qquad k=1,...,5,\n\\end{eqnarray}\nwhere $V_m^{[1]}$ ($V_m^{[2]}$) is the number of words that appear $m$ times in the first (second) half.\nFor the halves (\\ref{kush}) is written as \n\\begin{eqnarray}\n\\label{kusho1}\n&&{\\sum}_{m=1}^{f^{[\\ell]}_1 N\/2}\\, V^{[\\ell]}_m=n_\\ell, \\\\\n&&{\\sum}_{m=1}^{f^{[\\ell ]}_1N\/2} \\, mV^{[\\ell]}_m=N\/2,\n\\qquad \\ell=1,2,\n\\label{kusho2}\n\\end{eqnarray}\nwhere $n_1$ ($n_2$) is the number of different words in the first (second) half, and $f^{[1]}_1$ ($f^{[2]}_1$)\nis the frequency of the most frequent word in the first (second) half. \n}\n\nLet $V_m^{[1]}$ ($V_m^{[2]}$) is the number of words that appear $m$ times in the first (second) half.\nFor defining rare words we focused on \n\\begin{eqnarray}\n\\label{hh}\nh_\\ell(\\kappa) \\equiv \\sum_{m=1}^\\kappa V_m^{[\\ell]}\\qquad \\ell=1,2,\\qquad \\kappa=1,...,5,\n\\end{eqnarray}\ni.e. on words that appear up to $\\kappa$ times. We choose to work with different $\\kappa$'s to ensure that\nour results are robust with respect to varying the definition of ``rare''.\nFor both datasets we observed that in the majority of cases the number\nof rare words in the first half is larger than the number of rare words \nin the second half [see Tables~\\ref{rare_156} and \\ref{rare_350}]:\n\\begin{eqnarray}\n\\label{cov}\nh_1(\\kappa)>h_2(\\kappa),\\qquad \\kappa=1,...,5.\n\\end{eqnarray}\nWe confirmed via the $5\\sigma$ of the W-test that the probability to get (\\ref{cov}) due to random reasons is\nnegligible. \n\nEq.~(\\ref{cov}) suggests that the first half uses more rare words, but\nsuch a conclusion is incomplete, since the two halves have different\nnumbers of distinct words. Denote them as $n_1$ and $n_2$, for the first\nand second half respectively; cf.~(\\ref{udosh}). \nNote that\n\\begin{eqnarray}\n\\label{kusho1}\n{\\sum}_{m=1}^{f^{[\\ell]}_1 N\/2}\\, V^{[\\ell]}_m=n_\\ell, \n\\qquad \\ell=1,2,\n\\label{kusho2}\n\\end{eqnarray}\nwhere $f^{[1]}_1$ ($f^{[2]}_1$)\nis the frequency of the most frequent word in the first (second) half. \nHence in addition to (\\ref{cov}) it is necessary to consider normalized\nquantities, i.e.\n\\begin{eqnarray}\n\\label{boris} \nh_1(\\kappa)\/n_1>h_2(\\kappa)\/n_2, \\qquad \\kappa=1,...,5.\n\\end{eqnarray}\nRelation (\\ref{boris}) does hold statistically; see\nTable~\\ref{rare_156} for the first dataset and Table~\\ref{rare_350} for the\nsecond dataset. \n\nHence the first half has more rare words both in absolute and relative\nterms; see (\\ref{udosh}, \\ref{boris}). These differences between the\nhalves disappear after random shuffling of texts; see Tables\n\\ref{shuffle_156} and \\ref{shuffle_350}. \n\nNote that yet another possibility to define rare words comes from relations\n\\begin{eqnarray}\n\\label{kus}\n{\\sum}_{m=1}^{f^{[\\ell ]}_1N\/2} \\, mV^{[\\ell]}_m=N\/2,\n\\qquad \\ell=1,2.\n\\end{eqnarray}\nEq.~(\\ref{kus}) invites to compare the normalized quantities\n$\\frac{2}{N}\\sum_{m=1}^\\kappa m V_m^{[1]}$ with $\\frac{2}{N}\\sum_{m=1}^\\kappa m V_m^{[2]}$ for $\\kappa=1,...,5$.\nWe carried out this comparison and the results (for percentages and $W$-values) are very similar to (\\ref{cov}), i.e.\nwe obtain\n\\begin{eqnarray}\n\\label{kookoo1}\n\\sum_{m=1}^\\kappa m V_m^{[1]} > \\sum_{m=1}^\\kappa m V_m^{[2]}, \\qquad \\kappa=1,...,5,\n\\end{eqnarray}\nin the same statistical sense as (\\ref{cov}).\n\n\n\\subsection{Common words}\n\nBoth halves of a text have certain common words, e.g. non-common words of the \nsecond half are those that are not met in the first half. Let the number of \ncommon words in a given text is denoted by $C$. Our first result is that for each half\nthe common words are less numerous than non-common ones:\n\\begin{eqnarray}\n\\label{urartu}\nC1\/2,\\qquad c_2>1\/2.\n\\end{eqnarray}\nRelations (\\ref{van}) hold without exclusions for all text we studied.\nInequalities (\\ref{urartu}, \\ref{van}) are well expected, since common\nwords include functional words, which are frequent, but not numerous\n\\cite{pre}. \n\nOur next finding does indicate on a difference between\ntwo halves, and is therefore less expected: the overall frequency of common\nwords is larger in the second half [see Tables \\ref{n_156} and\n\\ref{n_350}]:\n\\begin{eqnarray}\n\\label{cc}\nc_1s_{\\rm inverted}.\n\\end{eqnarray}\nThe percentage of (\\ref{gel}) is remarkably high: it holds for $>97\\%$\ncases in both of our datasets; see Table \\ref{inversion}. \nEq.~(\\ref{gel}) is confirmed via the zip \ncompression method; see Table \\ref{inversion}. \n\nRecall that previous applications of the LZ-complexity in texts\n\\cite{lande,debowski1,debowski2} assumed that the LZ-complexity captures\ncorrelations between different text symbols (letters, words {\\it etc}).\nRelation (\\ref{gel}) can be explained by noting that $C_{\\rm LZ}$ focuses on\nshort range correlations of letters, which may be lost after inversion\nof words. To illustrate this point, let $\\ell_1\\ell_2\\ell_3 $ and\n$\\kappa_1\\kappa_2\\kappa_3 $ are two consecutive 3-letter words. Their\norder in ${\\rm T}$ [in ${\\rm T}_{\\rm inverted}$] is\n$\\ell_1\\ell_2\\ell_3\\,\\,\\kappa_1\\kappa_2\\kappa_3 $\n[$\\kappa_1\\kappa_2\\kappa_3\\,\\,\\ell_1\\ell_2\\ell_3$]. Now if there are\ncorrelations between $\\ell_3$ and $\\kappa_1$ in ${\\rm T}$, and such\ncorrelations are accounted for in $C_{\\rm LZ}$, then in ${\\rm T}_{\\rm\ninverted}$ such correlations will have a longer range, and will not be\nseen in $C_{\\rm LZ}$ \\footnote{\\label{syntagma}Work is in progress to\nunderstand whether such directional correlations are related to\nsyntagmatic correlations between words of the text well-known \nin linguistics \\cite{sure,sahlgren}. Qualitatively, these are correlations\nalong the text determined by co-occurence of words or letters\n\\cite{sure,sahlgren}. They are conceptually different from paradigmatic\nrelations between the words, where two words having such a relation tend to\nappear in the same context, i.e. in the same surrounding of words.}. \n\nNote that instead of inverting texts at the level of words we also\ninverted them at the level of letters: put the last letter as the first\none {\\it etc}. We saw that out of this letter inversion the\ncompressibility does change, i.e. there is more into the LZ-complexity\nthan just short-range correlations of letters. However, no clear\nindications emerged on the analogue of (\\ref{gel}) or on its inverse.\nThe results differ from one dataset to another and from one compression\nmethod to another. \n\n \n\\comment{\nRecall how the lossless compression methods (including zip)\nwork \\cite{cover}: they look for various repeating patterns (within some\nlocal window of consecutive symbols) and code more frequent patterns by\nshorter codewords. Hence sequences that contain more repeating patters\nare more compressible. }\n\n\\subsection{Compressibility of two halves}\n\nTable \\ref{zip} shows that the relative compressibility of the\nfirst half is statistically smaller, i.e. it is compressed less than the second half:\n\\begin{eqnarray}\n\\label{sss}\ns_1\\mu_2,\n\\end{eqnarray}\nand that this effect gets stronger|both in terms of the percentage of\ncases and the value of $W$ in (\\ref{gnu2}), when the the set $\\Omega$ in\n(\\ref{ort}) is restricted to common words of both halves; see Tables\n\\ref{mu_156} and \\ref{mu_350}. \n\n\\section{Features that do not show statistically significant\ndifferences between two halves}\n\\label{nego}\n\nSo far we mostly concentrated on one level of textual hierarchy, i.e.\nwords. Letters are on the hierarchy level below that of words. For the\ntotal number of letters $L$ in each half the statistical evidence we got\nis weaker, since the percentage of cases, where $L_1>L_2$ (i.e. the\nfirst half has a larger overall number of letters than the second half)\nand the W-statistics for $L_1>L_2$ are close to their critical values;\nsee Tables \\ref{n_156} and \\ref{n_350}. Moreover, in one of our datasets\nthe W-test is passed, while in the other it is not. However, there is a\nweak, but a definite evidence for the validity of $L_1>L_2$. First, after\na random shuffling of texts, the percentage of cases where $L_1>L_2$ holds\ndrops down from its value $\\simeq 0.58$ (for original texts) to $\\simeq 0.5$\nfor shuffled texts; see Tables \\ref{shuffle_156} and \\ref{shuffle_350}. Second, \n$L_1>L_2$ shows significant correlations with (\\ref{udosh}) and (\\ref{cov});\nsee Tables \\ref{corr_156} and \\ref{corr_350}. Third, the relation $L_1>L_2$\nholds in average for both datasets: \n\\begin{eqnarray}\n\\label{guppi}\n\\frac{1}{156}\\sum_{k=1}^{156} (L_1^{[k]}-L_2^{[k]})= 146.26,\\qquad\n\\frac{1}{350}\\sum_{k=1}^{350} (L_1^{[k]}-L_2^{[k]})= 340.97.\n\\end{eqnarray}\n\n\\comment{Further distinction between words of the text can be made via\ntheir average length (in letters): content words|which express specific\nmeaning|are normally longer than functional words that mostly serve for\nestablishing grammatic connections \\cite{zipf}. }\n\nFor hierarchy levels higher than that of words, our results are\nnegative, i.e. they do not indicate on a statistically significant\ndifference between the halves. The number of sentences $\\sigma$ does not\nshow significant differences; see Tables \\ref{n_156} and \\ref{n_350}.\nHere we (conventionally) defined the sentence as the shortest sequence\nof words located in between of any of the following symbols: comma, dot,\nsemicolon, question mark, exclamation mark. We also studied the number\nof sentences, when the comma is excluded from the above list (not shown\nin tables). This did not change our conclusion. \n\nWe also calculated the full distribution of sentences over the\nlength (measured in words): $\\kappa_\\alpha$ the fraction of sentences with\nword-length $\\alpha$ ($\\sum_\\alpha\\kappa_\\alpha=1$). Two specific\ncharacteristics of this distribution were looked at: the average\n$\\overline{\\alpha}$, dispersion $\\overline{\\Delta(\\alpha^2)}$ and\nentropy $\\varepsilon$:\n\\begin{gather}\n\\label{deviation}\n\\overline{\\alpha}={\\sum}_\\alpha\\kappa_\\alpha\\alpha, \\qquad\n\\overline{\\Delta(\\alpha^2)}\n={\\sum}_\\alpha\\kappa_\\alpha\n(\\alpha-\\overline{\\alpha})^2.\n\\end{gather}\nNone of these quantities shown a statistically significant difference\nbetween the halves; see Tables \\ref{n_156} and \\ref{n_350}. Another\nlevel of the textual hierarchy is the one containing paragraphs.\nDenoting the number of paragraphs as $\\rho$, we saw that there is no\nstatistical evidence in favor of $\\rho_1>\\rho_2$ or $\\rho_1<\\rho_2$; see\nTables \\ref{n_156} and \\ref{n_350}. Our results on Yule's constant that\ndescribes the repetitiveness of words (see Appendix \\ref{yu} details of\nthe definition) also do not indicate on a significant difference between\nthe halves. \n\n\\section{Outlook}\n\nWe proposed a set of relations between statistical features of the two\nhalves of a meaningful text; see Table~\\ref{tab0} for a summary of our\nresults. The validity of these relations is statistical, i.e. the\nmajority of them holds with $5\\sigma$ significance of the Wilcoxon test;\nsee Appendix \\ref{tablo}. No understanding of the text (or even knowing\nits writing system) is needed for checking these relations. We\nexplicitly confirmed that all these relations disappear after a random\npermutation of words in the text. \n\nWe conjecture that these relations between the halves are connected to a\nspecific, information-carrying structure of the text, where the\ninformation is introduced (defined) in the first half, and then is processed\nin the second half. Such a structure is anticipated in text linguistics,\nwhere the flow of the text narrative is conventionally separated between\nthe exposition and the denouement, which are typically located in the\nfirst and second halves, respectively \\cite{hutchins,valgina,hasan}.\nThis is however a qualitative concept, and hence the connection between\nour results and the exposition-denouement is stated as a\nhypothesis. Work is currently in progress for designing specific tests\nfor checking the hypothesis. \n\nPractically, knowing whether a string of symbols is a meaningful text or\nnot can be useful in cryptography, fraud detection and historical\nanalysis. The latter can refer to inferring whether a given text in\nunknown writing system is meaningful or asemic \\cite{writing_systems}.\nOne interesting application relates to the CETI problem (communication\nwith extra-terrestial intelligence) \\cite{ceti}. Here the code of a\npotential signal is completely unknown, but it can be plausibly\nconjectured that a meaningful signal has similar differences between the\nhalves. (Fractioning into ``words'' is is possible, once the ``space\nsymbol'' is identified as the most frequent symbol in the text\n\\cite{ceti}.)\n\nSome of our results were sporadically observed in literatures.\nRef.~\\cite{minn} emphasized the translation invariance of books, but\nstill noted on a concrete text that its last part has less rare words.\nRef.~\\cite{lrc_manin} noted the following (non-topical) differences\nbetween the first and the second halves of {\\it Moby Dick} (by H.\nMelville): {\\it (1.)} The word {\\it is} is more frequent than {\\it was}\nin the first half, but less frequent in the second half. {\\it (2.)} The\nratio of articles {\\it the} to {\\it a} is larger in the second half,\nwhich may mean that the second half makes more concrete statements.\n\n\\comment{for PRE version:\nAfter the initial version of this paper, one of its unknown referees\ninformed us that he\/she counted the number of rare words in the\nfirst\/second halves of {\\it all} of Project Gutenberg books that have\nbetween 20000 and 40000 words in total (within the lengths of the texts\nchosen by us). For a total of 10364 books, 5649 (54 \\%) had more rare\nwords in the first half compared to the second half. We note that this\nresult has a statistical validity higher than $5\\sigma$, since $0.5 +\n5\/(2 \\sqrt{10364})\\approx 0.5246$. For our situation the corresponding\npercentages of rare words differences are larger by some 10-15 \\%, see\nTables \\ref{rare_156} and \\ref{rare_350}, because we excluded from\nconsideration all those books that are {\\it a priori} meaningless to\ndivide into two halves (hence such texts contribute into the noise):\nbiographies, poems, collections of short stories or essays (in\nparticular, folk stories), lectures, proceedings, letters. }\n\nOur results concerning the compressibility features|in particular, the\nresult in section \\ref{invo} on the compressibility decrease under\ninverting the word order|are especially worth studying in more detail.\nWhile we focused on the Lempel-Ziv complexity and the related zip method\nfor defining the compressibility, it is known that the Lempel-Ziv complexity\nfor long but finite sequences has a drawback of not capturing the real\nrandomness, i.e. not agreeing with the Kolmogorov complexity\n\\cite{cuba}. Hence more refined compression methods are to be studied in\nfuture, e.g. the Huffman coding that is algorithmically slower, but does\ncapture the notion of randomness. It is also important to clarify whether\nthe compressibility difference between the original and inverted text \ncan serve for quantifying syntagmatic correlations between the words; \ncf.~Footnote \\ref{syntagma}.\n\nAnother important open problem relates to modeling the above effects. A\nsuperficial modeling would be possible via altering the existing\nsequential text-generating models (see \\cite{simon,zane,kanter} for\nexample) such that e.g. they generate less rare words towards the end of\nthe text. But we should warn the reader against such quick attempts.\nFirst, the main drawback of sequential models is that they do not\ndescribe sufficiently well the distribution of rare words \\cite{minn},\nwhich was so far possible only via non-sequential statistical models\n\\cite{pre}. So a good model should predicts {\\it both} the distribution\nof rare words and their difference between the halves. Second, the\nexample of the Zipf law|with its numerous models and explanations\n\\cite{shrejder,li,simon,zane,kanter,hill,pre,liu,baek,vakarin,dover,latham,mandelbrot,mandel,manin}|shows\nthat excessive modeling even with a reasonable quantitative agreement is\nnot at all a guarantee for understanding the actual meaning of a complex\ntextual phenomenon. Hence at the present stage of our understanding we\nwant to concentrate on conceptual issues (and novel tests) relating\nstatistics to meaningfulness, more than to develop a purely statistical\nmodel for explaining above results. \n\n\\section*{Acknowledgements}\n\nW. Deng was partially supported by the Fundamental Research Funds for\nthe Central Universities, the Program of Introducing Talents of\nDiscipline to Universities under grant no. B08033, and National Natural\nScience Foundation of China (Grant Nos. 11505071 and 11905163). A.E. Allahverdyan was\nsupported by SCS of Armenia, grants No. 18RF-015 and No. 18T-1C090. \n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction\\label{sec:intro}}\n\nThe color-magnitude diagram is a powerful diagnostic of galaxy\nevolution and formation. The presence, as early as $z \\sim 1.5$, of a\nprominent and low-scatter, `red-sequence' (RS) in galaxy clusters\nplaces useful constraints on the possible evolutionary pathways in\ngalaxy color and luminosity \\citep*{Mullisetal05, Stanfordetal05,\n Stanfordetal06, Belletal04, Faberetal07}. The red colors of the\nprimarily early-type RS galaxies are due to the observed filters\nspanning the 4000\\AA\\ spectral break. The universality and prominence\nof the RS in appropriately chosen filters have been used to discover\nhigh-redshift clusters \\citep*[e.g.,][]{RCS}. Moreover, the defining\ncharacteristic of galaxy clusters, i.e., the large numbers of galaxies\nall at the same redshift, allows the slope and intrinsic scatter of\nthe RS to be measured with great precision. Based on studies of\ngalaxy clusters at $z < 1.3$, the slope of the RS does not appear to\nevolve and therefore is more likely the by-product of the\nmass-metallicity relation as observed in local galaxy samples\n\\citep*[e.g.,][]{Tremontietal04} rather than the result of a mass-age\ntrend. The scatter, however, is likely due to the fractional age\ndifferences between the RS galaxies \\citep*[e.g.,][]{Blakesleeetal03}.\nBy constructing a set of model galaxies with different star-formation\nhistories and timescales it is possible to fit for the mean epoch of\nlast significant star-formation by matching the intrinsic scatter of\nthe RS. Such studies at $z \\sim 1$ have derived formation redshifts\nof $z_{\\rm form} \\sim 2.0 - 2.5$ \\citep*[e.g.,][]{HalfHubble06,\n vanDokkumvanderMarel07a}. At redshifts beyond $z \\sim 1.5$,\nhowever, the 4000\\AA-break moves into the near-infrared and galaxy\nclusters, and therefore the RS, have not been observed closer to the\ninferred epoch of formation for early-type galaxies. Hence, to\nuncover the younger or forming red-sequence at higher redshifts\nrequires deep near-infrared imaging of suspected (or, preferably,\nconfirmed) protocluster fields.\n\n\\begin{figure*}[t]\n\\plotone{f1.ps}\n\\caption{$J_{110} - H_{160}$ vs. $H_{160}$ color-magnitude diagram for\n the MRC 1138-262 NICMOS field (open black circles). The large\n yellow star is the radio galaxy itself. The blue background points\n are from the NICMOS data of the Ultra Deep Field and Hubble Deep\n Field North. The deep fields cover 2.5x the area of our\n observations. Also shown are the spectroscopically confirmed\n $H\\alpha$ (orange filled circles) and Lyman-$\\alpha$ (green filled\n circles) emitters. The three dot-dash lines show where the CMRs of\n lower redshift clusters would lie under different assumptions. The\n top line is the Coma cluster with no evolution, simply redshifted to\n $z=2.16$ and observed through the NICMOS filters. The next line\n down is the same but for the $z=1.24$ cluster RDCS1252. Finally, if\n we passively de-evolve RDCS1252 to redshift $z=2.16$ (almost exactly\n two Gyr), assuming a median age for the 1252 galaxies of about 3 Gyr\n (or $z_{\\rm form} \\sim 3$), we get the third line.\\label{fig:CMD}}\n\\end{figure*}\n\nWe have undertaken a NICMOS imaging program to study the red galaxy\npopulation in a protocluster at $z=2.16$. Broad and narrow-band\nimaging, both in the optical and near-infrared, of the field\nsurrounding the powerful radio galaxy MRC 1138-262 ($z=2.16$) have\nidentified more than 100 candidate companion galaxies. This target\nserved as the proof-of-concept for the successful VLT Large Program\nsummarized in Venemans et al. (2007)\\nocite{Venemansetal07}. There\nare surface-overdensities of both line-emitting candidates\n(Lyman-$\\alpha$ and H$\\alpha$), X-ray point sources, sub-mm selected\ngalaxies and red optical--near-infrared galaxies\n\\citep*{Pentericcietal02,KurkPhD,Kurketal04,Kurketal04b,Croftetal05,Stevensetal03}.\nFifteen of the Ly$\\alpha$ and 9 of the H$\\alpha$ emitters have now\nbeen spectroscopically confirmed to lie at the same redshift as the\nradio galaxy. The $I-K$-selected Extremely Red Objects (EROs; $I-K >\n4.3$ Vega) seem concentrated around the RG but have no spectroscopic\nredshifts at this time. However, by obtaining deep images through the\nNICMOS $J_{110}$ and $H_{160}$ filters, which effectively span the\n4000\\AA-break at $z=2.16$, accurate and precise colors and basic\nmorphological parameters can be measured for the red galaxy\npopulation. In this paper we present the first results from this\nproject. The article is organized as follows: in Section\n\\S\\ref{sec:obs} we describe the data and their reductions, in Section\n\\S\\ref{sec:overdensity} we present the comparison between the red\ngalaxy counts in this field and in deep field data, in Section\n\\S\\ref{sec:RS} we present the full color-magnitude diagram and our\nfits to the ``red sequence.'' We use a $(\\Omega_{\\Lambda},\\Omega_{M})\n= (0.73,0.27)$, $H_{0} = 71$ ${\\rm km}$ ${\\rm s^{-1}}$ ${\\rm\n Mpc^{-1}}$ cosmology throughout. At $z=2.16$ one arcsecond is\nequivalent to 8.4 kpc. All magnitudes are referenced to the AB system\n\\citep*{AB} unless otherwise noted.\n\n\\section{Observations, Data Reductions and Photometry\\label{sec:obs}}\n\n\\begin{figure}\n\\plotone{f2.ps}\n\\caption{$g_{435} - I_{814}$ vs. $J_{110}-H_{160}$ color-color\n diagram using the ACS and NICMOS data. Arrows represent limits\n where the galaxy is only detected in a single band for that color.\n Filled circles indicate spectroscopically-confirmed Lyman-$\\alpha$\n (green) and H-$\\alpha$ (orange) emitting protocluster members. The\n yellow star is the radio galaxy. The blue, green and red grids\n indicate the regions occupied by galaxies with an\n exponentially-decaying star-formation rate $\\tau = 0.15$ Gyr (red),\n $\\tau = 0.4$ Gyr (green) and a $\\tau = 1000$ Gyr (blue) at $z=2.16$\n for three ages (0.1, 1 and 3 Gyr) and two different extinctions\n ($E(B-V) = 0.0, 1.0$).\\label{fig:colorcolor}}\n\\end{figure}\n\nThe NICMOS instrument on-board {\\it HST} is capable of deep\nnear-infrared imaging more quickly than from the ground but with a\nrelatively small field-of-view ($51\\arcsec \\times 51\\arcsec$). In the\ncase of MRC~1138-262, we know that galaxies are overdense on the scale\nof a few arcminutes \\citep*{Kurketal04, Croftetal05} and are thus\nwell-suited for observations with NICMOS camera 3 on {\\it HST}. We\nused 30 orbits of {\\it HST} time to image 10 of the 24 confirmed\nmembers and $\\sim 70$ of the candidate (narrow-band excess sources and\nEROs) protocluster members in both the $J_{110}$ and $H_{160}$\nfilters. We used seven pointings of NICMOS camera 3 in both filters\nand one additional pointing in $H_{160}$ alone. This single\n`outrigger' $H_{160}$ pointing was included to obtain rest-frame\noptical morphological information for a small concentration of\ncandidate members. These observations reach an AB limiting magnitude\n($m_{10\\sigma}$; 10$\\sigma$, $0\\farcs5$ diameter circular aperture) of\n$m_{10\\sigma}=24.9$~mag in $J_{110}$ and $m_{10\\sigma}=25.1$~mag in\n$H_{160}$. The same field was imaged in the $g_{475}$\n($m_{10\\sigma}=27.5$~mag) and $I_{814}$ ($m_{10\\sigma}=26.8$~mag)\nfilters using the Wide-Field Channel of the Advanced Camera for\nSurveys on {\\it HST} as part of a Guaranteed Time program (\\# 10327;\nMiley et al. 2006\\nocite{Mileyetal06}). These optical data are useful\nfor their higher angular resolution and their coverage of the\nrest-frame far-UV, thus extending the observed SEDs of candidate\nprotocluster members to shorter wavelengths where young stars and\non-going star-formation dominate the emitted spectrum. In particular,\nthe $g_{475}$ and $I_{814}$ data allow us to partially differentiate\nobscured star-formation from evolved stellar populations in the\ncandidate RS galaxies.\n\nThe NICMOS images were reduced using the on-the-fly reductions from\nthe {\\it HST} archive, the IRAF task PEDSKY and the dither\/drizzle\npackage to combine the images in a mosaic. The dither offsets were\ncalculated using image cross-correlation and were refined with one\nfurther iteration of cross-correlation. Alignment of the pointings\nrelative to each other was accomplished using a rebinned version of\nthe ACS $I_{814}$ image as a reference. The final mosaic has a pixel\nscale of $0\\farcs1$. Galaxies were selected using the $H_{160}$-band\nimage for detection within SExtractor \\citep*{SExtractor}. We used a\n$2.2\\sigma$ detection threshold with a minimum connected area of 10\npixels. We also corrected the NICMOS data for the count-rate\ndependent non-linearity \\citep*{CPSNONLINEAR}. Total galaxy\nmagnitudes were estimated by using the MAG\\_AUTO values from\nSExtractor.\n\nThe $J_{110} - H_{160}$ colors were determined by running SExtractor\n\\citep*{SExtractor} in two-image mode using the $H_{160}$ image for\nobject detection and isophotal apertures. The $J_{110}$ image was\nPSF-matched to the $H_{160}$ band. The resulting colors and\nmagnitudes are shown in Figure~\\ref{fig:CMD}. For galaxies which are\nnot detected at $2\\sigma$ significance in the $J_{110}$-band (those to\nthe right of the thick dashed line, representing $J_{110, tot} >\n26.7$, in Fig.~\\ref{fig:CMD}) we consider the color to be a lower\nlimit.\n\nWe also measured similarly PSF-matched, isophotal colors using the two\nACS bands and have used them to construct a $g_{475}-I_{814}$ versus\n$J_{110}-H_{160}$ color-color diagram (Figure~\\ref{fig:colorcolor}).\nWe compared these colors to those of model SEDs for different ages,\nstar-formation histories and dust extinctions. Using the 2007 Charlot\n\\& Bruzual\\nocite{BC} population synthesis models we have constructed\nspectral energy distributions for galaxies with an\nexponentially-decaying star-formation rate with time constants of\n$\\tau = 0.15, 0.4, 1000.0$ Gyr (the red, green and blue grids in\nFig.~\\ref{fig:colorcolor} respectively). Each model's colors are\ncalculated for ages of 0.1, 1 and 3 Gyr and for $E(B-V) = 0.0$ and\n$1.0$. Aging of the population moves primarily the $J_{110}-H_{160}$\ncolor to the red while the dust extinction significantly reddens the\n$g_{475}-I_{814}$ color. From this analysis it appears that the\n$\\tau=0.4$ Gyr model represents well the colors of a majority of the\nred $J_{110}-H_{160}$ galaxies.\n\nTo extend the wavelength coverage for the protocluster galaxies we\nalso incorporated ground-based $U_{n}$-band data from LRIS-B on the\nKeck telescope, $K_{s}$-band imaging from VLT\/ISAAC and three band\nIRAC imaging (the 3.6, 4.5 and 5.8 $\\mu$m bands) from the {\\it Spitzer\n Space Telescope}. The Keck $U$-band data (PI W. van Breugel) were\nobtained in late January and early February of 2003. The ISAAC data\n(PI G. Miley) were taken in Period 73 in service mode. The {\\it\n Spitzer} data are from the IRAC Guaranteed Time program (PI\nG. Fazio, Program \\#17). We have smoothed the imaging data for all\nbands, apart from the IRAC data, to match the resolution of the\n$U_{n}$-band image (approximated by a FWHM~$\\sim 1\\arcsec$ Gaussian).\nWe then used SExtractor to measure galaxy magnitudes within a\n$0\\farcs5$ radius circular aperture for each smoothed image. To\nincorporate the IRAC data, which has much poorer angular resolution,\nwe derived aperture magnitudes which were then corrected to match the\nsmoothed data. These aperture corrections were derived using the\nphotometric curves-of-growth for 20 stars in the field. The resulting\ncatalog was used to generate photometric redshift estimates as\ndescribed below in Section \\S~\\ref{sec:photz}.\n\n\\begin{figure}\n\\plotone{f3.ps}\n\\caption{upper panel: Distribution of high-confidence ($>95\\%$)\n photometric redshifts and their selection function assuming a\n uniform $N(z)$ for our model template galaxies (yellow curve) for\n $1.1 \\leq (J_{110}-H_{160}) \\leq 2.1$ galaxies in the MRC 1138-262\n NICMOS field. The peak between $z=2.1$ and $z=2.4$ is statistically\n highly significant. middle panel: Sum of redshift probability\n distributions for all the galaxies in the upper panel. 38.5\\% of\n the total probability is contained in the redshift interval from 2\n to 2.3. lower 2 panels: two examples of the probability distribution\n function for individual galaxies\\label{fig:photzRS}}\n\\end{figure}\n\n\\section{Photometric Redshifts\\label{sec:photz}}\n\nWe have used the ACS ($g_{475}$, $I_{814}$), NICMOS ($J_{110}$,\n$H_{160}$), ground-based $U_{n}$-band from Keck\/LRIS-B, $K_{S}$-band\nimaging from VLT\/ISAAC and {\\it Spitzer}\/IRAC imaging to estimate\nphotometric redshifts for our $H_{160}$-band selected sample. We\ninput a catalog of aperture galaxy magnitudes, based on the matched,\nsmoothed images described above, into the Bayesian photometric\nredshift code (BPZ) of Ben{\\'{\\i}}tez (2000)\\nocite{BPZ} using a\nuniform prior. We felt that the default prior, based on optical\ngalaxy selection and spectroscopy in the HDF-N, would not necessarily\nrepresent the redshift distribution for our near-infrared selected\ngalaxies. We generated our own extensive set of template spectral\nenergy distributions using the models of Charlot \\& Bruzual\n(2007)\\nocite{CB07}. All these SEDs are $\\tau$ models with values for\n$\\tau = [0.15, 0.4, 1.0, 2.0, 1000.0]$ Gyr and ages $=[0.05, 0.1, 0.5,\n1.0, 2.0, 3.0]$ Gyr. We also included models with internal dust\nextinction ranging from $E(B-V) = [0.0, 0.1, 0.3, 0.5, 0.75, 1.0]$ mag\nand metallicity of $(Z\/Z_{\\odot}) = [0.3, 1.0, 2.5]$.\nWe focused particular attention on the $J_{110}-H_{160}$ selected\nsurface-overdensity. In the upper panel of Figure~\\ref{fig:photzRS}\nwe present the high confidence ($> 95\\%$) photo-$z$ distribution for\nthe NIR-color selected ($1.1 \\leq (J_{110} - H_{160}) \\leq 2.1$)\nsubsample. We ran extensive simulations by redshifting our template\nset, adding appropriate photometric errors and using BPZ to recover\nthe redshifts. The yellow curve represents the redshift selection\nfunction for this color cut, template set and filters assuming that\nthese model galaxies follow a uniform $N(z)$ over this redshift\ninterval. The simulation results were free of significant systematic\nerrors and the random errors are estimated to be $\\delta z\/z \\sim\n0.1$. Based on these SED fits, the approximate luminosity-weighted\nages of the red galaxies lie between 1 and 2.5 Gyrs and their stellar\nmasses are of order a few $\\times 10^{10} M_{\\odot}$. These stellar\nmasses are reasonable as are the absolute magnitudes (see\nFigure~\\ref{fig:CMRfits}). More detailed SED modeling is deferred to\na future paper.\n\nThere is a clear excess of galaxies between $z=2.0$ and $z=2.5$. For\neach galaxy fit by BPZ we have generated the full redshift probability\ndistribution. In the lower panel of Fig.~\\ref{fig:photzRS} we show\nthe $H_{160}$-band weighted-average of these probability\ndistributions. There is a clear peak (containing 38.5\\% of the total\nprobability compared to only 17\\% of the total selection function in\nthe same redshift interval) between $z=2.0$ and $z=2.3$, consistent\nwith the significant peak in the redshift histogram itself.\n\n\\section{NICMOS Galaxy Morphologies\\label{sec:morf}}\n\n\\begin{figure}[b]\n\\plotone{f4.ps}\n\\caption{Distribution of inferred stellar ages (in terms of $\\tau$)\n for both the concentrated ($n\\geq2.5$, red line) and diffuse\n ($n<2.5$, blue line) galaxies which are well-resolved in the NICMOS\n data. Constant star-formation models, for which the $e$-folding\n time is infinite, are placed at the left-hand edge of the plot. The\n blue and red distributions are quite different. Of particular note\n is that the most evolved galaxies generally have high $n$ while the\n low $n$ galaxies dominate the star-forming\n population.\\label{fig:tAn}}\n\\end{figure}\n\nNICMOS camera 3 provides good angular resolution over its entire\nfield-of-view. The FWHM of the PSF in our final mosaic is $\\approx\n0\\farcs27$. To exploit this resolution we have used the GALFIT code\n\\citep*{GALFIT} to fit analytic S{\\'e}rsic surface-brightness profiles\n\\citep*{Sersic} to all the $H_{160} \\leq 24.5$ sources in our\n$H_{160}$-band mosaic. A model point-spread function was created for\neach of these galaxies individually by generating a TinyTim simulated\nPSF \\citep*{TINYTIM} at the galaxies' positions in each exposure and\nthen drizzling these PSFs together in exactly the same fashion as for\nthe data themselves (see Zirm et al. 2007\\nocite{Zirmetal07}). We\nrestricted the S{\\'e}rsic index, $n$, to be between 1 and 5. We will\npresent a full analysis of the morphologies of these galaxies in a\nfuture paper. For the current work, we use these derived sizes and\nprofile shapes to assist us in selecting the morphological\n``early-type'' members of the red galaxy population. \n\n\\begin{figure*}[t]\n\\plotone{f5.ps}\n\\caption{Histogram of the color distributions for the 1138 and deep\n fields (blue). The deep field data has been normalized by total\n area to the 1138 data. Note the clear excess of red galaxies in the\n 1138 field. At $1.1 \\leq (J_{110} - H_{160}) \\leq 2.1$ (horizontal\n dotted lines) for galaxies brighter than the 2$\\sigma$\n $J_{110}$-band limit (dashed line) there is an overdensity of a\n factor of 6.2 in the 1138 field. \\label{fig:hist}}\n\\end{figure*}\n\nIn Figure~\\ref{fig:tAn} we show the distribution of galaxy ages\nderived via these SED fits as parametrized by the $\\tau$ value for\nthe best-fitting model for those galaxies with high and low S{\\'e}rsic\nindex ($n \\geq 2.5$, red line, and $n < 2.5$, blue line). It is clear\nthat while there is substantial overlap between these distributions\nthey are not identical and that they differ in the sense that one\nmight expect, namely, that the concentrated galaxies appear to be\ncomprised of older stellar populations. This trend gives us some\nconfidence in trying to select the ``early-type'' galaxies using these\ndata which is important for our discussion of the color-magnitude\nrelation in Section~\\ref{sec:RS}.\n\t\n\\section{Surface Overdensity of Red Galaxies\\label{sec:overdensity}}\n\n\\begin{figure*}[t]\n\\plotone{f6.ps}\n\\caption{Linear fits to the rest-frame $U-B$ (Vega) color-magnitude\n diagrams for three different sub-samples of the $H_{160}$-band\n selected NICMOS sample. Panel (A) shows the fit (solid line) and\n intrinsic scatter ($1\\sigma$; dotted lines) for a sample selected to\n have $1.1 \\leq (J_{110} - H_{160}) \\leq 2.1$. The crossed-out point\n are those which are rejected as outliers in more than half of the\n realizations (see \\S\\ref{sec:RS}). Both the observed and intrinsic\n scatter are smaller than the initial color cut. Panel (B) shows the\n fit and intrinsic scatter for a photometric redshift selected sample\n with $2.0 < z_{\\rm phot} < 2.5$. The stars indicate galaxies whose\n preferred photometric template has an age $< 4\\tau$, while circles\n represent galaxies with older than this limit. Panel (C) shows the\n fit and scatter for those galaxies which meet the same redshift cut\n but also are well-resolved with a high S{\\'e}rsic index ($n > 2.5$)\n and best-fit by an age $\\geq 4\\tau$ template.\\label{fig:CMRfits}}\n\\end{figure*}\n\nTo compare this protocluster field to more generic `blank' field data\nwe have compiled catalogs for the public NICMOS data in both the\nHubble Deep Field North (HDF-N) and the Ultra Deep Field (UDF).\nFigure~\\ref{fig:CMD} shows the $J_{110} - H_{160}$ color-magnitude\ndiagram (open black circles) and the color distributions for both the\nMRC~1138-262 and the combined HDF-N and UDF galaxy catalogs (blue\ncircles). The deep field data were also $H_{160}$-band selected. The\narea of the two deep fields is roughly 2.5 times the area of our\nprotocluster observations. We have applied no correction to the deep\nfield number counts to account for clustering in those fields. The\ncolor histogram in Figure~\\ref{fig:hist} shows the area-normalized\ngalaxy counts from the two deep fields (blue line) and from the 1138\nfield to the same ($2\\sigma$) limiting magnitude of $J_{110} =\n26.7$~mag (AB). The red dashed line shows the difference between the\ntwo color distributions. It is clear that the radio galaxy field is\noverdense in red galaxies by a large factor. For sources with colors\nbetween $1.1 \\leq (J_{110} - H_{160}) \\leq 2.1$, the\nhorizontal(vertical) dotted lines in\nFig.\\ref{fig:CMD}(\\ref{fig:hist}), and brighter than our\n$J_{110}$-band $2\\sigma$ limit ($26.7$), we calculate an\narea-normalized overdensity of $6.2$ when compared to the deep fields\ndata. We note that the exact value of the measured overdensity is\nrather sensitive to systematic color offsets between the protocluster\nand deep field data. A redward shift of 0.05 for the deep field\ngalaxies would lower the measured overdensity to $5.0$. However, we\nare confident that these systematic offsets remain small ($< 0.05$\nmag) since we have used the same instrument, filters, selection\ntechnique and photometric code with very similar input parameters for\nboth the deep field and 1138 datasets. Looking back at\nFigure~\\ref{fig:colorcolor} we can see that many of the\nspectroscopically-confirmed line emitters (filled blue circles) and\nred galaxies in the overdensity are well-represented by the $\\tau =\n0.4$ Gyr model (green lines) at different ages and extinctions.\n\n\n\nThis current work is not the first to observe red galaxies in this\nfield. Kurk et al. (2004)\\nocite{Kurketal04} identified a small\n($\\sim 1.5\\times$) surface overdensity of extremely red objects (EROs;\n$I-K > 4.3$ Vega magnitudes) using ground-based $I$ and $K$ band data.\nMany of these EROs are also identified as red in the NICMOS\n$J_{110}-H_{160}$ color. More recently Kodama et\nal. (2007)\\nocite{Kodamaetal07} observed this field using the\nwide-field NIR imager, MOIRCS, on the Subaru telescope. These authors\nfound several bright (presumably massive) red galaxies over a wider\nfield-of-view but to shallower depths than the NICMOS data presented\nhere. 24 of their color-selected protocluster candidates are within\nour NICMOS mosaic. 23 of the 24 are identified in our data as being\nred in $J_{110} - H_{160}$. Furthermore, 18 of the 94 galaxies which\nsatisfy our color criteria (and have $J_{110}<26.67$) are also\nidentified by Kodama et al. as protocluster candidates. The much\nlarger number of red galaxies in the NICMOS data is primarily due to\nfainter galaxies detected at high significance in our deeper data.\n\n\\section{An Emergent Red-Sequence?\\label{sec:RS}}\n\nTo study the colors and magnitudes of these galaxies in more detail\nand to possibly identify a red-sequence in the 1138 field we have\nsplit the galaxies into three sub-samples defined by $J_{110}-H_{160}$\ncolor, photometric redshift and morphology (S{\\'e}rsic index). The\nfirst sample (Sample A) comprises all 56 galaxies with $1.1 \\leq\n(J_{110}-H_{160}) \\leq 2.1$ and $H_{160} < 24.5$ and includes the\nradio galaxy itself. Sample B is made up of all 28 galaxies with a\nrobust photometric redshift between 2.0 and 2.5 and $J_{110}-H_{160} >\n0.75$ and $H_{160} < 26.0$. This liberal color cut is included to\nselect galaxies which comprise the large observed surface-overdensity.\nFinally, sample C contains seven galaxies with the same photometric\nredshift cut but which also have well-resolved $H_{160}$-band\nsurface-brightness profiles with S{\\'e}rsic index $n > 2.5$. All of\nthese galaxies' SEDs are also best-fit by models with relatively\nlittle on-going star-formation. We use a limit of (age $\\geq 4 \\times\n\\tau$, cf. Grazian et al 2007 \\nocite{Grazianetal07}). Therefore,\nsample C mimics the color, morphological and photometric redshift\nselection of early-type galaxies in clusters at $z \\mathrel{\\spose{\\lower 3pt\\hbox{$\\mathchar\"218$} 1$. The\nphotometry, photo-$z$s and sizes of the sample C galaxies are listed\nin Table~\\ref{tab:RS}, their rest-frame color-magnitude distributions\nare shown in Figure~\\ref{fig:CMRfits} and the two-dimensional spatial\ndistribution of the Sample A galaxies is plotted in\nFigure~\\ref{fig:spatial}. We note that because the measured\noverdensity is a factor of 6, we statistically expect one of every\nseven sample A galaxies to be a field galaxy. However, this should\nnot effect our results significantly.\n\nFor these three sample selections we have fit a line and measured the\nintrinsic scatter about that best-fit line (see\nFig.~\\ref{fig:CMRfits}). For comparison to lower redshift galaxy\nclusters we have transformed our observed $J_{110}-H_{160}$ color and\n$H_{160}$ magnitudes into rest-frame $U-B$ and $B$ (Vega),\nrespectively, using the following expressions:\n\\begin{equation}\n(U-B)_{\\rm rest} = 0.539 \\times (J_{110}-H_{160})_{\\rm obs} - 0.653\n\\end{equation}\n\n\\begin{equation}\n M_{B, {\\rm rest}} = H_{160,{\\rm obs}} - 0.170 \\times (J_{110}-H_{160})_{\\rm obs} - 43.625\n\\end{equation}\n\nThe small color corrections used in these relations were derived using\na family of $\\tau$-models with a range of ages (0.1-12 Gyr), $\\tau$\n(0.1-5 Gyr) and three metallicities ($0.4, 1$ and $2.5 Z_{\\odot}$).\n\n\\begin{figure*}[t]\n\\epsscale{0.8}\n\\plotone{f7.ps}\n\\caption{Spatial distribution of galaxies (red circles) relative to\n the radio galaxy MRC~1138-262 (yellow star) for the color-selected\n sample defined in the text (A) and shown in the first panel of\n Fig.~\\ref{fig:CMRfits}. The irregular black outline encloses the\n coverage of the 7 NICMOS camera 3 pointing with both $J_{110}$ and\n $H_{160}$ imaging.\\label{fig:spatial}}\n\\end{figure*}\n\nTo fit the ``CMR'' we used a bootstrap re-sampling technique to\nestimate the error on the fitted slope. Then, by assuming that all\nthe red galaxies lie on this fit line, we ran Monte Carlo realizations\nof the contribution of the photometric errors to the observed color\nscatter about the fit line, i.e., by fixing a color-magnitude relation\nwe calculate the measurement scatter with zero intrinsic scatter. We\nthen calculate the intrinsic scatter by subtracting (in quadrature)\nthe estimated measurement scatter from the observed scatter. We show\nthese fits (solid line) and the intrinsic scatters (dotted lines) for\nthe three samples (A, B and C) in Figure~\\ref{fig:CMRfits}. The fits\nto both Sample's A and B have nearly identical rest-frame $U-B$\nslopes, $0.027$ and $0.026$ respectively, and intrinsic scatters\n($0.10$ and $0.12$). While these slopes are comparable to those found\nfor the well-populated lower redshift cluster CMRs, the intrinsic\nscatters are considerably higher. However, the scatter measured for\nthe eight galaxy Sample C is comparable to that of the lower redshift\nsamples but with a much steeper slope ($0.130$). When these scatters\nare compared to model predictions based on lower redshift clusters\n(specifically RDCS 1252.9-2927 at $z=1.24$; Gobat et al. 2008) we find\nthat the 1138 protocluster has lower than predicted scatter. This may\nsuggest that the 1138 protocluster is in a more advanced evolutionary\nstate than RDCS 1252 was at $z=2.2$.\n\nWe have calculated three representative color-magnitude relations for\ncomparison to the colors and magnitudes of the red galaxies (three\ndot-dash lines in Fig.~\\ref{fig:CMD}). We have taken two lower\nredshift clusters, Coma at $z=0.023$ and RDCS~1252.9-2927 at $z=1.24$,\nand transformed them to the observed filters and $z=2.16$ under the\nassumption that the colors do not evolve. In this no evolution case\n(the two dot-dash lines in Fig.~\\ref{fig:CMD}) the CMRs appear at the\nred edge of the observed overdensity. There is almost exactly 2 Gyr\nof cosmic time between $z=2.16$ and $z=1.24$ in our adopted cosmology.\nFrom \\citet{Blakesleeetal03} we know that the median redshift of last\nsignificant star-formation for the RDCS~1252 galaxies is between\n$z=2.7-3.6$. Therefore, if we observe those galaxies at $z=2.16$ they\nwill be significantly younger and hence bluer. In fact, this\npassively de-evolved line (bluest dot-dash line in Fig.~\\ref{fig:CMD},\nlabeled `$z_{\\rm form} \\sim 3$') does fall blueward of the red galaxy\noverdensity. We discuss the implications for these comparisons in\nSection~\\ref{sec:conclusions}.\n\nWe have also translated the Kodama et al. ground-based $J-K$ colors to\nour NICMOS filters assuming all the red galaxies lie at $z=2.16$.\nThese bright galaxies also fall along the passively de-evolved line\nwith the radio galaxy. We have used our suite of SED models to\nestimate the color transformation from their ground-based $J-K_{S}$ to\nour NICMOS $J_{110}-H_{160}$ color. Roughly, the Kodama et al. bright\nred galaxies fall where the RDCS~1252 passive line crosses our color\ncut at $J_{110}-H_{160} = 1.1$. This result hints at a possible\nbi-modality in the red galaxy population of this protocluster.\nNamely, that there are faint red galaxies that are inconsistent with\npassively-evolving cluster members either due to large amounts of\ndust, or due to higher redshifts of formation but that the more\nluminous protocluster members may have already finished forming and\nseem consistent with passive evolution to the present-day.\n\n\n\\section{Discussion\\label{sec:conclusions}}\n\nWe have identified a (6.2$\\times$) surface-overdensity and a\ncorresponding photometric redshift `spike' of red $J_{110}-H_{160}$\ngalaxies which are likely associated with a known protocluster at\n$z=2.16$. The optical-NIR spectral energy distributions of these\nsources suggest that they comprise both evolved galaxies as well as\ndust-obscured star-forming galaxies. Based on our SED fits from the\nphotometric redshift determinations, the approximate\nluminosity-weighted ages of these sources lie between 1 and 2.5 Gyrs\nand their stellar masses are of order a few $\\times 10^{10}\nM_{\\odot}$. Detailed modeling of the SEDs for the protocluster\npopulation, along with their morphologies, is reserved for a future\npaper.\n\nComparison with the CMRs of lower redshift clusters shows that the red\ngalaxy overdensity primarily lies blueward of the no-evolution\npredictions. That the red galaxies in 1138 are also redder than the\n$z_{\\rm form} \\sim 3$ case suggests both that there are galaxies with\nsignificant dust content, an assertion supported by the SED fits, and\nalso that they were perhaps formed at higher redshift than the\nRDCS1252 galaxies. Of course, without a classical, low-scatter\nred-sequence to use as a baseline there remains considerable\nuncertainty in the age of the population as a whole. The results of\nSteidel et al. (2005)\\nocite{Steideletal05} suggest that protocluster\ngalaxies are older than their ``field'' counterparts at $z \\sim 2.3$\nand that these ages and stellar masses were broadly consistent with\nevolution to lower redshift cluster galaxies. However, their\nprotocluster members were all UV-selected and star-forming. With\nfuture spectroscopy of our red galaxy sample it will be possible to\nsee if these differences persist when looking at a more varied galaxy\nsample.\n\nFor three samples of galaxies drawn from the full $H_{160}$-band\nselected dataset we have fit a color-magnitude relation and estimated\nthe intrinsic scatter. The CMR at $z=2.16$ is not as well-defined as\nat $z \\sim 1$ or 0. For sample C, made up of 8 galaxies, the color,\nbest-fit spectral template, morphology and photo-$z$ all point towards\nthem being (proto-)elliptical galaxies within the protocluster. For\nthis small sample, the estimated intrinsic scatter is rather low and\nmay suggest that these galaxies represent the forming red-sequence in\nthis protocluster. The slope of this relation is extremely steep\ncompared to lower redshift clusters. The slope of the CMR is\ngenerally assumed to be a manifestation of the mass-metallicity\nrelation and would therefore flatten at higher redshift. The major\ncaveat regarding the steep slope of Sample C is that none of these\ngalaxies are spectroscopically confirmed protocluster members.\nTherefore, this ``relation'' may just be a random, although somewhat\nunlikely, coincidence rather than a nascent CMR. However, further\ndeep NIR imaging coverage of this field is required to identify\nadditional members of this proto-elliptical galaxy class.\n\n\\acknowledgments\n\nSupport for program \\# 10404 was provided by NASA through a grant\n(GO-10404.01-A) from the Space Telescope Science Institute, which is\noperated by the Association of Universities for Research in Astronomy,\nInc., under NASA contract NAS 5-26555. The work of SAS was performed\nin part under the auspices of the U.S. Department of Energy, National\nNuclear Security Administration by the University of California,\nLawrence Livermore National Laboratory under contract\nNo. W-7405-Eng-48. JK is financially supported by the DFG, grant SFB\n439. WvB acknowledges support for radio galaxy studies at UC Merced,\nincluding the work reported here, with the Hubble Space Telescope and\nthe Spitzer Space Telescope via NASA grants HST \\# 10127, SST \\#\n1264353, SST \\# 1265551, SST \\# 1279182.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nQuantum cohomology, or more general, higher genus Gromov-Witten invariants are defined for symplectic and algebraic manifolds in e.g. \\cite{Ruan96},\\cite{Beh97}, \\cite{LT98}. As a kind of basic varieties, quantum cohomology of the smooth complete intersections in projective spaces has been studied intensively. A kind of (numerical) mirror symmetry for Fano and Calabi-Yau complete intersections has been established (\\cite{Giv96}), which expresses the small J-function in terms of hypergeometric series. From small J-functions one can reconstruct all genus zero Gromov-Witten invariants with only \\emph{ambient cohomology classes} as insertions, i.e. the classes obtained by intersecting linear spaces with the complete intersections. The invariants with \\emph{primitive cohomology classes} as insertions are not well-understood as those with ambient insertions; for 3 point invariants see \\cite{Bea95}. But they are necessary for understanding for example, Dubrovin's conjecture on the relation between quantum cohomology and derived categories. \n\nFrom now on, we call genus zero primary (i.e. without $\\psi$-classes) Gromov-Witten invariants \\emph{correlators}, for brevity. \nIn \\cite{Hu15} we studied the big quantum cohomology of Fano complete intersections, especially the correlators with primitive insertions. One of our basic tool is the Monodromy group of the whole family of complete intersections of a given multidegree. The Zariski closure $\\overline{G}$ of such monodromy groups are either orthogonal groups, symplectic groups, or finite groups. If $\\overline{G}$ is a finite group, we call the complete intersections \\emph{exceptional}, and otherwise \\emph{non-exceptional}. There are only three kinds of exceptional complete intersections: the cubic surfaces, the quadric hypersurfaces, and the even dimensional complete intersections of two quadric hypersurfaces. The first two kinds has been somewhat well-studied, in the sense that all correlators can be effectively computed, and the semisimplicity (of the corresponding Frobenius manifold) is proved. \n\nIn this paper we studied the big quantum cohomology of the even dimensional complete intersections of two quadric hypersurfaces. We will also call them even $(2,2)$-complete intersections for short. To understand our approach and its limitations, let us recall more details in \\cite{Hu15}. The basic idea is still to use WDVV equations\n\\begin{equation}\\label{eq-WDVV-intro}\n\\sum_e\\sum_f\t(\\partial_{t^a}\\partial_{t^b}\\partial_{t^e}F)g^{ef}(\\partial_{t^f}\\partial_{t^c}\\partial_{t^d}F)=\\sum_e\\sum_f(\\partial_{t^a}\\partial_{t^c}\\partial_{t^e}F)g^{ef}(\\partial_{t^f}\\partial_{t^b}\\partial_{t^d}F).\n\\end{equation}\n The most direct way to use WDVV to get recursions is to use the leading terms. Namely, selecting a monomial $t^I$, where $I$ is a multi-index, and extracting the coefficients of $t^I$ on both sides, we get an equation of the form\n \\begin{eqnarray}\n \t&&\\mathrm{Coeff}_{t^I}(\\partial_{t^a}\\partial_{t^b}\\partial_{t^e}F)g^{ef}(\\partial_{t^f}\\partial_{t^c}\\partial_{t^d}F)(0)\n \t+\t(\\partial_{t^a}\\partial_{t^b}\\partial_{t^e}F)(0)g^{ef}\\mathrm{Coeff}_{t^I}(\\partial_{t^f}\\partial_{t^c}\\partial_{t^d}F)\\nonumber\\\\\n \t&&-\\mathrm{Coeff}_{t^I}(\\partial_{t^a}\\partial_{t^c}\\partial_{t^e}F)g^{ef}(\\partial_{t^f}\\partial_{t^b}\\partial_{t^d}F)(0)\n \t-(\\partial_{t^a}\\partial_{t^c}\\partial_{t^e}F)(0)g^{ef}\\mathrm{Coeff}_{t^I}(\\partial_{t^f}\\partial_{t^b}\\partial_{t^d}F)\\nonumber\\\\\n \t&=& \\mbox{combinations of coefficients of lower order terms}.\n \\end{eqnarray}\n Here we have adopted Einstein's summation convention, i.e. omitting the summation notations of the repeated indices $e$ and $f$. We call the resulted recursions \\emph{essentially linear recursions}. Such recursions are effective only when the correlators of length 3 have good properties. For non-exceptional complete intersections and the even $(2,2)$-complete intersections, the correlators with primitive insertions cannot provide a recursion to compute correlators of arbitrary lengths, essentially due to the monodromy reason. For non-exceptional ones, we remedy this by considering the symmetric reduction of the WDVV equations. From the classical theory of polynomial invariants of orthogonal groups or symplectic groups, the dependence of the generating function $F$ on the variables dual to the primitive classes are encoded in only one variable $s$. When the dimension $n$ of the complete intersection $X$ is even, $s$ is defined as\n \\begin{equation}\n \ts=\\frac{1}{2}\\sum (t^i)^2,\n \\end{equation}\n where $t^i$ are dual variables of an orthonormal basis of $H^n_{\\mathrm{prim}}(X)$. The main novelty of \\cite{Hu15} is that we found that the coefficient of $s^k$ in $F$ can be recursively computed by solving quadratic equations of one variable with two equal roots. For a precise conjectural algorithm and partial results on this we refer the reader to \\cite[Section 1]{Hu15}. There is only one case that is excluded in this algorithm: the cubic hypersurfaces. For cubic hypersurfaces we showed in \\cite{Hu15} that the generating function $F$ can be computed by an essentially linear recursion with respect to the variable $s$, using the symmetric-reduced WDVV equations. Note that $s$ is quadratic in $t^i$'s. So this means that we can compute $F$ by a recursion on the \\emph{sub-leading terms} of (\\ref{eq-WDVV-intro}) and the monodromy symmetries. As (even dimensional) cubic hypersurfaces and the even $(2,2)$-complete intersections are the only even dimensional complete intersections of Fano index $n-1$ (recall $n=\\dim X$), it is reasonable to expect an analogy between them.\n\nThe monodromy group of the family of even $n$-dimensional $(2,2)$-complete intersections, or more precisely, it image in $\\mathrm{Aut}(H^n_{\\mathrm{prim}}(X))$, is the Weyl group $D_{n+3}$. Moreover the lattice $H^n_{\\mathrm{prim}}(X)\\cap H^n(X;\\mathbb{Z})$ with the Poincar\u00e9 pairing is isomorphic to a $D_{n+3}$-lattice with $(-1)^{\\frac{n}{2}}$ times its standard inner product. We can choose a basis $\\alpha_1,\\dots,\\alpha_{n+3}$ of the lattice $H^n_{\\mathrm{prim}}(X)\\cap H^n(X;\\mathbb{Z})$ which maps to the $D_{n+3}$ roots under such an isomorphism, and then define a specific orthonormal basis $\\epsilon_1,\\dots,\\epsilon_{n+3}$ of $H^n_{\\mathrm{prim}}(X)$. Let $\\mathsf{h}_i$ be the $i$-th cup product of the hyperplane class $\\mathsf{h}$. Let $t^0,\\dots,t^{n},t^{n+1},\\dots,t^{2n+3}$ be the basis dual to $1,\\mathsf{h},\\dots,\\mathsf{h}_n,\\epsilon_1,\\dots,\\epsilon_{n+3}$, then the generating function of correlators of $X$ is a function of $t^0,\\dots,t^n$ and $s_1,\\dots,s_{n+3}$, where\n\\begin{equation}\\label{eq-invariantsOf-typeD-intro}\n\ts_{i}=\\begin{cases}\n\t\\vspace{0.2cm}\n\t\\frac{1}{(2i)!}\\sum_{j=n+1}^{2n+3}(t^j)^{2i},& \\mbox{for}\\ 1\\leq i\\leq n+2,\\\\\n\t\\prod_{j=n+1}^{2n+3}t^j,& i=n+3.\n\t\\end{cases}\n\\end{equation}\nInstead of doing the reduction of WDVV with the $D_{n+3}$ symmetry, we are going to find recursions based on the sub-leading terms of the WDVV equations, as mentioned above. For this we need first compute all the correlators of length 4. The correlators of length 4 with at most 2 primitive insertions are computed in \\cite{Hu15}. For the correlators of length 4 with 4 primitive insertions, we have a uniform partial result in \\cite[Section 9]{Hu15} which holds for all complete intersections of Fano index $n-1$, which is obtained by an application of Zinger's reduced genus 1 Gromov-Witten invariants \\cite{Zin08}. Combining this result with the above monodromy reason and some integrality, we obtain:\n\n\\begin{theorem}\\label{thm-4points-fanoIndex-even(2,2)-intro}(= Theorem \\ref{thm-4points-fanoIndex-even(2,2)})\nLet $X$ be an even dimensional complete intersection of two quadrics in $\\mathbb{P}^{n+2}$, with $n\\geq 4$. Then\n\\begin{equation}\\label{eq-4points-fanoIndex-even(2,2)-intro}\n\t\\frac{\\partial^2 F}{(\\partial s_1)^2}(0)=1,\\ \\frac{\\partial F}{\\partial s_2}(0)=-2.\n\\end{equation}\nEquivalently, for $1\\leq a,b\\leq n+3$,\n\\begin{equation}\\label{eq-4points-fanoIndex-even(2,2)-ab-intro}\n\t\\langle \\epsilon_a,\\epsilon_a,\\epsilon_b,\\epsilon_b\\rangle_{0,1}=1.\n\\end{equation}\n\\end{theorem}\nUsing Theorem \\ref{thm-4points-fanoIndex-even(2,2)-intro}, a careful study of the sub-leading terms of the WDVV equations leads to the following.\n\\begin{theorem}[Reconstruction]\\label{thm-reconstruction-even(2,2)-intro}(= Theorem \\ref{thm-reconstruction-even(2,2)})\nLet $X$ be an even dimensional complete intersection of two quadrics in $\\mathbb{P}^{n+2}$, with $n\\geq 4$.\nWith the knowledge of the 4-point invariants, all the invariants can be reconstructed from the WDVV, the deformation invariance, and the \\emph{special correlator}\n\\begin{equation}\\label{eq-intro-length(n+3)Invariant-even(2,2)}\n\t\\langle \\epsilon_{1},\\dots,\\epsilon_{n+3}\\rangle_{0,n+3,\\frac{n}{2}}.\n\\end{equation}\n\\end{theorem}\n\nWe tried some more intricate study of WDVV, e.g. some equations that a priori may give quadratic equations of the special correlator (\\ref{eq-intro-length(n+3)Invariant-even(2,2)}). But such equations in examples turn out to be trivial. We conjecture that WDVV does not give any new information on the special correlator; see Conjecture \\ref{conj-specialCorrelator-free} for a precise statement. We can show a weaker result: \n\\begin{lemma}\\label{lem-specialCorrelator-freedomOfSign}\nFor even $(2,2)$-complete intersections, the WDVV equations and the knowledge of correlators of length 4 can at most determine the special correlator with a freedom of signs, unless it vanishes.\n\\end{lemma}\nThe proof is trivial: we change the signs of the basis $\\alpha_1,\\dots,\\alpha_{n+3}$. The resulted basis is still allowable as above. Then each correlator will change by a sign $(-1)^k$, where $k$ is the number of primitive insertions. \n\nWe refer the reader to Remark \\ref{rem:choiceOfBasis}, and Section \\ref{sec:conjecturesOnSpecialCorrelator}, on the choice of the basis. In Section \\ref{sec:explictD-Lattice} we give an explicit construction of the basis, which we take as our standard choice.\n\n\nNevertheless, Theorem \\ref{thm-4points-fanoIndex-even(2,2)-intro} and \\ref{thm-reconstruction-even(2,2)-intro} are enough to imply the following.\n\n\\begin{theorem}[Analyticity and Semisimplicity]\\label{thm-intro-convergence-semisimplicity}\nThere exists an open (in the classical topology) neighborhood of the origin of $\\mathbb{C}^{2n+4}$, on which the generating function $F(t^0,\\dots,t^{2n+3})$ is analytic and defines a semisimple Frobenius manifold.\n\\end{theorem}\nThe analyticity follows from a bound of the correlators, which follows from an induction based on an algorithm given by the proof of Theorem \\ref{thm-reconstruction-even(2,2)-intro}. Let $\\widetilde{E}$ be the matrix of the big quantum multiplication by the Euler field $E$. We show the semisimplicity by showing that \n\\begin{equation}\\label{eq-statement-cutoffEulerField-distinctEigenValues}\n\t\\mbox{the cutoff of $\\widetilde{E}$ at order 2 has pairwise distinct eigenvalues.}\n\\end{equation}\nI would like to provide a prophetic view why this is possible (see also \\cite[Remark 3.2]{Hu15}). If the generating function $F$ has continuous symmetries such as the orthogonal or symplectic groups in the cases of non-exceptional complete intersections, one can construct a family of (normalized) Euler fields for the associated Frobenius manifold $\\mathcal{M}$. But semisimple Frobenius manifold has a unique normalized Euler field \\cite[Theorem I.3.6]{Man99}. So such $\\mathcal{M}$ cannot be generically semisimple. For the same reason, if one wants to show the Frobenius manifold associated to the generating function $F$ of quantum cohomology of an even $(2,2)$-complete intersection $X$ is semisimple, one needs to use the expansion of $F$ to a certain order such that it has no continuous symmetry. This explains why the small quantum cohomology of $X$ is not semisimple: the cutoff of $F$ at order 3 is a function of $t^0,\\dots,t^n$ and \n\\[\ns_1=\\sum_{i=n+1}^{2n+3}(t^i)^2.\n\\]\nSo it has symmetries from the orthogonal group $\\mathrm{O}\\big(H^n_{\\mathrm{prim}}(X)\\big)$. The degree 4 form \n\\[\ns_2=\\sum_{i=n+1}^{2n+3}(t^i)^4\n\\]\nhas only finitely many automorphisms; this is a classical theorem of Jordan \\cite{Jor1880}. So we can expect that the information of correlators of length 4 implies the semisimplicity. To capture the full information of correlators of length 4 we need only the cutoff of the matrix at an order $\\geq 2$.\n\nIn view of Kuznetsov's resulte \\cite[Corollary 5.7]{Kuz08}, the semisimplicity statement above confirms a part of Dubrovin' conjecture \\cite{Dub98} which predicts the equivalence of the existence of full exceptional collections in $D^b(\\mathrm{Coh}(X))$ and the generic semisimplicity of the Frobenius manifold of the quantum cohomology of $X$.\\\\\n\n\n\nBy an analogy to a vanishing conjecture \\cite[Conjecture 10.26]{Hu15} of the coefficient of $s^{n+1}$ in $F$ for cubic hypersurfaces, we make the following conjecture on the special correlator. For details see Section \\ref{sec:conjecturesOnSpecialCorrelator}. \n\\begin{conjecture}\\label{conj-unknownCorrelator-Even(2,2)-intro}\nFor an even $n$ dimensional complete intersection of two quadrics in $\\mathbb{P}^{n+2}$, let $\\varepsilon_1,\\dots,\\varepsilon_{n+3}$ be the basis of $H^n_{\\mathrm{prim}}(X)$ defined in Section \\ref{sec:explictD-Lattice}. Then \n\\begin{equation}\\label{eq-unknownCorrelator-Even(2,2)-intro}\n\t\\langle \\varepsilon_{1},\\dots,\\varepsilon_{n+3}\\rangle_{0,n+3,\\frac{n}{2}}=\\frac{(-1)^{\\frac{n}{2}}}{2}.\n\\end{equation}\n\\end{conjecture}\nHere $\\varepsilon_1,\\dots,\\varepsilon_{n+3}$ is an unnormalized orthogonal basis, see (\\ref{eq-normalizedOrthonormalBasis}).\n\nI must confess that this analogy is not so strong to make us confident about the validity. We can only show Conjecture \\ref{conj-unknownCorrelator-Even(2,2)-intro} in dimension 4.\n\\begin{theorem}\\label{thm-unknownCorrelator-Even(2,2)-4dim-intro}(= Corollary \\ref{cor-unknownCorrelator-Even(2,2)-4dim})\nFor any $4$-dimensional complete intersections of two quadrics in $\\mathbb{P}^{6}$, \n\\begin{equation}\\label{eq-unknownCorrelator-Even(2,2)-4dim-intro}\n\t\\langle \\epsilon_{1},\\dots,\\epsilon_{7}\\rangle_{0,7,2}=\\frac{1}{2}.\n\\end{equation}\n\\end{theorem}\nThe proof is essentially a computation \\emph{from the first principle}. Namely, we find explicit cycles $S_{0,1},S_{1,2},\\dots,S_{5,6},S_{6,0}$ in $X$ such that (\\ref{eq-unknownCorrelator-Even(2,2)-4dim-intro}) is equivalent to\n\\begin{equation}\\label{eq-enumerativeCorrelator-Dim4-intro}\n\t \\langle [S_{0,1}],[S_{1,2}],[S_{2,3}],[S_{3,4}],[S_{4,5}],[S_{5,6}],[S_{6,0}]\\rangle_{0,7,2}=1.\n\\end{equation}\nThen we show (\\ref{eq-enumerativeCorrelator-Dim4-intro}) by the definition of Gromov-Witten invariants: we count curves that intersect with the given cycles, and use deformation theory to compute the intersection multiplicity of the image of the virtual fundamental cycle and cycles in $X^7$ (see Definition \\ref{def-enumerativeCorrelator}). As a byproduct we obtain an enumerative result with somewhat classical flavor.\n\\begin{theorem}\\label{thm-countingConics-4dim-general-intro}(= Theorem \\ref{thm-countingConics-4dim-general})\nFor general 4-dimensional smooth complete intersections $X$ of two quadrics in $\\mathbb{P}^6$, there exists exactly one smooth conic that meets each of the 2-planes $S_{i,i+1}$ in $X$ for $0\\leq i\\leq 6$.\n\\end{theorem}\n\nLet us emphasize the significance of the computation of the special correlator. In view of our previous results and conjectures (see \\cite[Conjecture 1.15 and Table 1]{Hu15}), in the algorithmic sense the special correlator is the only genus 0 Gromov-Witten invariant of complete intersections, at least the even dimensional ones, that we have no definite way to compute. But see Remark \\ref{rem:compute-specialCorrelator-fromHigherGenusInv} for a possible approach via higher genus Gromov-Witten invariants with ambient insertions, as a byproduct of Theorem \\ref{thm-intro-convergence-semisimplicity}.\\\\\n\n\n\nWe are now in a position to make an additional remark on the semisimplicity.\nThere seems to be a folklore conjecture that if a smooth projective variety has generically semisimple quantum cohomology, then there exists finitely many correlators that can determine all correlators by WDVV and the Euler field. One might be temped to make a stronger and, quantitative, conjecture: if the values of the correlators of length $\\leq k$ suffice to imply the generic semisimplicity, then they determine all correlators. Our Theorem \\ref{thm-reconstruction-even(2,2)-intro} confirms the first conjecture in the case of the even $(2,2)$-complete intersections. On the other hand, by the \nstatement (\\ref{eq-statement-cutoffEulerField-distinctEigenValues}), Lemma \\ref{lem-specialCorrelator-freedomOfSign} and Theorem \\ref{thm-unknownCorrelator-Even(2,2)-4dim-intro}, the 4-dimensional $(2,2)$-complete intersections are counterexamples to the latter one. From Example \\ref{example-f(6)} and \\ref{example-f(8)} and integrality, one sees that the special correlator of the 6 dimensional and 8 dimensional $(2,2)$-complete intersections also do not vanish, so they are also counterexamples.\\\\\n\n\n\n\n\n\n\n\n\n\n\nIn the Appendix, we present an algorithm based on the proof of Theorem \\ref{thm-reconstruction-even(2,2)-intro} to compute the primary genus 0 Gromov-Witten invariants of an even $(2,2)$-complete intersection of dimension $\\geq 4$, with the special correlator (\\ref{eq-intro-length(n+3)Invariant-even(2,2)}) as an unknown. The algorithm is implemented in our Macaulay2 package \n\\texttt{QuantumCohomologyFanoCompleteIntersection}.\nMoreover we write also a Macaulay2 package\n \\texttt{ConicsOn4DimIntersectionOfTwoQuadrics} \n for the computations in Section \\ref{sec:EnumerativeGeometry-Even(2,2)}. The reader can find the packages in\n\n\\url{https:\/\/github.com\/huxw06\/Quantum-cohomology-of-Fano-complete-intersections}\n\n\\begin{comment}\nThe paper is organized as follows.\n\nIn Section \\ref{sec:preliminaries} we recall basic properties of Gromov-Witten invariants that we will use in this paper. Then we recall the results on the primitive cohomology lattice of even $(2,2)$-complete intersections.\n\\end{comment}\n\n\\vspace{0.2cm}\n\n\n\\emph{Acknowledgement}\\quad \n I am grateful to Hua-Zhong Ke for enlightening discussions. I also thank Huai-Liang Chang, Weiqiang He, Giosu\u00e8 Muratore, Maxim Smirnov, and Jinxing Xu for discussions on various related topics. \n \n This work is supported by NSFC 11701579.\n\n\n\n\n\\section{Preliminaries}\\label{sec:preliminaries}\n\\subsection{Properties of Gromov-Witten invariants}\nWe recall the definition of the Gromov-Witten invariants, and their properties that we need to use in this paper. Our main reference is\n\\cite[Chapter VI]{Man99}. All schemes in this paper are over $\\mathbb{C}$.\n\nLet $X$ be a smooth projective scheme over $\\mathbb{C}$ of dimension $n$. Let $k\\in \\mathbb{Z}_{\\geq 0}$, and $\\beta\\in H_2(X;\\mathbb{Z})\/\\mathrm{tor}$. The stack $\\overline{\\mathcal{M}}_{g,k}(X,\\beta)$ of stable maps of degree $\\beta$ from $k$-point genus $g$ marked semistable curves to $X$ is a proper Deligne-Mumford stack and carries a virtual fundamental class (\\cite{BF97}, \\cite{LT98}) $[\\overline{\\mathcal{M}}_{g,k}(X,\\beta)]^{\\mathrm{vir}}$ of dimension $(1-g)(n-3)+k+c_1(T_X)\\cdot \\beta$. For each $1\\leq i\\leq k$, the section $\\sigma_i$ pulls back the relative cotangent line bundle of the universal curve to form a line bundle on $\\overline{\\mathcal{M}}_{g,k}(X,\\beta)$, whose first Chern class is denoted by $\\psi_i$; moreover there is an associated \\emph{evaluation map} $\\mathrm{ev}_i=f\\circ \\sigma_i$, where $f$ is the universal stable map. For $\\gamma_1,\\dots,\\gamma_k\\in H^*(X;\\mathbb{Q})$ and $a_1,\\dots,a_k\\in \\mathbb{Z}_{\\geq 0}$, there is an associated \\emph{Gromov-Witten invariant}\n\\begin{equation}\n\t\\langle \\psi_1^{a_1}\\gamma_1,\\dots,\\psi_k^{a_k}\\gamma_k\\rangle_{g,k,\\beta}^X:=\\int_{[\\overline{\\mathcal{M}}_{g,k}(X,\\beta)]^{\\mathrm{vir}}}\\prod_{i=1}^{k}\\psi_i^{a_i}\\mathrm{ev}_i^*\\gamma_i\\in \\mathbb{Q}.\n\\end{equation}\nA term like $\\psi_i^{a_i}\\gamma_i$ in $\\langle \\psi_1^{a_1}\\gamma_1,\\dots,\\psi_k^{a_k}\\gamma_k\\rangle_{g,k,\\beta}$ is called an \\emph{insertion} of this invariant. We say $(g,k,\\beta)$ is in the \\emph{stable range} if either $2g-2+k>0$ or $\\beta$ is a nonzero effective curve class.\nIt is convenient to use simplified notations in the following occasions:\n\\begin{enumerate}\n\t\\item[(i)] The superscript $X$ will be omitted when it is obvious;\n\n\t\\item[(iii)] the subscript $k$ might be dropped when $k$ is obvious from the expression;\n\t\\item[(iv)] the subscript $\\beta$ might be dropped when it can be uniquely determined by the insertions and the following condition (\\ref{eq-Dim}), which is always the case for Fano complete intersections in projective spaces.\n\\end{enumerate}\n\nThe GW invariants $\\langle \\psi_1^{a_1}\\gamma_1,\\dots,\\psi_k^{a_k}\\gamma_k\\rangle_{g,k,\\beta}$ with $a_1=\\dots=a_k=0$ are called \\emph{primary}. In this paper we only deal with primary genus 0 invariants, although some results depends on the computation of certain non-primary invariants and genus 1 invariants in \\cite{Hu15}. For brevity we call the genus 0 primary Gromov-Witten invariants \\emph{correlators}.\n\n\nFor simplicity we assume $H^{\\mathrm{odd}}(X;\\mathbb{Q})=0$, which is the case for even dimensional $(2,2)$-complete intersections. We denote $H^*(X)=H^*(X;\\mathbb{C})$. \nFor two cohomology classes $\\gamma_1$ and $\\gamma_2$, we denote the Poincar\u00e9 pairing by\n\\begin{equation}\n\t(\\gamma_1,\\gamma_2):=\\int_X \\gamma_1\\cup \\gamma_2.\n\\end{equation}\nIf a basis $\\gamma_0,\\dots,\\gamma_N$ of $H^*(X)$ is given, we denote $g_{a,b}=(\\gamma_a,\\gamma_b)$ for $0\\leq a,b\\leq N$, and set $(g^{a,b})_{0\\leq a,b\\leq N}$ to be the inverse matrix of $(g_{a,b})_{0\\leq a,b\\leq N}$.\nLet $T^0,\\dots,T^N$ be dual basis with respect to $\\gamma_0,\\dots,\\gamma^N$, then the genus $g$ generating function is defined as\n \\begin{equation}\\label{eq-def-generatingFunction}\n \t\\mathcal{F}_g(T^0,\\dots,T^{N})=\\sum_{k\\geq 0} \\sum_{\\beta} \\frac{1}{k!}\\big\\langle \\sum_{i=0}^N \\gamma_i T^i,\\dots,\\sum_{i=0}^N \\gamma_i T^i\\big\\rangle_{g,k,\\beta},\n \\end{equation}\nwhere the invariants outside of the stable range are defined to be zero, by convention. Here we have implicitly use that $X$ is Fano, so that with fixed insertions there is finitely many $\\beta$ such that the invariant does not vanish. We denote $F= \\mathcal{F}_0$. \n\nWe record some standard properties of GW invariants as follows.\n\\begin{enumerate}\n\t\\item[(i)] Degree 0 correlators:\n\t\t\\begin{gather}\\label{eq-Deg0}\\tag{Deg0}\n\t\t\\langle \\gamma_1,\\dots,\\gamma_k\\rangle_{g,k,0}=\n\t\t\\begin{cases}\n\t\t\\int_{X}\\gamma_1\\cup \\gamma_2\\cup \\gamma_3,& \\mbox{if}\\ g=0, k=3;\\nonumber\\\\\n\t\t-\\frac{1}{24}\\int_X \\gamma_1\\cup c_{n-1}(T_X), & \\mbox{if}\\ g=1, k=1,\\\\\n\t\t0, & \\mbox{if}\\ 2g-2+k\\geq 2.\n\t\t\\end{cases}\n\t\t\\end{gather}\n\t\\item[(ii)]\tSuppose each $\\gamma_i$ has pure real degree $|\\gamma_i|$, then there is the dimension constraint:\n\t\t\\begin{equation}\\label{eq-Dim}\\tag{Dim}\n\t\t\t\\langle \\gamma_{1},\\dots, \\gamma_{k}\\rangle_{g,k,\\beta}=0\\ \\mbox{unless}\\\n\t\t\t\\sum_{i=1}^k \\frac{1}{2}|\\gamma_{b_k}|=(1-g)(n-3)+k+c_1(T_X)\\cap \\beta.\n\t\t\\end{equation}\n\t\\item[(iii)] The $S_n$-equivariance:\n\t\t\\begin{equation}\\label{eq-Sym}\\tag{Sym}\n\t\t\t\\langle \\gamma_{1},\\dots, \\gamma_{i-1}, \\gamma_{i},\\dots, \\gamma_{b_k}\\rangle_{g,k,\\beta}\n\t\t\t=\\langle \\gamma_{1},\\dots, \\gamma_{i}, \\gamma_{i-1},\\dots,\\gamma_{b_k}\\rangle_{g,k,\\beta}.\n\t\t\\end{equation}\n\t\\item[(iv)]\n\t\t\tThe divisor equation: for $\\gamma\\in H^2(X)$,\n\t\t\t\\begin{gather}\\label{eq-Div}\\tag{Div}\n\t\t\t\t\\langle \\gamma_1,\\dots,\\gamma_k,\\gamma\\rangle_{g,k+1,\\beta}=(\\gamma\\cap \\beta)\\langle \\gamma_1,\\dots,\\gamma_k\\rangle_{g,k,\\beta}.\n\t\t\t\\end{gather}\n\t\t\n\t\\item[(v)] Fundamental class axiom:\n\t\t\t\\begin{gather}\\label{eq-FCA}\\tag{FCA}\n\t\t\t\t\\langle 1, \\gamma_{1},\\dots,\\gamma_{k-1}\\rangle_{g,k,\\beta}=\n\t\t\t\t\\begin{cases}\n\t\t\t\t(\\gamma_1,\\gamma_2),& \\mbox{if}\\ g=0, k=3, \\beta=0;\\\\\n\t\t\t\t0, & \\mbox{if}\\ 3g-3+k\\geq 1\\ \\mbox{or}\\ \\beta\\neq 0.\n\t\t\t\t\\end{cases}\n\t\t\t\\end{gather}\n\t\\item[(vi)] If a basis $\\gamma_0,\\dots,\\gamma_N$ of $H^*(X)$ and $T^0,\\dots,T^N$ is the dual basis, then we have the WDVV equations, for $0\\leq a,b\\leq N$:\n\t\t\\begin{gather}\\label{eq-WDVV}\\tag{WDVV}\n\t\t\\sum_{e=0}^N \\sum_{f=0}^N \\frac{\\partial^3 F}{\\partial T^a \\partial T^b\\partial T^e}g^{ef}\\frac{\\partial^3 F}{\\partial T^f \\partial T^c\\partial T^d}\n\t\t=\\sum_{e=0}^N \\sum_{f=0}^N \\frac{\\partial^3 F}{\\partial T^a \\partial T^c\\partial T^e}g^{ef}\\frac{\\partial^3 F}{\\partial T^f \\partial T^b\\partial T^d}.\n\t\t\\end{gather}\n\n\\end{enumerate}\n\n\n\nNow suppose $\\gamma_0,\\dots,\\gamma_N$ is a basis of $H^*(X)$ such that each $\\gamma_i$ has a pure degree, and let $T^0,\\dots,T^N$ be the dual basis have pure degrees. Let\n\\[\nc_1(T_X)=\\sum_{i=0}^N a_i \\gamma_i.\n\\]\nOf course $a_i=0$ unless $|\\gamma_i|=2$. The Euler vector field is defined as\n\\begin{equation}\\label{eq-EV-0}\n\tE=\\sum_{i=0}^N(1-\\frac{|\\gamma_i|}{2})\\frac{\\partial }{\\partial T^i}+\\sum_{i=0}^N a_i \\frac{\\partial }{\\partial T^i}.\n\\end{equation}\nThen (\\ref{eq-Dim}) and the divisor equation (\\ref{eq-Div}) imply \n\\begin{gather}\\label{eq-EulerVectorField}\\tag{EV}\n\tEF=(3-n)F+ \\sum_{i=0}^N a_i \\frac{\\partial }{\\partial T^i} c,\n\\end{gather}\nwhere $c$ is the classical cubic intersection form \n\\begin{eqnarray}\\label{eq-cubicIntersectionForm}\nc(t_0, \\cdots, t^{n+m})=\\sum_a\\sum_b\\sum_c\\frac{t^{a}t^{b}t^{c}}{6}\\int_{X}\\gamma_a \\gamma_b \\gamma_c.\n\\end{eqnarray}\n\nThe \\emph{big quantum product} is defined as\n\\begin{equation}\n \t\\gamma_a\\star \\gamma_b=\\sum_{e}\\sum_f\\frac{\\partial^3 \\mathsf{F}}{\\partial T^a \\partial T^b\\partial T^e}g^{ef}\\gamma_f,\n \\end{equation} \n and the \\emph{small quantum product} is defined as\n\\begin{equation}\n \t\\gamma_a\\diamond \\gamma_b=\\gamma_a\\star \\gamma_b|_{T^0=\\dots=T^N=0}.\n \\end{equation} \n\n\n\n\\subsection{Monodromy group and the \\texorpdfstring{$D_{n+3}$}{D{n+3}} lattice}\\label{sec:monodromy-lattice}\nLet $n\\geq 4$ be even. \nLet $X$ be a complete intersection of two quadric hypersurfaces in $\\mathbb{P}^{n+2}$. Then $H^*(X)=H^{\\mathrm{even}}(X)$, and the Fano index of $X$ is $n-1$. Denote the hyperplane class of $X$ by $\\mathsf{h}$, and\n\\begin{equation}\n\t\\mathsf{h}_{i}=\\underbrace{\\mathsf{h}\\cup\\dots\\cup\\mathsf{h}}_{i\\ \\mathsf{h}'s}.\n\\end{equation}\n\nLet $V=\\mathbb{R}^{n+3}$ be the Euclidean space with the standard inner product. Let $\\varepsilon_1,\\dots,\\varepsilon_{n+3}$ be an orthonormal basis, and let\n\\begin{equation}\\label{eq-roots-D}\n\t\\begin{cases}\n\t\\alpha_i=\\varepsilon_{i}-\\varepsilon_{i+1}\\ \\mbox{for}\\ 1\\leq i\\leq n+2,\\\\\n\t\\alpha_{n+3}=\\varepsilon_{n+2}+\\varepsilon_{n+3}.\n\t\\end{cases}\n\\end{equation} \nThe Weyl group $D_{n+3}\\subset \\mathrm{GL}(n+3,\\mathbb{R})$ is generated the reflections with respect to the $\\alpha_i$'s. If one writes vectors in $\\mathbb{R}^{n+3}$ in terms of the coordinates according to the basis $\\varepsilon_1,\\dots,\\varepsilon_{n+3}$, i.e. \n\\[\n\\mathbf{v}=(v_1,\\dots,v_{n+3})=\\sum_{i=1}^{n+3}v_i \\varepsilon_i,\n\\]\nthen the group $D_{n+3}$ coincides with the group generated by the permutations of the coordinates, and the change of signs\n\\[\n(v_1,\\dots,v_{n+1},v_{n+2},v_{n+3})\\mapsto (v_1,\\dots,v_{n+1},-v_{n+2},-v_{n+3}).\n\\]\nBy \\cite[\\S 5]{Del73}, the monodromy group of the whole family of smooth complete intersections of two quadrics in $\\mathbb{P}^{n+2}$ is isomorphic to $D_{n+3}$, and in this way the primitive cohomology $H^n_{\\mathrm{prim}}(X)$ is a standard representation of $D_{n+3}$, and the integral lattice $H^n_{\\mathrm{prim}}(X)\\cap H^n(X;\\mathbb{Z})$ is generated by the roots $\\alpha_i$'s of $D_{n+3}$. In other words, there is an isomorphism\n\\begin{equation}\\label{eq-isomorphism-primCoh-even(2,2)}\n\tV\\otimes_{\\mathbb{R}}\\mathbb{C}\\xrightarrow{\\sim} H^n_{\\mathrm{prim}}(X),\n\\end{equation}\nvia which lattice in $V$ generated by $\\alpha_1,\\dots,\\alpha_{n+3}$ is mapped onto $H^n_{\\mathrm{prim}}(X)\\cap H^n(X;\\mathbb{Z})$.\n\\begin{proposition}\\label{prop-lattice-sign}\n We equip $H^n_{\\mathrm{prim}}(X)$ with the inner product $(-1)^{\\frac{n}{2}}(.,.)$, where $(.,.)$ is the Poincar\u00e9 pairing. Then (\\ref{eq-isomorphism-primCoh-even(2,2)}) becomes an isometry. \n\\end{proposition}\n\\begin{proof}\nThe Poincar\u00e9 pairing is invariant under monodromies. As we will recall in the following, the degree 2 polynomial invariant of the standard representation of $D_{n+3}$ is generated by the Poincar\u00e9 pairing. By the Hodge index theorem, $(-1)^{\\frac{n}{2}}(.,.)$ is positive definite. So the inner product on $H^n_{\\mathrm{prim}}(X)$ induced from $V$ by (\\ref{eq-isomorphism-primCoh-even(2,2)}) coincides with $(-1)^{\\frac{n}{2}}(.,.)$ up to a positive constant multiple. Since the discriminant of the lattice $\\alpha_1,\\dots,\\alpha_{n+3}$ equals 4, we are left to show that the discriminant of the lattice $H^n_{\\mathrm{prim}}(X)\\cap H^n(X;\\mathbb{Z})$ is $\\pm 4$. First we note that there exists $\\frac{n}{2}$-planes in $X$ (\\cite[Corollary 3.3]{Rei72}), whose intersection number with $\\mathsf{h}_{n\/2}$ is 1, from which it follows that the sub-lattice $L=H^*_{\\mathrm{amb}}(X)\\cap H^*(X;\\mathbb{Z})$ is generated by $\\mathsf{h}_{n\/2}$. \nSince $H^n(X;\\mathbb{Z})$ is unimodular and $(\\mathsf{h}_{n\/2},\\mathsf{h}_{n\/2})=4$, by \\cite[Prop. 5.3.3]{Kit93} the discriminant of $L^{\\perp}=H^n_{\\mathrm{prim}}(X)\\cap H^n(X;\\mathbb{Z})$ is $\\pm 4$. So we are done.\n\\end{proof}\n\n\\begin{corollary}\n$H^{n}(X;\\mathbb{Z})$ is generated by the classes of $\\frac{n}{2}$-planes in $X$.\n\\end{corollary}\n\\begin{proof}\nBy \\cite[Theorem 3.14]{Rei72}, the discriminant of the lattice of the classes of $\\frac{n}{2}$-planes in $X$ is $\\pm 4$, which coincides with that of the lattice $H^{n}(X;\\mathbb{Z})$ (see the proof of Proposition \\ref{prop-lattice-sign}).\n\\end{proof}\n\n\n Via (\\ref{eq-isomorphism-primCoh-even(2,2)}), we identify the vectors $\\varepsilon_i$ and $\\alpha_i$ with their images in $H^n_{\\mathrm{prim}}(X)$. Moreover we define, for $1\\leq i\\leq n+3$,\n\\begin{equation}\\label{eq-normalizedOrthonormalBasis}\n\t\\epsilon_i=\\begin{cases}\n\t\\varepsilon_i, & \\mbox{if}\\ n\\equiv 0 \\mod 4;\\\\\n\t\\sqrt{-1}\\varepsilon_i, & \\mbox{if}\\ n\\equiv 2 \\mod 4.\n\t\\end{cases}\n\\end{equation}\nThen $\\epsilon_1,\\dots,\\epsilon_{n+3}$ is an \\emph{orthonormal basis} of $H^n_{\\mathrm{prim}}(X)$.\n\n\n Let $t^{n+1},\\dots,t^{2n+3}$ be the basis of $H^*_{\\mathrm{prim}}(X)^{\\vee}$ dual to $\\epsilon_1,\\dots,\\epsilon_{n+3}$. By the invariant theory of Weyl groups \\cite[\\S 3.12]{Hum90}, \nthe polynomial invariants of $D_{n+3}$ are generated by $s_{1},\\dots,s_{n+3}$, where\n\\begin{equation}\\label{eq-invariantsOf-typeD-1}\n\ts_{i}=\\frac{1}{(2i)!}\\sum_{j=n+1}^{2n+3}(t^j)^{2i},\\ \\mbox{for}\\ 1\\leq i\\leq n+2,\n\\end{equation}\nand\n\\begin{equation}\\label{eq-invariantsOf-typeD-2}\n\ts_{n+3}=\\prod_{j=n+1}^{2n+3}t^j.\n\\end{equation}\nMoreover, $s_1,\\dots,s_{n+3}$ are algebraically independent. As a consequence of the deformation invariance of Gromov-Witten invariants, we have:\n\\begin{theorem}\\label{thm-monodromy-evenDim(2,2)}\nThe genus $g$ generating function $\\mathcal{F}_g$ of $X$ can be written in a unique way as a series of $s_1,\\dots,s_{n+3}$.\n\\end{theorem}\nOne can see this by directly quoting the definition of Gromov-Witten invariants via symplectic geometry. For an algebraic proof using \\cite[Theorem 4.2]{LT98}, see \\cite[Corollary 3.2]{Hu15}.\n\nWe introduce some notations that will be used throughout this paper. We denote the genus 0 generating function by $F$. The basis dual to $1,\\mathsf{h},\\dots,\\mathsf{h}_{n},\\epsilon_1,\\dots,\\epsilon_{n+3}$ is denoted by $t^0,\\dots,t^{2n+3}$. Denote the small quantum multiplication by $\\diamond$. Let \n\\begin{equation}\n\t\\tilde{\\mathsf{h}}_{i}=\t\\underbrace{\\mathsf{h}\\diamond\\dots\\diamond\\mathsf{h}}_{i\\ \\mathsf{h}'s}.\n\\end{equation}\nThe basis dual to $1,\\tilde{\\mathsf{h}},\\dots,\\tilde{\\mathsf{h}}_{n},\\epsilon_1,\\dots,\\epsilon_{n+3}$ is denoted by $\\tau^0,\\dots,\\tau^{n+3}$.\n\n\\begin{remark}\\label{rem:choiceOfBasis}\nThe choice of the isomorphism from a $D_{n+3}$-lattice to $\\big(H^n_{\\mathrm{prim}}(X)\\cap H^n(X;\\mathbb{Z}),(-1)^{\\frac{n}{2}}(.,.)\\big)$, or equivalently, the choice of $\\varepsilon_1,\\dots,\\varepsilon_{n+3}$ in \n$H^n_{\\mathrm{prim}}(X)$ as above, is not unique. Different choices differ by automorphisms of the $D_{n+3}$-lattice.\n\nBy \\cite[Theorem 1]{KM83}, the automorphism group, which we denote by $G_{n+3}$, of a $D_{n+3}$-lattice is generated by $D_{n+3}$ and the automorphism group of the $D_{n+3}$ Dykin diagram. Since $n>0$, the latter group is $\\mathbb{Z}\/2 \\mathbb{Z}$. Then $G_{n+3}$ is the semidirect product $D_{n+3}\\rtimes \\mathbb{Z}\/2 \\mathbb{Z}$, and $G_{n+3}$ is generated by $D_{n+3}$ and the map which fixes $\\alpha_i$ for $1\\leq i\\leq n+1$ and interchanges $\\alpha_{n+2}$ and $\\alpha_{n+3}$. Combined this with the description of the $D_{n+3}$ action on $V$ recalled at the beginning of this section, we can also describe $G_{n+3}$ as generated by $D_{n+3}$ and the map $-1$ which sends $\\varepsilon_i$ to $-\\varepsilon_i$ for $1\\leq i\\leq n+3$.\n\nAs a consequence of Theorem \\ref{thm-monodromy-evenDim(2,2)}, two choices of the basis $\\varepsilon_1,\\dots,\\varepsilon_{n+3}$ which differ by an action by $g\\in D_{n+3}$ do not affect the values of the correlators and also the generating functions $\\mathcal{F}_g$. But two choices which differ by an action by the map $-1$, do affect, for example the value of the correlator\n\\[\n\\langle \\varepsilon_1,\\dots,\\varepsilon_{n+3}\\rangle_{0,n+3,\\frac{n}{2}}.\n\\]\nIn Section \\ref{sec:explictD-Lattice} we will give an explicit choice of the $D_{n+3}$-roots $\\alpha_i$ and the basis $\\varepsilon_i$.\n\\end{remark}\n\n\n\\section{Correlators of length at most 4}\\label{sec:correlators-lengt-atMost4}\nIn this section, we fix a smooth complete intersection $X$ of two quadrics in $\\mathbb{P}^{n+2}$, where $n$ is even and $\\geq 4$. \n\n\\subsection{Recap of known results}\\label{sec:knownResults-correlators}\nBy Theorem \\ref{thm-monodromy-evenDim(2,2)}, we expand $F$ as a series of $s_1,\\dots,s_{n+3}$; in particular we denote the constant term of this expansion by $F^{(0)}$, and the coefficient of $s_1$ by $F^{(1)}$. Then $F^{(0)}$ is the generating function of genus 0 primary GW invariants with only \\emph{ambient} insertions, and $F^{(1)}$ is the generating function of genus 0 primary GW invariants with exactly two primitive insertions. By \\cite[Example 6.11]{Hu15},\n\\begin{equation}\\label{eq-tauTot}\n\\begin{cases}\n\\tau^0=t^0-4t^{n-1},\\\\\n\\tau^1=t^1-12t^{n},\\\\\n\\tau^i=t^i\\ \\mbox{for}\\ i\\geq 2.\n\\end{cases}\n\\end{equation}\nThen the information of the correlators of length 3 and length 4 with at most two primitive insertions are encoded in (see \\cite[Lemma 6.5]{Hu15})\n\\begin{eqnarray}\\label{eq-qp1.5}\n\\frac{\\partial F^{(0)}}{\\partial \\tau^a \\partial\\tau^b \\partial\\tau^c}(0)=\\left\\{\n\\begin{array}{cc}\n16^{\\frac{a+b+c-n}{n-1}} \\times 4, & \\mathrm{if}\\ \\frac{a+b+c-n}{n-1}\\in \\mathbb{Z}_{\\geq 0}; \\\\\n0, & \\mathrm{otherwise}.\n\\end{array}\n\\right.\n\\end{eqnarray}\nand (see \\cite[Example 6.11]{Hu15})\n\\begin{equation}\\label{eq-F122-tau}\n\\mathsf{F}^{(1)}(\\tau)\n=\\tau^0-2\\sum_{\\begin{subarray}{c}1\\leq i,j\\leq n\\\\\ni+j=n\\end{subarray}\n}\n\\tau^i \\tau^{j}\n-16\\tau^{n-1}\\tau^{n}\n+O\\big((\\tau)^3\\big),\n\\end{equation}\nor equivalently\n\\begin{eqnarray}\\label{eq-F^122}\n\\mathsf{F}^{(1)}\n=t^0-4t^{n-1}-2\\sum_{i=1}^{n-1}t^{i}t^{n-i}-16t^{n-1}t^{n}+O((t)^3).\n\\end{eqnarray}\n\nAs in \\cite[Section 6.1]{Hu15}, we define two transition matrices $M$ and $W$ by\n\\begin{eqnarray}\\label{eq-transform1}\n\\mathsf{h}_i=\\sum_{j=0}^n M_{i}^{j}\\tilde{\\mathsf{h}}_{j},\\\n\\tilde{\\mathsf{h}}_i=\\sum_{j=0}^n W_{i}^{j}\\mathsf{h}_{j},\\ \\mbox{for } 0\\leq i\\leq n,\n\\label{eq-transform1-2}\n\\end{eqnarray}\nor equivalently\n\\begin{eqnarray}\\label{eq-transform2}\n\\tau^i=\\sum_{j=0}^n M_j^i t^j,\\\nt^i=\\sum_{j=0}^n W_j^i \\tau^j.\n\\end{eqnarray}\nThen by (\\ref{eq-tauTot}), \n\\begin{equation}\\label{eq-tTotau}\n\\begin{cases}\nt^0=\\tau^0+4 \\tau^{n-1},\\\\\nt^1=\\tau^1+12 \\tau^{n},\\\\\nt^i=\\tau^i\\ \\mbox{for}\\ i\\geq 2.\n\\end{cases}\n\\end{equation}\nLet $\\gamma_0,\\dots,\\gamma_{2n+3}$ be the basis $1,\\mathsf{h},\\dots,\\mathsf{h}_n,\\epsilon_1,\\dots,\\epsilon_{n+3}$. For $0\\leq e,f\\leq 2n+3$, we define\n\\[\ng_{ef}=(\\gamma_e,\\gamma_f),\n\\]\nand define $(g^{ef})$ to be the inverse matrix of $(g_{ef})_{0\\leq e,f\\leq 2n+3}$. Let $\\widetilde{\\gamma}_0,\\dots,\\widetilde{\\gamma}_{2n+3}$ be the basis\n\\[\n1,\\tilde{\\mathsf{h}},\\dots,\\tilde{\\mathsf{h}}_n,\\epsilon_1,\\dots,\\epsilon_{n+3}.\n \\] For $0\\leq e,f\\leq 2n+3,$ we define\n\\[\n\\eta_{ef}=(\\widetilde{\\gamma}_e,\\widetilde{\\gamma}_f),\n\\]\nand define $(\\eta^{ef})$ to be the inverse matrix of $(\\eta_{ef})_{0\\leq e,f\\leq 2n+3}$. Then by \\cite[Lemma 6.2]{Hu15},\n\\begin{equation}\\label{eq-etaInversePairing-even(2,2)}\n\t\\eta^{ef}=\\begin{cases}\n\t-4,& \\mbox{if}\\ e+f=1;\\\\\n\t\\frac{1}{4},& \\mbox{if}\\ e+f=n,\\\\\n\t\\delta_{e,f},& \\mbox{if}\\ n+1\\leq e,f\\leq 2n+3,\\\\\n\t0,& \\mbox{otherwise}.\n\t\\end{cases}\n\\end{equation}\n\n\n\\subsection{Some preparatory computations}\\label{sec:preparatoryComputation-even(2,2)}\nIn this section we compute several genus 0 GW invariants of $X$ that are needed in the proof of Theorem \\ref{thm-4points-fanoIndex-even(2,2)}. We follow the notations in Section \\ref{sec:monodromy-lattice}. For convenience in summations we denote the basis $\\gamma_0,\\dots,\\gamma_{2n+3}$ also by $1,\\mathsf{h},\\dots,\\mathsf{h}_{n},\\epsilon_1,\\dots,\\epsilon_{n+3}$, and use Einstein's summation convention, where the summation is from $0$ to $2n+3$.\n\nIn principle one can do the symmetric reduction of the WDVV equation for $X$. But the computation is quite complicated and we have not completed it. The following computations use the deformation invariance and the monodromy group in the same spirit as the symmetric reduction.\n\n \\begin{lemma}\\label{lem-3point-inv-even(2,2)}\n\\begin{equation}\\label{eq-3point-inv-even(2,2)}\n\t\\langle \\mathsf{h}_{n-1},\\mathsf{h}_{n-1},\\mathsf{h}_n\\rangle_{0,2}=192.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nBy (\\ref{eq-tauTot}),\n\\begin{eqnarray*}\n&& \\langle \\mathsf{h}_{n-1},\\mathsf{h}_{n-1},\\mathsf{h}_n\\rangle_{0,2}=\\frac{\\partial^3 F}{\\partial t^{n-1}\\partial t^{n-1}\\partial t^n}(0)\\\\\n&=& \\sum_{0\\leq i,j,k\\leq n}\\frac{\\partial \\tau^i}{\\partial t^{n-1}}\\frac{\\partial \\tau^j}{\\partial t^{n-1}}\\frac{\\partial \\tau^k}{\\partial t^{n}}\\frac{\\partial^3 F}{\\partial \\tau^i \\partial \\tau^j\\partial \\tau^k}(0)\\\\\n&=&\\Big(\\big(-4\\frac{\\partial}{\\partial \\tau^0}+\\frac{\\partial}{\\partial \\tau^{n-1}}\\big)\n\\big(-4\\frac{\\partial}{\\partial \\tau^0}+\\frac{\\partial}{\\partial \\tau^{n-1}}\\big)\n\\big(-12\\frac{\\partial}{\\partial \\tau^1}+\\frac{\\partial}{\\partial \\tau^{n}}\\big)F\\Big)|_{\\tau=0}.\n\\end{eqnarray*}\nThen from (\\ref{eq-qp1.5}) we get (\\ref{eq-3point-inv-even(2,2)}).\n\\end{proof}\n\n\n\\begin{lemma}\\label{lem-5point-withTwoPrim-inv-even(2,2)}\n\\begin{equation}\\label{eq-5point-withTwoPrim-inv-even(2,2)}\n\t\\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_n,\\mathsf{h}_{n-1},\\mathsf{h}_n\\rangle_{0,3}=-192.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nBy (\\ref{eq-WDVV}) for $1\\leq a\\neq b\\leq n+3$,\n\\begin{eqnarray}\\label{lem-5point-withTwoPrim-inv-even(2,2)-1}\n&& \\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_n,\\mathsf{h}_{n-1},\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_b,\\epsilon_b\\rangle_0\n+\\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_n,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\mathsf{h}_{n-1},\\epsilon_b,\\epsilon_b\\rangle_0\\nonumber\\\\\n&&+\\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_{n-1},\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\mathsf{h}_n,\\epsilon_b,\\epsilon_b\\rangle_0\n+\\langle \\epsilon_a,\\epsilon_a,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\mathsf{h}_n,\\mathsf{h}_{n-1},\\epsilon_b,\\epsilon_b\\rangle_0\\nonumber\\\\\n&=& \\langle \\epsilon_a,\\epsilon_b,\\mathsf{h}_n,\\mathsf{h}_{n-1},\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_a,\\epsilon_b\\rangle_0\n+\\langle \\epsilon_a,\\epsilon_b,\\mathsf{h}_n,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\mathsf{h}_{n-1},\\epsilon_a,\\epsilon_b\\rangle_0\\nonumber\\\\\n&&+\\langle \\epsilon_a,\\epsilon_b,\\mathsf{h}_{n-1},\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\mathsf{h}_n,\\epsilon_a,\\epsilon_b\\rangle_0\n+\\langle \\epsilon_a,\\epsilon_b,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\mathsf{h}_n,\\mathsf{h}_{n-1},\\epsilon_a,\\epsilon_b\\rangle_0.\n\\end{eqnarray}\nBy (\\ref{eq-Dim}), (\\ref{eq-FCA}) and Theorem \\ref{thm-monodromy-evenDim(2,2)}, the LHS of (\\ref{lem-5point-withTwoPrim-inv-even(2,2)-1}) equals\n\\begin{eqnarray}\\label{lem-5point-withTwoPrim-inv-even(2,2)-2}\n&& \\frac{1}{4}\\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_n,\\mathsf{h}_{n-1},\\mathsf{h}_n\\rangle_{0,3} \n+\\frac{1}{4}\\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_n,\\mathsf{h}_{n-1},\\mathsf{h}\\rangle_{0,2} \\langle \\mathsf{h}_{n-1},\\epsilon_b,\\epsilon_b\\rangle_{0,1}\\nonumber\\\\\n&&+\\frac{1}{4}\\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_n,\\mathsf{h}_{n-1}\\rangle_{0,2} \\langle \\mathsf{h},\\mathsf{h}_{n-1},\\epsilon_b,\\epsilon_b\\rangle_{0,1}\n+\\frac{1}{4}\\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_{n-1},\\mathsf{h}\\rangle_{0,1} \\langle \\mathsf{h}_{n-1},\\mathsf{h}_n,\\epsilon_b,\\epsilon_b\\rangle_{0,2}\\nonumber\\\\\n&&+\\frac{1}{4}\\langle \\mathsf{h}_n,\\mathsf{h}_n,\\mathsf{h}_{n-1},\\epsilon_b,\\epsilon_b\\rangle_{0,3}\n+\\frac{1}{4}\\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_{n-1}\\rangle_{0,1}\\langle \\mathsf{h},\\mathsf{h}_n,\\mathsf{h}_{n-1},\\epsilon_b,\\epsilon_b\\rangle_{0,2},\n\\end{eqnarray}\nand the RHS of (\\ref{lem-5point-withTwoPrim-inv-even(2,2)-1}) is 0. \nBy (\\ref{eq-F^122}) and (\\ref{eq-Div}),\n\\begin{equation}\\label{lem-5point-withTwoPrim-inv-even(2,2)-3}\n\t\\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_{n-1},\\mathsf{h}\\rangle_{0,1}=\\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_{n-1}\\rangle_{0,1}=-4,\n\\end{equation}\nand\n\\begin{equation}\\label{lem-5point-withTwoPrim-inv-even(2,2)-4}\n\t\\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_n,\\mathsf{h}_{n-1}\\rangle_{0,2}=-16.\n\\end{equation}\nThen by (\\ref{eq-3point-inv-even(2,2)}), (\\ref{lem-5point-withTwoPrim-inv-even(2,2)-3}) and (\\ref{lem-5point-withTwoPrim-inv-even(2,2)-4}), (\\ref{lem-5point-withTwoPrim-inv-even(2,2)-1}) reads\n\\begin{eqnarray*}\n&& \\frac{1}{4}\\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_n,\\mathsf{h}_{n-1},\\mathsf{h}_n\\rangle_{0,3} \n+\\frac{1}{4}\\times 2\\times (-16)\\times (-4)\n+\\frac{1}{4}\\times (-16)\\times (-4)\\\\\n&&+\\frac{1}{4}\\times (-4)\\times(-16)\n+\\frac{1}{4}\\langle \\mathsf{h}_n,\\mathsf{h}_n,\\mathsf{h}_{n-1},\\epsilon_b,\\epsilon_b\\rangle_{0,3}\n+\\frac{1}{4}\\times(-4)\\times 2\\times(-16)=0.\n\\end{eqnarray*}\nBy Theorem \\ref{thm-monodromy-evenDim(2,2)}, \n\\begin{equation}\n\t\\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_n,\\mathsf{h}_{n-1},\\mathsf{h}_n\\rangle_{0,3}=\\langle \\mathsf{h}_n,\\mathsf{h}_n,\\mathsf{h}_{n-1},\\epsilon_b,\\epsilon_b\\rangle_{0,3}.\n\\end{equation}\nSo we obtain (\\ref{eq-5point-withTwoPrim-inv-even(2,2)}).\n\\end{proof}\n\n\n\\begin{lemma}\\label{lem-5point-inv-even(2,2)-1}\n\\begin{equation}\\label{eq-5point-inv-even(2,2)-1}\n\t\\langle \\epsilon_a,\\epsilon_a,\\epsilon_b,\\epsilon_b,\\mathsf{h}_n\\rangle_{0,2}=4-4\\langle \\epsilon_a,\\epsilon_b,\\epsilon_a,\\epsilon_b\\rangle_{0,1}.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nBy (\\ref{eq-WDVV}), for $1\\leq a\\neq b\\leq n+3$,\n\\begin{eqnarray*}\n&& \\langle \\epsilon_a,\\mathsf{h}_n,\\epsilon_a,\\epsilon_b,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_b,\\mathsf{h}_{n-1}\\rangle_0\n+\\langle \\epsilon_a,\\mathsf{h}_n,\\epsilon_a,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_b,\\epsilon_b,\\mathsf{h}_{n-1}\\rangle_0\\\\\n&&+\\langle \\epsilon_a,\\mathsf{h}_n,\\epsilon_b,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_a,\\epsilon_b,\\mathsf{h}_{n-1}\\rangle_0\n+\\langle \\epsilon_a,\\mathsf{h}_n,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_a,\\epsilon_b,\\epsilon_b,\\mathsf{h}_{n-1}\\rangle_0\\\\\n&=& \\langle \\epsilon_a,\\epsilon_b,\\epsilon_a,\\epsilon_b,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\mathsf{h}_n,\\mathsf{h}_{n-1}\\rangle_0\n+\\langle \\epsilon_a,\\epsilon_b,\\epsilon_a,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_b,\\mathsf{h}_n,\\mathsf{h}_{n-1}\\rangle_0\\\\\n&&+\\langle \\epsilon_a,\\epsilon_b,\\epsilon_b,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_a,\\mathsf{h}_n,\\mathsf{h}_{n-1}\\rangle_0\n+\\langle \\epsilon_a,\\epsilon_b,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_a,\\epsilon_b,\\mathsf{h}_n,\\mathsf{h}_{n-1}\\rangle_0.\n\\end{eqnarray*}\nThen (\\ref{eq-Dim}), (\\ref{eq-FCA}) and Theorem \\ref{thm-monodromy-evenDim(2,2)} yield\n\\begin{eqnarray}\\label{eq-lem-5point-inv-even(2,2)-1-1}\n&& \\langle \\epsilon_a,\\mathsf{h}_n,\\epsilon_a,\\epsilon_b,\\epsilon_b\\rangle_{0,2} \\langle \\epsilon_b,\\epsilon_b,\\mathsf{h}_{n-1}\\rangle_{0,1}\n+\\frac{1}{4}\\langle \\epsilon_a,\\mathsf{h}_n,\\epsilon_a,\\mathsf{h}_{n-1}\\rangle_{0,2} \\langle \\mathsf{h},\\epsilon_b,\\epsilon_b,\\mathsf{h}_{n-1}\\rangle_{0,1}\\nonumber\\\\\n&=&\\frac{1}{4} \\langle \\epsilon_a,\\epsilon_b,\\epsilon_a,\\epsilon_b,\\mathsf{h}\\rangle_{0,1} \\langle \\mathsf{h}_{n-1},\\mathsf{h}_n,\\mathsf{h}_{n-1}\\rangle_{0,2}\n+\\langle \\epsilon_a,\\epsilon_b,\\epsilon_a,\\epsilon_b\\rangle_{0,1} \\langle \\epsilon_b,\\epsilon_b,\\mathsf{h}_n,\\mathsf{h}_{n-1}\\rangle_{0,2}\\nonumber\\\\\n&&+\\langle \\epsilon_a,\\epsilon_b,\\epsilon_b,\\epsilon_a\\rangle_{0,1} \\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_n,\\mathsf{h}_{n-1}\\rangle_{0,2}.\n\\end{eqnarray}\nThen by (\\ref{eq-3point-inv-even(2,2)}), (\\ref{lem-5point-withTwoPrim-inv-even(2,2)-3}) and (\\ref{lem-5point-withTwoPrim-inv-even(2,2)-4}), (\\ref{eq-lem-5point-inv-even(2,2)-1-1}) reads\n\\begin{eqnarray*}\n&& -4\\langle \\epsilon_a,\\mathsf{h}_n,\\epsilon_a,\\epsilon_b,\\epsilon_b\\rangle_{0,2}\n+\\frac{1}{4} (-16)(-4)\\\\\n&=&\\frac{1}{4} \\langle \\epsilon_a,\\epsilon_b,\\epsilon_a,\\epsilon_b\\rangle_{0,1} \\times 192\n-16\\langle \\epsilon_a,\\epsilon_b,\\epsilon_a,\\epsilon_b\\rangle_{0,1} -16\\langle \\epsilon_a,\\epsilon_b,\\epsilon_b,\\epsilon_a\\rangle_{0,1}.\n\\end{eqnarray*}\nSo we get (\\ref{eq-5point-inv-even(2,2)-1}).\n\\end{proof}\n\n\\begin{lemma}\\label{lem-5point-inv-even(2,2)-2}\n\\begin{equation}\\label{eq-5point-inv-even(2,2)-2}\n\t \\langle \\epsilon_a,\\mathsf{h}_n,\\epsilon_a,\\epsilon_a,\\epsilon_a\\rangle_{0,2} \n= 12-4\\langle \\epsilon_a,\\epsilon_a,\\epsilon_a,\\epsilon_a\\rangle_{0,1}.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nBy (\\ref{eq-WDVV}), for $1\\leq a\\leq n+3$,\n\\begin{eqnarray}\\label{eq-lem-5point-inv-even(2,2)-1}\n&& \\langle \\epsilon_a,\\mathsf{h}_n,\\epsilon_a,\\epsilon_a,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_a,\\mathsf{h}_{n-1}\\rangle_0\n+2\\langle \\epsilon_a,\\mathsf{h}_n,\\epsilon_a,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_a,\\epsilon_a,\\mathsf{h}_{n-1}\\rangle_0\\nonumber\\\\\n&&+\\langle \\epsilon_a,\\mathsf{h}_n,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_a,\\epsilon_a,\\epsilon_a,\\mathsf{h}_{n-1}\\rangle_0\\nonumber\\\\\n&=& \\langle \\epsilon_a,\\epsilon_a,\\epsilon_a,\\epsilon_a,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\mathsf{h}_n,\\mathsf{h}_{n-1}\\rangle_0\n+2\\langle \\epsilon_a,\\epsilon_a,\\epsilon_a,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_a,\\mathsf{h}_n,\\mathsf{h}_{n-1}\\rangle_0\\nonumber\\\\\n&&+\\langle \\epsilon_a,\\epsilon_a,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_a,\\epsilon_a,\\mathsf{h}_n,\\mathsf{h}_{n-1}\\rangle_0,\n\\end{eqnarray}\nThen (\\ref{eq-Dim}), (\\ref{eq-FCA}) and Theorem \\ref{thm-monodromy-evenDim(2,2)} yield\n\\begin{eqnarray*}\n&& \\langle \\epsilon_a,\\mathsf{h}_n,\\epsilon_a,\\epsilon_a,\\epsilon_a\\rangle_{0,2} \\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_{n-1}\\rangle_{0,1}\n+2\\cdot\\frac{1}{4}\\langle \\epsilon_a,\\mathsf{h}_n,\\epsilon_a,\\mathsf{h}_{n-1}\\rangle_{0,2} \\langle \\mathsf{h},\\epsilon_a,\\epsilon_a,\\mathsf{h}_{n-1}\\rangle_{0,1}\\\\\n&=&\\frac{1}{4} \\langle \\epsilon_a,\\epsilon_a,\\epsilon_a,\\epsilon_a,\\mathsf{h}\\rangle_{0,1}\n\\langle \\mathsf{h}_{n-1},\\mathsf{h}_n,\\mathsf{h}_{n-1}\\rangle_{0,2}\n+2\\langle \\epsilon_a,\\epsilon_a,\\epsilon_a,\\epsilon_a\\rangle_{0,1} \\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_n,\\mathsf{h}_{n-1}\\rangle_{0,2}\\\\\n&&+\\frac{1}{4}\\langle \\epsilon_a,\\epsilon_a,1\\rangle_{0,0} \\langle \\mathsf{h}_n,\\epsilon_a,\\epsilon_a,\\mathsf{h}_n,\\mathsf{h}_{n-1}\\rangle_{0,3}\n+\\frac{1}{4}\\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_{n-1}\\rangle_{0,1} \\langle \\mathsf{h},\\epsilon_a,\\epsilon_a,\\mathsf{h}_n,\\mathsf{h}_{n-1}\\rangle_{0,2}.\n\\end{eqnarray*}\nBy (\\ref{eq-3point-inv-even(2,2)}), (\\ref{lem-5point-withTwoPrim-inv-even(2,2)-3}), (\\ref{lem-5point-withTwoPrim-inv-even(2,2)-4}), (\\ref{eq-5point-withTwoPrim-inv-even(2,2)}) and (\\ref{eq-Div}), (\\ref{eq-lem-5point-inv-even(2,2)-1}) reads\n\\begin{eqnarray*}\n&& -4\\langle \\epsilon_a,\\mathsf{h}_n,\\epsilon_a,\\epsilon_a,\\epsilon_a\\rangle_{0,2} \n+2\\cdot\\frac{1}{4} (-16)(-4)\\\\\n&=& 16 \\langle \\epsilon_a,\\epsilon_a,\\epsilon_a,\\epsilon_a\\rangle_{0,1}\n+\\frac{1}{4}\\cdot (-192)\n+\\frac{1}{4}\\times (-4)\\times 2\\times (-16),\n\\end{eqnarray*}\nso we obtain (\\ref{eq-5point-inv-even(2,2)-2}).\n\\end{proof}\n\n\n\\subsection{Correlators of length 4}\\label{sec:correlators-length4}\nBy Theorem \\ref{thm-monodromy-evenDim(2,2)} and the results recalled in Section \\ref{sec:knownResults-correlators}, all the correlators of length 4 other than the ones with only primitive insertions are computed. To compute the latter ones, we begin to recall a result on the length 4 correlators with only primitive insertions. Let $X$ be a smooth complete intersection of dimension $n$ and of Fano index $n-1$. Then $X$ is a cubic hypersurface i.e. the multidegree $\\mathbf{d}=3$, or a complete intersection of two quadrics i.e. the multidegree $\\mathbf{d}=(2,2)$. Let $m=\\dim H^*_{\\mathrm{prim}}(X)$, and $\\gamma_{n+1},\\dots,\\gamma_{n+m}$ be a basis of $H^*_{\\mathrm{prim}}(X)$. Then by \\cite[Proposition 9.12]{Hu15}, \n\\begin{equation}\\label{eq-4points-sum}\n\\sum_{e=n+1}^{n+m}\\sum_{f=n+1}^{n+m}\\langle \\gamma_b, \\gamma_e, g^{ef}\\gamma_f,\\gamma_c\\rangle_{0,1}\n=g_{bc}\\cdot\\begin{cases}\n\\frac{(-2)^{n+2}+8}{3},&\n \\mbox{if}\\ \\mathbf{d}=3;\\\\\n (-1)^n(n+1)+2,\n &\n \\mbox{if}\\ \\mathbf{d}=(2,2)\n\\end{cases}\n\\end{equation}\nWhen $X$ is a cubic hypersurface, or $X$ is an odd dimensional complete intersection of two quadrics, the Zariski closure of the monodromy group is the orthogonal group or the symplectic group, according to the parity of the dimension. Then as we have seen in \\cite[Theorem 9.13]{Hu15}, (\\ref{eq-4points-sum}) suffices to give all the length 4 correlators of $X$. For the exceptional case that $X$ is an even dimensional complete intersection of two quadrics, we need some ad hoc computations. \n\\begin{proposition}\\label{prop-4points-fanoIndex-even(2,2)}\nLet $X=X_n(2,2)$ of even dimension $n\\geq 4$. Then\n\\begin{equation}\\label{eq-4points-fanoIndex-even(2,2)-0}\n\t(n+5)\\frac{\\partial^2 F}{(\\partial s_1)^2}(0)+\\frac{\\partial F}{\\partial s_2}(0)\n\t=n+3.\n\\end{equation}\n\\end{proposition}\n\\begin{proof}\nTake $\\gamma_{n+1},\\dots,\\gamma_{2n+3}$ in \\ref{eq-4points-sum} to be the orthonormal basis $\\epsilon_1,\\dots,\\epsilon_{n+3}$ of $H^*_{\\mathrm{prim}}(X)$. Then\n\\[\n\\sum_{e=n+1}^{n+m}\\sum_{f=n+1}^{n+m}\\langle \\gamma_b, \\gamma_e, g^{ef}\\gamma_f,\\gamma_c\\rangle_{0,1}\n=\\sum_{e=1}^{n+3}\\langle \\epsilon_b, \\epsilon_c,\\epsilon_e,\\epsilon_e\\rangle_{0,1}.\n\\]\nBy (\\ref{eq-invariantsOf-typeD-1}) and the definition (\\ref{eq-def-generatingFunction}) of the generating function $F$, one finds\n\\begin{equation}\\label{eq-prop-4points-fanoIndex-even(2,2)-1}\n\\langle \\epsilon_b,\\epsilon_b,\\epsilon_e,\\epsilon_e\\rangle=\\begin{cases}\n\\frac{\\partial^2 F}{(\\partial s_1)^2}(0),& \\mbox{if}\\ 1\\leq e\\leq n+3\\ \\mbox{and}\\ e\\neq b,\\\\\n3\\frac{\\partial^2 F}{(\\partial s_1)^2}(0)+\\frac{\\partial F}{\\partial s_2}(0), & \\mbox{if}\\ e=b.\n\\end{cases}\n\\end{equation}\nSo\n\\begin{equation}\\label{eq-4points-sum-even(2,2)-1}\n\\sum_{e=n+1}^{n+m}\\sum_{f=n+1}^{n+m}\\langle \\gamma_b, \\gamma_e, g^{ef}\\gamma_f,\\gamma_c\\rangle_{0,1}\n=(n+5)\\frac{\\partial^2 F}{(\\partial s_1)^2}(0) \\delta_{b,c}+\\frac{\\partial F}{\\partial s_2}(0)\\delta_{b,c}.\n\\end{equation}\nOn the other hand by (\\ref{eq-4points-sum}) one finds\n\\begin{equation}\\label{eq-4points-sum-even(2,2)-2}\n\\sum_{e=n+1}^{n+m}\\sum_{f=n+1}^{n+m}\\langle \\gamma_b, \\gamma_e, g^{ef}\\gamma_f,\\gamma_c\\rangle_{0,1}\n=(n+3)\\delta_{b,c}.\n\\end{equation}\nComparing (\\ref{eq-4points-sum-even(2,2)-1}) and (\\ref{eq-4points-sum-even(2,2)-2}) we get (\\ref{eq-4points-fanoIndex-even(2,2)-0}).\n\\end{proof}\n\nUsing (\\ref{eq-WDVV}) and Theorem \\ref{thm-monodromy-evenDim(2,2)}, we can get a quadratic equation for $\\frac{\\partial^2 F}{(\\partial s_1)^2}(0)$ and $\\frac{\\partial F}{\\partial s_2}(0)$. Then we determine the correct solution by the integrality of certain invariants.\n\n\\begin{lemma}\\label{lem-4points-even(2,2)-integrality}\nLet $X=X_n(2,2)$. Let $\\alpha_i\\in H^n_{\\mathrm{prim}}(X)$ be the primitive classes corresponding to simple roots of $D_{n+3}$ as (\\ref{eq-roots-D}), for $1\\leq i\\leq n+3$. Then\n\\begin{equation}\\label{eq-4points-even(2,2)-integrality}\n\\langle \\alpha_{i},\\alpha_{j},\\alpha_{k},\\alpha_{l}\\rangle_{0,1}\\in \\mathbb{Z},\\ \\mbox{for}\\ 1\\leq i,j,k,l\\leq n+3.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nBy \\cite[Proposition 13.9]{Lew99}, for a general member $X$ in the family of $n$-dimensional smooth complete intersections of two quadrics, the Fano variety of lines $\\overline{\\mathcal{M}}_{0,0}(X,1)$\nis a smooth scheme of dimension $2n-4$. It follow that $\\overline{\\mathcal{M}}_{0,0}(X,1)$ has the expected dimension. Moreover one directly sees that $\\overline{\\mathcal{M}}_{0,0}(X,1)$ has no stacky point. Since $\\alpha_i\\in H^n(X;\\mathbb{Z})$, we have (\\ref{eq-4points-even(2,2)-integrality}) for such $X$. Then (\\ref{eq-4points-even(2,2)-integrality}) holds for all $n$-dimensional smooth complete intersections of two quadrics, by the deformation invariance. \n\\end{proof}\n\n\\begin{remark}\\label{rem:integreality-symplectic}\nOne can also directly apply the integrality of genus 0 Gromov-Witten invariants of with integral classes as insertions, of semipositive symplectic manifolds (e.g. \\cite[Theorem A]{Ruan96}).\n\\end{remark}\n\n\\begin{theorem}\\label{thm-4points-fanoIndex-even(2,2)}\nLet $X$ be an even dimensional complete intersection of two quadrics in $\\mathbb{P}^{n+2}$, with $n\\geq 4$. Then\n\\begin{equation}\\label{eq-4points-fanoIndex-even(2,2)}\n\t\\frac{\\partial^2 F}{(\\partial s_1)^2}(0)=1,\\ \\frac{\\partial F}{\\partial s_2}(0)=-2.\n\\end{equation}\nEquivalently, for $1\\leq a,b\\leq n+3$,\n\\begin{equation}\\label{eq-4points-fanoIndex-even(2,2)-ab}\n\t\\langle \\epsilon_a,\\epsilon_a,\\epsilon_b,\\epsilon_b\\rangle_{0,1}=1.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nBy (\\ref{eq-WDVV}), for $n+1\\leq a\\neq b\\leq 2n+3$,\n\\begin{eqnarray}\\label{eq-thm-4points-fanoIndex-even(2,2)-1}\n&&\\sum_{e=0}^{2n+3}\\sum_{f=0}^{2n+3}\\big(\\langle \\epsilon_a,\\epsilon_a,\\epsilon_a,\\epsilon_a,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_b,\\epsilon_b\\rangle_0+2\\langle \\epsilon_a,\\epsilon_a,\\epsilon_a,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_a,\\epsilon_b,\\epsilon_b\\rangle_0\n\\nonumber\\\\\n&&+\\langle \\epsilon_a,\\epsilon_a,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_a,\\epsilon_a,\\epsilon_b,\\epsilon_b\\rangle_0\\big)\\nonumber\\\\\n&=&\\sum_{e=0}^{2n+3}\\sum_{f=0}^{2n+3}\\big(2 \\langle \\epsilon_a,\\epsilon_b,\\epsilon_a,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_a,\\epsilon_a,\\epsilon_b\\rangle_0\n+2 \\langle \\epsilon_a,\\epsilon_b,\\epsilon_a,\\epsilon_a,\\gamma_e\\rangle_0 g^{ef}\\langle \\gamma_f,\\epsilon_a,\\epsilon_b\\rangle_0\\big).\\nonumber\\\\\n\\end{eqnarray}\nBy Theorem \\ref{thm-monodromy-evenDim(2,2)} and (\\ref{eq-prop-4points-fanoIndex-even(2,2)-1}), the RHS of (\\ref{eq-thm-4points-fanoIndex-even(2,2)-1}) equals\n\\begin{eqnarray}\\label{eq-thm-4points-fanoIndex-even(2,2)-2}\n2 \\langle \\epsilon_a,\\epsilon_b,\\epsilon_a,\\epsilon_b\\rangle_{0,1} \\langle \\epsilon_b,\\epsilon_a,\\epsilon_a,\\epsilon_b\\rangle_{0,1}=2\\big(\\frac{\\partial^2 F}{\\partial s_1^2}(0)\n\\big)^2.\n\\end{eqnarray}\nBy (\\ref{eq-Dim}), (\\ref{eq-FCA}) and Theorem \\ref{thm-monodromy-evenDim(2,2)}, the LHS (\\ref{eq-thm-4points-fanoIndex-even(2,2)-1}) equals\n\\begin{eqnarray}\\label{eq-thm-4points-fanoIndex-even(2,2)-3}\n&&\\frac{1}{4}\\langle \\epsilon_a,\\epsilon_a,\\epsilon_a,\\epsilon_a,\\mathsf{h}\\rangle_{0,1} \\langle \\mathsf{h}_{n-1},\\epsilon_b,\\epsilon_b\\rangle_{0,1}\n+\\frac{1}{4}\\langle \\epsilon_a,\\epsilon_a,\\epsilon_a,\\epsilon_a,\\mathsf{h}_n\\rangle_{0,2} \\langle 1,\\epsilon_b,\\epsilon_b\\rangle_{0,0}\\nonumber\\\\\n&&+2\\langle \\epsilon_a,\\epsilon_a,\\epsilon_a,\\epsilon_a\\rangle_{0,1} \\langle \\epsilon_a,\\epsilon_a,\\epsilon_b,\\epsilon_b\\rangle_{0,1}\n+\\frac{1}{4}\\langle \\epsilon_a,\\epsilon_a,1\\rangle_{0,0} \\langle \\mathsf{h}_{n},\\epsilon_a,\\epsilon_a,\\epsilon_b,\\epsilon_b\\rangle_{0,2}\\nonumber\\\\\n&&+\\frac{1}{4}\\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_{n-1}\\rangle_{0,1} \\langle \\mathsf{h},\\epsilon_a,\\epsilon_a,\\epsilon_b,\\epsilon_b\\rangle_{0,1}.\n\\end{eqnarray}\nSince $\\epsilon_a$ and $\\epsilon_b$ are chosen to be orthonormal, we have\n\\begin{equation}\\label{eq-thm-4points-fanoIndex-even(2,2)-4}\n\t1=\\langle 1,\\epsilon_a,\\epsilon_a\\rangle_{0,0}=\\langle 1,\\epsilon_b,\\epsilon_b\\rangle_{0,0}.\n\\end{equation}\nBy (\\ref{eq-F^122}),\n\\begin{equation}\\label{eq-thm-4points-fanoIndex-even(2,2)-5}\n\t\\langle \\epsilon_a,\\epsilon_a,\\mathsf{h}_{n-1}\\rangle_{0,1}=\\langle \\epsilon_b,\\epsilon_b,\\mathsf{h}_{n-1}\\rangle_{0,1}=-4.\n\\end{equation}\nBy (\\ref{eq-prop-4points-fanoIndex-even(2,2)-1}),\n\\begin{equation}\\label{eq-thm-4points-fanoIndex-even(2,2)-6}\n\t\\langle \\epsilon_a,\\epsilon_a,\\epsilon_a,\\epsilon_a\\rangle_{0,1}=3\\frac{\\partial^2 F}{\\partial s_1^2}(0)+\\frac{\\partial F}{\\partial s_2}(0).\n\\end{equation}\nUsing (\\ref{eq-Div}), by (\\ref{eq-thm-4points-fanoIndex-even(2,2)-4}), (\\ref{eq-thm-4points-fanoIndex-even(2,2)-5}), (\\ref{eq-thm-4points-fanoIndex-even(2,2)-6}),\n and (\\ref{eq-5point-inv-even(2,2)-1}), (\\ref{eq-5point-inv-even(2,2)-2}) in Section \\ref{sec:preparatoryComputation-even(2,2)}, (\\ref{eq-thm-4points-fanoIndex-even(2,2)-3}) equals\n\\begin{eqnarray*}\n 6\\big(\\frac{\\partial^2 F}{\\partial s_1^2}(0)\\big)^2+2\\frac{\\partial F}{\\partial s_2}(0)\\frac{\\partial^2 F}{\\partial s_1^2}(0)-8\\frac{\\partial^2 F}{\\partial s_1^2}(0)\n-2 \\frac{\\partial F}{\\partial s_2}(0)+4.\n\\end{eqnarray*}\nSo (\\ref{eq-thm-4points-fanoIndex-even(2,2)-1}) yields\n\\begin{equation}\\label{eq-thm-4points-fanoIndex-even(2,2)-7}\n\t4\\big(\\frac{\\partial^2 F}{\\partial s_1^2}(0)\\big)^2+2\\frac{\\partial F}{\\partial s_2}(0)\\frac{\\partial^2 F}{\\partial s_1^2}(0)-8\\frac{\\partial^2 F}{\\partial s_1^2}(0)\n-2 \\frac{\\partial F}{\\partial s_2}(0)+4=0.\n\\end{equation}\nSubstituting (\\ref{eq-4points-fanoIndex-even(2,2)-0}) into (\\ref{eq-thm-4points-fanoIndex-even(2,2)-7}),\nwe obtain\n\\begin{equation*}\n\t\\big((n+3)\\frac{\\partial^2 F}{\\partial s_1^2}(0)-(n+1)\\big)(\\frac{\\partial^2 F}{\\partial s_1^2}(0)-1)=0.\n\\end{equation*}\nThus\n\\begin{equation*}\n\t\\frac{\\partial^2 F}{\\partial s_1^2}(0)=\\frac{n+1}{n+3}\\ \\mbox{or}\\ 1.\n\\end{equation*}\nRecall the simple roots (\\ref{eq-roots-D}). We have\n\\begin{equation*}\n\\langle \\alpha_{1},\\alpha_{1},\\alpha_{n+3},\\alpha_{n+3}\\rangle_{0,1}=\\frac{\\partial^2 F}{\\partial s_1^2}(0)\\cdot (\\alpha_{1},\\alpha_{1})(\\alpha_{n+3},\\alpha_{n+3})=4 \\frac{\\partial^2 F}{\\partial s_1^2}(0).\n\\end{equation*}\nBut by Lemma \\ref{lem-4points-even(2,2)-integrality}, $\\langle \\alpha_{1},\\alpha_{1},\\alpha_{n+3},\\alpha_{n+3}\\rangle_{0,1}\\in \\mathbb{Z}$. Since $n$ is even, $\\frac{4(n+1)}{n+3}$ is never an integer. Hence\n\\begin{equation}\n\t\\frac{\\partial^2 F}{\\partial s_1^2}(0)=1,\n\\end{equation}\nand by (\\ref{eq-4points-fanoIndex-even(2,2)-0}) we get $\\frac{\\partial F}{\\partial s_2}(0)=-2$.\nThe formula (\\ref{eq-4points-fanoIndex-even(2,2)-ab}) follows then from (\\ref{eq-prop-4points-fanoIndex-even(2,2)-1}).\n\\end{proof}\n\n\n\n\\section{A reconstruction theorem}\\label{sec:reconstructionTheorem}\nIn this section, we fix a smooth complete intersection $X$ of two quadrics in $\\mathbb{P}^{n+2}$, where $n$ is even and $\\geq 4$. \nThe aim of this section is to show that, with the results in Section \\ref{sec:correlators-lengt-atMost4}, besides a special GW invariant, we can compute all genus zero GW invariants of $X$ by (\\ref{eq-Dim}), (\\ref{eq-EulerVectorField}), (\\ref{eq-WDVV}) and the deformation invariance. We begin with an easy observation.\nBy (\\ref{eq-Dim}), a length $k$ genus 0 GW invariant of $X$ with only primitive insertions is zero unless\n\\begin{equation}\n\tk\\cdot \\frac{n}{2}=n-3+k+\\beta\\cdot(n-1),\n\\end{equation}\ni.e.\n\\[\n\\beta=\\beta(k):=\\frac{\\frac{k(n-2)}{2}-n+3}{n-1}.\n\\]\nThis is an integer if and only if $n-1$ divides $k-4$. In particular\n\\begin{equation}\n\t \\beta(4)=1,\\ \\beta(n+3)=\\frac{n}{2},\\ \\beta(2n+2)=n-1.\n\\end{equation}\n\nFor the brevity of expressions, we introduce some notations. For $0\\leq j\\leq 2n+3$, we set \n\\[\n\\partial_{t^j}=\\frac{\\partial}{\\partial t^j}.\n\\]\nFor $I=\\{i_0,i_1,\\dots,i_{2n+3}\\}\\in \\mathbb{Z}_{\\geq 0}^{2n+4}$, we define\n\\[\n\\partial_{t^I}=(\\partial_{t^0})^{i_0}\\circ\\dots\\circ (\\partial_{t^{2n+3}})^{i_{2n+3}}.\n\\]\nFor $0\\leq j\\leq 2n+3$, let $e_j$ be the $(j+1)$-th unit vector in $\\mathbb{Z}_{\\geq 0}^{2n+4}$. So\n\\[\n\\partial_{t^{I+e_j}}=\\partial_{t^I}\\circ \\partial_{t^j}.\n\\]\nWe apply similar notations to the coordinates $\\tau^0,\\dots,\\tau^{2n+3}$. \n\nFor $I=(i_0,\\dots,i_{2n+3})$ in $\\mathbb{Z}_{\\geq 0}^{2n+4}$, we set $|I|=\\sum_{k=0}^{2n+3}i_k$, and $I!=\\prod_{k=0}^{2n+3}i_k!$. For $I=(i_0,\\dots,i_{2n+3})$ and $J=(j_0,\\dots,j_{2n+3})$ in $\\mathbb{Z}_{\\geq 0}^{2n+4}$, we say $I\\leq J$ if and only if $i_k\\leq j_k$ for $0\\leq k\\leq 2n+3$. We denote an zero vector $(0,\\dots,0)$ also by $0$ when no confusion arises. Moreover we define\n\\begin{equation}\\label{eq-binomOfLists}\n\\binom{I}{J}=\\prod_{k=0}^{2n+3} \\binom{i_k}{j_k}.\n\\end{equation}\n\n\nIn the following subsections we adopt Einstein's summation convention, where the range of the indices runs over $0,\\dots,2n+3$.\n\n\\subsection{Elimination of ambient classes}\n By \\cite[Theorem 3.1]{KM94}, a correlator with only ambient insertions can be computed from the length 3 correlators with only ambient correlators and (\\ref{eq-WDVV}). In \\cite[Appendix D]{Hu15} we present an explicit algorithm in the $\\tau$-coordinates for any Fano complete intersections in projective spaces.\n The system of $\\tau$-coordinates has the advantage that the linear recursion of the highest order terms in the WDVV equations is quite simple. The cost is that the expression of the Euler field becomes complicated. For the even dimensional intersections of two quadrics in consideration, by (\\ref{eq-tauTot}), the $\\tau$-coordinates are very close to the $t$-coordinates. Nevertheless we still work in $\\tau$-coordinates in this section. \n We will show how to eliminate an ambient class in any correlator by (\\ref{eq-EulerVectorField}) and (\\ref{eq-WDVV}). \n \n\\begin{lemma}\\label{lem-recursion-EulerVecField-even(2,2)}\nLet $I\\in \\mathbb{Z}_{\\geq 0}^{2n+4}$, and suppose $|I|\\geq 4$. Then\n\\begin{eqnarray}\\label{eq-recursion-EulerVecField-even(2,2)}\n\\partial_{\\tau^1}\\partial_{\\tau^I}F(0)\n&=& \\frac{\\sum_{j=0}^n(j-1)i_j+(\\frac{n}{2}-1)\\sum_{j=n+1}^{2n+3}i_j+3-n}{n-1}\\partial_{\\tau^I}F(0)\\nonumber\\\\\n&&-12i_n\\partial_{\\tau^1}\\partial_{\\tau^{I-e_n}}F(0).\n\\end{eqnarray}\n\\end{lemma}\n\\begin{proof}\nBy (\\ref{eq-tauTot}) and (\\ref{eq-tTotau}), we write the Euler vector field (\\ref{eq-EV-0}) in the $\\tau$-coordinates:\n\\begin{eqnarray}\\label{eq-EulerField-tau-Coordinates}\nE&=& \\sum_{i=0}^{n}(1-i)t^{i}\\frac{\\partial}{\\partial t^i}+\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})t^{i}\\frac{\\partial}{\\partial t^i}+(n-1)\\frac{\\partial}{\\partial t^1}\\nonumber\\\\\n\\begin{comment}\n&=&\\sum_{i=0}^{n}\\sum_{j=0}^n\\sum_{k=0}^n\n(1-i)W_j^i M_{i}^k \\tau^j\\frac{\\partial}{\\partial \\tau^k}\n+\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\frac{\\partial}{\\partial \\tau^i}\n+(n-1)\\frac{\\partial}{\\partial \\tau^1}\\nonumber\\\\\n&=&\\sum_{i=0}^{n}(1-i)\\tau^i\\frac{\\partial}{\\partial \\tau^i}+W_{n-1}^0 \\tau^{n-1}\\frac{\\partial}{\\partial \\tau^{0}}+(2-n)M_{n-1}^0\\tau^{n-1}\\frac{\\partial}{\\partial \\tau^{0}}\n+(1-n)M_n^1 \\tau^n\\frac{\\partial}{\\partial t^1}\n\\nonumber\\\\\n&&+\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\frac{\\partial}{\\partial \\tau^i}\n+(n-1)\\frac{\\partial}{\\partial \\tau^1}\\nonumber\\\\\n&=&\\sum_{i=0}^{n}(1-i)\\tau^i\\frac{\\partial}{\\partial \\tau^i}+4 \\tau^{n-1}\\frac{\\partial}{\\partial \\tau^{0}}-4(2-n)\\tau^{n-1}\\frac{\\partial}{\\partial \\tau^{0}}\n+(1-n)(-12) \\tau^n\\frac{\\partial}{\\partial \\tau^1}\n\\nonumber\\\\\n&&+\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\frac{\\partial}{\\partial \\tau^i}\n+(n-1)\\frac{\\partial}{\\partial \\tau^1}\\nonumber\\\\\n\\end{comment}\n&=&\\sum_{i=0}^{n}(1-i)\\tau^i\\frac{\\partial}{\\partial \\tau^i}+(4n-4) \\tau^{n-1}\\frac{\\partial}{\\partial \\tau^{0}}\n\t+(12n-12) \\tau^n\\frac{\\partial}{\\partial \\tau^1}\\nonumber\\\\\n&&+\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\frac{\\partial}{\\partial \\tau^i}\n+(n-1)\\frac{\\partial}{\\partial \\tau^1}.\n\\end{eqnarray}\nRecall (\\ref{eq-EulerVectorField})\n\\[\nE F=(3-n)F+(n-1)\\partial_{t^1}c.\n\\]\nSince $|I|\\geq 4$, we have\n\\begin{eqnarray*}\n&&(n-1)\\partial_{\\tau^1}\\partial_{\\tau^I}F(0)\\\\\n&=&-\\sum_{j=0}^n(1-j)i_j\\partial_{\\tau^I}F(0)-(12n-12)i_n\\partial_{\\tau^1}\\partial_{\\tau^{I-e_n}}F(0)\\\\\n&&-\\sum_{j=n+1}^{2n+3}(1-\\frac{n}{2})i_j \\partial_{\\tau^I}F(0)+(3-n)\\partial_{\\tau^I}F(0)\\\\\n&=& \\big(-\\sum_{j=0}^n(1-j)i_j-\\sum_{j=n+1}^{2n+3}(1-\\frac{n}{2})i_j+3-n\\big)\\partial_{\\tau^I}F(0)\\\\\n&&-(12n-12)i_n\\partial_{\\tau^1}\\partial_{\\tau^{I-e_n}}F(0).\n\\end{eqnarray*}\nSo we obtain (\\ref{eq-recursion-EulerVecField-even(2,2)}).\n\\end{proof}\n\n\\begin{lemma}\\label{lem-recursion-ambient-simplified-even(2,2)}\nLet $I\\in \\mathbb{Z}^{2n+4}_{\\geq 0}$.\nLet $n+1\\leq a,b\\leq 2n+3$, and $2\\leq i\\leq n$. Then\n\\begin{eqnarray}\\label{eq-recursion-ambient-simplified-even(2,2)}\n&&\\partial_{\\tau^i}\\partial_{\\tau^I}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\nonumber\\\\\n&=& \n4 \\delta_{a,b}\\partial_{\\tau^1}^2\\partial_{\\tau^{i-1}}\\partial_{\\tau^I}F(0)\t\\nonumber\\\\\n&&-\\frac{1}{4} \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|\\end{subarray}}\\sum_{e=0}^n\n\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{i-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\n\t\\partial_{\\tau^{n-e}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\nonumber\\\\\n&&- \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|\\end{subarray}}\\sum_{e=n+1}^{2n+3}\n\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{i-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\n\t\\partial_{\\tau^{e}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\nonumber\\\\\t\n&&+\t\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\n\t\\sum_{e=0}^n\n\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\n\t\\partial_{\\tau^{n-e}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{i-1}}\\partial_{\\tau^b}F(0)\\nonumber\\\\\n&&+\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\n\t\\sum_{e=n+1}^{2n+3}\n\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\n\t\\partial_{\\tau^{e}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{i-1}}\\partial_{\\tau^b}F(0).\t\n\\end{eqnarray}\n\\end{lemma}\n\\begin{proof}\nConsider the WDVV equation \n\\begin{equation}\\label{eq-lem-recursion-ambient-simplified-even(2,2)-WDVV-1}\n\t(\\partial_{\\tau^1}\\partial_{\\tau^{i-1}}\\partial_{\\tau^{e}}F)\\eta^{ef}(\\partial_{\\tau^f}\\partial_{\\tau^{a}}\\partial_{\\tau^{b}}F)\n\t=(\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^{e}}F)\\eta^{ef}(\\partial_{\\tau^f}\\partial_{\\tau^{i-1}}\\partial_{\\tau^{b}}F).\n\\end{equation}\nWe apply the differential operator $\\partial_{\\tau^I}$ to (\\ref{eq-lem-recursion-ambient-simplified-even(2,2)-WDVV-1}), and then take the constant terms of both sides. We obtain\n\\begin{eqnarray}\\label{eq-lem-recursion-ambient-simplified-even(2,2)-WDVV-2}\n &&\\sum_{0\\leq J\\leq I}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{i-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\nonumber\\\\\n&=&\\sum_{0\\leq J\\leq I}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{i-1}}\\partial_{\\tau^b}F(0).\n\\end{eqnarray}\n Note that \n\\[\n\\partial_{\\tau^j}\\partial_{\\tau^{k}}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\\]\nis the structure constant of the small quantum multiplication. By definition of $\\tilde{\\mathsf{h}}_{i}$ we have\n\\[\n\\tilde{\\mathsf{h}}_{1}\\diamond \\tilde{\\mathsf{h}}_{i-1}=\\tilde{\\mathsf{h}}_{i},\n\\]\nand thus\n\\[\n\\partial_{\\tau^1}\\partial_{\\tau^{i-1}}\\partial_{\\tau^e}F(0)\\eta^{ef}=\\delta_{i,f}.\n\\]\nSo the LHS of (\\ref{eq-lem-recursion-ambient-simplified-even(2,2)-WDVV-2}) equals\n\\begin{eqnarray}\\label{eq-lem-recursion-ambient-simplified-even(2,2)-WDVV-3}\n&& \\partial_{\\tau^1}\\partial_{\\tau^{i-1}}\\partial_{\\tau^e}F(0)\\eta^{ef}\\partial_{\\tau^f}\\partial_{\\tau^I}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\nonumber\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{i-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\nonumber\\\\\n&=& \\partial_{\\tau^i}\\partial_{\\tau^I}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\n\t+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{i-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0).\n\\end{eqnarray}\nThe RHS of (\\ref{eq-lem-recursion-ambient-simplified-even(2,2)-WDVV-2}) equals\n\\begin{eqnarray*}\n&& \\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^e}F(0)\\eta^{ef}\\partial_{\\tau^f}\\partial_{\\tau^I}\\partial_{\\tau^b}\\partial_{\\tau^{i-1}}F(0)\n\t+\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^I}\\partial_{\\tau^e}F(0)\\eta^{ef}\\partial_{\\tau^f}\\partial_{\\tau^b}\\partial_{\\tau^{i-1}}F(0)\\\\\n&& +\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{i-1}}\\partial_{\\tau^b}F(0)\\\\\n&\\stackrel{\\mbox{\\footnotesize{by} }(\\ref{eq-etaInversePairing-even(2,2)})}{=}&\n\\partial_{\\tau^1}\\partial_{\\tau^{a}}^2F(0)\\partial_{\\tau^I}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^{i-1}}F(0)\n\t+\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^b}\\partial_{\\tau^I}F(0)\\partial_{\\tau^b}^2\\partial_{\\tau^{i-1}}F(0)\\\\\n&& +\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{i-1}}\\partial_{\\tau^b}F(0).\n\\end{eqnarray*}\nBy (\\ref{eq-F122-tau}),\n\\[\n\\partial_{\\tau^1}\\partial_{\\tau^{a}}^2F(0)=0=\\partial_{\\tau^b}^2\\partial_{\\tau^{i-1}}F(0).\n\\]\t\nSo the RHS of (\\ref{eq-lem-recursion-ambient-simplified-even(2,2)-WDVV-2}) equals\n\\begin{equation}\\label{eq-lem-recursion-ambient-simplified-even(2,2)-WDVV-4}\n\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{i-1}}\\partial_{\\tau^b}F(0).\n\\end{equation}\nPutting together (\\ref{eq-lem-recursion-ambient-simplified-even(2,2)-WDVV-2}), (\\ref{eq-lem-recursion-ambient-simplified-even(2,2)-WDVV-3}) and (\\ref{eq-lem-recursion-ambient-simplified-even(2,2)-WDVV-4}) yields\n\\begin{eqnarray}\\label{eq-recursion-ambient-even(2,2)}\n\\partial_{\\tau^i}\\partial_{\\tau^I}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\n&=&- \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{i-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\nonumber\\\\\n&&+\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{i-1}}\\partial_{\\tau^b}F(0).\n\\end{eqnarray}\nFinally applying (\\ref{eq-etaInversePairing-even(2,2)}) to the RHS of (\\ref{eq-recursion-ambient-even(2,2)}) we obtain (\\ref{eq-recursion-ambient-simplified-even(2,2)}).\n\\end{proof}\n\n\n\n\n\\subsection{Elimination of primitive classes}\n\\begin{comment}\n\\begin{lemma}\\label{lem-recursion-primitive-aabb-even(2,2)}\nLet $I=(i_{n+1},\\dots,i_{2n+3})$ be a list indicating the number of insertions of $\\epsilon_{n+1},\\dots,\\epsilon_{2n+3}$. Suppose $n+1\\leq a\\leq 2n+3$.\nThen\n\\begin{eqnarray}\\label{eq-recursion-primitive-aabb-even(2,2)-3}\n&&\\partial_{t^n}\\partial_{t^I}\\partial_{t^a}^2F(0)=4|I|\\partial_{t^{a}}^2\\partial_{t^I}F(0)-12\\partial_{t^1}\\partial_{t^{a}}^2\\partial_{t^I}F(0)\\nonumber\\\\\n&&-\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{n-1}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^a}^2F(0) \\nonumber\\\\\n&&+\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{a}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^{n-1}}\\partial_{t^a}F(0).\n\\end{eqnarray}\n\\end{lemma}\n\\begin{proof}\nWe make use of the WDVV equation\n\\begin{equation}\\label{eq-WDVV-recursion-primitive-aabb-even(2,2)-3}\n\t(\\partial_{t^1}\\partial_{t^{n-1}}\\partial_{t^e}F)g^{ef}(\\partial_{t^f}\\partial_{t^a}\\partial_{t^a}F)=(\\partial_{t^1}\\partial_{t^a}\\partial_{t^e}F)g^{ef}(\\partial_{t^f}\\partial_{t^{n-1}}\\partial_{t^a}F).\n\\end{equation}\nThe coefficient of $t^I$ of the LHS of (\\ref{eq-WDVV-recursion-primitive-aabb-even(2,2)-3}), after multiplying $I!$, is\n\\begin{eqnarray*}\n&& \\sum_{0\\leq J\\leq I}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{n-1}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^a}^2F(0)\\\\\n&=& \\partial_{t^1}\\partial_{t^{n-1}}\\partial_{t^e}F(0)g^{ef}\\partial_{t^f}\\partial_{t^I}\\partial_{t^a}^2F(0)\n\t+\\sum_{j=n+1}^{2n+3}i_j \\partial_{t^1}\\partial_{t^{n-1}}\\partial_{t^j}^2F(0)\\partial_{t^I}\\partial_{t^a}^2F(0)\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{n-1}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^a}^2F(0)\\\\\n&=& \\partial_{t^n}\\partial_{t^I}\\partial_{t^a}^2F(0)+8\\partial_{t^1}\\partial_{t^I}\\partial_{t^a}^2F(0)-4|I|\\partial_{t^{a}}^2\\partial_{t^I}F(0)\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{n-1}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^a}^2F(0).\n\\end{eqnarray*}\nThe coefficient of $t^I$ of the RHS of (\\ref{eq-WDVV-recursion-primitive-aabb-even(2,2)-3}), after multiplying $I!$, is\n\\begin{eqnarray*}\n&& \\sum_{0\\leq J\\leq I}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{a}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^{n-1}}\\partial_{t^a}F(0)\\\\\n&=& \\partial_{t^1}\\partial_{t^{a}}\\partial_{t^e}F(0)g^{ef}\\partial_{t^f}\\partial_{t^I}\\partial_{t^a}\\partial_{t^{n-1}}F(0)\n\t+\\partial_{t^1}\\partial_{t^{a}}\\partial_{t^I}\\partial_{t^e}F(0)g^{ef}\\partial_{t^f}\\partial_{t^a}\\partial_{t^{n-1}}F(0)\\\\\n&& +\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{a}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^{n-1}}\\partial_{t^a}F(0)\\\\\n&=& \\partial_{t^1}\\partial_{t^{a}}^2F(0)\\partial_{t^I}\\partial_{t^a}^2\\partial_{t^{n-1}}F(0)\n\t+\\partial_{t^1}\\partial_{t^{a}}^2\\partial_{t^I}F(0)\\partial_{t^a}^2\\partial_{t^{n-1}}F(0)\\\\\n&& +\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{a}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^{n-1}}\\partial_{t^a}F(0)\\\\\n&=& -4 \t\\partial_{t^1}\\partial_{t^{a}}^2\\partial_{t^I}F(0)\t\n\t+\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{a}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^{n-1}}\\partial_{t^a}F(0).\n\\end{eqnarray*}\nSo (\\ref{eq-recursion-primitive-aabb-even(2,2)-3}) follows.\n\\end{proof}\n\\end{comment}\n\n\\begin{comment}\n\\begin{lemma}\\label{lem-recursion-primitive-aabb-even(2,2)}\nLet $I=(i_{n+1},\\dots,i_{2n+3})$ be a list indicating the number of insertions of $\\epsilon_{n+1},\\dots,\\epsilon_{2n+3}$. Suppose $n+1\\leq a\\leq 2n+3$.\nThen\n\\begin{eqnarray}\\label{eq-recursion-primitive-aabb-even(2,2)-3}\n&&\\partial_{\\tau^n}\\partial_{\\tau^I}\\partial_{\\tau^a}^2F(0)=4|I|\\partial_{\\tau^{a}}^2\\partial_{\\tau^I}F(0)\\nonumber\\\\\n&&-\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}^2F(0) \\nonumber\\\\\n&&+\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^a}F(0).\n\\end{eqnarray}\n\\end{lemma}\n\\begin{proof}\nWe make use of the WDVV equation\n\\begin{equation}\\label{eq-WDVV-recursion-primitive-aabb-even(2,2)-3}\n\t(\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^e}F)\\eta^{ef}(\\partial_{\\tau^f}\\partial_{\\tau^a}\\partial_{\\tau^a}F)=(\\partial_{\\tau^1}\\partial_{\\tau^a}\\partial_{\\tau^e}F)\\eta^{ef}(\\partial_{\\tau^f}\\partial_{\\tau^{n-1}}\\partial_{\\tau^a}F).\n\\end{equation}\nThe coefficient of $\\tau^I$ of the LHS of (\\ref{eq-WDVV-recursion-primitive-aabb-even(2,2)-3}), after multiplying $I!$, is\n\\begin{eqnarray*}\n&& \\sum_{0\\leq J\\leq I}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}^2F(0)\\\\\n&\\stackrel{\\mbox{by (\\ref{eq-F122-tau}) and (\\ref{eq-etaInversePairing-even(2,2)})} }{=}& \\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^e}F(0)\\eta^{ef}\\partial_{\\tau^f}\\partial_{\\tau^I}\\partial_{\\tau^a}^2F(0)\n\t+\\sum_{k=n+1}^{2n+3}i_k \\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^k}^2F(0)\\partial_{\\tau^I}\\partial_{\\tau^a}^2F(0)\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}^2F(0)\\\\\n&\\stackrel{\\mbox{by (\\ref{eq-F122-tau})}}{=}& \\partial_{\\tau^n}\\partial_{\\tau^I}\\partial_{\\tau^a}^2F(0)-4|I|\\partial_{\\tau^{a}}^2\\partial_{\\tau^I}F(0)\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}^2F(0).\n\\end{eqnarray*}\nThe coefficient of $\\tau^I$ of the RHS of (\\ref{eq-WDVV-recursion-primitive-aabb-even(2,2)-3}), after multiplying $I!$, is\n\\begin{eqnarray*}\n&& \\sum_{0\\leq J\\leq I}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^a}F(0)\\\\\n&=& \\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^e}F(0)\\eta^{ef}\\partial_{\\tau^f}\\partial_{\\tau^I}\\partial_{\\tau^a}\\partial_{\\tau^{n-1}}F(0)\n\t+\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^I}\\partial_{\\tau^e}F(0)\\eta^{ef}\\partial_{\\tau^f}\\partial_{\\tau^a}\\partial_{\\tau^{n-1}}F(0)\\\\\n&& +\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^a}F(0)\\\\\n&\\stackrel{\\mbox{by (\\ref{eq-F122-tau})}}{=}& \\partial_{\\tau^1}\\partial_{\\tau^{a}}^2F(0)\\partial_{\\tau^I}\\partial_{\\tau^a}^2\\partial_{\\tau^{n-1}}F(0)\n\t+\\partial_{\\tau^1}\\partial_{\\tau^{a}}^2\\partial_{\\tau^I}F(0)\\partial_{\\tau^a}^2\\partial_{\\tau^{n-1}}F(0)\\\\\n&& +\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^a}F(0)\\\\\n&\\stackrel{\\mbox{by (\\ref{eq-F122-tau})}}{=}& \t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^a}F(0).\n\\end{eqnarray*}\nSo (\\ref{eq-recursion-primitive-aabb-even(2,2)-3}) follows.\n\\end{proof}\n\\end{comment}\n\n\\begin{comment}\n\\begin{proposition}\\label{proposition-recursion-primitive-aabb-even(2,2)}\nLet $I=(i_{n+1},\\dots,i_{2n+3})$ be a list indicating the number of insertions of $\\epsilon_{n+1},\\dots,\\epsilon_{2n+3}$. Suppose $n+1\\leq a\\neq b\\leq 2n+3$.\nThen\n\\begin{eqnarray}\\label{eq-recursion-primitive-aabb-even(2,2)}\n&&(\\frac{2|I|-4}{n-1}-2i_b)\\partial_{t^{a}}^2\\partial_{t^I}F(0)\n\t+(\\frac{2|I|-4}{n-1}-2i_a)\\partial_{t^{b}}^2\\partial_{t^I}F(0)\\nonumber\\\\\n&=&\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{n-1}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^a}^2F(0) \\nonumber\\\\\n&&- \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{a}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^{n-1}}\\partial_{t^a}F(0)\\nonumber\\\\\t\n&&+\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{n-1}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^b}^2F(0) \\nonumber\\\\\n&&- \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{b}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^{n-1}}\\partial_{t^b}F(0)\\nonumber\\\\\n&&- \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{t^a}^2\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^b}^2F(0)\\nonumber\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{t^a}\\partial_{t^b}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^a}\\partial_{t^b}F(0).\t\n\\end{eqnarray}\t\n\\end{proposition}\n\\begin{proof}\nWe make use of the WDVV equation\n\\begin{equation}\\label{eq-WDVV-aabb}\n\t(\\partial_{t^a}\\partial_{t^a}\\partial_{t^e}F)g^{ef}(\\partial_{t^f}\\partial_{t^b}\\partial_{t^b}F)=(\\partial_{t^a}\\partial_{t^b}\\partial_{t^e}F)g^{ef}(\\partial_{t^f}\\partial_{t^a}\\partial_{t^b}F).\n\\end{equation}\nThe coefficient of $t^I$ of the LHS of (\\ref{eq-WDVV-aabb}), after multiplying $I!$, is\n\\begin{eqnarray*}\n&& \\sum_{0\\leq J\\leq I}\\binom{I}{J}\\partial_{t^a}^2\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^b}^2F(0)\\\\\n&=& \\partial_{t^a}^2\\partial_{t^I}\\partial_{t^1}F(0)\\frac{1}{4}\\partial_{t^{n-1}}\\partial_{t^b}^2F(0)+\n\t\\partial_{t^a}^2\\partial_{t^I}\\partial_{t^n}F(0)\\frac{1}{4}\\partial_{t^{0}}\\partial_{t^b}^2F(0)\\\\\n&&\t+\\partial_{t^a}^2\\partial_{t^0}F(0)\\frac{1}{4}\\partial_{t^{n}}\\partial_{t^I}\\partial_{t^b}^2 F(0)+\n\t\\partial_{t^a}^2\\partial_{t^{n-1}}F(0)\\frac{1}{4}\\partial_{t^{1}}\\partial_{t^I}\\partial_{t^b}^2F(0)\\\\\n&&+ \\sum_{j=n+1}^{2n+3}\ti_j\\partial_{t^a}^2\\partial_{t^I}F(0)\\partial_{t^{j}}^2\\partial_{t^b}^2F(0)\n\t+ \\sum_{j=n+1}^{2n+3}\ti_j\\partial_{t^a}^2\\partial_{t^{j}}^2F(0)\\partial_{t^I}\\partial_{t^b}^2F(0)\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{t^a}^2\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^b}^2F(0)\t\\\\\n&=& -\\partial_{t^a}^2\\partial_{t^I}\\partial_{t^1}F(0)+\n\t\\frac{1}{4}\\partial_{t^a}^2\\partial_{t^I}\\partial_{t^n}F(0)\n\t+\\frac{1}{4}\\partial_{t^{n}}\\partial_{t^I}\\partial_{t^b}^2 F(0)\n\t-\\partial_{t^{1}}\\partial_{t^I}\\partial_{t^b}^2F(0)\\\\\n&&+ |I|\\partial_{t^a}^2\\partial_{t^I}F(0)\n\t+ |I|\\partial_{t^I}\\partial_{t^b}^2F(0)\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{t^a}^2\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^b}^2F(0).\t\n\\end{eqnarray*}\nThe coefficient of $t^I$ of the RHS of (\\ref{eq-WDVV-aabb}), after multiplying $I!$, is\n\\begin{eqnarray*}\n&& \\sum_{0\\leq J\\leq I}\\binom{I}{J}\\partial_{t^a}\\partial_{t^b}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^a}\\partial_{t^b}F(0)\\\\\n&=& i_a\\partial_{t^a}\\partial_{t^b}\\partial_{t^{I-e_a}}\\partial_{t^b}F(0)\\partial_{t^{b}}\\partial_{t^a}\\partial_{t^a}\\partial_{t^b}F(0)\n\t+i_b\\partial_{t^a}\\partial_{t^b}\\partial_{t^{I-e_b}}\\partial_{t^a}F(0)\\partial_{t^{a}}\\partial_{t^b}\\partial_{t^a}\\partial_{t^b}F(0)\\\\\n&&+ i_a\\partial_{t^a}\\partial_{t^b}\\partial_{t^a}\\partial_{t^b}F(0)\\partial_{t^{b}}\\partial_{t^{I-e_a}}\\partial_{t^a}\\partial_{t^b}F(0)\t\n\t+ i_b\\partial_{t^a}\\partial_{t^b}\\partial_{t^b}\\partial_{t^a}F(0)\\partial_{t^{a}}\\partial_{t^{I-e_b}}\\partial_{t^a}\\partial_{t^b}F(0)\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{t^a}\\partial_{t^b}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^a}\\partial_{t^b}F(0)\\\\\t\n&=& i_a\\partial_{t^{I}}\\partial_{t^b}^2F(0)\n\t+i_b\\partial_{t^{I}}\\partial_{t^a}^2F(0)\n\t+ i_a\\partial_{t^{I}}\\partial_{t^b}^2F(0)\t\n\t+ i_b\\partial_{t^{I}}\\partial_{t^a}^2F(0)\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{t^a}\\partial_{t^b}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^a}\\partial_{t^b}F(0)\\\\\t\n&=& 2 i_a\\partial_{t^{I}}\\partial_{t^b}^2F(0)\n\t+2i_b\\partial_{t^{I}}\\partial_{t^a}^2F(0)\n\t+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{t^a}\\partial_{t^b}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^a}\\partial_{t^b}F(0).\n\\end{eqnarray*}\nSo (\\ref{eq-WDVV-aabb}) yields\n\\begin{eqnarray}\\label{proposition-recursion-primitive-aabb-even(2,2)-1}\n&& -\\partial_{t^a}^2\\partial_{t^I}\\partial_{t^1}F(0)+\n\t\\frac{1}{4}\\partial_{t^a}^2\\partial_{t^I}\\partial_{t^n}F(0)\n\t+\\frac{1}{4}\\partial_{t^{n}}\\partial_{t^I}\\partial_{t^b}^2 F(0)\n\t-\\partial_{t^{1}}\\partial_{t^I}\\partial_{t^b}^2F(0)\\nonumber\\\\\n&&+ |I|\\partial_{t^a}^2\\partial_{t^I}F(0)\n\t+ |I|\\partial_{t^I}\\partial_{t^b}^2F(0)\n\t-2 i_a\\partial_{t^{I}}\\partial_{t^b}^2F(0)\n\t-2i_b\\partial_{t^{I}}\\partial_{t^a}^2F(0)\\nonumber\\\\\n&=&- \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{t^a}^2\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^b}^2F(0)\\nonumber\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{t^a}\\partial_{t^b}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^a}\\partial_{t^b}F(0).\n\\end{eqnarray}\nWe make manipulations on the LHS of (\\ref{proposition-recursion-primitive-aabb-even(2,2)-1}). Since\n\\[\n\\frac{n}{2}\\times (2+|I|)-(n-3+2+|I|)=(\\frac{n}{2}-1)|I|+1,\n\\]\nby (Div) we have\n\\begin{equation}\\label{proposition-recursion-primitive-aabb-even(2,2)-2}\n\\partial_{t^a}^2\\partial_{t^I}\\partial_{t^1}F(0)\n=\\frac{(\\frac{n}{2}-1)|I|+1}{n-1}\\partial_{t^a}^2\\partial_{t^I}F(0).\n\\end{equation}\nSo by (\\ref{proposition-recursion-primitive-aabb-even(2,2)-2}) and (\\ref{eq-recursion-primitive-aabb-even(2,2)-3}) we get\n\\begin{eqnarray}\\label{proposition-recursion-primitive-aabb-even(2,2)-4}\n&& -\\partial_{t^a}^2\\partial_{t^I}\\partial_{t^1}F(0)+\n\t\\frac{1}{4}\\partial_{t^a}^2\\partial_{t^I}\\partial_{t^n}F(0)\n\t+\\frac{1}{4}\\partial_{t^{n}}\\partial_{t^I}\\partial_{t^b}^2 F(0)\n\t-\\partial_{t^{1}}\\partial_{t^I}\\partial_{t^b}^2F(0)\\nonumber\\\\\n&&+ |I|\\partial_{t^a}^2\\partial_{t^I}F(0)\n\t+ |I|\\partial_{t^I}\\partial_{t^b}^2F(0)\n\t-2 i_a\\partial_{t^{I}}\\partial_{t^b}^2F(0)\n\t-2i_b\\partial_{t^{I}}\\partial_{t^a}^2F(0)\\nonumber\\\\\n&=&\t-\\partial_{t^a}^2\\partial_{t^I}\\partial_{t^1}F(0)+\n\t|I|\\partial_{t^{a}}^2\\partial_{t^I}F(0)-3\\partial_{t^1}\\partial_{t^{a}}^2\\partial_{t^I}F(0)\\nonumber\\\\\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{n-1}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^a}^2F(0) \\nonumber\\\\\n&&+ \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{a}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^{n-1}}\\partial_{t^a}F(0)\\nonumber\\\\\t\n&&\t+|I|\\partial_{t^{b}}^2\\partial_{t^I}F(0)-3\\partial_{t^1}\\partial_{t^{b}}^2\\partial_{t^I}F(0)\\nonumber\\\\\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{n-1}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^b}^2F(0) \\nonumber\\\\\n&&+ \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{b}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^{n-1}}\\partial_{t^b}F(0)\\nonumber\\\\\n&&\t-\\partial_{t^{1}}\\partial_{t^I}\\partial_{t^b}^2F(0)\n\t+ |I|\\partial_{t^a}^2\\partial_{t^I}F(0)\n\t+ |I|\\partial_{t^I}\\partial_{t^b}^2F(0)\n\t-2 i_a\\partial_{t^{I}}\\partial_{t^b}^2F(0)\n\t-2i_b\\partial_{t^{I}}\\partial_{t^a}^2F(0)\\nonumber\\\\\n&=&\t2(|I|-i_b)\\partial_{t^{a}}^2\\partial_{t^I}F(0)-4\\partial_{t^1}\\partial_{t^{a}}^2\\partial_{t^I}F(0)\n\t+2(|I|-i_a)\\partial_{t^{b}}^2\\partial_{t^I}F(0)-4\\partial_{t^1}\\partial_{t^{b}}^2\\partial_{t^I}F(0)\\nonumber\\\\\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{n-1}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^a}^2F(0) \\nonumber\\\\\n&&+ \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{a}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^{n-1}}\\partial_{t^a}F(0)\\nonumber\\\\\t\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{n-1}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^b}^2F(0) \\nonumber\\\\\n&&+ \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{b}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^{n-1}}\\partial_{t^b}F(0)\\nonumber\\\\\t\n&=&(2|I|-2i_b-4\\times\\frac{(\\frac{n}{2}-1)|I|+1}{n-1})\\partial_{t^{a}}^2\\partial_{t^I}F(0)\\nonumber\\\\\n&&+(2|I|-2i_a-4\\times\\frac{(\\frac{n}{2}-1)|I|+1}{n-1})\\partial_{t^{b}}^2\\partial_{t^I}F(0)\\nonumber\\\\\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{n-1}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^a}^2F(0) \\nonumber\\\\\n&&+ \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{a}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^{n-1}}\\partial_{t^a}F(0)\\nonumber\\\\\t\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{n-1}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^b}^2F(0) \\nonumber\\\\\n&&+ \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{b}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^{n-1}}\\partial_{t^b}F(0)\\nonumber\\\\\t\n&=& (\\frac{2|I|-4}{n-1}-2i_b)\\partial_{t^{a}}^2\\partial_{t^I}F(0)\n\t+(\\frac{2|I|-4}{n-1}-2i_a)\\partial_{t^{b}}^2\\partial_{t^I}F(0)\\nonumber\\\\\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{n-1}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^a}^2F(0) \\nonumber\\\\\n&&+ \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{a}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^{n-1}}\\partial_{t^a}F(0)\\nonumber\\\\\t\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{n-1}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^b}^2F(0) \\nonumber\\\\\n&&+ \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{t^{b}}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^{n-1}}\\partial_{t^b}F(0).\t\n\\end{eqnarray}\nThen (\\ref{eq-recursion-primitive-aabb-even(2,2)}) follows from (\\ref{proposition-recursion-primitive-aabb-even(2,2)-1}) and (\\ref{proposition-recursion-primitive-aabb-even(2,2)-4}).\n\\end{proof}\n\\end{comment}\n\n\n\\begin{lemma}\\label{lem-recursion-primitive-abcc-even(2,2)}\nLet $I=(i_{n+1},\\dots,i_{2n+3})$ be a list indicating the number of insertions of $\\epsilon_{n+1},\\dots,\\epsilon_{2n+3}$. Suppose $n+1\\leq a,b\\leq 2n+3$.\nThen\n\\begin{eqnarray}\\label{eq-recursion-primitive-abcc-even(2,2)-3}\n&&\\partial_{\\tau^n}\\partial_{\\tau^I}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\n=4|I|\\partial_{\\tau^{a}}\\partial_{\\tau^b}\\partial_{\\tau^I}F(0)\\nonumber\\\\\n&&-\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0) \\nonumber\\\\\n&&+\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^b}F(0).\n\\end{eqnarray}\n\\end{lemma}\n\\begin{proof}\nWe make use of the WDVV equation\n\\begin{equation}\\label{eq-WDVV-recursion-primitive-abcc-even(2,2)-3}\n\t(\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^e}F)\\eta^{ef}(\\partial_{\\tau^f}\\partial_{\\tau^a}\\partial_{\\tau^b}F)=(\\partial_{\\tau^1}\\partial_{\\tau^a}\\partial_{\\tau^e}F)\\eta^{ef}(\\partial_{\\tau^f}\\partial_{\\tau^{n-1}}\\partial_{\\tau^b}F).\n\\end{equation}\nThe coefficient of $\\tau^I$ of the LHS of (\\ref{eq-WDVV-recursion-primitive-abcc-even(2,2)-3}), after multiplying $I!$, is\n\\begin{eqnarray*}\n&& \\sum_{0\\leq J\\leq I}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\\\\n&\\stackrel{\\mbox{by (\\ref{eq-F122-tau}) and (\\ref{eq-etaInversePairing-even(2,2)})} }{=}& \\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^e}F(0)\\eta^{ef}\\partial_{\\tau^f}\\partial_{\\tau^I}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\n\t+\\sum_{j=n+1}^{2n+3}i_j \\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^j}^2F(0)\\partial_{\\tau^I}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\\\\n&\\stackrel{\\mbox{by (\\ref{eq-F122-tau}) and $\\tilde{\\mathsf{h}}_1\\diamond \\tilde{\\mathsf{h}}_{n-1}=\\tilde{\\mathsf{h}}_{n}$ }}{=}& \\partial_{\\tau^n}\\partial_{\\tau^I}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)-4|I|\\partial_{\\tau^{a}}\\partial_{\\tau^b}\\partial_{\\tau^I}F(0)\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0).\n\\end{eqnarray*}\nThe coefficient of $\\tau^I$ of the RHS of (\\ref{eq-WDVV-recursion-primitive-abcc-even(2,2)-3}), after multiplying $I!$, is\n\\begin{eqnarray*}\n&& \\sum_{0\\leq J\\leq I}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^b}F(0)\\\\\n&=& \\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^e}F(0)\\eta^{ef}\\partial_{\\tau^f}\\partial_{\\tau^I}\\partial_{\\tau^b}\\partial_{\\tau^{n-1}}F(0)\n\t+\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^I}\\partial_{\\tau^e}F(0)\\eta^{ef}\\partial_{\\tau^f}\\partial_{\\tau^b}\\partial_{\\tau^{n-1}}F(0)\\\\\n&& +\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^b}F(0)\\\\\n&\\stackrel{\\mbox{by (\\ref{eq-F122-tau})}}{=}& \\partial_{\\tau^1}\\partial_{\\tau^{a}}^2F(0)\\partial_{\\tau^I}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^{n-1}}F(0)\n\t+\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^b}\\partial_{\\tau^I}F(0)\\partial_{\\tau^b}^2\\partial_{\\tau^{n-1}}F(0)\\\\\n&& +\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^b}F(0)\\\\\n&\\stackrel{\\mbox{by (\\ref{eq-F122-tau})}}{=}&\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^b}F(0).\n\\end{eqnarray*}\nSo (\\ref{eq-recursion-primitive-abcc-even(2,2)-3}) follows.\n\\end{proof}\n\n\n\\begin{proposition}\\label{proposition-recursion-primitive-aabb-even(2,2)}\nLet $I=(i_{n+1},\\dots,i_{2n+3})$ be a list indicating the number of insertions of $\\epsilon_{n+1},\\dots,\\epsilon_{2n+3}$. Suppose $n+1\\leq a, b\\leq 2n+3$ and $a\\neq b$.\nThen\n\\begin{eqnarray}\\label{eq-recursion-primitive-aabb-even(2,2)}\n&&(\\frac{2|I|-4}{n-1}-2i_b)\\partial_{\\tau^{a}}^2\\partial_{\\tau^I}F(0)\n\t+(\\frac{2|I|-4}{n-1}-2i_a)\\partial_{\\tau^{b}}^2\\partial_{\\tau^I}F(0)\\nonumber\\\\\n&=&\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}^2F(0) \\nonumber\\\\\n&&- \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^a}F(0)\\nonumber\\\\\t\n&&+\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^b}^2F(0) \\nonumber\\\\\n&&- \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{b}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^b}F(0)\\nonumber\\\\\n&&- \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}^2\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^b}^2F(0)\\nonumber\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0).\t\n\\end{eqnarray}\t\n\\end{proposition}\n\\begin{proof}\nWe make use of the WDVV equation\n\\begin{equation}\\label{eq-WDVV-aabb}\n\t(\\partial_{\\tau^a}\\partial_{\\tau^a}\\partial_{\\tau^e}F)\\eta^{ef}(\\partial_{\\tau^f}\\partial_{\\tau^b}\\partial_{\\tau^b}F)=(\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^e}F)\\eta^{ef}(\\partial_{\\tau^f}\\partial_{\\tau^a}\\partial_{\\tau^b}F).\n\\end{equation}\nThe coefficient of $\\tau^I$ of the LHS of (\\ref{eq-WDVV-aabb}), after multiplying $I!$, is\n\\begin{eqnarray*}\n&& \\sum_{0\\leq J\\leq I}\\binom{I}{J}\\partial_{\\tau^a}^2\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^b}^2F(0)\\\\\n&\\stackrel{\\mbox{by (\\ref{eq-F122-tau})}}{=}& \\partial_{\\tau^a}^2\\partial_{\\tau^I}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^b}^2F(0)\n\t+\\partial_{\\tau^a}^2\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I}}\\partial_{\\tau^b}^2F(0)\\\\\n&&+ \\sum_{k=n+1}^{2n+3}\ti_k\\partial_{\\tau^a}^2\\partial_{\\tau^I}F(0)\\partial_{\\tau^{k}}^2\\partial_{\\tau^b}^2F(0)\n\t+ \\sum_{k=n+1}^{2n+3}\ti_k\\partial_{\\tau^a}^2\\partial_{\\tau^{k}}^2F(0)\\partial_{\\tau^I}\\partial_{\\tau^b}^2F(0)\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}^2\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^b}^2F(0)\t\\\\\n&\\stackrel{\\mbox{by (\\ref{eq-F122-tau}), \n\t\t\t\t(\\ref{eq-etaInversePairing-even(2,2)}), \n\t\t\t\tand (\\ref{eq-4points-fanoIndex-even(2,2)-ab})}\n\t\t\t\t}{=}& -4\\partial_{\\tau^a}^2\\partial_{\\tau^I}\\partial_{\\tau^1}F(0)+\n\t\\frac{1}{4}\\partial_{\\tau^a}^2\\partial_{\\tau^I}\\partial_{\\tau^n}F(0)\n\t+\\frac{1}{4}\\partial_{\\tau^{n}}\\partial_{\\tau^I}\\partial_{\\tau^b}^2 F(0)\n\t-4\\partial_{\\tau^{1}}\\partial_{\\tau^I}\\partial_{\\tau^b}^2F(0)\\\\\n&&+ |I|\\partial_{\\tau^a}^2\\partial_{\\tau^I}F(0)\n\t+ |I|\\partial_{\\tau^I}\\partial_{\\tau^b}^2F(0)\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}^2\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^b}^2F(0).\t\n\\end{eqnarray*}\nThe coefficient of $\\tau^I$ of the RHS of (\\ref{eq-WDVV-aabb}), after multiplying $I!$, is\n\\begin{eqnarray*}\n&& \\sum_{0\\leq J\\leq I}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\\\\n&\\begin{subarray}{c}\\mbox{by (\\ref{eq-F122-tau}) and} \\\\ \\mbox{Theorem \\ref{thm-monodromy-evenDim(2,2)}}\\\\ =\\end{subarray}& i_a\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^{I-e_a}}\\partial_{\\tau^b}F(0)\\partial_{\\tau^{b}}\\partial_{\\tau^a}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\n\t+i_b\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^{I-e_b}}\\partial_{\\tau^a}F(0)\\partial_{\\tau^{a}}\\partial_{\\tau^b}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\\\\n&&+ i_a\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\partial_{\\tau^{b}}\\partial_{\\tau^{I-e_a}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\t\n\t+ i_b\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^b}\\partial_{\\tau^a}F(0)\\partial_{\\tau^{a}}\\partial_{\\tau^{I-e_b}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\\\\t\n&\\stackrel{\\mbox{by (\\ref{eq-4points-fanoIndex-even(2,2)-ab})}}{=}& i_a\\partial_{\\tau^{I}}\\partial_{\\tau^b}^2F(0)\n\t+i_b\\partial_{\\tau^{I}}\\partial_{\\tau^a}^2F(0)\n\t+ i_a\\partial_{\\tau^{I}}\\partial_{\\tau^b}^2F(0)\t\n\t+ i_b\\partial_{\\tau^{I}}\\partial_{\\tau^a}^2F(0)\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\\\\t\n&=& 2 i_a\\partial_{\\tau^{I}}\\partial_{\\tau^b}^2F(0)\n\t+2i_b\\partial_{\\tau^{I}}\\partial_{\\tau^a}^2F(0)\\\\\n&&\t+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0).\n\\end{eqnarray*}\nSo (\\ref{eq-WDVV-aabb}) yields\n\\begin{eqnarray}\\label{proposition-recursion-primitive-aabb-even(2,2)-1}\n&& -4\\partial_{\\tau^a}^2\\partial_{\\tau^I}\\partial_{\\tau^1}F(0)+\n\t\\frac{1}{4}\\partial_{\\tau^a}^2\\partial_{\\tau^I}\\partial_{\\tau^n}F(0)\n\t+\\frac{1}{4}\\partial_{\\tau^{n}}\\partial_{\\tau^I}\\partial_{\\tau^b}^2 F(0)\n\t-4\\partial_{\\tau^{1}}\\partial_{\\tau^I}\\partial_{\\tau^b}^2F(0)\\nonumber\\\\\n&&+ |I|\\partial_{\\tau^a}^2\\partial_{\\tau^I}F(0)\n\t+ |I|\\partial_{\\tau^I}\\partial_{\\tau^b}^2F(0)\n\t-2 i_a\\partial_{\\tau^{I}}\\partial_{\\tau^b}^2F(0)\n\t-2i_b\\partial_{\\tau^{I}}\\partial_{\\tau^a}^2F(0)\\nonumber\\\\\n&=&- \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}^2\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^b}^2F(0)\\nonumber\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0).\n\\end{eqnarray}\nWe make manipulations on the LHS of (\\ref{proposition-recursion-primitive-aabb-even(2,2)-1}). Since\n\\[\n\\frac{n}{2}\\times (2+|I|)-(n-3+2+|I|)=(\\frac{n}{2}-1)|I|+1,\n\\]\nby (\\ref{eq-tTotau}) and (\\ref{eq-Div}) we have\n\\begin{equation}\\label{proposition-recursion-primitive-aabb-even(2,2)-2}\n\\partial_{\\tau^a}^2\\partial_{\\tau^I}\\partial_{\\tau^1}F(0)\n=\\partial_{t^a}^2\\partial_{t^I}\\partial_{t^1}F(0)\n=\\frac{(\\frac{n}{2}-1)|I|+1}{n-1}\\partial_{\\tau^a}^2\\partial_{\\tau^I}F(0).\n\\end{equation}\nSo from (\\ref{eq-recursion-primitive-abcc-even(2,2)-3}) we obtain\n\\begin{eqnarray}\\label{proposition-recursion-primitive-aabb-even(2,2)-4}\n&& -4\\partial_{\\tau^a}^2\\partial_{\\tau^I}\\partial_{\\tau^1}F(0)+\n\t\\frac{1}{4}\\partial_{\\tau^a}^2\\partial_{\\tau^I}\\partial_{\\tau^n}F(0)\n\t+\\frac{1}{4}\\partial_{\\tau^{n}}\\partial_{\\tau^I}\\partial_{\\tau^b}^2 F(0)\n\t-4\\partial_{\\tau^{1}}\\partial_{\\tau^I}\\partial_{\\tau^b}^2F(0)\\nonumber\\\\\n&&+ |I|\\partial_{\\tau^a}^2\\partial_{\\tau^I}F(0)\n\t+ |I|\\partial_{\\tau^I}\\partial_{\\tau^b}^2F(0)\n\t-2 i_a\\partial_{\\tau^{I}}\\partial_{\\tau^b}^2F(0)\n\t-2i_b\\partial_{\\tau^{I}}\\partial_{\\tau^a}^2F(0)\\nonumber\\\\\n&=&\t-4\\partial_{\\tau^a}^2\\partial_{\\tau^I}\\partial_{\\tau^1}F(0)+\n\t|I|\\partial_{\\tau^{a}}^2\\partial_{\\tau^I}F(0)\\nonumber\\\\\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}^2F(0) \\nonumber\\\\\n&&+ \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{t^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^a}F(0)\\nonumber\\\\\t\n&&\t+|I|\\partial_{\\tau^{b}}^2\\partial_{\\tau^I}F(0)\n-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^b}^2F(0) \\nonumber\\\\\n&&+ \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{b}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^b}F(0)\\nonumber\\\\\n&&\t-4\\partial_{\\tau^{1}}\\partial_{\\tau^I}\\partial_{\\tau^b}^2F(0)\n\t+ |I|\\partial_{\\tau^a}^2\\partial_{\\tau^I}F(0)\n\t+ |I|\\partial_{\\tau^I}\\partial_{\\tau^b}^2F(0)\n\t-2 i_a\\partial_{\\tau^{I}}\\partial_{\\tau^b}^2F(0)\n\t-2i_b\\partial_{\\tau^{I}}\\partial_{\\tau^a}^2F(0)\\nonumber\\\\\n&=&\t2(|I|-i_b)\\partial_{\\tau^{a}}^2\\partial_{\\tau^I}F(0)-4\\partial_{\\tau^1}\\partial_{\\tau^{a}}^2\\partial_{\\tau^I}F(0)\n\t+2(|I|-i_a)\\partial_{\\tau^{b}}^2\\partial_{\\tau^I}F(0)-4\\partial_{\\tau^1}\\partial_{\\tau^{b}}^2\\partial_{\\tau^I}F(0)\\nonumber\\\\\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}^2F(0) \\nonumber\\\\\n&&+ \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^a}F(0)\\nonumber\\\\\t\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^b}^2F(0) \\nonumber\\\\\n&&+ \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{b}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^b}F(0)\\nonumber\\\\\t\n&\\stackrel{\\mbox{by (\\ref{proposition-recursion-primitive-aabb-even(2,2)-2})}}{=}&(2|I|-2i_b-4\\times\\frac{(\\frac{n}{2}-1)|I|+1}{n-1})\\partial_{\\tau^{a}}^2\\partial_{\\tau^I}F(0)\\nonumber\\\\\n&&+(2|I|-2i_a-4\\times\\frac{(\\frac{n}{2}-1)|I|+1}{n-1})\\partial_{\\tau^{b}}^2\\partial_{\\tau^I}F(0)\\nonumber\\\\\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}^2F(0) \\nonumber\\\\\n&&+ \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^a}F(0)\\nonumber\\\\\t\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^b}^2F(0) \\nonumber\\\\\n&&+ \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{b}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^b}F(0)\\nonumber\\\\\t\n&=& (\\frac{2|I|-4}{n-1}-2i_b)\\partial_{\\tau^{a}}^2\\partial_{\\tau^I}F(0)\n\t+(\\frac{2|I|-4}{n-1}-2i_a)\\partial_{\\tau^{b}}^2\\partial_{\\tau^I}F(0)\\nonumber\\\\\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}^2F(0) \\nonumber\\\\\n&&+ \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^a}F(0)\\nonumber\\\\\t\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^b}^2F(0) \\nonumber\\\\\n&&+ \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{b}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^b}F(0).\t\n\\end{eqnarray}\nThen (\\ref{eq-recursion-primitive-aabb-even(2,2)}) follows from (\\ref{proposition-recursion-primitive-aabb-even(2,2)-1}) and (\\ref{proposition-recursion-primitive-aabb-even(2,2)-4}).\n\\end{proof}\n\n\n\n\n\\begin{proposition}\\label{proposition-recursion-primitive-abcc-even(2,2)}\nLet $I=(i_{n+1},\\dots,i_{2n+3})$ be a list indicating the number of insertions of $\\epsilon_{n+1},\\dots,\\epsilon_{2n+3}$. Suppose $n+1\\leq a, b,c\\leq 2n+3$ and $a,b,c$ are pairwise distinct. Then\n\\begin{eqnarray}\\label{eq-recursion-primitive-abcc-even(2,2)}\n &&(\\frac{2|I|-4}{n-1}-2i_c)\\partial_{\\tau^{a}}\\partial_{\\tau^b}\\partial_{\\tau^I}F(0)\\nonumber\\\\\t\n&=&\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0) \\nonumber\\\\\n&&-\\frac{1}{4}\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^b}F(0)\\nonumber\\\\\n&&-\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^c}^2F(0)\\nonumber\\\\\n&& +\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^c}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^b}\\partial_{\\tau^c}F(0).\n\\end{eqnarray}\n\\end{proposition}\n\\begin{proof}\nWe make use of the WDVV equation\n\\begin{equation}\\label{eq-WDVV-abcc}\n\t(\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^e}F)\\eta^{ef}(\\partial_{\\tau^f}\\partial_{\\tau^c}\\partial_{\\tau^c}F)=(\\partial_{\\tau^a}\\partial_{\\tau^c}\\partial_{\\tau^e}F)\\eta^{ef}(\\partial_{\\tau^f}\\partial_{\\tau^b}\\partial_{\\tau^c}F).\n\\end{equation}\nThe coefficient of $\\tau^I$ of the LHS of (\\ref{eq-WDVV-abcc}), after multiplying $I!$, is\n\\begin{eqnarray*}\n&& \\sum_{0\\leq J\\leq I}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^c}^2F(0)\\\\\n&\\begin{subarray}{c}\\mbox{by (\\ref{eq-F122-tau}) and} \\\\ \\mbox{Theorem \\ref{thm-monodromy-evenDim(2,2)}}\\\\ =\\end{subarray}& \\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^I}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^c}^2F(0)\n\t+\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I}}\\partial_{\\tau^c}^2F(0)\\\\\n&&+ \\sum_{k=n+1}^{2n+3}\ti_k\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^I}F(0)\\partial_{\\tau^{k}}^2\\partial_{\\tau^c}^2F(0)\n\t+ i_a\\partial_{\\tau^a}^2\\partial_{\\tau^{b}}^2F(0)\\partial_{\\tau^{I-e_a}}\\partial_{\\tau^b}\\partial_{\\tau^c}^2F(0)\\\\\n&&+ i_b\\partial_{\\tau^a}^2\\partial_{\\tau^{b}}^2F(0)\\partial_{\\tau^{I-e_b}}\\partial_{\\tau^a}\\partial_{\\tau^c}^2F(0)\t\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^c}^2F(0)\t\\\\\t\n&=& \\partial_{t^a}\\partial_{t^b}\\partial_{t^I}\\partial_{t^1}F(0)\\frac{1}{4}\\partial_{t^{n-1}}\\partial_{t^c}^2F(0)+\n\t\\partial_{t^a}\\partial_{t^b}\\partial_{t^I}\\partial_{t^n}F(0)\\frac{1}{4}\\partial_{t^{0}}\\partial_{t^c}^2F(0)\\\\\n&&+ \\sum_{j=n+1}^{2n+3}\ti_j\\partial_{t^a}\\partial_{t^b}\\partial_{t^I}F(0)\\partial_{t^{j}}^2\\partial_{t^c}^2F(0)\n\t+ i_a\\partial_{t^a}^2\\partial_{t^{b}}^2F(0)\\partial_{t^{I-e_a}}\\partial_{t^b}\\partial_{t^c}^2F(0)\\\\\n&&+ i_b\\partial_{t^a}^2\\partial_{t^{b}}^2F(0)\\partial_{t^{I-e_b}}\\partial_{t^a}\\partial_{t^c}^2F(0)\t\\\\\n&& +\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{t^a}\\partial_{t^b}\\partial_{t^J}\\partial_{t^e}F(0)g^{ef}\n\t\\partial_{t^{f}}\\partial_{t^{I-J}}\\partial_{t^c}^2F(0)\\\\\n&\\stackrel{\\mbox{by (\\ref{eq-F122-tau}), \n\t\t\t\t(\\ref{eq-etaInversePairing-even(2,2)}), \n\t\t\t\tand (\\ref{eq-4points-fanoIndex-even(2,2)-ab})}\n\t\t\t\t}{=}& - 4\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^I}\\partial_{\\tau^1}F(0)\n\t+\\frac{1}{4}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^I}\\partial_{\\tau^n}F(0)\\\\\n&&+ |I|\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^I}F(0)\n\t+ i_a\\partial_{\\tau^{I-e_a}}\\partial_{\\tau^b}\\partial_{\\tau^c}^2F(0)\n\t+ i_b\\partial_{\\tau^{I-e_b}}\\partial_{\\tau^a}\\partial_{\\tau^c}^2F(0)\\\\\n&& +\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^c}^2F(0).\t\n\\end{eqnarray*}\nThe coefficient of $\\tau^I$ of the RHS of (\\ref{eq-WDVV-abcc}), after multiplying $I!$, is\n\\begin{eqnarray*}\n&& \\sum_{0\\leq J\\leq I}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^c}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^b}\\partial_{\\tau^c}F(0)\\\\\n&\\begin{subarray}{c}\\mbox{by (\\ref{eq-F122-tau}) and} \\\\ \\mbox{Theorem \\ref{thm-monodromy-evenDim(2,2)}}\\\\ =\\end{subarray}& i_b\\partial_{\\tau^a}\\partial_{\\tau^c}\\partial_{\\tau^{I-e_b}}\\partial_{\\tau^c}F(0)\\partial_{\\tau^{c}}\\partial_{\\tau^b}\\partial_{\\tau^b}\\partial_{\\tau^c}F(0)\n\t+i_c\\partial_{\\tau^a}\\partial_{\\tau^c}\\partial_{\\tau^{I-e_c}}\\partial_{\\tau^b}F(0)\\partial_{\\tau^{b}}\\partial_{\\tau^c}\\partial_{\\tau^b}\\partial_{\\tau^c}F(0)\\\\\n&&+ i_a\\partial_{\\tau^a}\\partial_{\\tau^c}\\partial_{\\tau^a}\\partial_{\\tau^c}F(0)\\partial_{\\tau^{c}}\\partial_{\\tau^{I-e_a}}\\partial_{\\tau^b}\\partial_{\\tau^c}F(0)\t\n\t+ i_c\\partial_{\\tau^a}\\partial_{\\tau^c}\\partial_{\\tau^c}\\partial_{\\tau^a}F(0)\\partial_{\\tau^{a}}\\partial_{\\tau^{I-e_c}}\\partial_{\\tau^b}\\partial_{\\tau^c}F(0)\\\\\n&& +\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^c}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^b}\\partial_{\\tau^c}F(0)\t\\\\\n&\\stackrel{\\mbox{by (\\ref{eq-4points-fanoIndex-even(2,2)-ab})}}{=}& i_b\\partial_{\\tau^a}\\partial_{\\tau^c}^2\\partial_{\\tau^{I-e_b}}F(0)\n\t+2i_c\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^{I}}F(0)\n\t+ i_a\\partial_{\\tau^b}\\partial_{\\tau^{c}}^2\\partial_{\\tau^{I-e_a}}F(0)\\\\\n&& +\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^c}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{t^b}\\partial_{\\tau^c}F(0).\t\n\\end{eqnarray*}\nSo (\\ref{eq-WDVV-abcc}) yields\n\\begin{eqnarray}\\label{proposition-recursion-primitive-abcc-even(2,2)-1}\n&&-4\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^I}\\partial_{\\tau^1}F(0)\n\t+\\frac{1}{4}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^I}\\partial_{\\tau^n}F(0)\n\t+ (|I|-2i_c)\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^I}F(0)\\nonumber\\\\\n&=&-\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^c}^2F(0)\\nonumber\\\\\n&& +\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^c}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^b}\\partial_{\\tau^c}F(0)\n\\end{eqnarray}\nBy (\\ref{eq-tTotau}) and (\\ref{eq-Div}),\n\\begin{equation}\\label{proposition-recursion-primitive-abcc-even(2,2)-2}\n\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^I}\\partial_{\\tau^1}F(0)\n=\\partial_{t^a}\\partial_{t^b}\\partial_{t^I}\\partial_{t^1}F(0)\n=\\frac{(\\frac{n}{2}-1)|I|+1}{n-1}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^I}F(0).\n\\end{equation}\nSo (\\ref{eq-recursion-primitive-abcc-even(2,2)-3}) we obtain\n\\begin{eqnarray}\\label{proposition-recursion-primitive-abcc-even(2,2)-4}\n&&-4\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^I}\\partial_{\\tau^1}F(0)\n\t+\\frac{1}{4}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^I}\\partial_{\\tau^n}F(0)\n\t+ (|I|-2i_c)\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^I}F(0)\\nonumber\\\\\n&=&-4\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^I}\\partial_{\\tau^1}F(0)\n\t+|I|\\partial_{\\tau^{a}}\\partial_{\\tau^b}\\partial_{\\tau^I}F(0)\\nonumber\\\\\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0) \\nonumber\\\\\n&&+\\frac{1}{4}\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^b}F(0)\\nonumber\\\\\n&& + (|I|-2i_c)\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^I}F(0)\\nonumber\\\\\t\n&\\stackrel{\\mbox{by (\\ref{proposition-recursion-primitive-abcc-even(2,2)-2})}}{=}& (2|I|-2i_c-4\\times\\frac{(\\frac{n}{2}-1)|I|+1}{n-1})\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^I}F(0)\\nonumber\\\\\t\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0) \\nonumber\\\\\n&&+\\frac{1}{4}\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^b}F(0)\\nonumber\\\\\n&=& (\\frac{2|I|-4}{n-1}-2i_c)\\partial_{\\tau^{a}}\\partial_{\\tau^b}\\partial_{\\tau^I}F(0)\\nonumber\\\\\t\n&&-\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0) \\nonumber\\\\\n&&+\\frac{1}{4}\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^b}F(0).\n\\end{eqnarray}\nThen (\\ref{eq-recursion-primitive-abcc-even(2,2)}) follows from (\\ref{proposition-recursion-primitive-abcc-even(2,2)-1}) and (\\ref{proposition-recursion-primitive-abcc-even(2,2)-4}).\n\\end{proof}\n\n\n\n\\subsection{A recursion with an unknown correlator}\n\n\\begin{theorem}\\label{thm-reconstruction-even(2,2)}\nWith the knowledge of the 4-point invariants, all the invariants can be reconstructed from the WDVV, the deformation invariance, and the correlator\n\\begin{equation}\\label{eq-specialLength(n+3)Invariant-even(2,2)}\n\t\\langle \\epsilon_{1},\\dots,\\epsilon_{n+3}\\rangle_{0,n+3,\\frac{n}{2}}.\n\\end{equation}\n\\end{theorem}\n\\begin{proof}\nBy Theorem \\ref{thm-monodromy-evenDim(2,2)}, a correlator with exactly one primitive insertion vanishes. By (\\ref{eq-FCA}) a correlator of length $\\geq 4$ having an insertion $1$ vanishes. So using Lemma \\ref{lem-recursion-EulerVecField-even(2,2)} and \\ref{lem-recursion-ambient-simplified-even(2,2)}, one can explicitly write a correlator of length $\\geq 5$ with at least one ambient insertion and one primitive insertion as a combination of correlators of smaller lengths.\n\n\nSuppose $I=(i_0,\\dots,i_{2n+3})\\in \\mathbb{Z}_{\\geq 0}^{2n+3}$ and $|I|>4$. What we said above means that if there exists $i_j>0$ for some $j\\leq n$, then $\\partial_{t^I}F(0)$ can be expressed as a combination of correlators with not larger length and less ambient insertions. Repeating this process our task is reduced to compute invariants with only primitive insertions. So we can assume that $I=(0,\\dots,0,i_{n+1},\\dots,i_{2n+3})$, and $|I|>4$. If there exists pairwise distinct $a,b,c$ in $\\{n+1,\\dots,2n+3\\}$ such that \n\\begin{equation}\\label{eq-condition-recursion-even(2,2)}\n\ti_a,i_b>0, \\mbox{and}\\ |I|-4-(n-1)i_c\\neq 0,\n\\end{equation}\nthen using (\\ref{eq-recursion-primitive-abcc-even(2,2)}), $\\partial_{t^I}F(0)$ can be expressed as a combination of invariants with not larger length and less primitive insertions. If there does not exist such $a,b,c$, then one of the following two cases happen:\n\\begin{enumerate}\n \t\\item[(i)] all $i_j$ but one vanish;\n \t\\item[(ii)] all $i_j>0$ and are equal, and $|I|-4-(n-1)i_j=0$ for all $n+1\\leq j\\leq 2n+3$.\n \\end{enumerate} \n In the case (i) one uses (\\ref{eq-recursion-primitive-aabb-even(2,2)}) to reduce to the case that there exists at least two nonvanishing components in $I$. In the case (ii), \n\\[\ni_{j}(n+3)-4=i_{j}(n-1)\n\\]\nfor all $n+1\\leq j\\leq 2n+3$, which implies $i_{n+1}=i_{n+2}=\\dots=i_{2n+3}=1$, and thus corresponds to (\\ref{eq-specialLength(n+3)Invariant-even(2,2)}).\n\\end{proof}\n\nThe following observation will be used in the proof of Theorem \\ref{thm-convergence}.\n\\begin{remark}\\label{rem:recursion-boundOfIndex}\nIn the proof of Theorem \\ref{thm-reconstruction-even(2,2)}, if the cases (i) and (ii) do not happen, we can take $a,b,c$ such that $i_a$ and $i_b$ are the biggest two components in $I$ and $i_c$ is the smallest one in $I$, and thus\n\\begin{equation}\\label{eq-requirementOnic}\n\ti_c\\leq \\frac{|I|}{n+3}. \n\\end{equation}\nIn fact, when $|I|>n+3$, \n\\[\n|I|-4>(n-1)\\cdot \\frac{|I|}{n+3},\n\\]\nso the condition (\\ref{eq-condition-recursion-even(2,2)}) is satisfied. \nWhen $|I|\\leq n+3$, from Theorem \\ref{thm-monodromy-evenDim(2,2)} it follows that the correlator $\\partial_{\\tau^I}F(0)$ has $i_c=0$ unless it is zero or $I=(1,\\dots,1)$. Then the condition (\\ref{eq-condition-recursion-even(2,2)}) is again satisfied. The requirement (\\ref{eq-requirementOnic}) will be used in the proof of Theorem \\ref{thm-convergence}.\n\\end{remark}\n\n\\begin{remark}{}\nWe call (\\ref{eq-specialLength(n+3)Invariant-even(2,2)}) the \\emph{special correlator} of $X$.\nThe proof of Theorem \\ref{thm-reconstruction-even(2,2)} provides an effective algorithm to compute any correlators of $X$, where we regard the special correlator as an indeterminate number. \nWe implement the algorithm as a Macaulay2 package. For further information see Appendix \\ref{sec:algorithm}.\n\\end{remark}\n\n\\subsection{Conjectures on the special correlator}\\label{sec:conjecturesOnSpecialCorrelator}\nWe made some attempts to extract equations on the special correlator of $X$ from the WDVV equation and Theorem \\ref{thm-monodromy-evenDim(2,2)}. All the relations that we found involving the special correlator turn out to be a trivial equation. I guess that this is always true (unfortunately!). More precisely:\n\\begin{conjecture}\\label{conj-specialCorrelator-free}\nSet the special correlator to be an indeterminate $z$. Let $F(t_0,\\dots,t_{2n+3};z)$ be the generating function of primary genus 0 Gromov-Witten invariants of $X$ computed by Algorithm \\ref{algorithm-correlator-even(2,2)} induced by the proof of Theorem \\ref{thm-reconstruction-even(2,2)}. Then $F(t_0,\\dots,t_{2n+3};z)$ satisfies (\\ref{eq-WDVV}) and the conclusion of Theorem \\ref{thm-monodromy-evenDim(2,2)}.\n\\end{conjecture}\nThis means that to compute the special correlator of $X$ one needs to introduce new tools. In the following of this section I try to make a speculation on the value of the special correlators, by making a comparison to the non-exceptional complete intersections of Fano index $n-1$, i.e. the cubic hypersurfaces and the odd dimensional complete intersections of two quadrics as we recalled in the beginning of Section \\ref{sec:correlators-length4}. Recall that for the non-exceptional complete intersections, a correlator of odd length with only primitive insertions vanishes. So to make comparisons we need first find an appropriate correlator of $X$, which has even length, and express it in terms of the special correlator.\n\\begin{conjecture}\\label{conj-unknownCorrelator-Even(2,2)-quadraticEquation}\nFor even $n$ dimensional complete intersections of two quadrics in $\\mathbb{P}^{n+2}$, \n\\begin{equation}\\label{eq-unknownCorrelator-Even(2,2)-quadraticEquation-normalized}\n\\langle \\epsilon_{1},\\epsilon_1,\\dots,\\epsilon_{n+1},\\epsilon_{n+1}\\rangle_{0,2n+2,n-1}=2^{n-3}\\big((\\langle \\epsilon_{1},\\dots,\\epsilon_{n+3}\\rangle_{0,n+3,\\frac{n}{2}})^2-\\frac{1}{4}\\big).\n\\end{equation}\nEquivalently,\n\\begin{equation}\\label{eq-unknownCorrelator-Even(2,2)-quadraticEquation}\n\\langle \\varepsilon_{1},\\varepsilon_1,\\dots,\\varepsilon_{n+1},\\varepsilon_{n+1}\\rangle_{0,2n+2,n-1}=2^{n-3}\\big((\\langle \\varepsilon_{1},\\dots,\\varepsilon_{n+3}\\rangle_{0,n+3,\\frac{n}{2}})^2-\\frac{(-1)^{\\frac{n}{2}}}{4}\\big).\n\\end{equation}\n\\end{conjecture}\nWe verify this conjecture of $n=4,6$ and $8$; see Appendix \\ref{sec:algorithm}. In \\cite[Conjecture 10.26]{Hu15} we conjectured that for cubic hypersurfaces the correlators of the same form as the LHS of (\\ref{eq-unknownCorrelator-Even(2,2)-quadraticEquation-normalized}) (equivalently, the the LHS of (\\ref{eq-unknownCorrelator-Even(2,2)})) vanish. I guess that the same vanishing holds for $X$ when $n\\equiv 0\\mod 4$.\n Note that $\\langle \\varepsilon_{1},\\dots,\\varepsilon_{n+3}\\rangle_{0,n+3,\\frac{n}{2}}\\in \\mathbb{Q}$ and thus the RHS of (\\ref{eq-unknownCorrelator-Even(2,2)-quadraticEquation}) cannot vanish when $n\\equiv 2\\mod 4$.\n This is the reason that we limit the guess to the dimension $n\\equiv 0\\mod 4$, and leads to the following conjecture. \n\nBefore stating the conjecture, we need to recall Remark \\ref{rem:choiceOfBasis} that there are choices of the basis $\\varepsilon_1,\\dots,\\varepsilon_{n+3}$. All the previous statements are independent of such choices, while the following conjecture on the value of the special correlator does depends on. \n\n\\begin{conjecture}\\label{conj-unknownCorrelator-Even(2,2)}\nFor an even $n$ dimensional complete intersection of two quadrics in $\\mathbb{P}^{n+2}$, let $\\varepsilon_1,\\dots,\\varepsilon_{n+3}$ be the basis of $H^n_{\\mathrm{prim}}(X)$ defined in Section \\ref{sec:explictD-Lattice}. Then \n\\begin{equation}\\label{eq-unknownCorrelator-Even(2,2)}\n\t\\langle \\varepsilon_{1},\\dots,\\varepsilon_{n+3}\\rangle_{0,n+3,\\frac{n}{2}}=\\frac{(-1)^{\\frac{n}{2}}}{2},\n\\end{equation}\nand \n\\begin{equation}\n\t\\langle \\varepsilon_{1},\\varepsilon_1,\\dots,\\varepsilon_{n+1},\\varepsilon_{n+1}\\rangle_{0,2n+2,n-1}=\n\t\\begin{cases}\n\t0,& \\mbox{if}\\ n\\equiv 0 \\mod 4,\\\\\n\t2^{n-4}, & \\mbox{if}\\ n\\equiv 2 \\mod 4.\n\t\\end{cases}\n\\end{equation}\n\\end{conjecture}\nWe will prove the $n=4$ case in Section \\ref{sec:EnumerativeGeometry-Even(2,2)}.\nAt this stage we have no further evidence for Conjecture \\ref{conj-unknownCorrelator-Even(2,2)}.\nThe values in (\\ref{eq-unknownCorrelator-Even(2,2)}) is quite speculative, at least when $n\\equiv 2 \\mod 4$. For the reason for our choice of the sign in (\\ref{eq-unknownCorrelator-Even(2,2)}), we refer the reader to Example \\ref{example-f(6)}.\n\n\\section{Convergence of the generating function}\nIn this section, we still fix a smooth complete intersection $X$ of two quadrics in $\\mathbb{P}^{n+2}$, where $n$ is even and $\\geq 4$. \nUsing Algorithm \\ref{algorithm-correlator-even(2,2)} induced by the proof of Theorem \\ref{thm-reconstruction-even(2,2)} we will show that the generating function $F$ has a positive convergence radius. So $F$ is an analytic function rather than only a formal series. The following theorem is a verification of \\cite[Conjecture 1]{Zin14} for $X$.\n\\begin{theorem}\\label{thm-convergence}\nLet $\\gamma_0,\\dots,\\gamma_{n},\\gamma_{n+1},\\dots,\\gamma_{2n+3}$ be a basis of $H^*(X)$. Then there exists a constant $C>0$ such that \n\\begin{equation}\\label{eq-convergence-0}\n\t\\langle \\gamma_{i_1},\\dots,\\gamma_{i_k}\\rangle_{0,k,\\beta}\\leq k! C^{k}\n\\end{equation}\nfor all $k\\geq 0$, and $0\\leq i_1,\\dots,i_k\\leq 2n+3$.\n\\end{theorem}\n\n\n\\begin{lemma}\\label{lem-inequalityOfBinomialOfLists}\nFor any $M\\leq |I|$, \n\\begin{equation}\\label{eq-inequalityOfBinomialOfLists}\n\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ |J|\\leq M\\end{subarray}}\\binom{I}{J}\\leq \\binom{|I|}{M}.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nThis follows from the enumerative meaning of both sides.\n\\end{proof}\n\\begin{lemma}\\label{lem-inequality-1}\nFor $n\\geq 4$,\n\\begin{equation}\\label{eq-inequality-1}\n\t\\sum_{k=2}^{n-2}\\frac{n(n-1)}{k(k-1)(n-k)(n-k-1)}<4.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\nWe compute\n\\begin{eqnarray*}\n&&\\sum_{k=2}^{n-2}\\frac{n(n-1)}{k(k-1)(n-k)(n-k-1)}\\\\\n&=& \\sum_{k=2}^{n-2}\\frac{n-1}{(k-1)(n-k)(n-k-1)}+\\sum_{k=2}^{n-2}\\frac{n-1}{k(k-1)(n-k-1)}\\\\\n&=& \\sum_{k=2}^{n-2}\\frac{1}{(n-k)(n-k-1)}+\\sum_{k=2}^{n-2}\\frac{1}{(k-1)(n-k-1)}\\\\\n&&+\\sum_{k=2}^{n-2}\\frac{1}{(k-1)(n-k-1)}+\\sum_{k=2}^{n-2}\\frac{1}{k(k-1)}\\\\\n&=& 2\\sum_{k=2}^{n-2}\\frac{1}{k(k-1)}+2\\sum_{k=2}^{n-2}\\frac{1}{(k-1)(n-k-1)}.\n\\end{eqnarray*}\nThen we estimate the two sums separately. For the first sum,\n\\[\n\\sum_{k=2}^{n-2}\\frac{1}{k(k-1)}=\\sum_{k=2}^{n-2}(\\frac{1}{k-1}-\\frac{1}{k})=1-\\frac{1}{n-2}.\n\\]\nFor the second sum we have\n\\begin{eqnarray*}\n&&\\sum_{k=2}^{n-2}\\frac{1}{(k-1)(n-k-1)}\\\\\n&=& 2\\sum_{k=2}^{\\frac{n}{2}-1}\\frac{1}{(k-1)(n-k-1)}+\\frac{1}{(\\frac{n}{2}-1)^2}\\\\\n&\\leq & 2\\sum_{k=2}^{\\frac{n}{2}-1}\\frac{1}{(k-1)\\frac{n}{2}}+\\frac{1}{(\\frac{n}{2}-1)^2}\\\\\n&\\leq& \\frac{4}{n}\\big(1+\\log(\\frac{n}{2}-2)\\big)+\\frac{1}{(\\frac{n}{2}-1)^2}.\n\\end{eqnarray*}\nIt follows that\n\\begin{eqnarray*}\n\\sum_{k=2}^{n-2}\\frac{n(n-1)}{k(k-1)(n-k)(n-k-1)}\n\\leq 2-\\frac{2}{n-2}+\\frac{8}{n}\\big(1+\\log(\\frac{n}{2}-2)\\big)+\\frac{2}{(\\frac{n}{2}-1)^2}<4.\n\\end{eqnarray*}\n\\end{proof}\n\n\n\\begin{proof}[Proof of Theorem \\ref{thm-convergence}]\nThe statement is independent of the choice of the basis $\\gamma_0,\\dots,\\gamma_{2n+3}$. \nWe take the basis $1,\\tilde{\\mathsf{h}}_1,\\dots,\\tilde{\\mathsf{h}}_n,\\epsilon_1,\\dots,\\epsilon_{n+3}$. Then (\\ref{eq-convergence-0}) is equivalent to the existence of $C>0$ such that\n\\begin{equation}\\label{eq-convergence-01}\n\t|\\partial_{\\tau^I}F(0)|\\leq |I|! C^{|I|}\n\\end{equation}\nfor all $I\\in \\mathbb{Z}_{\\geq 0}^{2n+4}$. \nWithout loss of generality, we can assume that (\\ref{eq-convergence-0}) holds for $k\\leq K$, where $K$ is an arbitrary chosen natural number, and prove (\\ref{eq-convergence-0}) inductively for all $k$. We note that the wanted statement is equivalent to the existence of $C>0$ such that \n\\begin{equation}\\label{eq-convergence-02}\n\t|\\partial_{\\tau^I}F(0)|\\leq (|I|-5)! C^{|I|-5}\n\\end{equation}\n for all $I\\in \\mathbb{Z}_{\\geq 0}^{2n+4}$. By \\cite[Theorem 1]{Zin14}, (\\ref{eq-convergence-0}), equivalently (\\ref{eq-convergence-02}), holds for correlators with only ambient classes; one can also find a simple proof of this fact in \\cite[Remark D.13]{Hu15}. In the following we show (\\ref{eq-convergence-02}) by induction on $|I|$ and the number of primitive insertions. Suppose (\\ref{eq-convergence-02}) holds for $5\\leq |I|\\leq k$. \nBy (\\ref{eq-recursion-EulerVecField-even(2,2)}),\n\\begin{eqnarray*}\n|\\partial_{\\tau^1}\\partial_{\\tau^I}F(0)|\n&=&\\big| \\frac{\\sum_{j=0}^n(j-1)i_j+(\\frac{n}{2}-1)\\sum_{j=n+1}^{2n+3}i_j+3-n}{n-1}\\partial_{\\tau^I}F(0)\\nonumber\\\\\n&&-12i_n\\partial_{\\tau^1}\\partial_{\\tau^{I-e_n}}F(0)\\big|\\\\\n&\\leq & |I|\\cdot (|I|-5)! C^{|I|-5} +12|I|\\cdot (|I|-5)! C^{|I|-5}.\n\\end{eqnarray*}\nThus replacing $C$ by a constant $C>65$ if necessary, we have\n\\begin{eqnarray*}\n|\\partial_{\\tau^1}\\partial_{\\tau^I}F(0)|\\leq (|I|-4)! C^{|I|-4}.\n\\end{eqnarray*}\nIn the following of the proof we use a temporary convention\n\\begin{equation}\n\tk!=1\\ \\mbox{for}\\ k<0.\n\\end{equation}\nThen by (\\ref{eq-recursion-ambient-even(2,2)}) and (\\ref{eq-etaInversePairing-even(2,2)}), for $2\\leq i\\leq n$ and $n+1\\leq a,b\\leq 2n+3$,\n\\begin{eqnarray*}\n&&|\\partial_{\\tau^i}\\partial_{\\tau^I}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)|\\\\\n&\\leq &\\Big|\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{i-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\Big|\\nonumber\\\\\n&&+\t\\Big|\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{i-1}}\\partial_{\\tau^b}F(0)\\Big|\\\\\n&\\leq & (2n+6)\\times 2\\times\t4\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J} (|J|-2)! C^{|J|-2} (|I|-|J|-2)! C^{|I|-|J|-2}\\\\\n&=& 8(2n+6) (|I|-2)! C^{|I|-4} \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\frac{|J|-2)!(|I|-|J|-2)!}{(|I|-2)!}.\n\\end{eqnarray*}\nBy Lemma \\ref{lem-inequalityOfBinomialOfLists} and Lemma \\ref{lem-inequality-1},\n\\begin{eqnarray*}\n&&\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\frac{|J|-2)!(|I|-|J|-2)!}{(|I|-2)!}\\\\\n&\\leq & \\sum_{M=1}^{|I|}\\binom{|I|}{M}\\frac{(M-2)!(|I|-M-2)!}{(|I|-2)!}\\\\\n&=& 2\\times \\frac{|I|}{|I|-2}+1+\\sum_{M=2}^{|I|-2}\\frac{|I|(|I|-1)}{M(M-1)(|I|-M)(|I|-M-1)}<9.\n\\end{eqnarray*}\nSo if $C^2>72(2n+6)$, we have\n\\begin{eqnarray*}\n&&|\\partial_{\\tau^i}\\partial_{\\tau^I}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)|\\\\\n&<& 72(2n+6) (|I|-2)! C^{|I|-4} <(|I|-2)! C^{|I|-2}.\n\\end{eqnarray*}\n\n\n\n\n\nBy (\\ref{eq-recursion-primitive-abcc-even(2,2)}), \n\\begin{eqnarray*}\n &&\\big|(\\frac{2|I|-4}{n-1}-2i_c)\\partial_{\\tau^{a}}\\partial_{\\tau^b}\\partial_{\\tau^I}F(0)\\big|\\nonumber\\\\\t\n&=&\\Big|\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0) \\nonumber\\\\\n&&-\\frac{1}{4}\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^b}F(0)\\nonumber\\\\\n&&-\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^c}^2F(0)\\nonumber\\\\\n&& +\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^c}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^b}\\partial_{\\tau^c}F(0)\\Big|\\\\\n&\\leq & 4\\times (2n+6)\\times 4 \t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J} (|J|-2)!C^{|J|-2} (|I|-|J|-2)!C^{|I|-|J|-2}\\\\\n&= & 16(2n+6) (|I|-2)! C^{|I|-4}\t\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J} \\frac{(|J|-2)! (|I|-|J|-2)!}{(|I|-2)!}\\\\\n&<& 144(2n+6) (|I|-2)! C^{|I|-4},\n\\end{eqnarray*}\nBy Remark \\ref{rem:recursion-boundOfIndex}, we can assume\n\\begin{equation}\n\ti_c\\leq \\frac{|I|+2}{n+3}. \n\\end{equation}\nThen \n\\[\n\\frac{2|I|-4}{n-1}-2i_c\\geq \\frac{8(|I|-n-1)}{(n-1)(n+3)},\n\\]\nand thus\n\\begin{eqnarray*}\n &&\\big|\\partial_{\\tau^{a}}\\partial_{\\tau^b}\\partial_{\\tau^I}F(0)\\big|\\nonumber\\\\\t\n&<& 144(2n+6) (|I|-3)! C^{|I|-4}\\cdot (n-1)(n+3) \\frac{|I|-2}{8(|I|-n-1)}.\n\\end{eqnarray*}\nWhen $|I|\\geq 2n+2$, \n\\[\n\\frac{|I|-2}{8(|I|-n-1)}<\\frac{1}{4}.\n\\]\nSo when $|I|\\geq 2n+2$ and \n\\[\nC>\\big(36(2n+6)(n-1)(n+3)\\big)^2,\n\\]\nwe have\n\\begin{equation}\\label{eq-convergence-primitive-abcc}\n\t\\big|\\partial_{\\tau^{a}}\\partial_{\\tau^b}\\partial_{\\tau^I}F(0)\\big|<(|I|-3)! C^{|I|-\\frac{7}{2}}.\n\\end{equation}\n\nBy (\\ref{eq-recursion-primitive-aabb-tau-simplified-even(2,2)}) and (\\ref{eq-convergence-primitive-abcc}), when $i_a=|I|$,\n\\begin{eqnarray*}\n&&(\\frac{2|I|-4}{n-1})\\partial_{\\tau^{a}}^2\\partial_{\\tau^I}F(0)\\nonumber\\\\\n&\\leq &\\Big|(\\frac{2|I|-4}{n-1}-2|I|)\\partial_{\\tau^{b}}^2\\partial_{\\tau^I}F(0)\\Big|\\\\\n&&+\\Big|\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\n\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}^2F(0) \\nonumber\\\\\n&&- \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{a}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^a}F(0)\\nonumber\\\\\t\n&&+\\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{n-1}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^b}^2F(0) \\nonumber\\\\\n&&- \\frac{1}{4}\\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|-1\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^1}\\partial_{\\tau^{b}}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^{n-1}}\\partial_{\\tau^b}F(0)\\nonumber\\\\\n&&- \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}^2\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^b}^2F(0)\\nonumber\\\\\n&&+ \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 2\\leq |J|\\leq |I|-2\\end{subarray}}\\binom{I}{J}\\partial_{\\tau^a}\\partial_{\\tau^b}\\partial_{\\tau^J}\\partial_{\\tau^e}F(0)\\eta^{ef}\n\t\\partial_{\\tau^{f}}\\partial_{\\tau^{I-J}}\\partial_{\\tau^a}\\partial_{\\tau^b}F(0)\\Big|\\\\\n&\\leq & 2|I|\\cdot(|I|-3)! C^{|I|-\\frac{7}{2}}\\\\\n&&\t+ 6\\times(2n+6)\\times 4\\times \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J} (|J|-2)!C^{|J|-2} (|I|-|J|-2)!C^{|I|-|J|-2}\\\\\n&\\leq & 2|I|\\cdot (|I|-3)! C^{|I|-\\frac{7}{2}}\\\\\n&&+ 24(2n+6) (|I|-2)! C^{|I|-4} \\sum_{\\begin{subarray}{c}0\\leq J\\leq I\\\\ 1\\leq |J|\\leq |I|\\end{subarray}}\\binom{I}{J} \\frac{(|J|-2)! (|I|-|J|-2)!}{(|I|-2)!}\\\\\n&\\leq & (|I|-3)! C^{|I|-3}\\big(\\frac{2|I|}{C^{\\frac{1}{2}}}+\\frac{216(2n+6)(|I|-2)}{C}\\big),\n\\end{eqnarray*}\nso\n\\begin{eqnarray*}\n&&\\partial_{\\tau^{a}}^2\\partial_{\\tau^I}F(0)\n\\leq (|I|-3)! C^{|I|-3}\n\\big(\\frac{|I|(n-1)}{(|I|-2)C^{\\frac{1}{2}}}+\\frac{108(2n+6)(n-1)}{C}\\big)<(|I|-3)! C^{|I|-3}\n\\end{eqnarray*}\nwhen \n\\begin{equation}\\label{eq-convergence-estimateC}\n\\frac{|I|(n-1)}{(|I|-2)C^{\\frac{1}{2}}}+\\frac{108(2n+6)(n-1)}{C}<1.\n\\end{equation}\nWe choose $C$ such that (\\ref{eq-convergence-02}) holds for $|I|<2n+1$, and such that \n\\[\nC>\\max\\{65,\\sqrt{72(2n+6)},\\big(36(2n+6)(n-1)(n+3)\\big)^2\n\\}\n\\]\nand (\\ref{eq-convergence-estimateC}) holds. Then by Algorithm \\ref{algorithm-correlator-even(2,2)} and the above estimates, (\\ref{eq-convergence-02}) holds for all $I$.\n\\end{proof}\n\n\\begin{corollary}\\label{cor-convergence}\nThere exists an open neighborhood of 0 in $\\mathbb{C}^{2n+4}$ on which the generating function $F$ is an analytic function of $t^0,\\dots,t^{2n+3}$.\n\\end{corollary}\n\n\\section{Semisimplicity}\nIn this section, we still fix a smooth complete intersection $X$ of two quadrics in $\\mathbb{P}^{n+2}$, where $n$ is even and $\\geq 4$. By Corollary \\ref{cor-convergence}, there exists an open neighborhood $U$ of $0\\in \\mathbb{C}^{2n+4}$, on which the generating function $F$ defines a Frobenius manifold $\\mathcal{M}_X$. In this section we show\n\\begin{theorem}\\label{thm-semisimplicity}\nThe Frobenius manifold $\\mathcal{M}_X$ is (generically) semisimple.\n\\end{theorem}\n The strategy is to show that at a general point of $U$, the multiplication by the Euler vector field $E$ has only simple eigenvalues. We work in the $\\tau$-coordinates. We use Einstein's summation convention, where the range of the indices runs over $0,\\dots,2n+3$.\n\n\\subsection{Quantum multiplication by the Euler vector field}\nIn $\\tau$-coordinates, using (\\ref{eq-EulerField-tau-Coordinates}), the big quantum multiplication by the Euler vector field $E$ is\n\\begin{eqnarray*}\n E\\star \\partial_{\\tau^j}&=&\\sum_{i=0}^{n}(1-i)\\tau^i(\\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^e}F)\\eta^{ef}\\partial_{\\tau^f}+(4n-4) \\tau^{n-1}\\partial_{\\tau^{j}}\\\\\n&&\t+(12n-12) \\tau^n(\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^e}F)\\eta^{ef}\\partial_{\\tau^f}\n+\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}(\\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^e}F)\\eta^{ef}\\partial_{\\tau^f}\\\\\n&&+(n-1)(\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^e}F)\\eta^{ef}\\partial_{\\tau^f}.\n\\end{eqnarray*}\nDenote by $\\widetilde{\\mathcal{E}}$ the matrix of the big quantum multiplication by $E$ in the basis $\\partial_{\\tau^0},\\dots,\\partial_{\\tau^{2n+3}}$.\nThen by (\\ref{eq-etaInversePairing-even(2,2)}) we get\n\\begin{eqnarray*}\n\\widetilde{\\mathcal{E}}_j^0&=&(4n-4) \\tau^{n-1}\\delta_{j}^0-4\\sum_{i=0}^{n}(1-i)\\tau^i \\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^1}F\n+\\frac{1}{4}\\sum_{i=0}^{n}(1-i)\\tau^i \\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^{n}}F\\\\\n&&\t-48(n-1) \\tau^n\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^1}F\n+3(n-1) \\tau^n\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^n}F\\\\\n&&-4\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^1}F\n+\\frac{1}{4}\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^{n}}F\\\\\n&&-4(n-1)\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^1}F\n+\\frac{1}{4}(n-1)\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^n}F,\n\\end{eqnarray*}\n\\begin{eqnarray*}\n\\widetilde{\\mathcal{E}}_j^1&=&(4n-4) \\tau^{n-1}\\delta_{j}^1-4\\sum_{i=0}^{n}(1-i)\\tau^i \\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^0}F\n+\\frac{1}{4}\\sum_{i=0}^{n}(1-i)\\tau^i \\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^{n-1}}F\\\\\n&&\t-48(n-1) \\tau^n\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^0}F\n+3(n-1) \\tau^n\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^{n-1}}F\\\\\n&&-4\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^0}F\n+\\frac{1}{4}\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^{n-1}}F\\\\\n&&-4(n-1)\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^0}F\n+\\frac{1}{4}(n-1)\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^{n-1}}F,\n\\end{eqnarray*}\nand for $2\\leq k\\leq n$,\n\\begin{eqnarray*}\n\\widetilde{\\mathcal{E}}_j^k&=&(4n-4) \\tau^{n-1}\\delta_{j}^k\n+\\frac{1}{4}\\sum_{i=0}^{n}(1-i)\\tau^i \\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^{n-k}}F\\\\\n&&+3(n-1) \\tau^n\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^{n-k}}F\n+\\frac{1}{4}\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^{n-k}}F\\\\\n&&+\\frac{1}{4}(n-1)\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^{n-k}}F,\n\\end{eqnarray*}\nand for $n+1\\leq k\\leq 2n+3$,\n\\begin{eqnarray*}\n\\widetilde{\\mathcal{E}}_j^k&=&(4n-4) \\tau^{n-1}\\delta_{j}^k\n+\\sum_{i=0}^{n}(1-i)\\tau^i \\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^{k}}F\\\\\n&&+12(n-1) \\tau^n\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^{k}}F\n+\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^{k}}F\\\\\n&&+(n-1)\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^{k}}F.\n\\end{eqnarray*}\n\n\n\n\n\\subsection{2nd order cutoff of the quantum multiplication by Euler vector field}\nTaking the 2nd order cutoff of the matrix $\\widetilde{\\mathcal{E}}$, and taking $\\tau^i=0$ for $0\\leq i\\leq n$, we denote the resulted matrix by $\\mathcal{E}$, i.e.\n\\[\n\\mathcal{E}(\\tau_{n+1},\\dots,\\tau_{2n+3})=\\widetilde{\\mathcal{E}}(0,\\dots,0,\\tau_{n+1},\\dots,\\tau_{2n+3})\n+o(\\tau^2).\n\\]\n Then\n\\begin{eqnarray}\\label{eq-cutoff-EulverVectorField-1}\n\\mathcal{E}_j^0&=&-4\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^1}F\n+\\frac{1}{4}\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^{n}}F\\nonumber\\\\\n&&-4(n-1)\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^1}F\n+\\frac{1}{4}(n-1)\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^n}F\\nonumber\\\\\n&=& \\begin{cases}\n-4(1-\\frac{n}{2})\\cdot (-4)\\sum_{i=n+1}^{2n+3}(\\tau^i)^2\n+\\frac{1}{4}(1-\\frac{n}{2})\\cdot (-64)\\sum_{i=n+1}^{2n+3}(\\tau^i)^2&\\\\\n-4(n-1)\\cdot(-4)\\cdot \\frac{1}{2}\\sum_{i=n+1}^{2n+3}(\\tau^i)^2 &\\\\\n+\\frac{1}{4}(n-1)\\cdot \\big(2\\times (-64)-12\\times (-4)\\big)\\cdot \\frac{1}{2}\\sum_{i=n+1}^{2n+3}(\\tau^i)^2\n,& \\mbox{if}\\ j=n-1,\\\\\n0 & \\mbox{if otherwise},\\\\\n\\end{cases}\\nonumber\\\\\n&=& \\begin{cases}\n-2(n-1)\\sum_{i=n+1}^{2n+3}(\\tau^i)^2,& \\mbox{if}\\ j=n-1,\\\\\n0 & \\mbox{if otherwise}.\n\\end{cases}\n\\end{eqnarray}\n\\begin{eqnarray}\\label{eq-cutoff-EulverVectorField-2}\n\\mathcal{E}_j^1&=&-4\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^0}F\n+\\frac{1}{4}\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^{n-1}}F\\nonumber\\\\\n&&-4(n-1)\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^0}F\n+\\frac{1}{4}(n-1)\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^{n-1}}F\\nonumber\\\\\n&=&\\begin{cases}\nn-1, & \\mbox{if}\\ j=0,\\\\\n\\frac{1}{4}\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\cdot(-4 \\tau^i)\n+\\frac{1}{4}(n-1)\\cdot(-4)\\times \\frac{1}{2}\\sum_{i=n+1}^{2n+3}(\\tau^i)^2,& \\mbox{if}\\ j=1,\\\\\n-16(n-1)+\\frac{1}{4}(n-1)\\times 64,& \\mbox{if}\\ j=n-1,\\\\\n\\frac{1}{4}(1-\\frac{n}{2})\\cdot (-64)\\sum_{i=n+1}^{2n+3}(\\tau^i)^2 &\\\\\n+\\frac{1}{4}(n-1)\\cdot \\big(2\\times (-64)-12\\times (-4)\\big)\\cdot \\frac{1}{2}\\sum_{i=n+1}^{2n+3}(\\tau^i)^2\n& \\mbox{if}\\ j=n,\\\\\n-4(1-\\frac{n}{2})\\tau^j+\\frac{1}{4}(n-1)\\cdot(-4 \\tau^j),&\\mbox{if}\\ n+1\\leq j\\leq 2n+3,\n\\end{cases}\\nonumber\\\\\n&=&\\begin{cases}\nn-1, & \\mbox{if}\\ j=0,\\\\\n-\\frac{1}{2}\\sum_{i=n+1}^{2n+3}(\\tau^{i})^2\n,& \\mbox{if}\\ j=1,\\\\\n0,& \\mbox{if}\\ j=n-1,\\\\\n(-2n-6)\\sum_{i=n+1}^{2n+3}(\\tau^{i})^2,& \\mbox{if}\\ j=n,\\\\\n(n-3)\\tau^j,&\\mbox{if}\\ n+1\\leq j\\leq 2n+3.\n\\end{cases}\n\\end{eqnarray}\nFor $2\\leq k\\leq n-1$,\n\\begin{eqnarray}\\label{eq-cutoff-EulverVectorField-3}\n\\mathcal{E}_j^k&=&\\frac{1}{4}\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^{n-k}}F\n+\\frac{1}{4}(n-1)\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^{n-k}}F\\nonumber\\\\\n&=&\\begin{cases}\nn-1, & \\mbox{if}\\ j=k-1,\\\\\n\\frac{1}{4}\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\cdot(-4 \\tau^i)\n+\\frac{1}{4}(n-1)\\cdot (-4)\\cdot\\frac{1}{2}\\sum_{i=n+1}^{2n+3}(\\tau^{i})^2,& \\mbox{if}\\ j=k,\\\\\n16(n-1), & \\mbox{if}\\ (j,k)=(n,2),\\\\\n0,\\mbox{otherwise},\n\\end{cases}\\nonumber\\\\\n&=&\\begin{cases}\nn-1, & \\mbox{if}\\ j=k-1,\\\\\n-\\frac{1}{2}\\sum_{i=n+1}^{2n+3}(\\tau^{i})^2,& \\mbox{if}\\ j=k,\\\\\n16(n-1), & \\mbox{if}\\ (j,k)=(n,2),\\\\\n0,\\mbox{otherwise}.\n\\end{cases}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\\label{eq-cutoff-EulverVectorField-4}\n\\mathcal{E}_j^n&=&\\frac{1}{4}\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^{0}}F\n+\\frac{1}{4}(n-1)\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^{0}}F\\nonumber\\\\\n&=&\\begin{cases}\nn-1,& \\mbox{if}\\ j=n-1,\\\\\n0, & \\mbox{if}\\ 0\\leq j\\leq n-2\\ \\mbox{of}\\ j=n,\\\\\n\\frac{2-n}{8}\\tau^j,& \\mbox{if}\\ n+1\\leq j\\leq 2n+3.\n\\end{cases}\n\\end{eqnarray}\nFor $n+1\\leq k\\leq 2n+3$,\n\\begin{eqnarray}\\label{eq-cutoff-EulverVectorField-5}\n\\mathcal{E}_j^k&=&\\sum_{i=n+1}^{2n+3}(1-\\frac{n}{2})\\tau^{i}\\partial_{\\tau^i}\\partial_{\\tau^j}\\partial_{\\tau^{k}}F\n+(n-1)\\partial_{\\tau^1}\\partial_{\\tau^j}\\partial_{\\tau^{k}}F\\nonumber\\\\\n&=&\\begin{cases}\n(1-\\frac{n}{2})\\tau^k,& \\mbox{if}\\ j=0,\\\\\n-4(n-1)\\tau^k, & \\mbox{if}\\ j=n-1,\\\\\n0,& \\mbox{if}\\ 1\\leq j\\leq n-2\\ \\mbox{or}\\ j=n,\\\\\n(2-n)\\tau^j \\tau^k+(n-1)\\tau^j \\tau^k, & \\mbox{if}\\ n+1\\leq j\\neq k\\leq 2n+3,\\\\\n(1-\\frac{n}{2})\\sum_{i=n+1}^{2n+3}(\\tau^{i})^2+(n-1)\\cdot\\frac{1}{2}\\sum_{i=n+1}^{2n+3}(\\tau^{i})^2, & \\mbox{if}\\ j=k,\n\\end{cases}\\nonumber\\\\\n&=&\\begin{cases}\n(1-\\frac{n}{2})\\tau^k,& \\mbox{if}\\ j=0,\\\\\n-4(n-1)\\tau^k, & \\mbox{if}\\ j=n-1,\\\\\n0,& \\mbox{if}\\ 0\\leq j\\leq n-2\\ \\mbox{or}\\ j=n,\\\\\n\\tau^j \\tau^k, & \\mbox{if}\\ n+1\\leq j\\neq k\\leq 2n+3,\\\\\n\\frac{1}{2}\\sum_{i=n+1}^{2n+3}(\\tau^{i})^2, & \\mbox{if}\\ j=k.\n\\end{cases}\n\\end{eqnarray}\n\n\\subsection{The characteristic polynomial}\n\n\n\\begin{comment}\n{\\footnotesize\n\\begin{equation*}\n\t\\begin{pmatrix}\n\t\\begin{matrix}\n\t0 & n-1 & 0 & 0 & \\dots & 0 & 0 \\\\\n\t0 & -\\frac{s}{2} & n-1 & 0 & \\dots & 0 & 0 \\\\\n\t0 & 0 & -\\frac{s}{2} & n-1 & \\dots & 0 & 0 \\\\\n\t\\vdots & \\vdots & \\vdots & \\vdots& \\vdots & \\vdots& \\vdots \\\\\n\t0 & 0 & \\dots & 0 & \\dots & n-1 & 0 \\\\\n\t-2(n-1)s & 0 & \\dots & \\dots & \\dots& -\\frac{s}{2} & n-1 \\\\\n\t0 & (-2n-6)s & 16(n-1) & 0 & \\dots& 0 & 0 \n\t\\end{matrix} & \\hspace*{-\\arraycolsep}\\vline\\hspace*{-\\arraycolsep} & \n\t\\begin{matrix}\n\t (1-\\frac{n}{2})\\tau^{n+1} & \\dots & (1-\\frac{n}{2})\\tau^{2n+3} \\\\\n\t 0 & \\dots & 0 \\\\\n\t 0 & \\dots & 0 \\\\\n\t \\vdots& \\vdots& \\vdots\\\\\n\t 0 & \\dots& 0\\\\\n\t (4-4n)\\tau^{n+1} & \\dots & (4-4n)\\tau^{2n+3}\\\\\n\t 0 & \\dots & 0 \n\t \\end{matrix} \\\\\n\t \\hline \n\t \\begin{matrix}\n\t0 & (n-3)\\tau^{n+1} & 0 & 0 & \\dots & 0 & \\frac{2-n}{8}\\tau^{n+1} \\\\\n\t0 & (n-3)\\tau^{n+2} & \\dots & \\dots & \\dots & 0 & \\frac{2-n}{8}\\tau^{n+2} \\\\\n\t\\vdots & \\vdots & \\vdots & \\vdots& \\vdots & \\vdots& \\vdots \\\\\n\t0 & (n-3)\\tau^{2n+3} & 0 & \\dots & \\dots & 0 & \\frac{2-n}{8}\\tau^{2n+3} \n\t\\end{matrix} & \\hspace*{-\\arraycolsep}\\vline\\hspace*{-\\arraycolsep} &\n\t\\begin{matrix}\n\t \\frac{s}{2} & \\dots & \\tau^{n+1}\\tau^{2n+3}\\\\\n\t \\tau^{n+1}\\tau^{n+1} & \\dots & \\tau^{n+2}\\tau^{2n+2} \\\\\n\t\\vdots& \\vdots& \\vdots\\\\\n\t\\tau^{n+1}\\tau^{2n+3} & \\dots & \\frac{s}{2}\n\t\\end{matrix}\n\t\\end{pmatrix}\n\\end{equation*}\n}\n\\end{comment}\n\nLet $\\mathcal{P}_n(z,\\tau^{n+1},\\dots,\\tau^{2n+3})$ be the characteristic polynomial of $\\widetilde{\\mathcal{E}}$ at $(0,\\dots,0,\\tau_{n+1},\\tau_{n+2},\\dots,\\tau_{2n+3})$, and let $P_n(z,\\tau^{n+1},\\dots,\\tau^{2n+3})$ be the characteristic polynomial of $\\mathcal{E}$ at $(\\tau_{n+1},\\tau_{n+2},\\dots,\\tau_{2n+3})$.\nFor brevity of expressions we define (recall (\\ref{eq-invariantsOf-typeD-1}))\n\\[\ns=2s_1=\\sum_{i=n+1}^{2n+3}(\\tau^{i})^2.\n\\]\nDenote by $E_{i,j}$ the elementary matrix whose only nonzero entry is $1$ at the position $(i,j)$.\n\\begin{proposition}\\label{prop-charPoly-EV-2ndOrder}\n\\begin{eqnarray}\\label{eq-charPoly-EV-2ndOrder}\n&&P_n(z,\\tau^{n+1},\\dots,\\tau^{2n+3})\\nonumber\\\\\n&=&\\bigg((n-1)^{n-1}\\big(-\\frac{(n-1)(n-2)^2 s^2}{4}+2(n-1)(n-4)sz\n-4(n-5)z^2\\big)\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\\nonumber\\\\\n&&+4(n-1)^n s z-16(n-1)^{n-1}z^2+z^2(z+\\frac{s}{2})^{n-1}\n -z^2(z+\\frac{s}{2})^{n-1}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2})\\bigg)\\nonumber\\\\\n&&\\cdot \\prod_{i=n+1}^{2n+3}\\big(z-\\frac{s}{2}+(\\tau^i)^2\\big).\n\\end{eqnarray}\n\\end{proposition}\n\\begin{proof}\nBy (\\ref{eq-cutoff-EulverVectorField-1})-(\\ref{eq-cutoff-EulverVectorField-5}), we have\n{\\footnotesize\n\\begin{equation*}\n\\mathcal{E}=\\begin{pNiceArray}{ccccccc|ccc}\n\t0 & n-1 & 0 & 0 & \\dots & 0 & 0 & (1-\\frac{n}{2})\\tau^{n+1} & \\dots & (1-\\frac{n}{2})\\tau^{2n+3} \\\\\n\t0 & -\\frac{s}{2} & n-1 & 0 & \\dots & 0 & 0 & 0 & \\dots & 0 \\\\\n\t0 & 0 & -\\frac{s}{2} & n-1 & \\dots & 0 & 0 & 0 & \\dots & 0 \\\\\n\t\\vdots & \\vdots & \\vdots & \\vdots& \\vdots & \\vdots& \\vdots & \t \\vdots& \\vdots& \\vdots\\\\\n\t0 & 0 & \\dots & 0 & \\dots & n-1 & 0 & \t 0 & \\dots& 0\\\\\n\t-2(n-1)s & 0 & \\dots & \\dots & \\dots& -\\frac{s}{2} & n-1 & (4-4n)\\tau^{n+1} & \\dots & (4-4n)\\tau^{2n+3}\\\\\n\t0 & (-2n-6)s & 16(n-1) & 0 & \\dots& 0 & 0 & 0 & \\dots & 0 \\\\\n\t\\hline\n\t0 & (n-3)\\tau^{n+1} & 0 & 0 & \\dots & 0 & \\frac{2-n}{8}\\tau^{n+1} & \\frac{s}{2} & \\dots & \\tau^{n+1}\\tau^{2n+3}\\\\ \n\t0 & (n-3)\\tau^{n+2} & \\dots & \\dots & \\dots & 0 & \\frac{2-n}{8}\\tau^{n+2} & \\tau^{n+1}\\tau^{n+2} & \\dots & \\tau^{n+2}\\tau^{2n+2} \\\\\n\t\\vdots & \\vdots & \\vdots & \\vdots& \\vdots & \\vdots& \\vdots & \t\\vdots& \\vdots& \\vdots\\\\\n\t0 & (n-3)\\tau^{2n+3} & 0 & \\dots & \\dots & 0 & \\frac{2-n}{8}\\tau^{2n+3} & \t\\tau^{n+1}\\tau^{2n+3} & \\dots & \\frac{s}{2}\n \t\\end{pNiceArray}.\n\\end{equation*}\n}\nIn the following we index the rows and columns by $0\\leq i\\leq 2n+3$. In the above we have blocked the matrix $\\mathcal{E}$ by the first $n+1$ rows and first $n+1$ columns.\nWe perform several similarity transforms of $\\mathcal{E}$.\nLet \n\\[\n\\mathcal{E}_1=\\mathrm{Diag}(\\underbrace{1,\\dots,1}_{n+1},\\tau^{n+1},\\dots,\\tau^{2n+3})\\cdot\n\\mathcal{E}\\cdot \\mathrm{Diag}(\\underbrace{1,\\dots,1}_{n+1},\\frac{1}{\\tau^{n+1}},\\dots,\\frac{1}{\\tau^{2n+3}}). \n\\]\nThe effect is the $i$-th row $\\times \\tau^i$, the $i$-th column $\\times \\frac{1}{\\tau^i}$, for $n+1\\leq i\\leq 2n+3$.\nThen\n{\\footnotesize\n\\begin{equation*}\n\\mathcal{E}_1=\\begin{pNiceArray}{ccccccc|ccc}\n\t0 & n-1 & 0 & 0 & \\dots & 0 & 0 & 1-\\frac{n}{2} & \\dots & 1-\\frac{n}{2} \\\\\n\t0 & -\\frac{s}{2} & n-1 & 0 & \\dots & 0 & 0 & 0 & \\dots & 0 \\\\\n\t0 & 0 & -\\frac{s}{2} & n-1 & \\dots & 0 & 0 & 0 & \\dots & 0 \\\\\n\t\\vdots & \\vdots & \\vdots & \\vdots& \\vdots & \\vdots& \\vdots & \t \\vdots& \\vdots& \\vdots\\\\\n\t0 & 0 & \\dots & 0 & \\dots & n-1 & 0 & \t 0 & \\dots& 0\\\\\n\t-2(n-1)s & 0 & \\dots & \\dots & \\dots& -\\frac{s}{2} & n-1 & 4-4n & \\dots & 4-4n\\\\\n\t0 & (-2n-6)s & 16(n-1) & 0 & \\dots& 0 & 0 & 0 & \\dots & 0 \\\\\n\t\\hline\n\t0 & (n-3)(\\tau^{n+1})^2 & 0 & 0 & \\dots & 0 & \\frac{2-n}{8}(\\tau^{n+1})^2 & \\frac{s}{2} & \\dots & (\\tau^{n+1})^2\\\\ \n\t0 & (n-3)(\\tau^{n+2})^2 & \\dots & \\dots & \\dots & 0 & \\frac{2-n}{8}(\\tau^{n+2})^2 & (\\tau^{n+2})^2 & \\dots & (\\tau^{n+2})^2 \\\\\n\t\\vdots & \\vdots & \\vdots & \\vdots& \\vdots & \\vdots& \\vdots & \t\\vdots& \\vdots& \\vdots\\\\\n\t0 & (n-3)(\\tau^{2n+3})^2 & 0 & \\dots & \\dots & 0 & \\frac{2-n}{8}(\\tau^{2n+3})^2 & \t(\\tau^{2n+3})^2 & \\dots & \\frac{s}{2}\n \t\\end{pNiceArray}.\n\\end{equation*}\n}\n\nLet \n\\[\n\\mathcal{E}_2=(I_{2n+4}-\\frac{8(n-3)}{n-2}E_{n,1})\\cdot \\mathcal{E}_1\\cdot (I_{2n+4}+\\frac{8(n-3)}{n-2}E_{n,1}).\n\\]\nThe effect is\n\\begin{eqnarray*}\n&&\\mbox{1st column}=>\\mbox{1st column}+\\frac{8(n-3)}{n-2}\\times \\mbox{$n$-th column},\\\\\n&&\\mbox{$n$-th row}=>\\mbox{$n$-th row}-\\frac{8(n-3)}{n-2}\\times \\mbox{$1$st row}.\n\\end{eqnarray*}\nThen\n{\\footnotesize\n\\begin{equation*}\n\\mathcal{E}_2=\\begin{pNiceArray}{ccccccc|ccc}\n\t0 & n-1 & 0 & 0 & \\dots & 0 & 0 & 1-\\frac{n}{2} & \\dots & 1-\\frac{n}{2} \\\\\n\t0 & -\\frac{s}{2} & n-1 & 0 & \\dots & 0 & 0 & 0 & \\dots & 0 \\\\\n\t0 & 0 & -\\frac{s}{2} & n-1 & \\dots & 0 & 0 & 0 & \\dots & 0 \\\\\n\t\\vdots & \\vdots & \\vdots & \\vdots& \\vdots & \\vdots& \\vdots & \t \\vdots& \\vdots& \\vdots\\\\\n\t0 & 0 & \\dots & 0 & \\dots & n-1 & 0 & \t 0 & \\dots& 0\\\\\n\t-2(n-1)s & \\frac{8(n-3)(n-1)}{n-2} & 0 & \\dots & \\dots& -\\frac{s}{2} & n-1 & 4-4n & \\dots & 4-4n\\\\\n\t0 & \\frac{-2n^2+2n}{n-2} s & \\frac{8(n-1)^2}{n-2} & 0 & \\dots& 0 & 0 & 0 & \\dots & 0 \\\\\n\t\\hline\n\t0 & 0 & 0 & 0 & \\dots & 0 & \\frac{2-n}{8}(\\tau^{n+1})^2 & \\frac{s}{2} & \\dots & (\\tau^{n+1})^2\\\\ \n\t0 & 0 & \\dots & \\dots & \\dots & 0 & \\frac{2-n}{8}(\\tau^{n+2})^2 & (\\tau^{n+2})^2 & \\dots & (\\tau^{n+2})^2 \\\\\n\t\\vdots & \\vdots & \\vdots & \\vdots& \\vdots & \\vdots& \\vdots & \t\\vdots& \\vdots& \\vdots\\\\\n\t0 & 0 & 0 & \\dots & \\dots & 0 & \\frac{2-n}{8}(\\tau^{2n+3})^2 & \t(\\tau^{2n+3})^2 & \\dots & \\frac{s}{2}\n \t\\end{pNiceArray}.\n\\end{equation*}\n}\nFor $i=n+1,\\dots,2n+2$, we make the following transformation \n\\[\nU=> (I_{2n+4}+E_{2n+3,i})\\cdot U\\cdot (I_{2n+4}-E_{2n+3,i}). \n\\]\nThe effect is, for $i=n+1,\\dots,2n+2$,\n\\begin{eqnarray*}\n&&\\mbox{$i$-th column}=>\\mbox{$i$-th column}- \\mbox{$(2n+3)$-th column},\\\\\n&&\\mbox{$(2n+3)$-th row}=> \\mbox{$(2n+3)$-th row}+\\mbox{$i$-th row}.\n\\end{eqnarray*}\nThen let\n\\[\n\\mathcal{E}_4=(I_{2n+4}-\\frac{8}{n-2}E_{n,2n+3})\\cdot \\mathcal{E}_4\\cdot (I_{2n+4}+\\frac{8}{n-2}E_{n,2n+3}).\n\\]\nThe effect is\n\\begin{eqnarray*}\n&&\\mbox{$(2n+3)$-th column}=>\\mbox{$(2n+3)$-th column}+\\frac{8}{n-2}\\times \\mbox{$n$-th column},\\\\\n&&\\mbox{$n$-th row}=>\\mbox{$n$-th row}-\\frac{8}{n-2}\\times \\mbox{$(2n+3)$-th row}.\n\\end{eqnarray*}\nThe matrices $\\mathcal{E}_3$ and $\\mathcal{E}_4$ are presented in the next page.\n\n\\begin{landscape}\n\\begin{eqnarray*}\n\\mathcal{E}_3=\\begin{pNiceArray}{ccccccc|cccc}\n\t0 & n-1 & 0 & 0 & \\dots & 0 & 0 & 0 & \\dots & 0 & 1-\\frac{n}{2} \\\\\n\t0 & -\\frac{s}{2} & n-1 & 0 & \\dots & 0 & 0 & 0 & \\dots & 0 & 0 \\\\\n\t0 & 0 & -\\frac{s}{2} & n-1 & \\dots & 0 & 0 & 0 & \\dots & 0 & 0 \\\\\n\t\\vdots & \\vdots & \\vdots & \\vdots& \\vdots & \\vdots& \\vdots & \\vdots &\t \\vdots& \\vdots& \\vdots\\\\\n\t0 & 0 & \\dots & 0 & \\dots & n-1 & 0 & \t 0 & \\dots& 0 & 0\\\\\n\t-2(n-1)s & \\frac{8(n-3)(n-1)}{n-2} & \\dots & \\dots & \\dots& -\\frac{s}{2} & n-1 & 0 & \\dots & 0 & 4-4n\\\\\n\t0 & \\frac{-2n^2+2n}{n-2} s & \\frac{8(n-1)^2}{n-2} & 0 & \\dots& 0 & 0 & 0 & \\dots & 0 & 0 \\\\\n\t\\hline\n\t0 & 0 & 0 & 0 & \\dots & 0 & \\frac{2-n}{8}(\\tau^{n+1})^2 & \\frac{s}{2}-(\\tau^{n+1})^2 & \\dots & 0 & (\\tau^{n+1})^2\\\\ \n\t0 & 0 & \\dots & \\dots & \\dots & 0 & \\frac{2-n}{8}(\\tau^{n+2})^2 & 0 & \\ddots & 0 & (\\tau^{n+2})^2 \\\\\n\t\\vdots & \\vdots & \\vdots & \\vdots& \\vdots & \\vdots& \\vdots & \t\\vdots& & \\vdots & \\vdots\\\\\n\t0 & \\dots & \\dots & \\dots & \\dots & 0 & \\frac{2-n}{8}(\\tau^{2n+2})^2 & 0 & \\dots & \\frac{s}{2}-(\\tau^{2n+2})^2 & (\\tau^{2n+2})^2 \\\\\n\t0 & 0 & 0 & \\dots & \\dots & 0 & \\frac{2-n}{8}s & (\\tau^{2n+3})^2-(\\tau^{n+1})^2 & \\dots & (\\tau^{2n+3})^2-(\\tau^{2n+2})^2 & \\frac{3s}{2}-(\\tau^{2n+3})^2 \n \t\\end{pNiceArray}.\n\\end{eqnarray*}\n\n\n\\begin{eqnarray*}\n\\mathcal{E}_4=\\begin{pNiceArray}{ccccccc|cccc}\n\t0 & n-1 & 0 & 0 & \\dots & 0 & 0 & 0 & \\dots & 0 & 1-\\frac{n}{2} \\\\\n\t0 & -\\frac{s}{2} & n-1 & 0 & \\dots & 0 & 0 & 0 & \\dots & 0 & 0 \\\\\n\t0 & 0 & -\\frac{s}{2} & n-1 & \\dots & 0 & 0 & 0 & \\dots & 0 & 0 \\\\\n\t\\vdots & \\vdots & \\vdots & \\vdots& \\vdots & \\vdots& \\vdots & \\vdots &\t \\vdots& \\vdots& \\vdots\\\\\n\t0 & 0 & \\dots & 0 & \\dots & n-1 & 0 & \t 0 & \\dots& 0 & 0\\\\\n\t-2(n-1)s & \\frac{8(n-3)(n-1)}{n-2} & \\dots & \\dots & \\dots& -\\frac{s}{2} & n-1 & 0 & \\dots & 0 & -\\frac{4(n-1)(n-4)}{n-2}\\\\\n\t0 & \\frac{-2n^2+2n}{n-2} s & \\frac{8(n-1)^2}{n-2} & 0 & \\dots& 0 & s & -\\frac{8\\big((\\tau^{2n+3})^2-(\\tau^{n+1})^2\\big)}{n-2} & \\dots & -\\frac{8\\big((\\tau^{2n+3})^2-(\\tau^{2n+2})^2\\big)}{n-2} & -\\frac{8}{n-2}\\big(\\frac{s}{2}-(\\tau^{2n+3})^2\\big) \\\\\n\t\\hline\n\t0 & 0 & 0 & 0 & \\dots & 0 & \\frac{2-n}{8}(\\tau^{n+1})^2 & \\frac{s}{2}-(\\tau^{n+1})^2 & \\dots & 0 & 0 \\\\ \n\t0 & 0 & \\dots & \\dots & \\dots & 0 & \\frac{2-n}{8}(\\tau^{n+2})^2 & 0 & \\ddots & 0 & 0 \\\\\n\t\\vdots & \\vdots & \\vdots & \\vdots& \\vdots & \\vdots& \\vdots & \t\\vdots& & \\vdots & \\vdots\\\\\n\t0 & \\dots & \\dots & \\dots & \\dots & 0 & \\frac{2-n}{8}(\\tau^{2n+2})^2 & 0 & \\dots & \\frac{s}{2}-(\\tau^{2n+2})^2 & 0 \\\\\n\t0 & 0 & 0 & \\dots & \\dots & 0 & \\frac{2-n}{8}s & (\\tau^{2n+3})^2-(\\tau^{n+1})^2 & \\dots & (\\tau^{2n+3})^2-(\\tau^{2n+2})^2 & \\frac{s}{2}-(\\tau^{2n+3})^2 \n \t\\end{pNiceArray}.\n\\end{eqnarray*}\n\\end{landscape}\n\nWe block \n\\[\n\\mathcal{E}_4=\\begin{pmatrix}\nA & B \\\\\nC & D \n\\end{pmatrix}\n\\]\nas indicated above. Compute the characteristic polynomial by the formula\n\\begin{eqnarray*}\n&& P_n(z,\\tau^{n+1},\\dots,\\tau^{2n+3})\\\\\n&=& \\det \\begin{pmatrix}\nz I-A & -B \\\\\n-C & zI- D \n\\end{pmatrix}\n=\\det \\begin{pmatrix}\nz I-A - B(zI-D)^{-1}C & 0 \\\\\n-C & zI- D \n\\end{pmatrix}.\n\\end{eqnarray*}\nSince\n\\[\n(zI-D)^{-1}=\n\\begin{pmatrix}\n\\frac{1}{z-\\frac{s}{2}+(\\tau^{n+1})^2} & 0 & \\dots & 0 \\\\\n0 & \\ddots & & 0\\\\\n0 & & \\ddots & 0\\\\\n\\frac{(\\tau^{2n+3})^2-(\\tau^{n+1})^2}{(z-\\frac{s}{2}+(\\tau^{n+1})^2)(z-\\frac{s}{2}+(\\tau^{2n+3})^2)} & \\dots & \n\\frac{(\\tau^{2n+3})^2-(\\tau^{2n+2})^2}{(z-\\frac{s}{2}+(\\tau^{2n+2})^2)(z-\\frac{s}{2}+(\\tau^{2n+3})^2)} & \\frac{1}{z-\\frac{s}{2}+(\\tau^{2n+3})^2}\n\\end{pmatrix}\n\\]\nwe get\n\\begin{gather*}\nB(zI-D)^{-1}C=\\\\\n\\begin{pNiceArray}{c|c} \n\\Block{6-1}<\\large>{0_{{\\scriptscriptstyle (n+1)\\times n}}} & \n{\\scriptstyle\n\\frac{(n-2)^2}{16}\\big(\\sum_{i=n+1}^{2n+2}\\frac{\\big((\\tau^{2n+3})^2-(\\tau^i)^2\\big)(\\tau^i)^2}{\\big(z-\\frac{s}{2}+(\\tau^{i})^2\\big)\\big(z-\\frac{s}{2}+(\\tau^{2n+3})^2\\big)}+\\frac{s}{z-\\frac{s}{2}+(\\tau^{2n+3})^2}\\big) }\\\\\n \\hspace*{1cm} & {\\scriptstyle\n0} \\\\ \n\\hspace*{1cm} & \\vdots \\\\\n \\hspace*{1cm} & {\\scriptstyle\n0} \\\\ \n \\hspace*{1cm} & \\vspace{1cm} {\\scriptstyle\n \\frac{(n-2)(n-4)}{2}\\big(\\sum_{i=n+1}^{2n+2}\\frac{\\big((\\tau^{2n+3})^2-(\\tau^i)^2\\big)(\\tau^i)^2}{\\big(z-\\frac{s}{2}+(\\tau^{i})^2\\big)\\big(z-\\frac{s}{2}+(\\tau^{2n+3})^2\\big)}+\\frac{s}{z-\\frac{s}{2}+(\\tau^{2n+3})^2}\\big) } \\\\ \n\\hspace*{1cm} & {\\scriptstyle\n\\sum_{i=n+1}^{2n+2}\\frac{\\big((\\tau^{2n+3})^2-(\\tau^i)^2\\big)(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^{i})^2}+\\big(\\frac{s}{2}-(\\tau^{2n+3})^2\\big) \\big(\\sum_{i=n+1}^{2n+2}\\frac{\\big((\\tau^{2n+3})^2-(\\tau^i)^2\\big)(\\tau^i)^2}{(z-\\frac{s}{2}+(\\tau^{i})^2)(z-\\frac{s}{2}+(\\tau^{2n+3})^2)}+\\frac{s}{z-\\frac{s}{2}+(\\tau^{2n+3})^2}\\big) }\n\\end{pNiceArray},\n\\end{gather*}\nwhere $0_{(n+1)\\times n}$ stands for a 0-matrix of size $(n+1)\\times n$.\nUsing \n\\[\n\\sum_{i=n+1}^{2n+2}\\frac{\\big((\\tau^{2n+3})^2-(\\tau^i)^2\\big)(\\tau^i)^2}{\\big(z-\\frac{s}{2}+(\\tau^{i})^2\\big)\\big(z-\\frac{s}{2}+(\\tau^{2n+3})^2\\big)}+\\frac{s}{z-\\frac{s}{2}+(\\tau^{2n+3})^2}\n=\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2},\n\\]\nand\n\\begin{eqnarray*}\n&&\\sum_{i=n+1}^{2n+2}\\frac{\\big((\\tau^{2n+3})^2-(\\tau^i)^2\\big)(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^{i})^2}\\\\\n&&+\\big(\\frac{s}{2}-(\\tau^{2n+3})^2\\big) \\big(\\sum_{i=n+1}^{2n+2}\\frac{\\big((\\tau^{2n+3})^2-(\\tau^i)^2\\big)(\\tau^i)^2}{(z-\\frac{s}{2}+(\\tau^{i})^2)(z-\\frac{s}{2}+(\\tau^{2n+3})^2)}+\\frac{s}{z-\\frac{s}{2}+(\\tau^{2n+3})^2}\\big)\\\\\n&=& z\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2} -s,\n\\end{eqnarray*}\nwe get\n\\begin{eqnarray*}\nB(zI-D)^{-1}C=\\begin{pNiceArray}{c|c} \n\\Block{6-1}<\\large>{0_{{\\scriptscriptstyle\n(n+1)\\times n}}} & \n\\frac{(n-2)^2}{16}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2} \\\\\n \\hspace*{1cm} & 0 \\\\ \n\\hspace*{1cm} & \\vdots \\\\\n \\hspace*{1cm} & 0 \\\\ \n \\hspace*{1cm} & \\frac{(n-2)(n-4)}{2}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2} \\\\ \n\\hspace*{1cm} & z\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2} -s \n\\end{pNiceArray},\n\\end{eqnarray*}\nand thus\n\\begin{gather*}\nzI-A-B(zI-D)^{-1}C=\\\\\n\\begin{pNiceArray}{ccccccc}\n\tz & 1-n & 0 & 0 & \\dots & 0 & -\\frac{(n-2)^2}{16}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2} \\\\\n\t0 & z+\\frac{s}{2} & 1-n & 0 & \\dots & 0 & 0 \\\\\n\t0 & 0 & z+\\frac{s}{2} & 1-n & \\dots & 0 & 0 \\\\\n\t\\vdots & \\vdots & \\vdots & \\vdots& \\vdots & \\vdots& \\vdots \\\\\n\t0 & 0 & \\dots & 0 & \\dots & 1-n & 0 \\\\\n\t2(n-1)s & -\\frac{8(n-3)(n-1)}{n-2} & \\dots & \\dots & \\dots& z+\\frac{s}{2} & 1-n-\\frac{(n-2)(n-4)}{2}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2} \\\\\n\t0 & \\frac{2n^2-2n}{n-2} s & -\\frac{8(n-1)^2}{n-2} & 0 & \\dots& 0 & z\\big(1-\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\\big) \n \t\\end{pNiceArray}.\n\\end{gather*}\nWe compute the determinant of $zI-A-B(zI-D)^{-1}C$ by expanding the determinant according to the last two columns and then the first column.\nSo the characteristic polynomial is\n\\begin{eqnarray*}\n&&\\det \\big(zI-A-B(zI-D)^{-1}C\\big)\\\\\n&=&\\Big(\\frac{(n-2)^2}{16}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\n\\cdot(2n-2)s\\\\\n&& +z\\big(1-n-\\frac{(n-1)(n-4)}{2}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\n\\big)\\Big)\\cdot (1-n)\\\\\n&&\\cdot \\big((-1)^{n-3}\\frac{2n^2-2n}{n-2}s\\cdot (1-n)^{n-3}\n+(-1)^{n-2}(-\\frac{8(n-1)^2}{n-2})\\cdot (1-n)^{n-4}(z+\\frac{s}{2})\\big)\\\\\n&&+z(1-\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2})\\big(\n(-1)^{n-1}(2n-2)s\\cdot (1-n)^{n-1}\\\\\n&&+(-1)^n(-\\frac{8(n-3)(n-1)}{n-2})\\cdot (1-n)^{n-2}z+z(z+\\frac{s}{2})^{n-1}\n\\big).\n\\end{eqnarray*}\nWe simplify its expression as follows, where the change in each step is indicated in \\textcolor{blue}{blue}. Recall that $n$ is even.\n\\begin{eqnarray*}\n&&\\det \\big(zI-A-B(zI-D)^{-1}C\\big)\\\\\n&=&\\Big(-\\frac{(n-2)^2 \\textcolor{blue}{s}}{\\textcolor{blue}{8}}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2} \n+z\\big(1+\\frac{n-4}{2}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\n\\big)\\Big)\\cdot \\textcolor{blue}{(1-n)^2}\\\\\n&&\\cdot \\big(\\frac{2n(n-1)^{\\textcolor{blue}{n-2}}}{n-2}s\n-\\frac{8(n-1)^{\\textcolor{blue}{n-2}}}{n-2})(z+\\frac{s}{2})\\big)\\\\\n&&+z(1-\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2})\\big(\n2(n-1)^{\\textcolor{blue}{n}} s\n-\\frac{8(n-3)(n-1)^{\\textcolor{blue}{n-1}}}{n-2}z+z(z+\\frac{s}{2})^{n-1}\n\\big)\\\\\n&=&(n-1)^2\\Big(-\\frac{(n-2)^2 s}{8}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\n+z\\big(1+\\frac{n-4}{2}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\n\\big)\\Big)\\\\\n&&\\cdot \\big(\\textcolor{blue}{2(n-1)^{n-2}s}-\\frac{8(n-1)^{n-2}}{n-2}z\\big)\\\\\n&&+z(1-\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2})\\big(\n2(n-1)^n s\n-\\frac{8(n-3)(n-1)^{n-1}}{n-2}z+z(z+\\frac{s}{2})^{n-1}\n\\big)\\\\\n&=&(n-1)^{\\textcolor{blue}{n}}\\Big(-\\frac{(n-2)^2 s}{8}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\n+z\\big(1+\\frac{n-4}{2}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\n\\big)\\Big)\\\\\n&&\\cdot \\big(2s-\\frac{8}{n-2}z\\big)\\\\\n&&+z(1-\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2})\\big(\n2(n-1)^n s\n-\\frac{\\textcolor{blue}{8(n-1)}(n-1)^{n-1}}{n-2}z\n\\big)\\\\\n&& +z(1-\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2})\\big(\n\\frac{\\textcolor{blue}{16}(n-1)^{n-1}}{n-2}z+z(z+\\frac{s}{2})^{n-1}\n\\big)\\\\\n&=&(n-1)^n\\Big(-\\frac{(n-2)^2 s}{8}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\n+z\\big(1+\\frac{n-4}{2}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\n\\big)\\Big)\\\\\n&&\\cdot \\big(2s-\\frac{8}{n-2}z\\big)\\\\\n&&+\\textcolor{blue}{(n-1)^n} z(1-\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2})\\big(\n2 s -\\frac{8}{n-2}z\\big)\\\\\n&& +z^2(1-\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2})\\big(\n\\frac{16(n-1)^{n-1}}{n-2}+(z+\\frac{s}{2})^{n-1}\n\\big)\\\\\n&=&(n-1)^n\\big(2s-\\frac{8}{n-2}z\\big)\\Big(-\\frac{(n-2)^2 s}{8}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\n+\\textcolor{blue}{2z+\\frac{n-6}{2}z}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\\Big)\\\\\n&& +z^2(1-\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2})\\big(\n\\frac{16(n-1)^{n-1}}{n-2}+(z+\\frac{s}{2})^{n-1}\n\\big)\\\\\n&=&(n-1)^n\\big(2s-\\frac{8}{n-2}z\\big)\\textcolor{blue}{\\Big(}-\\frac{(n-2)^2 s}{8}\n+\\frac{n-6}{2}z\\textcolor{blue}{\\Big)}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\\\\\n&&+(n-1)^n\\big(2s-\\frac{8}{n-2}z\\big)2z\\\\\n&& +z^2(1-\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2})\\big(\n\\frac{16(n-1)^{n-1}}{n-2}+(z+\\frac{s}{2})^{n-1}\n\\big)\\\\\n&=&(n-1)^n\\big(s-\\frac{\\textcolor{blue}{4}}{n-2}z\\big)\\Big(-\\frac{(n-2)^2 s}{\\textcolor{blue}{4}}\n+(n-6)z\\Big)\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\\\\\n&&+\\textcolor{blue}{4}(n-1)^n\\big(s-\\frac{4}{n-2}z\\big)z+\\textcolor{blue}{z^2\\big(\n\\frac{16(n-1)^{n-1}}{n-2}+(z+\\frac{s}{2})^{n-1}\\big)}\n\\\\\n&& -z^2\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2})\\big(\n\\frac{16(n-1)^{n-1}}{n-2}+(z+\\frac{s}{2})^{n-1}\n\\big)\\\\\n&=&(n-1)^n\\textcolor{blue}{\\big(-\\frac{(n-2)^2 s^2}{4}+(2n-8)sz-\n\\frac{4(n-6)}{n-2}z^2\\big)}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\\\\\n&&+4(n-1)^n s z\\textcolor{blue}{-16(n-1)^{n-1}z^2}+z^2(z+\\frac{s}{2})^{n-1}\n\\\\\n&& -z^2\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2})\\big(\n\\frac{16(n-1)^{n-1}}{n-2}+(z+\\frac{s}{2})^{n-1}\n\\big)\\\\\n&=&(n-1)^{\\textcolor{blue}{n-1}}\\big(-\\frac{(n-1)(n-2)^2 s^2}{4}+2(n-1)(n-4)sz\\\\\n&&\\textcolor{blue}{-\\frac{16}{n-2}z^2}-\\frac{4(n-1)(n-6)}{n-2}z^2\\big)\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\\\\\n&&+4(n-1)^n s z-16(n-1)^{n-1}z^2+z^2(z+\\frac{s}{2})^{n-1}\n-z^2\\textcolor{blue}{(z+\\frac{s}{2})^{n-1}}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\\\\\n\\begin{comment}&=&(n-1)^{n-1}\\big(-\\frac{(n-1)(n-2)^2 s^2}{4}+2(n-1)(n-4)sz\n-\\frac{\\textcolor{blue}{4(n^2-7n+10)}}{n-2}z^2\\big)\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\\\\\n&&+4(n-1)^n s z-16(n-1)^{n-1}z^2+z^2(z+\\frac{s}{2})^{n-1}\n -z^2(z+\\frac{s}{2})^{n-1}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\\\\\n\\end{comment}\n&=&(n-1)^{n-1}\\big(-\\frac{(n-1)(n-2)^2 s^2}{4}+2(n-1)(n-4)sz\n-4\\textcolor{blue}{(n-5)}z^2\\big)\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\\\\\n&&+4(n-1)^n s z-16(n-1)^{n-1}z^2+z^2(z+\\frac{s}{2})^{n-1}\n -z^2(z+\\frac{s}{2})^{n-1}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}.\n\\end{eqnarray*}\nHence we obtain (\\ref{eq-charPoly-EV-2ndOrder}).\n\\end{proof}\n\n\n\\begin{comment}\nWhen $s=0$,\n\\begin{eqnarray*}\n&&P_n(z,\\tau^{n+1},\\dots,\\tau^{2n+3})\\\\\n&=&z^2\\bigg(\n-4(n-5)(n-1)^{n-1}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z+(\\tau^i)^2}\n-16(n-1)^{n-1}+z^{n-1}\\\\\n&&-z^{n-1}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z+(\\tau^i)^2})\\bigg)\n\\cdot \\prod_{i=n+1}^{2n+3}\\big(z+(\\tau^i)^2\\big).\n\\end{eqnarray*}\nIn particular, when $\\tau^i=\\zeta_{n+3}^{i-n}$, \n\\begin{eqnarray*}\nP_n(z,\\tau^{n+1},\\dots,\\tau^{2n+3})\n&=&z^2\\big(z^{2n+2}-16(n-1)^{n-1}z^{n+3}-(n+2)z^{n-1}\\\\\n&&-4(n-1)^{n-1}(n^2-2n-11)\n\\big).\n\\end{eqnarray*}\n\\end{comment}\n\n\\subsection{Proof of semisimplicity}\n\n\\begin{proposition}\\label{prop-charPoly-EV-2ndOrder-simpleRoots}\nFor general $(\\tau^{n+1},\\dots,\\tau^{2n+3})\\in \\mathbb{C}^{n+3}$, the polynomial in $z$\n\\begin{eqnarray}\\label{eq-charPoly-EV-2ndOrder-1}\n&&\\mathcal{P}_n(z,\\tau^{n+1},\\dots,\\tau^{2n+3})=\nP_n(z,\\tau^{n+1},\\dots,\\tau^{2n+3})+o(\\tau^2)\n\\nonumber\\\\\n&=&\\bigg((n-1)^{n-1}\\big(-\\frac{(n-1)(n-2)^2 s^2}{4}+2(n-1)(n-4)sz\n-4(n-5)z^2\\big)\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2}\\nonumber\\\\\n&&+4(n-1)^n s z-16(n-1)^{n-1}z^2+z^2(z+\\frac{s}{2})^{n-1}\n -z^2(z+\\frac{s}{2})^{n-1}\\sum_{i=n+1}^{2n+3}\\frac{(\\tau^i)^2}{z-\\frac{s}{2}+(\\tau^i)^2})\\bigg)\\nonumber\\\\\n&&\\cdot \\prod_{i=n+1}^{2n+3}\\big(z-\\frac{s}{2}+(\\tau^i)^2\\big)+o(\\tau^2)\n\\end{eqnarray}\nhas only simple roots.\n\\end{proposition}\n\\begin{proof}\nWe have\n\\begin{eqnarray}\nP_n(z,0,\\dots,0)=z^{n+5}(-16(n-1)^{n-1}+z^{n-1}).\n\\end{eqnarray}\nIt has $n-1$ nonzero simple roots and a root of multiplicity $n+5$ at 0. We are going to find a (germ of) line $L$ on the $\\tau$-space starting at 0, parametrized by $\\theta$, and find the branches of solutions on this ray. Let the line $L$ be, say $\\tau^i=\\alpha_i \\theta$, where $\\alpha_i\\in \\mathbb{C}$, for $n+1\\leq i\\leq 2n+3$ are temporarily not specified. By the theory of Puiseux expansion, near the origin of $L$, there are solutions in the form \n\\begin{equation}\n\tz=\\sum_{i=1}^{\\infty}a_{\\frac{i}{M}}\\theta^{\\frac{i}{M}}\n\\end{equation}\nwhere $M$ is a natural number. The index $\\frac{i}{M}$ of the least possible nonzero coefficient $a_{\\frac{i}{M}}$ can be read out from the expression (\\ref{eq-charPoly-EV-2ndOrder}). It is also given by the \\emph{Newton polygon}; indeed, regarding the restriction of $P_n$ on $L$ as a polynomial $P_n(\\theta,z)$ the pair of indices that gives the first slope of the Newton polygon is $(0,n+5)$ and $(2,n+4)$. So the slope is $2$, and we get the expansion \n\\[\nz=a_2 \\theta^2+o(\\theta^{2}).\n\\]\nTo show that $\\mathcal{P}_n$ has $n+5$ distinct branches of solutions of $z$ at 0, we need only to show that the equation \n\\begin{eqnarray}\n\t&&\\big(-\\frac{(n-1)(n-2)^2 s}{4}+2(n-1)(n-4)sz-4(n-5)z^2\\big)\\sum_{i=0}^{n+2}\\frac{\\tau^2_i}{z-\\frac{s}{2}+\\tau^2_i}\\nonumber\\\\\n&&+4(n-1) s z -16z^2=0\n\\end{eqnarray}\nfor $a_2$ has $n+5$ simple roots, for generic choices of $\\tau^{n+1},\\dots,\\tau^{2n+3}$. This follows from the following Proposition \\ref{prop-secondOrderCoeff-distinctRoots}.\n\\end{proof}\n \n\n\n\n\\begin{lemma}\\label{lem-simpleRoots}\nWhen $n\\geq 3$ and $a\\in \\mathbb{Q}$, the equation $z^n-az+1=0$ has only simple roots.\n\\end{lemma}\n\\begin{proof}\n$z^n-az+1$ and $nz^{n-1}-a$ have common factors only if $a=\\frac{n}{n-1}\\cdot (n-1)^{\\frac{1}{n}}$, which is not rational when $n\\geq 3$.\n\\end{proof}\n\n\\begin{proposition}\\label{prop-secondOrderCoeff-distinctRoots}\nLet $n\\geq 4$ be an even natural number.\nLet $\\varsigma=\\varsigma(y_0,\\dots,y_{n+2})=y_0+\\dots+y_{n+2}$. Then for general $(y_0,\\dots,y_{n+2})\\in \\mathbb{C}^{n+3}$, the equation \n\\begin{eqnarray}\\label{eq-secondOrderCoeff-distinctRoots}\n&&\t\\Big(\\big(-\\frac{(n-1)(n-2)^2 \\varsigma^2}{4}+2(n-1)(n-4)\\varsigma z\n-4(n-5)z^2\\big)\\sum_{i=0}^{n+2}\\frac{y_i}{z-\\frac{\\varsigma}{2}+y_i}\\nonumber\\\\\n&&+4(n-1) \\varsigma z-16z^2\\Big)\\cdot \\prod_{i=0}^{n+2}\\big(z-\\frac{\\varsigma}{2}+y_i\\big)=0\n\\end{eqnarray}\nfor $z$ has $n+5$ simple roots.\n\\end{proposition}\n\\begin{proof}\nLet $\\zeta_{n+3}=e^{2\\pi \\sqrt{-1}}$ be a $(n+3)$-th root of unity, and set \n\\begin{equation}\n\ty_i=\\zeta_{n+3}^i+y.\n\\end{equation}\nIt suffices to show that for general $y\\in \\mathbb{C}$, (\\ref{eq-secondOrderCoeff-distinctRoots}) has only simple roots.\nSince\n\\begin{equation*}\n\t\\prod_{i=0}^{n+2}(z+\\zeta_{n+3}^i)=z^{n+3}+1,\n\\end{equation*}\nand \n\\begin{equation*}\n\t\\sum_{i=0}^{n+2}\\frac{1}{z+\\zeta_{n+3}^i}=\\frac{n+3}{z^{n+3}+1},\n\\end{equation*}\nwe have\n\\begin{equation*}\n\t\\prod_{i=0}^{n+2}(z-\\frac{s}{2}+y_i)=(z-\\frac{s}{2}+y)^{n+3}+1,\n\\end{equation*}\nand\n\\begin{equation*}\n\t\\sum_{i=0}^{n+2}\\frac{y_i}{z-\\frac{s}{2}+y_i}\n\t=n+3-(z-\\frac{s}{2})\\sum_{i=0}^{n+2}\\frac{1}{z-\\frac{s}{2}+y_i}\n\t=(n+3)(1-\\frac{z-\\frac{s}{2}}{(z-\\frac{s}{2}+y)^{n+3}+1}).\n\\end{equation*}\nSo (\\ref{eq-secondOrderCoeff-distinctRoots}) reads\n\\begin{eqnarray}\\label{eq-prop-secondOrderCoeff-distinctRoots-1}\n\t&&\\big(-\\frac{(n-1)(n-2)^2(n+3)^2y^2}{4}+2(n-1)(n-4)(n+3)yz\n-4(n-5)z^2\\big)\\nonumber\\\\\n&&\\cdot(n+3)((z-\\frac{n+1}{2}y)^{n+3}+1-z+\\frac{n+3}{2}y) \\nonumber\\\\\n&&+\\big(4(n-1)(n+3)y z-16z^2\\big)\\big((z-\\frac{n+1}{2}y)^{n+3}+1\\big)\n=0.\n\\end{eqnarray}\nWhen $y=0$, this equation is\n\\begin{equation}\n\t4z^2\\big((- n^2+2 n+11) z^{n+3}+( n^2-2 n-15)z+(- n^2 +2 n +11) \\big)=0,\n\\end{equation}\ni.e.\n\\begin{equation}\n\t4(- n^2+2 n+11)z^2\\big( z^{n+3}-\\frac{n^2-2 n-15}{n^2-2 n-11}z+1 \\big)=0.\n\\end{equation}\nBy Lemma \\ref{lem-simpleRoots}, the second factor has only simple roots.\nWe study the branches $z=z(y)$ with $z(0)=0$. They can be written in the form of Puiseux series\n\\[\nz=a_{\\frac{1}{2}}y^{\\frac{1}{2}}+a_1 y+O(y^{\\frac{3}{2}}).\n\\]\nExpanding (\\ref{eq-prop-secondOrderCoeff-distinctRoots-1}), the equation for $a_{\\frac{1}{2}}$ is\n\\[\n0=-4(n-5)(n+3)a_{\\frac{1}{2}}^2-16a_{\\frac{1}{2}}^2=0\n\\]\nso $a_{\\frac{1}{2}}=0$. One can also argue by using Newton polygon.\nThen the equation for $a_1$ is\n\\begin{eqnarray*}\n\t&&(n+3)\\big(-\\frac{(n-1)(n-2)^2(n+3)^2}{4}+2(n-1)(n-4)(n+3)a_1\n-4(n-5)a_1^2\\big)\\\\\n&&+\\big(4(n-1)(n+3)a_1-16a_1^2\\big)=0.\n\\end{eqnarray*}\nThis is a quadratic equation with discriminant equal to\n\\[\n64 (-1 + n) (2 + n) (3 + n)^2.\n\\]\nSo we have two distinct branches with $z(0)=0$.\n\\end{proof}\n\n\\begin{proof}[Proof of Theorem \\ref{thm-semisimplicity}]\nBy Proposition \\ref{prop-charPoly-EV-2ndOrder} and \\ref{prop-charPoly-EV-2ndOrder-simpleRoots}, the Euler field $E$ has pairwise distinct eigenvalues on a general point in an open neighborhood of 0. This means that $\\mathcal{M}_X$ is semisimple in the sense of \\cite[Definition 3.1]{Dub99}. Then by \\cite[Theorem 3.1]{Dub99}, $\\mathcal{M}_X$ is (generically) semisimple in the usual sense.\n\\end{proof}\n\n\n\\section{Enumerative geometry on even dimensional intersections of two quadrics}\\label{sec:EnumerativeGeometry-Even(2,2)}\nIn this section we study the special correlator \n\\begin{equation}\\label{eq-specialCorrelator}\n\t\\langle \\epsilon_{1},\\dots,\\epsilon_{n+3}\\rangle^X_{0,n+3,\\frac{n}{2}}\n\\end{equation}\nby relating its value to enumerative geometry of $X$. Then we prove the $n=4$ case of Conjecture \\ref{conj-unknownCorrelator-Even(2,2)}. \nWe begin by recalling some facts about the even dimensional intersections of two quadrics from \\cite{Rei72}. Let $\\lambda_0,\\dots,\\lambda_{n+2}\\in\\mathbb{C}$ be pairwise distinct. Let \n\\begin{equation}\\label{eq-definingEquationOfQi}\n\t\\varphi_1(Y_0,\\dots,Y_{n+2})=\\sum_{i=0}^{n+2}Y_i^2,\\\n\t\\varphi_2(Y_0,\\dots,Y_{n+2})=\\sum_{i=0}^{n+2} \\lambda_i Y_i^2,\n\\end{equation}\nand $X=\\{\\varphi_1=\\varphi_2=0\\}\\subset \\mathbb{P}^{n+2}$. By \\cite[Prop. 2.1]{Rei72}, every smooth complete intersections of two quadrics can be obtained in this way by choosing appropriate coordinates on $\\mathbb{P}^{n+2}$.\n\\begin{lemma}\nFor $0\\leq k\\leq \\frac{n}{2}$, let $P_k$ be the point in $\\mathbb{P}^{n+2}$ whose $i$-th homogeneous coordinate is\n\\begin{equation}\n\\frac{\\lambda_i^k}{\\sqrt{\\prod_{\\begin{subarray}{c}0\\leq j\\leq n+2\\\\ j\\neq i\\end{subarray}}\n(\\lambda_i- \\lambda_j)}}. \n\\end{equation}\nThen\n\\begin{equation}\\label{eq-vanish-varphiPk}\n\t\\varphi_1(P_k)=\\varphi_2(P_k)=0,\n\\end{equation}\nand $P_0,\\dots,P_{\\frac{n}{2}}$ span an $\\frac{n}{2}$-plane contained in $X$. Here we choose a root $\\sqrt{\\lambda_i- \\lambda_j}$ uniformly in the expressions of $P_0,\\dots,P_k$, for every pair ${i,j}$, for $0\\leq i\\neq j\\leq n+2$.\n\\end{lemma}\n\\begin{proof}\nDirectly check (\\ref{eq-vanish-varphiPk}) using \n\\[\n\\frac{\\lambda^l}{\\prod_{\\begin{subarray}{c}0\\leq j\\leq n+2\\\\ j\\neq i\\end{subarray}}(\\lambda_i- \\lambda_j)}=0\n\\]\nfor $0\\leq l\\leq n+1$.\n\\end{proof}\n\nWe denote this $\\frac{n}{2}$-plane by $S$. Then $S$ is defined by the following linear equations:\n\\begin{equation}\n\\sum_{i=0}^{n+2}\\frac{\\lambda_i^k}{\\sqrt{\\prod_{\\begin{subarray}{c}0\\leq j\\leq n+2\\\\ j\\neq i\\end{subarray}}\n(\\lambda_i- \\lambda_j)}}Y_i=0,\\ \\mbox{for}\\ 0\\leq k\\leq \\frac{n}{2}+1.\n\\end{equation}\nMake a change of coordinates\n\\begin{equation}\\label{eq-Wcoordinates}\n\tW_i=\\frac{Y_i}{\\sqrt{\\prod_{\\begin{subarray}{c}0\\leq j\\leq n+2\\\\ j\\neq i\\end{subarray}}\n(\\lambda_i- \\lambda_j)}}.\n\\end{equation}\nThen $S$ is defined by\n\\begin{equation}\\label{eq-definingEquations-WCoordinates}\n\t\\sum_{i=0}^{n+2}\\lambda_i^k W_i=0,\\ \\mbox{for}\\ 0\\leq k\\leq \\frac{n}{2}+1.\n\\end{equation}\n\nRecall from \\cite[Lemma 3.10]{Rei72}:\n\\begin{lemma}\\label{lem-intersectionNumber-middelDimension}\n\\begin{enumerate}\n\t\\item[(i)] Let $T$ be a $\\frac{n}{2}$-plane contained in $X$. Then $[T]\\cdot \\mathsf{h}_{n\/2}=1$.\n\t\\item[(ii)] Let $T_1$ and $T_2$ be $\\frac{n}{2}$-planes contained in $X$.\n\tSuppose $\\dim(T_1\\cap T_2)=r$, then \n\t\\begin{equation}\n\t\t\t[T_1]\\cdot [T_2]=(-1)^{r}(\\lfloor\\frac{r}{2}\\rfloor+1).\n\t\\end{equation}\n\\end{enumerate}\n\\end{lemma}\n\n\\begin{notation}\nFor integers $a\\leq b$ let $[a,b]$ be the set of integers $c$ satisfying $a\\leq c\\leq b$. \nFor a subset $I\\subset [0,n+2]$, let $S_I$ be the $\\frac{n}{2}$-plane obtained by reversing the sign of the $i$-th homogeneous coordinate of the points on $S$ for all $i\\in I$. Denote the complement of $I$ by $C(I)$. Then $S_I=S_{C(I)}$. \n\\end{notation}\nBy \\cite[Theorem 3.8]{Rei72}, every $\\frac{n}{2}$-planes of $X$ is equal to $S_I$ for some $I\\subset [0,n+2]$.\nIn particular, let $S_i=S_{\\{i\\}}$. Let $\\varsigma_i=[S_i]$, the homology class of $S_i$. Let $\\varsigma=[S]$. By \\cite[Lemma 3.13]{Rei72},\n\\[\n\\mathsf{h}_{\\frac{n}{2}},\\varsigma_0,\\varsigma_1,\\dots,\\varsigma_{n+2}\n\\]\nis a basis of $H^{\\frac{n}{2}}(X;\\mathbb{Q})$, and \n\\begin{equation}\n\t\\varsigma=\\frac{\\frac{n}{2}+1}{n+1}\\mathsf{h}_{n\/2}-\\frac{1}{n+1}\\sum_{i=0}^{n+2}\\varsigma_i.\n\\end{equation}\nFor $I\\subset [0,n+2]$, let $a(I)=(x_0,\\dots,x_{n+2})\\in \\mathbb{Z}^{n+3}$ such that \n\\[\n x_i=\\begin{cases}\n 1,& \\mbox{if}\\ i\\not\\in I,\\\\\n -1,& \\mbox{if}\\ i\\in I.\n \\end{cases}\n \\] \n For $I$ and $J \\subset [0,n+2]$, suppose $a(I)=(x_0,\\dots,x_{n+2})$ and $a(J)=(y_0,\\dots,y_{n+2})$. Define\n \\[\n a(I,J):=(x_0 y_0,\\dots,x_{n+2}y_{n+2}).\n \\]\n \\begin{lemma}\\label{lem-intersectionDimension}\nLet $m(I,J)$ be the number of $-1$ in the $(n+3)$-tuple $a(I,J)$. Then\n\\begin{equation}\n\t\\dim S_I\\cap S_J=\\begin{cases}\n\t\\frac{n}{2}-m(I,J),& \\mbox{if}\\ m(I,J)\\leq \\frac{n}{2},\\\\\n\t-1,& \\mbox{if}\\ m(I,J)=\\frac{n}{2}+1\\ \\mbox{or}\\ \\frac{n}{2}+2,\\\\\n\tm(I,J)-\\frac{n}{2}-3,& \\mbox{if}\\ m(I,J)\\geq \\frac{n}{2}+3,\n\t\\end{cases}\t\n\\end{equation}\nwhere $\\dim S_I\\cap S_J=-1$ means that the intersection is empty.\n \\end{lemma}\n\\begin{proof}\nOne can show this using directly the defining equations (\\ref{eq-definingEquations-WCoordinates}) of $S$, or from \\cite[Theorem 3.8]{Rei72}.\n\\end{proof} \nBy Lemma \\ref{lem-intersectionNumber-middelDimension} and Lemma \\ref{lem-intersectionDimension}, \n\\begin{equation}\\label{eq-intersectionNumber-Si-Sj}\n\t\\varsigma_i\\cdot \\varsigma_j=\\begin{cases}\n\t(-1)^{\\frac{n}{2}-2}(\\lfloor \\frac{\\frac{n}{2}-2}{2}\\rfloor+1), & \\mbox{if}\\ i\\neq j,\\\\\n\t(-1)^{\\frac{n}{2}}(\\lfloor \\frac{n}{4}\\rfloor+1),& \\mbox{if}\\ i=j.\n\t\\end{cases}\n\\end{equation}\n\n\n\n\\subsection{An explicit orthonormal basis}\\label{sec:explictD-Lattice}\nFor $1\\leq i\\leq n+3$, we define\n \\begin{equation}\n\t\\varepsilon_i=\\varsigma_{i-1}-\\frac{1}{n+1}\\sum_{i=0}^{n+2}\\varsigma_{i}+\\frac{1}{2(n+1)}\\mathsf{h}_{n\/2}.\n\\end{equation}\n\\begin{lemma}\\label{lem-expressingEpsilonbyMiddleDimPlanes}\n\\begin{equation}\n\t\\varsigma_{i-1}=\\varepsilon_i-\\frac{1}{2}\\sum_{i=1}^{n+3}\\varepsilon_i+\\frac{\\mathsf{h}_{n\/2}}{4},\\ \\mbox{for}\\ 1\\leq i\\leq n+3,\n\\end{equation}\n\\begin{equation}\n\t\\varsigma=\\frac{1}{4}\\mathsf{h}_{n\/2}+\\frac{1}{2}\\sum_{i=1}^{n+3}\\varepsilon_i.\n\\end{equation}\n\\end{lemma}\n\\begin{lemma}\\label{lem-intersectionNumber-epsilon}\nFor $1\\leq i\\leq n+3$,\n\\begin{equation}\\label{eq-intersectionNumber-epsilon-h}\n\\varepsilon_i\\cdot \\mathsf{h}_{n\/2}=0,\n\\end{equation}\n\\begin{equation}\\label{eq-intersectionNumber-epsiloni-self}\n\t\\varepsilon_i \\cdot \\varepsilon_i=(-1)^{\\frac{n}{2}},\n\\end{equation}\nfor $i\\neq j$,\n\\begin{equation}\\label{eq-intersectionNumber-epsiloni-epsilonj}\n\t\\varepsilon_i \\cdot \\varepsilon_j=0.\n\\end{equation}\n\\end{lemma}\n\\begin{proof}\n(\\ref{eq-intersectionNumber-epsilon-h}) follows from Lemma \\ref{lem-intersectionNumber-middelDimension} (i).\nFor (\\ref{eq-intersectionNumber-epsiloni-self}) and (\\ref{eq-intersectionNumber-epsiloni-epsilonj}) we use Lemma \\ref{lem-intersectionNumber-middelDimension} (i) and (\\ref{eq-intersectionNumber-Si-Sj}): for $1\\leq i\\neq j\\leq n+3$,\n\\begin{eqnarray*}\n&& \\varepsilon_i \\cdot \\varepsilon_j\\\\\n&=& \\big(1-\\frac{2(n+2)}{n+1}+\\frac{(n+3)(n+2)}{(n+1)^2}\\big)\\cdot (-1)^{\\frac{n}{2}-2}(\\lfloor \\frac{\\frac{n}{2}-2}{2}\\rfloor+1)\\\\\n&&+ \\big(-\\frac{2}{n+1}+\\frac{n+3}{(n+1)^2}\\big)\\cdot (-1)^{\\frac{n}{2}}(\\lfloor \\frac{n}{4}\\rfloor+1)\\\\\n&&+ 2(1-\\frac{n+3}{n+1})\\frac{1}{2(n+1)}+\\frac{1}{4(n+1)^2}\\cdot 4\\\\\n&=& \\begin{cases}\n\\frac{n+3}{(n+1)^2}\\cdot \\frac{n}{4}-\\frac{n-1}{(n+1)^2}\\cdot (\\frac{n}{4}+1)-\\frac{1}{(n+1)^2},& \\mbox{if}\\ n\\equiv 0 \\mod 4,\\\\\n-\\frac{n+3}{(n+1)^2}\\cdot \\frac{n-2}{4}+\\frac{n-1}{(n+1)^2}\\cdot (\\frac{n-2}{4}+1)-\\frac{1}{(n+1)^2},& \\mbox{if}\\ n\\equiv 2 \\mod 4,\\\\\n\\end{cases}\\\\\n&=& 0,\n\\end{eqnarray*}\nand\n\\begin{eqnarray*}\n&& \\varepsilon_i \\cdot \\varepsilon_i\\\\\n&=& \\big(-\\frac{2(n+2)}{n+1}+\\frac{(n+3)(n+2)}{(n+1)^2}\\big)\\cdot (-1)^{\\frac{n}{2}-2}(\\lfloor \\frac{\\frac{n}{2}-2}{2}\\rfloor+1)\\\\\n&&+ \\big(1-\\frac{2}{n+1}+\\frac{n+3}{(n+1)^2}\\big)\\cdot (-1)^{\\frac{n}{2}}(\\lfloor \\frac{n}{4}\\rfloor+1)\\\\\n&&+ 2(1-\\frac{n+3}{n+1})\\frac{1}{2(n+1)}+\\frac{1}{4(n+1)^2}\\cdot 4\\\\\n&=& \\begin{cases}\n\\frac{(-n+1)(n+2)}{(n+1)^2}\\cdot \\frac{n}{4}+\\frac{n^2+n+2}{(n+1)^2}\\cdot (\\frac{n}{4}+1)-\\frac{1}{(n+1)^2},& \\mbox{if}\\ n\\equiv 0 \\mod 4,\\\\\n-\\frac{(-n+1)(n+2)}{(n+1)^2}\\cdot \\frac{n-2}{4}-\\frac{n^2+n+2}{(n+1)^2}\\cdot (\\frac{n-2}{4}+1)-\\frac{1}{(n+1)^2},& \\mbox{if}\\ n\\equiv 2 \\mod 4,\\\\\n\\end{cases}\\\\\n&=& (-1)^{\\frac{n}{2}}.\n\\end{eqnarray*}\n\\end{proof}\nWe define \n\\begin{equation}\\label{eq-roots-D-usingExplicitEpsilonClass}\n\t\\begin{cases}\n\t\\alpha_i=\\varepsilon_{i}-\\varepsilon_{i+1}\\ \\mbox{for}\\ 1\\leq i\\leq n+2,\\\\\n\t\\alpha_{n+3}=\\varepsilon_{n+2}+\\varepsilon_{n+3}.\n\t\\end{cases}\n\\end{equation} \nThen from Lemma \\ref{lem-expressingEpsilonbyMiddleDimPlanes} we have\n\\begin{equation}\\label{eq-roots-D-usingExplicitMiddleDimPlanes}\n\t\\begin{cases}\n\t\\alpha_i=\\varsigma_{i-1}-\\varsigma_{i}\\ \\mbox{for}\\ 1\\leq i\\leq n+2,\\\\\n\t\\alpha_{n+3}=\\varsigma_{n+1}+\\varsigma_{n+2}+ 2\\varsigma-\\mathsf{h}_{n\/2}.\n\t\\end{cases}\n\\end{equation} \nSo $\\alpha_i\\in H^*(X;\\mathbb{Z})$. Using Lemma (\\ref{lem-intersectionNumber-epsilon}), and comparing (\\ref{eq-roots-D-usingExplicitEpsilonClass}) and (\\ref{eq-roots-D}), one sees that the group generated by the reflections with respecto $\\alpha_i$'s is the Weyl group $D_{n+3}$. By the Picard-Lefschetz formula, one can take the class $\\alpha_i$ in Section \\ref{sec:monodromy-lattice} to be $\\alpha_i$ defined here. This justifies the notations. Moreover, because of (\\ref{eq-intersectionNumber-epsiloni-self}), we define an orthonormal basis $\\epsilon_i$ of $H^*_{\\mathrm{prim}}(X)$ exactly as (\\ref{eq-normalizedOrthonormalBasis}).\n\n\n\\begin{comment}\nLet $g_i$ be the map that reverses the sign of $Y_i$. Then $g_i$ is an automorphism of $X$, and \n\\[\ng_i(S_{j})=\\begin{cases}\nS_{i,j},& \\mbox{if}\\ i\\neq j,\\\\\nS,& \\mbox{if}\\ i=j.\n\\end{cases}\n\\]\nSo for $i\\neq j-1$,\n\\begin{eqnarray*}\n&& g_{i*}(\\varepsilon_j)=\\varsigma_{j-1,i}-\\frac{1}{n+1}\\sum_{k\\neq i}\\varsigma_{k,i}-\\frac{1}{n+1}\\varsigma+\\frac{1}{2(n+1)}\\mathsf{h}_{n\/2}\\\\\n&=&(\\mathsf{h}_{n\/2}-\\varsigma-\\varsigma_{j-1}-\\varsigma_{i})-\\frac{1}{n+1}\\sum_{k\\neq i}(\\mathsf{h}_{n\/2}-\\varsigma-\\varsigma_{k}-\\varsigma_{i})-\\frac{1}{n+1}\\varsigma+\\frac{1}{2(n+1)}\\mathsf{h}_{n\/2}\\\\\n&=&-\\varsigma_{j-1}+\\frac{1}{n+1}\\varsigma_{i}+\\frac{1}{n+1}\\sum_{k\\neq i}\\varsigma_k-\\frac{1}{2(n+1)}\\mathsf{h}_{n\/2}\\\\\n&=& -\\varepsilon_j,\n\\end{eqnarray*}\nand \n\\begin{eqnarray*}\n&& g_{i*}(\\varepsilon_{i+1})=\\varsigma-\\frac{1}{n+1}\\sum_{k\\neq i}\\varsigma_{k,i}-\\frac{1}{n+1}\\varsigma+\\frac{1}{2(n+1)}\\mathsf{h}_{n\/2}\\\\\n&=&\\varsigma-\\frac{1}{n+1}\\sum_{k\\neq i}(\\mathsf{h}_{n\/2}-\\varsigma-\\varsigma_{k}-\\varsigma_{i})-\\frac{1}{n+1}\\varsigma+\\frac{1}{2(n+1)}\\mathsf{h}_{n\/2}\\\\\n&=&2\\varsigma+\\frac{n+2}{n+1}\\varsigma_{i}+\\frac{1}{n+1}\\sum_{k\\neq i}\\varsigma_k-\\frac{2n+3}{2(n+1)}\\mathsf{h}_{n\/2}\\\\\n&=& \\varsigma_{i}-\\frac{1}{n+1}\\sum_{k\\neq i}\\varsigma_k+\\frac{1}{2(n+1)}\\mathsf{h}_{n\/2}=\\varepsilon_{i+1}.\n\\end{eqnarray*}\n\n\\begin{eqnarray*}\n&& (g_j\\circ g_i)_* (\\varsigma_k)=\\varsigma_{i,j,k}\\\\\n&=& -(\\mathsf{h}_{n\/2}-2\\varsigma-\\varsigma_i-\\varsigma_j-\\varsigma_k)\\\\\n&=& -\\mathsf{h}_{n\/2}+(\\frac{n+2}{n+1}\\mathsf{h}_{n\/2}-\\frac{2}{n+1}\\sum_{l=0}^{n+2}\\varsigma_l)+\\varsigma_i+\\varsigma_j+\\varsigma_k\\\\\n&=& \\varsigma_i+\\varsigma_j+\\varsigma_k-\\frac{2}{n+1}\\sum_{l=0}^{n+2}\\varsigma_l+\\frac{1}{n+1}\\mathsf{h}_{n\/2}\n\\end{eqnarray*}\n\\end{comment}\n\n\n\n\n\\subsection{A potentially enumerative correlator}\\label{sec:enumerativeCorrelator}\nWe begin with giving a working definition of \\emph{enumerative correlators}.\n\\begin{definition}\\label{def-enumerativeCorrelator}\nLet $n\\geq 4$ be an even natural number, and $X$ be an $n$-dimensional smooth complete intersection of two quadrics in $\\mathbb{P}^{n+2}$. Denote the $i$-th projection from $X^{n+3}$ to $X$ by $q_i$. Consider the product of the evaluation morphisms\n\\begin{equation}\n\t\\mathrm{ev}_1\\times\\cdots \\mathrm{ev}_{n+3}: \\overline{\\mathcal{M}}_{0,n+3}(X,\\frac{n}{2})\\rightarrow X^{n+3}.\n\\end{equation}\nLet $I_1,\\dots, I_{n+3}\\subset [0,n+2]$. We say that the correlator \n\\begin{equation}\\label{eq-enumerativeCorrelator-0}\n\t\\langle \\varsigma_{I_1},\\dots,\\varsigma_{I_{n+3}}\\rangle\n\\end{equation}\nis \\emph{enumerative} if there exists an irreducible component $M$ of $\\overline{\\mathcal{M}}_{0,n+3}(X,\\frac{n}{2})$ satisfying the following:\n\\begin{enumerate}\n\t\\item[(i)] $\\dim M$ equals the expected dimension.\n\t\\item[(ii)] The cycles $(\\mathrm{ev}_1\\times\\cdots \\mathrm{ev}_{n+3})(M)$ and $q_1^{-1} S_{I_1}$,\\dots, $q_{n+3}^{-1}S_{I_{n+3}}$ intersect \\emph{properly}, i.e. the dimension of their (scheme theoretic) intersection is 0.\n\t\\item[(iii)] Each irreducible component of $\\overline{\\mathcal{M}}_{0,n+3}(X,\\frac{n}{2})$ other than $M$ has empty intersection with $q_1^{-1} S_{I_1}$,\\dots, $q_{n+3}^{-1} S_{I_n+3}$. \n\\end{enumerate}\n\\end{definition}\n\nThis definition is of course not standard in any sense. It just facilitates the following presentation. \n\nOur strategy to compute the special correlator by the enumerative geometry of $X$ consists of three steps:\n\\begin{enumerate}\n \t\\item Select $I_1,\\dots, I_{n+3}\\subset [0,n+2]$, such that the correlator (\\ref{eq-enumerativeCorrelator-0}) is enumerative.\n\t\\item Express (\\ref{eq-enumerativeCorrelator-0}) in terms of the special correlator, using Lemma \\ref{lem-expressingEpsilonbyMiddleDimPlanes}.\n\t\\item Solve the corresponding enumerative problem by counting curves. More precisely, compute the intersection multiplicities of the intersection \n\t\\begin{equation}\n\t\t(\\mathrm{ev}_1\\times\\cdots \\mathrm{ev}_{n+3})_*[M]\\cap q_1^{*} [S_{I_1}]\\cap\\dots\\cap q_{n+3}^{*} [S_{I_{n+3}}]\n\t\\end{equation}\n\t in the condition (ii) above. \n \\end{enumerate} \n\n \\begin{example}\\label{exp-nonEnumerative-correlator}\n The correlator \n \\begin{equation}\n \t\\langle \\varsigma_{0},\\dots,\\varsigma_{n+3}\\rangle\n \\end{equation}\n should not be enumerative in general. For example, let $n=4$. Then the intersection $S\\cap S_i$ is a line, for $0\\leq i\\leq 6$. The moduli space of conics on $S$ passing through the seven lines has a positive dimension. So there are infinitely many conics passing through $S_0,\\dots,S_6$. Then the conditions (ii) and (iii) in Definition \\ref{def-enumerativeCorrelator} cannot be true simultaneously.\n \\end{example}\n\nStimulated by Example \\ref{exp-nonEnumerative-correlator}, we propose a possibly enumerative correlator.\nLet $S_{[i,i+k-1]}$ be the $\\frac{n}{2}$-plane $S_{i,i+1,\\dots,i+k-1}$ in $X$, where we understand the indices $i$ in the subscript in mod $n+3$ sense. For example, when $n=4$, $S_{[6,6+1]}=S_{6,0}$, $S_{[5,5+3]}=S_{5,6,0,1}$.\n\\begin{lemma}\\label{lem-uniqueS}\n$S$ is the only $\\frac{n}{2}$-plane in $X$ that has non-empty intersections with each of $S_{[i,i+\\frac{n}{2}-1]}$, for $0\\leq i\\leq n+2$. Moreover $S$ meets $S_{[i,i+\\frac{n}{2}-1]}$ at exactly one point.\n\\end{lemma}\n\\begin{proof}\nThe second statement follows from Lemma \\ref{lem-intersectionDimension}.\n\nWe show the first statement.\nLet $I\\subset [0,n+2]$, and suppose $a(I)=(x_0,\\dots,x_{n+2})$. Let $J_i=[i,i+\\frac{n}{2}-1]$ in the mod $n+3$ sense. By Lemma \\ref{lem-intersectionDimension}, the first statement is equivalent to that when $a(I)\\neq (1,1,\\dots,1)$ or $(-1,-1,\\dots,-1)$, there exists $i$ such that $a(I,J_i)=\\frac{n}{2}+1$ or $\\frac{n}{2}+2$, or equivalently the sum of all components of $a(I,J_i)$ is equal to $\\pm 1$. Put\n\\[\np(x_0,\\dots,x_{n+2})=\\sum_{i=0}^{n+2}\\big(a(I,J_i)^2-1\\big).\n\\]\nWe regard $p(x_0,\\dots,x_{n+2})$ as a polynomial of indeterminates $x_0,\\dots,x_{n+2}$.\nThen we are left to show that when $(x_0,\\dots,x_{n+2})\\in \\{1,-1\\}^{n+3}$ and $(x_0,\\dots,x_{n+2})\\neq (1,1,\\dots,1)$ or $(-1,-1,\\dots,-1)$, \n\\begin{equation}\n\tp(x_0,\\dots,x_{n+2})=0.\n\\end{equation}\nPut \n\\[\ny_i=\\sum_{j\\in J_i}x_j.\n\\]\nThen\n\\begin{equation}\n\ta(I,J_i)=\\sum_{i=0}^{n+2}x_i-2y_i.\n\\end{equation}\nSo $p(x_0,\\dots,x_{n+2})$ is manifestly symmetric in $y_i$. It follows that $p(x_0,\\dots,x_{n+2})$ is symmetric in $x_i$, rather than only cyclicly symmetric. So it suffices to show the statement for all $I\\subset[0,n+2]$ of the form $I=[0,k]$ for some $0\\leq k0$ and $\\sum_{k=0}^{2n+3}i_k>3$ then return 0. (By (\\ref{eq-FCA}))\n\t\\item If $i_1>0$ and $\\sum_{k=0}^{2n+3}i_k>3$ then apply (\\ref{eq-recursion-EulerVecField-even(2,2)}).\n\t\\item If $\\sum_{k=2}^{n}i_k>0$ and $\\sum_{k=0}^{2n+3}i_k\\geq 2$ then apply \n\t(\\ref{eq-recursion-ambient-simplified-even(2,2)}).\n\t\\item If $\\sum_{k=n+1}^{2n+3}i_k=1$, return 0. (By Theorem \\ref{thm-monodromy-evenDim(2,2)})\n\n\t\\item If $\\sum_{k=n+1}^{2n+3}i_k=3$, return 0. (By Theorem \\ref{thm-monodromy-evenDim(2,2)})\n\t\\item If $\\sum_{k=0}^{n}i_k=0$ and $\\sum_{k=n+1}^{2n+3}i_k=4$, then if the nonzero components of $(i_{n+1},\\dots,i_{2n+3})$ is a 4, or two 2's, then return 1, otherwise return 0. (By Theorem \\ref{thm-monodromy-evenDim(2,2)} and Theorem \\ref{thm-reconstruction-even(2,2)})\n\t\\item If $\\sum_{k=0}^{n}i_k=0$ and $i_k=1$ for all $n+1\\leq k\\leq 2n+3$, return an indeterminate \\texttt{x}, which stands for the unknown special correlator (\\ref{eq-specialLength(n+3)Invariant-even(2,2)}).\n\t\\item If $\\sum_{k=0}^{n}i_k=0$, $\\sum_{k=n+1}^{2n+3}i_k>4$ and there is only one nonzero component $i_a$ in $(i_{n+1},\\dots,i_{2n+3})$, we rearrange $I$ such that $I=(0,\\dots,0,i_{2n+3})$, and take $b=n+1$ so that $i_b=0$. Then apply (\\ref{eq-recursion-primitive-aabb-tau-simplified-even(2,2)}) to $a=2n+3$, $b=n+1$, and $I=I-2e_a$, and thus return the RHS of (\\ref{eq-recursion-primitive-aabb-tau-simplified-even(2,2)}) divided by a nonzero number.\n\t\\item We define a function \\texttt{indexTriple}. The input is $(i_{n+1},\\dots,i_{2n+3})$. Then we check that neither of the conditions\n\t\\begin{enumerate}\n\t \t\\item[(i)] there is only one nonzero component in $(i_{n+1},\\dots,i_{2n+3})$\n\t \t\\item[(ii)] $i_k=1$ for all $n+1\\leq k\\leq 2n+3$\n\t \\end{enumerate}\n\tis true. When this is the case, the output is the indices $a,b,c$ as given in Remark \\ref{rem:recursion-boundOfIndex}. This produce in a definite way the indices $a,b,c$ in the proof of Theorem \\ref{thm-reconstruction-even(2,2)}.\n\t\\item If $\\sum_{k=0}^{n}i_k=0$, $\\sum_{k=n+1}^{2n+3}i_k>4$, and the neither of the conditions (i) and (ii) in the last step is true, we apply \\texttt{indexTriple} to $(i_{n+1},\\dots,i_{2n+3})$ to get $(a,b,c)$. Then we apply (\\ref{eq-recursion-primitive-abcc-even(2,2)}) to $(a,b,c)$ and $I=I-e_a-e_b$, and thus return the RHS of (\\ref{eq-recursion-primitive-abcc-even(2,2)}) divided by a nonzero number. (As we have shown in the proof of Theorem \\ref{thm-reconstruction-even(2,2)} that the coefficient on the LHS of (\\ref{eq-recursion-primitive-abcc-even(2,2)}) is nonzero in this case.)\n\\end{enumerate}\n\t \t\n\n\n\\end{algorithm}\nWe implemented Algorithm \\ref{algorithm-correlator-even(2,2)} by the command \n\\[\n\\mbox{\\texttt{correlatorEvenIntersecTwoQuadricsRecursionInTauCoord}}\n\\]\n in the Macaulay2 package \\texttt{QuantumCohomologyFanoCompleteIntersection}. The input of this command is a list $\\{n,I\\}$, and the output is $\\partial_{\\tau^I}F(0)$. For example, running\n\\[\n\\mbox{\\texttt{correlatorEvenIntersecTwoQuadricsRecursionInTauCoord} $\\{4,\\{0,0,7,0,0,0,0,0,0,0,0,0\\}\\}$}\n\\]\nreturns 46656, so $\\langle\\mathsf{h}_2,\\mathsf{h}_2,\\mathsf{h}_2,\\mathsf{h}_2,\\mathsf{h}_2,\\mathsf{h}_2,\\mathsf{h}_2\\rangle=46656$. Running\n\\[\n\\mbox{\\texttt{correlatorEvenIntersecTwoQuadricsRecursionInTauCoord} $\\{4,\\{0,0,5,0,0,2,0,0,0,0,0,0\\}\\}$}\n\\]\nreturns $-624$, so $\\langle \\mathsf{h}_2,\\mathsf{h}_2,\\mathsf{h}_2,\\mathsf{h}_2,\\mathsf{h}_2,\\epsilon_1,\\epsilon_1\\rangle=-624$. Similarly one can get the correlators in Lemma \\ref{lem-correlatorsOfLength7-4dim}. The special correlator (\\ref{eq-specialLength(n+3)Invariant-even(2,2)}) is denoted by \\texttt{x} in this package. For example running\n\\begin{multline*}\n\\texttt{correlatorEvenIntersecTwoQuadricsRecursionInTauCoord}\\\\ \\{6,\\{0,0,0,0,0,0,0,0,2,2,2,2,2,2,2,0,0\\}\\}\n\\end{multline*}\nreturns $8\\mbox{\\texttt{x}}^2-2$. One can check Conjecture \\ref{conj-unknownCorrelator-Even(2,2)-quadraticEquation} in this way.\n\n\n\\end{appendices}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction\nInflation theory has claimed that the origin of the large-scale structure of the universe and temperature fluctuations in the cosmic microwave background radiations is quantum fluctuations. Remarkably, the inflation theory also predicts the existence of primordial gravitational waves stemming from the\nquantum fluctuations of the spacetime (relic gravitons). After the discovery\nof gravitational waves from a black hole binary system~\\cite{LIGOScientific:2016aoc},\nthe detection of primordial gravitational waves has been\nthe most important research objective~\\cite{Kawamura:2011zz,Amaro-Seoane:2012aqc}.\n\nThe notable nature of primordial gravitational waves is their quantum origin. If the relic gravitons were found, it would strongly support the inflationary universe. The finding of the relic gravitons would also give a hint of quantum gravity. Hence, it is extremely important to explore\nthe quantum nature of the primordial gravitational waves.\n\nIt is well known that the generation of relic gravitons\ncan be interpreted as the squeezing process of a quantum state during inflation~\\cite{Grishchuk:1989ss,Grishchuk:1990bj,Albrecht:1992kf,Polarski:1995jg}. Since the degree of squeezing is extremely high, the quantum state is highly entangled between two modes with opposite wave number vectors due to conservation of momentum. The squeezed state of the relic gravitons is a key for \nproving the nonclassicality of primordial gravitational waves. \nIn fact, the squeezed gravitons can significantly enhance the quantum noise in interferometers~\\cite{Parikh:2020nrd,Kanno:2020usf,Parikh:2020kfh,Parikh:2020fhy,Kanno:2021gpt}. \nHence, we need to show the degree of the squeezing generated during inflation survives under the decoherence processes in the evolution of the universe. So far, the decoherence process due to short wavelength modes of a field has been investigated~\\cite{Lombardo:2005iz,Martineau:2006ki,Burgess:2006jn,Nelson:2016kjm}. However, it is argued \nthat the decoherence obtained by tracing out the short wavelength modes is false decoherence~\\cite{2011arXiv1110.2199U}. \nThus, it is worth studying different decoherence processes.\n\n \nIn this paper, as a source of the decoherence, we assume the presence of a sizable magnetic field at the beginning of inflation. We then consider conversion process of the squeezed gravitons into photons during inflation in the case of minimal coupling between gravitons and photons~\\cite{Gertsenshtein:1962,Raffelt:1987im,Chen:1994ch,Cillis:1996qy}. \nThe squeezed state of gravitons may turn into the squeezed state of photons due to the graviton-photon conversion. \nHence, it is important to clarify to what extent the squeezing of the relic gravitons survives at present. \nThe purpose of this paper is to compute the \ndegree of squeezing parameters of graviton and photon and cross squeezing parameter between gravitons and photons during inflation.\n\nThe paper is organized as follows: In section 2, we derive basic equations for analyzing the conversion process of gravitons into photons during inflation. \nIn section 3, we explain the perturbative formalism for solving a coupled system between gravitons and photons in order to obtain\nthe time evolution of mode functions.\nIn section 4, we derive Bogoliubov transformations\ndue to the squeezing process in the presence of primordial magnetic fields. \nIn section 5, we deduce formulae for the squeezing parameters and \n reveal the time evolution of the squeezing parameters numerically and analytically. \nWe also discuss the implications of our results.\nThe final section is devoted to the conclusion.\n\n\n\\section{Graviton-photon conversion during inflation\n\n\n\nWe represent the graviton in a spatially flat expanding background by the tensor mode perturbation in the three-dimensional metric, \n\\begin{eqnarray}\nds^2=a^2(\\eta)\\left[-d\\eta^2+\\left(\\delta_{ij}+h_{ij}\\right)dx^idx^j\\right]\\,,\n\\end{eqnarray}\nwhere $\\eta$ is the conformal time and the metric perturbation $h_{ij}$ satisfies the transverse traceless conditions $h_{ij}{}^{,j}=h^i{}_i=0$. The spatial indices $i,j,k,\\cdots$ are raised and lowered by $\\delta^{ij}$ and $\\delta_{k\\ell}$.\n\nThe Einstein-Hilbert action and the action for the electromagnetic field is given by\n\\begin{eqnarray}\nS=S_g+S_A=\\frac{M_{\\rm pl}^2}{2}\\int d^4x \\sqrt{-g}\n\\,\nR-\\frac{1}{4}\\int d^4x \\sqrt{-g}\\,\nF^{\\mu\\nu}\nF_{\\mu\\nu}\n\\label{original action}\\,,\n\\end{eqnarray}\nwhere $M_{\\rm pl}=1\/\\sqrt{8\\pi G}$ is the Planck mass. The gauge field $A_\\mu$ represents the photon and the field strength is defined by $F_{\\mu\\nu}=\\partial_\\mu A_{\\nu}-\\partial_\\nu A_{\\mu}$.\nExpanding the Einstein-Hilbert action up to the second order in perturbations $h_{ij}$, we find\n\\begin{eqnarray}\n\\delta S_g=\\frac{M_{\\rm pl}^2}{8}\\int d^4x\\,a^2\\left[\nh^{ij\\prime}\\,h_{ij}^\\prime-h^{ij,k}h_{ij,k}\n\\right]\\,.\n\\label{action:g}\n\\end{eqnarray}\nHere, a prime denotes the derivative with respect to the conformal time. The action for the photon up to second order in perturbations $A_i$ reads\n\\begin{eqnarray}\n\\delta S_A=\\frac{1}{2}\\int d^4x\\left[A_i^{\\prime\\, 2}-A_{k,i}^2\\right]\\,,\n\\label{action:A}\n\\end{eqnarray}\nwhere the photon field satisfies the Coulomb gauge $A_0=0$ and $A^i{}_{,i}=0$.\nThe action for the interaction between the graviton and the photon up to second order in perturbations $h_{ij}, A^i$ is found to be\n\\begin{eqnarray}\n\\delta S_{\\rm I}=\\int d^4x \\left[\n\\varepsilon_{i\\ell m}B_m h^{ij}\\left(\\partial_j A_\\ell\n-\\partial_\\ell A_j\\right)\n\\right]\\,.\n\\label{action:I}\n\\end{eqnarray}\nNote that $B_m=\\varepsilon_{mj\\ell}\\,\\partial_j A_\\ell$ is a constant background magnetic field that we assumed the presence at the beginning of inflation.\n\nAt quadratic order, it is convenient to expand $h_{ij}(\\eta,x^i)$ and $A_i(\\eta,x^i)$ in the Fourier modes,\n\\begin{align}\n&h_{ij}(\\eta,x^i)=\\frac{2}{M_{\\rm pl}} \\sum_{P}\\frac{1}{(2\\pi)^{3\/2}} \\int d^3 k\\,h^{P}_{\\bm k}(\\eta)\\, e_{ij}^{P}(\\bm{k})\\,e^{i\\bm{k}\\cdot\\bm{x}}\\,,\\\\\n&A_i(\\eta,x^i)=\\sum_{P} \\frac{\\pm i}{(2\\pi)^{3\/2}}\n\\int d^3 k\\,A^{P}_{\\bm k}(\\eta)\\,e_i^{P}(\\bm{k})\\, e^{i\\bm{k}\\cdot\\bm{x}},\n\\end{align}\nwhere three-vectors are denoted by bold math type and $e_{ij}^{P}(\\bm{k})$ and $e_i^{P}(\\bm{k})$ are the polarization tensors and vectors for the ${\\bm k}$ mode respectively normalized as $e^{ijP}(\\bm{k})e_{ij}^{Q}(\\bm{k})=\\delta^{PQ}$ and $e^{iP}(\\bm{k}) e_i^{Q}(\\bm{k})=\\delta^{PQ}$ with $P,Q=+,\\times$.\nUsing the canonical variable $y^P_{\\bm k}(\\eta)=a(\\eta)h_{\\bm k}^{P}(\\eta)$,\nwe can rewrite the quadratic actions~(\\ref{action:g}), (\\ref{action:A}) and (\\ref{action:I}) as\n\\begin{eqnarray}\n\\delta S_g&=&\\frac{1}{2}\\sum_P\\int d^3 k\\,d\\eta\\left[\\,\n|y_{\\bm k}^{P\\,\\prime}|^2\n-k^2|y_{\\bm k}^P|^2\n-\\frac{a^\\prime}{a}y_{\\bm k}^{P}y_{-\\bm k}^{P\\,\\prime}\n-\\frac{a^\\prime}{a}y_{-\\bm k}^{P}y_{\\bm k}^{P\\,\\prime}\n+\\left(\\frac{a^\\prime}{a}\\right)^2|y_{\\bm k}^P|^2\n\\right]\\,,\n\\label{action:g2}\\\\\\\n\\delta S_A&=&\\frac{1}{2}\\sum_P\\int d^3 k\\,d\\eta\\left[\\,\n|A_{\\bm k}^{P\\,\\prime}|^2-k^2|A_{\\bm k}^P|^2\n\\,\\right]\\,,\n\\label{action:A2}\\\\\n\\delta S_I&=&\\frac{2}{M_{\\rm pl}}\\sum_{P,Q}\\int d^3 k\\,d\\eta\\,\\frac{1}{a}\\left[\n\\varepsilon_{i\\ell m}\\,B_m\\,y_{\\bm k}^PA_{-\\bm k}^Q\n\\,e_{ij}^P(\\bm k)\\Bigl\\{ik_\\ell\\,e_{j}^Q(-\\bm k)-ik_j\\,e_{\\ell }^Q(-\\bm k)\\Bigr\\}\\right]\n\\label{action:I2}\\,,\n\\end{eqnarray}\nwhere $k=|\\bm k|$. Polarization vectors $e^{i+}, e^{i\\times}$ and a vector $k^i\/k$ constitute an orthnormal basis.\nWithout loss of generality, we assume the constant background magnetic field is in the ($k^i, e^{i \\times}$)-plane.\n\\begin{figure}[H]\n\\centering\n \\includegraphics[keepaspectratio, scale=0.6]{Configuration.pdf}\n \\renewcommand{\\baselinestretch}{3}\n \\caption{Configuration of the polarization vector ${\\bm e}^P(\\bm k)$, wave number ${\\bm k}$, and background magnetic field ${\\bm B}$.}\n \\label{Configuration}\n \\end{figure}\n\\noindent\nThe polarization tensors can be written in terms of polarization vectors \n$e^{i+}$ and $e^{i\\times}$\n as\n\\begin{align}\n &e_{ij}^+(\\bm{k})\n =\\frac{1}{\\sqrt{2}} \\Bigl\\{\n e^+_i(\\bm{k}) e^+_j(\\bm{k})-e^\\times_i(\\bm{k}) e^\\times_j(\\bm{k})\n \\Bigr\\}\\,,\\\\\n &e_{ij}^\\times(\\bm{k})\n =\\frac{1}{\\sqrt{2}} \n \\Bigl\\{\n e^+_i(\\bm{k}) e^\\times_j(\\bm{k})+e^\\times_i(\\bm{k}) e^+_j(\\bm{k})\n \\Bigr\\}\\, .\n \n\\end{align}\nIn the following, we assume\n\\begin{eqnarray}\ne_i^\\times(-\\bm{k})=-e_i^\\times(\\bm{k})\\,.\n\\end{eqnarray}\nThe action (\\ref{action:I2}) is then reduced into\n\\begin{eqnarray}\n\\delta S_I&=&\\frac{\\sqrt{2}}{M_{\\rm pl}}\\int d^3k\\,d\\eta\\,\\frac{1}{a}\n\\left[\\,\\lambda(\\bm k)\\,y_{\\bm k}^+(\\eta)\\,A_{-\\bm k}^+(\\eta)+\\lambda(\\bm k)\\,y_{\\bm k}^\\times(\\eta)\\,A_{-\\bm k}^\\times(\\eta)\\,\\right]\\,,\n\\label{action:I3}\n\\end{eqnarray}\nwhere we defined the coupling between graviton and photon as \n\\begin{align}\n\\lambda(\\bm{k})\\equiv\\frac{\\sqrt{2}}{M_{\\rm pl}}\n\\varepsilon^{i\\ell m}\\,e_i^+\\,k_\\ell\\,B_m\\,.\n\\label{coupling}\n\\end{align}\nHere, the conditions for the graviton and photon to be real read,\n$h_{-\\bm k}^{+,\\times}(\\eta)=h_{\\bm k}^{*\\,+,\\times}(\\eta)$ and $A_{-\\bm k}^{+,\\times}(\\eta)=-A_{\\bm k}^{*\\,+,\\times}(\\eta)$\\,.\nBelow, we focus on the plus polarization and omit the index $P$ unless there may be any confusion. \n\nIn the case of de Sitter space, the scale factor is given by $a(\\eta)=-1\/(H\\eta)$ where $-\\infty<\\eta<0$.\nThe variation of the actions (\\ref{action:g2}), (\\ref{action:A2}) and (\\ref{action:I3}) with respect to the graviton and the photon fields gives\n\\begin{align}\n &y_{\\bm{k}}''+\\left(k^2-\\frac{2}{\\eta^2}\\right)y_{\\bm{k}}=\\lambda H \\eta A_{\\bm{k}}\\,,\n \\label{eom:graviton}\n \\\\\n &A_{\\bm{k}}''+k^2A_{\\bm{k}}=\\lambda H \\eta\\,y_{\\bm{k}}\\,.\n \\label{eom:photon}\n\\end{align}\nIf we define the Lagrangian in the actions (\\ref{action:g2}) and (\\ref{action:A2}) by $\\delta S_g=\\int d\\eta\\,L_g$ and $\\delta S_A=\\int d\\eta\\,L_A$, the conjugate momenta of graviton $p_{\\bm k}$ and photon $\\pi_{\\bm k}$ are respectively given by\n\\begin{align}\n &p_{\\bm{k}}(\\eta)=\\frac{\\partial L_g}{\\partial y^\\prime_{-\\bm k}}=y_{\\bm{k}}'(\\eta)+\\frac{1}{\\eta}y_{\\bm{k}}(\\eta) \\ , \n \\label{p}\\\\\n &\\pi_{\\bm{k}}(\\eta)=\\frac{\\partial L_A}{\\partial A^\\prime_{-\\bm k}}=A_{\\bm{k}}'(\\eta) \\ .\n \\label{pi}\n\\end{align}\nNow we promote variables $y_{\\bm k}(\\eta), A_{\\bm k}(\\eta)$ and their momenta $p_{\\bm k}(\\eta), \\pi_{\\bm k}(\\eta)$ into operators. The annihilation operator for the graviton is expressed by canonical variables as\n\\begin{eqnarray}\n\\hat{a}_y(\\eta,{\\bm k})=\\sqrt{\\frac{k}{2}}\\hat{y}_{\\bm k}(\\eta)+\\frac{i}{\\sqrt{2k}}\\hat{p}_{\\bm k}(\\eta)\\,.\n\\label{y:annihi}\n\\end{eqnarray}\nIn the same way, the annihilation operator for photon is given by\n\\begin{eqnarray}\n\\hat{a}_A(\\eta,{\\bm k})=\\sqrt{\\frac{k}{2}}\\hat{A}_{\\bm k}(\\eta)+\\frac{i}{\\sqrt{2k}}\\hat{\\pi}_{\\bm k}(\\eta)\\,.\n\\label{A:annihi}\n\\end{eqnarray}\nThe commutation relations $[\\hat{a}_y(\\eta,{\\bm k}),\\hat{a}^\\dag_y(\\eta,-{\\bm k}^\\prime)]=\\delta({\\bm k}+{\\bm k}^\\prime)$ and $[\\hat{a}_A(\\eta,{\\bm k}),\\hat{a}^\\dag_A(\\eta,-{\\bm k}^\\prime)]=\\delta({\\bm k}+{\\bm k}^\\prime)$ guarantee the canonical commutation relations $[y_{\\bm k}(\\eta),p_{{\\bm k}^\\prime}(\\eta)]=i\\delta({\\bm k}-{\\bm k}^\\prime)$ and $[A_{\\bm k}(\\eta),\\pi_{{\\bm k}^\\prime}(\\eta)]=i\\delta({\\bm k}-{\\bm k}^\\prime)$.\nNotice that the annihilation operator becomes time dependent through the time dependence of canonical variables. Thus, the vacuum defined by $\\hat{a}(\\eta,{\\bm k})|0\\rangle =0$ is time dependent as well and the vacuum in this formalism turns out to be defined at every moment.\n\n\n\nIn this paper, we suppose $B_m\/M_{\\rm pl}\\ll 1$ so that the coupling between graviton and photon~(\\ref{coupling}) is weak . Then we solve the Eqs.~(\\ref{eom:graviton}) and (\\ref{eom:photon}) iteratively up to the second order in $y_{\\bm{k}}$ and $A_{\\bm{k}}$ in the next section.\n\n\\section{Time evolution of mode functions\n\nUsing the basic equations presented in the previous section, we perturbatively derive mode functions in this section.\n\n\n\\subsection{Zeroth order}\n\nBy letting $\\lambda=0$ in Eqs.~(\\ref{eom:graviton}) and (\\ref{eom:photon}), the equations of the zeroth order approximation become\n\\begin{align}\n &\\hat{y}_{\\bm{k}}^{(0)\\prime\\prime}+\\left(k^2-\\frac{2}{\\eta^2}\\right)\\hat{y}_{\\bm{k}}^{(0)}=0\\,,\n \\label{GWeqs}\n \\\\\n &\\hat{A}_{\\bm{k}}^{(0)\\prime\\prime}+k^2\\hat{A}_{\\bm{k}}^{(0)}=0 \\, ,\n \\label{EMeqs}\n\\end{align}\nwhere the superscript $(0)$ denotes the zeroth order. The solutions of the above equations are\n\\begin{align}\n &\\hat{y}_{\\bm k}^{(0)}(\\eta)=u_{\\bm k}^{(0)}(\\eta) ~\\hat{c}\n +u_{\\bm k}^{(0)*}(\\eta)~\\hat{c}^\\dagger\\,, \n \\label{0th:graviton}\\\\\n &\\hat{A}_{\\bm k}^{(0)}(\\eta)=v_{\\bm k}^{(0)}(\\eta) ~\\hat{d}\n +v_{\\bm k}^{(0)*}(\\eta)~\\hat{d}^\\dagger,\n \\label{0th:photon}\n\\end{align}\nwhere $\\hat{c}$\\,($\\hat{d}$) and its conjugate \n$\\hat{c}^\\dag$($\\hat{d}^\\dag$) are constant operators of integration. We choose the properly normalized positive frequency mode in the remote past as a basis, which is expressed as\n\\begin{align}\n &u_{\\bm k}^{(0)}(\\eta)=\\frac{1}{\\sqrt{2k}} \\biggl(1-\\frac{i}{k\\eta}\\biggr) e^{-ik\\eta}\n \\,,\\qquad\n v_{\\bm k}^{(0)}(\\eta)=\\frac{1}{\\sqrt{2k}} e^{-ik\\eta}.\n\\end{align}\n\n\n\\subsection{First order}\n\nInserting the solutions of zeroth order approximation (\\ref{0th:graviton}) and (\\ref{0th:photon}) into the r.h.s of Eqs.~(\\ref{eom:graviton}) and (\\ref{eom:photon}) as the source terms, the equations of the first order approximation are written as\n\\begin{align}\n &\\hat{y}_{\\bm{k}}^{(1)\\prime\\prime}+\\left(k^2-\\frac{2}{\\eta^2}\\right)\\hat{y}_{\\bm{k}}^{(1)}\n =\\lambda H \\eta \\hat{A}_{\\bm{k}}^{(0)}\\,,\n \\label{GWeq1}\n \\\\\n &\\hat{A}_{\\bm{k}}^{(1)\\prime\\prime}+k^2\\hat{A}_{\\bm{k}}^{(1)}=\\lambda H \\eta \\hat{y}_{\\bm{k}}^{(0)} \\,.\n \\label{EMeq1}\n\\end{align}\nThe effect of photon comes in Eq.~(\\ref{GWeq1}). Using the Green function\n\\begin{eqnarray}\nG_{\\rm dS}(\\eta,\\eta')=\n\\frac{1}{2ik} \\biggl(1+\\frac{i}{k\\eta'}\\biggr)\n\\biggl(1-\\frac{i}{k\\eta}\\biggr)\ne^{-ik(\\eta-\\eta')}\n-\\frac{1}{2ik} \n\\biggl(1-\\frac{i}{k\\eta'}\\biggr)\n\\biggl(1+\\frac{i}{k\\eta}\\biggr)\ne^{ik(\\eta-\\eta')}\\,,\n\\end{eqnarray}\nwe obtain the solution as\n\\begin{align}\n \\hat{y}^{(1)}_{\\bm k}(\\eta)\n &=-\\int_{\\eta_i}^\\eta d\\eta'\n G_{\\rm dS} (\\eta,\\eta')\n \\lambda H \\eta'\\hat{A}^{(0)}_{\\bm k}(\\eta')\\nonumber\\\\\n &=-\\int_{\\eta_i}^\\eta d\\eta'\n G_{\\rm dS} (\\eta,\\eta')\n \\lambda H \\eta' \n v^{(0)}_{\\bm k}(\\eta')~\\hat{d}\n -\\int_{\\eta_i}^\\eta d\\eta'\n G_{\\rm dS} (\\eta,\\eta')\n \\lambda H\\eta'\n v^{(0)*}_{\\bm k}(\\eta') \n ~\\hat{d}^\\dagger \\nonumber\\\\\n &\\equiv u^{(1)}_{\\bm k}(\\eta)~\\hat{d}+u^{(1)*}_{\\bm k}(\\eta)~\\hat{d}^\\dagger\\,,\n \\label{1st:graviton}\n\\end{align}\nwhere $\\eta_i$ is an initial time. From the first line to the second line we used Eq.~(\\ref{0th:photon}). In the last line, we defined the first order correction due to the source of photon to the positive frequency mode of graviton by\n\\begin{align}\n u^{(1)}_{\\bm k}(\\eta)&\\equiv-\\int_{\\eta_i}^\\eta d\\eta'\n G_{\\rm dS} (\\eta,\\eta')\n \\lambda H \\eta' \n v^{(0)}_{\\bm k}(\\eta') \\ .\n \\label{u1Green}\n\\end{align}\nAfter integration, we have\n\\begin{eqnarray}\nu^{(1)}_{\\bm k}(\\eta)&=&\n\\frac{ \\lambda H }{8 \\sqrt{2} \\eta k^{9\/2}} \n\\left[e^{- i k \\eta } \n\\Bigl\\{2 i \\eta ^3 k^3+\\eta k \n\\Bigl(2 \\eta_i k (2-i \\eta_i k)+3 i\\Bigr)\n-2 \\eta_i k \n(\\eta_i k+2 i)\n+3\\Bigr\\}\n\\right.\\nonumber\\\\\n&&\\left.\n-e^{ i k (\\eta -2\\eta_i ) } \n(\\eta k+i) (2 \\eta_i k-3 i)\n\\right] \\ .\n\\label{u1}\n\\end{eqnarray}\n\nSimilarly, the effect of graviton comes in Eq.~(\\ref{EMeq1}).\nBy using the Green function \n\\begin{eqnarray}\nG_{\\rm M}(\\eta,\\eta')=-\\frac{1}{k} \\sin{k(\\eta-\\eta')}\\,,\n\\end{eqnarray}\nwe have\n\\begin{align}\n \\hat{A}^{(1)}_{\\bm k}(\\eta)\n &=-\\int_{\\eta_i}^\\eta d\\eta'\n G_{\\rm M} (\\eta,\\eta')\n \\lambda H \\eta' \\hat{y}^{(0)}_{\\bm k}(\\eta')\\nonumber\\\\\n &=-\\int_{\\eta_i}^\\eta d\\eta' G_{\\rm M}(\\eta,\\eta')\n \\lambda H \\eta' u_{\\bm k}^{(0)}(\\eta)~\\hat{c}\n -\\int_{\\eta_i}^\\eta d\\eta' G_{\\rm M}(\\eta,\\eta')\n \\lambda H \\eta' u_{\\bm k}^{(0)*}(\\eta)~\\hat{c^\\dagger}\\nonumber\\\\\n &\\equiv v^{(1)}_{\\bm k}(\\eta)~\\hat{c}+v^{(1)*}_{\\bm k}(\\eta)~\\hat{c}^\\dagger \\ ,\n \\label{1st:photon}\n\\end{align}\nwhere we used Eq.~(\\ref{0th:graviton}) from the first line to the second line. We also defined the first order correction due to the source of graviton to the positive frequency mode of photon in the third line by\n\\begin{align}\n v^{(1)}_{\\bm k}(\\eta)&\\equiv-\\int_{\\eta_i}^\\eta d\\eta'\n G_{\\rm M} (\\eta,\\eta')\n \\lambda H \\eta' \n u^{(0)}_{\\bm k}(\\eta') \\ .\n \\label{v1Green}\n\\end{align}\nMore explicitly, the above is written as\n\\begin{eqnarray}\nv^{(1)}_{\\bm k}(\\eta)\n&=&\n\\frac{ \\lambda H }{8 \\sqrt{2} k^{7\/2}}\n\\left[e^{- i k \\eta} \\Bigl\\{2 i k^2 (\\eta^2 -\\eta_i^2) +k (6 \\eta -4 \\eta_i)-3 i\\Bigr\\}\n+e^{ i k(\\eta -2\\eta_i ) } (-2 \\eta_i k+3 i)\\right] \\ .\\nonumber\\\\\n\\label{v1} \n\\end{eqnarray}\n\n\n\n\\subsection{Second order}\n\nBy plugging the solution of the first order approximation (\\ref{1st:graviton}) and (\\ref{1st:photon}) into the r.h.s of Eqs.~(\\ref{eom:graviton}) and (\\ref{eom:photon}) as the source terms, the equations of the second order approximation are\n\\begin{align}\n &y_{\\bm{k}}^{(2)\\prime\\prime}+\\left(k^2-\\frac{2}{\\eta^2}\\right)y_{\\bm{k}}^{(2)}\n =\\lambda H \\eta A_{\\bm{k}}^{(1)}\\,,\n \\label{GWeq2}\n \\\\\n &A_{\\bm{k}}^{(2)\\prime\\prime}+k^2A_{\\bm{k}}^{(2)}=\\lambda H \\eta\\,y_{\\bm{k}}^{(1)} \\,.\n \\label{EMeq2}\n\\end{align}\nAt this order, the effect of graviton itself comes in Eq.~(\\ref{GWeq2}). The solution is written by the Green function $G_{\\rm dS}$ such as\n\\begin{align}\n \\hat{y}^{(2)}_{\\bm k}(\\eta)\n &=-\\int_{\\eta_i}^\\eta d\\eta'\n G_{\\rm dS} (\\eta,\\eta')\n \\lambda H \\eta' \\hat{A}^{(1)}_{\\bm k} (\\eta') \\nonumber\\\\\n &=-\\int_{\\eta_i}^\\eta d\\eta'\n G_{\\rm dS} (\\eta,\\eta')\n \\lambda H \\eta' v^{(1)}_{\\bm k}(\\eta')~\\hat{c}\n -\\int_{\\eta_i}^\\eta d\\eta'\n G_{\\rm dS} (\\eta,\\eta')\n \\lambda H \\eta' v^{(1)*}_{\\bm k}(\\eta')~\\hat{c}^\\dagger\n \\nonumber\\\\\n &=-\\int_{\\eta_i}^{\\eta} d\\eta' G_{\\rm dS} (\\eta,\\eta')\n \\lambda H \\eta' \\Biggl(-\\int_{\\eta_i}^{\\eta'}d\\eta'' G_{\\rm M}(\\eta',\\eta'')\\lambda H \\eta''u^{(0)}_{\\bm k}(\\eta'') \\Biggr)~ \\hat{c} \\nonumber\\\\\n &\\qquad -\\int_{\\eta_i}^{\\eta} d\\eta' G_{\\rm dS} (\\eta,\\eta')\n \\lambda H \\eta' \\Biggl(-\\int_{\\eta_i}^{\\eta'}d\\eta''G_{\\rm M}(\\eta',\\eta'') \\lambda H \\eta''u^{(0)*}_{\\bm k}(\\eta'') \\Biggr)~ \\hat{c}^\\dagger \\nonumber\\\\\n &\\equiv u^{(2)}_{\\bm k}(\\eta)~\\hat{c}+u^{(2)*}_{\\bm k}(\\eta)~\\hat{c}^\\dagger, \n\\end{align}\nwhere we used Eqs.~(\\ref{1st:photon}) and (\\ref{v1Green}) in the second and the third lines respectively. In the last line, we defined\n\\begin{align}\n u^{(2)}_{\\bm k}(\\eta)\n &\\equiv\n -\\int_{\\eta_i}^\\eta d\\eta' G_{\\rm dS} (\\eta,\\eta')\n \\lambda H \\eta'\n v^{(1)}_{\\bm k}(\\eta')\n \\nonumber\\\\\n &=\\int_{\\eta_i}^\\eta d\\eta' G_{\\rm dS} (\\eta,\\eta')\n \\lambda H \\eta'\n \\int_{\\eta_i}^{\\eta'} d\\eta'' G_{\\rm M}(\\eta',\\eta'') \n \\lambda H \\eta'' u^{(0)}_{\\bm k} (\\eta'').\n \\label{u2Green}\n\\end{align}\nBy performing the integration, the explicit form of the $u^{(2)}_{\\bm k}(\\eta)$ is found to be\n\\begin{eqnarray}\nu^{(2)}_{\\bm k}(\\eta)\n&=&\n-\\frac{ \\lambda^2 H^2}{192 \\sqrt{2} \\eta k^{15\/2}}\\nonumber\\\\\n&&\\times\n\\Biggl[\n3 e^{ i k ( \\eta -2\\eta_i)} \n\\biggl\\{2 \\eta^3 k^3 (-3-2 i \\eta_i k)+\\eta k \\biggl(-35+2 \\eta_i k (\\eta_i k (11+2 i \\eta_i k)-23 i)\\bigg)\\nonumber\\\\\n&& \\qquad +2 \\eta_i k (23+\\eta_i k (-2 \\eta_i k+11 i))-35 i\n\\biggr\\}\\nonumber\\\\\n&&\n+e^{- i k \\eta }\n\\Biggl\\{k \n\\Bigl(-105 \\eta +72 \\eta_i+6 \\eta k^4 \\left(\\eta^2-\\eta_i^2\\right)^2-2 i k^3 \\left(17 \\eta^4-12 \\eta^3 \\eta_i-8 \\eta\\eta_i^3+3 \\eta_i^4\\right)\\nonumber\\\\\n&& \\qquad +k^2 \\left(16 \\eta_i^3-52 \\eta^3\\right)+72 i \\eta\\eta_i k\n\\Bigr)\n+105 i\n\\Biggr\\}\n\\Biggr] \\ .\n\\label{u2}\n\\end{eqnarray}\nSimilarly, the effect of photon itself comes in Eq.~(\\ref{EMeq2}) and the solution is given by\n\\begin{align}\n \\hat{A}^{(2)}_{\\bm k}(\\eta)\n &=-\\int_{\\eta_i}^\\eta d\\eta' G_{\\rm M} (\\eta,\\eta')\n \\lambda H \\eta'~\\hat{y}^{(1)}_{\\bm k} (\\eta') \\nonumber\\\\\n &=-\\int_{\\eta_i}^\\eta d\\eta' G_{\\rm M} (\\eta,\\eta')\n \\lambda H \\eta'u^{(1)}_{\\bm k}(\\eta')~\\hat{d} \n -\\int_{\\eta_i}^\\eta d\\eta' G_{\\rm M} (\\eta,\\eta')\n \\lambda H \\eta'~u^{(1)*}_{\\bm k}(\\eta')~\\hat{d}^\\dagger \\nonumber\\\\\n &=-\\int_{\\eta_i}^\\eta d\\eta' G_{\\rm M}(\\eta,\\eta')\n \\lambda H \\eta'~\n \\Biggl( \n -\\int_{\\eta_i}^{\\eta'} d\\eta'' G_{\\rm dS}(\\eta',\\eta'') \n \\lambda H \\eta'' v^{(0)}_{\\bm k} (\\eta'')\n \\Biggr) \\hat{d}\n \\nonumber\\\\\n &\\qquad-\\int_{\\eta_i}^\\eta d\\eta' G_{\\rm M} (\\eta,\\eta')\n \\lambda H \\eta'~\n \\Biggl( \n -\\int_{\\eta_i}^{\\eta'} d\\eta'' G_{\\rm dS}(\\eta',\\eta'') \n \\lambda H \\eta'' v^{(0)*}_{\\bm k} (\\eta'')\n \\Biggr) \\hat{d}^\\dagger\n \\nonumber\\\\\n &=v^{(2)}_{\\bm k}(\\eta)~\\hat{d}+v^{(2)*}_{\\bm k}(\\eta)~\\hat{d}^\\dagger,\n\\end{align}\nwhere we used Eqs.~(\\ref{1st:graviton}) in the second line and (\\ref{u1Green}) in the third line and in the last line. We defined\n\\begin{align}\n v^{(2)}_{\\bm k}(\\eta)\n &\\equiv\n -\\int_{\\eta_i}^\\eta d\\eta' G_{\\rm M} (\\eta,\\eta')\n \\lambda H \\eta'\n u^{(1)}_{\\bm k}(\\eta')\n \\nonumber\\\\&=\\int_{\\eta_i}^\\eta d\\eta' G_{\\rm M}(\\eta,\\eta')\n \\lambda H \\eta'\n \\int_{\\eta_i}^{\\eta'} d\\eta'' G_{\\rm dS}(\\eta',\\eta'') \n \\lambda H \\eta'' v^{(0)}_{\\bm k} (\\eta'')\\,.\n \\label{v2Green}\n\\end{align}\nThe integral of the above reduces to\n\\begin{eqnarray}\nv^{(2)}_{\\bm k}(\\eta)\n&=&\n-\\frac{ \\lambda^2 H^2 }{64 \\sqrt{2} k^{13\/2}}\\nonumber\\\\\n&&\\times\n\\Biggl[\ne^{- i k \\eta } \n\\Biggl(2 k^4 \\left(\\eta^2-\\eta_i^2\\right)^2-4 i \\eta k^3 (\\eta -\\eta_i) (\\eta +3 \\eta_i)\\nonumber\\\\\n&& \\qquad +12 \\eta_i k^2 (\\eta_i-2 \\eta )-12 i k (\\eta -2 \\eta_i)-15\n\\Biggr)\\nonumber\\\\\n&&\\qquad +e^{ i k ( \\eta -2 \\eta_i)} \n\\left(2 k (3+2 i \\eta_i k) \\left(\\eta_i^2 k-\\eta (\\eta k+3 i)\\right)+6 i \\eta_i k+15\n\\right)\n\\Biggr] \\ .\n\\label{v2}\n\\end{eqnarray}\n \n\n\\section{Bogoliubov transformations\n\nBy solving Eqs.(\\ref{eom:graviton}) and (\\ref{eom:photon}) iteratively up to the second order, we can take into account the backreaction of graviton and photon respectively. For the graviton,\nthe field and its conjugate momentum are now given by\n\\begin{eqnarray}\n\\hspace{-1cm}\n\\hat{y}_{\\bm k}(\\eta)&=&\\Bigl(u^{(0)}_{\\bm k}+u^{(2)}_{\\bm k}\\Bigr)\\,\\hat{c}\n+u^{(1)}_{\\bm k}\\,\\hat{d}\n+{\\rm h.c.} \\,,\n\\\\\n\\label{pGWsol}\n\\hat{p}_{\\bm k}(\\eta)&=&\\Bigl(u^{(0)\\,\\prime}_{\\bm k}+u^{(2)\\,\\prime}_{\\bm k}\\Bigr)\\,\\hat{c}\n+u^{(1)\\,\\prime}_{\\bm k}\\,\\hat{d}\n+\\frac{1}{\\eta}\\Bigl\\{\\left(\nu^{(0)}_{\\bm k}+u^{(2)}_{\\bm k}\n\\right)\\hat{c}+u^{(1)}_{\\bm k}\\,\\hat{d}\\Bigr\\}+{\\rm h.c.}\\,,\n\\end{eqnarray}\nwhere we used Eq.~(\\ref{p}) and h.c. represents Hermitian conjugate. For the photon,\nthe field and its conjugate momentum become\n\\begin{eqnarray}\n\\hat{A}_{\\bm k}(\\eta)&=&\\Bigl(v^{(0)}_{\\bm k}+v^{(2)}_{\\bm k}\\Bigr)\\,\\hat{d}\n+v^{(1)}_{\\bm k}\\,\\hat{c}\n+{\\rm h.c.}\\,,\\\\\n\\hat{\\pi}_{\\bm k}(\\eta)&=&\n\\Bigl(v^{(0)\\,\\prime}_{\\bm k}+v^{(2)\\,\\prime}_{\\bm k}\\Bigr)\\,\\hat{d}\n+v^{(1)\\,\\prime}_{\\bm k}\\,\\hat{c}\n+{\\rm h.c.}\\,,\n\\label{pEMsol}\n\\end{eqnarray}\nwhere we used Eq.~(\\ref{pi}).\nThen the annihilation operators for the graviton and photon are obtained by using Eqs.~(\\ref{y:annihi}) and (\\ref{A:annihi}) such as\n\\begin{align}\n \\hat{a}_{y}(\\eta,\\bm{k})\n &= \\Bigl(\n \\psi_{p}^{(0)}\n +\\psi_{p}^{(2)}\n \\Bigr)\\hat{c}\n +\\Bigl(\n \\psi_{m}^{(0)*}\n +\\psi_{m}^{(2)*}\n \\Bigr)\\hat{c}^\\dagger\n +\\psi_{ p}^{(1)} \\hat{d}\n +\\psi_{m}^{(1)*} \\hat{d}^\\dagger\\,,\n \\label{y:annihifull}\n \\\\\n \\hat{a}_{A}(\\eta,\\bm{k})\n &= \\Bigl(\n \\phi_{p}^{(0)}\n +\\phi_{p}^{(2)}\n \\Bigr)\\hat{d}\n +\\Bigl(\n \\phi_{ m}^{(0)*}\n +\\phi_{ m}^{(2)*}\n \\Bigr)\\hat{d}^\\dagger\n +\\phi_{ p}^{(1)} \\hat{c}\n +\\phi_{ m}^{(1)*} \\hat{c}^\\dagger\\,.\n \\label{A:annihifull}\n\\end{align}\nHere, we defined new variables\n\\begin{align}\n&\\psi_{p}^{(j)} \n =\\sqrt{\\frac{k}{2}} u^{(j)}_{\\bm k}(\\eta)\n +\\frac{i}{\\sqrt{2k}}\n \\Bigl(u^{(j)\\prime}_{\\bm k}(\\eta)+\\frac{1}{\\eta}u^{(j)}_{\\bm k} (\\eta)\n \\Bigr) ,\\label{psip}\\\\\n&\\psi_{ m}^{(j)}\n =\\sqrt{\\frac{k}{2}} u^{(j)}_{\\bm k}(\\eta)\n -\\frac{i}{\\sqrt{2k}}\n \\Bigl(u^{(j)\\prime}_{\\bm k}(\\eta)+\\frac{1}{\\eta}u^{(j)}_{\\bm k}(\\eta)\n \\Bigr), \\label{psim}\\\\\n&\\phi_{ p}^{(j)}\n =\\sqrt{\\frac{k}{2}} v^{(j)}_{\\bm k}(\\eta)\n +\\frac{i}{\\sqrt{2k}} v_{\\bm k}^{(j)\\prime}(\\eta),\n \\label{phip}\\\\\n&\\phi_{ m}^{(j)} \n =\\sqrt{\\frac{k}{2}} v^{(j)}_{\\bm k}(\\eta)\n -\\frac{i}{\\sqrt{2k}} v_{\\bm k}^{(j)\\prime}(\\eta),\n \\label{phim}\n\\end{align}\nwhere $j=0,1,2$ denotes the order of perturbations.\n\nWe see that all mode functions other than the zeroth order given in Eqs.~(\\ref{u1Green}), (\\ref{v1Green}) (\\ref{u2Green}) and (\\ref{v2Green}) vanish at the initial time $\\eta_i$. Thus only the zeroth order of the above Eqs.~(\\ref{psip}) $\\sim$ (\\ref{phim}) remains at the initial time. This means that annihilation operators in Eqs.(\\ref{y:annihifull}) and (\\ref{A:annihifull}) at the initial time are expressed by the zeroth order variables\n\\begin{align}\n \\hat{a}_{y}(\\eta_i,\\bm{k})\n &=\\left(1-\\frac{i}{2k\\eta_i}\\right)e^{-ik\\eta_i}\\,\\hat{c}\n +\\frac{i}{2k\\eta_i} e^{ik\\eta_i}\\,\\hat{c}^\\dagger,\n \\label{ycRel}\n \\\\\n \\hat{a}_A(\\eta_i,\\bm{k})\n &=e^{-ik \\eta_i} \\hat{d}.\n\\label{AdRel}\n\\end{align}\nCombining Eqs. (\\ref{ycRel}) and (\\ref{AdRel}) with their complex conjugate, we can express the $\\hat{c}$ and $\\hat{d}$ by the initial creation and annihilation \noperators as\n\\begin{eqnarray}\n \\hat{c} &=& \\left( 1+\\frac{i}{2k\\eta_i}\\right)e^{ik\\eta_i}\\,\\hat{a}_y(\\eta_i,\\bm{k} )\n -\\frac{i}{2k\\eta_i} e^{ik\\eta_i}\\,\\hat{a}_y^\\dagger (\\eta_i,-\\bm{k}) \\ , \\\\\n \\hat{d} &=& e^{ik\\eta_i}\\,\\hat{a}_A(\\eta_i,\\bm{k}) \\ .\n\\end{eqnarray}\nPlugging the above back into Eqs.(\\ref{y:annihifull}) and (\\ref{A:annihifull}), the time evolution of annihilation operator of graviton is described by the Bogoliubov transformation in the form\n\\begin{align}\n &\\hat{a}_y(\\eta,\\bm{k})=\n \\Biggl[\n \\biggl(\n \\psi_{p}^{(0)}\n +\\psi_{p}^{(2)}\n \\biggr)\n \\Bigl( 1+\\frac{i}{2k\\eta_i} \\Bigr)\n e^{ik\\eta_i} \n +\\biggl(\n \\psi_{m}^{(0)*}\n +\\psi_{ m}^{(2)*}\n \\biggr) \\frac{i}{2k\\eta_i} e^{-ik\\eta_i} \n \\Biggr]\n \\hat{a}_y(\\eta_i,\\bm{k})\\nonumber\\\\\n &\\hspace{1.5cm}\n +\\Biggl[\n \\biggl(\n \\psi_{p}^{(0)}\n +\\psi_{p}^{(2)}\n \\biggr)\\Bigl(-\\frac{i}{2k\\eta_i} \\Bigr) e^{ik\\eta_i} \n +\\biggl(\n \\psi_{m}^{(0)*}\n +\\psi_{ m}^{(2)*}\n \\biggr)\n \\Bigl(1-\\frac{i}{2k\\eta_i}\\Bigr)\n e^{-ik\\eta_i} \\Biggr]\\hat{a}_y^\\dagger(\\eta_i,-\\bm{k})\n \\nonumber\\\\\n &\\hspace{1.5cm}\n +\\psi_{p}^{(1)} e^{ik\\eta_i}\n \\hat{a}_A (\\eta_i,\\bm{k})\n +\\psi_{m}^{(1)*} e^{-ik\\eta_i}\n \\hat{a}_A^\\dagger(\\eta_i,-\\bm{k}),\n \\label{y:bogoliubov1}\n\\end{align}\nand the time evolution of annihilation operator of photon is expressed by the Bogoliubov transformation such as\n\\begin{align}\n &\\hat{a}_A(\\eta,\\bm{k})=\n \\Biggl( \n \\phi_{ p}^{(1)}\n \\Bigl(1+\\frac{i}{2k\\eta_i} \\Bigr)\n e^{ik\\eta_i} \n + \\phi_{ m}^{(1)*}\n \\frac{i}{2k\\eta_i}\n e^{-ik\\eta_i} \n \\Biggr)\\hat{a}_y(\\eta_i,\\bm{k})\\nonumber\\\\\n &~~~~~~~~~~~\n +\\Biggl(\n -\\phi_{ p}^{(1)} \\frac{i}{2k\\eta_i}\n e^{ik\\eta_i}\n +\\phi_{ m}^{(1)*}\n \\Bigl(1-\\frac{i}{2k\\eta_i} \\Bigr)\n e^{-ik\\eta_i}\n \\Biggr) \\hat{a}_y^\\dagger (\\eta_i,-\\bm{k})\n \\nonumber\\\\\n &~~~~~~~~~~~\n +\\Bigl(\n \\phi_{p}^{(0)}+\\phi_{p}^{(2)}\n \\Bigr)e^{ik\\eta_i} \\hat{a}_{A}(\\eta_i,\\bm{k})\n +\\Bigl(\n \\phi_{m}^{(0)*}+\\phi_{m}^{(2)*}\n \\Bigr)e^{-ik\\eta_i} \\hat{a}_A^\\dagger(\\eta_i,-\\bm{k}).\n \\label{A:bogoliubov1}\n\\end{align}\nThese Bogoliubov transformations show the particle production during inflation and the mixing between graviton and photon.\n\nIt is useful to use a matrix form for later calculations. \nIn fact, the Bogoliubov transofomation (\\ref{y:bogoliubov1}) and (\\ref{A:bogoliubov1}) and their conjugate can be accommodated into the simple $4\\times 4$ matrix form $M$\n\\begin{eqnarray}\n\\begin{pmatrix}\na_y(\\eta)\\\\\na_y^{\\dagger}(\\eta)\\\\\na_A(\\eta)\\\\\na_A^{\\dagger}(\\eta)\\\\\n\\end{pmatrix}\n=M\n\\begin{pmatrix}\na_y(\\eta_i)\\\\\na_y^{\\dagger}(\\eta_i)\\\\\na_A(\\eta_i)\\\\\na_A^{\\dagger}(\\eta_i)\\\\\n\\end{pmatrix}\n=\n\\left\\{\n\\begin{pmatrix}\nA_{0}& 0\\\\\n0 &D_{0}\\\\\n\\end{pmatrix} \n+\n\\begin{pmatrix}\n0 &B_{1}\\\\\nC_{1}& 0\\\\\n\\end{pmatrix} \n+\n\\begin{pmatrix}\nA_{2}& 0\\\\\n0 & D_{2}\\\\\n\\end{pmatrix} \n\\right\\}\n\\begin{pmatrix}\na_y(\\eta_i)\\\\\na_y^{\\dagger}(\\eta_i)\\\\\na_A(\\eta_i)\\\\\na_A^{\\dagger}(\\eta_i)\\\\\n\\end{pmatrix}\\ .\n\\nonumber\n\\hspace{-6mm}\\\\\n\\label{bogoliubov}\n\\end{eqnarray}\nHere, the zeroth order Bogoliubov transformation consists of $2\\times 2$ matrices $A_0$ and $D_0$ given by \n\\begin{eqnarray}\nA_0=\n\\begin{pmatrix}\nK^* & -L^* \\\\\n-L & K \\\\\n\\end{pmatrix}\n\\ ,\\qquad\nD_0=\n\\begin{pmatrix}\ne^{ik\\,(\\eta-\\eta_i)} & 0\\\\\n0 & e^{-ik\\,(\\eta-\\eta_i)} \\\\\n\\end{pmatrix} \\ ,\n\\end{eqnarray}\nwhere we defined \n\\begin{eqnarray}\nK&=&\\left( 1+\\frac{i}{2k\\eta}\\right)\\left( 1-\\frac{i}{2k\\eta_i}\\right)e^{ik(\\eta-\\eta_i)} \n-\\frac{1}{4k^2\\eta\\eta_i }e^{-ik(\\eta-\\eta_i)}\\ ,\\\\\nL&=&-\\frac{i}{2k\\eta_i }\\left( 1+\\frac{i}{2k\\eta}\\right) e^{ik(\\eta-\\eta_i)} \n+\\frac{i}{2k\\eta }\\left( 1+\\frac{i}{2k\\eta_i}\\right) e^{-ik(\\eta-\\eta_i)} \\ .\n\\end{eqnarray}\nThe first order Bogoliubov transofomation is written by $2\\times 2$ matrices $B_1$ and $C_1$ such as\n\\begin{eqnarray}\nB_1=\n\\begin{pmatrix}\ne^{ik\\eta_i}\\psi_{p}^{(1)} & e^{-ik\\eta_i}\\psi_{m}^{(1)*}\\\\\n e^{ik\\eta_i}\\psi_{m}^{(1)}& e^{-ik\\eta_i}\\psi_{p}^{(1)*} \\\\\n\\end{pmatrix}\n\\end{eqnarray}\nand \n\\begin{eqnarray}\nC_1=\n\\begin{pmatrix}\n\\left( 1+\\frac{i}{2k\\eta_i}\\right)e^{ik\\eta_i}\\phi_{p}^{(1)}+\\frac{i}{2k\\eta_i} e^{-ik\\eta_i}\\phi_{m}^{(1)*} & \\left( 1-\\frac{i}{2k\\eta_i}\\right)e^{-ik\\eta_i}\\phi_{m}^{(1)*}-\\frac{i}{2k\\eta_i} e^{ik\\eta_i}\\phi_{p}^{(1)} \\\\\n \\left( 1+\\frac{i}{2k\\eta_i}\\right)e^{ik\\eta_i}\\phi_{m}^{(1)}+\\frac{i}{2k\\eta_i} e^{-ik\\eta_i}\\phi_{p}^{(1)*}& \\left( 1-\\frac{i}{2k\\eta_i}\\right)e^{-ik\\eta_i}\\phi_{p}^{(1)*}-\\frac{i}{2k\\eta_i} e^{ik\\eta_i}\\phi_{m}^{(1)} \\\\\n\\end{pmatrix}\\ .\n\\end{eqnarray}\nFinally, the second order Bogoliubov transformation $A_2$ and $D_2$ are\n\\begin{eqnarray}\nA_2=\n\\begin{pmatrix}\n\\left( 1+\\frac{i}{2k\\eta_i}\\right)e^{ik\\eta_i}\\psi_{p}^{(2)}+\\frac{i}{2k\\eta_i} e^{-ik\\eta_i}\\psi_{m}^{(2)*} & \\left( 1-\\frac{i}{2k\\eta_i}\\right)e^{-ik\\eta_i}\\psi_{m}^{(2)*}-\\frac{i}{2k\\eta_i} e^{ik\\eta_i}\\psi_{p}^{(2)} \\\\\n \\left( 1+\\frac{i}{2k\\eta_i}\\right)e^{ik\\eta_i}\\psi_{m}^{(2)}+\\frac{i}{2k\\eta_i} e^{-ik\\eta_i}\\psi_{p}^{(2)*} & \\left( 1-\\frac{i}{2k\\eta_i}\\right)e^{-ik\\eta_i}\\psi_{p}^{(2)*}-\\frac{i}{2k\\eta_i} e^{ik\\eta_i}\\psi_{m}^{(2)}\\\\\n\\end{pmatrix}\\ \n\\end{eqnarray}\nand\n\\begin{eqnarray}\nD_2=\n\\begin{pmatrix}\ne^{ik\\eta_i}\\phi_{p}^{(2)} & e^{-ik\\eta_i}\\phi_{m}^{(2)*}\\\\\ne^{ik\\eta_i}\\phi_{m}^{(2)} & e^{-ik\\eta_i}\\phi_{p}^{(2)*} \\\\\n\\end{pmatrix} \\ .\n\\end{eqnarray}\n\n\\section{Time evolution of squeezing parameters\nIn the previous section, we obtained the Bogoliubov transformation that mix the operators $\\hat{a}_y(\\eta)$, $\\hat{a}_A(\\eta)$\nand their Hermitian conjugates $\\hat{a}^\\dagger_y(\\eta)$, $\\hat{a}^\\dagger_A(\\eta)$.\nNote that the initial Bunch-Davies state is defined by\n\\begin{eqnarray}\n\\hat{a}_y(\\eta_i,\\bm{k}) |{\\rm BD}\\rangle= \\hat{a}_A(\\eta_i,\\bm{k}) |{\\rm BD}\\rangle =0\\,.\n\\label{BD}\n\\end{eqnarray}\nIn order to impose these conditions,\nwe need to invert the Bogoliubov transformations (\\ref{y:bogoliubov1}) and (\\ref{A:bogoliubov1})\ninto the form\n\\begin{eqnarray}\n\\hat{a}_y(\\eta_i,\\bm{k})\n&=&\\alpha_y\\,\\hat{a}_y (\\eta,\\bm{k})+ \\beta_y\\,\\hat{a}_y^\\dagger(\\eta,-\\bm{k})\n + \\gamma_A\\,\\hat{a}_A(\\eta,\\bm{k})+ \\delta_A\\,\\hat{a}_A^\\dagger(\\eta,-\\bm{k})\\,,\n \\label{y:invert}\\\\\n\\hat{a}_A(\\eta_i,\\bm{k})\n&=&\\gamma_y\\,\\hat{a}_y(\\eta,\\bm{k})+ \\delta_y\\,\\hat{a}_y^\\dagger(\\eta,-\\bm{k}) \n +\\alpha_A\\,\\hat{a}_A (\\eta,\\bm{k})+ \\beta_A\\,\\hat{a}_A^\\dagger(\\eta,-\\bm{k})\\,,\n \\label{A:invert}\n\\end{eqnarray}\nwhere $\\alpha_y$, $\\beta_y$, $\\gamma_A$, $\\delta_A$, $\\gamma_y$, $\\delta_y$, $\\alpha_A$ and $\\beta_A$ are the Bogoliubov coefficients and we will find these coefficients in the next subsection. \n\n\\subsection{Inversion of the Bogoliubov transformation}\n\nThe matrix $M$ in Eq.~(\\ref{bogoliubov}) can be expanded perturbatively as\n\\begin{eqnarray}\n M = M^{(0)}+M^{(1)}+M^{(2)}\n =M^{(0)}\\left[ 1+ M^{(0)-1}M^{(1)}\n +M^{(0)-1}M^{(2)}\\right] \\ ,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\nM^{(0)}=\n\\begin{pmatrix}\nA_{0}& 0\\\\\n0 &D_{0}\\\\\n\\end{pmatrix} \n\\,,\\qquad\nM^{(1)}=\n\\begin{pmatrix}\n0 &B_{1}\\\\\nC_{1}& 0\\\\\n\\end{pmatrix} \n\\,,\\qquad\nM^{(2)}=\n\\begin{pmatrix}\nA_{2}& 0\\\\\n0 & D_{2}\\\\\n\\end{pmatrix}\n\\,.\n\\end{eqnarray}\nThen the inverse of the $M$ is given by\n\\begin{eqnarray}\n M^{-1} \n =\\left[ 1- M^{(0)-1}M^{(1)}\n -M^{(0)-1}M^{(2)}\n + M^{(0)-1}M^{(1)}M^{(0)-1}M^{(1)} \\right]M^{(0)-1}\\ .\n\\end{eqnarray}\nUsing the above general formula, the inverse of the $M$ is obtained in the form\n\\begin{eqnarray}\nM^{-1}=\n\\begin{pmatrix}\nA_0^{-1}-A_0^{-1}A_2 A_0^{-1}\n+A_0^{-1}B_{1}D_0^{-1}C_1A_0^{-1}& -A_0^{-1}B_{1}D_0^{-1}\\\\\n-D_0^{-1}C_1 A_0^{-1}& D_0^{-1}-D_0^{-1}D_{2}D_0^{-1}\n+D_0^{-1}C_{1}A_0^{-1}B_1D_0^{-1}\n\\label{inverseM}\n\\end{pmatrix} \\,.\n\\nonumber\n\\hspace{-5mm}\\\\\n\\end{eqnarray}\nWe see that $A_0^{-1}$ and $D_0^{-1}$ are necessary to calculate the elements of the $M^{-1}$. They are given by\n\\begin{eqnarray}\nA_0^{-1}=\n\\begin{pmatrix}\nK & L^* \\\\\nL & K^* \\\\\n\\end{pmatrix} \n\\ , \\qquad\nD_0^{-1}=\n\\begin{pmatrix}\ne^{-ik\\,(\\eta-\\eta_i)} & 0\\\\\n0 & e^{ik\\,(\\eta-\\eta_i)} \\\\\n\\end{pmatrix} \\ .\n\\end{eqnarray}\nFrom Eqs.~(\\ref{y:invert}) and (\\ref{A:invert}), the $M^{-1}$ is also written as\n\\begin{eqnarray}\nM^{-1}=\n\\begin{pmatrix}\n\\alpha_y & \\beta_y & \\gamma_A & \\delta_A \\\\\n\\beta_y^* & \\alpha_y^* & \\delta_A^* & \\gamma_A^* \\\\\n\\gamma_y & \\delta_y & \\alpha_A & \\beta_A \\\\\n\\delta_y^* & \\gamma_y^* & \\beta_A^* & \\alpha_A^* \\\\\n\\end{pmatrix} \\,,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n && \\alpha_y =\\alpha_y^{(0)} +\\alpha_y^{(2)} \\,,\\qquad\n \\beta_y =\\beta_y^{(0)} +\\beta_y^{(2)} \\,,\\qquad\n \\gamma_A =\\gamma_A^{(1)} \\,,\\qquad\n \\delta_A =\\delta_A^{(1)} \\,,\n \\label{expand1}\\\\\n&& \\alpha_A =\\alpha_A^{(0)} +\\alpha_A^{(2)} \\,,\\qquad\n \\beta_A = \\beta_A^{(2)} \\,,\\qquad\n \\gamma_y =\\gamma_y^{(1)} \\,,\\qquad\n \\delta_y =\\delta_y^{(1)} \\ .\n \\label{expand2}\n\\end{eqnarray}\nThe zeroth order elements are given by\n\\begin{eqnarray}\n&& \\alpha^{(0)}_y= \\left( 1+\\frac{i}{2k\\eta}\\right)\\left( 1-\\frac{i}{2k\\eta_i}\\right)e^{ik(\\eta-\\eta_i)} \n-\\frac{1}{4k^2\\eta\\eta_i }e^{-ik(\\eta-\\eta_i)} \\ ,\\\\\n&& \\beta^{(0)}_y = \\frac{i}{2k\\eta_i }\\left( 1-\\frac{i}{2k\\eta}\\right) e^{-ik(\\eta-\\eta_i)} \n-\\frac{i}{2k\\eta }\\left( 1-\\frac{i}{2k\\eta_i}\\right) e^{ik(\\eta-\\eta_i)} \\ , \\\\\n&& \\alpha^{(0)}_A =e^{ik(\\eta-\\eta_i)} \\ ,\\qquad\n \\beta^{(0)}_A = 0\\ .\n\\end{eqnarray}\nThe first order elements are written as\n\\begin{eqnarray}\n \\gamma^{(1)}_A &=& -\\left( K\\psi_{p}^{(1)}\n+L^*\\psi_{m}^{(1)}\\right)e^{ik\\eta}\\ ,\\\\\n \\delta^{(1)}_A &=& -\\left(K\\psi_{m}^{(1)*}\n+L^*\\psi_{p}^{(1)*}\\right)\ne^{-ik\\eta} \\ , \\\\\n \\gamma^{(1)}_y &=& -K \\left[\\left( 1+\\frac{i}{2k\\eta_i}\\right)e^{ik\\eta}\\phi_{p}^{(1)}\n+\\frac{i}{2k\\eta_i} e^{ik(\\eta-2\\eta_i)}\\phi_{m}^{(1)*} \\right]\\nonumber\\\\\n&&-L \\left[\\left( 1-\\frac{i}{2k\\eta_i}\\right)e^{ik(\\eta-2\\eta_i)}\\phi_{m}^{(1)*}\n-\\frac{i}{2k\\eta_i} e^{ik\\eta}\\phi_{p}^{(1)} \\right]\\ ,\\\\\n \\delta^{(1)}_y &=& -L^* \\left[\\left( 1+\\frac{i}{2k\\eta_i}\\right)e^{ik\\eta}\\phi_{p}^{(1)}\n+\\frac{i}{2k\\eta_i} e^{ik(\\eta-2\\eta_1)}\\phi_{m}^{(1)*} \\right]\\nonumber\\\\\n&&-K^* \\left[\\left( 1-\\frac{i}{2k\\eta_i}\\right)e^{ik(\\eta-2\\eta_i)}\\phi_{m}^{(1)*}\n-\\frac{i}{2k\\eta_i} e^{ik\\eta}\\phi_{p}^{(1)} \\right] \\ .\n\\end{eqnarray}\nThe second order are \n\\begin{eqnarray}\n \\alpha^{(2)}_y &=& -K\\left(KA_{11}+L^* A_{21}\\right)\n-L\\left(KA_{12}+L^* A_{22}\\right) \\nonumber\\\\\n&&+(C_{11} K+C_{12} L)\\left(K\\psi_{p}^{(1)}\n+L^*\\psi_{m}^{(1)}\\right)e^{ik\\eta}\\nonumber\\\\\n&&+(C_{21} K+C_{22} L)\\left(K\\psi_{m}^{(1)*}\n+L^*\\psi_{p}^{(1)*}\\right)e^{-ik\\eta}\\ , \\\\\n \\beta^{(2)}_y &=& -L^*\\left(KA_{11}+L^* A_{21}\\right)\n-K^* \\left(KA_{12}+L^* A_{22}\\right) \\nonumber\\\\\n&&+(C_{11} L^* +C_{12} K^*)\\left(K \\psi_{p}^{(1)}\n+L^*\\psi_{m}^{(1)}\\right)e^{ik\\eta}\\nonumber\\\\\n&&+(C_{21} L^*+C_{22} K^*)\\left(K\\psi_{m}^{(1)*}\n+L^*\\psi_{p}^{(1)*}\\right)e^{-ik\\eta}\\ ,\\\\ \n \\alpha^{(2)}_A &=& - e^{ik(2\\eta-\\eta_i)}\\phi_{p}^{(2)} \\nonumber\\\\\n&& +(C_{11} K+C_{12} L)e^{ik(2\\eta - \\eta_i)}\\psi_{p}^{(1)}\n+(C_{11} L^* +C_{12} K^*)e^{ik(2\\eta - \\eta_i)}\\psi_{m}^{(1)}\n \\ , \\\\\n \\beta^{(2)}_A &=&-e^{-ik\\eta_i}\\phi_{m}^{(2)*}\\nonumber\\\\\n&& +(C_{11} K+C_{12} L) e^{-ik\\eta_i }\\psi_{m}^{(1)*}\n+(C_{11} L^* +C_{12} K^*)e^{-ik\\eta_i}\\psi_{p}^{(1)*}\\ ,\n\\end{eqnarray}\nwhere we have defined\n\\begin{eqnarray}\nA_{11}&=&\\psi_{p}^{(2)}\\left(1+\\frac{i}{2k\\eta_i}\\right)e^{ik\\eta_i}+\\psi_{m}^{(2)*}\\frac{i}{2k\\eta_i}e^{-ik\\eta_i}\\,,\\\\\nA_{12}&=&-\\psi_{p}^{(2)}\\frac{i}{2k\\eta_i}e^{ik\\eta_i}+\\psi_{m}^{(2)*}\\left(1-\\frac{i}{2k\\eta_i}\\right)e^{-ik\\eta_i}\\,,\\\\\nA_{21}&=&\\psi_{p}^{(2)*}\\frac{i}{2k\\eta_i}e^{-ik\\eta_i}+\\psi_{m}^{(2)}\\left(1+\\frac{i}{2k\\eta_i}\\right)e^{ik\\eta_i}\\,,\\\\\nA_{22}&=&\\psi_{p}^{(2)*}\\left(1-\\frac{i}{2k\\eta_i}\\right)e^{-ik\\eta_i}-\\psi_{m}^{(2)}\\frac{i}{2k\\eta_i}e^{ik\\eta_i}\\,,\n\\end{eqnarray}\nand\n\\begin{eqnarray}\nC_{11}&=&\\phi_{p}^{(1)}\\left(1+\\frac{i}{2k\\eta_i}\\right)e^{ik\\eta_i}+\\phi_{m}^{(1)*}\\frac{i}{2k\\eta_i}e^{-ik\\eta_i}\\,,\\\\\nC_{12}&=&\\phi_{m}^{(1)*}\\left(1-\\frac{i}{2k\\eta_i}\\right)e^{-ik\\eta_i}-\\phi_{p}^{(1)}\\frac{i}{2k\\eta_i}e^{ik\\eta_i}\\,,\\\\\nC_{21}&=&\\phi_{p}^{(1)*}\\frac{i}{2k\\eta_i}e^{-ik\\eta_i}+\\phi_{m}^{(1)}\\left(1+\\frac{i}{2k\\eta_i}\\right)e^{ik\\eta_i}\\,,\\\\\nC_{22}&=&\\phi_{p}^{(1)*}\\left(1-\\frac{i}{2k\\eta_i}\\right)e^{-ik\\eta_i}-\\phi_{m}^{(1)}\\frac{i}{2k\\eta_i}e^{ik\\eta_i}\\,.\n\\end{eqnarray}\n\n\n\\subsection{Squeezing operator}\nIn the previous subsection, we obtained the Bogoliubov coefficients of Eqs.~(\\ref{y:invert}) and (\\ref{A:invert}) up to the second order. If we apply the Eqs.~(\\ref{y:invert}) and (\\ref{A:invert}) to the definition of the Bunch-Davies vacuum~(\\ref{BD}) and use the relations $[\\hat{a}_y(\\eta,{\\bm k}),\\hat{a}^\\dag_y(\\eta,-{\\bm k}^\\prime)]=\\delta({\\bm k}+{\\bm k}^\\prime)$\\,, $[\\hat{a}_A(\\eta,{\\bm k}),\\hat{a}^\\dag_A(\\eta,-{\\bm k}^\\prime)]=\\delta({\\bm k}+{\\bm k}^\\prime)$ and $[\\hat{a}_y(\\eta,{\\bm k}),\\hat{a}_A(\\eta,-{\\bm k}^\\prime)]=0$, the Bunch-Davies vacuum can be written by using squeezing parameters $\\Lambda,\\Xi$ and $\\Omega$ such as \n\\begin{eqnarray}\n|{\\rm BD}\\rangle = \\prod_{k=-\\infty}^{\\infty}\\exp\\left[\\frac{\\Lambda}{2}\\,\n\\hat{a}_y^\\dag (\\eta,\\bm{k}) \\hat{a}_y^\\dag (\\eta,-\\bm{k})+\\Xi\\,\\hat{a}_y^\\dag (\\eta,\\bm{k}) \\hat{a}_A^\\dag (\\eta,-\\bm{k})\n+\\frac{\\Omega}{2}\\,\\hat{a}_A^\\dag (\\eta,\\bm{k}) \\hat{a}_A^\\dag (\\eta,-\\bm{k})\\right]|0\\rangle,\\nonumber\n\\hspace{-5mm}\\\\\n\\end{eqnarray}\nwhere $|0\\rangle$ is the instantaneous vacuum defined by\n\\begin{eqnarray}\n\\hat{a}_y(\\eta,{\\bm k}) |0\\rangle=\\hat{a}_A(\\eta,{\\bm k}) |0\\rangle =0 \\,.\n\\end{eqnarray}\nThis describes a four mode squeezed state of pairs of graviton $y$ and photon $A$. In a different context, a four-mode squeezed state of two free massive scalar fields is discussed in~\\cite{Albrecht:2014aga,Kanno:2015ewa}.\nIf we expand the exponential function in Taylor series, we find\n\\begin{eqnarray}\n|{\\rm BD}\\rangle =\n\\prod_{\\bm k} \\sum_{p\\,,q\\,,r=0}^{\\infty}\n \\frac{\\Lambda^p\\,\\Xi^q\\,\\Omega^r}{2^{p+r}p!\\,q!\\,r!} \n |p+q \\rangle_{y,{\\bm k}} \\otimes |p \\rangle_{y,-{\\bm k}} \\otimes |r \\rangle_{A,{\\bm k}} \\otimes |q+r \\rangle_{A,-{\\bm k}}\\,.\n\\end{eqnarray}\nThis is a four-mode squeezed state which consists of an infinite number of entangled particles in the ${\\cal H}_{y,{\\bm k}}\\otimes{\\cal H}_{y,{-\\bm k}}\\otimes{\\cal H}_{A,{\\bm k}}\\otimes{\\cal H}_{A,-{\\bm k}}$ space. \nIn particular, in the highly squeezing limit $\\Lambda\\,,\\Xi\\,,\\Omega\\rightarrow 1$, the Bunch-Davies vacuum becomes the maximally entangled state from the point of view of the instantaneous vacuum. \n\nNow we find the squeezing parameters.\nThe condition $\\hat{a}_y(\\eta_i,{\\bm k})|{\\rm BD}\\rangle=0$ of Eq.~(\\ref{BD}) yields\n\\begin{eqnarray}\n \\alpha_y \\Lambda +\\beta_y +\\gamma_A \\Xi =0 \\ , \\qquad\n \\alpha_y \\Xi +\\gamma_A \\Omega +\\delta_A =0\\,,\n\\end{eqnarray}\nand another condition $\\hat{a}_A(\\eta_i,{\\bm k}) |{\\rm BD}\\rangle=0$ of Eq.~(\\ref{BD}) gives \n\\begin{eqnarray}\n \\alpha_A \\Xi +\\gamma_y \\Lambda +\\delta_y =0 \\ , \\qquad\n \\alpha_A \\Omega +\\beta_A +\\gamma_y \\Xi =0\\,.\n\\end{eqnarray}\nThen, we obtain the three squeezing parameters $\\Lambda,\\Xi$ and $\\Omega$ of the form\n\\begin{eqnarray}\n \\Lambda= \\frac{\\gamma_A\\delta_y -\\beta_y \\alpha_A}{\\alpha_y \\alpha_A -\\gamma_y\\gamma_A} \\ , \\qquad\n \\Xi= \\frac{\\beta_y \\gamma_y - \\alpha_y \\delta_y}{\\alpha_y \\alpha_A -\\gamma_y\\gamma_A} \\ , \\qquad\n \\Omega= \\frac{\\gamma_y\\delta_A -\\beta_A \\alpha_y}{\\alpha_y \\alpha_A -\\gamma_y\\gamma_A}\\,.\n \\label{squeezingparameters}\n\\end{eqnarray}\nWe have four relations for three parameters $\\Lambda,\\Xi$ and $\\Omega$. The remaining relation is turned out to be guaranteed by the commutation relation:\n\\begin{eqnarray}\n [\\hat{a}_y(\\eta,{\\bm k}) \\ , \\hat{a}_A(\\eta,{\\bm k}) ]\n = -\\alpha_A\\delta_A +\\beta_A \\gamma_A\n -\\gamma_y \\beta_y +\\alpha_y \\delta_y =0\\,.\n\\end{eqnarray}\nThus, we find that Eq.~(\\ref{squeezingparameters}) is the unique solution.\nSince the Bogoliubov coefficients are given up to the second order as in Eqs.~(\\ref{expand1}) and (\\ref{expand2}),\nthe squeezing parameters can be expanded up to the second order such as\n\\begin{eqnarray}\n && \\Lambda= -\\frac{\\beta_y^{(0)}}{\\alpha_y^{(0)}} \n \\left[1-\\frac{\\alpha_y^{(2)}}{\\alpha_y^{(0)}}\n + \\frac{\\beta_y^{(2)}}{\\beta_y^{(0)}}\n + \\frac{\\gamma_y^{(1)}\\gamma_A^{(1)}}{\\alpha_y^{(0)}\\alpha_A^{(0)}}\n -\\frac{\\gamma_A^{(1)}\\delta_y^{(1)}}{\\beta_y^{(0)}\\alpha_A^{(0)}}\n \\right]\\ , \\\\\n && \\Xi= \\frac{\\beta_y^{(0)}\\gamma_y^{(1)}}{\\alpha_y^{(0)}\\alpha_A^{(0)}} \n - \\frac{\\delta_y^{(1)}}{\\alpha_A^{(0)}}\\ , \\\\\n && \\Omega= \\frac{\\delta_A^{(1)}\\gamma_y^{(1)}}{\\alpha_y^{(0)}\\alpha_A^{(0)}} \n - \\frac{\\beta_A^{(2)}}{\\alpha_A^{(0)}}\\ .\n\\end{eqnarray}\nIn this way, we obtained the squeezing parameters perturbatively up to the second order. We will discuss the behaviour of the squeezing of graviton $\\Lambda$, the squeezing of mixing between graviton and photon $\\Xi$ and the squeezing of photon $\\Omega$ in the next section.\n\n\n\\subsection{Numerical and analytical results}\n\nThe results of numerical calculations for the amplitude and the phase of the squeezing parameters $\\Lambda$, $\\Xi$, and $\\Omega$ are plotted in FIGs.~\\ref{SqueezingA2}, \\ref{PhaseA}, \\ref{SqueezingB2}, \\ref{PhaseB}, \\ref{SqueezingC2}, and \\ref{PhaseC}, respectively, where we normalized the scale factor at the end of inflation as $a(\\eta_f)=1$.\nThe evolution of the amplitude of $\\Lambda$ in FIG. \\ref{SqueezingA2} shows graviton is squeezed, that is, graviton pair production occurs during inflation ($\\eta<0$).\nWe see that sub-horizon modes oscillates rapidly and no graviton pair production seems to occur before horizon exit. In the presence of coupling with magnetic fields ($\\lambda\\neq 0$), the amplitude of oscillation is relatively small as represented by blue line. After horizon exit, the oscillation ceases and graviton pair production starts to occur and eventually $\\Lambda$ becomes one. This means that almost maximum entangled pair of graviton are produced. This behavior does not change even for $\\lambda\\neq 0$.\nThe evolution of phase of $\\Lambda$ is plotted in FIG. \\ref{PhaseA}, in which we see the phase converges to zero. This is consistent with the result\nin \\cite{Polarski:1995jg}.\nThe time evolution of the amplitude of $\\Xi$ in FIG. \\ref{SqueezingB2} shows that one of pair of gravitons is converted to a photon and graviton-photon pair production occurs. \nWe see that some amount of pair-production occurs when the mode leaves the horizon but the graviton-photon pair production decreases rapidly by the end of inflation.\nThe evolution of the phase of $\\Xi$ plotted in FIG. \\ref{PhaseB} is found to oscillate rapidly but eventually becomes constant after horizon exit. The similar behavior appears in the evolution of phase of $\\Lambda$ in FIG.~{\\ref{PhaseA}}.\nHowever, the final\nphase is found to depend on the initial condition in this case.\nThe time evolution of the amplitude of $\\Omega$ in FIG. \\ref{SqueezingC2} tells us that photon is squeezed, that is, graviton pair production is converged to photon pair production. \nInterestingly, photon pair production occurs rapidly only at the initial time and no more production occurs after that. \nThe behavior of the phase evolution of $\\Omega$ \nin FIG.\\ref{PhaseC} is similar to that of $\\Xi$.\n\\begin{figure}[H]\n\\centering\n \\includegraphics[width=\\textwidth]{SqueezingA2.pdf}\n \\renewcommand{\\baselinestretch}{3}\n \\caption{Squeezing parameter of graviton pair $\\Lambda$ during inflation as a function of the scale factor $a(\\eta)$. We set $\\lambda=5\\times10^{-13}{\\rm GeV}^2$(blue line) and $\\lambda=0~{\\rm GeV}^2$(yellow line). \n Other parameters are set as $k=10^2{\\rm GeV}$, $H = 10^{14}{\\rm GeV}$, $\\eta_i=-2{\\rm GeV}^{-1}$, $\\eta_{f} =-10^{-14}{\\rm GeV}^{-1}$, $a(\\eta_i)=(2\\times 10^{14})^{-1}$, and $a(\\eta_{f})=1$. The red grid line shows the scale factor $a=1.59...\\times10^{-13}$ at the time of horizon exit $\\eta=-2\\pi\/k$.}\n \\label{SqueezingA2}\n \\end{figure}\n \n \n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.55\\textwidth]{PhaseA.pdf}\n\\renewcommand{\\baselinestretch}{1.2}\n\\caption{The phase of the squeezing parameter of graviton pair $\\Lambda(a)$ during inflation as function of the scale factor $a(\\eta)$. We set $\\lambda=5\\times10^{-13}{\\rm GeV}^2$(blue line) and $\\lambda=0~{\\rm GeV}^2$(yellow line). \nOther parameters are set as $k=10^2{\\rm GeV}$, $H=10^{14}{\\rm GeV}$, $\\eta_i=-2{\\rm GeV}^{-1}$, $\\eta_f =-10^{-14}{\\rm GeV}^{-1}$, $a(\\eta_i)=(2\\times 10^{14})^{-1}$, and $a(\\eta_f)=1$. }\n\\label{PhaseA}\n\\end{figure}\n \n\n\n\n\nNow, we investigate the behavior of those squeezing parameters for $k\\eta\\ll 1$ and $k\\eta_i \\gg 1$ analytically. The leading and sub-leading terms of $\\Lambda$ and $\\Xi$ can be calculated as\n\\begin{align}\n\\Lambda=1+ {\\cal O} \\left(\\frac{\\lambda^2 H^2\\eta_i^2 }{k^4 }\\right)\\,,\\qquad\n\\Xi=0+ {\\cal O} \\left(\\frac{\\lambda H \\eta}{k^2 }\\right)\\,.\n\\label{lambdaxi}\n\\end{align}\nWe find that sub-leading terms of $\\Lambda$ and $\\Xi$ are negligibly small near the end of inflation and which is consistent with the numerical results in FIGs.~\\ref{SqueezingA2} and \\ref{SqueezingB2}. This result tells us that the conversion from graviton pair production to graviton-photon pair production is hard to occur.\nFor the squeezing parameter $\\Omega$, we find\n\\begin{align}\n\\Omega= i e^{2ik \\eta_i} \\frac{5\\lambda^2H^2 \\eta_i^3 }{32k^3}\\,.\n\\label{main}\n\\end{align}\nIf we use the numerical values $\\lambda=5\\times10^{-13}\\,{\\rm GeV}^2$, \n $k=10^2\\,{\\rm GeV}$, $H=10^{14}\\,{\\rm GeV}$, and $\\eta_i=-2\\,{\\rm GeV}^{-1}$\n, we find $|\\Omega| \\sim 0.003$ and which agrees with the numerical result in FIG. \\ref{SqueezingC2}. Thus only small amount of conversion from graviton pair production to photon pair production occurs at the end of inflation.\nThese results support the validity of our iterative method to derive squeezing parameters.\n\n\n\n\\begin{figure}[H]\n\\centering\n \\includegraphics[width=0.55\\textwidth]{SqueezingB2.pdf}\n \\renewcommand{\\baselinestretch}{1.2}\n \\caption{Squeezing parameter of graviton-photon pair $\\Xi$ during inflation as a function of the scale factor $a$. We set $\\lambda=5\\times10^{-13}{\\rm GeV}^2$ (blue line). \n Other parameters are set as $k=10^2{\\rm GeV}$, $H=10^{14}{\\rm GeV}$, $\\eta_i=-2{\\rm GeV}^{-1}$, $\\eta_f =-10^{-14}{\\rm GeV}^{-1}$, $a(\\eta_i)=(2\\times 10^{14})^{-1}$, and $a(\\eta_f)=1$. }\n \\label{SqueezingB2}\n \\end{figure}\n \n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=\\textwidth]{PhaseB.pdf}\n\\renewcommand{\\baselinestretch}{1.2}\n\\caption{The phase of the squeezing parameter of photon pair $\\Xi(a)$ during inflation as a function of the scale factor $a(\\eta)$. We set $\\lambda=5\\times10^{-13}{\\rm GeV}^2$, $k=10^2{\\rm GeV}$, $H=10^{14}{\\rm GeV}$, $\\eta_i=-2{\\rm GeV}^{-1}$, $\\eta_f =-10^{-14}{\\rm GeV}^{-1}$, $a(\\eta_i)=(2\\times 10^{14})^{-1}$, and $a(\\eta_f)=1$. }\n\\label{PhaseB}\n\\end{figure}\n \n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=0.55\\textwidth]{SqueezingC2.pdf}\n\\renewcommand{\\baselinestretch}{1.2}\n\\caption{Squeezing parameter of photon pair $\\Omega$ during inflation as a function of the scale factor $a(\\eta)$. We set $\\lambda=5\\times10^{-13}{\\rm GeV}^2$, $k=10^2{\\rm GeV}$, $H=10^{14}{\\rm GeV}$, $\\eta_i=-2{\\rm GeV}^{-1}$, $\\eta_f =-10^{-14}{\\rm GeV}^{-1}$, $a(\\eta_i)=(2\\times 10^{14})^{-1}$, and $a(\\eta_f)=1$. }\n\\label{SqueezingC2}\n\\end{figure}\n\n\n\n\\begin{figure}[H]\n\\centering\n\\includegraphics[width=\\textwidth]{PhaseC.pdf}\n\\renewcommand{\\baselinestretch}{1.2}\n\\caption{The phase of the squeezing parameter of photon pair $\\Omega$ during inflation as a function of the scale factor $a(\\eta)$. We set $\\lambda=5\\times10^{-13}{\\rm GeV}^2$, $k=10^2{\\rm GeV}$, $H=10^{14}{\\rm GeV}$, $\\eta_i=-2{\\rm GeV}^{-1}$, $\\eta_f =-10^{-14}{\\rm GeV}^{-1}$, $a(\\eta_i)=(2\\times 10^{14})^{-1}$, and $a(\\eta_f)=1$. }\n\\label{PhaseC}\n\\end{figure}\n\n\n\n\n\nLet us discuss implications of our numerical and analytical results. If the squeezing of graviton decreases as time evolves, it implies that the decoherence of graviton occurs. \nHowever, we found that \nthe squeezing parameter of graviton pair increases and becomes $\\Lambda\\rightarrow 1$, so it seems that the decoherence is hard to occur.\nThis behavior can be understood as follows.\nSince the effective coupling $\\lambda H\\eta$ between graviton and photon in Eqs.~(\\ref{eom:graviton}) and (\\ref{eom:photon}) decreases and eventually becomes negligible as $\\eta\\rightarrow 0$ during inflation, practically graviton-photon conversion stops.\nEven after the graviton-photon conversion stops, the squeezing process of graviton pair continues as time evolves during inflation, so the squeezing of graviton pair $\\Lambda$ continues to grow irrespective of the presence of the magnetic field as shown in FIG.~\\ref{SqueezingA2}. \nNext, from FIG.~\\ref{SqueezingB2}, we see the squeezing parameter of graviton-photon pair vanishes $\\Xi\\rightarrow 0$ as time evolves. This is consistent with Eq.~(\\ref{lambdaxi}).\nThis is because the graviton-photon pair production is possible only in the presence of magnetic fields\ndue to spin conservation. In our setup, however, the energy density of the background magnetic field decreases proportional to $a(\\eta)^{-4}$ as the universe expands. \nHence, the rapid decay of magnetic fields lead to the rapid decay of $\\Xi$.\nFinally, we consider the evolution of the squeezing parameter of photon pair $\\Omega$. By using the coupling constant $\\lambda \\simeq Bk \/M_{\\rm pl}$ in Eq.~(\\ref{coupling}) and the scale factor at the initial time $a_i \\equiv -1\/(H\\eta_i)$, the $\\Omega$ reads\n\\begin{eqnarray}\n\\Omega\\simeq \\frac{B^2}{a_i^4 M_{\\rm pl}^2 H^2}\n\\frac{1}{k\\eta_i} \\ .\n\\end{eqnarray}\nThe first factor is the ratio of the energy density of the background magnetic field at the time $\\eta_i$\nto that of the inflaton field. The second factor is the ratio of the mode of graviton to the Hubble radius. In order to have inflation, the energy density of the magnetic field has to be smaller than that of inflaton fields, that is, $B^2\/a_i^4\\ll M_{\\rm pl}^2H^2$. And all modes of graviton is inside horizon initially, that is, $1\/k\\ll \\eta_i$. Hence, the $\\Omega$\nnever exceeds unity after time evolution, which is consistent with FIG.~\\ref{SqueezingC2}. \nMoreover, since graviton-photon conversion stops, \nthe squeezing of photon pair $\\Omega$ converges to a constant value as shown in FIG.~\\ref{SqueezingC2}. \n\n \n\\section{Conclusion\n\nThe relic gravitons are expected to be squeezed during inflation. In that case, quantum noise induced by them can be significantly enhanced in current interferometers. \nHowever, we need to properly take into account the decoherence of the relic gravitons during cosmic history.\nAs a first modest step in this direction, we assumed the presence of a sizable magnetic field at the beginning of inflation. If the squeezing of graviton decreases as time evolves, it implies that the decoherence of graviton occurs.\nSo, we studied the conversion processes of the squeezed gravitons into photons during inflation in the case of minimal coupling between gravitons and photons.\nWe solved the dynamical evolution of a coupled system of graviton and photon \nperturbatively. We numerically plotted\nthe squeezing parameters for the system of graviton and photon. \nFIG.~\\ref{SqueezingA2} showed that magnetic fields do not affect the graviton squeezing parameter. \nIn FIG.~\\ref{SqueezingB2},\nwe numerically checked the parameter of squeezed graviton-photon pair $\\Xi$ and found that the $\\Xi$ rapidly decays at the end of inflation.\nThis fact was confirmed analytically in Eq.~(\\ref{lambdaxi}). \nWe found that the rapid decay of the initial presence of magnetic fields leads to the rapid decay of the $\\Xi$.\nIn FIG.~\\ref{SqueezingC2}, \nwe depicted the squeezing parameter of the photon. It turned out that the amount of squeezed photon produced by the conversion was tiny. \nWe derived an analytic formula for the\nsqueezing parameter of photons $\\Omega$\nand found that the degree of squeezing is at a few percent at most.\n\n\nSince we found that gravitons are robust against the decoherence caused by the cosmological magnetic field, we could expect to find squeezed relic gravitons through quantum noise induced by them in interferometers~\\cite{Parikh:2020nrd,Kanno:2020usf,Parikh:2020kfh,Parikh:2020fhy,Kanno:2021gpt}.\nWe should note that the analysis in our paper can also be applicable to the dark magnetic field models~\\cite{Masaki:2018eut} based on the dark photon scenario~\\cite{Caputo:2021eaa}.\n\nThere are several directions to be pursued. \nIt would be intriguing to follow up the evolution of the squeezed relic gravitons up to the radiation-dominated \nand matter-dominated eras. If we could show the absence of decoherence of the squeezed relic gravitons, the robustness of them would be proved.\nIt would also be interesting to study the case that the primordial magnetic fields persist against the cosmic no-hair theorem during inflation~\\cite{Kanno:2009ei}.\nOn top of gravitons, the squeezing occurs for the light axion dark matter fields~\\cite{Kuss:2021gig,Kanno:2021vwu}.\nThe decoherence of axion fields due to magnetic fields can be discussed in a similar way.\nWe leave these issues for future work.\n\n\n\n\n\n\n\n\n\n\\section*{Acknowledgments}\nS.\\ K. was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number JP22K03621.\nJ.\\ S. was in part supported by JSPS KAKENHI Grant Numbers JP17H02894, JP17K18778, JP20H01902, JP22H01220.\nK.\\ U. was supported by the Japan Society for the Promotion of Science (JSPS) KAKENHI Grant Number 20J22946.\n\n\\printbibliography\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\\label{sec:Intro}\nThe well known Lady\\v zenskaja-Babu\\v ska-Brezzi (LBB) condition is a particular instance of the\nso-called\ndiscrete inf--sup condition which is necessary and sufficient for the well-posedness of discrete\nsaddle point problems\narising from discretization via Galerkin methods. If ${\\bf X}_h$ denotes the discrete velocity space and\n$M_h$ the discrete\npressure space, then the LBB condition for the Stokes problem states that there is a constant\n$c$ independent of the discretization parameter $h$ such that\n\\begin{equation}\n c \\| q_h \\|_{L^2} \\leq \\sup_{v_h \\in {\\bf X}_h} \n \\frac{ \\int_\\Omega (\\DIV v_h)\\, q_h}{\\|v_h\\|_{{\\bf H}^1}},\n \\quad \\forall q_h \\in M_h.\n\\tag{LBB}\n\\label{eq:LBB}\n\\end{equation}\nThe reader is referred to \\cite{GR86} for the basic theory on saddle point problems on Banach spaces\nand \ntheir numerical analysis. Simply put, this condition sets a structural restriction on the discrete\nspaces so that\nthe continuous level property that the divergence operator is closed and surjective, see\n\\cite{MR82m:26014,MR1880723}, is preserved uniformly with respect to the discretization parameter.\n\nIn the literature the following condition, which we shall denote the generalized LBB condition, is\nalso assumed\n\\begin{equation}\nc \\| \\GRAD q_h \\|_{{\\bf L}^2} \\leq\n \\sup_{v_h \\in {\\bf X}_h}\n \\frac{ \\int_\\Omega (\\DIV v_h)\\, q_h }{ \\| v_h \\|_{{\\bf L}^2} },\n \\quad \\forall q_h \\in M_h,\n\\tag{GLBB}\n\\label{eq:wLBB}\n\\end{equation}\nhere and throughout we assume $M_h \\subset H^1(\\Omega)$. By properly defining a discrete gradient\noperator, the case of discontinuous pressure spaces can be analyzed with similar arguments to those\nthat we shall present. Condition \\eqref{eq:wLBB}, for example, was used by Guermond\n(\\cite{MR2210084,MR2334774}) to show that approximate solutions to the three-dimensional Navier\nStokes equations constructed using the Faedo-Galerkin method\nconverge to a suitable, in the sense of Scheffer, weak solution.\nOn the basis of \\eqref{eq:wLBB}, the same author has also built (\\cite{MR2520170}) an\n${\\bf H}^s$-approximation theory for the Stokes problem, $0\\leq s \\leq1$.\nOlshanski{\\u\\i}, in \\cite{MR2833487}, under the assumption that the spaces satisfy \\eqref{eq:wLBB}\ncarries out a multigrid analysis for the Stokes problem. Finally, Mardal et al.\\@\\xspace, \n\\cite{schoberlwinther}, use a weighted inf--sup condition to analyze preconditioning techniques\nfor singularly perturbed Stokes problems (see Section \\ref{sec:section5} below).\n\nIt is not difficult to show that, on quasi-uniform meshes, \\eqref{eq:wLBB} implies \\eqref{eq:LBB}, see\n\\cite{MR2210084}. We include the proof of this result below for completeness.\nThe question that naturally arises is whether the converse holds. Recall that a well-known result of\nFortin \\cite{BF91}\nshows that the inf--sup condition \\eqref{eq:LBB} is equivalent to the existence of a so-called Fortin\nprojection that is stable in ${{ \\bf H}^1_0 (\\Omega)}$. In this work, under the assumption that the mesh is \nshape regular and quasi-uniform, we will show that \\eqref{eq:wLBB} is equivalent\nto the existence of a Fortin projection that has ${\\bf L}^2$-approximation properties. Moreover,\nwhen the domain is such that the solution to the Stokes problem \npossesses ${\\bf H}^2$-regularity, we will prove that \n\\eqref{eq:wLBB} is in fact equivalent to \\eqref{eq:LBB}, again on quasi-uniform meshes.\n\nThe work by Girault and Scott (\\cite{MR1961943}) must be mentioned \nwhen dealing with the construction of Fortin projection operators with ${\\bf L}^2$-approximation properties.\nThey have constructed such operators for many commonly \nused inf--sup stable spaces, one notable exception being the lowest order Taylor-Hood element \nin three dimensions.\nHowever, \\eqref{eq:wLBB} has been shown to hold for the lowest order Taylor-Hood\nelement directly \\cite{MR2210084}.\nOur results then can be applied to show that, \\eqref{eq:wLBB} is satisfied \nby almost all inf--sup stable finite element spaces, regardless of the smoothness of the domain.\n\nThis work is organized as follows.\nSection~\\ref{sec:prel} introduces the notation and assumptions we shall work with. Condition\n\\eqref{eq:wLBB} is discussed in Section~\\ref{sec:wLBB}. In Section~\\ref{sec:Equiv} we actually show the\nequivalence of conditions \\eqref{eq:LBB} and \\eqref{eq:wLBB}, provided the domain is smooth enough.\nA weighted inf--sup condition related to uniform preconditioning of the time-dependent Stokes problem\nis presented in Section~\\ref{sec:section5}, where we show that \\eqref{eq:wLBB} implies it.\nSome concluding remarks are provided in Section~\\ref{sec:conclusion}.\n\n\\section{Preliminaries}\n\\label{sec:prel}\nThroughout this work, we will denote by $\\Omega \\subset \\mathbb R^d$ with $d=2$ or $3$ an open bounded\ndomain with Lipschitz boundary. If additional smoothness of the domain is needed, it will be specified\nexplicitly.\n${{ L}^2 (\\Omega)}$, ${{ H}^{1}(\\Omega)}$ and ${{ H}^1_0 (\\Omega)}$ denote, respectively, the usual Lebesgue and Sobolev spaces.\nWe denote by ${L^2_{\\scriptscriptstyle\\!\\int\\!=0} (\\Omega)}$ the set of functions in ${{ L}^2 (\\Omega)}$ with mean zero.\nVector valued spaces will be denoted by bold characters.\n\nWe introduce a conforming triangulation ${\\mathcal T}_h$ of $\\Omega$ which we assume shape-regular and\nquasi-uniform in the sense of \\cite{BF91}. The size of the cells in the triangulation is characterized by \n$h>0$. We introduce finite dimensional spaces ${\\bf X}_h \\subset {{ \\bf H}^1_0 (\\Omega)}$ and \n$M_h \\subset {L^2_{\\scriptscriptstyle\\!\\int\\!=0} (\\Omega)} \\cap {{ H}^{1}(\\Omega)}$ which are\nconstructed, for instance using finite elements,\non the triangulation ${\\mathcal T}_h$. For these spaces, the inverse inequalities\n\\begin{equation}\n \\| v_h \\|_{{\\bf H}^1} \\leq c h^{-1} \\| v_h \\|_{{\\bf L}^2}, \\quad \\forall v_h \\in {\\bf X}_h,\n\\label{eq:invX}\n\\end{equation}\nand\n\\begin{equation}\n \\| q_h \\|_{H^1} \\leq c h^{-1} \\| q_h \\|_{L^2}, \\quad \\forall q_h \\in M_h,\n\\label{eq:invM}\n\\end{equation}\nhold, see \\cite{BF91}. Here and in what follows we denote by $c$ will a constant that is independent of $h$. \n\nWe shall denote by ${\\mathcal C}_h : {{ \\bf H}^1_0 (\\Omega)} \\rightarrow {\\bf X}_h$ the so-called \nScott-Zhang interpolation operator (\\cite{SZ90}) onto the velocity space and we recall that\n\\begin{equation}\n \\| v - {\\mathcal C}_h v \\|_{{\\bf L}^2} + h \\|{\\mathcal C}_h v\\|_{{\\bf H}^1} \\leq c h \\| v \\|_{{\\bf H}^1}, \\quad \\forall v\n\\in {{ \\bf H}^1_0 (\\Omega)}.\n\\label{eq:SZprop}\n\\end{equation}\nand\n\\begin{equation}\n \\| v - {\\mathcal C}_h v\\|_{{\\bf H}^1} \\leq c h \\| v \\|_{{\\bf H}^2}, \\quad \\forall v \\in {{ \\bf H}^1_0 (\\Omega)} \\cap {{ \\bf H}^2 (\\Omega)}\n\\label{eq:SZh1}\n\\end{equation}\nThe Scott-Zhang interpolation operator onto the pressure space\n${\\mathcal I}_h: {L^2_{\\scriptscriptstyle\\!\\int\\!=0} (\\Omega)} \\rightarrow M_h$ can be defined analogously\nand satisfies similar stability and approximation properties.\nWe shall denote by $\\pi_h: {{ \\bf L}^2 (\\Omega)} \\rightarrow {\\bf X}_h$ the ${\\bf L}^2$-projection onto ${\\bf X}_h$ and\nby $\\Pi_0 : {{ L}^2 (\\Omega)} \\rightarrow {{ L}^2 (\\Omega)}$ the $L^2$-projection operator onto the space\nof piecewise constant functions, i.e.,\\@\\xspace\n\\[\n \\Pi_0 q = \\sum_{T \\in {\\mathcal T}_h} \\frac1{|T|}\\left(\\int_T q\\right)\\chi_T, \\quad \\forall q \\in {{ L}^2 (\\Omega)}.\n\\]\n\nFor one result below we shall require full ${\\bf H}^2$-regularity of the solution to the Stokes problem:\n\n\\begin{assumption}\\label{assumption1}\nThe domain $\\Omega$ is such that for any $f\\in {{ \\bf L}^2 (\\Omega)}$, the solution\n$(\\psi,\\theta) \\in {{ \\bf H}^1_0 (\\Omega)} \\times {L^2_{\\scriptscriptstyle\\!\\int\\!=0} (\\Omega)}$ to the Stokes problem\n\\begin{equation}\n\\label{eq:contstokes}\n \\begin{dcases}\n -\\LAP \\psi + \\GRAD \\theta = f, & \\text{in } \\Omega, \\\\\n \\DIV \\psi = 0, & \\text{in } \\Omega, \\\\\n \\psi = 0, & \\text{on } \\partial\\Omega,\n \\end{dcases}\n\\end{equation}\nsatisfies the following estimate:\n\\begin{equation}\n \\| \\psi \\|_{{\\bf H}^2} + \\| \\theta \\|_{H^1} \\leq c \\| f \\|_{{\\bf L}^2}.\n\\label{eq:Cattabriga}\n\\end{equation}\n\\end{assumption}\n\nAssumption~\\ref{assumption1} is known to hold in two and three dimensions ($d=2,3$) whenever\n$\\Omega$ is convex or of class ${\\mathcal C}^{1,1}$, see \\cite[Theorem 6.3]{MR977489}.\n\nBy suitably defining a discrete gradient operator acting on the pressure space, the proofs for \ndiscontinuous pressure spaces can be carried out with similar arguments.\n\nWe introduce the definition of a Fortin projection.\n\\begin{definition}\n\\label{def:Fortin}\nAn operator ${\\mathcal F}_h : {{ \\bf H}^1_0 (\\Omega)} \\rightarrow {\\bf X}_h$ is called a Fortin projection if ${\\mathcal F}_h^2 =\n{\\mathcal F}_h$ and\n\\begin{equation}\n \\int_\\Omega \\DIV(v-{\\mathcal F}_h v)q_h = 0, \\quad \\forall v \\in {{ \\bf H}^1_0 (\\Omega)}, \\quad \\forall q_h \\in M_h.\n\\label{eq:Fortin}\n\\end{equation}\n\\end{definition}\n\nWe shall be interested in Fortin projections ${\\mathcal F}_h$ that satisfy the condition:\n\\begin{equation}\n \\| {\\mathcal F}_h v \\|_{{\\bf H}^1} \\leq c \\| v \\|_{{\\bf H}^1}, \\quad \\forall v \\in {{ \\bf H}^1_0 (\\Omega)},\n\\tag{FH1}\n\\label{eq:fh1}\n\\end{equation}\nor\n\\begin{equation}\n \\| v - {\\mathcal F}_h v \\|_{{\\bf L}^2} \\leq c h \\| v\\|_{{\\bf H}^1}, \\quad \\forall v \\in {{ \\bf H}^1_0 (\\Omega)}.\n\\tag{FL2}\n\\label{eq:fl2}\n\\end{equation}\n\nLet us remark that the approximation property \\eqref{eq:fl2} implies ${\\bf H}^1$-stability.\n\\begin{lem}\n\\label{lem:fl2implfh1}\nIf an operator ${\\mathcal F}_h : {{ \\bf H}^1_0 (\\Omega)} \\rightarrow {\\bf X}_h$ satisfies \\eqref{eq:fl2} then it is ${\\bf H}^1$-stable, i.e., \\eqref{eq:fh1} is satisfied.\n\\end{lem}\n\\begin{proof}\nThe proof relies on the stability and approximation properties \\eqref{eq:SZprop} of the Scott-Zhang operator\nand on the inverse estimate \\eqref{eq:invX}, for if $v \\in {{ \\bf H}^1_0 (\\Omega)}$,\n\\begin{align*}\n \\| {\\mathcal F}_h v \\|_{{\\bf H}^1} &\\leq \\| {\\mathcal F}_h v - {\\mathcal C}_h v \\|_{{\\bf H}^1} + c \\| v \\|_{{\\bf H}^1}\n \\leq c h^{-1} \\| {\\mathcal F}_h v - {\\mathcal C}_h v \\|_{{\\bf L}^2} + c \\| v \\|_{{\\bf H}^1} \\\\\n &\\leq ch^{-1} \\| v - {\\mathcal F}_h v \\|_{{\\bf L}^2} + ch^{-1}\\| v - {\\mathcal C}_h v \\|_{{\\bf L}^2} + c \\|v\\|_{{\\bf H}^1}.\n\\end{align*}\nConclude using the ${\\bf L}^2$-approximation properties of the operators ${\\mathcal F}_h$ and ${\\mathcal C}_h$.\n\\end{proof}\n\n\\begin{rem}\nGirault and Scott, \\cite{MR1961943}, explicitly constructed a Fortin projection \nthat satisfies \\eqref{eq:fh1} and \\eqref{eq:fl2} for many\ncommonly used spaces. In fact, they showed that the approximation is local, i.e.,\\@\\xspace\n\\[\n \\| {\\mathcal F}_h v - v \\|_{{\\bf L}^2(T)}+ h_T\\| {\\mathcal F}_h v -v\\|_{{\\bf H}^1(T)} \\leq c h_T \\| v\n\\|_{{\\bf H}^1({\\mathcal N}(T))}, \\quad \\forall v \\in {{ \\bf H}^1_0 (\\Omega)}\n \\text{ and } \\forall T \\in {\\mathcal T}_h,\n\\]\nwhere ${\\mathcal N}(T)$ is a patch containing $T$.\nIn particular, they have shown the existence of this projection for the Taylor-Hood elements in two \ndimensions. In three dimensions they proved this result for all the Taylor-Hood elements except the \nlowest order case. \n\\end{rem}\n\nIn this work we shall prove the implications\n\\[\n \\xymatrix{\n \\eqref{eq:LBB} \\ar@{<=>}[r] \\ar@{<=}[d]\n &\\exists {\\mathcal F}_h \\mbox{\\, s.t.\\,} \\eqref{eq:Fortin} \\text{ and } \\eqref{eq:fh1} & \\\\\n \\eqref{eq:wLBB} \\ar@{<=>}[r]\n &\\exists {\\mathcal F}_h \\mbox{\\, s.t.\\,} \\eqref{eq:Fortin} \\text{ and } \\eqref{eq:fl2} & \\eqref{eq:LBB} \\text{ and \nAssumption } \\ref{assumption1} \\ar@{=>}[l] \n }\n\\]\nthus showing that, in our setting, all these conditions are indeed equivalent. \nThe top equivalence is\nwell-known, see \\cite{BF91,GR86,MR2050138}. The left implication is also known (see\n\\cite{MR2210084}), for completeness we show\nthis in Theorem~\\ref{thm:wlbbimpllbb}. The bottom implications, although simple to prove, seem to be\nnew. \n\n\\section{The Generalized LBB Condition}\n\\label{sec:wLBB}\nLet us begin by noticing that the generalized LBB condition \\eqref{eq:wLBB} is actually a statement\nabout coercivity of the ${\\bf L}^2$-projection on gradients of functions in the pressure space. Namely,\n\\eqref{eq:wLBB} is equivalent to\n\\begin{equation}\\label{GLBBb}\n \\| \\pi_h \\GRAD q_h \\|_{{\\bf L}^2} \\geq c \\| \\GRAD q_h \\|_{{\\bf L}^2}, \\quad \\forall q_h \\in M_h.\n\\end{equation}\n\nIt is well known that \\eqref{eq:wLBB} implies \\eqref{eq:LBB}. For completeness we present the proof.\nWe\nbegin with a perturbation result.\n\n\\begin{lem}\n\\label{lem:verfurth}\nThere exists a constant $c$ independent of $h$ such that, for all $q_h \\in M_h$, the following holds:\n\\[\n c \\| q_h \\|_{L^2} \\leq \n \\sup_{v_h \\in {\\bf X}_h} \\frac{ \\int_\\Omega (\\DIV v_h)\\, q_h }{\\| \\GRAD v_h \\|_{{\\bf L}^2} }\n + h \\| \\GRAD q_h \\|_{{\\bf L}^2}.\n\\]\n\\end{lem}\n\\begin{proof}\nThe proof relies on the properties \\eqref{eq:SZprop} of the Scott-Zhang interpolation operator ${\\mathcal C}_h$,\n\\begin{align*}\n c \\|q_h\\|_{L^2} &\\leq \\sup_{v \\in {{ \\bf H}^1_0 (\\Omega)}} \\frac{ \\int_\\Omega (\\DIV v)\\, \\, q_h }{\\| \\GRAD v\n\\|_{{\\bf L}^2} }\n \\leq \n \\sup_{ v \\in {{ \\bf H}^1_0 (\\Omega)}} \\frac{ \\int_\\Omega (\\DIV\\,{\\mathcal C}_h v)\\, q_h }{\\| \\GRAD ({\\mathcal C}_h v)\n\\|_{{\\bf L}^2} }\n +\n \\sup_{ v \\in {{ \\bf H}^1_0 (\\Omega)}}\\frac{\\int_\\Omega\\big(\\DIV\\left(v - {\\mathcal C}_h v \\right)\\big)q_h }{\\|\\GRAD\nv\\|_{{\\bf L}^2} }\n \\\\ &\\leq\n \\sup_{ v_h \\in {\\bf X}_h} \\frac{ \\int_\\Omega (\\DIV v_h) \\, q_h }{\\| \\GRAD v_h \\|_{{\\bf L}^2} }\n + \\sup_{ v \\in {{ \\bf H}^1_0 (\\Omega)}} \\frac{ \\int_\\Omega\\left(v - {\\mathcal C}_h v \\right){\\cdot}\\GRAD q_h}{\\|\\GRAD\nv\\|_{{\\bf L}^2}},\n\\end{align*}\nconclude using \\eqref{eq:SZprop}.\n\\end{proof}\nOn the basis of Lemma~\\ref{lem:verfurth} we can readily show that \\eqref{eq:wLBB} implies \\eqref{eq:LBB}. Again,\nthis result is not new and we only include the proof for completeness.\n\\begin{thm}\n\\label{thm:wlbbimpllbb}\n\\eqref{eq:wLBB} implies \\eqref{eq:LBB}.\n\\end{thm}\n\\begin{proof}\nSince we assumed that $M_h \\subset {L^2_{\\scriptscriptstyle\\!\\int\\!=0} (\\Omega)} \\cap {{ H}^{1}(\\Omega)}$, the proof is straightforward:\n\\[\n \\sup_{ v_h \\in {\\bf X}_h} \\frac{ \\int_\\Omega (\\DIV v_h)\\, q_h }{\\|\\GRAD v_h \\|_{{\\bf L}^2} }\n =\n \\sup_{ v_h \\in {\\bf X}_h} \\frac{ \\int_\\Omega v_h {\\cdot} \\GRAD q_h }{\\|\\GRAD v_h \\|_{{\\bf L}^2} }\n \\geq\n \\frac{ \\int_\\Omega \\pi_h \\GRAD q_h {\\cdot}\\GRAD q_h }{\\|\\GRAD \\pi_h \\GRAD q_h \\|_{{\\bf L}^2} }\n =\n \\frac{ \\|\\pi_h \\GRAD q_h \\|_{{\\bf L}^2}^2 }{ \\| \\GRAD \\pi_h \\GRAD q_h \\|_{{\\bf L}^2} }\n \\geq\n c h \\|\\pi_h \\GRAD q_h \\|_{{\\bf L}^2}\n\\]\nwhere, in the last step, we used the inverse inequality \\eqref{eq:invX}.\nThis, in conjunction with Lemma~\\ref{lem:verfurth} and the characterization \\eqref{GLBBb}, implies the result.\n\\end{proof}\n\nLet us now show that the generalized LBB condition \n\\eqref{eq:wLBB} is equivalent to the existence of a Fortin operator satisfying \\eqref{eq:fl2}.\nWe begin with a modification of a classical result.\n\n\\begin{lem}\n\\label{lem:tartarwithh}\nFor all $p\\in {{ H}^{1}(\\Omega)}$ there is $v\\in{{ \\bf H}^1_0 (\\Omega)}$ such that\n\\[\n \\DIV v = p - \\Pi_0 p, \\qquad v|_{\\partial T}=0 \\quad \\forall T\\in {\\mathcal T}_h,\n\\]\nand\n\\[\n \\| v \\|_{{\\bf L}^2} \\leq c \\left( \\sum_{T \\in {\\mathcal T}_h} h_T^4 \\| \\GRAD p \\|_{{\\bf L}^2(T)}^2 \\right)^{1\/2}.\n\\]\n\\end{lem}\n\\begin{proof}\nLet $p\\in {{ H}^{1}(\\Omega)}$ and $T \\in {\\mathcal T}_h$. Clearly,\n\\[\n \\int_T p - \\Pi_0 p = 0.\n\\]\nA classical result (\\cite{MR82m:26014,MR1846644,GR86,MR1880723}) implies that there is a \n$v_T \\in {\\bf H}^1_0(T)$ with\n$ \\DIV v_T = p - \\Pi_0 p$\nin $T$ and\n\\begin{equation}\n \\| \\GRAD v_T \\|_{{\\bf L}^2(T)} \\leq c \\| p - \\Pi_0 p \\|_{L^2(T)}.\n\\label{eq:saves}\n\\end{equation}\nGiven that the mesh is assumed to be shape regular, by mapping to the reference element it is seen\nthat\nthe constant in the last inequality does not depend on $T \\in {\\mathcal T}_h$.\n\nLet $v \\in {{ \\bf H}^1_0 (\\Omega)}$ be defined as $ v|_T = v_T$ for all $T$ in ${\\mathcal T}_h$. By construction,\n\\[\n \\DIV v = p - \\Pi_0 p, \\quad \\text{\\ae in } \\Omega.\n\\]\nMoreover,\n\\[\n \\| v \\|_{{\\bf L}^2}^2 = \\sum_{T \\in {\\mathcal T}_h} \\| v \\|_{{\\bf L}^2(T)}^2\n \\leq c \\sum_{T \\in {\\mathcal T}_h} h_T^2 \\| \\GRAD v \\|_{{\\bf L}^2(T)}^2\n \\leq c \\sum_{T \\in {\\mathcal T}_h} h_T^2 \\| p - \\Pi_0 p \\|_{L^2(T)}^2\n \\leq c \\sum_{T \\in {\\mathcal T}_h} h_T^4 \\| \\GRAD p \\|_{{\\bf L}^2(T)}^2.\n\\]\nThe first equality is by definition; then we applied the Poincar\\'e-Friedrichs inequality (since\n$v|_T = v_T \\in {\\bf H}^1_0(T)$); next we used the properties of the function $v_T$ and the\napproximation properties of the projector $\\Pi_0$.\n\\end{proof}\n\nWith this result at hand we can prove the following.\n\n\\begin{thm}\n\\label{thm:fl2implwlbb}\nIf there exists a Fortin operator ${\\mathcal F}_h$ that satisfies \\eqref{eq:fl2}, then\n\\eqref{eq:wLBB} holds.\n\\end{thm}\n\\begin{proof}\nLet $q_h \\in M_h$. Using the properties of the operator $\\Pi_0$ and \nthe local analogue of\nthe inverse inequality \\eqref{eq:invM}, we get\n\\[\n \\| \\GRAD q_h \\|_{{\\bf L}^2}^2 \n = \\sum_{T \\in {\\mathcal T}_h} \\left\\| \\GRAD \\left(q_h - \\Pi_0 q_h \\right) \\right\\|_{{\\bf L}^2(T)}^2\n \\leq \\sum_{T \\in {\\mathcal T}_h} \\frac1{h_T^2} \\| q_h - \\Pi_0 q_h \\|_{{\\bf L}^2(T)}^2\n \\leq \\frac{c}{h^2} \\| q_h - \\Pi_0 q_h \\|_{{\\bf L}^2}^2.\n\\]\nFrom Lemma~\\ref{lem:tartarwithh} we know there exists $v \\in {{ \\bf H}^1_0 (\\Omega)}$ with $\\DIV v = q_h - \\Pi_0 q_h$\nand\n\\[\n \\| v \\|_{{\\bf L}^2} \\leq c h^2 \\| \\GRAD q_h \\|_{{\\bf L}^2},\n\\]\nhence\n\\[\n \\| \\GRAD q_h \\|_{{\\bf L}^2}^2 \\leq \\frac{c}{h^2} \\| q_h - \\Pi_0 q_h \\|_{L^2}^2\n = \\frac{c}{h^2} \\int_\\Omega (\\DIV v) \\, (q_h-\\Pi_0 q_h)\n = \\frac{c}{h^2} \\int_\\Omega (\\DIV v) \\, q_h,\n\\]\nwhere the last inequality follows from integration by parts over each $T$ and using the fact that $v|_{\\partial T} = 0$ (see Lemma~\\ref{lem:tartarwithh}).\n\nUsing the existence of the operator ${\\mathcal F}_h$,\n\\[\n \\| \\GRAD q_h \\|_{{\\bf L}^2}^2 \\le \\frac{c}{h^2} \\int_\\Omega (\\DIV \\,{\\mathcal F}_h v) q_h\n \\leq \\left( \\sup_{ w_h \\in {\\bf X}_h} \\frac{ \\int_\\Omega (\\DIV w_h) \\, q_h }{ \\| w_h \\|_{{\\bf L}^2}\n}\\right) \n \\frac{c}{h^2}\\| {\\mathcal F}_h v \\|_{{\\bf L}^2}.\n\\]\nIt remains to show that \n\\[\n \\| {\\mathcal F}_h v \\|_{{\\bf L}^2} \\leq c h^2 \\| \\GRAD q_h \\|_{{\\bf L}^2}.\n\\]\nFor this purpose, we use the approximation property \\eqref{eq:fl2} and\nLemma~\\ref{lem:tartarwithh}\n\\[\n \\| {\\mathcal F}_h v \\|_{{\\bf L}^2} \\leq \\| {\\mathcal F}_h v - v\\|_{{\\bf L}^2} + \\|v \\|_{{\\bf L}^2}\n \\leq ch \\| \\GRAD v \\|_{{\\bf L}^2} + c h^2 \\|\\GRAD q_h \\|_{{\\bf L}^2}\n \\leq ch^2 \\| \\GRAD q_h \\|_{{\\bf L}^2},\n\\]\nwhere the last inequality holds because of \\eqref{eq:saves}.\n\\end{proof}\n\nThe converse of Theorem~\\ref{thm:fl2implwlbb} is given in the following.\n\n\\begin{thm}\n\\label{thm:wlbbimplfl2}\nIf \\eqref{eq:wLBB} holds, then there exists a Fortin projector ${\\mathcal F}_h$ that satisfies\n\\eqref{eq:fl2}.\n\\end{thm}\n\\begin{proof}\nLet $v\\in{{ \\bf H}^1_0 (\\Omega)}$. Define $(z_h, p_h) \\in {\\bf X}_h \\times M_h$ as the solution of\n\\begin{equation}\n \\begin{dcases}\n \\int_\\Omega z_h{\\cdot} w_h -\\int_\\Omega p_h \\DIV w_h = \\int_\\Omega v{\\cdot} w_h, & \\forall w_h \\in {\\bf X}_h, \\\\\n \\int_\\Omega q_h \\DIV z_h = \\int_\\Omega q_h \\DIV v, & \\forall q_h \\in M_h.\n \\end{dcases}\n\\label{eq:l2stokes}\n\\end{equation}\nNotice that \\eqref{eq:wLBB} provides precisely necessary and sufficient conditions for this problem\nto have a\nunique solution.\n\nDefine ${\\mathcal F}_h v := z_h$ we claim that this is indeed a Fortin projection that satisfies\n\\eqref{eq:fl2}.\nBy construction, \\eqref{eq:Fortin} holds (see the second equation in \\eqref{eq:l2stokes}). To show\nthat this is indeed a\nprojection, assume that $v=v_h \\in {\\bf X}_h$ in \\eqref{eq:l2stokes}, setting $w_h = z_h - v_h$ we\nreadily obtain that\n\\[\n \\| z_h - v_h \\|_{{\\bf L}^2}^2 =0.\n\\]\nIt remains to show the approximation properties of this operator. We begin by noticing that\n\\eqref{eq:wLBB} implies\n\\begin{equation}\n c \\| \\GRAD p_h \\|_{{\\bf L}^2} \\leq \\sup_{w_h \\in {\\bf X}_h} \\frac{ \\int_\\Omega p_h \\DIV w_h }{ \\| w_h\n\\|_{{\\bf L}^2}}\n \\leq \\sup_{w_h \\in {\\bf X}_h} \\frac{ \\int_\\Omega (v-{\\mathcal F}_h v){\\cdot} w_h }{ \\| w_h \\|_{{\\bf L}^2}}\n \\leq \\| v - {\\mathcal F}_h v \\|_{{\\bf L}^2},\n\\label{eq:boundgradp}\n\\end{equation}\nwhere we used \\eqref{eq:l2stokes}. To obtain the approximation property \\eqref{eq:fl2} we use the\nScott-Zhang interpolation operator ${\\mathcal C}_h$,\n\\begin{align*}\n \\| {\\mathcal F}_h v - v \\|_{{\\bf L}^2}^2 &= \\int_\\Omega ( {\\mathcal C}_h v - v ){\\cdot}( {\\mathcal F}_h v - v )\n + \\int_\\Omega ( {\\mathcal F}_h v - {\\mathcal C}_h v ){\\cdot}( {\\mathcal F}_h v - v ) \\\\\n & \\leq\n \\| {\\mathcal C}_h v - v \\|_{{\\bf L}^2}\\| {\\mathcal F}_h v - v \\|_{{\\bf L}^2}\n + \\int_\\Omega ( {\\mathcal F}_h v - {\\mathcal C}_h v ){\\cdot}( {\\mathcal F}_h v - v ).\n\\end{align*}\nWe bound the first term using the approximation property \\eqref{eq:SZprop} of ${\\mathcal C}_h$. To bound\nthe second\nterm we use problem \\eqref{eq:l2stokes} with $w_h = {\\mathcal F}_h v - {\\mathcal C}_h v$, then\n\\[\n \\int_\\Omega ( {\\mathcal F}_h v - {\\mathcal C}_h v ){\\cdot}( {\\mathcal F}_h v - v )\n = \\int_\\Omega p_h \\DIV( {\\mathcal F}_h v - {\\mathcal C}_h v )\n = \\int_\\Omega p_h \\DIV( v - {\\mathcal C}_h v )\n = -\\int_\\Omega \\GRAD p_h {\\cdot} ( v - {\\mathcal C}_h v ),\n\\]\nwe conclude applying the Cauchy-Schwarz inequality and using \\eqref{eq:boundgradp}.\n\\end{proof}\n\n\\section{Smooth Domains}\n\\label{sec:Equiv}\nHere we show that, provided \\eqref{eq:LBB} holds and, moreover,\nthe domain $\\Omega$ is such that Assumption~\\ref{assumption1} is satisfied, then \\eqref{eq:fl2}\nholds and hence \\eqref{eq:wLBB} holds as well. This is shown in the following.\n\n\\begin{thm}\n\\label{thm:lbbimplwlbb}\nAssume the domain $\\Omega$ is such that the solution to\n\\eqref{eq:contstokes} possesses ${\\bf H}^2$-elliptic regularity, i.e.,\\@\\xspace Assumption \\ref{assumption1}\nholds. Then \\eqref{eq:LBB} implies that there is a Fortin operator\n${\\mathcal F}_h$ that satisfies \\eqref{eq:fl2}.\n\\end{thm}\n\\begin{proof}\nLet $v\\in {{ \\bf H}^1_0 (\\Omega)}$. Define $(z_h,p_h) \\in {\\bf X}_h \\times M_h$ as the solution to the discrete Stokes\nproblem\n\\begin{equation}\n \\begin{dcases}\n \\int_\\Omega \\GRAD z_h {:} \\GRAD w_h -\\int_\\Omega p_h \\DIV w_h = \\int_\\Omega \\GRAD v {:} \\GRAD w_h, & \\forall w_h\n\\in {\\bf X}_h, \\\\\n \\int_\\Omega q_h \\DIV z_h = \\int_\\Omega q_h \\DIV v, & \\forall q_h \\in M_h,\n \\end{dcases}\n\\label{eq:stokes}\n\\end{equation}\nwhere, in \\eqref{eq:stokes}, the colon is used to denote the tensor product of matrices. Notice that \\eqref{eq:LBB} implies that\nthis problem always has a unique solution.\n\nSet ${\\mathcal F}_h v := z_h$. Proceeding as in the proof of Theorem~\\ref{thm:wlbbimplfl2} we see that this is \nindeed a projection. Moreover, \\eqref{eq:Fortin} holds by construction.\nIt remains to\nshow that \\eqref{eq:fl2} is satisfied. To this end, analogously to the proof of\nTheorem~\\ref{thm:wlbbimplfl2},\nwe notice that \\eqref{eq:LBB} implies\n\\[\n \\| p_h \\|_{L^2} \\leq c \\| \\GRAD ({\\mathcal F}_h v - v )\\|_{{\\bf L}^2}.\n\\]\nWe now argue by duality. Let $\\psi$ and $\\phi$ solve\n\\eqref{eq:contstokes} with $f={\\mathcal F}_h v- v$. Assumption \\eqref{eq:Cattabriga} then implies\n\\begin{align*}\n \\| {\\mathcal F}_h v - v \\|_{{\\bf L}^2}^2 &= \\int_\\Omega ({\\mathcal F}_h v - v){\\cdot}(-\\LAP\\psi + \\GRAD \\theta) \\\\\n &= \\int_\\Omega \\GRAD({\\mathcal F}_h v -v ):\\GRAD(\\psi-{\\mathcal C}_h \\psi) -\\int_\\Omega(\\theta - {\\mathcal I}_h\n\\theta)\\,\\DIV({\\mathcal F}_h v -v )\\, \\\\\n & \\phantom{=}+ \\int_\\Omega \\GRAD({\\mathcal F}_h v -v ):\\GRAD({\\mathcal C}_h \\psi) - \\int_\\Omega ({\\mathcal I}_h \\theta)\\,\\DIV({\\mathcal F}_h v -v )\\,\n\\end{align*}\nNotice that since ${\\mathcal I}_h \\theta\\in M_h$, $\\int_\\Omega ({\\mathcal I}_h \\theta)\\,\\DIV({\\mathcal F}_h v -v )=0$.\nSince $\\DIV \\psi = 0$,\nusing \\eqref{eq:stokes}, the estimate for $p_h$, \\eqref{eq:SZh1} and \\eqref{eq:Cattabriga},\n\\[\n \\int_\\Omega \\GRAD({\\mathcal F}_h v -v ):\\GRAD({\\mathcal C}_h \\psi) = \\int_\\Omega p_h \\DIV ({\\mathcal C}_h \\psi-\\psi)\n \\leq ch \\| v - {\\mathcal F}_h v \\|_{{\\bf H}^1} \\| v - {\\mathcal F}_h v \\|_{{\\bf L}^2}.\n\\]\nA direct application of of \\eqref{eq:SZh1}, \\eqref{eq:SZprop} and \\eqref{eq:Cattabriga} allows us to\nobtain the following estimates:\n\\[\n \\int_\\Omega (\\theta - {\\mathcal I}_h \\theta)\\,\\DIV({\\mathcal F}_h v -v ) \n + \\int_\\Omega \\GRAD({\\mathcal F}_h v -v ){:}\\GRAD(\\psi-{\\mathcal C}_h \\psi)\n \\leq\n c h \\| {\\mathcal F}_h v - v \\|_{{\\bf L}^2}\\| v \\|_{{\\bf H}^1}\n\\]\n\nWe conclude using a stability estimate for \\eqref{eq:stokes}\n\\[\n \\| {\\mathcal F}_h v - v \\|_{{\\bf L}^2} \\leq c h \\| {\\mathcal F}_h v - v \\|_{{\\bf H}^1} \\leq c h \\| v \\|_{{\\bf H}^1},\n\\]\nwhich, given \\eqref{eq:LBB}, is uniform in $h$.\n\\end{proof}\n\n\\section{The Weighted LBB condition}\n\\label{sec:section5}\n\nIn relation to the construction of uniform preconditioners for\ndiscretizations of the time dependent Stokes problem, Mardal, Sch\\\"oberl and Winther, \\cite{schoberlwinther}, \nconsider the following inf--sup condition,\n\\begin{equation}\n c\\| q_h \\|_{H^1+\\epsilon^{-1} L^2} \\leq \n \\sup_{v_h \\in {\\bf X}_h} \\frac{ \\int_\\Omega \\DIV v_h q_h}{\\|v_h\\|_{{\\bf L}^2 \\cap \\epsilon {\\bf H}^1}}, \n \\quad \\forall q_h \\in M_h.\n\\label{eq:winf-sup}\n\\end{equation}\nwhere\n\\[\n\\| q \\|_{H^1+\\epsilon^{-1} L^2}^2= \\inf_{q_1+q_2=q} \n \\left( \\|q_1\\|_{H^1}^2+ \\epsilon^{-2} \\|q_2\\|_{L^2}^2 \\right),\n\\]\nand\n\\[\n \\|v\\|_{{\\bf L}^2 \\cap \\epsilon {\\bf H}^1}^2 = \\|v\\|_{{\\bf L}^2}^2 + \\epsilon^2 \\|v\\|_{{\\bf H}^1}^2.\n\\]\n\nBy constructing a Fortin projection operator that is ${\\bf L}^2$-bounded they have showed, on quasi-uniform meshes, \nthat the inf--sup condition \\eqref{eq:winf-sup} holds for the lowest order Taylor-Hood element in two dimension.\nIn addition, they proved the same result, on shape regular meshes, for the mini-element.\nHere, we show that \\eqref{eq:winf-sup} holds if we assume \\eqref{eq:wLBB}. A simple consequence of this \nresult is that, on quasi-uniform meshes, \\eqref{eq:winf-sup} holds for any order Taylor-Hood elements\nin two and three dimensions.\n\n\\begin{thm}\n\\label{thm:wlbbimplweightlbb}\nLet $\\Omega$ be star shaped with respect to ball.\nIf the spaces ${\\bf X}_h$ and $M_h$ are such that \\eqref{eq:wLBB} is satisfied,\nthen the inf--sup condition \\eqref{eq:winf-sup} holds with a constant that \ndoes not depend on $\\epsilon$ or $h$.\n\\end{thm}\n\\begin{proof}\nWe consider two cases: $\\epsilon \\ge h$ and $\\epsilon < h$.\n\nGiven that the domain $\\Omega$ is star shaped with respect to a ball, \nwe can conclude (\\cite{schoberlwinther}) that the following \ncontinuous inf--sup condition holds,\n\\begin{equation}\\label{conweight}\nc \\| q \\|_{H^1+\\epsilon^{-1} L^2} \\leq \\sup_{v \\in {{ \\bf H}^1_0 (\\Omega)}}\n \\frac{ \\int_\\Omega q\\, \\DIV v }{ \\|v\\|_{{\\bf L}^2 \\cap \\epsilon {\\bf H}^1}},\n \\quad \\forall q \\in {L^2_{\\scriptscriptstyle\\!\\int\\!=0} (\\Omega)},\n\\end{equation}\nwith a constant $c$ independent of $\\epsilon$.\n\n\nWe first assume that $\\epsilon \\ge h$. Using \\eqref{conweight} for $q_h \\in M_h$ we have, \n\\begin{align*}\n c \\| q_h \\|_{H^1+\\epsilon^{-1} L^2} &\\leq\n \\sup_{v \\in {{ \\bf H}^1_0 (\\Omega)} } \\frac{ \\int q_h\\,\\DIV v }{ \\| v \\|_{{\\bf L}^2 \\cap \\epsilon {\\bf H}^1 } } =\n \\sup_{v \\in {{ \\bf H}^1_0 (\\Omega)} } \\frac{ \\int_\\Omega q_h\\,\\DIV ({\\mathcal F}_h v) }{\\|{\\mathcal F}_h v\\|_{{\\bf L}^2 \\cap \\epsilon {\\bf H}^1}}\n \\frac{\\|{\\mathcal F}_h v\\|_{{\\bf L}^2 \\cap \\epsilon {\\bf H}^1}}{\\|v\\|_{{\\bf L}^2\\cap \\epsilon {\\bf H}^1} } \\\\\n &\\leq \\sup_{v_h \\in {\\bf X}_h} \\frac{ \\int_\\Omega q_h\\,\\DIV v_h }\n {\\|v_h\\|_{{\\bf L}^2 \\cap \\epsilon {\\bf H}^1} }\n \\sup_{ v \\in {{ \\bf H}^1_0 (\\Omega)}} \\frac{\\|{\\mathcal F}_h v\\|_{{\\bf L}^2 \\cap \\epsilon {\\bf H}^1}}{\\|v\\|_{{\\bf L}^2 \\cap \\epsilon {\\bf H}^1}},\n\\end{align*}\nwhere we used that, since \\eqref{eq:wLBB} holds, Theorem~\\ref{thm:wlbbimplfl2} shows that \nthere exists a Fortin operator ${\\mathcal F}_h$ that satisfies\n\\eqref{eq:Fortin}. By Lemma~\\ref{lem:fl2implfh1} and\nthe approximation properties \\eqref{eq:fl2} of the Fortin operator,\n\\begin{align*}\n \\|{\\mathcal F}_h v \\|_{{\\bf L}^2 \\cap \\epsilon {\\bf H}^1} &\\leq\n c \\left( \\|{\\mathcal F}_h v \\|_{{\\bf L}^2} + \\epsilon \\| {\\mathcal F}_h v \\|_{{\\bf H}^1} \\right) \\leq\n c \\left( \\| v \\|_{{\\bf L}^2} + \\|v - {\\mathcal F}_h v \\|_{{\\bf L}^2} + \\epsilon \\| v \\|_{{\\bf H}^1} \\right) \\\\\n &\\leq c \\left( \\|v\\|_{{\\bf L}^2} + (\\epsilon + h)\\| v \\|_{{\\bf H}^1} \\right)\n \\leq c \\left( \\|v\\|_{{\\bf L}^2} + 2\\epsilon\\| v \\|_{{\\bf H}^1} \\right)\n \\leq c\\| v \\|_{{\\bf L}^2 \\cap \\epsilon {\\bf H}^1},\n\\end{align*}\nwhere we used that $h \\leq \\epsilon$.\n\nOn the other hand, if $\\epsilon < h$ we use $q_1 = q_h$ and $q_2 = 0$ in the definition of the weighted norm for the pressure space.\nCondition \\eqref{eq:wLBB} then implies\n\\[\n \\| q_h \\|_{H^1+\\epsilon^{-1} L^2} \\leq\n c \\| \\GRAD q_h \\|_{{\\bf L}^2} \\leq c \\sup_{v_h \\in {\\bf X}_h} \\frac{ \\int_\\Omega q_h\\,\\DIV v_h }{ \\| v_h \\|_{{\\bf L}^2} }\n \\leq c \\sup_{v_h \\in {\\bf X}_h} \\frac{ \\int_\\Omega q_h\\, \\DIV v_h }{ \\| v_h \\|_{{\\bf L}^2 \\cap \\epsilon {\\bf H}^1} }\n \\sup_{v_h \\in {\\bf X}_h} \\frac{ \\| v_h \\|_{{\\bf L}^2 \\cap \\epsilon {\\bf H}^1} }{ \\| v_h \\|_{{\\bf L}^2} }.\n\\]\nBy the inverse inequality \\eqref{eq:invX},\n\\[\n \\frac{ \\| v_h \\|_{{\\bf L}^2 \\cap \\epsilon {\\bf H}^1} }{ \\| v_h \\|_{{\\bf L}^2} } \\leq\n c\\left( 1 + \\epsilon h^{-1} \\right).\n\\]\nConclude using that $\\epsilon < h $.\n\\end{proof}\n\n\\section{Concluding Remarks}\n\\label{sec:conclusion}\nThere seems to be one main drawback to our methods of proof. Namely, all our results rely heavily on\nthe fact that we have a quasi-uniform mesh. However, at the present moment we do not know whether this\ncondition can be removed. Finally, it will be interesting to see if \\eqref{eq:LBB} is in fact equivalent to\n\\eqref{eq:wLBB} on domains that do not satisfy the regularity assumption \n\\eqref{eq:Cattabriga} (e.g.\\@\\xspace non convex polyhedral domains).\n\nOn the other hand, it seems to us that condition \\eqref{eq:wLBB} must be regarded as the most important one.\nOur results show that, under the sole assumption that the mesh is quasi-uniform, this condition implies\nthe classical condition \\eqref{eq:LBB} (Theorem~\\ref{thm:wlbbimpllbb}). Moreover, as shown in \nTheorem~\\ref{thm:wlbbimplweightlbb}, this condition implies the weighted inf--sup condition \\eqref{eq:winf-sup} on quasi-uniform meshes.\n\n\\bibliographystyle{plain}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}