diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzznmvd" "b/data_all_eng_slimpj/shuffled/split2/finalzznmvd" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzznmvd" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nWilson loops are an important class of observables in gauge and string theory that, in particular, \n help clarifying the interpolation between the weak and strong coupling regimes from the AdS\/CFT perspective.\nIn several supersymmetric gauge theories it is possible to compute expectation values of \nBPS Wilson loops using \nlocalization in terms of matrix model integrals\n(see, e.g., \\cite{Pestun:2016zxk}). \n\nIn the maximally supersymmetric $\\mathcal N=4$ $SU(N)$ gauge theory \nthe corresponding matrix model can be solved for any gauge coupling and gauge group rank\n\\cite{Drukker:2000rr}.\\footnote{The same is true also for the $\\mathcal N=4$ theory with $SO(N)$ and $USp(N)$ gauge groups \n \\cite{Fiol:2014fla,Giombi:2020kvo}.}\n This allows, in particular, to study the strong coupling limit of the coefficients in the $1\/N$ expansion, \n providing a possibility to compare to the large tension limit of the coefficients in the expansion in powers of string coupling (genus) \n on the dual ${\\rm AdS}_5 \\times S^5~$ string theory side, and thus leading to highly non-trivial checks of AdS\/CFT duality \\cite{Giombi:2020mhz,Beccaria:2020ykg,Beccaria:2021alk}.\n \n\nLocalization method applies also \nto a large class of gauge theories with reduced $\\mathcal N=2$ supersymmetry, \n but the associated matrix models have \nnon-polynomial potentials and are not directly solvable. \nWhile developing the small $\\lambda$ expansion is straightforward, \n extracting the strong coupling limit of the gauge theory observables is a non-trivial problem. \nAt leading order in the large $N$ expansion \nthis requires a Wiener-Hopf analysis of the matrix model as first exploited in \\cite{Passerini:2011fe} for $\\mathcal N=2$ $SU(N)$ SYM \nwith $N_{F}=2N$ fundamental hypermultiplets, and later generalized to other superconformal Lagrangian models admitting a large $N$ limit \n\\cite{Russo:2013kea,Zarembo:2014ooa,Baggio:2014sna,Baggio:2015vxa,Baggio:2016skg,Fiol:2015mrp,Zarembo:2016bbk}. \nThe methods used at leading planar level are not, however, applicable to the analysis of the higher \n$1\/N$ corrections. \n\nAn interesting class of models where $1\/N$ corrections happen to \n be more tractable is that of $\\mathcal N=2$ superconformal gauge theories with $SU(N)$ gauge group \nwhose leading large $N$ limit is equivalent (in a particular ``common'' sector) to that of the $\\mathcal N=4$ SYM theory. \nOne such example is the $SU(N)\\times SU(N)$ quiver gauge theory \nwith bi-fundamental hypermultiplets and equal gauge couplings. \nThis $\\mathcal N=2$ theory may be interpreted as a $\\mathbb Z_{2}$ orbifold of the $\\mathcal N=4$ $SU(2N)$ SYM and is \ndual to superstring theory on AdS$_{5}\\times (S^{5}\/\\mathbb Z_{2})$ \\cite{Kachru:1998ys}. \n\n\n\nLet us first review some basic results about the expectation value $\\vev{\\mathcal W} $ of \n circular $\\frac{1}{2}$-BPS Wilson loop in $\\mathcal N=4$ SYM theory. \nIts planar limit is given by \nis \\cite{Erickson:2000af,Drukker:2000rr,Pestun:2007rz}\n\\ba\n\\label{1.1}\n\\vev{\\mathcal W}_0 = \n\\frac{2\\,N}{\\sqrt\\lambda}\\,I_{1}(\\sqrt\\lambda) = \\sqrt\\frac{2}{\\pi}\\,N\\,\\lambda^{-3\/4}\\,e^{\\sqrt\\lambda}\n\\Big[1+\\mathcal O\\Big(\\frac{1}{\\sqrt\\lambda}\\Big)\\Big] \\ , \\qquad \\qquad \\lambda = g^{2}_{_{\\rm YM}}\\,N \\ . \n\\ea\nThe relative weight of the leading $1\/N$ correction to $\\vev{\\mathcal W}$ with respect to the planar result may be \nrepresented as \n\\begin{equation}\n\\label{1.2}\n\\frac{\\vev{\\mathcal W}}{\\vev{\\mathcal W}_0} = 1+\\frac{1}{N^{2}}\\,q(\\lambda)+\\mathcal O\\Big(\\frac{1}{N^{4}}\\Big) \\ , \n\\end{equation}\nwhere the function $q(\\lambda)$ (and its analogs in the $\\mathcal N=2$ models with planar equivalence to $\\mathcal N=4$ SYM) \nwill be our main interest below. Starting with the general (Laguerre polynomial) expression for $\\vev{\\mathcal W} $ in $\\mathcal N=4$ $SU(N)$ SYM \\cite{Drukker:2000rr}\none finds for the leading terms in $q(\\lambda)$ at weak and strong coupling\\footnote{In what follows the label ``$\\mathcal N=4$'' will always refer to the $\\mathcal N=4$ $SU(N)$ SYM expression. } \n\\begin{equation}\n\\label{1.3}\n\\qquad q^{\\mathcal N=4}(\\lambda) = \\frac{\\lambda}{96}\\Big[\\frac{\\sqrt\\lambda\\,I_{2}(\\sqrt\\lambda)}{I_{1}(\\sqrt\\lambda)}-12\\Big] = \\begin{cases}\n-\\frac{1}{8}\\lambda+\\frac{1}{384}\\lambda^{2} -\\frac{1}{9216}\\,\\lambda^{3}+\\mathcal O(\\lambda^{4}), &\\qquad \\lambda \\to 0, \\\\\n\\frac{1}{96}\\lambda^{3\/2}-\\frac{9}{64}\\lambda+\\frac{1}{256}\\lambda^{1\/2}+\\mathcal O(1), &\\qquad \\lambda \\to \\infty \\ . \n\\end{cases}\n\\end{equation}\nThe $\\lambda^{3\/2}$ scaling of $q^{\\mathcal N=4}$ at $\\lambda\\gg 1$ has a string interpretation. The \n string coupling and tension for the dual string theory on AdS$_{5}\\times S^{5}$ are defined as \n\\begin{equation}\n\\label{1.4}\ng_{\\text{s}} = \\frac{\\lambda}{4\\pi N},\\qquad \\qquad T = \\frac{{L}^2}{2\\pi \\a'}=\\frac{\\sqrt\\lambda}{2\\pi}.\n\\end{equation}\nAs was argued in \\cite{Giombi:2020mhz}, the leading large $T$ dependence of the string theory \nexpectation value for $\\vev{\\mathcal W} $ at each order in $g_{\\text{s}}$ is also \n controlled by the Euler number, $\\chi=1-2p$, (or genus $p$) \n of the string world sheet, \\textit{i.e.} \n\\begin{equation}\\label{1.5}\n\\vev{\\mathcal W} =\\sum_{p=0}^{\\infty} \\vev{\\mathcal W}_p = e^{2\\pi\\,T}\\sum_{p=0}^{\\infty}c_{p}\\,\\Big(\\frac{g_{\\text{s}}}{\\sqrt T}\\Big)^{2p-1}\\Big[1+\\mathcal O\\Big(T^{-1}\\Big)\\Big] \\ .\n\\end{equation}\nWritten in terms of $N$ and $\\lambda$ in \\rf{1.4} this reads ($c_p'= { c_p \\over ( 8 \\pi)^{p-1\/2}}$)\n \\begin{equation}\n\\label{1.6}\n\\vev{\\mathcal W}^{} = N\\,e^{\\sqrt\\lambda}\\sum_{p=0}^{\\infty}\n c'_p \\frac{\\lambda^{\\frac{6p-3}{4}}}{N^{2p}}\\Big[1+\\mathcal O\\Big(\\frac{1}{\\sqrt\\lambda}\\Big)\\Big] \\ , \n\\end{equation}\nand thus \n matches the structure of the $1\/N$ expansion of the exact $\\mathcal N=4$ SYM result \\cite{Drukker:2000rr}. \n In particular, comparing to \\rf{1.2}, \n we have, in agreement with (\\ref{1.3}), \n\\begin{equation}\n\\label{1.7}\n\\frac{1}{N^{2}}\\,q^{\\mathcal N=4}(\\lambda)\\ \\stackrel{\\lambda\\gg 1}{\\sim}\\ \\frac{g_{\\text{s}}^{2}}{T}\\,\\ \\propto\\,\\ \\frac{\\lambda^{3\/2}}{N^{2}} \\ . \n\\end{equation}\nThe discussion in \\cite{Giombi:2020mhz} leading to \\rf{1.5} \n relied only on the fact that one expands near the AdS$_2$ minimal surface \nembedded into the AdS$_3$ part of AdS$_5$ space; thus it should apply not only to ${\\rm AdS}_5 \\times S^5~$ superstring but also \nto its closely related orbifold and orientifold modifications \n based on AdS$_5 \\times S'^5$ where $S'^5$ is locally a 5-sphere. \n Indeed, since the fluctuations of string world sheet fields \n related to $S'^5$ remain ``massless'', the \n reasoning \\cite{Giombi:2020mhz} determining the tension dependence from the \n way how the AdS$_5$ radius appears in the 1-loop (leading large $T$) string partition function should not change. \n \n In \\cite{Beccaria:2021ksw} it was argued that this should apply, in particular, \n to the orbifold ${\\rm AdS}_5\\times (S^{5}\/\\mathbb Z_{2})$ theory and evidence for the validity \n of \\rf{1.5},\\rf{1.6} was provided at the first non-trivial $1\/N^2$ order. \nTo recall, \nin the $SU(N)\\times SU(N)$ orbifold model, \nfor each of the two $SU(N)$ factors, it is possible to define the $\\frac{1}{2}$-BPS circular Wilson loops \ncoupled to the associated gauge and scalar fields and $\\vev{\\mathcal W_{1}} = \\vev{\\mathcal W_{2}}\\equiv\\vev{\\mathcal W}^{\\rm orb}$. \nAt the leading planar level one has \n$\\vev{\\mathcal W}^{\\rm orb}_{N\\to\\infty} = \\vev{\\mathcal W}^{\\mathcal N=4}_{N\\to\\infty} = \\vev{\\mathcal W}_0$\n \\cite{Rey:2010ry,Zarembo:2020tpf}.\\footnote{The strong coupling limit of the planar expectation value \n of a similar \n Wilson loop in quiver gauge theory with unequal \n gauge couplings was solved in \\cite{Zarembo:2020tpf}, see also \\cite{Mitev:2014yba,Mitev:2015oty,Ouyang:2020hwd}.}\nStarting from the corresponding localization matrix model representation for $\\vev{\\mathcal W}^{\\rm orb}$, \na numerical analysis of the $q^{\\rm orb}(\\lambda)$ function, defined as in (\\ref{1.3}), \nextrapolated to large $\\lambda$ values \n gave the following estimate \\cite{Beccaria:2021ksw} \n\\ba\\label{18}\nq^{\\rm orb}(\\lambda) \\ \\stackrel{\\lambda\\gg 1}{=} C \\, \\lambda^{\\eta}, \\qquad \n\\eta = 1.49(2), \\ \\qquad C\\simeq - 0.0049(5)\\ .\n\\ea\nThe value of the asymptotic exponent $\\eta$ is thus quite consistent with the string theory expectation 3\/2 in (\\ref{1.7}).\n\n\\\n\n\nIn this paper we shall consider another $\\mathcal N=2$ superconformal model where the structure of the large $N$, strong coupling expansion \n should be of the same universal form as in \\rf{1.5}. We shall confirm this\n expectation with\n\n an analytic argument for the strong-coupling scaling in \\rf{1.7}, in addition to \n numerical evidence based on high-precision extrapolation from the weak to strong coupling regimes.\n \n This theory \nis the $SU(N)$ gauge theory with $\\mathcal N=2$ vector multiplet coupled to two hypermultiplets -- \nin rank-2 symmetric and antisymmetric $SU(N)$ representations.\\footnote{It \n is one of the five cases of 4d $\\mathcal N=2$ superconformal theories with gauge group $SU(N)$ \n defined for an arbitrary value of $N$ \\cite{Howe:1983wj,Koh:1983ir}. \n} \nThis model admits a regular 't Hooft large $N$ expansion and \n its string theory dual \nis expected to be a particular orientifold of AdS$_{5}\\times S^{5} $\n type IIB superstring theory \\cite{Park:1999ep,Ennes:2000fu}.\nFor that reason in what follows we shall refer to this $\\mathcal N=2$ gauge theory as the ``orientifold theory''.\n\nTo recall, in $\\mathcal N=2$ gauge theories the $\\beta$-function for the gauge coupling has only the 1-loop contribution: \nin a model with $N_{F}$ hypermultiplets in the fundamental, $N_{S}$ in rank-2 symmetric, and $N_{A}$ in rank-2 antisymmetric representations one has \n$\\beta_{\\rm 1-loop} = 2N-N_{F}-N_{S}(N+2)-N_{A}(N-2)$. It thus vanishes for the \n orientifold theory where $N_F=0, \\ N_A=N_S=1$. \nLet us also mention for completeness that the 4d Weyl anomaly coefficients a and c \nfor an $\\mathcal N=2$ theory with $n_{V}$ vector and $n_{H}$ hyper multiplets\n are given by ${\\rm a} = \\tfrac{5}{24}\\,n_{V}+\\tfrac{1}{24}\\,n_{H}$,\\ ${\\rm c} = \\tfrac{1}{6}\\,n_{V}+\\tfrac{1}{12}\\,n_{H}$,\n\nso that in the present case with \n$n_{V}=N^{2}-1$ and $n_{H} = \\frac{1}{2}N(N+1)+\\frac{1}{2}N(N-1) = N^{2}$, we get \n${\\rm a} = \\tfrac{1}{4}N^{2}-\\tfrac{5}{24}$, \\ ${\\rm c} = \\tfrac{1}{4}N^{2}-\\tfrac{1}{6}$.\nThus a and c are equal at the leading $N^{2}$ order which is consistent\nwith the existence of a well defined holographic dual.\\footnote{Let us also note that it should be possible \nto reproduce the subleading terms in a and c on the dual orientifold string theory side by summing up \nthe 1-loop contributions of the ``massless'' $D=10$ supergravity fields \n (corresponding to short multiplets represented by towers of Kaluza-Klein modes) similarly to how that was done in the \n case of the $\\mathcal N=4$ $SU(N)$ SYM theory (where a=c$={1\\over 4} N^2- {1\\over 4}$) \\cite{Beccaria:2014xda} and \n some orbifold theories \\cite{Ardehali:2013xya}.}\n\n\n Our aim will be to consider the \n expectation value of the $\\frac{1}{2}$-BPS circular Wilson loop in the orientifold theory. \n At the leading large $N$ order it is the same as in the $\\mathcal N=4$ SYM theory in \\rf{1.1}.\\footnote{This is a \n manifestation of the planar equivalence between the orientifold theory and the $\\mathcal N=4$ SYM in the ``untwisted'' sector.\n For a detailed discussion of planar equivalence violations in ``odd'' sectors see \\cite{Beccaria:2020hgy}.}\n The main focus will be on the \n leading non-planar correction represented by the function $q^{\\rm orient}(\\lambda)$ defined as in (\\ref{1.2}).\n\n\n\n\nThe string dual of this $\\mathcal N=2$ model \nis the type IIB superstring theory defined on the orientifold \n AdS$_{5}\\times S^{5}\/G_{\\rm orient} $ \\cite{Ennes:2000fu}.\n\n Here $G_{\\rm orient} =\\mathbb Z_{2}^{\\rm orb} \\times \\mathbb Z_{2}^{\\rm orient}$, \n where $Z_{2}^{\\rm orient}$ in addition to the target space coordinate \n inversions (in directions transverse to the original D3-branes) \n involves the product \n world-sheet parity operator $\\Omega$ and $(-1)^{F_L}$. \n The compact part of the 10d space $S'^5= S^{5}\/G_{\\rm orient}$ is different from $S^5$ only by special identifications of the angular coordinates\n \\cite{Ennes:2000fu}:\n \n \\noindent\n \\ $ds'^2_5= d \\theta_1^2 +\\sin^2\\theta_1\\, d \\phi_3^2 + \\cos^2\\theta_1\\, ds'^2_3 , \\ \\ \\ \\ \n ds'^2_3 = d \\theta_2^2 +\\sin^2\\theta_2\\, d \\phi_2^2 + \\cos^2\\theta_2\\, d\\phi_1^2$, \\ \\\n \n \\noindent\n $\\theta_{1}\\equiv \\theta_1 + {\\pi \\over 2}, \\ \\theta_{2}\\equiv \\theta_2 + {\\pi \\over 2},\\ \\\n \\phi_1 \\equiv \\phi_1 + {\\pi \\over 2},\\ \\ \\phi_2 \\equiv \\phi_2 - {\\pi \\over 2},\\ \\\n \\phi_3 \\equiv \\phi_3 + {\\pi}$. \n \n \nThe dual string theory description of the circular Wilson loop is based again on the string partition function \n expanded near the AdS$_2$ minimal surface embedded in AdS$_5$. \n As the UV divergent part of the 1-loop fluctuation determinants \\cite{Drukker:2000ep} near this minimal surface\n should not be sensitive to the global identifications in the $S'^5$ part of the orientifold geometry, the argument in \n \\cite{Giombi:2020mhz} leading to the universal structure of strong-coupling expansion \n \\rf{1.5},\\rf{1.6} should apply not only to the original ${\\rm AdS}_5 \\times S^5~$ or orbifold theory considered in \\cite{Beccaria:2021ksw} \n but also to this orientifold theory as well. \n \n \n Below we shall provide evidence for this, i.e. for the validity of \\rf{1.5},\\rf{1.6}, \n on the dual orientifold gauge theory side \n by showing that the localization matrix model representation for the circular BPS Wilson loop \n implies that the $1\/N^2$ term in \\rf{1.2} indeed \n scales as in (\\ref{1.7}) at the leading order at strong coupling, \\textit{i.e.} \n \\begin{equation}\n \\label{1.9}\n q^{\\rm orient}(\\lambda) \\stackrel{\\lambda\\gg 1}{\\sim} \\lambda^{3\/2} \\ .\n \\end{equation}\n\n\\\n\nLet us briefly summarize our main results. The aim will be to present a detailed study of \nthe $1\/N^2$ coefficient in (\\ref{1.2}) in the orientifold theory, i.e. of $q^{\\rm orient}(\\lambda)$.\nAs in the orbifold theory discussed in \\cite{Beccaria:2021ksw}, from the \nmatrix model representation for the Wilson loop in the orientifold theory one can relate the difference \nbetween $ q^{\\rm orient}(\\lambda) $ and $ q^{\\mathcal N=4}(\\lambda)$\n\\begin{equation}\\label{1.10}\n\\Delta q(\\lambda) \\equiv q^{\\rm orient}(\\lambda) - q^{\\mathcal N=4}(\\lambda) \\ , \n\\end{equation}\nto the $N \\to \\infty$ limit of the difference of the corresponding free energies\\footnote{Note that while the individual free energies on $S^4$ are, in general, scheme-dependent, their difference $\\Delta F$ is scheme-independent.\nEarlier discussion of leading terms in perturbative expansion \nin Wilson loop and free energy in this theory was in \\cite{Fiol:2020bhf}.\n} \n\\ba\n\\label{1.11}\n\\Delta q(\\lambda) &= - \\frac{\\lambda^{2}}{4}\\frac{d}{d\\lambda}\\,\\Delta F(\\lambda)\\ , \n\\qquad \\qquad \\Delta F(\\lambda) \\equiv \\lim_{N\\to \\infty } \\Delta F(\\lambda; N )\\ , \\\\\n\\label{1.12}\n\\Delta F(\\lambda; N ) &\\equiv F^{\\rm orient}(\\lambda; N)-F^{\\mathcal N=4}(\\lambda; N) = -\\log\\frac{Z^{\\rm orient}(\\lambda; N)}{Z^{\\mathcal N=4}(\\lambda; N)} \\ . \n\\ea\nHere $Z^{\\rm orient}$ and $Z^{\\mathcal N=4}$ are the corresponding partition functions on $S^{4}$. \nThe function $\\Delta F(\\lambda)$ turns out to have the following weak coupling expansion \n\\ba\n \\Delta F(\\lambda) = \n&5 \\zeta _5 \\hat{\\lambda }^3-\\tfrac{105}{2} \\zeta _7 \\hat{\\lambda \n}^4+441 \\zeta _9 \\hat{\\lambda }^5 -(25 \\zeta _5^2+3465 \\zeta _{11}) \\hat{\\lambda }^6 \\nonumber \\\\ \n&\\ \\ \n+\\big(525 \\zeta _5 \\zeta _7+\\tfrac{3.6355}{8} \\zeta _{13}\\big) \n\\hat{\\lambda }^7 \n+\\cdots, \\qquad \\qquad \\hat\\lambda=\\frac{\\lambda}{8\\pi^{2}} \\ . \\label{1.13}\n\\ea\nExtracting the strong coupling expansion is much harder. Since in the $\\mathcal N=4$ SYM theory the matrix model representation \nimplies that $F^{\\mathcal N=4}= - \\frac{1}{2}\\,(N^2-1) \\log { \\lambda} $ \\cite{Russo:2012ay}, \ncombining \\rf{1.11},\\rf{1.12} and \\rf{1.9} we get, as in the orbifold theory case \\cite{Beccaria:2021ksw}, \n the following prediction for $F^{\\rm orient}$ \n\\begin{equation}\n\\label{1.14}\nF^{\\rm orient}(\\lambda; N) \\stackrel{\\lambda\\gg 1}{=} - \\frac{1}{2}\\,N^2 \\log { \\lambda} + \\big[ c_1\\, \\sqrt \\lambda + \\mathcal O(\\log \\lambda) \\big] \n+ \\mathcal O \\Big({1\\over N^2}\\Big) \\ . \n\\end{equation}\nThe leading $\\mathcal O(N^2)$ term in (\\ref{1.14}) is implied by the planar equivalence to \n the $SU(N)$ SYM theory and should follow from the leading type IIB supergravity term evaluated on \n ${\\rm AdS}_5\\times (S^{5}\/G_{\\rm orient})$.\\footnote{Planar equivalence also implies that like in the $\\mathcal N=4$ SYM theory \n this leading $N^2$ term should not get string $\\frac{1}{\\sqrt\\lambda}$ corrections: they should still vanish on\n ${\\rm AdS}_5\\times (S^{5}\/G_{\\rm orient})$.}\n \n\nBelow we will analytically derive the $ \\sqrt \\lambda $ term in \\rf{1.14} and thus in $\\Delta F$ finding that \n\\begin{equation}\\label{1.15}\n\\Delta F(\\lambda) \\stackrel{\\lambda\\gg 1}{=} \\frac{\\sqrt\\lambda}{2\\pi} \\quad\\to\\quad \n\\Delta q(\\lambda) \\stackrel{\\lambda\\gg 1}{=} -\\frac{\\lambda^{3\/2}}{16\\pi}+\\cdots\\ , \n\\end{equation}\nwhere we used (\\ref{1.11}).\nWe will also confirm this result by a high-precision resummation and extrapolation analysis of the weak-coupling expansion \nby numerical methods including a conformal-mapping \nimproved Pad\\'e analysis. \n\n\n Let us note that while the coefficient in strong-coupling limit of $q$ in ${\\cal N}=4 $ SYM case is positive, \n $q^{{\\cal N}=4}= { 1 \\over 96} \\lambda^{3\/2} +...$ (see \\rf{1.3}), it was found \n \\cite{Beccaria:2021ksw} to be negative in the ${\\cal N}=2$ orbifold theory \\rf{18}. The result in \\rf{1.15} implies that it is also negative in the orientifold theory, \n $q^{\\rm orient}= q^{{\\cal N}=4} + \\Delta q = ({ 1 \\over 96} - {1\\over 16\\pi} ) \\lambda^{3\/2} +...\\approx -0.00948 \\lambda^{3\/2} +... $.\n It would be interesting to understand the reason for this \n sign change on the dual string theory side where $q$ should be expressed in terms of the string partition function on the disc with one handle.\n \n \n\n\\\n\nThe rest of the paper is organised as follows. \nIn section 2 we shall describe the localization matrix model representation for the expectation value of the $\\frac{1}{2}$-BPS \n Wilson loop in the orientifold gauge theory. We shall then present the derivation of the relation \\rf{1.11} between the coefficient $\\Delta q$ \n of the $1\/N^2$ term \n in the ratio of the orientifold and ${\\cal N}=4$ SYM Wilson loops and the large $N$ limit $\\Delta F(\\lambda)$ of the difference of the corresponding \n free energies. This reduces the problem of determining the strong coupling limit of $\\Delta q$ to finding that of $\\Delta F$. \n \n In section 3 we shall first study $\\Delta F(\\lambda)$ at weak coupling and then \n find its explicit representation \\rf{3.22}, i.e. $ \\Delta F ={1 \\over 2} \\log \\det (1 + M) = -\\sum_{n=1}^\\infty {1\\over n} (-1)^n {\\rm tr\\,} M^n$, in terms of an infinite-dimensional matrix $M$ \\rf{3.23}. \n Each ${\\rm tr\\,} M^n$ term in $\\Delta F$ turns out to be of fixed order $n$ in products of $\\zeta$-function values \n when written in the weak-coupling expansion. In section 4 we shall study the first two ${\\rm tr\\,} M$ and ${\\rm tr\\,} M^2$ terms finding that \n at large $\\lambda$ one has ${\\rm tr\\,} M^n \\sim \\lambda^n$. \n \n The derivation of the strong coupling asymptotics \\rf{1.15} of the total $\\Delta F$ implying $\\Delta q \\sim \\lambda^{3\/2}$ \n is given in section 5. \n In section 6 we shall independently test this $\\Delta F \\sim \\lambda^{1\/2} $ scaling by two different numerical methods. \n\n Some technical details are delegated to appendices. \n \n \n\n\nThe methods used here may be \napplicable to other similar $\\mathcal N=2$ models. \nOne candidate is the $\\mathcal N=2$ superconformal $SU(N)$ gauge theory with $N_F=4$ fundamental and $N_A=2$ \n rank-2 antisymmetric hypermultiplets. In this case the dual string \n theory is expected to be again a IIB orientifold of AdS$_{5}\\times S^{5}$\nwhere $S^{5}$ is modded out by a $\\mathbb Z_{4}$ that mixes non-trivially the orbifold and orientifold \ntwists \\cite{Ennes:2000fu}.\\footnote{In this model the Weyl anomaly coefficients are \n${\\rm a}=\\frac{1}{4}N^2+\\frac{1}{8}N-\\frac{5}{24}$ and ${\\rm c}=\\frac{1}{4}N^2+\\frac{1}{4}N-\\frac{1}{6}$. \n The $\\mathcal O(N)$ terms in a and c should be possible to derive on the dual string theory side as in \n\n \\cite{Blau:1999vz} (see also \\cite{Aharony:1999rz,Naculich:2001xu}) using that here the background involves D7-branes wrapping AdS$_5$ and $S^3$ of $S'^5$ with $R^2$terms in the effective 8-dimensional world-volume theory.}\nHowever, the presence of fundamentals means that here the large $N$ expansion will go in powers of $1\/N$ rather than \n$1\/N^2$ and thus will be different in structure from \\rf{1.5},\\rf{1.6}.\n\n\n \n\\section{Matrix model representation and $1\/N^2$ correction to Wilson loop}\n\nThe field content of the orientifold theory \nis represented by the adjoint $\\mathcal N=2$ vector multiplet (gauge vector $A_{\\mu}$, a complex scalar $\\varphi$, and two Weyl fermions) and rank-2 symmetric and antisymmetric \n hypermultiplets (each containing two complex scalars and two Weyl fermions). \n The $\\frac{1}{2}$-BPS Wilson loop is defined in terms of the fields of the vector multiplet as \n\\begin{equation}\n\\label{2.1}\n\\mathcal W = {\\rm tr\\,}\\mathcal P\\,\\exp\\Big\\{g_{_{\\rm YM}}\\oint\\Big[i\\,A_{\\mu}(x) dx^{\\mu}+\\tfrac{1}{\\sqrt 2}\\big(\\varphi(x)+\\varphi^{+}(x)\\big)\\,ds\\Big]\\Big\\},\n\\end{equation}\nwhere the contour $x^{\\mu}(s)$ represents a circle of unit radius. \n\n\\def S_{\\rm int} {S_{\\rm int}}\n\nThe supersymmetric localization implies \nthat the partition function of this gauge theory \non a sphere $S^{4}$ of unit radius admits a \nrepresentation in terms of an integral over the eigenvalues $\\{m_{i}\\}_{i=1}^{N}$ of a traceless hermitian $ N \\times N$ matrix $m$ \\cite{Pestun:2007rz}\n\\ba\\label{2.2}\nZ^{\\rm orient} &\\equiv e^{-F^{\\rm orient} }= \\int \\mathcal Dm \\,e^{-S(m)}\\ , \\qquad\\qquad \\\\\n S(m) &= S_0(m) + S_{\\rm int}(m) \\ , \\ \\ \\ \\ \\quad S_0= { 8 \\pi^2 N\\over \\lambda} {\\rm tr\\,} m^2 \\ , \\qquad \\lambda = g^2_{_{\\rm YM}}N \\ , \\label{221}\\\\\n\\mathcal Dm &\\equiv \\prod_{i=1}^{N}dm_{i}\\,\\delta\\big(\\sum_{j}m_{j}\\big)\\,\\big[\\Delta(m)\\big]^{2}\\ , \\qquad \\qquad \n\\Delta(m) = \\prod_{ii$.}\n\\begin{equation}\\label{322}\nU_{ij} = \\frac{(-1)^{1-j}2^{1-i}\\sqrt{1+2j}\\,\\Gamma(2+2i)}{\\sqrt 3\\,\\Gamma(1+i-j)\\ \\Gamma(2+i+j)} \\ . \\end{equation}\nOne can then replace $\\widetilde M$ in (\\ref{3.18}) by $M$, getting \n\\begin{equation}\n\\label{3.22}\n\\Delta F(\\lambda) = \\frac{1}{2}{\\rm tr\\,} \\log(1+M) = \\frac{1}{2}\\sum_{n=1}^{\\infty}\\frac{(-1)^{n+1}}{n}\\,{\\rm tr\\,} M^{n}.\n\\end{equation}\nThe advantage of this form of $\\Delta F$ is that the matrix $M$ in \\rf{3.21} admits the following \nBessel function representation\n\\ba\n\\label{3.23}\nM_{ij}\n &= 8\\,(-1)^{i+j}\\,\\sqrt{2i+1}\\sqrt{2j+1}\\,\\int_{0}^{\\infty}\\frac{dt}{t}\\frac{e^{2\\pi\\,t}}{(e^{2\\pi\\,t}-1)^{2}}\\,J_{2i+1}(t\\,\\sqrt{\\lambda})\\,J_{2j+1}(t\\,\\sqrt{\\lambda}) \\ , \n\\ea\nwhich will prove to be useful in the analysis of the strong-coupling limit. \n\n\n\n\\section{Contributions to $\\Delta F$ of finite degree in $\\zeta$-function values}\n\nThe weak coupling expansion of $\\Delta F$ in (\\ref{3.5}) can be represented as \n\\begin{equation}\\label{4.1}\n\\Delta F = \\sum_{n=1}^{\\infty}\\Delta F^{(n)}\\ , \\qquad \\qquad \\Delta F^{(n)} = \\sum_{k_1, ..., k_n} c_{k_1 ... k_n} (\\lambda) \\ \\zeta_{k_1} ...\\zeta_{k_n}= \\frac{(-1)^{n+1}}{n}{\\rm tr\\,} M^{n}\\ , \n\\end{equation}\nwhere $\\Delta F^{(n)}$ is the total contribution of terms that are \nproducts of a fixed number $n$ of the Riemann $\\zeta$-function values. Equivalently, $ \\Delta F^{(n)} $ represents the \ncontribution of\nthe ${\\rm tr\\,} M^{n}$ term in \\rf{3.22} (cf. (\\ref{3.19})). \n\nHere we will study $\\Delta F^{(n)}$, \n computing, in particular, its leading strong coupling asymptotic expansion. \n We shall focus in detail on the $n=1$ term\nand then discuss \n the $n=2$ one.\n We will find that \n\\begin{equation}\n\\label{4.2}\n\\Delta F^{(n)}(\\lambda) \\stackrel{\\lambda\\gg 1}{=} C_{n}\\lambda^{n}+\\cdots.\n\\end{equation}\nIn the next section 5 \n we will compute all the coefficients $C_{n}$ in \\rf{4.2} and then \n evaluate the sum of all $\\Delta F^{(n)}$ thus determining the strong coupling asymptotics of $\\Delta F$.\n\n\n\n\\medskip\n Defining \n\\ba\n\\label{4.3}\nG(t,t') &\\equiv 8\\,\\sum_{i=1}^{\\infty}(2i+1)\\,J_{2i+1}(t)\\,J_{2i+1}(t') \n= -\\frac{4tt'}{t^{2}-t'^{2}}\\Big[t\\,J_{1}(t)\\,J_{2}(t')-t'\\,J_{2}(t)\\,J_{1}(t')\\Big]\\ ,\n\\ea\nwhich has also the following integral form \\cite{Tracy:1993xj}\n\\begin{equation}\n\\label{4.7}\nG(t,t') = 2\\,t\\,t'\\,\\int_{0}^{1}du\\,J_{2}(t\\sqrt u)\\,J_{2}(t'\\sqrt u)\\ , \n\\end{equation}\nwe may use \n the Bessel function representation of the matrix $M$ (\\ref{3.23}) to represent the traces of $M^n $ as the iterated integrals\n\\ba\n{\\rm tr\\,} M &= \\int_{0}^{\\infty}\\widehat{dt}\\,G(t\\sqrt\\lambda, t\\sqrt\\lambda), \\qquad\\qquad \\qquad \\widehat{dt} = \\frac{dt}{t}\\,\\frac{e^{2\\pi\\,t}}{(e^{2\\pi\\,t}-1)^{2}}, \\label{4.4} \\\\\n{\\rm tr\\,} M^{2} &= \\int_{0}^{\\infty}\\widehat{dt}\\,\\int_{0}^{\\infty}\\widehat{dt'} \\, G(t\\sqrt\\lambda, t'\\sqrt\\lambda)\\,G(t'\\sqrt\\lambda, t\\sqrt\\lambda), \\label{4.5} \\\\\n{\\rm tr\\,} M^{3} &= \\int_{0}^{\\infty}\\widehat{dt}\\,\\int_{0}^{\\infty}\\widehat{dt'} \\,\\int_{0}^{\\infty}\\widehat{dt''}\\, G(t\\sqrt\\lambda, t'\\sqrt\\lambda)\\,G(t'\\sqrt\\lambda, t''\\sqrt\\lambda)\\,G(t''\\sqrt\\lambda, t'\\sqrt\\lambda)\\ , \\ \\ \\ \\ {\\it etc.}\\label{4.6} \n\\ea\nWe remark that (\\ref{4.3}) coincides with the Tracy-Widom kernel \\cite{Tracy:1993xj} upon the change of variables $t = \\sqrt x$. \nIt remains to be clarified whether this is a coincidence or there is some deeper relation to eigenvalue statistics. This Bessel kernel also appears in the BES equation \\cite{Beisert:2006ez}\nand seems prevalent in integrable equations\/models.\n\n\n\\subsection{Term linear in $\\zeta_n$}\n\n$\\Delta F^{(1)}$ or ${\\rm tr\\,} M$ is just a single integral (\\ref{4.4}) and may be treated exactly. Using \\rf{4.3} we have \n\\ba\\label{4.8}\n\\Delta F^{(1)} &= \\frac{1}{2}{\\rm tr\\,} M = 4\\,\\int_{0}^{\\infty}\\frac{dt}{t}\\frac{e^{2\\pi\\,t}}{(e^{2\\pi\\,t}-1)^{2}}\\,\\sum_{i=1}^{\\infty}(2i+1)\\,\\big[J_{2i+1}(t\\,\\sqrt{\\lambda})\\big]^{2}.\n\\ea\nFrom the identity\n\\begin{equation}\n4\\sum_{i=1}^{\\infty}(2i+1)\\,\\big[J_{2i+1}(x)\\big]^{2} = x^{2}\\,\\big[J_{0}(x)\\big]^{2}+(x^{2}-4)\\,\\big[J_{1}(x)\\big]^{2},\n\\end{equation}\nwe obtain \n\\ba\n\\Delta F^{(1)} &= \\int_{0}^{\\infty}\\frac{dt\\,e^{2\\pi t}}{t(e^{2\\pi t}-1)^{2}}\\Big(t^{2}\\lambda\\,\\big[J_{0}(t\\,\\sqrt\\lambda)\\big]^{2}+(t^{2}\\lambda-4)\\,\\big[J_{1}(t\\sqrt\\lambda)\\big]^{2}\\Big) \\nonumber \\\\\n&= \\frac{\\lambda}{2\\pi}\\int_{0}^{\\infty}\\frac{dt}{e^{2\\pi t}-1}\\,\\Big(\\big[J_{0}(t\\sqrt\\lambda)\\big]^{2}-\\frac{8\\,J_{0}(t\\sqrt\\lambda)J_{1}(t\\sqrt\\lambda)}{t\\sqrt\\lambda}\n+\\frac{(12-t^{2}\\lambda)\\,\\big[J_{1}(t\\sqrt\\lambda)\\big]^{2}}{t^{2}\\lambda}\\Big), \\label{4.11}\n\\ea\nwhere we used integration by parts.\n This expression is exact and may be expanded at weak or strong coupling.\n\n\\paragraph{Weak coupling expansion:} Using \n\\begin{equation}\n\\label{4.12}\n\\int_{0}^{\\infty}dt\\,\\frac{e^{2\\pi t}}{(e^{2\\pi t}-1)^{2}}t^{2p+1} = \\,{(2p+1)!\\over (2\\pi)^{2p+2}} \\,\\zeta_{2p+1} \\ , \n\\end{equation}\n and expanding in $\\lambda$ we recover from (\\ref{4.11}) the first line in (\\ref{3.19}). \nOne can find the following all-order result\\footnote{This remarkably simple form of the coefficients follows from the relation\n\\begin{equation}\\notag\n\\big[J_{0}(t)\\big]^{2}-\\frac{8\\,J_{0}(t)J_{1}(t)}{t}\n+\\frac{(12-t^{2})\\,\\big[J_{1}(t)\\big]^{2}}{t^{2}} = \\frac{5}{192}\\,t^{4}\\,{}_{1}F_{2}(\\tfrac{7}{2}; 4,5; -t^{2}).\n\\end{equation}\n}\n\\begin{equation}\n\\label{4.13}\n\\Delta F^{(1)} = \\frac{4}{\\pi}\\sum_{k=2}^{\\infty}(-8)^{k}\\,\\frac{(k-1)\\,k\\,(k+2)\\,\\Gamma(k+\\frac{1}{2})\\Gamma(k+\\frac{3}{2})}{\\big[\\Gamma(k+3)\\big]^{2}}\\zeta_{2k+1}\\hat\\lambda^{k+1}.\n\\end{equation}\nBy the standard ratio test this shows that the radius of convergence is $\\pi^{2}$, as could be expected. \nIndeed, $\\lambda=\\pi^2$ is the radius of convergence of perturbative expansion in $\\mathcal N=4$ \nSYM theory in the planar limit (as suggested by the single-magnon dispersion relation, fixed by the superconformal symmetry \\cite{Beisert:2004hm,Beisert:2006ez}, or by the \nquantum algebraic curve approach \\cite{Gromov:2017blm}). The same is expected to apply also to the \n$\\mathcal N = 2$ superconformal theories\n(as was first observed in the mass-deformed $\\mathcal N=2^{*}$ theory \\cite{Russo:2013qaa}, and recently found also \nin the orbifold theory case \\cite{Beccaria:2021ksw}).\n\nExpanding the exponentials in the integral in (\\ref{4.11}) gives an alternative representation\nin terms of a sum of elliptic integrals\n\\begin{equation}\n\\label{4.14}\n\\Delta F^{(1)} = 2\\, \\sum_{n=1}^{\\infty}n\\,\\Big[-1-\\frac{8\\pi^{2}n^{2}-\\lambda}{2\\pi \\lambda}\\,\\mathbb{E}\\Big(-\\frac{\\lambda}{\\pi^{2}n^{2}}\\Big)\n+\\frac{8\\pi^{2}n^{2}+7\\lambda}{2\\pi \\lambda}\\,\\mathbb{K}\\Big(-\\frac{\\lambda}{\\pi^{2}n^{2}}\\Big)\\Big].\n\\end{equation}\nHere one sees explicit singularities at $\\lambda = -\\pi^{2}n^{2}$ where the argument of $\\mathbb K$ becomes unity.\n\n\n\\paragraph{Strong coupling expansion:}\n\nThe strong coupling (asymptotic) expansion of $\\Delta F^{(1)}$ may be computed by Mellin transform methods \\cite{zagier,flajolet}.\nDefining the Mellin transform $ \\widetilde{f}(s) = \\int_0^\\infty dx \\, x^{s-1}\\,f(x)\\,$ and considering the convolution \n\\begin{equation}\n\\label{4.15}\n(f\\star g) (x)= \\int_0^\\infty dt \\, f(t\\,x)\\,g(t),\n\\end{equation}\nwe have $(\\widetilde{f\\star g})(s) = \\widetilde{f}(s)\\,\\widetilde{g}(1-s)$.\nLet $\\alpha< s< \\beta$ be the fundamental strip of analyticity of $\\widetilde{f}(s) $.\nThe asymptotic expansion of $f(x)$ for $x\\to\\infty$ is obtained by \nlooking at the poles of $\\widetilde{f}(s)$ in the region $s\\ge \\beta$. Then the pole \n$\n\\frac{1}{(s-s_0)^N} $ in the Mellin transform leads to the term $ \\frac{(-1)^N}{(N-1)!}\\,\\frac{1}{x^{s_0}}\\,\\log^{N-1} x\n$ in the original function. \nIn our case, we can compare the right hand side of (\\ref{4.11}) with (\\ref{4.15}) as \n\\ba\nx &= \\sqrt\\lambda, \\qquad g(t) = \\frac{1}{4\\pi}\\frac{1}{e^{2\\pi t}-1}, \\qquad \nf(t) = \\big[J_{0}(t)\\big]^{2}-\\frac{8\\,J_{0}(t)J_{1}(t)}{t}\n+\\frac{(12-t^{2})\\,\\big[J_{1}(t)\\big]^{2}}{t^{2}}.\n\\ea\nThe Mellin transform is then \n\\begin{equation}\n(\\widetilde{f\\star g})(s) = \\frac{2^{-6+s} s (2+s) \\csc ^2(\\frac{\\pi s}{2}) \\Gamma (2-s)\\ \\zeta \n(s)}{\\big[\\Gamma (1-\\frac{s}{2})\\big]^2\\ \\Gamma (2-\\frac{s}{2})\\ \\Gamma \n(3-\\frac{s}{2})},\n\\end{equation}\nand the asymptotic expansion at strong coupling can be extracted from the poles at $s=0, 1, 2, \\dots$. This gives\n\\begin{equation}\n\\label{4.18}\n\\Delta F^{(1)} = \\frac{\\lambda}{16\\pi^{2}}-\\frac{\\sqrt\\lambda}{2\\pi^{2}}+\\frac{1}{6}+\\frac{\\sqrt\\lambda}{2\\pi^{7\/2}}\n\\sum_{p=1}^{\\infty}\n\\frac{\\Gamma(\\frac{5}{2}+p)\\Gamma(p-\\frac{1}{2})\\Gamma(p-\\frac{3}{2})}{\\Gamma(p)}\n\\frac{\\zeta_{2p+1}}{\\lambda^{p}}.\n\\end{equation}\n The infinite sum in \\rf{4.18} has zero radius of convergence, with factorially divergent coefficients.\\footnote{Let us note \n that replacing the $\\zeta$-values by the integral using (\\ref{4.12}) and doing the sum, we obtain another representation\n\\begin{equation}\\notag\n\\Delta F^{(1)} = \\frac{\\lambda}{16\\pi^{2}}-\\frac{\\sqrt\\lambda}{2\\pi^{2}}+\\frac{1}{6}+\\frac{2\\lambda}{\\pi^{3}}\\int_{0}^{\\infty}\\frac{dt}{e^{2t\\sqrt\\lambda}-1}\\Big[\\mathbb{K}(t^{2})-(1+8t^{2})\\,\\mathbb{E}(t^{2})\\Big].\n\\end{equation}\nThis integral has a logarithmic singularity at $t=1$ on the $t$ integration contour, and so should be understood as an average above and below the cut.\n}\nThe leading order $\\lambda$ term corresponds to the $n=1$ case of\n the general pattern (\\ref{4.2}).\n\n\nThe leading term in (\\ref{4.18}) can be derived more directly.\nWe can expand the \nintegrand in (\\ref{4.4}) at large $\\lambda$ and read off the coefficient of a suitable power of $\\lambda$ from a convergent integral\\footnote{This procedure works for the leading order; at subleading orders one gets \ndivergent integrals \nrequiring a more careful treatment.}\n\\ba\n\\label{4.19}\n{\\rm tr\\,} M &= \\int_{0}^{\\infty}\\frac{dt}{t}\\,\\frac{e^{2\\pi\\,t}}{(e^{2\\pi\\,t}-1)^{2}}\\,G(t\\sqrt\\lambda, t\\sqrt\\lambda) = \n\\int_{0}^{\\infty}\\frac{dt}{t}\\,\\frac{e^{2\\pi\\,t\/\\sqrt\\lambda}}{(e^{2\\pi\\,t\/\\sqrt\\lambda}-1)^{2}}\\,G(t, t) \\notag \\\\ & \n= \\frac{\\lambda}{2\\pi^{2}}\\int_{0}^{\\infty}\\frac{dt}{t}\\big[J_{2}(t)^{2}-J_{1}(t)J_{3}(t)\\big] +\\dots = \\frac{\\lambda}{8\\pi^{2}}+\\dots\\ . \n\\ea\nAs $\\Delta F^{(1)} = \\frac{1}{2}{\\rm tr\\,} M$ (cf. \\rf{4.8}), this result is thus in agreement with (\\ref{4.18}). \n\n\n\n\n\\subsection{Term quadratic in $\\zeta_n$}\n\n\nIn the case of $\\Delta F^{(2)} = -\\frac{1}{4}{\\rm tr\\,} M^{2}$ in \\rf{4.1} \n we can obtain an all-order weak coupling expansion in almost-closed form. \n Although it is not as explicit as\n(\\ref{4.13}) for $\\Delta F^{(1)}$, it may be used to generate a very large number of terms.\n Here we will present the final result, with details given in Appendix ~\\ref{app:M2}.\nLet us define the polynomials\n\\begin{equation}\n\\label{4.20}\nd_{\\ell}(x) = (-1)^{\\ell}\\sum_{p=0}^{\\ell}\\frac{P_{p}^{(2,-2p-5)}(1-2x^{2})\\ P_{\\ell-p}^{(2,-2\\ell + 2 p-5)}(1-2x^{2})}{4^{p+2}4^{\\ell-p+2}\\Gamma(p+3)\\Gamma(p+4)\\Gamma(\\ell-p+3)\\Gamma(\\ell-p+4)},\n\\end{equation}\nwhere $P^{(\\alpha, \\beta)}_{n}(x)$ are Jacobi polynomials. We may write $d_\\ell$ in the form \n\\begin{equation}\n\\label{4.21}\nd_{\\ell}(x) = x^{\\ell}\\mathop{\\sum_{m=m_{0}}}_{\\Delta m = 2}^{\\ell}a_{m}^{(\\ell)}(x^{m}+x^{-m}),\n\\end{equation}\nwhere $m_{0} = 0\/1$ if $\\ell$ is even\/odd and $m$ varies in steps of 2. \nThe weak coupling expansion of ${\\rm tr\\,} M^{2}$ can then be written in terms of sums with \n coefficients $a_{m}^{(\\ell)}$ that are easily computed from \\rf{4.20},(\\ref{4.21})\n\\begin{equation}\n\\label{4.22}\n{\\rm tr\\,} M^{2} = 8\\sum_{\\ell=0}^{\\infty}(2\\pi)^{-12-2\\ell}\\,\\lambda^{\\ell+6}\\,\\sum^\\ell_{m}a_{m}^{(\\ell)} \\Gamma(\\ell+6+m)\\Gamma(\\ell+6-m)\\, \\zeta_{\\ell+5+m}\\, \\zeta_{\\ell+5-m}.\n\\end{equation}\n\n\\paragraph{Leading term at strong coupling:} \nThe expansion (\\ref{4.22}) may not be used directly at strong coupling. Nevertheless, we succeed in applying the manipulation \nwe exploited in (\\ref{4.19}). Indeed, we have\n\\begin{equation}\n{\\rm tr\\,} M^{2} = \\lambda^{2}\\int_{0}^{\\infty}\\int_{0}^{\\infty}dt dt'\\ \\frac{[t' J_{1}(t')J_{2}(t)-t J_{1}(t)J_{2}(t')]^{2}}{\\pi^{4}tt'(t^{2}-t'^{2})^{2}}+\\dots\n\\end{equation}\nThe integrand is symmetric so we write\n\\begin{equation}\n\\int_{0}^{\\infty}\\int_{0}^{\\infty}dt dt' f(t,t') = 2\\int_{0}^{\\infty}dt\\int_{0}^{t}dt' f(t,t') = \n2\\int_{0}^{\\infty}dt\\,t\\,\\int_{0}^{1}dx f(t,tx).\n\\end{equation}\nDoing first the integral over $t$, we get\n\\ba\n\\label{4.25}\n{\\rm tr\\,} M^{2} &= \\lambda^{2}\\int_{0}^{1}dx\\, \\tfrac{x(15-7x^{2}-7x^{4}+15 x^{6})-3(1-x^{2})^{2}(5+6x^{2}+5x^{4})\\, \\text{arctanh}\\, x }{144\\pi^{6}x^{5}} +\\dots \n= \\frac{\\lambda^{2}}{192\\pi^{4}}+\\dots \\ . \n\\ea\nThis strong-coupling asymptotics \nfollows again the general pattern (\\ref{4.2}). \nA numerical test of this prediction will be discussed in section \\ref{sec:pade}.\n\n\n\n\\section{Strong coupling limit of $\\Delta F$: analytic derivation}\n\\label{sec:sc}\n\nLet us now generalize the derivation of strong-coupling limit to the full $\\Delta F$.\n The starting point will be the explicit form of \n\n the large $\\lambda$ expansion of the matrix $M$ in \\rf{3.23}. It can be found \n as in \\rf{4.15}--\\rf{4.18} using the Mellin transform.\nWe have \n\\ba\nM_{ij} = &8(-1)^{i+j}\\sqrt{(2i+1)(2j+1)}\\, N_{ij} \\ , \\label{68}\\\\\nN_{ij} \\equiv & \\sqrt\\lambda\\,(f\\star g_{ij})(\\sqrt\\lambda)\\ , \\qquad \\qquad \nf(t) = \\frac{e^{2\\pi t}}{(e^{2\\pi t}-1)^{2}}\\ ,\\qquad \\ \\ g_{ij}(t) = \\frac{1}{t}J_{2i+1}(t)J_{2j+1}(t).\n\\ea\nEvaluating the Mellin transforms and taking residues, we get the asymptotic expansion of $N_{ij}$ \n\\ba\nN_{ij} \\stackrel{\\lambda\\gg 1}{=} & \\Big[\\frac{\\delta_{ij}}{i(i+1)(2i+1)}+\\frac{\\delta_{i+1,j}}{(i+1)(2i+1)(2i+3)}+\\frac{\\delta_{i,j+1}}{i(2i-1)(2i+1)}\\Big]\\,\\frac{\\lambda}{64\\pi^{2}}\\notag\\\\\n& -\\frac{\\delta_{ij}}{24\\,(2i+1)}+\\frac{\\zeta_3 }{2\\pi^{2}}\\,\\cos\\big(\\pi(i-j)\\big)\\, \\frac{1}{\\sqrt\\lambda}+\\cdots. \\label{610}\n\\ea\nThen the leading strong-coupling part of $M$ may be written as \n\\ba\n& \\qquad \\qquad \\qquad M \\stackrel{\\lambda\\gg 1}{=} {\\lambda\\over 2 \\pi^2} {\\rm S}} \\def \\ss{{\\rm s}} \\def \\jj {{\\rm j} + ...\\ , \\label{58} \\\\\n& {\\rm S}} \\def \\ss{{\\rm s}} \\def \\jj {{\\rm j}_{ij} = {1 \\over 4} (-1)^{i+j}\\sqrt{\\frac{2j+1}{2i+1}}\\Big[\\frac{\\delta_{ij}}{i(i+1)}+\\frac{\\delta_{i+1,j}}{(i+1)(2i+3)}+\\frac{\\delta_{i,j+1}}{i(2i-1)}\\Big]\\ , \\label{59}\n\\ea\nwhere ${\\rm S}} \\def \\ss{{\\rm s}} \\def \\jj {{\\rm j}$ is a symmetric three-diagonal infinite-dimensional matrix. \nAs a result, we get \n\\begin{equation}\n\\label{5.1}\n{\\rm tr\\,} M^{n} \\stackrel{\\lambda\\gg 1}{=} b_{n}\\,\\Big(\\frac{\\lambda}{2\\pi^{2}}\\Big)^{n}+\\cdots, \\ \\ \\ \\ \\ \\ \\ \\ \\ \nb_{n} = {\\rm tr\\,} {\\rm S}} \\def \\ss{{\\rm s}} \\def \\jj {{\\rm j}^{n} \\ .\n\\end{equation}\nThe explicit values of the coefficients \n $b_n$ (related to $C_n$ in \\rf{4.2} as $C_n = {(-1)^{n+1}\\over n (2 \\pi^2)^n} b_n $)\n are given in Appendix \\ref{NEW}.\n\n\nRemarkably, ${\\rm S}} \\def \\ss{{\\rm s}} \\def \\jj {{\\rm j}$ in \\rf{59} is essentially the same (up to 1\/2) as the matrix appearing in Eq.~(2.7) of \\cite{ikebe}. \nIt follows from the analysis in \\cite{ikebe} that in the infinite matrix limit the eigenvalues $\\{\\ss_{1}, \\ss_{2}, \\dots\\}$ \nof ${\\rm S}} \\def \\ss{{\\rm s}} \\def \\jj {{\\rm j}$ are \n\\begin{equation}\n\\ss_{k} = \\frac{2}{\\jj_{1,k}^{2}} \\ ,\\qquad \\qquad k=1, 2, ..., \\label{510}\n\\end{equation} \nwhere $\\jj_{1,k}$ are the zeroes of the Bessel function $J_{1}(x)$.\n\\iffalse \n\\footnote{\nThis implies, in particular, that the \ngenerating function for the coefficients $b_n={\\rm tr\\,} {\\rm S}} \\def \\ss{{\\rm s}} \\def \\jj {{\\rm j}^n $, i.e. $\nb(x)=\\sum_{n=1}^{\\infty} b_{n}x^{n-1} $ is given by \n$b(x) = \\frac{1}{\\sqrt{2x}}\\frac{J_{2}(\\sqrt{2x})}{J_{1}(\\sqrt{2x})} $.}\n\\fi\nHence, we get the following remarkable relation\\footnote{This follows from the Weierstrass infinite product representation of the Bessel function in terms of its zeroes:\n\n$J_{\\nu}(z) = \\frac{(z\/2)^{\\nu}}{\\Gamma(\\nu+1)}\\prod_{n=1}^{\\infty}(1-\\frac{z^{2}}{{\\rm j}^2_{\\nu,n}})$, see for instance section 15.41 in \\cite{watson}.}\n\\begin{equation} \\label{555}\n \\det \\big(1 + \\frac{\\lambda}{2\\pi^{2} } {\\rm S}} \\def \\ss{{\\rm s}} \\def \\jj {{\\rm j} \\big) = \\prod_{k=1}^\\infty \\big( 1 + \\frac{\\lambda}{\\pi^{2} } { 1 \\over \\jj^2_{1,k}} \\big) \n= \\frac{2\\pi}{i\\sqrt\\lambda}\\,J_{1}\\Big(\\frac{i\\sqrt\\lambda}{\\pi}\\Big) = \\frac{2\\pi}{\\sqrt\\lambda}\\,I_{1}\\Big(\\frac{\\sqrt\\lambda}{\\pi}\\Big) \\ .\n\\end{equation} \nAs a result, we get for $\\Delta F$ in \\rf{3.22} \n\\ba\n\\label{5.10}\n\\Delta F &= \\frac{1}{2}\\log\\det (1+M) \\stackrel{\\lambda\\gg 1}{=} \\frac{1}{2} \\log\\det (1 + \\frac{\\lambda}{2\\pi^{2} } {\\rm S}} \\def \\ss{{\\rm s}} \\def \\jj {{\\rm j} ) \n= \\frac{1}{2}\\log\\bigg[\\frac{2\\pi}{\\sqrt\\lambda}\\,I_{1}\\Big(\\frac{\\sqrt\\lambda}{\\pi}\\Big)\\bigg]\\ \\stackrel{\\lambda\\gg 1}{=} \\ \\frac{\\sqrt\\lambda}{2\\pi}+\\cdots \\ . \n\\ea\nEq.\\rf{5.10} implies that $c_1$ in \\rf{1.14} is equal to $\\frac{1}{2\\pi}$.\nThen using \\rf{1.11} we obtain the following expression \\rf{1.15} for the strong-coupling limit of $\\Delta q$ \n\\begin{equation}\n\\Delta q(\\lambda) \\stackrel{\\lambda\\gg 1}{=} -\\frac{\\lambda^{3\/2}}{16\\pi}+\\cdots. \\label{5.11} \n\\end{equation}\n\n\\section{Numerical evaluation of $\\Delta F$: interpolation from small to large $\\lambda$}\n\nIn this final section we present various approaches to test the analytical result (\\ref{5.10}) for the strong coupling limit of $\\Delta F$ by numerical methods.\nWe will first consider the approach based on Pad\\'e approximants using as an input many terms \nin the weak coupling expansion\nof $\\Delta F$. \nThen, we will discuss a method based on a direct evaluation of $\\Delta F=\\frac{1}{2}{\\rm tr\\,}\\log(1+M)$ where the large $\\lambda$ limit of the infinite matrix $M$ is first replaced\nby its finite-size truncation.\n\n\n\\subsection{Pad\\'e-conformal method}\n\\label{sec:pade}\n\n\n \nWe begin with the small $\\lambda$ expansion of $\\Delta F$:\n\\begin{equation}\n\\Delta F(\\tilde\\lambda)=\\sum_k c_k \\tilde{\\lambda}^k\\ , \\qquad \\qquad \\qquad \\tilde{\\lambda} \\equiv 8\\hat\\lambda = \\frac{\\lambda}{\\pi^2}\\ . \n\\label{6.1}\n\\end{equation}\nThe particular definition of $\\tilde \\lambda$ is chosen so that the radius of convergence of the series in \\rf{6.1} is as close as possible to 1. \nThis is helpful for the numerical analysis, as it avoids the appearance of very large or very small coefficients at high order. \n\nThe technical goal is to extrapolate from small to \nlarge $\\lambda$, starting from a {\\it finite} number of terms in the weak coupling expansion. \n Optimal and near-optimal methods for such an extrapolation have been \nanalyzed recently in \\cite{Costin:2019xql,Costin:2020hwg,Costin:2020pcj}. The key information is some knowledge, either analytic or numerical, of the singularity \nstructure of the function $\\Delta F(\\tilde\\lambda)$. This information can be extracted numerically by suitable combinations of ratio tests, Pad\\'e approximants, and conformal maps. \n\nThe magnitude of the leading singularity is equal to the radius of convergence $R$, which can be found by a simple ratio test:\n\\begin{equation}\n\\left| \\frac{c_{k+1}}{c_k}\\right| \\to \\frac{1}{R}, \\qquad k\\to \\infty.\n\\label{6.2}\n\\end{equation}\nThe convergence of this ratio of successive coefficients to the inverse radius can be accelerated using Richardson acceleration \\cite{Bender-Orszag} (for example, for the ${\\rm tr\\,} M$ case see the left hand panel of Figure \\ref{fig:trm1-ratio} below).\n\n This permits an extremely precise numerical estimate of the radius of convergence, \nif it is not known analytically. For $\\Delta F=\\frac{1}{2}{\\rm tr}\\log(1+M)$, we will see that the leading singularity is at $\\tilde \\lambda \\approx -1$, i.e. \n$\\lambda\\approx -\\pi^2$.\nBy studying the subleading corrections to this ratio test limit one can determine the nature of the leading singularity, using Darboux's theorem, see Appendix \\ref{app:darboux}. \nFor this orientifold model the small $\\lambda$ expansion indicates that the leading singularity is logarithmic (see the right hand panel of Figure \\ref{fig:trm1-ratio} below). This is consistent with the exact analytical structure of individual ${\\rm tr\\,} M^{n}$ terms for finite $n$, see section \\ref{sec:sc}.\n\nA closely related method, which also yields information about the singularity structure is based on the use of a Pad\\'e approximant \\cite{Bender-Orszag,Baker}. Here one matches the finite number $K$ of terms of the expansion to the expansion of a ratio of polynomials $R_L$ and $Q_M$:\n\\begin{equation}\n\\mathcal P_{[L,M]}\\Big[\\sum_{k}^{K} c_k \\tilde{\\lambda}^k\\Big]=\\frac{R_L(\\tilde\\lambda)}{Q_M(\\tilde\\lambda)}+O(\\tilde\\lambda^{K+1})\\ . \n\\label{6.3}\n\\end{equation}\nSince it is an approximation in terms of rational functions, Pad\\'e only has poles as singularities, which are the zeros of the denominator polynomial $Q_M$. \nIf the truncated series is that of a function with branch point singularities, then Pad\\'e produces arcs of poles accumulating at the branch points.\\footnote{There is a deep connection to \nelectrostatics and potential theory, whereby (in this interpretation it is easiest to consider an expansion about infinity instead of about zero) in the $K\\to\\infty$ limit a Pad\\'e approximation\nproduces lines of poles that form a capacitor having minimal capacitance \\cite{Stahl,Costin:2020pcj}.} The practical implication of this is that if one has enough \nexpansion terms one can frequently distinguish between an isolated pole and a branch point simply by looking at the poles of a Pad\\'e approximant. \nIndeed, the left panel of \nFigure \\ref{fig:trm1-pade-poles} shows a line of Pad\\'e poles accumulating to the branch point at $\\tilde\\lambda=-1$. \n\nHowever, this reveals a fundamental problem with Pad\\'e, because these accumulating poles, which are trying to represent a branch cut, obscure possible higher \nsingularities which may be physical. This problem can be resolved by making a conformal map before making the Pad\\'e approximation \\cite{Costin:2019xql,Costin:2020hwg,Costin:2020pcj}. \nBased on the leading branch cut $(\\infty, -1]$ on the negative real $\\tilde{\\lambda}$ axis, as suggested by the Pad\\'e approximation in this case (see the left hand panel of Figure \\ref{fig:trm1-pade-poles}), one maps the expansion into the unit disk $|z|\\leq 1$:\n\\begin{equation}\n\\label{6.4}\nz=\\frac{\\sqrt{1+\\tilde\\lambda}-1}{\\sqrt{1+\\tilde\\lambda}+1}, \\qquad \\qquad \\tilde\\lambda=\\frac{4z}{(1-z)^2}.\n\\end{equation}\nWe re-expand $\\Delta F\\left(\\frac{4z}{(1-z)^2}\\right)$ in powers of $z$ to the same order $K$, and {\\bf then} construct a Pad\\'e approximant in terms of $z$.\\footnote{As a technical comment: \nwhen dealing with high order Pad\\'e approximants, numerical instabilities can arise due to close zeros and poles, also associated with large coefficients of the Pad\\'e polynomials. \nThis instability can be ameliorated by converting the Pad\\'e representation to a partial fraction expansion, which in principle is equivalent but in practice is more stable numerically.} \nInside the unit disk this expansion is convergent by construction, but further singularities along the line $\\tilde\\lambda\\in (\\infty, -1]$ will appear as singularities on the unit circle. \nIf these are branch points they will appear as the accumulation points of arcs of Pad\\'e poles. \n\nThe advantage of the conformal map is that collinear singularities in the $\\tilde\\lambda$ \nplane (which may be hidden under a line of accumulating poles) are separated to different points on the unit circle. See for example the right panel of \nFigure \\ref{fig:trm1-pade-poles}, which shows the \nleading singularity at $z=-1$, the conformal map image of $\\tilde\\lambda=-1$, but also clearly shows further singularities at the conformal map images of $\\tilde\\lambda=-4$, at $\\tilde\\lambda=-9$, and so on. \nThis numerical evidence suggests that the singularities are:\n\\begin{equation}\n\\label{6.5}\n{\\rm singularites}\\big(\\Delta F(\\lambda)\\big)=-l^2\\, \\pi^2\\ , \\qquad\\qquad l=1, 2, 3, \\dots\\ .\n\\end{equation}\nThe source of these singularities can be understood analytically from the study of ${\\rm tr\\,} M^{n}$ for finite $n$, and the singularity structure appears to be inherited by ${\\rm tr}\\,\\log(1+M)$.\n\nA further advantage of the conformal map is that it enhances the precision of the subsequent Pad\\'e extrapolation. To construct the Pad\\'e-conformal extrapolation\\footnote{This was applied to the \nBorel transform function in \\cite{Costin:2019xql,Costin:2020hwg}, but it can also be applied to any series with a finite radius of convergence \\cite{Costin:2020pcj}.} we make a Pad\\'e approximant \nin terms of $z$ and then evaluate it on the inverse map in (\\ref{6.4}). This introduces square roots; thus we are representing the function not just by rational approximations, but in a much \nwider class of functions. For branch point singularities the increase in precision can be quantified precisely using the asymptotics of orthogonal polynomials \\cite{Costin:2020hwg} and is \nquite dramatic, as is illustrated\n in Figures \\ref{fig:trm1-extrap} and \\ref{fig:trlog1pm-extrap} below.\n\n\\subsubsection{Example: ${\\rm tr\\,} M$}\n\nTo illustrate this Pad\\'e-conformal extrapolation technique, we first consider the expansions of ${\\rm tr\\,} M$ and ${\\rm tr\\,} M^{2}$, for which we can compare with analytic results found in section 4. \nBut we stress that the power of this method is in cases when such analytic comparisons are not available, and one is only presented with a truncated series, and possibly some physical \nintuition about the singularity structure. For ${\\rm tr\\,} M$ we have the exact expansion (\\textit{cf.} (\\ref{4.13}))\n\\begin{equation}\n\\label{6.6}\n{\\rm tr\\,} M =\\sum_{k=2}^\\infty \\frac{(-1)^k (k-1) k (k+2) \\, \\zeta_{2 k+1}\\, \n \\Gamma \\left(k+\\frac{1}{2}\\right) \\Gamma \\left(k+\\frac{3}{2}\\right)}{\\pi \\big[ \\Gamma (k+3)\\big]^2} \\, \\tilde\\lambda^{k+1}\\ .\n\\end{equation}\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[scale=.6, width=0.45\\textwidth]{FigRatioLeft.pdf}\n\\includegraphics[scale=.6, width=0.45\\textwidth]{FigRatioRight.pdf}\n\\caption{\\small ${\\rm tr\\,} M$ ratio test. On the left, we show in blue the ratio $c_{k+1}\/c_{k}$ that tends to $-1$. The orange line is obtained after applying a 5th order Richardson acceleration. On the right, we present\nthe same analysis for the Darboux indicator $\\pi k (-1)^{k}c_{k}$. }\n\\label{fig:trm1-ratio}\n\\end{center}\n\\end{figure}\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=0.4\\textwidth]{FigPolesLeft.pdf} \\hskip 0.15\\textwidth\n\\includegraphics[width=0.4\\textwidth]{FigPolesRight.pdf}\n\\caption{\\small Pad\\'e poles of ${\\rm tr\\,} M$ from 150 terms. On the left, we show the poles of the direct approximants. These poles lie on the negative real axis and accumulate to $\\tilde\\lambda=-1$. On the right\nwe show the poles in $z$-plane after application of the\nconformal transformation (\\ref{6.4}) followed by Pad\\'e. In this case collinear singularities on the line $\\tilde\\lambda\\in (-\\infty, -1]$ are separated and made visible as arcs converging to points on the unit circle in the $z$ plane. These agree with a similar analysis for the \nwhole $\\Delta F$, see (\\ref{6.5}).}\n\\label{fig:trm1-pade-poles}\n\\end{center}\n\\end{figure}\n\\begin{figure}[htb]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{FigTrMextrapolationLeft.pdf} \n\\includegraphics[width=0.45\\textwidth]{FigTrMextrapolationRight.pdf}\n\\caption{\\small Extrapolations of ${\\rm tr\\,} M(\\tilde\\lambda)\/\\tilde\\lambda$ compared to the analytic value $1\/8$ (blue line). The left and right plot differ only in the range of $\\tilde\\lambda$ values, $10^{4}$ on the left\nand $10^{6}$ on the right. The orange line is diagonal Pad\\'e of order 75, applied to the first 150 terms in the weak coupling expansion (\\ref{6.6}). The\ngreen line is the Pad\\'e-conformal extrapolation based on the transformation (\\ref{6.4}).}\n\\label{fig:trm1-extrap}\n\\end{center}\n\\end{figure}\n\\begin{figure}[h!]\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{FigDFextrapolationLeft.pdf}\n\\includegraphics[width=0.45\\textwidth]{FigDFextrapolationRight.pdf}\n\\caption{\\small Extrapolations of $\\Delta F\/\\sqrt{\\lambda}$ (plotted here as functions of $\\lambda$, not $\\tilde\\lambda$) compared to the asymptotic value $c_1= {1\\over 2\\pi}$ (blue line). In the left plot, in linear scale, the \norange line is the diagonal Pad\\'e approximant based on 150 terms of the full weak-coupling expansion, \\textit{i.e.} the extension of (\\ref{3.5}) to the order $\\lambda^{150}$. This Pad\\'e approximant breaks down shortly after $\\lambda=10^{4}$. \nThe green line is the Pad\\'e conformal result and it extends to much higher values of the coupling $\\lambda$. The left plot strongly supports the functional form $\\Delta F \\sim \\sqrt\\lambda$ at large $\\lambda$. \nThe convergence of the coefficient to the asymptotic value is steady but slow, as illustrated in the right panel on a logarithmic scale. See Section \\ref{sec:deltaf} for a more refined estimate of the overall coefficient.}\n\\label{fig:trlog1pm-extrap}\n\\end{center}\n\\end{figure}\nThe ratio $c_{k+1}\/c_k$ is plotted in the left panel of Figure \\ref{fig:trm1-ratio} based on the first 150 terms, indicating an alternating series with radius of convergence 1. \nThe fact that the leading singularity is logarithmic is shown by the fact that $c_k\\sim \\frac{(-1)^k}{k} \\times {\\rm constant}$ as $k\\to\\infty$. See the right panel in Figure \\ref{fig:trm1-ratio}.\nThe fact that the leading singularity is a branch point is also indicated by the Pad\\'e poles, which are shown in the left panel of Figure \\ref{fig:trm1-pade-poles}, accumulating along the negative real axis to the branch point at $\\tilde\\lambda=-1$.\n\nAfter the conformal map (\\ref{6.4}), followed by re-expansion to 150 terms in $z$, the poles of the resulting diagonal Pad\\'e approximant are shown in the right panel of the same figure. \nThis Figure indicates the existence of branch point singularities at the $z$ plane images of $\\tilde\\lambda=-1, -4,-9, -16$. The data becomes noisy at the conformal image of $-25$, \nwith unphysical poles appearing inside the unit disk. These can be resolved by taking more terms in the original expansion.\n\nWe now map this Pad\\'e approximant back to the physical $\\tilde\\lambda$ plane using the inverse conformal map in (\\ref{6.4}), and plot to large $\\lambda$. Figure \\ref{fig:trm1-extrap} compares the diagonal Pad\\'e extrapolation (orange curve), divided by $\\tilde\\lambda$, with the analytic large $\\tilde\\lambda$ limit of $\\frac{1}{8}$ (blue curve) and the Pad\\'e-conformal extrapolation (green curve). The first plot extends out to $\\tilde\\lambda=10^{4}$, while the second plot extends out to $\\tilde\\lambda=10^{6}$.\nNote that the Pad\\'e approximant eventually breaks down at $\\tilde\\lambda\\approx 1.5\\cdot 10^{4}$, while the Pad\\'e-conformal approximant extends much further to very large $\\tilde\\lambda$. \nWe stress that exactly the same input coefficient data was used in producing these two extrapolations, illustrating the dramatic effect of the conformal map.\n\n A similar analysis can be applied to \n${\\rm tr\\,} M^{2}$ where we do not have a simple closed form expression for the expansion coefficients, but there is a systematic way to expand to very high order\n (multiple hundreds of terms, \nsee (\\ref{4.22})).\nThe resulting structure is very similar to that for the ${\\rm tr\\,} M$ case discussed above, so we do not repeat the analogous plots.\n\n\\subsubsection{$\\Delta F$}\n\n\nLet us now consider the large $\\lambda$ extrapolation of the full $\\Delta F$. \n We begin with the small $\\lambda$ expansion discussed in section \\ref{sec:dF-weak}. \nWe generated 150 terms of this expansion, with 450 digit precision for the coefficients. The coefficients are sums of products of odd $\\zeta_{2k+1}$-values, but it is faster to work with finite but high precision coefficients. \nThe ratio test and Pad\\'e analysis again indicate a leading singularity at $\\lambda=-\\pi^2$, so we make the same conformal map (\\ref{6.4}) and subsequent Pad\\'e approximant \nand inverse map back to the physical $\\lambda$ plane. \n\nFigure \\ref{fig:trlog1pm-extrap} shows the result, and we again see that the Pad\\'e-conformal extrapolation extends to a much larger value of $\\lambda$.\nThis extrapolation shows that the functional form of the large $\\lambda$ behavior is (left panel of the figure)\n\\begin{equation}\n\\Delta F(\\lambda)=\\frac{1}{2} {\\rm tr} \\log(1+M) \\stackrel{\\lambda\\to + \\infty}{=}\nc_1 \\sqrt{\\lambda} \\ . \n\\end{equation}\nThis functional form matches the result of resumming the leading large $\\lambda$ terms of ${\\rm tr\\,} M^{n}$ in \\rf{5.10}, and the coefficients approximately agree. \n\nWe stress that the only input information used for this extrapolation from small $\\lambda$ to large $\\lambda$ was the list of 150 perturbative coefficients. \nTo get a better estimate of the result requires fitting the ratio $\\Delta F\/\\sqrt\\lambda$ and it is hard to support a specific functional form. The slow convergence shown in the right panel\nof Figure \\ref{fig:trlog1pm-extrap} should be\ndue to the expected logarithmic corrections in (\\ref{1.14}) if they do not happen to cancel in $\\Delta F$.\n\n\\subsection{Evaluation of $\\Delta F$ at large $\\lambda$ using truncation method}\n\\label{sec:deltaf}\n\nIn this subsection we use a complementary numerical method in order to extract the precise large $\\lambda$ behaviour of $\\Delta F$. \nStarting from the expansion (\\ref{610}), \nlet us denote by $M_K$ the $K\\times K$ matrix which is \nthe linear in $\\lambda$ part of $M$, truncated to the first $K$ rows and columns. Then \n\\begin{equation}\n\\label{6.12} \\Delta F(\\lambda) = \\lim_{K \\to \\infty} \\Delta F_K(\\lambda) \\ , \\qquad \\qquad \n\\Delta F_K(\\lambda) = \\frac{1}{2}{\\rm tr\\,}\\log(1+M_K).\n\\end{equation}\nTo determine the large $\\lambda$ behaviour of $\\Delta F(\\lambda)$, we need to take first $K\\to \\infty$, \n and then $\\lambda\\to \\infty$.\n \nTo bypass this double limit procedure, we will fix $K$, increase $\\lambda$ until the ratio $\\frac{\\Delta F_K(\\lambda)}{\\sqrt\\lambda}$ reaches a maximum\n\\begin{equation}\n\\label{6.13}\n\\mu_{K} = \\max_{\\lambda}\\frac{\\Delta F_K(\\lambda)}{\\sqrt\\lambda}\\ ,\n\\end{equation}\nand, finally, extrapolate $\\mu_{K}$ to $K\\to \\infty$.\n According to (\\ref{5.11}), the expected value is $c_1=\\frac{1}{2\\pi}$. The explicit numerical \nresults are collected in Figure \\ref{fig:trunc}. In the left panel we show the curves $\\frac{\\Delta F_K(\\lambda)}{\\sqrt\\lambda}$ for $K=20,40,60, \\dots 260$.\nFor each $K$ a maximum in \\rf{6.13} is reached at a value of $\\lambda$ that increases with $K$. The maximum value\n $\\mu_{K}$ is shown in the right panel\nof the figure and fitted by the dashed curve\n\\begin{equation}\n\\label{6.14}\n\\mu_{K}^{\\rm fit} = 0.158-\\frac{0.301}{\\sqrt K}+\\frac{0.290}{K},\n\\end{equation}\nthat empirically works very well. The estimated value of the coefficient of $\\sqrt\\lambda$ in (\\ref{5.11}) is thus $0.158$, which differs by less then 1\\%\nfrom the analytical prediction $\\frac{1}{2\\pi} \\simeq 0.159$.\n\\begin{figure}\n\\begin{center}\n\\includegraphics[width=0.45\\textwidth]{FigTruncLeft.pdf}\n\\includegraphics[width=0.45\\textwidth]{FigTruncRight.pdf}\n\\caption{\\small Analysis of $\\Delta F$ by considering a truncated leading order approximation of the matrix $M$. In the left panel \nwe plot the ratio $\\Delta F_K\/\\sqrt\\lambda$ where $F_K$ is defined in (\\ref{6.12}), and $K=20, 40, \\dots, 260$ from bottom to top.\nFor each $K$, there is a maximal value $\\mu_{K}$. In the right panel we plot $\\mu_{K}$ vs. $K$ (blue dots) and compute its best fit\n(orange dashed line) with a constant plus a leading $\\sim K^{-1\/2}$ and subleading $\\sim K^{-1}$ terms. The best fit\nparameters are in (\\ref{6.14}).\n}\n\\label{fig:trunc}\n\\end{center}\n\\end{figure}\n\n\n\\subsection*{Acknowledgements}\nWe would like to thank M. Bill\\`o, S. Giombi, A. Lerda and J. Russo for related discussions. \nMB was supported by the INFN grant GSS (Gauge Theories, Strings and Supergravity). \nGD was supported by the U.S. Department of Energy, Office of Science,\nOffice of High Energy Physics under Award Number DE-SC0010339.\nAAT was supported by the STFC grant ST\/T000791\/1.\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\n\n\n\n\n\n\n\n\n\\section{Introduction} \\label{Intro} A Brain-Computer Interfaces (BCI) \\cite{graimann2010brain,rahman2018efficient} is a message transmission framework, through which an individual can communicate for necessities by his or her brain signals, even absence of normal pathway of the computer system and a very effective device for the person with severe motor impairment \\cite{gandhi2014eeg,zhao2016ssvep}. It is pragmatic area, which has focused to the design and invent of neuron rooted means to endue solutions for disease prediction , communication and control \\cite{anderson1998multivariate}, \\cite{babiloni2000linear}, \\cite{keirn1990new}. On the ground of acquisition of the brain signal BCI is broadly divided in three categories in literature \\cite{kubler2000brain}, \\cite{schalk2008brain}, viz, invasive, semi-invasive (electrocorticography (ECoG)) and non-invasive(electroencephalography EEG). Economically nature \\cite{hewireless} and calibre to capture brain signals in a non-invasive fashion, EEG is a mostly preferred technique to aquire brain activity for BCI systems \\cite{akram2014p300}, \\cite{keirn1990new}. BCI systems can be used as a \\textit{Response to mental tasks} system, \\cite{bashashati2007survey}, which is perceived to be more practical for locomotive patients. The basic assumption of this type of system is that mental activities lead to produce task-originated patterns. The BCI system's success depends on the precision of classification assorted mental tasks. These tasks requires extractions of discriminative features from the raw EEG signal to distinguish different mental tasks \\cite{zhang2017classification}.\\\\\nIn previous studies, the researchers have utilized plenty approaches of feature extraction for better representation of the EEG signal for the classification process in the BCI domain, for example Band Power \\cite{pfurtscheller1997eeg}, amplitude values of EEG signals \\cite{kaper2004bci}, Power Spectral Density (PSD)\\cite{chiappa2004hmm,neshov2018classification}, \\cite{moore2003real}, \\cite{palaniappan2002new}, Autoregressive (AR) and Adaptive Autoregressive (AAR) parameters \\cite{penny2000eeg}, time-frequency and inverse model-based features \\cite{qin2004motor}, \\cite{kamousi2005classification},\\cite{congedo2006classification}. Wavelet Transform (WT) \\cite{mallat1989theory},\\cite{gupta2012three} and Empirical Mode Decomposition (EMD) \\cite{huang1998empirical}, \\cite{diez2009classification},\\cite{gupta2012relevant},\\cite{bajaj2012classification},\\cite{rutkowski2010emd}, \\cite{fine2010assessing},\\cite{mylonas2016modular} have been used to decompose non-stationary and non-linear EEG signals into smaller frequency components. However, both WT and EMD methods provide low-frequency resolution and may not handle efficiently different overlapping frequency bands \\cite{battista2007application}, \\cite{adamczak2010investigating} present in the EEG. On the other hand, power spectral analysis provides high-frequency resolution. The recording of EEG data occurs from multiple sensors\/channels. Hence, the EEG data contains huge number of features but the recording session of the person usually very small in number. That produces, a small number of data samples. Hence, it suffers the curse of dimensionality as the ratio of features and sample is very small \\cite{bellman1961adaptive}. To overcome this problem, reduction of the dimension using feature selection is suggested in literature \\cite{guyon2003introduction}. In spite of that, no in-depth study has ever been conducted about how to use power spectral features effectively with combination feature selection techniques in BCI the applications.\\\\\nIn this article we provide answers to the following questions:\n\\begin{enumerate}\n\\item Whether extraction of features using power spectral techniques helps in mental task classification.\n\\item Whether the further reduction in dimensionality of features using feature selection approaches improves the classification performance or not.\n \\item Is multivariate feature selection approach better than univariate feature selection approach?\n\\item Which conjunction of feature extraction and selection method performs best for mental task classification? \n\\end{enumerate}\nThus, this present work proposes a procedure of the determination of a compact collection of \\textit{relevant and non-redundant features} from the EEG signal in the two-phase approach. The first phase elaborates about the extraction of PSD features from the EEG signal using three different approaches. In the second level, a set of relevant and non-redundant features is sorted by multivariate filter feature selection approach. \n To investigate the performance of different combinations of PSD method and multivariate feature selection method, experiments are conducted on an open EEG data \\cite{keirn1990new} source. The performance is calculated in terms of classification accuracy and compared with a combination of PSD and a univariate filter feature selection method. \n In order to rank and compare multiple combinations of power spectral density and feature selection methods Ranking method and Friedman's statistical test were also performed. \\\\\n\n\nThe rest of the paper is organized as follows: The Power Spectral Estimation approach has been discussed briefly in Section ~\\ref{PSD}.The proposed approach to obtain minimal subset of relevant and non-redundant of the PSD features using multivariate feature selection methods is included in the Section ~\\ref{PA}. The Descriptive information, data and method results are presented in Section ~\\ref{EX}. In the final, Section ~\\ref{CON} conclusions and future work is discussed.\n\\section{Feature Extraction using Power Spectral Density}\\label{PSD} Power Spectral Density (PSD) is a measure of average power associated with any random sequence \\cite{stoica2005spectral}, which can be catalogued into three categories: (i) Non-parametric, (ii) Parametric and (iii) Subspace. The non-parametric methods are simple to compute and robust. Periodogram based estimation, Bartlett Window, Welch window and Blackman and Tuckey method are examples of this category. However, they do not provide the necessary frequency resolution due to their inability to extrapolate the finite length sequence for data points exceeding the signal length. Another drawback of this approach is spectral leakage \\cite{proakis2001digital}. To overcome the drawback of non-parametric methods, parametric method is suggested. The estimation of PSDs values from a given signal in parametric approaches are carried out by assuming that output of the linear system is driven by white noise and then parameters of the system are calculated. Examples are the \\textit{Yule-Walker autoregressive }(\\textit{AR}) \\textit{method} \\cite{pfurtscheller1998separability}, the \\textit{Burg method} \\cite{chiappa2004hmm}, Covariance and modified covariance etc. The commonly used parametric linear system model is the all-pole model which consists of a filter with all zeroes at the origin and occurs in the z-plane. The output produced by such a filter using white noise as input is an autoregressive (AR) process. Thus, these spectral estimation methods are also sometimes known as \\textit{AR methods}. The AR methods tend to aptly describe data spectrum that is \"peaky\", the data having PSDs value large at certain frequencies, e.g. speech data. Smoother estimates of the PSD are produced by parametric methods than non-parametric methods but are subject to error if the order of model is not chosen correctly. Sub-Space methods are often used when SNR is low. PSDs values are obtained concerning Eigen-decomposition of autocorrelation matrix. For line spectra or spectra having sinusoidal nature Subspace methods are better choice and are also effective in the recognition of sinusoids mixed in noise. However, the subspace methods suffer from the following:The method in all probability does not generate true PSD estimates; it does not store power which is required for processing between the time and frequency domains; and it flunks in getting back the autocorrelation series by computing the inverse Fourier transform of the frequency estimate.\\\\\nFor a given stationary random signal $\\mathbf{x}_m $, the PSD $P_{xx}$ is mathematically related to the autocorrelation sequence by Fourier transform, which regarding normalized frequency $f_{s}$, is given by,\n\\begin{equation} \n{\\ P}_{xx}\\left(f\\right)=\\frac{1}{f_s}\\sum^{\\infty }_{m=-\\infty }{R_{xx}}\\left(m\\right)e^{-\\frac{j2\\pi mf}{f_s}} \n\\end{equation} \nwhere $f_s$ is the sampling frequency. Fourier transform of the autocorrelation of the signal also gives the PSD. Using the inverse discrete-time Fourier transform from the PSD the correlation sequence can be derived as following:\n\\begin{equation} \nR_{xx}=\\int^{\\pi }_{-\\pi }{P_{xx}}\\left(\\omega \\right)e^{-j\\omega m}d\\omega =\\int^{{f_s}\/{2}}_{{-f_s}\/{2}}{P_{xx}}\\left(f\\right)e^{{j2\\pi f}\/{f_s}}df\n\\end{equation} \nThe average power of the sequence \\textit{x}${}_{n}$ over the entire Nyquist interval is represented by\n\\begin{equation} \nR_{xx}\\left(0\\right)=\\int^{\\pi }_{-\\pi }{P_{xx}}\\left(\\omega \\right)d\\omega =\\int^{{f_s}\/{2}}_{{-f_s}\/{2}}{P_{xx}}\\left(f\\right)df\n\\end{equation} \nFor a particular frequency band [$\\omega $${}_{1}$, $\\omega $${}_{2}$], $\\left( 0\\le {{\\omega }_{1}}\\le {{\\omega }_{2}}\\le \\pi \\right)$, the average power of a signal is given by:\n\\begin{equation} \\\n\\overline{P_{\\left[{\\omega }_1,{\\omega }_1\\right]}}=\\int^{{\\omega }_2}_{{\\omega }_1}{P_{xx}}\\left(\\omega \\right)d\\omega \n\\end{equation} \nIt can be seen from the above expression that ${{\\rm P}}_{{\\rm xx}}\\left({\\rm w}\\right)$ represents the power content of a signal in an \\textit{extremely small} frequency band, which is why it is known as the power spectral \\textit{density}. \n\\subsection{Welch Method}\nThis method falls under non-parametric approach. For a finite time duration random signal $\\mathbf{x}_m$ of $N$ interval length, PSD values are estimated with the help of a periodogram which is the squared modulus of discrete Fourier transform of the signal and is given by\n\\begin{equation} \n{\\ P}_{\\mathbf{x}\\mathbf{x}}\\left(f\\right)=\\ \\frac{1}{N}{\\left|\\mathbf{x}\\left(f\\right)\\right|}^2 \n\\end{equation} \nHere $f$ corresponds to the frequency of the sequence and $X(f)$ is the Fourier transform of the signal. A periodogram gives asymptotically non biased estimate of power spectrum. \n\nIn Welch method, $N$ length signal is divided into $K$ overlapped segments each of length $M$. The $i^{th}\\ $segment is given by,\n\\begin{equation} \n{{\\rm \\ \\ \\ \\ \\mathbf{x}}}_{{\\rm i}}\\left({\\rm n}\\right){\\rm =\\mathbf{x}}\\left({\\rm n+iD}\\right)\n\\end{equation} \nHere $n=0$\\dots $N-1$, $i=0$\\dots $K-1$ and $D$ is overlap segment. For this, a windowed segment periodogram is given by \\begin{equation}\n{ {P}^i_{XX}\\left(f\\right)=\\frac{1}{MU}{\\left|\\sum^{N-1}_{i=0}{w(n){{\\mathbf x}}_i(n)e^{-j2\\pi fn}}\\right|}^2\\ }\n\\end{equation}\n\nwhere $w(n)$ is the window function and $U$ is the power of the window function given by, \n\\begin{equation} \n{ U=\\frac{1}{M}\\sum^{M-1}_{n=0}{w^2\\left(n\\right)}} \n\\end{equation} \n\nThe average of $K$ periodograms depicts Welch power spectrum and is given by: \n\\begin{equation} \n{{P}^W_{XX}=\\frac{1}{K}\\sum^{K-1}_{i=0}{P^i_{XX}\\left(f\\right) }} \n\\end{equation} .\n\\subsection{Burg Method}\nThe Burg method \\cite{stoica2005spectral} is a parametric method of spectral analysis. The PSDs values can be obtained by finding $pth$ order coefficients of an AR process. A $pth$ order real valued AR signal $\\mathbf{x}(n)$ (with zero mean) at point $n$ is given by \\cite{palaniappan2002new}.\n\\begin{equation} \n\\mathbf{x}\\left(n\\right)=-\\sum^p_{m=1}{a_mx\\left(n-m\\right)}+e(n)\n\\end{equation} \nHere $a_m$ is AR coefficient of $x(n-m)$, $e(n)$ is the error term at point $n$ independent of past terms. Burg algorithm test to find the AR coefficient by applying more data points and minimizes the forward and backward prediction errors in the least squares sense \\cite{palaniappan2002new}, with the AR coefficients constrained to satisfy the Levinson-Durbin recursion. It provides high resolution for short data records. \nAfter finding AR coefficients by Burg Algorithm, PSD value $S(f)$ at frequency $f$ is given by:\n\\begin{equation} \n\\ S\\left(f\\right)=\\frac{S_e(f)}{{\\left|1+\\ \\sum^p_{i=1}{a_ie^{-j2\\pi fiT}}\\right|}^2} \n\\end{equation} \nHere $T$ is the sampling period and $S_e\\left(f\\right)$ is spectrum of error sequence which should be flat i.e. independent of frequency. One of foremost concern in AR modelling is the choice of order \\textit{p. }To determine $p$, several criterion such as final prediction error (FPE)\\cite{akaike1969fitting}, minimum description length \\cite{rissanen1983universal}, Akaike information criterion (AIC)\\cite{akaike1974new}, and autoregressive transfer function \\cite{parzen1957consistent} are proposed in literature. Among these, AIC is commonly used, which is given by\n\\begin{equation} \n AIC\\left(p\\right)=ln{\\sigma }^2_{wp}+\\frac{2p}{n} \n\\end{equation} \nwhere${\\ \\ \\sigma }^2_{wp}$ is estimated variance in linear prediction error. From Table ~ \\ref{Table-1}, it can be observed that AIC value is minimum for order 5 or 6. We have chosen p=6 in our experiments which is also suggested by Kerin \\& Aunon \\cite{keirn1990new}.\n\\begin{center}\n\\input{chapters\/table3\/arorder}\n\\end{center}\n\\subsection{Multiple Signal Classification (MUSIC)}\n Music is an orthogonal subspace decomposition method is based on Pisarenko idea \\cite{kia2007high} that allows the estimation of low Signal-to-Noise ratio (SNR) frequency components. This method is used to lowers the effect of noise in the analysed signal and finds the optimal frequency resolution in a dynamic signal \\cite{ubeyli2008implementing}. Subspace method assumes that any discrete time signal $s[n]$ is representable in the form of $m$ complex sinusoids with a noise $p[n]$ such that\n\\begin{equation} \ns\\left[n\\right]=\\sum^m_{i=1}{\\overline{A_i}\\ e^{j2\\pi f_i}}+p\\left[n\\right],\\ n=0,1,2,\\dots ,N-1 \n\\end{equation} \nwhere $\\overline{A_i}=\\left|A_i\\right|e^{{\\emptyset }_i}$ is magnitude of $i^{th}$ complex sinusoid, $m,N,f_i\\ and\\ {\\emptyset }_i$ are frequency signal dimension order, number of sample data, frequency and phase of $i^{th}$ complex sinusoid.\n\nThe autocorrelation matrix $\\mathbf{R}$ of signal $s[n]$ is given by:\n\\begin{equation} \n{\\mathbf R}=\\sum^m_{i=1}{{\\left|A_i\\right|}^2p\\left(f_i\\right)p^H\\left(f_i\\right)+\\ {\\sigma }^2{\\mathbf I}} \n\\end{equation} \nwhere $p\\left(f_i\\right)={\\left[1\\ e^{j2\\pi f_i}\\ e^{j4\\pi f_i\\ }\\dots \\ e^{2\\pi {\\left(N-1\\right)f}_i}\\right]}^T$ and ${\\sigma }^2$ is variance of white noise signal, H is hermitian transpose and I is the identity matrix.\n Therefore, it can be observed that $\\mathbf{R}$ is a composition of sum of signal and noise autocorrelation matrices such that\n\\begin{equation} \n{\\mathbf R}{\\mathbf =}{{\\mathbf R}}_{{\\mathbf s}}{\\mathbf +\\ }{\\sigma }^2{\\mathbf I} \n\\end{equation} \nPisarenko has noticed that variance of noise acts with the smallest eigenvalues of $\\mathbf{R}$. The orthogonality of the signal and noise subspace is given as\n\\begin{equation}\np{{({{f}_{i}})}^{H}}v(m+1)=0,i=1,2,...,m\n\\end{equation}\nwhere $v(m+1)$ is the eigenvector of noise in matrix $\\mathbf{R}$ with dimension of $(m+1)\\times(m+1)$\nThe estimation of PSD by Pisarnako is defined as\n\\begin{equation}\n{{P}_{Pisarnako}}=\\frac{1}{{{\\left| p{{\\left( {{f}_{i}} \\right)}^{H}}v\\left( m+1 \\right) \\right|}^{2}}}\n\\end{equation}\nPSD estimation by MUSIC gives better performance than Pisarenko due to addition of averaging of extra noise eigenvectors$(k=m+1,m+2,~\\ldots ,M)$.\nEstimation of PSDs by MUSIC is given by:\n\\begin{equation} \nP_{MUSIC}\\left(f\\right)=\\frac{1}{\\sum^M_{k=m+1}{{\\left|{p(f)}^Hv_{(k)}\\right|}^2}} \n\\end{equation} \nHere $~p{{(f)}^{H}}{{v}_{(k)}}=0$ for $k=1,\\ldots ,m$ using orthogonality of the signal and noise subspace. These PSD values have major peaks at the principal components only. The performance of Music depends on the dimension of the autocorrelation matrix $(M\\le N)$\n\\section{Proposed Approach-Feature Selection}\\label{PA}\nThe number of PSD values obtained using one of the given three methods from multiple channels would be large, otherwise the number of training samples available is in general relatively small. Hence the method suffers from curse-dimensionality problem \\cite{bellman1961adaptive}\n.In order to subdue this problem, there is a need to determine a minimal set of pertinent features which can improve classification accuracy of a learning system. This work has proposed an approach to find a minimal subset of relevant feature using multivariate feature selection methods.\n\nFeature selection \\cite{kohavi1997wrappers}, \\cite{guyon2003introduction} is one of the widely accepted approach to determine relevant features. In spite of available plenty of research works for the feature selection, not much work has been carried out in the domain of mental task classification. The filter and the wrapper approaches are the two major approaches of feature selection techniques. In filter approach, the step of selecting optimal features set is considered as one of the pre-processing steps of just before applying any machine learning algorithm. This approach adopts only inherent properties of the features and does not consider any virtue of any learning algorithm. Hence, it may not select the optimal feature set for the learning algorithm. Instead, the wrapper approach \\cite{kohavi1997wrappers} finds an optimal features subset, which is compatible with the given learning algorithm. The given classifier requires to be trained for each feature of set of the all features separately in the wrapper approach, which makes it more computational costly than filter approach. \\\\\nFilter approach is further partitioned in two categories on the basis of the way of opting features \\cite{guyon2003introduction}, as Univariate (single feature ranking) and Multivariate (feature subset ranking). Univariate method utilizes a scoring function for measuring relevance of the feature. Implementation of the Univariate method is very simple. In BCI field the researchers, \\cite{koprinska2010feature},\\cite{rodriguez2013efficient},\\cite{guerrero2010eeg},\\cite{murugappan2010classification} used univariate filter method. The performance of learning model usually improve with the help of reduced set of relevant features obtained by Univariate feature selection method. But it does not captures the correlation among the features. Hence there may be many redundant features in the subset of relevant feature which may take down the performance of learning model. Wrapper method, \\cite{bhattacharyya2014automatic}, \\cite{dias2010feature}, \\cite{keirn1990new} has been applied to obtain a subset of non-redundant features for the mental task classification. Due to high- dimensionality of feature of EEG data, wrapper approach is not feasible option for mental task classification as it will become more computationally expensive. Hence we have applied both uni-variate as well as multivariate filter feature selection algorithms.\\\\\nLet us assume we have a data matrix $\\mathbf{X}$, of $m$ rows,and $k+1$ columns, with data sample $\\mathbf{x}_{i}, i=1,\\ 2,\\ \\ldots, m$; containing features set $S=\\mathbf{f}_{1}, \\mathbf{f}_{2}, \\ldots \\mathbf{f}_{k}$ and class label $C_{1}, C_{2}, \\ldots C_{n}, \\text{where}\\ n\\leq m$.\n\\subsection{Uni-variate Feature selection}\n\\subsubsection{Pearson's Correlation}\nPearson's correlation coefficient (CORR), \\cite{pearson1920notes,dowdy2011statistics} is employed to determine linear relationship between two variable. CORR of \\textit{i}$^{th}$ feature vector (\\textbf{f}$_i$) with the class label vector (\\textbf{c}) is given by\n \\begin{eqnarray}\nCORR\\left( {{\\mathbf{f}}_{i}},\\mathbf{c} \\right)=\\frac{cov({{\\mathbf{f}}_{i}},\\mathbf{c})}{{{\\sigma }_{{{\\mathbf{f}}_{i}}}}{{\\sigma }_{c}}}=\\frac{E[({{\\mathbf{f}}_{i}}-{{\\mu }_{i}})(\\mathbf{c}-\\bar{c})}{{{\\sigma }_{{{\\mathbf{f}}_{i}}}}{{\\sigma }_{c}}}\n \\end{eqnarray}\nwhere $ i=1,\\ 2,\\ \\ldots ,\\ k$, ${\\sigma }_{{{\\mathbf f}}_{i}}{,\\ \\sigma }_{{\\mathbf c}}$ represent respectively the standard deviations of feature vector ${{\\mathbf f}}_{i}$ and ${\\mathbf c}$. $cov\\left({{\\mathbf f}}_{i}{\\mathbf ,\\ }{\\mathbf c}\\right)$ represents the covariance between ${{\\mathbf f}}_{i}$ and ${\\mathbf c}$, ${{\\mu }_{i}}=\\frac{1}{k}\\underset{i=1}{\\overset{k}{\\mathop \\sum }}\\,{{X}_{ik}}$ and $\\bar{c}=\\frac{1}{k}\\underset{i=1}{\\overset{k}{\\mathop \\sum }}\\,{{c}_{i}}$ are the mean of \\textbf{f}$_k$ and \\textbf{c} respectively.\n\nRange of $CORR\\left({{\\mathbf f}}_{i},{\\mathbf c}\\right)$ falls between -1 \\& +1. The value nearby to $|1|$, depicts the stronger linear relation among the prescribed variables while zero value implies no correlation between the two variables.\n\\subsubsection{Mutual Information}\nMutual information [MI] is a feature ranking method on basis of Shannon entropy, which determines relationship between two variables. MI of a feature vector \\textbf{f}$_i$ and the class vector $\\mathbf{c}$ can be calculated as\\cite{shannon1949mathematical}:\n\\begin{equation}\nI({\\mathbf{f_\\mathit{i}}},\\mathbf{c})=\\sum P({\\mathbf{f_\\mathit{i}}},\\mathbf{c})\\log\\frac{P({\\mathbf{f_\\mathit{i}}},\\mathbf{c})}{P({\\mathbf{f_\\mathit{i}}})P(\\mathbf{c})}\n\\end{equation}\n where $P({\\mathbf{f_\\mathit{i}}})$ and $P(\\mathbf{c})$ are the marginal probability distribution functions for random variables ${{\\mathbf f}}_{{\\mathit i}}$ and \\textbf{c} respectively and $P({\\mathbf{f_\\mathit{i}}},\\mathbf{c})$ is joint probability distribution. The most extreme estimation of MI demonstrates the higher reliance of the variable on the class label. The advantage of MI is that it can discover even the non-linear dependency between the attribute and the relating class label vector \\textbf{c}. \n \n\\subsubsection{Fisher Discriminant Ratio }\nFisher Discriminant Ratio (FDR) is a univariate filter feature selection technique which depends on the statistical virtue of the attributes or features. FDR (${{\\mathbf f}}_{{\\mathit i}}$) for $i^{th}$ features for two class $C_{1}$ and $C_{2}$ can be given as:\n\\begin{equation}\nFDR({{\\mathbf{f}}_{i}})=\\frac{{{({{\\mu }_{1(i)}}-{{\\mu }_{2(i)}})}^{^{2}}}}{\\sigma _{1(i)}^{2}+\\sigma _{2(i)}^{2}}\n\\end{equation}\nhere, ${\\mu}_{ 1(i)}$ and ${\\sigma}^{2}_{1(i)}$ are the mean and deviation of the data of class $C_{1}$ respectively for $i^{th}$ feature. \n\\subsubsection{Wilcoxon's Ranksum Test}\nWilcoxon Ranksum Test, suggested by \\cite{wilcoxon1945individual}, is a non-parametric statistical test, accomplishes between data of two classes on the basis of median of the samples having no prior knowledge of probability distribution.\n\n The statistical distinctness $t(\\mathbf{f}_i)$ of feature $\\mathbf{f}_i$ for known two classes, class $C_{1}$ and $C_{2}$ using Wilcoxon's statistics can be defined as \\cite{li2008gene}:\n\\begin{equation}\nt\\left( {{\\mathbf{f}}_{i}} \\right)=\\underset{l=1}{\\overset{{{N}_{i}}}{\\mathop \\sum }}\\,\\underset{m=1}{\\overset{{{N}_{j}}}{\\mathop \\sum }}\\,DF(({{X}_{li}}-{{X}_{mi}})\\le 0)\n\\end{equation}\nwhere $N_i$ and $N_j$ are the number of the data example in class $C_{1}$ and $C_{2}$ respectively, $DF$ is the logical discriminative mapping between two classes of data, which defines an estimation of 1 or 0 corresponding to true or false and $X_{li}$, is the expression values of $i^{th}$ feature for $l^{th}$ sample. The value of $t(\\mathbf{f}_i)$ lies between zero and $(N_i\\times N_j)$. The relevance of the feature can be fined as:\n\\begin{equation}\nR\\left( t({{\\mathbf{f}}_{i}}) \\right)=\\text{max}(t({{\\mathbf{f}}_{i}}),{{N}_{i}}\\times {{N}_{j}}-t({{\\mathbf{f}}_{i}}))\n\\end{equation}\n\\subsection{Multivariate Feature Selection}\nTime-efficient multivariate filter method picks a subset of features, which are relevant to the class label of data and independent from each other. Thus it up dues the limitations of both uni-variate and wrapper approaches. Thus we have opted most widely utilized multivariate filter methods by research community for the dimensionality reduction, are Bhattacharya distance measure \\cite{bhattacharyya1946measure}, Ratio of scatter matrices \\cite{devijver1982pattern}, Linear regression \\cite{park2007forward} and minimum Redundancy-Maximum Relevance (mRMR) \\cite{peng2005feature}. A brief discussion on the mentioned techniques is given below.\n\\subsubsection{Bhattacharaya Distance}\nIn literature, Bhattacharya distance is used for find similarity between two continuous or discrete probability distribution. It is a special case of Chernoff distance that provides similarity overlap of the distribution. For multivariate normal probability distribution, Chernoff Distance measure can be written as \\cite{chernoff1952measure}:\n\\begin{equation}\n\\begin{split}\nJ_c=\\frac{1}{2}\\beta(1-\\beta)(\\boldsymbol{\\mu}_{2}-\\boldsymbol{\\mu}_{1})^{T}[(1-\\beta)\\boldsymbol{\\Sigma}_{1}+\\beta\\boldsymbol{\\Sigma}_{2}]^{-1}(\\boldsymbol{\\mu}_{2}-\\boldsymbol{\\mu}_{1})+\\\\ \\frac{1}{2}log\\frac{\\left | (1-\\beta)\\boldsymbol{\\Sigma}_{1}+\\beta\\mathbf{\\Sigma}_{2} \\right |}{\\left |\\mathbf{\\Sigma}_{1} \\right |^{1-\\beta}\\left |\\mathbf{\\Sigma}_{2} \\right |^{\\beta}}\n\\end{split}\n\\end{equation}\nwhere ${\\boldsymbol{\\mu}}_{i}$ and $\\mathbf{\\Sigma}_{i}$ are mean vector and covariance matrix for class $C_{i}$ respectively($i$=1,2).\\\\When $\\beta$ is $\\frac{1}{2}$ then this distance is called as Bhattacharya distance (BD)\\cite{bhattacharyya1946measure}, which is given as \n\\begin{equation}\nJ_B=\\frac{1}{8}(\\boldsymbol{\\mu}_{2}-\\boldsymbol{\\mu}_{2})^{T}(\\boldsymbol{\\mu}_{2}-\\boldsymbol{\\mu}_{2})+\\frac{1}{2}log\\frac{(\\frac{\\left |\\boldsymbol{\\Sigma}_{1}+\\boldsymbol{\\Sigma}_{2}\\right |}{2})}{\\left | \\boldsymbol{\\Sigma}_{1} \\right |^{\\frac{1}{2}}\\left |\\boldsymbol{\\Sigma}_{2} \\right |^{\\frac{1}{2}}}\n\\end{equation}\n\\subsubsection{Ratio of Scatter Matrices}\nIn literature, the trace of ratio of scatter matrices (SR),is a measure of separability, as the trace of a scatter matrix is equal to the sum of the eigenvalues and therefore an indicator of the total variance in the data. How well features cluster around their class mean, as well as, how well they separate the class means. The scatter matrices, within-class scatter matrices,$\\mathbf{S}_{w}$, and between class scatter matrices, $\\mathbf{S}_{b}$, can be defined as\n\\begin{equation}\n\\mathbf{S}_{w}=\\sum_{i=1}^{c}P_{i}E[(\\mathbf{x}-\\boldsymbol{\\mu}_{i})^{T}(\\mathbf{x}-\\boldsymbol{\\mu}_{i})]\n\\end{equation}\n\\begin{equation}\n\\mathbf{S}_{b}=\\sum_{i=1}^{c}P_{i}(\\boldsymbol{\\mu}_{i}-\\mathbf{\\mu_{0}})^T(\\boldsymbol{\\mu}_{i}-\\mathbf{\\mu_{0}})\n\\end{equation}\nwhere $\\boldsymbol{\\mu}_{i}$, $P_{i}$ and $\\mu_{0}$ are mean vector of $i^{th}$ class data, prior probability of $i^{th}$ class data and global mean of data samples respectively.\\\\\nFrom the definitions of scatter matrices, the criterion value which has to be maximized, is given as:\n\\begin{equation}\nJ_{SR}=\\frac{trace(\\mathbf{S}_{b})}{trace(\\mathbf{S}_{w})}\n\\end{equation}\nWhen intra cluster distance is very small and the inter cluster distance is very large $J_{SR}$ takes the high value. The main advantage of this criterion that it is not subject any external parameters and assumptions of any probability density function. Also the measure $J_{SR}$ under linear transformation has the advantage of being invariant under linear transformation.\n\\subsubsection{Linear Regression}\nLinear regression is a statistical approach, which determines casual link of an independent variable upon a dependent variable. The class label of the data is recognized as the target dependent variable and the feature that affect the target is known as independent variable. There may be many features which can affect the class of the data, thus in such case multiple regression analysis would be more appropriate. A multiple regression model with $k$ independent features $\\mathbf{f_1}, \\mathbf{f_2}, \\ldots, \\mathbf{f_k}$ and a class variable $y$ can be written as \\cite{park2007forward};\n\\begin{equation}\ny_i=\\beta _{0}+\\beta _{1}X_{i1}+...+\\beta_{k}X_{ik}+\\zeta _{i}, i=1,2,...,n\n\\end{equation}\nwhere $\\beta_{0},\\beta_{1},...,\\beta_{k}$ is set of fixed values calculated by the class label $y$ and observed values of $\\mathbf{X}$ and $\\zeta _{i}$ is the error term. The sum of squared error (SSE) is given by\n\\begin{equation}\nSSE=\\sum_{i=1}^{n}(y_{i}-y_{i}^{p})^2\n\\end{equation}\nwhere $y_{i}$ and $y_{i}^{p}$ are observed and predicated values respectively. The lower value of SSE depicts preferable regression model. The total sum of squares (SSTO) can be calculated as:\n\\begin{equation}\nSSTO=\\sum_{i=1}^{n}(y_{i}-\\bar{y})^2\n\\end{equation}\nwhere $\\bar{y}$ is the mean value of $y_{i}, i=1,2,...,n$. The criterion function $J_{LR}$ is given as:\n\\begin{equation}\nJ_{LR}=R^2=1-\\frac{SSE}{SSTO}\n\\end{equation}\nThe value of $J_{LR}$ lies between $0$ and $1$. The feature for which the value of $J_{LR}$ is higher is selected.\n\\subsubsection{minimum Redundancy-Maximum Relevance}\nminimum-redundancy maximum-relevance (mRMR) \\cite{park2007forward} is based on mutual information to discover a subset of features that have minimum redundancy among themselves and maximum relevance with the class labels. mRMR uses mutual information $I(\\mathbf{f}_i,\\mathbf{f}_l)$ as a measure of similarity between two feature vector $\\mathbf{f}_i$ and $\\mathbf{f}_l$, which is given as pursues:\n\\begin{equation}\nI(\\mathbf{f}_{i},\\mathbf{f}_{l})=\\sum _{k,l}p(\\mathbf{f}_{k},\\mathbf{f}_{l})\\log(\\frac{p(\\mathbf{f}_{i},\\mathbf{f}_{l})}{p(\\mathbf{f}_{i})p(\\mathbf{f}_{l})})\n\\end{equation}\nwhere $p(\\mathbf{f}_{i})$,$p(\\mathbf{f}_{l})$ are the marginal probabilities of $k^{th}$ and $l^{th}$ features respectively and $p(\\mathbf{f}_{i},\\mathbf{f}_{l})$ is selected joint probability density. The relevance between the set of features S and the target class label vector $\\mathbf{c}$, denoted by $REL$, is expressed as:\n\\begin{equation}\nREL=\\frac{1}{\\left | S \\right |}\\sum_{\\mathbf{f}_i\\in S}I(\\mathbf{f}_i,\\mathbf{c})\n\\end{equation}\nThe average redundancy among features in the set $S$, denoted by $RED$, is defined as:\n\\begin{equation}\nRED=\\frac{1}{\\left | S \\right |^2}\\sum_{\\mathbf{f}_i,{\\mathbf{f}_l}\\in S}I(\\mathbf{f}_i,\\mathbf{f}_l)\n\\end{equation}\nwhere $S$ denotes the subset of features and $\\left | S \\right |$ denotes the number of features in set $S$. Minimum redundancy and maximum relevance is measured by:\n\\begin{eqnarray}\n\\begin{split}\nJ_{MID}=max(f_{i})[REL-RED]=\\\\ max(f_{i})\\left[\\frac{1}{\\left | S \\right |}\\sum_{\\mathbf{f}_i\\in S}I(\\mathbf{f}_i,c)-\\frac{1}{\\left | S \\right |^2}\\sum_{\\mathbf{f}_i,{\\mathbf{f}_l}\\in S}I(\\mathbf{f}_i,\\mathbf{f}_l)\\right]\n\\end{split}\n\\end{eqnarray}\nClearly, the maximum values of $J_{MID}$ can be achieved with minimum redundancy among features and maximum relevance with target vector.\n\\section{Results and Discussion}\\label{EX}\n\\subsection{Data}\n For the simulation of our proposed framework, we have utilized a freely available Mental Task Classification data-set which has been used first time in the work of(Keirn and Aunon, 1990). Seven subjects (person) participated in the recording of this EEG dataset, but we did not utilize of Subject 4 due to incomplete information. In the baseline task (relax: B) each subject was instructed to carry out five different mental tasks ; the mental Letter Composing task (L); the Non trivial Mathematical task (M); the Visualizing Counting of numbers written on a blackboard task (C), and the Geometric Figure Rotation task (R). Each recording consists of the five trials of each of above said five mental tasks. EEG signals are recorded from C3, C4, P3, P4, O1 and O2 electrode position with A1 and A2 as the reference electrode as shown in Figure ~\\ref{Fig-1}.\n\\begin{figure}\n\\centering\n\\includegraphics[width=.35\\textwidth]{new_elepla1}\n\\caption{Electrode placement of EEG recording adapted from\\cite{palaniappan2002new}.}\n\\label{Fig-1}\n\\end{figure}\n Each trial is recorded for 10 seconds duration recorded with the sampling rate of 250 per second, which resulted in 2500 samples points per trial.\n\nOverall flow of the proposed approach for mental task classification is shown in Figure ~\\ref{Flow}. The proposed approach consists four steps: segmentation feature extraction, feature selection and classification; to distinguish two different mental tasks. The main contribution of the work is employment of filter feature selection algorithm to enhance performance of learning algorithm for the classification of the mental tasks.\n\\subsection{Feature Formation}\nFor feature vector formulation, each trial data is pre-processed by decomposing into half-second segments, generating 20 fragments per trial for each subject. \n\\begin{figure}\n\\centering\n\\includegraphics[width=.35\\textwidth]{flow_diagram}\n\\caption{Flow Diagram of the proposed approach for mental task classification.}\n\\label{Flow}\n\\end{figure}\nThe extraction of features has been carried out from each signal using three different power spectral density approaches such as Welch, Burg, and MUSIC separately. A total of 52 PSD values are obtained from each channel. Combining PSD values of all six channels, each signal is represented regarding 312 PSD values. PSD values obtained for different tasks using Burg (parametric approach) for all six channels are shown in Figure ~\\ref{Fig-3},\n\\begin{figure*}\n \\begin{minipage}{.5\\textwidth}\n \\includegraphics[width=0.28\\textheight]{channel1_burg}\n \\end{minipage}\n \\begin{minipage}{.5\\textwidth}\n \\includegraphics[width=0.28\\textheight]{channel2_burg}\n \\end{minipage}\n \\begin{minipage}{.5\\textwidth}\n \\includegraphics[width=0.28\\textheight]{channel3_burg}\n \\end{minipage}\n \\begin{minipage}{.5\\textwidth}\n \\includegraphics[width=0.28\\textheight]{channel4_burg}\n \\end{minipage}\n \\begin{minipage}{.5\\textwidth}\n \\includegraphics[width=0.28\\textheight]{channel5_burg}\n \\end{minipage}\n \\begin{minipage}{.5\\textwidth}\n \\includegraphics[width=0.28\\textheight]{channel6_burg}\n \\end{minipage}\n\\caption{Comparison of features of different mental tasks using Burg method.}\n\\label{Fig-3}\n\\end{figure*}\nwhich shows that the extracting features from Burg PSD approach are effective in distinguishing different mental tasks. It can be also observed that PSD values at some frequency values differ considerably among different mental tasks (e.g. Frequency range of 6-9 Hz for channel C3, 6-13 Hz for channel C4, 6-13 Hz for channel P3, 6-16 Hz for channel P4, 6-9 Hz for channel O1 and 16-19 Hz for channel O2). This difference in PSD values can help in distinguishing different mental tasks. While PSD values at some frequency values take similar values (e.g. Frequency values above 15 Hz for C3, above 17 Hz for channel C4, above 13 Hz for channel O1, above 30 Hz for channel O2, above 20 Hz for channel P3 and above 22 Hz for channel P4) and cannot help in distinguishing different mental tasks. Similar observations are also noted for Welch and MUSIC methods. This suggests that all features (PSD values) are not relevant for mental task classification.\n\\subsection{Application of Uni-variate Feature Selection}\nTo determine relevant features that can distinguish different mental tasks, four different univariate methods: Correlation (Cor), Fisher Discriminant Ratio (FDR), Mutual Information (MI) and Wilcoxon's Rank Sum Test (Ranksum) are investigated in our experiment. FDR score corresponding to features obtained from each of the three PSD approaches to distinguish Baseline task from Count Task is shown from Figure ~\\ref{Fig-4} to Figure ~\\ref{Fig-6}.\n\\begin{figure*}\n\\centering\n\\includegraphics[width=.65\\textwidth]{fdr_welch_new}\n\\caption{Fisher Discriminant Ratio score for a pair of Baseline task and Count Task for features extracted using Welch.}\n\\label{Fig-4}\n\\end{figure*}\n\\begin{figure*}\n\\centering\n\\includegraphics[width=.65\\textwidth]{fdr_burg_new}\n\\caption{Fisher Discriminant Ratio score for a pair of Baseline task and Count Task for features extracted using Burg.}\n\\label{Fig-5}\n\\end{figure*}\n\\begin{figure*}\n\\centering\n\\includegraphics[width=.65\\textwidth]{fdr_music_new}\n\\caption{Fisher Discriminant Ratio score for a pair of Baseline task and Count Task for features extracted using MUSIC.}\n\\label{Fig-6}\n\\end{figure*}\n It can be seen from these figures that FDR score corresponding to few features is very high and very less for others. This suggests that some features are more relevant than others. Similar observations are also noted for other univariate methods and other pairs of tasks.\nFor all univariate feature selection methods, the top 25 -ranked features are incrementally added to develop the decision model using forward feature selection approach. Comparison of different methods is reported in terms of maximum average classification accuracy for top features of 10 runs of 10 cross-validations. We have used three well-known classifiers: Linear Discriminant Analysis (LDA), Quadratic Discriminant Analysis (QDA) and Support Vector Machine (SVM) in our experiments. Figure ~\\ref{Fig-7} shows a comparison of all combinations of three PSD approaches and four univariate methods with each of the three PSD approaches without any feature selection in terms of average classification accuracy (over six subjects for all combination of tasks).\n\\begin{figure*}\n\\centering\n\\includegraphics[width=.65\\textwidth]{univariate}\n\\caption{ Comparison of different combination of univariate methods and PSD methods in terms of classification accuracy.}\n\\label{Fig-7}\n\\end{figure*}\nWe can observe the following from Figure ~\\ref{Fig-7}\n\\begin{itemize}\n\\item In general, the classification accuracy of all the three PSD approaches improves with the use of univariate feature selection method with all three classifiers. \n\\item Among all combinations of PSD approaches, univariate methods, and classifiers, the maximum classification accuracy is achieved with the combination of Burg, FDR, and SVM.\n\\item Among four univariate feature selection methods, maximum classification accuracy is achieved with FDR.\n\\end{itemize}\n\\subsection{Application of Multivariate Feature Selection}\nFigure ~\\ref{Fig-8} shows a color map of correlation values among top 20 relevant features obtained using the combination of FDR and Burg method to distinguish Baseline task from Count Task. \n\\begin{figure*}\n\\centering\n\\includegraphics[width=.65\\textwidth]{coburgbc}\n\\caption{ Colormap of Correlation values for top 20 PSD features obtained using combination of FDR .}\n\\label{Fig-8}\n\\end{figure*}\nIt can be noted that some of the correlation values take a high value which depicts that such features are correlated (redundant) among themselves. Similar observations are also noted for other combinations of PSD approaches and univariate methods for another pair of tasks. This observation suggests the need to determine a subset of relevant and non-redundant features to further improve the performance of mental task classification.\nFor this, we used four well known multivariate methods: linear regression (LR), Bhattacharya distance (BD), Scatter Ratio (SR), Minimum Redundancy-Maximum Relevance (mRMR) to obtain minimal subset of non-redundant and relevant features using forward feature selection approach. Figure ~\\ref{Fig-9} shows a comparison of all combinations of three PSD approaches and four multivariate methods with the combination of PSD approaches and FDR (best performing univariate method) in terms of average classification accuracy.\n\\begin{figure*}\n\\centering\n\\includegraphics[width=.65\\textwidth]{multivariate}\n\\caption{ Comparison of all combinations of three PSD approaches and four multivariate methods with combination of PSD approaches and FDR in terms of average classification accuracy.}\n\\label{Fig-9}\n\\end{figure*}\n We can observe the following from Figure ~\\ref{Fig-9}\n\\begin{itemize}\n\\item Among all combinations of PSD approaches, multivariate feature selection methods and classifiers, the maximum classification accuracy is achieved with the combination of Burg, LR, and LDA.\n\\item The performance of all combination of PSD approaches and multivariate methods is better in comparison to the combination of PSD approaches and FDR for LDA and QDA in terms of classification accuracy. \n\\item The performance of MUSIC is worst among three PSD approaches with univariate as well as multivariate feature selection methods.\n\\end{itemize}\n\\subsection{The Rankings of Respective Combinations of Feature Extraction and Selection Methods}\nTo investigate the relational rank of both univariate and multivariate methods feature selection techniques in combination with a feature extraction method, we have utilized the robust ranking approach \\cite{adhikari2012performance}, on the ground of percentage gain in classification accuracy with respect to without applying any feature selection method \\cite{gupta2017fuzzy}.\\\\\nFigure ~\\ref{Fig-10} shows twenty-four combinations of FS-FXT methods which are the feature selection and extraction methods. These methods are compared on the basis of percentage gain in accuracy of the different combination of selection and extraction methods and their corresponding ranks. From the Figure ~\\ref{Fig-10}, we can observe that the combination of multivariate feature selection with all three feature extraction is ranked better in comparison to the combination of univariate feature selection and all three feature extraction methods except one combination (BD-MUSIC). Among all combination of selection and extraction methods, the combination of LR and Burg is best, whereas the team of MUSIC and Ranksum performs the worst.\n\\begin{figure*}\n\\centering\n\\includegraphics[width=1\\textwidth]{ranking_4chp.jpg}\n\\caption{ Ranking of different combinations of Feature Extraction and Selection methods}\n\\label{Fig-10}\n\\end{figure*}\n\\subsection{Friedman statistical test}\nWe have applied a non-parametric statistical test known as Friedman test in order to compare the statistically significant difference evolving in various combination of the feature selection and the PSD methods. \nFrom Table ~\\ref{Table_4-12}, it can be noted that almost (11 out of 12) all combinations of multivariate feature selection with PSD methods obtained better rank than the combination of univariate feature selection method and PSD methods.\n\\input{chapters\/table4\/freiedman_ranking}\nThe SEL-EXT pair performance is also examined with respect to a control method, i.e., the one that emerges with the lowest rank (combination of LR and Burg). \nIn the comparison of the control method with other 23 combinations of feature selection and feature extraction method, adjusted p-values \\cite{derrac2011practical} we computed in order to take into account the error accumulated and provide the correct correlation. A set of post-hoc procedures to provide correct correlation is defined in the literature. The adjusted p-values in the method are computed in order to find whether the control method shows any statistical difference when compared with the remaining methods. For pair-wise comparisons, the widely used post hoc methods to obtain adjusted p-values are \\cite{derrac2011practical}: Holm, Hochberg and Hommel procedures. Table ~\\ref{Table_4-13} illustrates adjusted p-values for the Hommel procedure.\n\\input{chapters\/table4\/control_methods}\nThe values in Table ~\\ref{Table_4-13} represents the p-value when the pair-wise comparison with control method(Burg+LR) is conducted. The bold values suggest the significant difference observed from the control method (Burg+LR) with the combinations at the significance level of $0.05$. This demonstrates that combination of Burg with LR performs significantly better than all combinations of univariate method and feature extraction methods. It also performs significantly better than few combinations of multivariate method and feature extraction method.\n\\section{Conclusion} \\label{CON}\nIn this paper, we have examined the performance of the combination of three different PSD approaches, with four well-known uni-variates as well as four very popular multi-variates, filter feature selection methods. The experimental findings demonstrate that the multivariate feature selection algorithms endue more distinguishable feature set for the mental task classification, compared with univariate feature selection approach. The outcome determined features for the mental task classification by a minimal subset of relevant and non-redundant features. Experimental results demonstrate significant improvement in classification accuracy utilizing the selected feature selection methods. It is observed that the performance of multivariate filter feature selection methods is, in general, better than univariate filter feature selection methods. The combination of Burg method, LR and Linear Discriminant Analysis(LDA) achieved maximum classification accuracy among all other combinations.\n\n Statistical tests also endorsed that the performance of the combination of Burg and the linear regression is notably different from the majority of the combinations. It has also been observed that for mental task classification multivariate feature selection approach works better than univariate feature selection approach in most of the cases with the conjunction of power spectral density approach. \n\n In the future, we would like to extract spectral density of different brain frequency separately. Since the comparisons and investigations have been done on binary mental task classification, we would, therefore, like to extend this approach for multi-class mental tasks classification. \n\\section*{Acknowledgements}\nThe first author would like to express his gratitude to the Council of Scientific \\& Industrial Research (CSIR), India, and acknowledge the financial support for the research work.\n\\bibliographystyle{IEEEtran}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\\noindent Many key results in algebraic geometry can be established using topological and combinatorial descriptions of a given variety, or of its degenerations and deformations. However, even with a clear combinatorial model on such a degeneration or deformation, determining which properties of the original variety can be controlled by combinatorics is still in general a difficult question.\n\nThis paper considers arbitrary smooth $3$-fold flopping contractions, which form a fundamental building block of the minimal model programme. Our main point is that, as far as their enumerative geometry is concerned, all such flopping contractions \\emph{are} combinatorial, provided we are content with describing only the shape of the enumerative invariants, rather than their precise values. This qualitative perspective allows us to extract, and prove rather easily, many fundamental results. We determine which curve classes give rise to non-zero invariants, then control how these invariants transform under flop, in a visually pleasing and satisfyingly combinatorial manner. Along the way it is necessary to enhance existing geometric structures, such as the movable cone.\n\n\n\\subsection{Gopakumar--Vafa: finite arrangements}\nLet $f \\colon \\EuScript{X} \\to \\mathop{\\rm Spec}\\nolimits \\EuScript{R}$ be a crepant resolution of a $3$-fold isolated cDV singularity, equivalently a germ of a smooth $3$-fold flopping contraction. The morphism $f$ contracts a finite collection $\\{ \\Curve_i \\subseteq \\EuScript{X} \\mid i \\in \\scrI^{\\kern 0.5pt\\mathrm{c}}\\}$ of complete curves to a point, and these freely generate the group of algebraic curve classes\n\\begin{equation*} \nA_1(\\EuScript{X}) = \\langle \\Curve_i \\mid i \\in \\scrI^{\\kern 0.5pt\\mathrm{c}} \\rangle_{\\mathbb{Z}}.\n\\end{equation*}\nGiven $\\upbeta \\in A_1(\\EuScript{X})$, Katz defines an associated Gopakumar--Vafa (GV) invariant \\cite{KatzGV} \n\\[\nn_{\\upbeta}=n_{\\upbeta}(\\EuScript{X}) \\in \\mathbb{Z}_{\\geq 0}\n\\]\nwhich we recall in Subsection~\\ref{GV section} below. It is an important invariant of flopping contractions, with close connections to other known invariants \\cite{DW1, TodaGV}.\n\n\\medskip\nOur first result determines those $\\upbeta$ for which $n_{\\upbeta} \\neq 0$. The description is direct and combinatorial, encoded in the associated finite hyperplane arrangement $\\EuScript{H}_\\EuScript{I}$ of Iyama and the second author \\cite{IyamaWemyssTits}. It turns out that this arrangement is the movable cone, and thus the GV invariants are to first approximation encoded by the walls of this cone. There is however a slight catch: \\emph{combinatorially} the walls carry multiplicities, and this data is not part of the definition of the movable cone. This multiplicity, which is new information (see Remark~\\ref{rem: movable via alg geom}), turns out to be the key to determining whether $n_\\upbeta\\neq 0$. \n\n\n\\medskip\nAs is standard, and recalled in Subsection~\\ref{sec: Dynkin recap}, slicing $\\EuScript{X} \\to \\mathop{\\rm Spec}\\nolimits \\EuScript{R}$ by a generic hyperplane section gives rise to a partial crepant resolution of an ADE surface singularity. From this slicing we thus obtain the Dynkin diagram $\\Delta$ of the ADE surface singularity, together with a subset $\\EuScript{I}$ of nodes: the full minimal resolution dominates the partial resolution, and $\\EuScript{I}$ are the curves which are contracted by this morphism.\n\n\\begin{example}\\label{example: intro}\nAs the running example, consider a two-curve smooth $3$-fold flop for which the corresponding Dynkin data is $\\Eeight{B}{P}{B}{B}{B}{B}{B}{P}$, where by convention $\\EuScript{I}$ equals the six black nodes. The Dynkin data gives rise to a finite \\emph{intersection arrangement} $\\EuScript{H}_{\\EuScript{I}}\\subseteq\\mathbb{R}^{|\\Delta|-|\\EuScript{I}|}=\\mathbb{R}^2$ \\cite[Section~3]{IyamaWemyssTits}. One method of calculating $\\EuScript{H}_{\\EuScript{I}}$ is to first restrict all 120 positive roots of $E_8$ to the subset $\\scrI^{\\kern 0.5pt\\mathrm{c}} = \\Delta \\setminus\\EuScript{I}$, and thus obtain the set \n\\[\n\\{ 01, 11, 21, 42, 31, 41, 10, 20, 30\\}.\n\\] \nThese so-called \\emph{restricted roots} give rise to hyperplanes in the dual space, where for example $42$ gives rise to the hyperplane $4x+2y=0$. The output is thus the following hyperplane arrangement, which we emphasise is constructed entirely from $\\EuScript{I}\\subseteq\\Delta$.\n\\begin{equation}\n\\begin{array}{cccc}\n\\begin{array}{c}\n\\begin{tikzpicture}[scale=0.5]\n\\draw[->,densely dotted] (180:2cm)--(0:2cm);\n\\node at (0:2.5) {$\\scriptstyle x$};\n\\draw[->,densely dotted] (-90:2cm)--(90:2cm);\n\\node at (90:2.5) {$\\scriptstyle y$};\n\\end{tikzpicture}\n\\end{array}\n&\n\\begin{array}{c}\n\\begin{tikzpicture}[scale=1]\n\\draw[line width=0.5 mm,Pink] (180:2cm)--(0:2cm);\n\\node at (180:2.2) {$\\scriptstyle 1$};\n\\draw[line width=0.5 mm,Orange] (135:2cm)--(-45:2cm);\n\\node at (135:2.2) {$\\scriptstyle 1$};\n\\draw[line width=0.5 mm, Blue] (116.57:2cm)--(-63.43:2cm);\n\\node at (116.57:2.2) {$\\scriptstyle 2$};\n\\draw[line width=0.5 mm, Green] (108.43:2cm)--(-71.57:2cm);\n\\node at (108.43:2.2) {$\\scriptstyle 1$};\n\\draw[line width=0.5 mm, Grey] (104.04:2cm)--(-75.96:2cm);\n\\node at (104.04:2.2) {$\\scriptstyle 1$};\n\\draw[line width=0.5 mm,Pink] (90:2cm)--(-90:2cm);\n\\node at (90:2.2) {$\\scriptstyle 3$};\n\\end{tikzpicture}\n\\end{array}&\n\\begin{array}{c}\n\\begin{tabular}{ccc}\n\\toprule\nRestricted Root&\\\\\n\\midrule\n$01$&$\\tikz\\draw[line width=0.5 mm, Pink] (0,0) -- (0.25,0);$\\\\\n$11$&$\\tikz\\draw[line width=0.5 mm, Orange] (0,0) -- (0.25,0);$\\\\\n$21, 42$&$\\tikz\\draw[line width=0.5 mm, Blue] (0,0) -- (0.25,0);$\\\\\n$31$&$\\tikz\\draw[line width=0.5 mm, Green] (0,0) -- (0.25,0);$\\\\\n$41$&$\\tikz\\draw[line width=0.5 mm, Grey] (0,0) -- (0.25,0);$\\\\\n$10, 20, 30$&$\\tikz\\draw[line width=0.5 mm, Pink] (0,-0.15) -- (0,0.15);$\\\\\n\\bottomrule\n\\end{tabular}\n\\end{array}\n\\end{array}\\label{running example finite}\n\\end{equation}\nNote that the restricted root $42$ gives rise to the hyperplane $2(2x+y)=0$, and so the blue diagonal $2x+y=0$ line carries the list $[1,2]$ of multiplicities. We write $2$ beside the blue line to emphasise this fact. Similarly, the line $x=0$ carries the list $[1,2,3]$ of multiplicities, as a consequence of $20$ and $30$.\n\n\\end{example}\n\nReturning to general flopping contractions $\\EuScript{X}\\to\\mathop{\\rm Spec}\\nolimits\\EuScript{R}$, since by construction the nodes in $\\Delta\\backslash\\EuScript{I}$ can be identified with the curves in $\\EuScript{X}$, after some natural identifications $\\EuScript{H}_{\\EuScript{I}}$ (with multiplicities) can be viewed inside $\\mathop{\\rm Pic}\\nolimits\\EuScript{X} \\otimes \\mathbb{R}$. So can the movable cone. After forgetting the multiplicities, $\\EuScript{H}_{\\EuScript{I}}$ is equal to the movable cone \\cite{Pinkham, HomMMP}.\n\n\nThe following is our first main result. It describes the non-zero GV invariants in an elementary combinatorial way, and asserts that it is the hyperplanes of $\\EuScript{H}_{\\EuScript{I}}$, counted with multiplicity, that control the non-zero GV invariants.\n\n\\begin{thm}[\\ref{cor: GV nonzero}]\\label{nonzero GV intro}\nFor $\\upbeta \\in A_1(\\EuScript{X})$ the GV invariant $n_{\\upbeta}$ is non-zero if and only if $\\upbeta$ is a restricted root.\n\\end{thm}\n\nThe theme of this paper is that the shape of the enumerative geometry of $\\EuScript{X}$ is controlled, in a very visual way, from this finite amount of initial data.\n\n\n\\subsection{Gromov--Witten: infinite arrangements}\\label{sec: GW intro}\nGiven any subset $\\EuScript{I}$ of a Dynkin diagram, the finite arrangement $\\EuScript{H}_{\\EuScript{I}}$ of the previous subsection has an infinite cousin $\\EuScript{H}_{\\EuScript{I}}^{\\mathsf{aff}}$. Given a restricted root $\\upbeta=(\\upbeta_i)_{i\\in\\scrI^{\\kern 0.5pt\\mathrm{c}}}$, the hyperplane $\\sum_{i \\in \\scrI^{\\kern 0.5pt\\mathrm{c}}} \\upbeta_i x_i = 0 $ appearing in the finite arrangement gets translated over the integers $\\mathbb{Z}$, to give an infinite family\n\\begin{equation}\n \\sum_{i \\in \\scrI^{\\kern 0.5pt\\mathrm{c}}} \\upbeta_i x_i \\in \\mathbb{Z}. \\label{int translates}\n\\end{equation}\nRepeating this over every restricted root results in an infinite arrangement of affine hyperplanes, written $\\EuScript{H}_{\\EuScript{I}}^{\\mathsf{aff}}$. Note that multiplicities on hyperplanes of $\\EuScript{H}_{\\EuScript{I}}$ result in more translations, as if say $2\\upbeta=(2\\upbeta_i)$ is also a restricted root, then its translates give rise to the family $ \\sum 2\\upbeta_i x_i \\in \\mathbb{Z}$, i.e.\\ to $ \\sum \\upbeta_i x_i \\in \\frac{1}{2}\\mathbb{Z}$. This is larger than \\eqref{int translates}.\n\nIn the running Example~\\ref{example: intro}, taking all the relevant translations of \\eqref{running example finite} results in the following $\\EuScript{H}^{\\mathsf{aff}}_{\\EuScript{I}}$.\n\\begin{equation}\n\\begin{array}{cc}\n\\begin{array}{c}\n\\begin{tikzpicture}[scale=0.5]\n\\draw[->,densely dotted] (0,-0.5)--($(0,-0.5)+(52.5:4.5cm)$);\n\\node at ($(0,-0.5)+(57.5:3.9cm)$) {$\\scriptstyle x$};\n\\draw[->,densely dotted] (0,-0.5)--(0,1.3);\n\\node at (90:1.75) {$\\scriptstyle y$};\n\\end{tikzpicture}\n\\end{array}\n&\n\\begin{array}{c}\n\\includegraphics[angle=0,width=7.5cm,height=4cm]{Y-figure0.pdf}\n\\end{array}\n\\end{array}\\label{running example infinite}\n\\end{equation}\n\n\nReturning to a general flopping contraction $\\EuScript{X}\\to\\mathop{\\rm Spec}\\nolimits\\EuScript{R}$, the next main result relates the Gromov--Witten (GW) theory of $\\EuScript{X}$ to the associated infinite arrangement $\\EuScript{H}^{\\mathsf{aff}}_{\\EuScript{I}}$. The GW invariants are virtual degrees of moduli spaces of stable maps, and provide a system of curve counts related to the more enumerative GV invariants by multiple cover formulae. The GW invariants form the structure constants for the quantum cohomology algebra, but for our purposes it is more convenient to package this information into a generating function, called the \\emph{quantum potential} (see Subsection~\\ref{sec: GW invariants} for details). \n\n\\begin{thm}[\\ref{cor: quantum is hyper}]\\label{GW=hyper intro} \nLet $\\EuScript{X}\\to\\mathop{\\rm Spec}\\nolimits\\EuScript{R}$ be a smooth $3$-fold flopping contraction. The pole locus of the quantum potential is the complexification of the infinite arrangement $\\EuScript{H}^{\\mathsf{aff}}_{\\EuScript{I}}$. \n\\end{thm}\n\\noindent By \\cite{HW2}, the complement of the complexified arrangement $\\EuScript{H}^\\mathsf{aff}_\\EuScript{I}$ forms the base of the Bridgeland stability covering map, for a natural compactly-supported subcategory of the derived category of $\\EuScript{X}$. Theorem~\\ref{GW=hyper intro} therefore connects quantum-cohomological Frobenius manifolds and spaces of stability conditions, a phenomenon which has been observed in other contexts \\cite{BridgelandNonCompact,IkedaQiu,McAuleyThesis}.\n\n\n\\subsection{Flops via simultaneous resolution}\nTo track the change of GV\/GW invariants under iterated flops requires us to first rework some of the theory of simultaneous resolutions, which may be of independent interest. \n Our new contribution is to use the wall crossing formula from \\cite{IyamaWemyssTits}, which indexes chambers of the movable cone by certain cosets, to construct iterated flops from simultaneous (partial) resolutions, and explain how the dual graph changes under flop. This completes work of Reid \\cite{Reid}, Pinkham \\cite{Pinkham}, and Katz--Morrison \\cite{KatzMorrison} in the 80s and 90s, rounding off a circle of ideas going back to Brieskorn \\cite{Brieskorn}. \n\n\nFor any Kleinian singularity $\\mathbb{C}^2\/G$, consider the corresponding Dynkin diagram $\\Delta$, root space $\\mathfrak{h}$, and Weyl group $W$. As is standard, $\\mathbb{C}^2\/G$ admits a versal deformation $\\mathop{\\rm Spec}\\nolimits\\EuScript{V}$ over the base $\\mathfrak{h}_{\\mathbb{C}}\/W$. For any subset $\\EuScript{I}\\subseteq\\Delta$, consider the parabolic subgroup $W_\\EuScript{I}\\colonequals \\langle s_i\\mid i\\in \\EuScript{I}\\rangle$, and take the pullback to obtain\n\\[\n\\begin{tikzpicture}\n\\node (U) at (2,-1.5) {$\\mathop{\\rm Spec}\\nolimits \\EuScript{V}_{\\EuScript{I}}$};\n\\node (V) at (4,-1.5) {$\\mathop{\\rm Spec}\\nolimits \\EuScript{V}$};\n\\node (PRes) at (2,-3) {$\\mathfrak{h}_{\\mathbb{C}}\/W_\\EuScript{I}$};\n\\node (Res) at (4,-3) {$\\mathfrak{h}_{\\mathbb{C}}\/W$.};\n\\draw[->] (U)--(V);\n\\draw[->] (V)--(Res);\n\\draw[->] (U)--node[right]{$\\scriptstyle \\mathsf{g}_{\\EuScript{I}}$}(PRes);\n\\draw[->] (PRes)--(Res);\n\\end{tikzpicture}\n\\]\nWhen $\\EuScript{I}=\\emptyset$, the parabolic $W_\\EuScript{I}=\\mathds{1}$, and in this case classically $\\mathop{\\rm Spec}\\nolimits\\EuScript{V}_\\emptyset$ admits a simultaneous resolution. \n\nAs is now standard, to describe smooth $3$-fold flops requires singular surface geometry, and so the ability to consider $\\EuScript{I}\\neq\\emptyset$ is crucial. By \\cite{KatzMorrison}, for each $\\EuScript{I}$ there is a preferred, or \\emph{standard} simultaneous partial resolution $\\mathsf{h}_{\\EuScript{I}}\\colon \\EuScript{Y}_\\EuScript{I}\\to\\mathop{\\rm Spec}\\nolimits\\EuScript{V}_\\EuScript{I}$ (see Subsection~\\ref{sec: sim partial res I}). Further, by \\emph{loc.\\ cit.\\ }all smooth flops can be constructed via appropriate classifying maps $\\upmu\\colon\\EuScript{D} \\mathrm{isc}\\to\\mathfrak{h}_{\\mathbb{C}}\/W_{\\EuScript{I}}$ from the formal disc to $\\mathfrak{h}_{\\mathbb{C}}\/W_{\\EuScript{I}}$ for some $\\EuScript{I}$, giving the following cartesian diagram\n\\[\n\\begin{tikzpicture}\n\\node (X) at (0,0) {$\\EuScript{X}$};\n\\node (Z) at (2,0) {$\\EuScript{Y}_{\\EuScript{I}}$};\n\\node (R) at (0,-1.5) {$\\mathop{\\rm Spec}\\nolimits \\EuScript{R}$};\n\\node (U) at (2,-1.5) {$\\mathop{\\rm Spec}\\nolimits \\EuScript{V}_\\EuScript{I}$};\n\\node (V) at (4,-1.5) {$\\mathop{\\rm Spec}\\nolimits \\EuScript{V}$};\n\\node (t) at (0,-3) {$\\EuScript{D} \\mathrm{isc}$};\n\\node (PRes) at (2,-3) {$\\mathfrak{h}_{\\mathbb{C}}\/W_{\\EuScript{I}}$};\n\\node (Res) at (4,-3) {$\\mathfrak{h}_{\\mathbb{C}}\/W$.};\n\\draw[->] (X)--(Z);\n\\draw[->] (X)--(R);\n\\draw[->] (Z)--node[right]{$\\scriptstyle \\mathsf{h}_{\\EuScript{I}}$}(U);\n\\draw[->] (R)--(U);\n\\draw[->] (U)--(V);\n\\draw[->] (V)--(Res);\n\\draw[->] (R)--(t);\n\\draw[->] (U)--node[right]{$\\scriptstyle \\mathsf{g}_{\\EuScript{I}}$}(PRes);\n\\draw[->] (t)--node[above]{$\\scriptstyle\\upmu$}(PRes);\n\\draw[->] (PRes)--(Res);\n\\end{tikzpicture}\n\\]\nThe question is, given $\\EuScript{X}\\to\\mathop{\\rm Spec}\\nolimits \\EuScript{R}$, how to construct the flop at a given curve from the classifying map $\\upmu$. This was solved in the case $\\EuScript{I}=\\emptyset$ by Reid \\cite{Reid}, but the general case is harder, since the subset $\\EuScript{I}$ changes under flop. Pinkham \\cite{Pinkham} counts only the number of simultaneous resolutions.\n\nWe solve this problem by appealing to the wall crossing combinatorics of \\cite{IyamaWemyssTits}. The key point is that when $\\EuScript{I}\\neq\\emptyset$, chambers in the movable cone are indexed by cosets, not by elements of the Weyl group. For any curve $\\Curve_i\\subseteq\\EuScript{X}$, set $w_i=\\ell_{\\EuScript{I}}\\ell_{\\EuScript{I}\\cup\\{i\\}} \\in W$ where $\\ell_\\Gamma$ denotes the longest element in the Weyl group $W_\\Gamma$. Then there is a unique subset $\\upomega_i(\\EuScript{I})\\subseteq\\Delta$, described explicitly in Section~\\ref{sec: flops via sim res}, for which $W_{\\EuScript{I}}w_i=w_iW_{\\upomega_i(\\EuScript{I})}$.\n\n\\begin{thm}[\\ref{thm: produce flop}]\nPost-composing $\\upmu$ with $w_i^{-1}\\colon \\mathfrak{h}_{\\mathbb{C}}\/W_{\\EuScript{I}}\\to\\mathfrak{h}_{\\mathbb{C}}\/W_{\\omega_i(\\EuScript{I})}$ and taking the pullback constructs the flop $\\EuScript{X}_i^+\\to\\mathop{\\rm Spec}\\nolimits\\EuScript{R}$ of the curve $\\Curve_i\\subset\\EuScript{X}$. In particular\n\\begin{enumerate}\n\\item $\\upomega_i(\\EuScript{I})$ is the dual graph of the exceptional locus of $\\EuScript{X}_i^+\\to\\mathop{\\rm Spec}\\nolimits\\EuScript{R}$.\n\\item All other crepant resolutions can be obtained from the fixed $\\upmu$ by post-composing with $x^{-1}\\colon\\mathfrak{h}_{\\mathbb{C}}\/W_{\\EuScript{I}}\\to\\mathfrak{h}_{\\mathbb{C}}\/W_{\\EuScript{K}}$ and pulling back along $\\EuScript{Y}_\\EuScript{K}$, as the pair $(x,\\EuScript{K})$ ranges over the (finite) indexing set $\\mathsf{Cham}(\\Delta,\\EuScript{I})$.\n\\end{enumerate}\n\\end{thm}\nWe describe the above in detail in Section~\\ref{sec: flops via sim res}, but emphasise here that everything is formed intrinsically from the Dynkin data, once $\\EuScript{X}\\to\\mathop{\\rm Spec}\\nolimits\\EuScript{R}$ and thus $\\upmu$ is fixed.\n\n\\subsection{Tracking fundamental regions}\nWith the above in hand, tracking the change in GV\/GW invariants under all possible flops becomes easy, and satisfyingly visual. In our running Example~\\ref{example: intro}, each of the 12 crepant resolutions $\\EuScript{X}_i\\to\\mathop{\\rm Spec}\\nolimits\\EuScript{R}$ admits a fundamental region in \\eqref{running example infinite}. As calibration, the fixed $\\EuScript{X}\\to\\mathop{\\rm Spec}\\nolimits \\EuScript{R}$ corresponds to the unit box in the purple axes\n\\begin{equation}\n\\begin{array}{cc}\n\\begin{array}{c}\n\\begin{tikzpicture}[scale=0.5]\n\\draw[->,densely dotted] (0,-0.5)--($(0,-0.5)+(52.5:4.5cm)$);\n\\node at ($(0,-0.5)+(57.5:3.9cm)$) {$\\scriptstyle x$};\n\\draw[->,densely dotted] (0,-0.5)--(0,1.3);\n\\node at (90:1.75) {$\\scriptstyle y$};\n\\end{tikzpicture}\n\\end{array}\n&\n\\begin{array}{c}\n\\includegraphics[angle=0,width=7.5cm,height=4cm]{Y-figure1.pdf}\n\\end{array}\n\\end{array}\\label{running example infinite fund region 1}\n\\end{equation}\nThe other 12 chambers in the movable cone generate similar fundamental regions as in \\eqref{running example infinite fund region 1} and this is illustrated in Figure~\\ref{figure1}.\n\\begin{figure}[h] \n\\[\n\\begin{array}{c}\n\\includegraphics[angle=0,width=7.5cm,height=6cm]{Y-figure2.pdf}\n\\end{array}\n\\]\n\\caption{The 12 fundamental regions corresponding to the 12 crepant resolutions.}\n\\label{figure1}\n\\end{figure}\n\nReassuringly, flopping a single curve turns out to correspond to the neighbouring region. Although Figure~\\ref{figure1} only illustrates the two-curve flop in the running Example~\\ref{example: intro}, similar things happen in full generality (see Section~\\ref{sec: applications}). The following is our third main result, which controls how GV invariants transform under iterated flops. As notation, set $\\EuScript{X}_i^+$ to be the scheme obtained from $\\EuScript{X}$ after flopping only a single curve $\\Curve_i$, and further write $n_{\\upbeta,\\EuScript{X}}$ for the GV invariant of curve class $\\upbeta$ in $\\EuScript{X}$. In what follows $\\mathsf{M}_i$ is an explicit matrix, defined in \\eqref{defn Mi}, that can be easily built using Dynkin combinatorics. \n\n\n\\begin{theorem}[\\ref{thm: GV under flop}]\\label{thm: GV translate intro}\nWith the notation as above,\n\\[\nn_{\\upbeta,\\EuScript{X}^+_i}=\n\\begin{cases}\nn_{\\upbeta,\\EuScript{X}}&\\mbox{if }\\upbeta\\in \\mathbb{Z} \\Curve_i\\\\\nn_{\\mathsf{M}_i \\upbeta, \\EuScript{X}}&\\mbox{else}.\n\\end{cases}\n\\]\n\\end{theorem} \n\\noindent Example~\\ref{ex: running example GV after flop} illustrates this in the case of the running Example~\\ref{example: intro}.\n\n\n\n\n\\subsection{Applications}\nThe above results have a series of corollaries. The first is a direct and explicit proof of the Crepant Transformation Conjecture for germs of $3$-fold flopping contractions. Indeed, combining Theorems~\\ref{GW=hyper intro} and \\ref{thm: GV translate intro} allows us to easily extract the following, which recovers the main result of \\cite{LiRuan}. We remark here that our simplified approach also gives more refined information, in the form of the explicit matrix $\\mathsf{N}_{i}=(\\mathsf{M}_i^{-1})^\\star$ which identifies the quantum potentials. Our proof also avoids the use of symplectic cuts, side-stepping the associated technical difficulties.\n\n\n\\begin{cor}[\\ref{thm: CTC}] Under the identification of the Novikov parameters given by the explicit matrix $\\mathsf{M}_i$ of Theorem~\\ref{thm: GV translate intro}, the quantum potentials of $\\EuScript{X}$ and $\\EuScript{X}_i^+$ coincide, up to a correction term which does not depend on the Novikov parameters, namely\n\\begin{equation} \\label{eqn: CTC intro} \n\\Phi^{\\EuScript{X}_i^+}_{\\mathsf{r}}\\big(\\upgamma_1, \\upgamma_2,\\upgamma_3\\big) - \\Phi^{\\EuScript{X}}_{\\mathsf{r}}(\\mathsf{N}_i \\upgamma_1,\\mathsf{N}_i \\upgamma_2,\\mathsf{N}_i \\upgamma_3) = -(\\upgamma_1 \\cdot \\Curve_i^+)(\\upgamma_2 \\cdot \\Curve_i^+)(\\upgamma_3\\cdot \\Curve_i^+) \\sum_{k \\geq 1} k^3 n_{k\\Curve_i,\\EuScript{X}} .\n\\end{equation}\nHere $\\mathsf{N}_i \\colon \\mathrm{H}^2(\\EuScript{X}_{i}^+;\\mathbb{C}) \\to \\mathrm{H}^2(\\EuScript{X};\\mathbb{C})$ is the dual of $\\mathsf{M}_i^{-1}$. The above identification holds after a specific analytic continuation in the quantum parameters. \n\\end{cor}\n\\noindent The correction terms on the right-hand side arise due to the non-compactness of $\\EuScript{X}$ (see Remarks~\\ref{rmk: no algebra} and \\ref{rmk: CTC non-compact}). The key point is that the quantum potential of $\\EuScript{X}_i^+$ can be effectively reconstructed from the quantum potential of $\\EuScript{X}$. Thus, whilst the GW invariants themselves are not combinatorial, their transformation across the flop \\emph{is} combinatorial, which is why we obtain such an elementary proof; compare \\cite{LiRuan} and \\cite{McLean}. \n\nThe explicit matrix $\\mathsf{N}_i$ turns out to have many different incarnations: it arises naturally as the image in K-theory of the Bridgeland's flop functor, but more interestingly it can be calculated using very simple Dynkin-style combinatorics (see Remark~\\ref{rem: matrix Mk}). However, simply by iterating and multiplying matrices, it is possible to obtain a direct isomorphism between the generating functions of any two crepant resolutions of $\\mathop{\\rm Spec}\\nolimits\\EuScript{R}$.\\medskip\n\n The second corollary is algebraic. The flopping contraction $\\EuScript{X}\\to\\mathop{\\rm Spec}\\nolimits \\EuScript{R}$ has an associated contraction algebra $\\mathrm{A}_{\\con}$ \\cite{DW1, DW3}, and it is known by Hua--Toda \\cite{HuaToda} for single curves, and Toda \\cite{TodaUtah} in general (see \\ref{thm: Toda dim formula}), that the dimension of the contraction algebra is determined as the weighted sum of GV invariants \n\\[\n\\dim_{\\mathbb{C}}\\mathrm{A}_{\\con}=\\sum_{\\upbeta\\in A_1(\\EuScript{X})}n_\\upbeta (\\upbeta\\cdot \\mathds{1})^2\n\\]\nwhere $\\upbeta\\cdot \\mathds{1}$ is the sum of the entries of $\\upbeta$. For any curve $\\Curve_i\\subseteq\\EuScript{X}$, the contraction algebra can be intrinsically mutated to obtain $\\upnu_i\\mathrm{A}_{\\con}$, and this is the contraction algebra for the flop $\\EuScript{X}^+_i\\to\\mathop{\\rm Spec}\\nolimits\\EuScript{R}$.\n \nIt is known \\cite{Dugas, AugustTiltingTheory} that $\\mathrm{A}_{\\con}$ and $\\upnu_i\\mathrm{A}_{\\con}$ are derived equivalent via a two-term tilting complex, but what is surprising here is that their dimension transforms in a very elementary manner, dictated by the K-theory of that derived equivalence. \n\\begin{cor}[\\ref{cor: Toda formula iterate}]\\label{cor: Toda formula iterate intro}\nUnder mutation at vertex $i$, \n\\[\n\\dim_{\\mathbb{C}}\\upnu_i\\mathrm{A}_{\\con}=\\sum_{\\upbeta\\in A_1(\\EuScript{X})}n_\\upbeta \\big(\\,(\\mathsf{M}^{-1}_i\\upbeta)\\cdot \\mathds{1}\\big)^2\n\\]\nwhere $\\mathsf{M}_i$ is the explicit matrix in Corollary~\\textnormal{\\ref{thm: GV translate intro}}. \n\\end{cor}\nThe above is remarkable: it says that not only are there just finitely many algebras in the derived equivalence class of the finite dimensional algebra $\\mathrm{A}_{\\con}$ (by \\cite{AugustFinite}), furthermore the dimensions of all the other algebras can be easily obtained combinatorially from the first. The proof of Corollary~\\ref{cor: Toda formula iterate intro} is slightly subtle, since it is not a priori clear that the GV invariants defined by Toda are the same as the GV invariants defined here, but this is all discussed in Appendix~\\ref{sec: n beta eq n beta}.\n\n\n\n\\subsection*{Acknowledgements} \nWe thank Tom Coates and Misha Feigin for helpful discussions on quantum cohomology, and Jenny August, Ben Davison, Okke van Garderen and Yukinobu Toda for wider discussions on GV invariants and contraction algebras.\n\n\\subsection*{Conventions} \nAll cDV singularities and related algebraic geometry takes place over $\\mathbb{C}$. Vector spaces will be over $\\mathbb{R}$, unless stated otherwise, and the complexification of a vector space $V$ will be written $V_{\\mathbb{C}}$.\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\\section{Root theory, deformations and perturbations}\n\\noindent Fix an isolated $3$-fold cDV singularity $\\mathop{\\rm Spec}\\nolimits\\EuScript{R}$ with a crepant resolution $f \\colon \\EuScript{X} \\to\\mathop{\\rm Spec}\\nolimits\\EuScript{R}$. There is a finite collection of exceptional complete curves in $\\EuScript{X}$, contracted to a point $p\\in \\mathop{\\rm Spec}\\nolimits\\EuScript{R}$, and such that $f$ restricts to an isomorphism on the complement. In other words, $f \\colon \\EuScript{X} \\to \\mathop{\\rm Spec}\\nolimits \\EuScript{R}$ is a germ of a smooth $3$-fold flopping contraction, and conversely every such germ arises in this way \\cite{Reid}.\n\n\\subsection{Elephants}\\label{sec: elephant}\nThe pullback along $f$ of a general hyperplane section through $p \\in \\mathop{\\rm Spec}\\nolimits\\EuScript{R}$ is a \\emph{partial} crepant resolution of an ADE surface singularity, the so-called general elephant \\cite[(1.14)]{Reid}\n\\begin{equation}\n\\begin{array}{c}\n\\begin{tikzpicture}\n\\node (A) at (0,0) {$\\EuScript{X}$};\n\\node (B) at (-2,0) {$Y$};\n\\node at (-3.4,-1) {$ \\mathbb{C}^2\/G\\cong$};\n\\node (b) at (-2,-1) {$\\mathop{\\rm Spec}\\nolimits \\EuScript{R}\/g$};\n\\node (a) at (0,-1) {$\\mathop{\\rm Spec}\\nolimits \\EuScript{R}$.};\n\\draw[->] (B)--(A);\n\\draw[->] (b)--(a);\n\\draw[->] (B)--(b);\n\\draw[->] (A)--(a);\n\\end{tikzpicture}\n\\end{array}\\label{elephant pullback}\n\\end{equation}\nLet $\\Delta$ be the Dynkin diagram associated to $\\mathbb{C}^2\/G$, and let the composition\n\\[\nZ \\to Y\\to \\mathbb{C}^2\/G\n\\]\nbe the full minimal resolution. By the McKay correspondence, the exceptional curves $\\Curve_i \\subseteq Z$ are indexed by the nodes $i \\in \\Delta$. We write\n\\[\n \\EuScript{I} \\subseteq \\Delta\n\\]\nfor the subset indexing those curves $\\Curve_i \\subseteq Z$ which are contracted by the morphism $Z\\to Y$, so that the complement $\\scrI^{\\kern 0.5pt\\mathrm{c}} = \\Delta \\setminus \\EuScript{I}$ indexes the curves that survive. In particular\n\\[\\{ \\Curve_i \\mid i \\in \\scrI^{\\kern 0.5pt\\mathrm{c}} \\}\\]\nforms the set of exceptional curves in both $Y$ and $\\EuScript{X}$, and the group $A_1(Y)=A_1(\\EuScript{X})$ is freely generated by their cycle classes.\n\n\\begin{notation}\\label{notation: Y_I}\nWrite $Y=Y_{\\EuScript{I}}$ for the partial resolution of $\\mathbb{C}^2\/G$ obtained from the full minimal resolution $Z$ by blowing down the curves in $\\EuScript{I}$.\n\\end{notation}\n\nThe geometry of $\\EuScript{X}$ will be studied by viewing it as the total space of a one-parameter deformation of $Y_\\EuScript{I}$. This requires detailed control over the associated root theory, which we establish in the following subsections.\n \n\\subsection{Root theory}\\label{sec: Dynkin recap}\nFor any Dynkin diagram $\\Delta$, let $\\mathfrak{h}$ be the $\\mathbb{R}$-vector space based by the set of simple roots $\\{\\upalpha_i \\mid i \\in \\Delta \\}$, so that\n\\[\n\\mathfrak{h}=\\bigoplus_{i\\in\\Delta}\\mathbb{R}\\upalpha_i ,\n\\]\nand write $\\Uptheta = \\mathfrak{h}^\\star$ for the dual. The Weyl group $W$ acts naturally on both $\\mathfrak{h}$ and $\\Uptheta$. For every positive root $\\upalpha \\in \\mathfrak{h}$, write $\\EuScript{D}_{\\upalpha}\\subseteq\\mathfrak{h}$ for the perpendicular hyperplane, and write $\\mathsf{H}_{\\upalpha} \\subseteq \\Theta$\nfor the dual hyperplane.\n\n\\begin{notation}\\label{notation: subset stuff}\nFor any subset $\\EuScript{I}\\subseteq \\Delta$, consider the following data.\n\\begin{enumerate}\n\\item The complement $\\scrI^{\\kern 0.5pt\\mathrm{c}} = \\Delta \\setminus \\EuScript{I}$.\n\\item The parabolic subgroup $W_{\\EuScript{I}}\\colonequals \\langle s_i\\mid i\\in\\EuScript{I}\\rangle \\subseteq W$.\n\\item The $\\mathbb{R}$-vector space $\\mathfrak{h}_{\\EuScript{I}}$ obtained as the quotient of $\\mathfrak{h}$ by the $\\mathbb{R}$-subspace spanned by $\\{ \\upalpha_i \\mid i \\in \\EuScript{I} \\}$. The associated quotient map will be written\n \\[\n \\uppi_{\\EuScript{I}} \\colon \\mathfrak{h} \\to \\mathfrak{h}_{\\EuScript{I}}.\n \\] \nNote that $\\mathfrak{h}_{\\EuScript{I}}$ has basis $\\{ \\uppi_{\\EuScript{I}}(\\upalpha_i) \\mid i \\in \\scrI^{\\kern 0.5pt\\mathrm{c}}\\}$ and may be identified with the subspace of $\\mathfrak{h}$ based by $\\{ \\upalpha_i \\mid i \\in \\scrI^{\\kern 0.5pt\\mathrm{c}} \\}$.\n\\item The \\emph{restricted positive roots} in $\\mathfrak{h}_\\EuScript{I}$, which are precisely the non-zero images of positive roots under $\\uppi_{\\EuScript{I}}$.\n\\item For $\\upvartheta_i\\in\\mathbb{R}$ with $i\\in\\Delta$, write $(\\upvartheta_i)=\\sum_{i\\in\\Delta}\\upvartheta_i\\upalpha_i^\\star$, and consider\n\\[\n\\Uptheta_{\\EuScript{I}}\\colonequals \\{ (\\upvartheta_i)\\in\\Uptheta\\mid \\upvartheta_i=0\\mbox{ for all }i\\in\\EuScript{I}\\}\\subseteq\\Uptheta.\n\\]\nThe reflecting hyperplanes in $\\Uptheta$ intersect $\\Uptheta_{\\EuScript{I}}$, and in this way $\\Uptheta_{\\EuScript{I}}$ inherits the structure of a finite hyperplane arrangement. Note that $\\Uptheta_{\\EuScript{I}}$ has basis $\\{\\upalpha_i^\\star\\mid i \\in \\scrI^{\\kern 0.5pt\\mathrm{c}}\\}$. Of course, $\\mathfrak{h}_{\\EuScript{I}}$ and $\\Uptheta_{\\EuScript{I}}$ are dual, and both have dimension $|\\scrI^{\\kern 0.5pt\\mathrm{c}}|$. \n \n\\item The set $\\mathsf{Cham}(\\Delta,\\EuScript{I})$ which indexes chambers of $\\Uptheta_\\EuScript{I}$ \\cite[1.8]{IyamaWemyssTits}. Combinatorially, $\\mathsf{Cham}(\\Delta,\\EuScript{I})$ can be defined as the set of all pairs $(x,\\EuScript{K})$ with $x\\in W$ and $\\EuScript{K}\\subseteq\\Delta$ for which $W_\\EuScript{I} x=xW_\\EuScript{K}$ and $\\mathrm{length}(x)=\\mathrm{min}\\{\\mathrm{length}(y)\\mid y\\in xW_\\EuScript{K}\\}$.\n \\end{enumerate}\n \\end{notation}\nFor a given restricted positive root $\\upbeta \\in \\mathfrak{h}_{\\EuScript{I}}$, there are in general many different positive roots $\\upalpha \\in \\mathfrak{h}$ such that $ \\uppi_{\\EuScript{I}}(\\upalpha)=\\upbeta$. The following result controls the possible lifts. It is a very mild generalisation of \\cite[Lemma~2.4]{BryanKatzLeung}, and will be used later to relate enumerative invariants to hyperplane arrangements.\n\n\\begin{lemma} \\label{lem: Dynkin combinatorics}\nFor any ADE Dynkin diagram $\\Delta$, and any subset $\\EuScript{I}\\subseteq\\Delta$, let $\\upalpha,\\upalpha^\\prime \\in \\mathfrak{h}$ be positive roots such that $\\uppi_\\EuScript{I}(\\upalpha), \\uppi_{\\EuScript{I}}(\\upalpha^\\prime) \\in \\mathfrak{h}_{\\EuScript{I}}$ are non-zero. Then the following are equivalent.\n\\begin{enumerate}\n\\item $\\uppi_{\\EuScript{I}}(\\upalpha)=\\uppi_{\\EuScript{I}}(\\upalpha^\\prime)$.\n\\item $\\upalpha$ and ${\\upalpha^\\prime}$ are identified under the action of $W_{\\EuScript{I}}$ on $\\mathfrak{h}$.\n\\item $\\EuScript{D}_\\upalpha$ and $\\EuScript{D}_{\\upalpha^\\prime}$ are identified under the action of $W_{\\EuScript{I}}$ on $\\mathfrak{h}$.\n\\end{enumerate}\n\\end{lemma}\n\\begin{proof} \nSince $\\EuScript{I}$ is fixed, to ease notation set $\\uppi=\\uppi_{\\EuScript{I}}$.\\medskip\n\n\\noindent (1)$\\Rightarrow$(2) This is the only difficult part, and is a direct case analysis. Consider first the $A_n$ root system, where the positive roots are precisely the connected chains of $1$s on the Dynkin graph\n\\[\n\\begin{array}{c@{\\hspace{2pt}}c@{\\hspace{2pt}}c@{\\hspace{2pt}}c@{\\hspace{2pt}}c@{\\hspace{1pt}}c@{\\hspace{2pt}}c@{\\hspace{2pt}}c@{\\hspace{2pt}}c@{\\hspace{2pt}}c@{\\hspace{2pt}}c@{\\hspace{2pt}}}\n\\upalpha_{ij}\\colonequals \\ 0&\\hdots &0&1& \\hdots &1&0&\\hdots&0 \n\\quad (1 \\leq i \\leq j \\leq n).\\\\[-0.5mm]\n&&& i && j\n\\end{array}\n\\]\nBy definition of the action of $W$ on $\\mathfrak{h}$, the reflection $s_i$ acts by\n\\begin{equation}\ns_i(\\upalpha_{ij})=\\upalpha_{i+1\\, j} \\qquad \\text{for $i,densely dotted] (U)--(V);\n\\draw[->] (V)--(Res);\n\\draw[->,densely dotted] (U)--(PRes);\n\\draw[->] (PRes)--(Res);\n\\end{tikzpicture}\n\\]\ngives $\\mathop{\\rm Spec}\\nolimits\\EuScript{W}\\to\\mathfrak{h}_{\\mathbb{C}}$, which admits a simultaneous resolution. Since $\\dim\\EuScript{W}\\geq 3$, there are in fact many such simultaneous resolutions, since minimal models are not unique.\n\nIn \\cite[Theorem 1]{KatzMorrison} Katz--Morrison construct a particular simultaneous resolution, from a particular $\\mathop{\\rm Spec}\\nolimits\\EuScript{V}\\to\\mathfrak{h}_{\\mathbb{C}}\/W$, for which positive roots and their hyperplanes control those curve classes that survive under deformation \\cite[Theorem 1(c)]{KatzMorrison}. We will recap this result in greater generality in Theorem~\\ref{thm: KM Theorem 1(c)} below, but for now write $\\EuScript{Z}\\to\\mathop{\\rm Spec}\\nolimits\\EuScript{W}$ for this preferred resolution. Katz--Morrison refer to their particular choice of $\\EuScript{Z}$ as the standard simultaneous resolution \\cite[Section~6]{KatzMorrison}.\n\n\n\\subsubsection{Simultaneous Partial Resolution} \nGiven any subset $\\EuScript{I}\\subseteq \\Delta$, consider the standard simultaneous resolution $\\EuScript{Z}\\to\\mathop{\\rm Spec}\\nolimits\\EuScript{W}$ from Subsection~\\ref{sec: pref sim res} above. As explained in \\cite[above Theorem 3]{KatzMorrison} following \\cite{Pinkham} it is possible to blow down $\\EuScript{Z}$ at the curves in $\\EuScript{I}$ and take the quotient by $W_{\\EuScript{I}}$ to obtain $\\EuScript{Y}_{\\EuScript{I}}$, which sits in the following commutative diagram\n\\begin{equation}\n\\begin{array}{c}\n\\begin{tikzpicture}\n\\node (ZZ) at (0,1.5) {$\\EuScript{Z}$};\n\\node (B) at (0,0) {$\\EuScript{Y}_\\EuScript{I}^\\dag$};\n\\node (Z) at (2,0) {$\\EuScript{Y}_{\\EuScript{I}}$};\n\\node (R) at (0,-1.5) {$\\mathop{\\rm Spec}\\nolimits \\EuScript{W}$};\n\\node (U) at (2,-1.5) {$\\mathop{\\rm Spec}\\nolimits \\EuScript{V}_{\\EuScript{I}}$};\n\\node (V) at (4,-1.5) {$\\mathop{\\rm Spec}\\nolimits \\EuScript{V}$};\n\\node (t) at (0,-3) {$\\mathfrak{h}_\\mathbb{C}$};\n\\node (PRes) at (2,-3) {$\\mathfrak{h}_{\\mathbb{C}}\/W_{\\EuScript{I}}$};\n\\node (Res) at (4,-3) {$\\mathfrak{h}_{\\mathbb{C}}\/W$};\n\\draw[->] (ZZ)--(B);\n\\draw[->] (B) -- (Z);\n\\draw[->] (B) -- (R);\n\\draw[->] (Z)--node[right]{$\\scriptstyle \\mathsf{g}_{\\EuScript{I}}$}(U);\n\\draw[->] (R)--(U);\n\\draw[->] (U)--(V);\n\\draw[->] (V)--(Res);\n\\draw[->] (R)--(t);\n\\draw[->] (U)--node[right]{$\\scriptstyle \\mathsf{h}_{\\EuScript{I}}$}(PRes);\n\\draw[->] (t)--(PRes);\n\\draw[->] (PRes)--(Res);\n\\end{tikzpicture}\n\\end{array}\\label{preferred sim partial res}\n\\end{equation}\nwith all squares cartesian. By construction, the fibre $(\\mathsf{h}_\\EuScript{I}\\circ \\mathsf{g}_{\\EuScript{I}})^{-1}(0)$ is the partial resolution of $\\mathbb{C}^2\/G$ obtained from the full minimal resolution by blowing down the curves in $\\EuScript{I}$. Namely, recalling Notation~\\ref{notation: Y_I}, $(\\mathsf{h}_\\EuScript{I}\\circ \\mathsf{g}_{\\EuScript{I}})^{-1}(0)=Y_{\\EuScript{I}}$. \n\nIn a similar way as in Subsection~\\ref{sec: pref sim res}, the middle morphism $\\mathsf{h}_\\EuScript{I}\\colon\\mathop{\\rm Spec}\\nolimits\\EuScript{V}_{\\EuScript{I}}\\to\\mathfrak{h}_{\\mathbb{C}}\/W_{\\EuScript{I}}$ admits simultaneous \\emph{partial} resolutions. Again these are not unique, however we will refer to the choice $\\EuScript{Y}_{\\EuScript{I}}\\to\\mathop{\\rm Spec}\\nolimits\\EuScript{V}_\\EuScript{I}$ constructed above as the standard simultaneous partial resolution associated to $\\EuScript{I}$. From our perspective, the point is that $\\EuScript{Y}_{\\EuScript{I}}$ is precisely the partial simultaneous resolution for which Theorem~\\ref{thm: KM Theorem 1(c)} below holds.\n\n\\subsection{Surface deformations via simultaneous partial resolution} \\label{sec: deformation theory}\nFix a subset $\\EuScript{I}\\subseteq\\Delta$ and consider the composition\n\\[ \\mathsf{s}_\\EuScript{I} = \\mathsf{h}_{\\EuScript{I}}\\circ \\mathsf{g}_{\\EuScript{I}}\\colon \\EuScript{Y}_{\\EuScript{I}}\\to \\mathfrak{h}_{\\mathbb{C}}\/W_{\\EuScript{I}}\\]\nfrom \\eqref{preferred sim partial res}. This is a versal deformation of the surface $Y_\\EuScript{I}$.\n\n\\begin{definition} \\label{def: standard disc locus}\nThe standard discriminant locus\n\\[\n\\EuScript{D}_{\\EuScript{I}} \\subseteq \\mathfrak{h}_{\\mathbb{C}}\/W_{\\EuScript{I}}\n\\]\nis the set of points $p \\in \\mathfrak{h}_{\\mathbb{C}}\/W_{\\EuScript{I}}$ such that the fibre $\\mathsf{s}_\\EuScript{I}^{-1}(p)$ contains a complete curve. \n\\end{definition}\nThere is a similar definition of a discriminant locus associated to any simultaneous partial resolution: the word standard in Definition~\\ref{def: standard disc locus} emphasises the choice made in \\eqref{preferred sim partial res}. The following discussion draws heavily on \\cite[Theorem 1]{KatzMorrison} as used in \\cite[Proposition 2.2]{BryanKatzLeung}, while also incorporating Lemma~\\ref{lem: Dynkin combinatorics} above to relate the resulting combinatorics to the enhanced movable cone. \n\nTo set notation, recall that $\\EuScript{D}_{\\upalpha}\\subseteq\\mathfrak{h}$ is the hyperplane perpendicular to $\\upalpha$, and let $\\EuScript{D}_{\\upalpha,\\mathbb{C}}\\subseteq \\mathfrak{h}_\\mathbb{C}$ denote its complexification. Recall from Notation~\\ref{notation: subset stuff} that for $\\EuScript{I} \\subseteq \\Delta$ there is a quotient map $\\uppi_\\EuScript{I} \\colon\\mathfrak{h}\\to\\mathfrak{h}_{\\EuScript{I}}$, where the vector space $\\mathfrak{h}_{\\EuScript{I}} $ has basis $\\{ \\uppi_{\\EuScript{I}}(\\upalpha_i) \\mid i \\in \\scrI^{\\kern 0.5pt\\mathrm{c}}\\}$, and that there is a natural identification\n\\begin{align} \n\\mathfrak{h}_{\\EuScript{I}} & \\cong A_1(Y_{\\EuScript{I}})_{\\mathbb{R}}\\label{eq:ident A with comb}\\\\\n\\uppi_{\\EuScript{I}}(\\upalpha_i) & \\mapsto \\Curve_i.\\nonumber\n\\end{align}\nEvery restricted positive root $\\uppi_\\EuScript{I}(\\upalpha) \\in \\mathfrak{h}_{\\EuScript{I}}$ has non-negative integer coefficients, and so may be interpreted as a curve class $\\uppi_\\EuScript{I}(\\upalpha)=\\upbeta \\in A_1(Y_{\\EuScript{I}})$. \n\n\n\\begin{thm}[Katz--Morrison] \\label{thm: KM Theorem 1(c)} For any subset $\\EuScript{I}\\subseteq\\Delta$, the following statements hold.\n\\begin{enumerate}\n\\item The standard discriminant locus $\\EuScript{D}_\\EuScript{I} \\subseteq \\mathfrak{h}_\\mathbb{C}\/W_\\EuScript{I}$ from Definition~\\ref{def: standard disc locus} decomposes as\n\\[\n\\EuScript{D}_{\\EuScript{I}}=\\bigcup_{\\uppi_{\\EuScript{I}}(\\upalpha)\\neq 0}\\,\\,\\EuScript{D}_{\\upalpha,\\mathbb{C}}\/W_\\EuScript{I}\n\\]\nwhere the union is over all positive roots $\\upalpha$ such that $\\uppi_{\\EuScript{I}}(\\upalpha)\\neq 0$. The irreducible components $\\EuScript{D}_{\\upalpha,\\mathbb{C}}\/W_\\EuScript{I} \\subseteq \\EuScript{D}_\\EuScript{I}$ are indexed by the restricted positive roots $\\uppi_\\EuScript{I}(\\upalpha)$.\n\\item For $p\\in\\EuScript{D}_{\\upalpha,\\mathbb{C}}\/W_\\EuScript{I}$ the fibre $\\mathsf{s}_\\EuScript{I}^{-1}(p)$ is a deformation of $Y_{\\EuScript{I}}$ containing a complete curve of class $\\upbeta \\colonequals \\uppi_\\EuScript{I}(\\upalpha)$. If in addition $p$ does not belong to any other component of $\\EuScript{D}_\\EuScript{I}$, then this is the only complete curve in $\\mathsf{s}_\\EuScript{I}^{-1}(p)$.\n\\item If $\\upbeta \\in A_1(Y_\\EuScript{I})$ is not a restricted positive root, then there are no deformations of $Y_\\EuScript{I}$ containing a complete curve of class $\\upbeta$.\n\\end{enumerate}\n\\end{thm}\n\\begin{proof} \n(1) If $\\EuScript{I}=\\emptyset$ then by \\cite[Theorem 1(3)]{KatzMorrison} (see also \\cite[Proposition~2.2]{BryanKatzLeung}) there is a decomposition of the standard discriminant locus\n\\[ \n\\EuScript{D}_\\emptyset = \\bigcup_\\upalpha \\EuScript{D}_{\\upalpha,\\mathbb{C}} \\subseteq \\mathfrak{h}_\\mathbb{C} \n\\]\nwhere the union is over all positive roots $\\upalpha$. The analogous decomposition for general $\\EuScript{I}$ follows by considering the standard simultaneous resolution $\\EuScript{Z}$ of the standard simultaneous partial resolution $\\EuScript{Y}_\\EuScript{I}$ from \\ref{sec: sim partial res I}\n\\begin{center}\\begin{tikzcd}\n\\EuScript{Z} \\ar[r] \\ar[d,\"\\mathsf{t}_\\EuScript{I}\"] & \\EuScript{Y}_\\EuScript{I} \\ar[d,\"\\mathsf{s}_\\EuScript{I}\"] \\\\\n\\mathfrak{h}_\\mathbb{C} \\ar[r,\"\\upphi_\\EuScript{I}\"] & \\mathfrak{h}_\\mathbb{C}\/W_\\EuScript{I}.\n\\end{tikzcd}\\end{center}\nThe map $\\EuScript{Z} \\to \\EuScript{Y}_\\EuScript{I}$ is given by blowing down $\\EuScript{Z}$ at the curves in $\\EuScript{I}$ and then taking the quotient by $W_\\EuScript{I}$.\n\nFixing a point $p \\in \\mathfrak{h}_\\mathbb{C}$, it follows that the fibre $\\mathsf{s}_\\EuScript{I}^{-1}(\\upphi_\\EuScript{I}(p))$ contains a complete curve if and only if the fibre $\\mathsf{t}_\\EuScript{I}^{-1}(p)$ contains a complete curve which is not blown down. Again by \\cite[Theorem 1(3)]{KatzMorrison}, the fibre $\\mathsf{t}_\\EuScript{I}^{-1}(p)$ contains a complete curve if and only if $p \\in \\EuScript{D}_{\\upalpha,\\mathbb{C}}$ for some positive root $\\upalpha$, and this curve is not blown down if and only if $\\uppi_\\EuScript{I}(\\upalpha) \\neq 0$. This produces the desired decomposition of $\\EuScript{D}_\\EuScript{I}$. It then follows from Lemma~\\ref{lem: Dynkin combinatorics} that the components $\\EuScript{D}_{\\upalpha,\\mathbb{C}}\/\\EuScript{W}_\\EuScript{I}$ are indexed by the restricted positive roots $\\uppi_\\EuScript{I}(\\upalpha)$, i.e. $\\EuScript{D}_{\\upalpha,\\mathbb{C}}\/W_{\\EuScript{I}}=\\EuScript{D}_{\\upalpha^\\prime,\\mathbb{C}}\/W_{\\EuScript{I}}$ if and only if $\\uppi_\\EuScript{I}(\\upalpha)=\\uppi_\\EuScript{I}(\\upalpha^\\prime)$.\n\n\\noindent (2)(3) First recall what it means for a curve in $\\mathsf{s}_\\EuScript{I}^{-1}(p)$ to have class $\\upbeta \\in A_1(Y_\\EuScript{I})$. The inclusion of the central fibre $\\mathsf{i}_0 \\colon \\mathsf{s}_\\EuScript{I}^{-1}(0) = Y_\\EuScript{I} \\hookrightarrow \\EuScript{Y}_\\EuScript{I}$ induces an isomorphism\n\\[ \n\\mathsf{i}_{0\\star} \\colon A_1(Y_\\EuScript{I}) \\xrightarrow{\\sim} A_1(\\EuScript{Y}_\\EuScript{I}).\n\\]\nNow consider an arbitrary fibre $\\mathsf{s}_\\EuScript{I}^{-1}(p)$ with inclusion $\\mathsf{i}_p \\colon \\mathsf{s}_\\EuScript{I}^{-1}(p) \\hookrightarrow \\EuScript{Y}_\\EuScript{I}$. If $\\Curve \\subseteq \\mathsf{s}_\\EuScript{I}^{-1}(p)$ is a complete curve, then $\\Curve$ has class $\\upbeta \\in A_1(Y_\\EuScript{I})$ if $\\mathsf{i}_{p\\star} \\Curve = \\mathsf{i}_{0\\star} \\upbeta$. The same definition applies to the full simultaneous resolution $\\EuScript{Z}$. Both (2) and (3) are known for $\\EuScript{Z}$ by \\emph{loc. cit.}, and the general case follows by tracking curve classes from $\\EuScript{Z}$ to $\\EuScript{Y}_\\EuScript{I}$.\n\\end{proof}\n\n\\begin{remark} \\label{rmk: proportional but distinct}\nConsider classes $\\upbeta,\\upbeta^\\prime \\in A_1(Y_{\\EuScript{I}})$ which are proportional but distinct, i.e.\\ $k\\upbeta = k^\\prime \\upbeta^\\prime$ for some distinct integers $k,k^\\prime \\geq 1$. If both $\\upbeta, \\upbeta^\\prime$ are restricted positive roots, then the lifts $\\upalpha,\\upalpha^\\prime$ will \\emph{not} be proportional, by the root system axioms. In particular, the corresponding components of the standard discriminant locus $\\EuScript{D}_{\\EuScript{I}}$ will be distinct. Every component of the discriminant locus therefore corresponds to a unique curve class $\\upbeta$. This is in contrast to the components of the hyperplane arrangement $\\EuScript{H}_\\EuScript{I} \\subseteq \\Uptheta_\\EuScript{I}$ from Subsection~\\ref{sec: hyperplane arrangements}. \\end{remark}\n\n\n\\begin{remark} \\label{rem: def of E}\nThe description of the standard discriminant locus $\\EuScript{D}_\\EuScript{I}$ in Theorem~\\ref{thm: KM Theorem 1(c)} is a union over positive roots $\\upalpha \\in \\mathfrak{h}$ such that $\\uppi_\\EuScript{I}(\\upalpha) \\neq 0$. The complementary union of hyperplane quotients\n\\[\n\\EuScript{E}_{\\EuScript{I}}=\\bigcup_{\\uppi_{\\EuScript{I}}(\\upalpha)= 0}\\,\\,\\EuScript{D}_{\\upalpha,\\mathbb{C}}\/W_\\EuScript{I}\n\\]\nparametrise points $p \\in \\mathfrak{h}_{\\mathbb{C}}\/W_{\\EuScript{I}}$ such that the fibre $\\mathsf{s}_\\EuScript{I}^{-1}(p)$ is singular. Clearly\n\\[ \n\\EuScript{D}_\\emptyset\/W_\\EuScript{I} = \\EuScript{D}_\\EuScript{I} \\cup \\EuScript{E}_\\EuScript{I}.\n\\]\nFor a generic point $p \\in \\EuScript{E}_\\EuScript{I}$, the fibre $\\mathsf{s}_\\EuScript{I}^{-1}(p)$ contains a single $A_1$ singularity. The locus $\\EuScript{E}_\\EuScript{I}$ will play a less central role than $\\EuScript{D}_{\\EuScript{I}}$. The translation between our notation and that of \\cite[Section~2]{BryanKatzLeung} is as follows: $\\EuScript{D}_{\\EuScript{I}} = D^{\\operatorname{curv}}, \\EuScript{E}_{\\EuScript{I}} = D^{\\operatorname{sing}}$ and $\\EuScript{D}_{\\EuScript{I}} \\cup \\EuScript{E}_{\\EuScript{I}} = D$.\n\\end{remark}\n\n\n\n\n\\subsection{$3$-fold perturbations via surface deformations} \\label{sec: deforming to Xt} \\label{sec: perturbing target}\nGiven the flopping contraction $\\EuScript{X}\\to\\mathop{\\rm Spec}\\nolimits\\EuScript{R}$, a choice of local equation for the hypersurface $\\mathop{\\rm Spec}\\nolimits\\EuScript{R}\/g \\subseteq \\mathop{\\rm Spec}\\nolimits \\EuScript{R}$ produces a flat family $\\mathop{\\rm Spec}\\nolimits \\EuScript{R} \\to \\EuScript{D} \\mathrm{isc}$ over a formal disc, with central fibre an ADE surface singularity. By composition this produces a flat family $\\EuScript{X} \\to \\EuScript{D} \\mathrm{isc}$ with central fibre the partial resolution $Y\\cong Y_{\\EuScript{I}}$ of the ADE singularity \\cite{Reid}. This exhibits $\\EuScript{X}$, respectively $\\mathop{\\rm Spec}\\nolimits\\EuScript{R}$, as the total space of a one-parameter deformation of the surface $Y_{\\EuScript{I}}$, respectively $\\mathop{\\rm Spec}\\nolimits \\EuScript{R}\/g$. These deformations are induced by an associated classifying map\n\\[\n\\upmu \\colon \\EuScript{D} \\mathrm{isc} \\to \\mathfrak{h}_{\\mathbb{C}}\/W_{\\EuScript{I}}\n\\]\nwhere $\\mathfrak{h}_{\\mathbb{C}}\/W_{\\EuScript{I}}$ is as described in the previous subsection, and the contraction $\\EuScript{X} \\to \\mathop{\\rm Spec}\\nolimits \\EuScript{R}$ is obtained from the simultaneous partial resolution of Subsection~\\ref{sec: sim partial res I} by base change\n\\begin{center}\\begin{tikzcd}\n\\EuScript{X} \\ar[d,\"f\" left] \\ar[r] & \\EuScript{Y}_{\\EuScript{I}} \\ar[d,\"\\mathsf{g}_{\\EuScript{I}}\"] \\\\\n\\mathop{\\rm Spec}\\nolimits \\EuScript{R} \\ar[r] \\ar[d] & \\mathop{\\rm Spec}\\nolimits \\EuScript{V}_\\EuScript{I} \\ar[d,\"\\mathsf{h}_\\EuScript{I}\"] \\\\\n\\EuScript{D} \\mathrm{isc} \\ar[r,\"\\upmu\"] & \\mathfrak{h}_\\mathbb{C}\/W_\\EuScript{I}.\n\\end{tikzcd}\\end{center}\nThe central fibre of $\\mathop{\\rm Spec}\\nolimits\\EuScript{R} \\to \\EuScript{D} \\mathrm{isc}$ is the ADE singularity corresponding to $\\Delta$, so $\\upmu(0) = 0$. On the other hand, since $\\EuScript{X}$ contains no complete curves outside of $Y$ the map $\\upmu$ does not intersect the discriminant locus away from the origin, so $\\upmu^{-1}(\\EuScript{D}_\\EuScript{I}) = \\upmu^{-1}(0) = 0$. \n\nAs in \\cite[below Lemma 2.7]{BryanKatzLeung} there exists a one-parameter perturbation of $\\upmu$\n\\[ \n(\\upmu_t)_{t\\in [0,\\upvarepsilon]} \\colon \\EuScript{D} \\mathrm{isc} \\times [0,\\upvarepsilon] \\to \\mathfrak{h}_{\\mathbb{C}}\/W_{\\EuScript{I}} \n\\]\nsuch that $\\upmu_0=\\upmu$ and for $t \\neq 0$ the following transversality condition is satisfied\n\\begin{equation}\\label{condition transversality of muprime} \n\\text{$\\upmu_{t}$ intersects $\\EuScript{D}_{\\EuScript{I}}\\cup \\EuScript{E}_{\\EuScript{I}}$ transversely and away from codimension two strata}\n \\end{equation}\nwhere $\\EuScript{E}_{\\EuScript{I}}$ is defined in Remark~\\ref{rem: def of E}. Furthermore, making $\\upvarepsilon$ smaller if necessary, we can assume that $\\upmu_t^{-1}(\\EuScript{D}_{\\EuScript{I}})$ is bounded away from the boundary of $ \\EuScript{D} \\mathrm{isc}$.\n\nThe spaces $\\EuScript{X}_{t\\neq 0}$ give generic perturbations of the target $\\EuScript{X}_0=\\EuScript{X}$. The following is well-known, and will be used to reduce the enumerative geometry of $\\EuScript{X}_t$, locally, to that of the Atiyah flop.\n \n \\begin{lemma} \nFor any $t \\neq 0$, the total space $\\EuScript{X}_t$ of the family of surfaces associated to $\\upmu_t$ is a smooth $3$-fold. Further, every complete curve in $\\EuScript{X}_t$ is isolated, smooth, and rational, with normal bundle isomorphic to $\\EuScript{O}_{\\mathbb{P}^1}(-1)\\oplus \\EuScript{O}_{\\mathbb{P}^1}(-1)$. \n\\end{lemma}\n \\begin{proof} \n This is essentially \\cite[Proposition 2.2]{BryanKatzLeung}, which it itself is extracted from the proof of \\cite[Theorem 1]{KatzMorrison}. \\end{proof}\n \nWrite $\\mathfrak{X}$ for the $4$-dimensional total space of the entire family $(\\upmu_t)_{t}$. Then as explained in \\cite[Section~3]{Wilson}, pulling back along inclusions of fibres induces isomorphisms\n\\begin{equation}\n\\mathrm{H}^2(Y_{\\EuScript{I}};\\mathbb{Z})\\xleftarrow{\\cong} \\mathrm{H}^2(\\EuScript{X};\\mathbb{Z})\\xleftarrow{\\cong} \\mathrm{H}^2(\\mathfrak{X};\\mathbb{Z})\\xrightarrow{\\cong} \\mathrm{H}^2(\\EuScript{X}_t;\\mathbb{Z})\\label{eqn: lift line bundles}\n\\end{equation}\nfor any $t$. Any class $L$ in $\\mathrm{H}^2(\\EuScript{X},\\mathbb{Z})\\cong\\mathop{\\rm Pic}\\nolimits(\\EuScript{X})$ thus induces an invertible sheaf $\\EuScript{L}$ on $\\mathfrak{X}$ with $\\EuScript{L}|_{\\EuScript{X}_0}=L$. Similarly, pushing forward curve classes along the inclusion of fibres induces isomorphisms\n \\begin{equation}\n A_1(Y_{\\EuScript{I}})\\xrightarrow{\\cong} A_1(\\EuScript{X}) \\xrightarrow{\\cong} A_1(\\mathfrak{X}) \\xleftarrow{\\cong} A_1(\\EuScript{X}_t) \\label{eq:ident A three ways}\n \\end{equation}\n for any $t$. Given $\\upbeta \\in A_1(\\EuScript{X})$ we abuse notation and let $\\upbeta \\in A_1(\\EuScript{X}_t)$ denote the image of $\\upbeta$ under the composition of the natural isomorphisms above. Further, combining \\eqref{eq:ident A three ways} and \\eqref{eq:ident A with comb} it makes sense to ask when curve classes are restricted roots.\n \n \\begin{cor}\\label{secret reason why nonzero}\\label{lem: properties of Xt}\\label{cor: curves on perturbation}\nFix $t \\neq 0$ and $\\upbeta \\in A_1(\\EuScript{X})$ non-zero. Then the following statements hold:\n\\begin{enumerate}\n\\item If $\\upbeta$ is not a restricted positive root, i.e. there does not exist a positive root $\\upalpha$ with $\\uppi_\\EuScript{I}(\\upalpha)=\\upbeta$, then there is no complete curve in $\\EuScript{X}_t$ of class $\\upbeta$.\n\\item If $\\upbeta$ is a restricted positive root, with $\\uppi_{\\EuScript{I}}(\\upalpha)=\\upbeta$, then the number of complete curves in $\\EuScript{X}_t$ of class $\\upbeta$ is equal to $|\\upmu_t^{-1}(\\EuScript{D}_{\\upalpha,\\mathbb{C}}\/W_\\EuScript{I})|$ and is always $\\geq 1$.\n\\end{enumerate}\n\\end{cor}\n\\begin{proof} This follows from Theorem~\\ref{thm: KM Theorem 1(c)}, the only new claim being that $|\\upmu_t^{-1}(\\EuScript{D}_{\\upalpha,\\mathbb{C}}\/W_\\EuScript{I})| \\geq 1$. We observed above that $\\upmu(0)=0$. Since $\\upmu=\\upmu_0$ and $0 \\in \\EuScript{D}_{\\upalpha,\\mathbb{C}}\/W_\\EuScript{I}$ for every positive root $\\upalpha$, it follows that $|\\upmu_0^{-1}(\\EuScript{D}_{\\upalpha,\\mathbb{C}}\/W_\\EuScript{I})| \\geq 1$ for every positive root $\\upalpha$. This quantity does not change under a small perturbation of $\\upmu_0$ to $\\upmu_t$.\\end{proof}\n\n\\section{Curve counting and hyperplane arrangements}\\label{sec: enumerative invariants}\n\n\\noindent This section obtains structural results governing the systems of enumerative invariants attached to $\\EuScript{X}$. These turn out to be intimately related to the associated arrangement of hyperplanes from Subsection~\\ref{sec: hyperplane arrangements}, and this correspondence is then used to describe the pole locus of the associated quantum potential.\n\n\\subsection{Gopakumar--Vafa}\\label{GV section} \\label{sec: GV invariants} Curve counting invariants of $\\EuScript{X}$ can be be defined using the perturbed target $\\EuScript{X}_t$ (for some fixed $t \\neq 0$) constructed in Subsection~\\ref{sec: perturbing target}. Given a curve class $\\upbeta \\in A_1(\\EuScript{X})$ the associated genus-zero Gopakumar--Vafa (GV) invariant\n\\[ \nn_\\upbeta = n_{\\upbeta,\\EuScript{X}} \\in \\mathbb{Z}_{\\geq 0}\n\\]\nis defined as the number of complete curves in $\\EuScript{X}_t$ of class $\\upbeta$. By Corollary~\\ref{lem: properties of Xt} this is zero if $\\upbeta$ is not a restricted positive root, and otherwise is equal to the number of intersection points of $\\upmu_t$ with the appropriate component of the discriminant locus, i.e.\n\\begin{equation} \\label{eqn: nbeta equals intersection with hyperplane} n_{\\upbeta} = |\\upmu_t^{-1}(\\EuScript{D}_{\\upalpha,\\mathbb{C}}\/W_\\EuScript{I})| \\end{equation}\nwhere $\\upalpha$ is any positive root with $\\uppi_{\\EuScript{I}}(\\upalpha)=\\upbeta$. This number is independent of the choice of small perturbation $\\upmu_t$.\n\nIn what follows, for a curve class $\\upbeta$ consider the dual hyperplane $\\mathsf{H}_\\upbeta\\subseteq \\Uptheta_\\EuScript{I}$.\n\\begin{corollary} \\label{cor: GV nonzero}\nIf $\\upbeta \\in A_1(\\EuScript{X})$ then $n_\\upbeta$ is non-zero if and only if $\\upbeta$ is a restricted positive root, equivalently if and only if $\\mathsf{H}_\\upbeta$ belongs to the enhanced finite arrangement $\\EuScript{H}_\\EuScript{I}$.\n\\end{corollary}\n\\begin{proof} \nThis follows immediately from Corollary~\\ref{cor: curves on perturbation}, together with the definition of enhanced finite arrangement in Subsection~\\ref{sec: hyperplane arrangements}.\n\\end{proof}\nNote that $\\mathsf{H}_\\upbeta$ and $\\mathsf{H}_{2\\upbeta}$ should be considered as different hyperplanes in the enhanced finite arrangement. See also the discussion in Subsection~\\ref{sec: hyperplane arrangements}, and Remark~\\ref{rmk: proportional but distinct}.\n\n\\begin{remark} It follows from Corollary~\\ref{cor: GV nonzero} that there are only finitely many non-zero GV invariants. There is already a known range outside of which the GV invariants are guaranteed to vanish. Indeed, every simple root $i \\in \\Delta$ has an associated length $\\updelta_i$, given by the coefficient of $\\upalpha_i$ in the maximal root, and writing $\\upbeta = \\Sigma_i m_i \\Curve_i$ it is known that $n_{\\upbeta} = 0$ unless $m_i \\leq \\updelta_i$ for all $i$. However, this bound is far from sharp, while Corollary~\\ref{cor: GV nonzero} provides a precise characterisation.\n\\end{remark}\n\n\\subsection{Gromov--Witten} \\label{sec: GW invariants} We refer to \\cite[Section~7]{CoxKatz} for an introduction to Gromov--Witten theory. For every non-zero curve class $\\upbeta \\in A_1(\\EuScript{X})$ there is an associated genus-zero Gromov--Witten (GW) invariant\n\\[\n N_\\upbeta=N_{\\upbeta,\\EuScript{X}} \\in \\mathbb{Q}\n \\]\ndefined as the virtual degree of the corresponding moduli space of stable maps to $\\EuScript{X}$. By deformation invariance this coincides with the virtual degree of the moduli space of stable maps to $\\EuScript{X}_t$ for $t \\neq 0$, as constructed in Subsection~\\ref{sec: deforming to Xt}. \n\nThe latter space decomposes as a disjoint union of spaces of stable maps to $\\mathbb{P}^1$, and applying the Aspinwall--Morrison multiple cover formula \\cite{AspinwallMorrison,VoisinAspinwallMorrison} for the local invariants of $\\EuScript{O}_{\\mathbb{P}^1}(-1)\\oplus \\EuScript{O}_{\\mathbb{P}^1}(-1)$ gives the following relationship between the GW invariants and the GV invariants from Subsection~\\ref{sec: GV invariants}, namely \n\\begin{equation} \nN_\\upbeta = \\sum_{d | \\upbeta} \\dfrac{n_{\\upbeta\/d}}{d^3}\\label{eqn: multiple cover formula}.\n\\end{equation}\nMore generally, given $k \\geq 0$ and homogeneous classes $\\upgamma_1,\\ldots,\\upgamma_k \\in \\mathrm{H}^\\star(\\EuScript{X};\\mathbb{C})$, the associated GW invariant with cohomological insertions at marked points is defined to be\n\\[\n\\langle \\upgamma_1,\\ldots,\\upgamma_k \\rangle^{\\EuScript{X}}_{0,k,\\upbeta} \\colonequals \\int_{[\\ol{\\mathcal{M}}_{0,k}(\\EuScript{X},\\upbeta)]^{\\operatorname{virt}}} \\prod_{i=1}^k \\ev_i^\\star \\upgamma_i \n\\]\nand provides a virtual count of rational curves in $\\EuScript{X}$ of class $\\upbeta$ passing through the cycles $\\upgamma_1,\\ldots,\\upgamma_k$. Note that in particular $N_{\\upbeta} = \\langle \\rangle^{\\EuScript{X}}_{0,0,\\upbeta}$. Since $\\EuScript{X}$ is a Calabi--Yau $3$-fold, the invariant vanishes unless the input data satisfies the dimension constraint\n\\begin{equation}\\label{eqn: dimension constraint} \\sum_{i=1}^k \\deg \\upgamma_i = 2k.\\end{equation}\nThe cohomology of $\\EuScript{X}$ is well-understood, see e.g.\\ \\cite[5.2]{CaibarThesisPaper}. In particular\n\\begin{equation*} \n\\mathrm{H}^0(\\EuScript{X}; \\mathbb{C}) = \\mathbb{C} \\cdot \\mathds{1},\\quad \\mathrm{H}^1(\\EuScript{X}; \\mathbb{C})=0,\\quad \\mathrm{H}^2(\\EuScript{X}; \\mathbb{C}) = \\mathop{\\rm Pic}\\nolimits \\EuScript{X} \\otimes \\mathbb{C}.\n\\end{equation*}\nMoreover, as we work in the complete local setting, by e.g.\\ \\cite[3.4.4]{VdB1d} $\\mathop{\\rm Pic}\\nolimits\\EuScript{X}$ is dual to the group $A_1(\\EuScript{X})$ of curve classes, since there is a basis of divisor classes $\\mathop{\\rm Pic}\\nolimits\\EuScript{X} = \\langle D_i \\mid i \\in \\scrI^{\\kern 0.5pt\\mathrm{c}} \\rangle_{\\mathbb{Z}}$ which satisfies $D_i \\cdot \\Curve_j = \\updelta_{ij}$.\n\nGiven a GW invariant $\\langle \\upgamma_1,\\ldots,\\upgamma_k \\rangle^{\\EuScript{X}}_{0,k,\\upbeta}$, if any $\\upgamma_i=\\mathds{1}$ then the invariant vanishes by the string equation. It follows from \\eqref{eqn: dimension constraint} that the invariant vanishes unless each $\\upgamma_i \\in \\mathrm{H}^2(\\EuScript{X};\\mathbb{C})$. But then the $\\upgamma_i$ are divisors, and the $k$-pointed invariants with divisorial insertions are related to the $0$-pointed invariants by the divisor equation\n\\[ \n\\langle D_{j_1}, \\ldots, D_{j_k} \\rangle^{\\EuScript{X}}_{0,k,\\upbeta} = \\left( \\prod_{i=1}^k D_{j_i}\\cdot \\upbeta \\right) N_{\\upbeta}.\n\\]\nIn this way, the non-zero GW invariants are controlled entirely by the $N_{\\upbeta}$, which by \\eqref{eqn: multiple cover formula} are controlled entirely by the GV invariants $n_{\\upbeta}$. The latter constitutes a finite list of numbers.\n\n\n\\subsection{Quantum cohomology} \\label{sec: quantum potential} As is well known, the GW invariants form the structure constants for quantum cohomology. The information defining quantum cohomology is equivalent to the \\emph{quantum potential}, defined in our setting as\n\\begin{equation} \\label{eqn: quantum potential} \n\\Phi^{\\EuScript{X}}_{\\mathsf{t}}( \\upgamma_1,\\upgamma_2,\\upgamma_3) \\colonequals \\sum_{\\substack{\\upbeta \\in A_1(\\EuScript{X})\\\\ \\upbeta \\neq 0}} \\,\\sum_{k \\geq 0} \\,\\,\\dfrac{1}{k!} \\langle \\upgamma_1,\\upgamma_2,\\upgamma_3, \\mathsf{t} , \\ldots, \\mathsf{t} \\rangle^{\\EuScript{X}}_{0,k+3,\\upbeta}. \\end{equation}\nHere we exclude the case $\\upbeta = 0$ from the sum, since for non-compact $\\EuScript{X}$ such invariants are not defined (see Remark~\\ref{rmk: no algebra} below). We view \\eqref{eqn: quantum potential} as a family of multilinear maps\n\\[\n\\Phi^{\\EuScript{X}}_\\mathsf{t} \\colon \\mathrm{H}^\\star(\\EuScript{X};\\mathbb{C})^{\\otimes 3} \\to \\mathbb{C}\t\n\\]\nparametrised by the formal variable $\\mathsf{t} \\in \\mathrm{H}^\\star(\\EuScript{X};\\mathbb{C})$. By the earlier dimension arguments, we see that the quantum potential only depends on the component of $\\mathsf{t}$ in the $\\mathrm{H}^2(\\EuScript{X};\\mathbb{C})$ direction. Thus we may assume $\\mathsf{t} \\in \\mathrm{H}^2(\\EuScript{X};\\mathbb{C})$ and write\n\\[\n\\mathsf{t} = (\\mathsf{t}_i)_{i \\in \\scrI^{\\kern 0.5pt\\mathrm{c}}} = \\sum_{i \\in \\scrI^{\\kern 0.5pt\\mathrm{c}}} \\mathsf{t}_i D_i.\n\\]\nThe parameter space for the quantum potential is thus co-ordinatised by the $\\mathsf{t}_i$. An alternative co-ordinate system is given by the Novikov parameters, defined by\n\\[ \n\\mathsf{q}_i \\colonequals \\exp(\\mathsf{t}_i).\n\\]\nThe following result resembles expressions appearing in earlier work \\cite{MorrisonKaehler, Wilson}, but is more explicit, being given in terms of canonical bases for $\\mathrm{H}^2(\\EuScript{X};\\mathbb{C})$ and $A_1(\\EuScript{X})$. This refined information will allow us in Corollary~\\ref{cor: quantum is hyper} to pinpoint the non-vanishing terms using Dynkin combinatorics, and in Corollary~\\ref{thm: CTC} to track the change in quantum potential under iterated flops.\n\n\\begin{theorem} \\label{thm: structure of quantum potential} The quantum potential has a natural analytic continuation over the parameter space, given as a finite sum of terms indexed by the non-vanishing GV invariants\n\\begin{equation} \\label{eqn: rational expression for quantum potential} \\Phi^{\\EuScript{X}}_\\mathsf{t}(\\upgamma_1,\\upgamma_2,\\upgamma_3) = \\sum_{\\upbeta=(m_i)} n_\\upbeta (\\upgamma_1 \\cdot \\upbeta) (\\upgamma_2 \\cdot \\upbeta) (\\upgamma_3 \\cdot \\upbeta) \\dfrac{\\prod_{i \\in \\scrI^{\\kern 0.5pt\\mathrm{c}}} \\mathsf{q}_i^{m_i}}{1-\\prod_{i \\in \\scrI^{\\kern 0.5pt\\mathrm{c}}} \\mathsf{q}_i^{m_i}}. \\end{equation}\n\\end{theorem}\n\\noindent The sum is over non-zero curve classes\n\\[\\upbeta=(m_i)_{i \\in \\scrI^{\\kern 0.5pt\\mathrm{c}}} = \\sum m_i \\Curve_i \\in A_1(\\EuScript{X}).\\]\nEach term is a cubic polynomial in the input variables, multiplied by a specific rational function in the Novikov parameters, and weighted by the GV invariant $n_{\\upbeta}$. We will also use the term `quantum potential' to refer to this analytic continuation.\n\n\\begin{proof} Write the formal parameter $\\mathsf{t}$ and the curve class $\\upbeta$ as sums\n\\[\n\\mathsf{t} = \\sum_{i \\in \\scrI^{\\kern 0.5pt\\mathrm{c}}} \\mathsf{t}_i D_i, \\qquad\n\\upbeta = \\sum_{i \\in \\scrI^{\\kern 0.5pt\\mathrm{c}}} m_i \\Curve_i.\n\\]\nApplying the divisor equation together with the multiple cover formula \\eqref{eqn: multiple cover formula} then gives\n\\begin{align*}\n\\Phi^{\\EuScript{X}}_{\\mathsf{t}}(\\upgamma_1,\\upgamma_2,\\upgamma_3) & = \\sum_{\\upbeta} \\,\\sum_{k \\geq 0} \\,\\,\\dfrac{1}{k!} \\langle \\upgamma_1,\\upgamma_2,\\upgamma_3, \\mathsf{t} , \\ldots, \\mathsf{t} \\rangle^{\\EuScript{X}}_{0,k+3,\\upbeta}\\\\\n& = \\sum_{\\upbeta} \\langle \\upgamma_1,\\upgamma_2,\\upgamma_3 \\rangle^{\\EuScript{X}}_{0,3,\\upbeta} \\,\\big(\\textstyle\\sum_{k\\geq 0} \\tfrac{(\\mathsf{t} \\cdot \\upbeta)^k}{k!} \\big)\\\\\n& = \\sum_{\\upbeta} N_\\upbeta (\\upgamma_1 \\cdot \\upbeta) (\\upgamma_2 \\cdot \\upbeta) (\\upgamma_3 \\cdot \\upbeta) \\exp\\!\\big( \\textstyle\\sum_{i \\in \\scrI^{\\kern 0.5pt\\mathrm{c}}} m_i \\mathsf{t}_i\\big) \\\\\n& = \\sum_{\\upbeta} n_{\\upbeta} \\sum_{d \\geq 1} \\dfrac{1}{d^3} (\\upgamma_1 \\cdot d\\upbeta)(\\upgamma_2 \\cdot d\\upbeta)(\\upgamma_3 \\cdot d\\upbeta) \\exp\\!\\big( \\textstyle\\sum_{i \\in \\scrI^{\\kern 0.5pt\\mathrm{c}}} dm_i \\mathsf{t}_i\\big) \\\\\n& = \\sum_{\\upbeta} n_\\upbeta (\\upgamma_1 \\cdot \\upbeta) (\\upgamma_2 \\cdot \\upbeta) (\\upgamma_3 \\cdot \\upbeta) \\sum_{d \\geq 1} \\exp\\!\\big( \\textstyle\\sum_{i \\in \\scrI^{\\kern 0.5pt\\mathrm{c}}} m_i \\mathsf{t}_i\\big)^d \\\\\n& = \\sum_{\\upbeta} n_\\upbeta (\\upgamma_1 \\cdot \\upbeta) (\\upgamma_2 \\cdot \\upbeta) (\\upgamma_3 \\cdot \\upbeta) \\sum_{d \\geq 1} \\big( \\Pi_{i \\in \\scrI^{\\kern 0.5pt\\mathrm{c}}} \\mathsf{q}_i^{m_i}\\big)^d.\n\\end{align*}\nNote that this sum is finite, since $n_\\upbeta=0$ for all but finitely many $\\upbeta$ (Corollary~\\ref{cor: GV nonzero}). For fixed inputs $(\\upgamma_1,\\upgamma_2,\\upgamma_3)$, the above is a formal power series in the Novikov parameters. It is the Taylor series for the following rational function, expanded about the point $(\\mathsf{q}_i)_i = (0,\\ldots,0)$, equivalently $(\\mathsf{t}_i)_i = (-\\infty, \\ldots, -\\infty)$, \n\\begin{align*} \n\\Phi^{\\EuScript{X}}_\\mathsf{t}(\\upgamma_1,\\upgamma_2,\\upgamma_3) = \\sum_\\upbeta n_\\upbeta (\\upgamma_1 \\cdot \\upbeta) (\\upgamma_2 \\cdot \\upbeta) (\\upgamma_3 \\cdot \\upbeta) \\dfrac{\\prod_{i \\in \\EuScript{I}} \\mathsf{q}_i^{m_i}}{1-\\prod_{i \\in \\EuScript{I}} \\mathsf{q}_i^{m_i}}.\n\\end{align*}\nThis expression provides a natural analytic continuation of the quantum potential beyond the radius of convergence $\\{ |\\mathsf{q}_i| < 1 \\mid i \\in \\scrI^{\\kern 0.5pt\\mathrm{c}} \\}$.\n\\end{proof}\n\nRecall that after combining \\eqref{eq:ident A three ways} with \\eqref{eq:ident A with comb} we can ask which curve classes are restricted roots. \n\\begin{corollary}\\label{cor: quantum is hyper}\nUnder the uniformly rescaled co-ordinates $\\mathsf{p}_i \\colonequals \\mathsf{t}_i\/2\\uppi\\sqrt{-1}$ on $\\mathrm{H}^2(\\EuScript{X};\\mathbb{C})$, the pole locus of the quantum potential is given by\n\\[\n\\bigcup_{\\upbeta=(m_i)}\\big\\{\\textstyle\\sum_{i \\in \\scrI^{\\kern 0.5pt\\mathrm{c}}} m_i \\mathsf{p}_i \\in \\mathbb{Z} \\big\\}\n\\]\nwhere the union is over all restricted positive roots $\\upbeta$. This is precisely the complexification of $\\EuScript{H}^{\\mathsf{aff}}_{\\EuScript{I}}$ under the natural identification $\\mathrm{H}^2(\\EuScript{X};\\mathbb{R})\\cong\\Uptheta_{\\EuScript{I}}$ dual to \\eqref{eq:ident A with comb}.\n\\end{corollary}\n\\begin{proof} \nThe pole locus of \\eqref{eqn: rational expression for quantum potential} is the union of loci in the parameter space given by\n\\[ \n\\prod_{i \\in \\scrI^{\\kern 0.5pt\\mathrm{c}}} \\mathsf{q}_i^{m_i} = 1 \\\n\\Leftrightarrow \\ \\sum_{i \\in \\scrI^{\\kern 0.5pt\\mathrm{c}}} m_i \\mathsf{t}_i \\in 2 \\uppi \\sqrt{-1} \\cdot \\mathbb{Z} \\\n\\Leftrightarrow \\ \\sum_{i \\in \\scrI^{\\kern 0.5pt\\mathrm{c}}} m_i \\mathsf{p}_i \\in \\mathbb{Z}\n\\]\nwhere $\\upbeta=(m_i)_{i \\in \\scrI^{\\kern 0.5pt\\mathrm{c}}}=\\sum m_i \\Curve_i$ is such that $n_\\upbeta$ is non-zero. The first statement then follows from Corollary~\\ref{cor: GV nonzero}, since $n_\\upbeta$ is non-zero if and only if $\\upbeta$ is a restricted positive root. The second is an immediate consequence of the definition of $\\EuScript{H}^{\\mathsf{aff}}_{\\EuScript{I}}$ in \\eqref{eqn: infinite arrangement}.\n\\end{proof}\n\n\n\\begin{example}\nFor a single-curve flop with $\\EuScript{I}=\\Eseven{B}{B}{P}{B}{B}{B}{B}$, the complexification of $\\EuScript{H}^{\\mathsf{aff}}_{\\EuScript{I}}$ is \n\\def4{4}\n\\[\n\\begin{tikzpicture}[scale=0.9]\n\\draw[densely dotted,->] ($(-4,0)+(-0.66,0)$) -- ($(4,0)+(0.66,0)$);\n\\node at ($(4,0)+(1,0)$) {$\\scriptstyle\\mathbb{R}$};\n\\draw[densely dotted,->] (0,-1) -- (0,1);\n\\node at (0,1.2) {$\\scriptstyle\\mathrm{i}\\mathbb{R}$};\n\n{\\foreach \\i in {-2,-1,0,1,2}\n\\filldraw[fill=white,draw=black] (4*1\/2*\\i,0) circle (2pt);\n}\n{\\foreach \\i in {-2,-1,1,2}\n\\filldraw[fill=white,draw=black] (4*1\/3*\\i,0) circle (2pt);\n}\n{\\foreach \\i in {-3,-2,-1,1,2,3}\n\\filldraw[fill=white,draw=black] (4*1\/4*\\i,0) circle (2pt);\n}\n\n\\node at (-4,-0.25) {$\\scriptstyle -1$};\n\n\\node at (0,-0.25) {$\\scriptstyle 0$};\n\n\\node at (4*0.25,-0.3) {$\\scriptstyle \\frac{1}{4}$};\n\\node at (4*1\/3,-0.3) {$\\scriptstyle \\frac{1}{3}$};\n\\node at (4*1\/2,-0.3) {$\\scriptstyle \\frac{1}{2}$};\n\\node at (4*2\/3,-0.3) {$\\scriptstyle \\frac{2}{3}$};\n\\node at (4*3\/4,-0.3) {$\\scriptstyle \\frac{3}{4}$};\n\\node at (4,-0.25) {$\\scriptstyle 1$};\n\\end{tikzpicture}\n\\]\nextended to infinity in both directions. The non-zero GV invariants are $n_{k\\Curve}$ for $1 \\leq k \\leq 4$.\n\\end{example}\n\n\\begin{example}\nIn the running Example~\\ref{example: intro}, namely a two-curve flop with $\\EuScript{I}=\\Eeight{B}{P}{B}{B}{B}{B}{B}{P}$, the complexification of $\\EuScript{H}^{\\mathsf{aff}}_{\\EuScript{I}}$ is \nthe complexification of the real arrangement in \\eqref{running example infinite}.\n\\end{example}\n\n\\begin{remark} \\label{rmk: no algebra} The definition of the quantum cohomology algebra requires a perfect pairing on cohomology in order to raise indices, but since here $\\EuScript{X}$ is non-compact, such a pairing does not exist. This technical issue is often circumvented by localising to a torus-fixed locus which is compact, see e.g.\\ \\cite{BryanGholampour,CoatesIritaniJiang}. Since our geometries do not always carry a suitable torus action, instead we simply equate `quantum cohomology' with the data of the quantum potential \\eqref{eqn: quantum potential}, as this is consistent with other approaches \\cite{LiRuan}. In cases where a natural quantum cohomology algebra can be defined, our results apply equally well to that algebra. The only modification required is to reinstate the $\\upbeta=0$ terms in the quantum potential, which encode the given perfect pairing. \\end{remark}\n\n\n\n\\section{Flops via simultaneous partial resolutions}\\label{sec: flops via sim res}\n\n\\noindent This section constructs flops, and describes how their dual graph changes, via simultaneous partial resolutions, completing work of Pinkham \\cite{Pinkham}. As a consequence we obtain an explicit change-of-basis matrix, which in Subsection~\\ref{sec: GV under flop} is used to track the change of GV invariants under iterated flop.\n\nThe construction requires some more Dynkin notation, so for a subset $\\EuScript{I}\\subseteq\\Delta$, $j\\in \\EuScript{I}$ and $i\\in \\scrI^{\\kern 0.5pt\\mathrm{c}}$ write\n\\[\n\\EuScript{I}+i= \\EuScript{I}\\cup \\{i\\}\\quad\\mbox{and}\\quad\n\\EuScript{I}-j= \\EuScript{I}\\setminus \\{j\\}.\n\\]\nFurther, to every Dynkin diagram $\\Gamma$ is an associated \\emph{Dynkin involution}, which we will denote $\\upiota_\\Gamma$. For Type $A_n$ and $E_6$ this is the obvious reflection, for $E_7$ and $E_8$ it is trivial, and for $D_n$ the behaviour depends on the parity of $n$, see e.g.\\ \\cite[(1.2.B)]{IyamaWemyssTits}. If $\\Gamma$ is a disjoint union of Dynkin diagrams, then $\\upiota_\\Gamma$ by definition acts separately on each component. Further, if $\\Delta$ is ADE, and $\\Gamma$ is a subset of $\\Delta$, then automatically the subgraph $\\Gamma$ is a disjoint union of ADE diagrams, and so there is an associated $\\upiota_\\Gamma$.\n\n\\begin{notation}[\\hspace{1sp}{\\cite[1.16]{IyamaWemyssTits}}]\nFor $i\\in\\scrI^{\\kern 0.5pt\\mathrm{c}}$, the wall crossing $\\upomega_i(\\EuScript{I})$ is defined by the rule\n\\[\n\\upomega_i(\\EuScript{I}) \\colonequals \\EuScript{I}+i-\\upiota_{\\EuScript{I}+i}(i) \\subseteq \\Delta.\n\\]\n\\end{notation}\n\n\\begin{example}\\label{ex: changing dual graph}\nConsider the running Example~\\ref{example: intro}, namely $\\EuScript{I}=\\Eeight{B}{P}{B}{B}{B}{B}{B}{P}$, where by convention $\\EuScript{I}$ equals the six black dots. There are two choices for $i\\in \\scrI^{\\kern 0.5pt\\mathrm{c}}$, namely the two pink nodes. Let $i$ be the rightmost. Then $\\EuScript{I}+i$ equals the black dots in the following.\n\\[\n\\EuScript{I}+i=\n\\begin{array}{c}\n\\begin{tikzpicture}[scale=0.21]\n\\node at (0,0) [B] {};\n\\node at (1,0) [P] {};\n\\node at (2,0) [B] {};\n\\node at (2,1) [B] {};\n\\node at (3,0) [B] {};\n\\node at (4,0) [B] {};\n\\node at (5,0) [B] {};\n\\node at (6,0) [B] {};\n\\draw[densely dotted] (1.5,1.5)--(2.5,1.5) -- (2.5,0.5)--(6.5,0.5)--(6.5,-0.5)--(1.5,-0.5)--cycle;\n\\draw[densely dotted] (-0.5,0.5)--(0.5,0.5) --(0.5,-0.5)--(-0.5,-0.5)--cycle;\n\\draw[<->,bend right] (6,0.5) to (2.5,1.5);\n\\end{tikzpicture}\n\\end{array}\n\\]\nThe black dots form $A_1\\times A_6$, and so applying the Dynkin involution $\\upiota_{\\EuScript{I}+i}$ illustrated, we see that $\\upiota_{\\EuScript{I}+i}(i)$ is the top node. Thus, for this choice of $i$, \n\\[\n\\upomega_i(\\EuScript{I})=\\EuScript{I}+i-\\upiota_{\\EuScript{I}+i}(i)=\\Eeight{B}{P}{B}{P}{B}{B}{B}{B}.\n\\]\n\\end{example}\n\nConsider now the fixed flopping contraction $\\EuScript{X}\\to\\mathop{\\rm Spec}\\nolimits\\EuScript{R}$ which slices under \\eqref{elephant pullback} to give $Y_{\\EuScript{I}} \\to \\mathbb{C}^2\/G$. Pick a flopping curve $\\Curve_i$ in $\\EuScript{X}$. This corresponds to a choice of $i\\in\\scrI^{\\kern 0.5pt\\mathrm{c}}$, so we can form $\\upomega_i(\\EuScript{I})$. In what follows consider $\\ell_\\EuScript{I}\\ell_{\\EuScript{I}+i}$, where $\\ell_\\EuScript{I}$ and $\\ell_{\\EuScript{I}+i}$ are the longest elements in the parabolic subgroups $W_\\EuScript{I}$ and $W_{\\EuScript{I}+i}$ respectively. \n\n\\begin{lemma}\\label{lem: induced iso A}\nThe action by $\\ell_\\EuScript{I}\\ell_{\\EuScript{I}+i}$ induces an isomorphism $\\mathfrak{h}_{\\mathbb{C}}\/W_{\\upomega_i(\\EuScript{I})}\\to \\mathfrak{h}_{\\mathbb{C}}\/W_{\\EuScript{I}}$.\n\\end{lemma}\n\\begin{proof}\nWriting $w=\\ell_\\EuScript{I}\\ell_{\\EuScript{I}+i}$, the point is that $wW_{\\upomega_i(\\EuScript{I})}=W_{\\EuScript{I}}w$ (see e.g.\\ \\cite[1.20(1)(a)]{IyamaWemyssTits}). The action by $w$ is an isomorphism $\\mathfrak{h}_\\mathbb{C}\\to\\mathfrak{h}_\\mathbb{C}$, and under this isomorphism any orbit $(W_{\\upomega_i(\\EuScript{I})}) p$ gets sent to $w (W_{\\upomega_i(\\EuScript{I})} )p=(W_{\\EuScript{I}}) w p$, which is an orbit under $W_{\\EuScript{I}}$.\n\\end{proof}\nNote that $\\ell_{\\upomega_i(\\EuScript{I})}\\ell_{\\EuScript{I}+i}\\ell_\\EuScript{I}\\ell_{\\EuScript{I}+i}=1$ \\cite[1.2(3)]{IyamaWemyssTits}, and thus $\\ell_{\\upomega_i(\\EuScript{I})}\\ell_{\\EuScript{I}+i}\\colon \\mathfrak{h}_{\\mathbb{C}}\/W_{\\EuScript{I}}\\to \\mathfrak{h}_{\\mathbb{C}}\/W_{\\upomega_i(\\EuScript{I})}$ is the inverse map. Recall the notation in \\eqref{preferred sim partial res}. Using the universal property of the pullback gives a non-obvious isomorphism between $\\EuScript{V}_{\\EuScript{I}}$ and $\\EuScript{V}_{\\upomega_i(\\EuScript{I})}$, which sits in the following commutative diagram.\n\\def-1{-1}\n\\def-1{-1}\n\\[\n\\begin{tikzpicture}[>=stealth,scale=1.2]\n\\node[black!70!white] (U) at (2,-1.5) {$\\mathop{\\rm Spec}\\nolimits \\EuScript{V}_{\\upomega_i(\\EuScript{I})}$};\n\\node[black!70!white] (V) at (4,-1.5) {$\\mathop{\\rm Spec}\\nolimits \\EuScript{V}$};\n\\node[black!70!white] (PRes) at (2,-3) {$\\mathfrak{h}_{\\mathbb{C}}\/W_{\\upomega_i(\\EuScript{I})}$};\n\\node[black!70!white] (Res) at (4,-3) {$\\mathfrak{h}_{\\mathbb{C}}\/W$};\n\\draw[black!70!white,->] (U)--(V);\n\\draw[black!70!white,->] (V)--(Res);\n\\draw[black!70!white,->] (U)--node[gap,pos=0.76]{\\phantom .}(PRes);\n\\draw[black!70!white,->] (PRes)--node[gap,pos=0.31]{\\phantom .}(Res);\n\\node (fU) at ($(2,-1.5)+(-1,-1)$) {$\\mathop{\\rm Spec}\\nolimits \\EuScript{V}_{\\EuScript{I}}$};\n\\node (fV) at ($(4,-1.5)+(-1,-1)$) {$\\mathop{\\rm Spec}\\nolimits \\EuScript{V}$};\n\\node (fPRes) at ($(2,-3)+(-1,-1)$) {$\\mathfrak{h}_{\\mathbb{C}}\/W_{\\EuScript{I}}$};\n\\node (fRes) at ($(4,-3)+(-1,-1)$) {$\\mathfrak{h}_{\\mathbb{C}}\/W$};\n\\draw[->] (fU)--(fV);\n\\draw[->] (fV)--(fRes);\n\\draw[->] (fU)--(fPRes);\n\\draw[->] (fPRes)--(fRes);\n\\draw[double] (fRes)--(Res);\n\\draw[double] (fV)--(V);\n\\draw[<-] (fPRes)--node[pos=0.35,right]{$\\scriptstyle \\ell_{\\EuScript{I}}\\ell_{\\EuScript{I}+i}\\cdot$}(PRes);\n\\draw[<-] (fU)--node[right]{$\\scriptstyle \\sim$}(U);\n\\end{tikzpicture}\n\\]\n\n\nNow let $\\EuScript{Y}_\\EuScript{I}$, respectively $\\EuScript{Y}_{\\upomega_i(\\EuScript{I})}$, be the standard simultaneous resolution associated to $\\EuScript{I}$, respectively $\\upomega_i(\\EuScript{I})$. As explained in Subsection~\\ref{sec: deforming to Xt}, $\\EuScript{X}$ can be obtained from $\\upmu\\colon\\EuScript{D} \\mathrm{isc}\\to\\mathfrak{h}_{\\mathbb{C}}\/W_{\\EuScript{I}}$ by pulling back $\\EuScript{Y}_\\EuScript{I}$. Hence, setting $\\upnu=( \\ell_{\\upomega_i(\\EuScript{I})}\\ell_{\\EuScript{I}+i} )\\circ\\upmu=(\\ell_\\EuScript{I}\\ell_{\\EuScript{I}+i})^{-1}\\circ\\upmu$ and pulling back to $\\EuScript{Y}_{\\upomega_i(\\EuScript{I})}$ constructs a variety $\\EuScript{X}^+_i$, sitting within the following commutative diagram.\n\\begin{equation}\n\\begin{array}{c}\n\\begin{tikzpicture}[>=stealth,scale=1.2]\n\\node[black!70!white] (X) at (0,0) {$\\EuScript{X}_i^+$};\n\\node[black!70!white] (Z) at (2,0) {$\\EuScript{Y}_{\\upomega_i(\\EuScript{I})}$};\n\\node[black!70!white] (R) at (0,-1.5) {$\\mathop{\\rm Spec}\\nolimits \\EuScript{R}'$};\n\\node[black!70!white] (U) at (2,-1.5) {$\\mathop{\\rm Spec}\\nolimits \\EuScript{V}_{\\upomega_i(\\EuScript{I})}$};\n\\node[black!70!white] (V) at (4,-1.5) {$\\mathop{\\rm Spec}\\nolimits \\EuScript{V}$};\n\\node[black!70!white] (t) at (0,-3) {$\\EuScript{D} \\mathrm{isc}$};\n\\node[black!70!white] (PRes) at (2,-3) {$\\mathfrak{h}_{\\mathbb{C}}\/W_{\\upomega_i(\\EuScript{I})}$};\n\\node[black!70!white] (Res) at (4,-3) {$\\mathfrak{h}_{\\mathbb{C}}\/W$};\n\\draw[black!70!white,->] (X)--(Z);\n\\draw[black!70!white,->] (X)--node[gap,pos=0.75]{\\phantom .}(R);\n\\draw[black!70!white,->] (Z)--(U);\n\\draw[black!70!white,->] (R)--(U);\n\\draw[black!70!white,->] (U)--(V);\n\\draw[black!70!white,->] (V)--(Res);\n\\draw[black!70!white,->] (R)--node[gap,pos=0.72]{\\phantom .}(t);\n\\draw[black!70!white,->] (U)--node[gap,pos=0.76]{\\phantom .}(PRes);\n\\draw[black!70!white,->] (t)--node[gap,pos=0.735]{\\phantom .}node[gap,pos=0.4]{$\\scriptstyle\\upnu$}(PRes);\n\\draw[black!70!white,->] (PRes)--node[gap,pos=0.31]{\\phantom .}(Res);\n\\node (fX) at (-1,-1) {$\\EuScript{X}$};\n\\node (fZ) at ($(2,0)+(-1,-1)$) {$\\EuScript{Y}_{\\EuScript{I}}$};\n\\node (fR) at ($(0,-1.5)+(-1,-1)$) {$\\mathop{\\rm Spec}\\nolimits \\EuScript{R}$};\n\\node (fU) at ($(2,-1.5)+(-1,-1)$) {$\\mathop{\\rm Spec}\\nolimits \\EuScript{V}_{\\EuScript{I}}$};\n\\node (fV) at ($(4,-1.5)+(-1,-1)$) {$\\mathop{\\rm Spec}\\nolimits \\EuScript{V}$};\n\\node (ft) at ($(0,-3)+(-1,-1)$) {$\\EuScript{D} \\mathrm{isc}$};\n\\node (fPRes) at ($(2,-3)+(-1,-1)$) {$\\mathfrak{h}_{\\mathbb{C}}\/W_{\\EuScript{I}}$};\n\\node (fRes) at ($(4,-3)+(-1,-1)$) {$\\mathfrak{h}_{\\mathbb{C}}\/W$};\n\\draw[->] (fX)--(fZ);\n\\draw[->] (fX)--(fR);\n\\draw[->] (fZ)--(fU);\n\\draw[->] (fR)--(fU);\n\\draw[->] (fU)--(fV);\n\\draw[->] (fV)--(fRes);\n\\draw[->] (fR)--(ft);\n\\draw[->] (fU)--(fPRes);\n\\draw[->] (ft)--node[above]{$\\scriptstyle\\upmu$}(fPRes);\n\\draw[->] (fPRes)--(fRes);\n\\draw[double] (fRes)--(Res);\n\\draw[double] (fV)--(V);\n\\draw[<-] (fPRes)--node[pos=0.35,right]{$\\scriptstyle \\ell_{\\EuScript{I}}\\ell_{\\EuScript{I}+i}\\cdot$}(PRes);\n\\draw[<-] (fU)--node[right]{$\\scriptstyle \\sim$}(U);\n\\draw[double] (ft)--(t);\n\\draw[<-] (fR)--node[right]{$\\scriptstyle \\sim$}(R);\n\\end{tikzpicture}\\label{create flop comm diagram}\n\\end{array}\n\\end{equation}\nComposing the map $\\EuScript{X}^+_i\\to\\mathop{\\rm Spec}\\nolimits\\EuScript{R}'$ with the isomorphism $\\mathop{\\rm Spec}\\nolimits\\EuScript{R}'\\to\\mathop{\\rm Spec}\\nolimits\\EuScript{R}$ yielding a morphism $\\EuScript{X}^+_i\\to\\mathop{\\rm Spec}\\nolimits \\EuScript{R}$.\n\n\n\\begin{thm}\\label{thm: produce flop} \\label{thm: flop by sim res}\nWith notation as above, $\\EuScript{X}_i^+\\to\\mathop{\\rm Spec}\\nolimits\\EuScript{R}$ is the flop of $\\EuScript{X}$ at the curve $\\Curve_i$. In particular, the following statements hold.\n\\begin{enumerate}\n\\item $\\upomega_i(\\EuScript{I}) \\subseteq \\Delta$ is the Dynkin data associated to the flopping contraction $\\EuScript{X}_i^+\\to\\mathop{\\rm Spec}\\nolimits\\EuScript{R}$.\n\\item All other crepant resolutions of $\\mathop{\\rm Spec}\\nolimits\\EuScript{R}$ can be obtained from the fixed $\\upmu$ by post-composing with $x^{-1}\\colon\\mathfrak{h}_{\\mathbb{C}}\/W_{\\EuScript{I}}\\to\\mathfrak{h}_{\\mathbb{C}}\/W_{\\EuScript{K}}$ and pulling back along $\\EuScript{Y}_\\EuScript{K}$, as the pair $(x,\\EuScript{K})$ ranges over the (finite) indexing set $\\mathsf{Cham}(\\Delta,\\EuScript{I})$ of Notation~\\ref{notation: subset stuff}.\n\\end{enumerate}\n\\end{thm}\n\\begin{proof}\n Since the exceptional locus of $\\mathsf{g}_\\EuScript{I}\\colon\\EuScript{Y}_\\EuScript{I}\\to\\mathop{\\rm Spec}\\nolimits\\EuScript{V}_\\EuScript{I}$ has codimension two, $\\mathop{\\rm Cl}\\nolimits(\\EuScript{V}_{\\EuScript{I}})\\cong\\mathop{\\rm Cl}\\nolimits(\\EuScript{Y}_\\EuScript{I})$. But $\\EuScript{Y}_\\EuScript{I}$ is smooth, so the latter is isomorphic to $\\mathop{\\rm Pic}\\nolimits(\\EuScript{Y}_\\EuScript{I})$, which in turn is isomorphic to $\\mathbb{Z}^{|\\scrI^{\\kern 0.5pt\\mathrm{c}}|}$ based by divisors dual to the $|\\scrI^{\\kern 0.5pt\\mathrm{c}}|$ curves above the origin. Choosing this basis, we may write $\\mathop{\\rm Cl}\\nolimits(\\EuScript{V}_{\\EuScript{I}})\\cong\\bigoplus_{j\\in\\scrI^{\\kern 0.5pt\\mathrm{c}}} \\mathbb{Z}e_j^\\star$, and further by definition $\\EuScript{Y}_\\EuScript{I}$ is obtained by blowing up any element in the characteristic cone\n\\[\nC_{\\EuScript{I}}\\colonequals\\{ (z_j)\\mid z_j>0\\mbox{ for all }j\\}\\subseteq \\,\\,\\bigoplus_{j\\in\\scrI^{\\kern 0.5pt\\mathrm{c}}} \\mathbb{Z}e_j^\\star.\n\\] \nApplying a similar analysis to the standard simultaneous resolution $\\EuScript{Z}\\to\\mathop{\\rm Spec}\\nolimits\\EuScript{W}$, we have $\\mathop{\\rm Cl}\\nolimits(\\EuScript{W})\\cong\\bigoplus_{j\\in\\Delta} \\mathbb{Z}e_j^\\star$, and since by construction $Y_{\\EuScript{I}}$ is obtained from $\\EuScript{Z}$ by simultaneously blowing down curves, it is clear that\n\\begin{equation}\n\\begin{array}{c}\n\\begin{tikzpicture}\n\\node (A) at (0,0) {$\\mathop{\\rm Cl}\\nolimits(\\EuScript{V}_\\EuScript{I})$};\n\\node (B) at (2.5,0) {$\\mathop{\\rm Cl}\\nolimits(\\EuScript{W})$};\n\\node (a) at (0,-1.25) {$\\bigoplus_{j\\in\\scrI^{\\kern 0.5pt\\mathrm{c}}} \\mathbb{Z}e_j^\\star$};\n\\node (b) at (2.5,-1.25) {$\\bigoplus_{j\\in\\Delta} \\mathbb{Z}e_j^\\star$};\n\\draw[->] (A)--node[left]{$\\scriptstyle \\sim$}(a);\n\\draw[->] (B)--node[left]{$\\scriptstyle \\sim$}(b);\n\\draw[right hook->] (A)--(B);\n\\draw[right hook->] (a)--(b);\n\\end{tikzpicture}\n\\end{array}\\label{class group inclusions}\n\\end{equation}\nwhere the bottom morphism is the obvious inclusion induced by the inclusion $\\scrI^{\\kern 0.5pt\\mathrm{c}} \\subseteq \\Delta$. There is a natural identification of $\\mathop{\\rm Cl}\\nolimits(\\EuScript{V}_\\EuScript{I})$, respectively $\\mathop{\\rm Cl}\\nolimits(\\EuScript{W}_\\EuScript{I})$, with the lattice inside $\\Theta_\\EuScript{I}$, respectively $\\Theta$.\n\nBy the universal property of the pullback, the action of any $s_j\\in W$ on $\\mathfrak{h}_{\\mathbb{C}}$ induces an automorphism of $\\EuScript{W}$\n\\[\n\\begin{tikzpicture}[>=stealth,scale=1.2];\n\\node[black!70!white] (U) at (2,-1.5) {$\\mathop{\\rm Spec}\\nolimits \\EuScript{W}$};\n\\node[black!70!white] (V) at (4,-1.5) {$\\mathop{\\rm Spec}\\nolimits \\EuScript{V}$};\n\\node[black!70!white] (PRes) at (2,-3) {$\\mathfrak{h}_{\\mathbb{C}}$};\n\\node[black!70!white] (Res) at (4,-3) {$\\mathfrak{h}_{\\mathbb{C}}\/W$};\n\\draw[black!70!white,->] (U)--(V);\n\\draw[black!70!white,->] (V)--(Res);\n\\draw[black!70!white,->] (U)--node[gap,pos=0.74]{\\phantom .}(PRes);\n\\draw[black!70!white,->] (PRes)--node[gap,pos=0.6]{\\phantom .}(Res);\n\\node (fU) at ($(2,-1.5)+(-1,-1)$) {$\\mathop{\\rm Spec}\\nolimits \\EuScript{W}$};\n\\node (fV) at ($(4,-1.5)+(-1,-1)$) {$\\mathop{\\rm Spec}\\nolimits \\EuScript{V}$};\n\\node (fPRes) at ($(2,-3)+(-1,-1)$) {$\\mathfrak{h}_{\\mathbb{C}}$};\n\\node (fRes) at ($(4,-3)+(-1,-1)$) {$\\mathfrak{h}_{\\mathbb{C}}\/W$};\n\\draw[->] (fU)--(fV);\n\\draw[->] (fV)--(fRes);\n\\draw[->] (fU)--(fPRes);\n\\draw[->] (fPRes)--(fRes);\n\\draw[double] (fRes)--(Res);\n\\draw[double] (fV)--(V);\n\\draw[<-] (fPRes)--node[pos=0.4,right]{$\\scriptstyle s_j\\cdot$}(PRes);\n\\draw[<-] (fU)--node[right]{$\\scriptstyle \\sim$}(U);\n\\end{tikzpicture}\n\\]\nThe effect on homology of $\\EuScript{Z}$ is via the Weyl reflection $s_j$ on roots \\cite{Reid}, and thus the action on $\\mathop{\\rm Cl}\\nolimits(\\EuScript{W})$ is the dual, namely the action of the Weyl reflection $s_j$ on coroots. \n\nComposing these $s_j$, we can consider the action of $\\ell_\\EuScript{I}\\ell_{\\EuScript{I}+i}$. We already know that the bottom right square in \\eqref{create flop comm diagram} commutes, and hence the bottom two squares in the following are well defined, and commute.\n\\[\n\\begin{array}{c}\n\\begin{tikzpicture}[>=stealth,scale=1.2]\n\\node[black!70!white] (R) at (0,-1.5) {$\\mathop{\\rm Spec}\\nolimits \\EuScript{W}$};\n\\node[black!70!white] (U) at (2,-1.5) {$\\mathop{\\rm Spec}\\nolimits \\EuScript{V}_{\\upomega_i(\\EuScript{I})}$};\n\\node[black!70!white] (V) at (4,-1.5) {$\\mathop{\\rm Spec}\\nolimits \\EuScript{V}$};\n\\node[black!70!white] (t) at (0,-3) {$\\mathfrak{h}_{\\mathbb{C}}$};\n\\node[black!70!white] (PRes) at (2,-3) {$\\mathfrak{h}_{\\mathbb{C}}\/W_{\\upomega_i(\\EuScript{I})}$};\n\\node[black!70!white] (Res) at (4,-3) {$\\mathfrak{h}_{\\mathbb{C}}\/W$};\n\\draw[black!70!white,->] (R)--(U);\n\\draw[black!70!white,->] (U)--(V);\n\\draw[black!70!white,->] (V)--(Res);\n\\draw[black!70!white,->] (R)--node[gap,pos=0.74]{\\phantom .}(t);\n\\draw[black!70!white,->] (U)--node[gap,pos=0.76]{\\phantom .}(PRes);\n\\draw[black!70!white,->] (t)--node[gap,pos=0.77]{\\phantom .}(PRes);\n\\draw[black!70!white,->] (PRes)--node[gap,pos=0.31]{\\phantom .}(Res);\n\\node (fR) at ($(0,-1.5)+(-1,-1)$) {$\\mathop{\\rm Spec}\\nolimits \\EuScript{W}$};\n\\node (fU) at ($(2,-1.5)+(-1,-1)$) {$\\mathop{\\rm Spec}\\nolimits \\EuScript{V}_{\\EuScript{I}}$};\n\\node (fV) at ($(4,-1.5)+(-1,-1)$) {$\\mathop{\\rm Spec}\\nolimits \\EuScript{V}$};\n\\node (ft) at ($(0,-3)+(-1,-1)$) {$\\mathfrak{h}_{\\mathbb{C}}$};\n\\node (fPRes) at ($(2,-3)+(-1,-1)$) {$\\mathfrak{h}_{\\mathbb{C}}\/W_{\\EuScript{I}}$};\n\\node (fRes) at ($(4,-3)+(-1,-1)$) {$\\mathfrak{h}_{\\mathbb{C}}\/W$};\n\\draw[->] (fR)--(fU);\n\\draw[->] (fU)--(fV);\n\\draw[->] (fV)--(fRes);\n\\draw[->] (fR)--(ft);\n\\draw[->] (fU)--(fPRes);\n\\draw[->] (ft)--(fPRes);\n\\draw[->] (fPRes)--(fRes);\n\\draw[double] (fRes)--(Res);\n\\draw[double] (fV)--(V);\n\\draw[<-] (fPRes)--node[pos=0.4,right]{$\\scriptstyle \\ell_{\\EuScript{I}}\\ell_{\\EuScript{I}+i}\\cdot$}(PRes);\n\\draw[<-] (fU)--node[right]{$\\scriptstyle \\sim$}(U);\n\\draw[<-] (ft)--node[right]{$\\scriptstyle\\ell_\\EuScript{I}\\ell_{\\EuScript{I}+i}\\cdot$}(t);\n\\draw[<-] (fR)--node[right]{$\\scriptstyle \\sim$}(R);\n\\end{tikzpicture}\n\\end{array}\n\\]\nThe universal property of pullbacks give the induced isomorphisms in the top squares. The top left square induces the following bottom commutative square on class groups, and the top square is obtained by applying \\eqref{class group inclusions} to both $\\EuScript{I}$ and to $\\upomega_i(\\EuScript{I})$. \n\\[\n\\begin{tikzpicture}[>=stealth,scale=1.2];\n\\node[black!70!white] (U) at (2,-1.5) {$\\bigoplus_{j\\in\\Delta} \\mathbb{Z}e_j^\\star$};\n\\node[black!70!white] (V) at (4.5,-1.5) {$\\bigoplus_{j\\in\\upomega_i(\\EuScript{I})^{\\mathrm{c}}} \\mathbb{Z}e_j^\\star$};\n\\node[black!70!white] (PRes) at (2,-3) {$\\mathop{\\rm Cl}\\nolimits(\\EuScript{W})$};\n\\node[black!70!white] (Res) at (4.5,-3) {$\\mathop{\\rm Cl}\\nolimits(\\EuScript{V}_{\\upomega_i(\\EuScript{I})})$};\n\\draw[black!70!white,left hook->] (V)--node[above]{$\\scriptstyle i_1$}(U);\n\\draw[black!70!white,<-] (V)--(Res);\n\\draw[black!70!white,<-] (U)--node[gap,pos=0.745]{\\phantom .}(PRes);\n\\draw[black!70!white,left hook->] (Res)--node[gap,pos=0.19]{\\phantom .}(PRes);\n\\node (fU) at ($(2,-1.5)+(-1,-1)$) {$\\bigoplus_{j\\in\\Delta} \\mathbb{Z}e_j^\\star$};\n\\node (fV) at ($(4.5,-1.5)+(-1,-1)$) {$\\bigoplus_{j\\in\\scrI^{\\kern 0.5pt\\mathrm{c}}} \\mathbb{Z}e_j^\\star$};\n\\node (fPRes) at ($(2,-3)+(-1,-1)$) {$\\mathop{\\rm Cl}\\nolimits(\\EuScript{W})$};\n\\node (fRes) at ($(4.5,-3)+(-1,-1)$) {$\\mathop{\\rm Cl}\\nolimits(\\EuScript{V}_\\EuScript{I})$};\n\\draw[left hook->] (fV)--node[pos=0.4,above]{$\\scriptstyle i_2$}(fU);\n\\draw[<-] (fV)--(fRes);\n\\draw[<-] (fU)--(fPRes);\n\\draw[left hook->] (fRes)--(fPRes);\n\\draw[<-] (fRes)--node[pos=0.4,right]{$\\scriptstyle \\ell_\\EuScript{I}\\ell_{\\EuScript{I}+i}\\cdot$}(Res);\n\\draw[densely dotted,<-] (fV)--(V);\n\\draw[<-] (fPRes)--node[pos=0.4,right]{$\\scriptstyle \\ell_\\EuScript{I}\\ell_{\\EuScript{I}+i}\\cdot$}(PRes);\n\\draw[<-] (fU)--node[pos=0.6,left]{$\\scriptstyle \\ell_\\EuScript{I}\\ell_{\\EuScript{I}+i}\\cdot$}(U);\n\\end{tikzpicture}\n\\]\nIn the above diagram, every non-hooked arrow is an isomorphism. By construction, the dotted arrow takes the characteristic cone \n\\[\nC_{\\upomega_i(\\EuScript{I})}\\colonequals\\{ (z_j)\\mid z_j>0\\mbox{ for all }j\\}\\subseteq \\bigoplus_{j\\in\\upomega_i(\\EuScript{I})^{\\mathrm{c}}} \\mathbb{Z}e_j^\\star\n\\]\nof $Y_{\\upomega_i(\\EuScript{I})}$ to the region $\\ell_{\\EuScript{I}}\\ell_{\\EuScript{I}+i}C_{\\upomega_i(\\EuScript{I})}\\subseteq \\bigoplus_{j\\in\\scrI^{\\kern 0.5pt\\mathrm{c}}} \\mathbb{Z}e_j^\\star$. Since $\\ell_{\\EuScript{I}}\\ell_{\\EuScript{I}+i}$ acts via the action on coroots, this matches the conventions in \\cite[Section~3]{IyamaWemyssTits}. It thus immediately follows from \\cite[1.20(1)(d)]{IyamaWemyssTits} that $\\ell_{\\EuScript{I}}\\ell_{\\EuScript{I}+i}C_{\\upomega_i(\\EuScript{I})}$ and $C_{\\EuScript{I}}$ are neighbouring regions, adjacent via the wall $z_i=0$. \n\nWe next restrict this information to $3$-folds. As in \\eqref{create flop comm diagram}, consider the following commutative diagram.\n\\begin{equation}\n\\begin{array}{c}\n\\begin{tikzpicture}[>=stealth,scale=1.2]\n\\node[black!70!white] (X) at (0,0) {$\\EuScript{X}_i^+$};\n\\node[black!70!white] (Z) at (2,0) {$\\EuScript{Y}_{\\upomega_i(\\EuScript{I})}$};\n\\node[black!70!white] (R) at (0,-1.5) {$\\mathop{\\rm Spec}\\nolimits \\EuScript{R}'$};\n\\node[black!70!white] (U) at (2,-1.5) {$\\mathop{\\rm Spec}\\nolimits \\EuScript{V}_{\\upomega_i(\\EuScript{I})}$};\n\\node[black!70!white] (t) at (0,-3) {$\\EuScript{D} \\mathrm{isc}$};\n\\node[black!70!white] (PRes) at (2,-3) {$\\mathfrak{h}_{\\mathbb{C}}\/W_{\\upomega_i(\\EuScript{I})}$};\n\\draw[black!70!white,->] (X)--(Z);\n\\draw[black!70!white,->] (X)--node[gap,pos=0.75]{\\phantom .}(R);\n\\draw[black!70!white,->] (Z)--(U);\n\\draw[black!70!white,->] (R)--(U);\n\\draw[black!70!white,->] (R)--node[gap,pos=0.72]{\\phantom .}(t);\n\\draw[black!70!white,->] (U)--node[gap,pos=0.76]{\\phantom .}(PRes);\n\\draw[black!70!white,->] (t)--node[gap,pos=0.735]{\\phantom .}node[gap,pos=0.4]{$\\scriptstyle\\upnu$}(PRes);\n\\node (fX) at (-1,-1) {$\\EuScript{X}$};\n\\node (fZ) at ($(2,0)+(-1,-1)$) {$\\EuScript{Y}_{\\EuScript{I}}$};\n\\node (fR) at ($(0,-1.5)+(-1,-1)$) {$\\mathop{\\rm Spec}\\nolimits \\EuScript{R}$};\n\\node (fU) at ($(2,-1.5)+(-1,-1)$) {$\\mathop{\\rm Spec}\\nolimits \\EuScript{V}_{\\EuScript{I}}$};\n\\node (ft) at ($(0,-3)+(-1,-1)$) {$\\EuScript{D} \\mathrm{isc}$};\n\\node (fPRes) at ($(2,-3)+(-1,-1)$) {$\\mathfrak{h}_{\\mathbb{C}}\/W_{\\EuScript{I}}$};\n\\draw[->] (fX)--(fZ);\n\\draw[->] (fX)--(fR);\n\\draw[->] (fZ)--(fU);\n\\draw[->] (fR)--(fU);\n\\draw[->] (fR)--(ft);\n\\draw[->] (fU)--(fPRes);\n\\draw[->] (ft)--node[above]{$\\scriptstyle\\upmu$}(fPRes);\n\\draw[<-] (fPRes)--node[pos=0.35,right]{$\\scriptstyle \\ell_{\\EuScript{I}}\\ell_{\\EuScript{I}+i}\\cdot$}(PRes);\n\\draw[<-] (fU)--node[right]{$\\scriptstyle \\sim$}(U);\n\\draw[double] (ft)--(t);\n\\draw[<-] (fR)--node[right]{$\\scriptstyle \\sim$}(R);\n\\end{tikzpicture}\n\\end{array}\\label{create flop comm diagram 2}\n\\end{equation}\nAs explained by Pinkham \\cite{Pinkham}, $\\mathop{\\rm Cl}\\nolimits(\\EuScript{R})\\cong \\mathop{\\rm Cl}\\nolimits(\\EuScript{V}_{\\EuScript{I}})$, and further $\\mathop{\\rm Cl}\\nolimits(\\EuScript{V}_\\EuScript{I})\\cong\\bigoplus_{j\\in\\scrI^{\\kern 0.5pt\\mathrm{c}}} \\mathbb{Z}e_j^\\star$ as explained above. Hence $\\mathop{\\rm Cl}\\nolimits(\\EuScript{R})\\cong \\bigoplus_{j\\in\\scrI^{\\kern 0.5pt\\mathrm{c}}} \\mathbb{Z}e_j^\\star$, and under \\emph{this} choice of basis $\\EuScript{X}$ is obtained as the blowup of the characteristic cone $C_\\EuScript{I}$. The same analysis holds for $\\EuScript{X}_i^+$, which is the blowup of the characteristic cone $C_{\\upomega_i(\\EuScript{I})}$ under the choice of basis $\\mathop{\\rm Cl}\\nolimits(\\EuScript{R}')\\cong \\bigoplus_{j\\in\\upomega_i(\\EuScript{I})^{\\mathrm{c}}} \\mathbb{Z}e_j^\\star$ induced from $\\mathop{\\rm Cl}\\nolimits(\\EuScript{R}')\\cong \\mathop{\\rm Cl}\\nolimits(\\EuScript{V}_{\\upomega_i(\\EuScript{I})})$.\n\nPulling across the middle horizontal plane in \\eqref{create flop comm diagram 2} it thus follows that the map $\\EuScript{X}_i^+\\to\\mathop{\\rm Spec}\\nolimits\\EuScript{R}$ is obtained by blowing up the region $\\ell_\\EuScript{I}\\ell_{\\EuScript{I}+i}C_{\\upomega_i(\\EuScript{I})}$ in $\\mathop{\\rm Cl}\\nolimits(\\EuScript{R})\\cong\\bigoplus_{j\\in\\scrI^{\\kern 0.5pt\\mathrm{c}}} \\mathbb{Z}e_j^\\star$. Since these are neighbouring regions, separated by the codimension one wall $e_i^\\star=0$, it is implicit in \\cite{Pinkham} (see also \\cite{HomMMP}) that $\\EuScript{X}_i\\to\\mathop{\\rm Spec}\\nolimits\\EuScript{R}$ is the flop at the curve $\\Curve_i$. Since $\\EuScript{X}^+_i$ is obtained from $\\EuScript{Y}_{\\upomega_i(\\EuScript{I})}$ via pullback, the statement on Dynkin data follows.\n\nThe final statement about all other crepant resolutions follows by iterating over all possible simple flops. Indeed, the finite indexing set $\\mathsf{Cham}(\\Delta,\\EuScript{I})$ is precisely the combinatorial object which indexes all the chambers \\cite[Section~1]{IyamaWemyssTits}, and each chamber $(x,\\EuScript{K})$ can be obtained from $(1,\\EuScript{I})$ by iteratively applying the wall crossing rule \\cite[1.20(2)]{IyamaWemyssTits}.\n\\end{proof}\n\n\n\n\n\\section{Applications}\\label{sec: applications}\n\n\\noindent As before, consider a smooth $3$-fold flopping contraction $\\EuScript{X}\\to\\mathop{\\rm Spec}\\nolimits\\EuScript{R}$. The main applications of the previous sections are to GV and GW invariants, the Crepant Transformation Conjecture, and to the associated contraction algebras.\n\n\n\\subsection{Tracking GV invariants under flop}\\label{sec: GV under flop}\nThe benefit of Theorem~\\ref{thm: produce flop} is that both $\\EuScript{X}$ and a flop $\\EuScript{X}_i^+$ can be perturbed using essentially the same classifying map, and thus their curve invariants can be easily compared. This requires three combinatorial results.\n\n\\begin{lemma}\\label{lem: induced iso B}\n$\\ell_{\\EuScript{I}}\\ell_{\\EuScript{I}+i}\\colon\\mathfrak{h}\\to\\mathfrak{h}$ induces an isomorphism $\\mathsf{M}_i \\colon \\mathfrak{h}_{\\upomega_i(\\EuScript{I})}\\to\\mathfrak{h}_{\\EuScript{I}}$.\n\\end{lemma}\n\\begin{proof}\nSet $w=\\ell_{\\EuScript{I}}\\ell_{\\EuScript{I}+i}$, then it suffices to prove that $w\\cdot$ restricts to an isomorphism between the subspace spanned by $\\{\\upalpha_j\\mid j\\in\\upomega_i(\\EuScript{I})\\}$ and the subspace spanned by $\\{\\upalpha_j\\mid j\\in\\EuScript{I}\\}$. To see this, recall that $\\ell_\\Gamma\\upalpha_i=-\\upalpha_{\\upiota_\\Gamma(i)}$ for all Dynkin $\\Gamma$ and all $i\\in\\Gamma$, where $\\upiota_\\Gamma$ is the Dynkin involution on $\\Gamma$.\n\nFor all $j\\in \\upomega_i(\\EuScript{I})\\subseteq\\EuScript{I}+i$, it follows that $\\ell_\\EuScript{I}\\ell_{\\EuScript{I}+i}\\upalpha_j=-\\ell_{\\EuScript{I}}\\upalpha_{\\upiota_{\\EuScript{I}+i}(j)}$. Now $\\upiota_{\\EuScript{I}+i}\\colon\\upomega_i(\\EuScript{I})\\to\\EuScript{I}$ is a bijection. Indeed, $\\upiota_{\\EuScript{I}+i}\\colon\\EuScript{I}+i\\to\\EuScript{I}+i$ is a bijection, sending $\\upiota_{\\EuScript{I}+i}(i)$ to $i$, and so removing these elements gives the claimed bijection. Hence for all $j\\in\\upomega_i(\\EuScript{I})$, it follows that $\\upiota_{\\EuScript{I}+i}(j)\\in\\EuScript{I}$ and so $\\ell_{\\EuScript{I}}\\upalpha_{\\upiota_{\\EuScript{I}+i}(j)}=-\\upalpha_{\\upiota_\\EuScript{I}\\upiota_{\\EuScript{I}+i}(j)}$. Combining gives $\\ell_\\EuScript{I}\\ell_{\\EuScript{I}+i}\\upalpha_j=\\upalpha_{\\upiota_\\EuScript{I}\\upiota_{\\EuScript{I}+i}(j)}$ for all $j\\in\\upomega_i(\\EuScript{I})$. Since $\\upiota_\\EuScript{I}\\upiota_{\\EuScript{I}+i}(j)\\in\\EuScript{I}$, this proves the claim.\n\\end{proof}\nWrite $\\mathsf{M}_i$ for the induced isomorphism in Lemma~\\ref{lem: induced iso B}, so that the following diagrams commute.\n\\begin{equation}\n\\begin{array}{c}\n\\begin{tikzpicture}\n\\node (A) at (0,0) {$\\mathfrak{h}$};\n\\node (B) at (2.5,0) {$\\mathfrak{h}$};\n\\node (a) at (0,-1.5) {$\\mathfrak{h}_{\\upomega_i(\\EuScript{I})}$};\n\\node (b) at (2.5,-1.5) {$\\mathfrak{h}_{\\EuScript{I}}$};\n\\draw[->] (A)--node[above]{$\\scriptstyle \\ell_{\\EuScript{I}}\\ell_{\\EuScript{I}+i}\\cdot$}(B);\n\\draw[->] (A)--node[left]{$\\scriptstyle \\uppi_{\\upomega_i(\\EuScript{I})}$}(a);\n\\draw[->] (B)--node[right]{$\\scriptstyle \\uppi_{\\EuScript{I}}$}(b);\n\\draw[->,densely dotted] (a)--node[above]{$\\scriptstyle \\mathsf{M}_i$}(b);\n\\end{tikzpicture}\n\\end{array}\n\\quad\n\\begin{array}{c}\n\\begin{tikzpicture}\n\\node (A) at (0,0) {$\\mathfrak{h}$};\n\\node (B) at (2.5,0) {$\\mathfrak{h}$};\n\\node (a) at (0,-1.5) {$\\mathfrak{h}_{\\EuScript{I}}$};\n\\node (b) at (2.5,-1.5) {$\\mathfrak{h}_{\\upomega_i(\\EuScript{I})}$};\n\\draw[->] (A)--node[above]{$\\scriptstyle \\ell_{\\upomega_i(\\EuScript{I})}\\ell_{\\EuScript{I}+i}\\cdot$}(B);\n\\draw[->] (A)--node[left]{$\\scriptstyle \\uppi_{\\EuScript{I}}$}(a);\n\\draw[->] (B)--node[right]{$\\scriptstyle \\uppi_{\\upomega_i(\\EuScript{I})}$}(b);\n\\draw[->,densely dotted] (a)--node[above]{$\\scriptstyle \\mathsf{M}_i^{-1}$}(b);\n\\end{tikzpicture}\n\\end{array}\n\\label{defn Mi}\n\\end{equation}\nIn essence, the bases of $\\mathfrak{h}_{\\upomega_i(\\EuScript{I})}$ and $\\mathfrak{h}_{\\EuScript{I}}$ really only differ at one element. Indeed, setting $e_j = \\uppi_\\EuScript{I}(\\upalpha_j)$ then $\\{ e_j \\mid j \\not\\in \\EuScript{I} \\}$ is a basis for $\\mathfrak{h}_\\EuScript{I}$. On the other hand for $\\mathfrak{h}_{\\omega_i(\\EuScript{I})}$ we abuse notation, setting $e_j=\\uppi_{\\upomega_i(\\EuScript{I})}(\\upalpha_j)$ whenever $j\\notin\\EuScript{I}+i$, and $e_i=\\uppi_{\\upomega_i(\\EuScript{I})}(\\upalpha_{\\upiota_{\\EuScript{I}+i}(i)})$. Then $\\{e_j,e_i\\mid j\\notin\\EuScript{I}+i\\}$ is a basis for $\\mathfrak{h}_{\\upomega_i(\\EuScript{I})}$.\n\\begin{lemma}\\label{lem: action of Mi}\nFor $i \\in \\scrI^{\\kern 0.5pt\\mathrm{c}}$ the action of $\\mathsf{M}_i$ is given in terms of the above bases as\n\\[\ne_k\n\\mapsto\n\\begin{cases}\ne_k+\\uplambda_{k}e_i&\\mbox{if }k\\notin\\EuScript{I}+i\\\\\n-e_i&\\mbox{if }k=i\n\\end{cases}\n\\]\nfor some $\\uplambda_k \\in \\mathbb{Z}_{\\geq 0}$.\n\\end{lemma}\n\\begin{proof}\nIn \\eqref{defn Mi}, given any $\\sum a_i\\upalpha_i$, since $\\ell_\\EuScript{I}\\ell_{\\EuScript{I}+i}$ consists only of reflections $s_i$ with $i\\in\\EuScript{I}+i$, the map $\\ell_\\EuScript{I}\\ell_{\\EuScript{I}+i}$ cannot change the coefficient of any $a_j$ with $j\\notin\\EuScript{I}+i$. The claim that the induced map $\\mathsf{M}_i$ sends $e_k\\mapsto e_k+\\uplambda_{k}e_i$ if $k\\notin\\EuScript{I}+i$ follows. We next claim that $\\uplambda_k$ is positive. In the decomposition of $\\ell_\\EuScript{I}\\ell_{\\EuScript{I}+i}\\upalpha_k$ into simple roots, there is at least some positive coefficient (namely the coefficient of $\\upalpha_k$, which is $1$). Hence all coefficients must be positive, in particular the coefficient of $\\upalpha_{\\upiota_{\\EuScript{I}+i}(i)}$. But under the induced map, this coefficient is what gives $\\uplambda_k$ in the claim. \n\nNow as in Lemma~\\ref{lem: induced iso B}, for all Dynkin $\\Gamma$ and all $i\\in\\Gamma$, $\\ell_\\Gamma\\upalpha_i=-\\upalpha_{\\upiota_\\Gamma(i)}$. Thus since $\\upiota_{\\EuScript{I}+i}(i)\\in\\EuScript{I}+i$, it follows that\n\\[\n\\mathsf{M}_ie_i\n\\stackrel{\\scriptstyle\\eqref{defn Mi}}{=}\n\\uppi_\\EuScript{I}(\\ell_\\EuScript{I}\\ell_{\\EuScript{I}+i}\\upalpha_{\\upiota_{\\EuScript{I}+i}(i)})\n=\\uppi_\\EuScript{I}(-\\ell_\\EuScript{I}\\upalpha_i)=\\uppi_\\EuScript{I}(-\\upalpha_i)=-e_i,\n\\]\nwhere we have used the fact that $\\ell_\\EuScript{I}$ only changes coefficients in $\\EuScript{I}$, and $\\uppi_\\EuScript{I}$ forgets these.\n\\end{proof}\n\n\n\\begin{cor}\\label{cor: action of Mi pos neg}\nIf $\\upbeta\\in\\mathfrak{h}_{\\upomega_i(\\EuScript{I})}$ is a restricted root, the following hold.\n\\begin{enumerate}\n\\item If $\\upbeta\\in\\mathbb{Z}e_i$, say $\\upbeta=ze_i$, then $\\mathsf{M}_i\\cdot(ze_i)=-ze_i$.\n\\item If $\\upbeta\\notin\\mathbb{Z}e_i$, then all entries of $\\mathsf{M}_i\\cdot\\upbeta$ are positive.\n\\end{enumerate}\n\\end{cor}\n\\begin{proof}\nThe first part is an immediate consequence of the $k=i$ case in Lemma~\\ref{lem: action of Mi}. For the second part, by Lemma~\\ref{lem: action of Mi} the only coefficient of $\\upbeta=\\sum\\upmu_ie_i$ that can change under $\\mathsf{M}_i$ is the coefficient on $e_i$. Hence, provided there is some other positive coefficient $\\upmu_k$, this survives under $\\mathsf{M}_i$, so $\\mathsf{M}_i\\cdot\\upbeta$ has at least one positive entry. Now by assumption $\\upbeta$ is a restricted root, say $\\uppi_{\\upomega_i(\\EuScript{I})}(\\upalpha)=\\upbeta$, Under \\eqref{defn Mi}, $w\\cdot\\upalpha$ is a root restricting to $\\mathsf{M}_i\\cdot \\upbeta$ and this root $w\\cdot\\upalpha$ must contain at least one positive coefficient, since $\\mathsf{M}_i\\cdot \\upbeta$ does. Hence all must be positive. In particular, all entries of $\\mathsf{M}_i\\cdot \\upbeta$ must also be positive.\n\\end{proof}\n\nThe following is one of our main results. In order to obtain a unified statement, set $|\\mathsf{M}_i\\cdot\\upbeta|$ to be the curve class obtained from $\\mathsf{M}_i\\cdot\\upbeta$ by making every coefficient positive.\n\n\\begin{theorem}\\label{thm: GV under flop}\nWith the notation as above, for any curve class $\\upbeta\\inA_1(\\EuScript{X}_i^+)\\cong\\mathfrak{h}_{\\upomega_i(\\EuScript{I})}$,\n\\begin{align*}\nn_{\\upbeta,\\EuScript{X}^+_i}&=\n\\begin{cases}\nn_{\\upbeta,\\EuScript{X}}&\\mbox{if }\\upbeta\\in \\mathbb{Z}e_i=\\mathbb{Z}\\Curve_i^+\\\\\nn_{\\kern 1pt\\mathsf{M}_i\\cdot \\upbeta, \\EuScript{X}}&\\mbox{else}\n\\end{cases}\\\\\n&=\nn_{\\kern 1pt|\\mathsf{M}_i\\cdot \\upbeta|, \\EuScript{X}}\n\\end{align*}\n\\end{theorem} \n\\begin{proof}\nWith respect to the notation in \\eqref{create flop comm diagram}, perturbing $\\upmu$ to $\\upmu_t$ gives, by composition, a perturbation of $\\upnu$ to $\\upnu_t$.\n\nSet $w=\\ell_{\\EuScript{I}}\\ell_{\\EuScript{I}+i}$. Then for any positive root $\\upalpha$ for which $\\uppi_{\\upomega_i(\\EuScript{I})}(\\upalpha)=\\upbeta$, \n\\begin{align*}\nn_{\\upbeta,\\EuScript{X}^+_i}&=|\\upnu_t^{-1}(\\EuScript{D}_{\\upalpha,\\mathbb{C}}\/W_{\\omega_i(\\EuScript{I})})|\\tag{by \\eqref{eqn: nbeta equals intersection with hyperplane} applied to $\\EuScript{X}_i^+$}\\\\\n&=|\\upmu_t^{-1}(\\EuScript{D}_{w\\cdot\\upalpha,\\mathbb{C}}\/W_\\EuScript{I})|.\\tag{since $\\upnu=( w^{-1}\\cdot )\\circ\\upmu$}\n\\end{align*}\nNow by \\eqref{defn Mi} we have $\\uppi_{\\EuScript{I}}(w\\cdot \\upalpha)=\\mathsf{M}_i\\circ\\uppi_{\\upomega_i(\\EuScript{I})}(\\upalpha)=\\mathsf{M}_i\\cdot \\upbeta$ and so $w\\cdot\\upalpha$ is a lift of $\\mathsf{M}_i\\cdot\\upbeta$, albeit not necessarily a positive one. \n\n\\noindent\n\\emph{Case 1.} If $\\upbeta\\notin\\mathbb{Z}\\Curve^+_i$, then by Corollary~\\ref{cor: action of Mi pos neg} all entries of $\\mathsf{M}_i\\cdot\\upbeta$ are positive, and further as argued in the proof, $w\\cdot\\upalpha$ is a positive root restricting to $\\mathsf{M}_i\\cdot \\upbeta$. \\eqref{eqn: nbeta equals intersection with hyperplane} then implies that $|\\upmu_t^{-1}(\\EuScript{D}_{w\\cdot\\upalpha,\\mathbb{C}}\/W_\\EuScript{I})|=n_{\\kern 1pt\\mathsf{M}_i\\cdot \\upbeta, \\EuScript{X}}$. \n\n\\noindent\n\\emph{Case 2.} If $\\upbeta\\in\\mathbb{Z}\\Curve^+_i$, then by Corollary~\\ref{cor: action of Mi pos neg}, $\\mathsf{M}_i\\cdot\\upbeta=-\\upbeta$. Arguing as above, it follows that $w\\cdot\\upalpha$ is negative root restricting to $\\mathsf{M}_i\\cdot\\upbeta=-\\upbeta$, and thus $-w\\cdot\\upalpha$ is positive root restricting to $\\upbeta$. But negating a root does not effect the hyperplane, and combining this fact with \\eqref{eqn: nbeta equals intersection with hyperplane} it follows that\n\\[\n|\\upmu_t^{-1}(\\EuScript{D}_{w\\cdot\\upalpha,\\mathbb{C}}\/W_\\EuScript{I})|=|\\upmu_t^{-1}(\\EuScript{D}_{-w\\cdot\\upalpha,\\mathbb{C}}\/W_\\EuScript{I})|=n_{\\upbeta,\\EuScript{X}}.\n\\]\nThis covers both cases. For the final equality, note in case 1 that $|\\mathsf{M}_i\\cdot\\upbeta|=\\mathsf{M}_i\\cdot\\upbeta$ since all coefficients are already positive, and in case 2 that $|\\mathsf{M}_i\\cdot\\upbeta|=|-\\upbeta|=\\upbeta$.\n\\end{proof}\n\n\n\n\\begin{example}\\label{ex: running example GV after flop}\nConsider the running Example~\\ref{example: intro}. Then after flop of the right pink curve, by Theorem~\\ref{thm: produce flop} and Example~\\ref{ex: changing dual graph} we obtain $\\upomega_i(\\EuScript{I})=\\Eeight{B}{P}{B}{Or}{B}{B}{B}{B}$. Hence the restricted roots, and thus curve classes giving nonzero GV invariants, on the flopped space $\\EuScript{X}_i^+$ are as follows, where the hyperplanes are drawn in $\\Uptheta_{\\upomega_i(\\EuScript{I})}$.\n\\[\n\\begin{array}{cccc}\n\\begin{array}{c}\n\\begin{tikzpicture}[scale=0.5]\n\\draw[->,densely dotted] (180:2cm)--(0:2cm);\n\\node at (0:2.5) {$\\scriptstyle x$};\n\\draw[->,densely dotted] (-90:2cm)--(90:2cm);\n\\node at (90:2.5) {$\\scriptstyle y$};\n\\end{tikzpicture}\n\\end{array}\n&\n\\begin{array}{c}\n\\begin{tikzpicture}[scale=1]\n\\draw[line width=0.5 mm,Pink] (180:2cm)--(0:2cm);\n\\node at (180:2.2) {$\\scriptstyle 1$};\n\\draw[line width=0.5 mm,Pink] (135:2cm)--(-45:2cm);\n\\node at (135:2.2) {$\\scriptstyle 3$};\n\\draw[line width=0.5 mm, Grey] (126.87:2cm)--(-53.13:2cm);\n\\node at (126.87:2.2) {$\\scriptstyle 1$};\n\\draw[line width=0.5 mm, Green] (123.69:2cm)--(-56.31:2cm);\n\\node at (123.69:2.2) {$\\scriptstyle 1$};\n\\draw[line width=0.5 mm, Blue] (116.55:2cm)--(-63.45:2cm);\n\\node at (116.55:2.2) {$\\scriptstyle 2$};\n\\draw[line width=0.5 mm,Orange] (90:2cm)--(-90:2cm);\n\\node at (90:2.2) {$\\scriptstyle 1$};\n\\end{tikzpicture}\n\\end{array}&\n\\begin{array}{c}\n\\begin{tabular}{ccc}\n\\toprule\nRestricted Root&\\\\\n\\midrule\n$01$&$\\tikz\\draw[line width=0.5 mm, Pink] (0,0) -- (0.25,0);$\\\\\n$11,22,33$&$\\tikz\\draw[line width=0.5 mm, Pink] (0,0) -- (0.25,0);$\\\\\n$43$&$\\tikz\\draw[line width=0.5 mm, Grey] (0,0) -- (0.25,0);$\\\\\n$32$&$\\tikz\\draw[line width=0.5 mm, Green] (0,0) -- (0.25,0);$\\\\\n$21,42$&$\\tikz\\draw[line width=0.5 mm,Blue] (0,0) -- (0.25,0);$\\\\\n$10$&$\\tikz\\draw[line width=0.5 mm, Orange] (0,-0.15) -- (0,0.15);$\\\\\n\\bottomrule\n\\end{tabular}\n\\end{array}\n\\end{array}\n\\]\nWrite $1$ for the leftmost pink node, $2$ for the orange node, and $2'$ for the rightmost pink node (in $\\EuScript{I}$). Under this wall crossing $\\ell_{\\EuScript{I}}\\ell_{\\EuScript{I}+i}$ is very large, however the morphism\n\\[\n\\mathsf{M}_i\\colon\\mathfrak{h}_{\\upomega_i(\\EuScript{I})}\\to\\mathfrak{h}_{\\EuScript{I}}\n\\]\nis easily described: in the notation of Lemma~\\ref{lem: action of Mi}, $\\uplambda_1=1$ and thus $\\mathsf{M}_i$ sends $\\upmu_1e_1+\\upmu_2e_2\\mapsto \\upmu_1e_1+(\\upmu_1-\\upmu_2)e_{2'}$. Under the dual transformation between the hyperplane arrangements in Example~\\ref{example: intro} and here, the pictures are drawn so that hyperplanes are sent to hyperplanes in such a way that the colours are preserved. \n\n\nIndeed, $\\mathsf{M}_i$ sends $01\\mapsto0-\\!1$, with all other restricted roots being permuted; e.g.\\ $31\\mapsto 32$. In particular, by Theorem~\\ref{thm: GV under flop} the GV invariants on $\\EuScript{X}_i^+$ can be obtained from the GV invariants on $\\EuScript{X}$ as follows\n\\[\n\\begin{tabular}{ccccc}\n\\toprule\nGV on $\\EuScript{X}^+_i$&GV on $\\EuScript{X}$&\\\\\n\\midrule\n$01$&$01$&$\\tikz\\draw[line width=0.5 mm, Pink] (0,0) -- (0.25,0);$\\\\\n$10$&$11$&$\\tikz\\draw[line width=0.5 mm, Orange] (0,0) -- (0.25,0);$\\\\\n$21,42$&$21, 42$&$\\tikz\\draw[line width=0.5 mm, Blue] (0,0) -- (0.25,0);$\\\\\n$32$&$31$&$\\tikz\\draw[line width=0.5 mm, Green] (0,0) -- (0.25,0);$\\\\\n$43$&$41$&$\\tikz\\draw[line width=0.5 mm, Grey] (0,0) -- (0.25,0);$\\\\\n$11,22,33$&$10, 20, 30$&$\\tikz\\draw[line width=0.5 mm, Pink] (0,0) -- (0.25,0);$\\\\\n\\bottomrule\n\\end{tabular}\n\\] \n\\end{example}\n\n\\begin{remark}\\label{rem: movable via alg geom}\nAs explained in the introduction, the finite arrangement $\\EuScript{H}_{\\EuScript{I}}$ equals the movable cone. The multiplicities of the restricted roots are assigned to each wall, and this enhancement is required in order to describe the curve-counting invariants. It is possible, albeit not a priori obvious, to enhance the movable cone without Dynkin combinatorics. Given a chamber corresponding to some crepant resolution $\\EuScript{X}^\\dag \\to\\mathop{\\rm Spec}\\nolimits\\EuScript{R}$ say, then the multiplicities on the walls of that chamber turn out to correspond to the lengths of all the individual single-curve contractions obtained from $\\EuScript{X}^\\dag$. The issue with this method is that, whilst it explains walls, it does not explain \\emph{hyperplanes}: it is not so clear that \\emph{every} chamber touching the hyperplane containing the said wall should be enriched with the same scheme-theoretic length. This geometric fact falls out from our approach. \n\\end{remark}\n\n\n\n\\subsection{Tracking fundamental regions}\\label{sec:track fund}\nThe previous subsection tracked GV invariants from $\\EuScript{X}$ to $\\EuScript{X}^+_i$. As with the movable cone, it is possible to fix $\\EuScript{X}$ and track all other crepant resolutions back to $\\EuScript{X}$. \n\nAs notation, recall that the fixed $\\EuScript{X}\\to\\mathop{\\rm Spec}\\nolimits\\EuScript{R}$ has an associated $\\Uptheta_\\EuScript{I}$ in Notation~\\ref{notation: subset stuff}, and recall from \\eqref{defn Mi} that there is a map $\\mathsf{M}_i^{-1}\\colon\\mathfrak{h}_{\\EuScript{I}}\\to\\mathfrak{h}_{\\upomega_i(\\EuScript{I})}$. Write\n\\[\n\\mathsf{N}_i\\colon \\Uptheta_{\\upomega_i(\\EuScript{I})}\\to\\Uptheta_\\EuScript{I}\n\\] \nfor the dual. Below, $\\Uptheta_\\EuScript{I}$ will be temporarily be written $\\Uptheta_{\\EuScript{X}}$, to allow for the flexibility of considering another crepant resolution $\\EuScript{Y}\\to\\mathop{\\rm Spec}\\nolimits\\EuScript{R}$ which has associated $\\Uptheta_{\\EuScript{Y}}$. \n\\begin{definition}\nLet $\\EuScript{Y}\\to\\mathop{\\rm Spec}\\nolimits\\EuScript{R}$ be a crepant resolution. Consider a chain of flops, each flopping a single irreducible curve, that links $\\EuScript{Y}$ to $\\EuScript{X}$, and the resulting maps\n\\[\n\\Uptheta_{\\EuScript{Y}}\\xrightarrow{\\mathsf{N}_{i_1}}\\hdots\\xrightarrow{\\mathsf{N}_{i_t}}\\Uptheta_\\EuScript{X}.\n\\]\nThe composition will be called the comparison map, and will be written $\\mathsf{N}\\colon\\Uptheta_\\EuScript{Y}\\to\\Uptheta_\\EuScript{X}$.\n\\end{definition}\nBy \\cite[4.8]{HW2} the comparison map $\\mathsf{N}$ is independent of the choice of chain of flops.\n\n\\begin{definition}\nGiven a crepant resolution $\\EuScript{Y}\\to\\mathop{\\rm Spec}\\nolimits\\EuScript{R}$, the fundamental region $\\mathsf{Fund}_\\EuScript{Y}$ of $\\Uptheta_\\EuScript{Y}$ is defined as the intersection of the infinite hyperplane arrangement inside $\\Uptheta_\\EuScript{Y}$ with the unit box $\\{ (\\upvartheta_i) \\mid 0\\leq \\upvartheta_i\\leq 1\\mbox{ for all }i\\}$.\n\\end{definition}\n\\begin{prop}\nFor any crepant resolution $\\EuScript{Y}\\to\\mathop{\\rm Spec}\\nolimits\\EuScript{R}$, $\\mathsf{N}(\\mathsf{Fund}_{\\EuScript{Y}})$ generates $\\Uptheta_\\EuScript{I}$ via translation. Furthermore two $\\mathsf{N}(\\mathsf{Fund}_{\\EuScript{X}_1})$ and $\\mathsf{N}(\\mathsf{Fund}_{\\EuScript{X}_2})$ share a codimension one wall if and only if $\\EuScript{X}_1$ and $\\EuScript{X}_2$ are connected by a flop at a single curve.\n\\end{prop}\n\\begin{proof}\nSince the axes belong to the finite hyperplane arrangement in $\\Uptheta_{\\EuScript{Y}}$, and the definition of the infinite arrangement involves translating this finite collection of hyperplanes over $\\mathbb{Z}$ or at worst $\\tfrac{1}{k}\\mathbb{Z}$ (see Subsections~\\ref{sec: GW intro} and \\ref{sec: hyperplane arrangements}), it is clear that the fundamental region $\\mathsf{Fund}_\\EuScript{Y}$ generates the arrangement in $\\Uptheta_\\EuScript{Y}$. The first statement then follows, since $\\mathsf{N}$ is known to preserve the infinite arrangements \\cite[Section~9]{IyamaWemyssTits}. Since the only codimension one wall that the fundamental regions can share belong to the finite arrangement, the last statement is really a statement on the movable cone, which is e.g.\\ \\cite[Sections~5--6]{HomMMP}. \n\\end{proof}\n\n\\begin{example}\nWrite $\\EuScript{Y}\\to\\mathop{\\rm Spec}\\nolimits\\EuScript{R}$ for the crepant resolution obtained after flop in Example~\\ref{ex: running example GV after flop}. Then the region $\\mathsf{N}(\\mathsf{Fund}_{\\EuScript{Y}})$ is illustrated below, where for clarity we have illustrated the images of the $x$ and $y$ co-ordinates in Example~\\ref{ex: running example GV after flop} under the map $\\mathsf{N}$.\n\\[\n\\begin{array}{cc}\n\\begin{array}{c}\n\\begin{tikzpicture}[scale=0.5]\n\\draw[->,densely dotted] (0,-0.5)--($(0,-0.5)+(52.5:4.5cm)$);\n\\node at ($(0,-0.5)+(60:4cm)$) {$\\scriptstyle \\mathsf{N}x$};\n\\node at (0,-0.6){};\n\\draw[->,densely dotted] (0,-0.5)--($(0,-0.5)+(33:3.25cm)$);\n\\node at ($(0,-0.5)+(40:3.25cm)$) {$\\scriptstyle \\mathsf{N}y$};\n\\end{tikzpicture}\n\\end{array}\n&\n\\begin{array}{c}\n\\includegraphics[angle=0,width=7.5cm,height=4cm]{Y-figure4.pdf}\n\\end{array}\n\\end{array}\n\\]\nIt is visually clear that both $\\mathsf{Fund}_{\\EuScript{X}}$ in \\eqref{running example infinite fund region 1} and $\\mathsf{N}(\\mathsf{Fund}_{\\EuScript{Y}})$ above individually generate $\\EuScript{H}_{\\EuScript{I}}^{\\mathsf{aff}}$, via translation, and that $\\mathsf{Fund}_{\\EuScript{X}}$ and $\\mathsf{N}(\\mathsf{Fund}_{\\EuScript{Y}})$ are different.\n\\end{example}\n\n\\begin{remark}\nThe above example gives a visual proof of the Crepant Transformation Conjecture of Subsection~\\ref{sec:CTC proof} below. The regions $\\mathsf{Fund}_{\\EuScript{X}}$ and $\\mathsf{N}(\\mathsf{Fund}_{\\EuScript{Y}})$ are different. But they generate the same object, namely $\\EuScript{H}_{\\EuScript{I}}^{\\mathsf{aff}}$, which by Corollary~\\ref{cor: quantum is hyper} is the pole locus of the GW quantum potential. Thus, although the curve invariants of $\\EuScript{X}$ and $\\EuScript{Y}$, captured in the fundamental regions, are technically different, after a change in variables (namely $\\mathsf{N}$) they can be compared, where they generate the same object. \n\\end{remark}\n\n\\begin{remark}\\label{rem: matrix Mk}\nThe matrix $\\mathsf{N}_i$ appears via moduli tracking in the HomMMP \\cite[5.4]{HomMMP}, and via the K-theory of contraction algebras \\cite[2.4]{AugustWemyss}. In contrast, $\\mathsf{M}_i$ from \\eqref{defn Mi} is the dual, and it arises from the change in dimension vector \\cite[5.4]{HomMMP}, or in the K-theory of projective modules \\cite[3.2]{HW2}. For more details, see \\cite[2.4]{AugustWemyss} and references therein.\n\\end{remark}\n\n\\subsection{Crepant Transformation Conjecture} \\label{sec:CTC proof}\nWe make no attempt at a comprehensive summary of the Crepant Transformation Conjecture (CTC), and instead refer the reader to \\cite{CoatesRuan, CoatesIritaniJiang, BryanGraber,YPLeeLectures}. For a pair of smooth varieties related via a sequence of flops, the conjecture asserts that their quantum potentials should coincide, under a suitable identification of (co)homologies and analytic continuation in the Novikov parameters. There has been extensive work on this conjecture within both algebraic and symplectic geometry \\cite{LiRuan, McLean, LLWMotives, LLW1, LLW2, LLQW3}.\n\nHere we prove the CTC for germs of isolated $3$-fold flops, as a direct application of the expression for the quantum potential in Theorem~\\ref{thm: structure of quantum potential} together with the construction of flops via simultaneous partial resolutions in Theorem~\\ref{thm: flop by sim res}. This gives the first algebraic-geometric proof of the CTC for flops of arbitrary type. \n\n\\medskip\nAs before, consider a curve $\\Curve_i \\subseteq \\EuScript{X}$ and let $\\EuScript{X}_i^+$ be the flop of $\\EuScript{X}$ at $\\Curve_i$. Recall that the following vector spaces are based by the sets of exceptional curves\n\\[ \n\\mathfrak{h}_{\\EuScript{I},\\mathbb{C}}= A_1(\\EuScript{X})_{\\mathbb{C}}= \\langle \\Curve_j \\mid j \\in \\scrI^{\\kern 0.5pt\\mathrm{c}} \\rangle_{\\mathbb{C}}, \\qquad \\mathfrak{h}_{\\upomega_i(\\EuScript{I}),\\mathbb{C}} = A_1(\\EuScript{X}_i^+)_{\\mathbb{C}} = \\langle \\Curve_j,\\Curve_i^+\\mid j\\notin\\EuScript{I}+i \\rangle_{\\mathbb{C}}. \n\\]\nwhere as in Lemma~\\ref{lem: action of Mi} we abuse notation by denoting the flopped curve $\\Curve_i^+$ instead of $\\Curve_{\\raisemath{10pt}{\\upiota_{\\EuScript{I}+i}(i)}}^+$. As explained in Subsection~\\ref{sec: GV under flop}, there is an explicit transformation matrix\n\\[ \\mathsf{M}_i \\colon A_1(\\EuScript{X}_i^+)_{\\mathbb{C}} \\to A_1(\\EuScript{X})_{\\mathbb{C}}.\\]\nThis is the complexification of the matrix $\\mathsf{M}_i$ from earlier, but we use the same symbol. Let $\\mathsf{N}_i$ be the matrix dual to $\\mathsf{M}_i^{-1}$ which can be viewed as a linear map\n\\[ \n\\mathsf{N}_i \\colon \\mathrm{H}^2(\\EuScript{X}_i^+;\\mathbb{C}) \\to \\mathrm{H}^2(\\EuScript{X};\\mathbb{C}) \n\\]\nwith the property that $\\mathsf{N}_i \\upgamma \\cdot \\upbeta = \\upgamma \\cdot \\mathsf{M}_i^{-1} \\upbeta$ for $\\upgamma \\in \\mathrm{H}^2(\\EuScript{X}_i^+;\\mathbb{C})$ and $\\upbeta \\in A_1(\\EuScript{X})_{\\mathbb{C}}$. We notate the Novikov co-ordinates on the parameter spaces for the quantum potentials by\n\\[\n\\begin{array}{ll}\n\\{ \\mathsf{q}_j \\mid j \\in \\scrI^{\\kern 0.5pt\\mathrm{c}}\\} &\\text{on $A_1(\\EuScript{X})_{\\mathbb{C}}$}, \\\\\n\\{ \\mathsf{r}_j \\mid j \\in \\scrI^{\\kern 0.5pt\\mathrm{c}} \\} & \\text{on $A_1(\\EuScript{X}_i^+)_{\\mathbb{C}}$.}\n\\end{array}\n\\]\nThe matrix $\\mathsf{M}_i^{-1}$ defines a monomial co-ordinate transformation relating the two sets of Novikov parameters. Writing monomials as\n\\[ \\mathsf{q}^{\\upbeta} \\colonequals \\prod_{j \\in \\scrI^{\\kern 0.5pt\\mathrm{c}}} \\mathsf{q}_j^{m_j},\\qquad \\mathsf{r}^{\\upbeta} \\colonequals \\prod_{j \\in \\scrI^{\\kern 0.5pt\\mathrm{c}}} \\mathsf{r}_j^{m_j}\\]\nthis is given by the equation\n\\begin{equation} \\label{eqn: transformation of Novikov parameters} \n\\mathsf{q}^{\\upbeta} = \\mathsf{r}^{\\mathsf{M}_i^{-1} \\upbeta}.\n\\end{equation}\nBy Lemma~\\ref{lem: action of Mi}, $\\mathsf{q}_i = \\mathsf{r}_i^{-1}$ whereas every other $\\mathsf{q}_{j}$ is a monomial in the $\\mathsf{r}_j$ with non-negative coefficients.\n\nLastly, recall the quantum potentials of $\\EuScript{X}$ and $\\EuScript{X}_i^+$ from Subsection~\\ref{sec: quantum potential}, which to ease notation will be written $\\Phi$ and $\\Phi^+$, namely\n\\begin{align*} \\Phi_{\\mathsf{q}}(\\upgamma_1,\\upgamma_2,\\upgamma_3) & \\colonequals \\Phi_{\\mathsf{q}}^{\\EuScript{X}}(\\upgamma_1,\\upgamma_2,\\upgamma_3), \\\\\n\\Phi_{\\mathsf{r}}^+(\\upgamma_1,\\upgamma_2,\\upgamma_3) & \\colonequals \\Phi_{\\mathsf{r}}^{\\EuScript{X}_i^+}(\\upgamma_1,\\upgamma_2,\\upgamma_3).\n\\end{align*}\nThe equation \\eqref{eqn: transformation of Novikov parameters} will be used to express the quantum potential of $\\EuScript{X}$ in the variables $\\mathsf{r}$, and this will be denoted $\\Phi_r(\\upgamma_1,\\upgamma_2,\\upgamma_3)$.\n\n\\begin{corollary}[Crepant Transformation Conjecture] \\label{thm: CTC} On the $\\mathsf{r}$ parameter space, the quantum potentials of $\\EuScript{X}$ and $\\EuScript{X}_i^+$ coincide, up to the following explicit term which does not depend on the Novikov parameters\n\\begin{equation} \\label{eqn: CTC} \\Phi^{+}_{\\mathsf{r}}\\big(\\upgamma_1, \\upgamma_2,\\upgamma_3\\big) - \\Phi_{\\mathsf{r}}(\\mathsf{N}_i \\upgamma_1,\\mathsf{N}_i \\upgamma_2,\\mathsf{N}_i \\upgamma_3) = -(\\upgamma_1 \\cdot \\Curve_i^+)(\\upgamma_2 \\cdot \\Curve_i^+)(\\upgamma_3\\cdot \\Curve_i^+) \\sum_{k \\geq 1} k^3 n_{k\\Curve_i,\\EuScript{X}}. \n\\end{equation}\n\\end{corollary}\n\\noindent\nMore precisely the pole loci are transformed into each other via \\eqref{eqn: transformation of Novikov parameters}, and away from these the analytic continuations constructed in Theorem~\\ref{thm: structure of quantum potential} coincide.\n\n\\begin{remark} \\label{rmk: CTC non-compact} The quantum potentials of $\\EuScript{X}$ and $\\EuScript{X}_i^+$ have no constant terms in their respective Novikov parameters, due to the absence of a perfect pairing on cohomology (see Remark~\\ref{rmk: no algebra}). However, the change of parameters \\eqref{eqn: transformation of Novikov parameters} introduces constant terms into $\\Phi_{\\mathsf{r}}(\\upgamma_1,\\upgamma_2,\\upgamma_3)$, which form the right-hand side of \\eqref{eqn: CTC}. It follows that the quantum potentials agree once these extraneous constant terms are removed from $\\Phi_{\\mathsf{r}}(\\upgamma_1,\\upgamma_2,\\upgamma_3)$. In particular, $\\Phi^+$ can be effectively reconstructed from $\\Phi$. In situations where an ordinary cup product can be defined, the additional terms on the right-hand side account for the defect between the cup products on $\\EuScript{X}$ and $\\EuScript{X}_i^+$, see e.g. \\cite[Subsection~4.3 and Equation~(4.4)]{MorrisonKaehler}.\n\\end{remark}\n\n\\begin{remark} \nThe expansion points for the quantum potentials differ, as\n\\[ (\\mathsf{r}_j)_{j \\in \\scrI^{\\kern 0.5pt\\mathrm{c}}} = (0,\\ldots,0) \\ \\Leftrightarrow \\ (\\mathsf{q}_j)_{j \\in \\scrI^{\\kern 0.5pt\\mathrm{c}}} = (0,\\ldots,0,\\infty,0,\\ldots,0)\\]\nwith $\\infty$ in the $i$th position. Thus, the term $\\Phi_{\\mathsf{r}}(\\mathsf{N}_i \\upgamma_1,\\mathsf{N}_i \\upgamma_2,\\mathsf{N}_i \\upgamma_3)$ is analytically continued from $\\mathsf{q}_i=0$ to $\\mathsf{q}_i=\\infty$, the analytic continuation occurring precisely in the Novikov parameter corresponding to the flopped curve.\n\\end{remark}\n\n\\begin{proof}[Proof of \\ref{thm: CTC}] \nWe explicitly match both sides, using our knowledge of the structure of the quantum potentials (Theorem~\\ref{thm: structure of quantum potential}) and the behaviour of the GV invariants under the flop (Theorem~\\ref{thm: GV under flop}).\n\nSeparating curve classes according to whether or not they are a multiple of $\\Curve_i$, the quantum potential for $\\EuScript{X}$ may be written as the sum of contributions\n\\begin{align*} \n\\Phi_{\\mathsf{q}}(\\mathsf{N}_i\\upgamma_1,\\mathsf{N}_i\\upgamma_2,\\mathsf{N}_i\\upgamma_3) & = \\mathsf{G}_{\\mathsf{q}}(\\mathsf{N}_i \\upgamma_1,\\mathsf{N}_i \\upgamma_2, \\mathsf{N}_i \\upgamma_3) + \\mathsf{H}_{\\mathsf{q}}(\\mathsf{N}_i \\upgamma_1,\\mathsf{N}_i \\upgamma_2, \\mathsf{N}_i \\upgamma_3) \n\\end{align*}\nwhere\n\\begin{align*}\n\\mathsf{G}_{\\mathsf{q}}(\\mathsf{N}_i \\upgamma_1,\\mathsf{N}_i \\upgamma_2, \\mathsf{N}_i \\upgamma_3) & =\\quad\\, \\sum_{k \\geq 1} n_{k\\Curve_i,\\EuScript{X}} \\, (\\mathsf{N}_i\\upgamma_1 \\cdot k\\Curve_i)(\\mathsf{N}_i\\upgamma_2 \\cdot k\\Curve_i) (\\mathsf{N}_i\\upgamma_3 \\cdot k\\Curve_i) \\dfrac{\\mathsf{q}_i^k}{1-\\mathsf{q}_i^k} \\\\ \n\\mathsf{H}_{\\mathsf{q}}(\\mathsf{N}_i \\upgamma_1,\\mathsf{N}_i \\upgamma_2, \\mathsf{N}_i \\upgamma_3) & = \\sum_{\\substack{\\upbeta \\in A_1(\\EuScript{X}) \\\\ \\upbeta \\neq k\\Curve_i}} n_{\\upbeta,\\EuScript{X}} \\, (\\mathsf{N}_i\\upgamma_1\\cdot\\upbeta)(\\mathsf{N}_i\\upgamma_2\\cdot\\upbeta)(\\mathsf{N}_i\\upgamma_3\\cdot\\upbeta) \\dfrac{\\mathsf{q}^\\upbeta}{1-\\mathsf{q}^\\upbeta}. \n\\end{align*}\nSimilarly, write the quantum potential of $\\EuScript{X}_i^+$ as\n\\begin{align*}\n\\Phi^{+}_{\\mathsf{r}}(\\upgamma_1,\\upgamma_2,\\upgamma_3) & = \\mathsf{G}^{+}_{\\mathsf{r}}(\\upgamma_1,\\upgamma_2,\\upgamma_3) +\\mathsf{H}^{+}_{\\mathsf{r}}(\\upgamma_1,\\upgamma_2,\\upgamma_3),\n\\end{align*}\nwhere\n\\begin{align*} \n\\mathsf{G}^{+}_{\\mathsf{r}}(\\upgamma_1,\\upgamma_2,\\upgamma_3) & = \\quad\\, \\sum_{k \\geq 1} n_{k\\Curve_i^+,\\EuScript{X}_i^+} \\, (\\upgamma_1 \\cdot k\\Curve_i^+)(\\upgamma_2 \\cdot k\\Curve_i^+) (\\upgamma_3 \\cdot k\\Curve_i^+) \\dfrac{\\mathsf{r}_i^{k}}{1-\\mathsf{r}_i^{k}} \\\\ \n\\mathsf{H}^{+}_{\\mathsf{r}}(\\upgamma_1,\\upgamma_2,\\upgamma_3) & = \\sum_{\\substack{\\upbeta \\in A_1(\\EuScript{X}) \\\\ \\upbeta \\neq k\\Curve_i}} n_{\\upbeta,\\EuScript{X}_i^+} \\, (\\upgamma_1\\cdot\\upbeta)(\\upgamma_2\\cdot\\upbeta)(\\upgamma_3\\cdot\\upbeta) \\dfrac{\\mathsf{r}^{\\upbeta}}{1-\\mathsf{r}^{\\upbeta}}. \n\\end{align*}\nWe begin with the $\\mathsf{G}$ terms. Using \\eqref{eqn: transformation of Novikov parameters} to change variables from $\\mathsf{q}$ to $\\mathsf{r}$ gives\n\\begin{align*} \n\\mathsf{G}_{\\mathsf{r}}(\\mathsf{N}_i\\upgamma_1,\\mathsf{N}_i\\upgamma_2,\\mathsf{N}_i\\upgamma_3) & = \\sum_{k \\geq 1} n_{k\\Curve_i,\\EuScript{X}} \\, (\\mathsf{N}_i\\upgamma_1 \\cdot k\\Curve_i)(\\mathsf{N}_i\\upgamma_2 \\cdot k\\Curve_i) (\\mathsf{N}_i\\upgamma_3 \\cdot k\\Curve_i) \\dfrac{\\mathsf{r}_i^{-k}}{1-\\mathsf{r}_i^{-k}} \\\\\n& = (\\upgamma_1 \\cdot \\Curve_i^+) (\\upgamma_2 \\cdot \\Curve_i^+) (\\upgamma_3 \\cdot \\Curve_i^+) \\sum_{k \\geq 1} k^3 n_{k\\Curve_i,\\EuScript{X}} \\dfrac{1}{1-r_i^k}. \n\\end{align*}\nwhere the second equality follows from $\\mathsf{N}_i \\upgamma \\cdot \\Curve_i = \\upgamma \\cdot \\mathsf{M}_i^{-1} \\Curve_i = - \\upgamma \\cdot \\Curve_i^+$ and the equality\n\\[ \n\\dfrac{r_i^{-k}}{1-r_i^{-k}} = \\dfrac{1}{r_i^k - 1}. \n\\]\nUsing $n_{k\\Curve_i,\\EuScript{X}} = n_{k\\Curve_i^+,\\EuScript{X}_i^+}$ by Theorem~\\ref{thm: GV under flop}, the difference $\\mathsf{G}^{+}_{\\mathsf{r}}(\\upgamma_1,\\upgamma_2,\\upgamma_3) - \\mathsf{G}_{\\mathsf{r}}(\\mathsf{N}_i\\upgamma_1,\\mathsf{N}_i\\upgamma_2,\\mathsf{N}_i\\upgamma_3)$ is equal to \n\\begin{align*} \n&\\phantom{=}(\\upgamma_1 \\cdot \\Curve_i^+)(\\upgamma_2 \\cdot \\Curve_i^+)(\\upgamma_3\\cdot \\Curve_i^+) \\sum_{k \\geq 1} k^3 n_{k\\Curve_i,\\EuScript{X}} \\left( \\dfrac{r_i^k}{1-r_i^k} - \\dfrac{1}{1-r_i^k} \\right) \\\\\n& = -(\\upgamma_1 \\cdot \\Curve_i^+)(\\upgamma_2 \\cdot \\Curve_i^+)(\\upgamma_3\\cdot \\Curve_i^+) \\sum_{k \\geq 1} k^3 n_{k\\Curve_i,\\EuScript{X}}. \n\\end{align*}\nWe next examine the $\\mathsf{H}$ terms. Note that for $\\upbeta \\in A_1(\\EuScript{X})$ we have $\\upbeta \\in \\mathbb{Z} \\Curve_i$ if and only if $\\mathsf{M}_i^{-1}\\upbeta \\in \\mathbb{Z} \\Curve_i^+$. Consequently\n\\begin{align*} \n\\mathsf{H}_{\\mathsf{r}}(\\mathsf{N}_i \\upgamma_1,\\mathsf{N}_i \\upgamma_2,\\mathsf{N}_i\\upgamma_3) & = \\sum_{\\substack{\\upbeta \\in A_1(\\EuScript{X}) \\\\ \\upbeta \\neq k\\Curve_i}} n_{\\upbeta,\\EuScript{X}} \\, (\\mathsf{N}_i\\upgamma_1\\cdot\\upbeta)(\\mathsf{N}_i\\upgamma_2\\cdot\\upbeta)(\\mathsf{N}_i\\upgamma_3\\cdot\\upbeta) \\dfrac{\\mathsf{r}^{\\mathsf{M}_i^{-1} \\upbeta}}{1-\\mathsf{r}^{\\mathsf{M}_i^{-1} \\upbeta}} \\\\\n& = \\sum_{\\substack{\\upbeta \\in A_1(\\EuScript{X}_i^+) \\\\ \\upbeta \\neq k\\Curve_i^+}} n_{\\mathsf{M}_i \\upbeta,\\EuScript{X}} \\, (\\mathsf{N}_i \\upgamma_1 \\cdot \\mathsf{M}_i \\upbeta)(\\mathsf{N}_i \\upgamma_2 \\cdot \\mathsf{M}_i \\upbeta)(\\mathsf{N}_i \\upgamma_3 \\cdot \\mathsf{M}_i \\upbeta) \\dfrac{r^{\\upbeta}}{1-r^{\\upbeta}} \\\\\n& = \\sum_{\\substack{\\upbeta \\in A_1(\\EuScript{X}_i^+) \\\\ \\upbeta \\neq k\\Curve_i^+}} n_{\\upbeta,\\EuScript{X}_i^+} \\, (\\upgamma_1 \\cdot \\upbeta)(\\upgamma_2 \\cdot \\upbeta)(\\upgamma_3 \\cdot \\upbeta) \\dfrac{r^{\\upbeta}}{1-r^{\\upbeta}} \\\\\n& = \\mathsf{H}^{+}_{\\mathsf{r}}(\\upgamma_1,\\upgamma_2,\\upgamma_3).\n\\end{align*}\nwhere the penultimate equality holds since $\\mathsf{N}_i \\upgamma \\cdot \\mathsf{M}_i \\upbeta = \\upgamma \\cdot \\mathsf{M}_i^{-1} \\mathsf{M}_i \\upbeta = \\upgamma \\cdot \\upbeta$ and $n_{\\mathsf{M}_i \\upbeta,\\EuScript{X}} = n_{\\upbeta,\\EuScript{X}_i^+}$ again by Theorem~\\ref{thm: GV under flop}. Combining the comparison of the $\\mathsf{G}$ terms with the comparison of the $\\mathsf{H}$ terms gives \\eqref{eqn: CTC}, as required.\n\\end{proof}\n\n\n\n\n\n\n\n\n\n\n\n\n\\subsection{The contraction algebra under flop}\nThe flopping contraction $\\EuScript{X}\\to\\mathop{\\rm Spec}\\nolimits \\EuScript{R}$ has an associated contraction algebra $\\mathrm{A}_{\\con}$, defined using noncommutative deformation theory \\cite{DW1, DW3}. After flopping a single curve $\\Curve_i$ to obtain $\\EuScript{X}_i^+\\to\\mathop{\\rm Spec}\\nolimits \\EuScript{R}$, noncommutative deformation theory associates to this another contraction algebra, written $\\upnu_i\\mathrm{A}_{\\con}$. The algebra $\\upnu_i\\mathrm{A}_{\\con}$ can be intrinsically obtained from $\\mathrm{A}_{\\con}$ via a certain mutation procedure, and in fact $\\mathrm{A}_{\\con}$ and $\\upnu_i\\mathrm{A}_{\\con}$ are derived equivalent \\cite{AugustTiltingTheory}. Both $\\mathrm{A}_{\\con}$ and $\\upnu_i\\mathrm{A}_{\\con}$ are finite dimensional algebras \\cite[2.13]{DW1}.\n\nFix the GV invariants $n_\\upbeta$ associated to $\\EuScript{X}\\to\\mathop{\\rm Spec}\\nolimits \\EuScript{R}$, then Toda's dimension formula (see \\ref{thm: Toda dim formula}) asserts that\n\\[\n\\dim_{\\mathbb{C}}\\mathrm{A}_{\\con}=\\sum_{\\upbeta\\in A_1(\\EuScript{X})}n_\\upbeta \\big(\\,\\upbeta\\cdot \\mathds{1}\\big)^2\n\\]\nIn many, but not all, cases the dimension of $\\mathrm{A}_{\\con}$ is in fact enough to recover the $n_\\upbeta$. The next result asserts that the numbers $n_\\upbeta$ associated to $\\mathrm{A}_{\\con}$, together with the matrix $\\mathsf{M}^{-1}_i$, completely determine the dimension of $\\upnu_i\\mathrm{A}_{\\con}$.\n\\begin{cor}\\label{cor: Toda formula iterate}\nUnder mutation at vertex $i$, equivalently flop at $\\Curve_i$,\n\\[\n\\dim_{\\mathbb{C}}\\upnu_i\\mathrm{A}_{\\con}=\\sum_{\\upbeta\\in A_1(\\EuScript{X})}n_\\upbeta \\big(\\,(\\mathsf{M}^{-1}_i\\upbeta)\\cdot \\mathds{1}\\big)^2\n\\]\nwhere $\\mathsf{M}_i$ is the explicit matrix in \\eqref{defn Mi}.\n\\end{cor}\n\\begin{proof}\nCombining previous results, it follows that\n\\begin{align*}\n\\dim_{\\mathbb{C}}\\upnu_i\\mathrm{A}_{\\con}\n&=\\sum_{\\upgamma\\in A_1(\\EuScript{X}_i^+)}n_{\\upgamma}(\\upgamma\\cdot \\mathds{1})^2\\tag{by \\ref{thm: Toda dim formula}}\\\\\n&=\\sum_{\\upbeta\\in A_1(\\EuScript{X})}n_{\\upbeta}\\big(\\,|\\mathsf{M}^{-1}_i\\upbeta|\\cdot \\mathds{1}\\big)^2\\tag{$\\upgamma=|\\mathsf{M}^{-1}_i\\upbeta|$ in Theorem~\\ref{thm: GV under flop}}\\\\\n&=\\sum_{\\upbeta\\in A_1(\\EuScript{X})}n_{\\upbeta}\\big(\\,(\\mathsf{M}_i^{-1}\\upbeta)\\cdot \\mathds{1}\\big)^2\\tag{by Corollary~\\ref{cor: action of Mi pos neg}}\n\\end{align*}\nwhere the point is that, by Corollary~\\ref{cor: action of Mi pos neg}, the sign issue doesn't matter once we square. \n\\end{proof}\nIn particular, it is possible to compute the dimension of $\\upnu_i\\mathrm{A}_{\\con}$ without first having to present it.\n\\begin{example}\nAs in \\cite{SmithWemyss}, consider the $cA_2$ example $\\EuScript{R}_k=\\frac{\\mathbb{C}[\\![u,v,x,y]\\!]}{uv-xy(x^k+y)}$ for $k\\geq 1$, and the specific crepant resolution $\\EuScript{X}\\to\\mathop{\\rm Spec}\\nolimits\\EuScript{R}_k$ described in \\cite[3.1]{SmithWemyss}, obtained first by blowing up $(u,y)$ then $(u,x)$. In this case, as explained in \\cite[Subsection~6.2]{SmithWemyss} $\\mathrm{A}_{\\con}$ can be presented as\n\\[\n\\begin{array}{c}\n\\begin{tikzpicture}[scale=0.8,>=stealth]\n\\node (A) at (0,0) [Bquiv] {};\n\\node (B) at (2,0) [Bquiv] {};\n\\draw[->, bend left] (A) to node[above]{$\\scriptstyle a$} (B);\n\\draw[->, bend left] (B) to node[below]{$\\scriptstyle b$} (A);\n\\end{tikzpicture}\n\\end{array}\n\\quad\n\\begin{array}{c}\n(ab)^ka=0=b(ab)^k.\n\\end{array}\n\\]\nWe can immediately read off the GV invariants, namely $n_{1,0}=1$, $n_{0,1}=1$, and $n_{1,1}=k$. Thus $\\dim_\\mathbb{C}\\mathrm{A}_{\\con}=n_{1,0}\\cdot (1+0)^2+n_{0,1}\\cdot (0+1)^2+n_{1,1}\\cdot (1+1)^2$, which equals $1+1+4k=4k+2$.\n\nWe now flop the right hand curve. In this Type $A$ example $\\mathsf{M}_i^{-1}$ sends $(1,0)\\mapsto (1,1)$, $(1,1)\\mapsto (1,0)$ and $(0,1)\\mapsto (0,-1)$. Thus, by Corollary~\\ref{cor: Toda formula iterate},\n\\[\n\\dim_\\mathbb{C}\\upnu_i\\mathrm{A}_{\\con}=n_{1,0}\\cdot (1+1)^2+n_{0,1}\\cdot (0+1)^2+n_{1,1}\\cdot (0-1)^2,\n\\] \nwhich equals $4+1+k=k+5$. In particular, $\\upnu_i\\mathrm{A}_{\\con}\\ncong\\mathrm{A}_{\\con}$ provided that $k\\geq 2$.\n\\end{example}\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\n\nToday, nearly 100 multi-planet systems have been reported, of which\nroughly $1\/3$ possess at least one ``moderate close-in planet'', that is, a planet with a semi-major axis between 0.1 and 0.5~AU\\footnote{http:\/\/exoplanet.eu}.\nPlanets in this range are supposed to undergo significant tidal interactions, resulting in slowly modified spins and orbits.\nHowever, for the typically assumed dissipation rates for gaseous planets, the spin of moderate close-in planets reaches an equilibrium state in only a few million years, while the orbital evolution can last for the entire age of the system (Gyr timescale).\n\nAmong two-planet systems, there is a special class whose semi-major ratio $a_1 \/ a_2 $ is lower than 0.1, the so-called ``hierarchical systems''. \nThis class counts at least 20 members ($1\/5$ of all multi-planet systems), and usually at least one of the planets' orbits is highly eccentric\\footnotemark[\\value{footnote}]. \nDuring the formation process, the orbital eccentricities can increase through gravitational scattering \\citep[e.g.][]{Nagasawa_etal_2008}, which is simultaneously responsible for an increase of the orbits' mutual inclination \\citep[e.g.][]{Chatterjee_etal_2008}.\nEvidently, smaller mass planets, that are as yet undetected, can exist in these systems, but for a semi-major axis larger than 0.1~AU the orbits usually present high values in the eccentricities, which reduces the stability areas for additional companions.\n\nHierarchical systems are particularly interesting from a dynamical point of view, because they can be stable with very eccentric and inclined orbits, which is an uncommon behavior.\nIn particular, they are very interesting when the inner planet is sufficiently close to the star to undergo tidal interactions, since the final outcome of the evolution can be in a configuration that is totally different from the initial one.\nBecause the semi-major ratio is small, low-order mean motion resonances cannot occur, which allows us to perform analytical approximations such as averaging the orbits over the mean anomalies.\nIn addition, tidal effects usually act over very long timescales and therefore\napproximate theories also allow one to accelerate the numerical simulations and to\nexplore the parameter space much more rapidly.\n\nSecular perturbation theories based on series expansions have been used for hierarchical systems. \nFor low eccentricity values, the expansion of the perturbation in eccentricity series is very efficient \\citep[e.g.][]{Wu_Goldreich_2002}, but this method is not appropriate for orbits that become very eccentric. An expansion in the ratio of the semi-major axis $a_1 \/ a_2 $ is then preferred, because exact expressions can be computed for the secular system \\citep[e.g.][]{Laskar_Boue_2010}. \nThe development to the second order in $a_1 \/ a_2 $, called the quadrupole approximation, was used by \\citet{Lidov_1961,Lidov_1962} and \\citet{Kozai_1962} for the restricted inner problem (the outer orbit is unperturbed).\nIn this case, the conservation of the normal component of the angular momentum enables the inner orbit to periodically exchange its eccentricity with inclination (the so-called Lidov-Kozai mechanism).\nFor planar problems, the series expansions in $a_1 \/ a_2 $ should be conducted to the octupole\norder \\citep[e.g.][]{Marchal_1990,\nFord_etal_2000,Laskar_Boue_2010}, because the quadrupole approximation \nfails to reproduce the eccentricity oscillations \\citep[e.g.][]{Lee_Peale_2003}. \nSince we do not have any restrictions for the eccentricities or for the mutual inclination, we need to expand the gravitational potential in $a_1 \/ a_2 $ to the octupole order.\n\nThe ultimate stage for tidal evolution is the spin synchronization and orbital circularization \\citep[e.g.][]{Correia_2009}.\nIndeed, the observed mean eccentricity for planets and binary stars with $ a_1 < 0.1 $~AU is close to zero within the observational limitations \\citep[e.g.][]{Pont_etal_2011}.\nAlthough tidal effects modify the spin on a much shorter timescale than they modify the orbit, synchronous rotation can only occur when the eccentricity is very close to zero: the rotation rate tends to be locked with the orbital speed at the periapsis, because tidal effects are stronger when the two bodies are closer to each other.\nIn addition, if there is a companion body, the eccentricity oscillates \\citep[e.g.][]{Mardling_2007, Laskar_etal_2012}, and the rotation rate of the planet shows variations that follow the eccentricity.\nThis is exactly what is observed for Mercury, whose average orbital eccentricity is around 0.2, and its rotation is captured in a 3\/2 spin-orbit resonance \\citep{Correia_Laskar_2004,Correia_Laskar_2012}.\nTherefore, spin and orbital evolution cannot be dissociated, and some unexpected behavior can be observed, such as a secular increase of the eccentricity \\citep[e.g.][]{Correia_etal_2011, Correia_etal_2012}.\n\nIn this paper we intend to intensify the study of hierarchical two-planet systems, in which the inner orbit undergoes tidal dissipation.\nWe present here another counterintuitive behavior, the inclination damping, which is also a consequence of the above-mentioned eccentricity pumping when the two orbits are not coplanar.\n\n\n\\section{Model}\n\n\\llabel{TheModel}\n\nWe considered a system consisting of a central star of mass $m_0$, an inner planet of mass $m_1$, and an outer companion of mass $m_2$.\nWe used Jacobi canonical coordinates, with $\\vv{r_1} $ being the position of\n$m_1$ relative to $m_0$, and $ \\vv{r_2} $ the position of $ m_2 $\nrelative to the center of mass of $ m_1 $ and $ m_0 $.\nWe assumed that the system is hierarchical, thus $|\\vv{r_1}| \\ll |\\vv{r_2}|$.\nFor simplicity, we express all the angles in the invariable plane of the system, i.e., the plane perpendicular to the total angular momentum\n\\vskip-.5cm\n\\begin{equation}\n\\vv{H} = \\vv{L} + \\vv{G}_1 + \\vv{G}_2 \\ , \\llabel{120329c}\n\\end{equation}\nwhere $\\vv{L}$ is the rotational angular momentum of the inner planet, and $\\vv{G}_i $ the orbital angular momentum of each body.\n\nThe inner planet is considered an oblate ellipsoid with gravity field coefficients given by $J_2$, rotating about the axis of maximal inertia (gyroscopic approximation), \nwith rotation rate $\\omega$, such that \\citep[e.g.][]{Lambeck_1988}\n\\vskip-.5cm\n\\begin{equation}\nJ_2 = k_2 \\frac{\\omega_{}^2 R_{}^3}{3 {\\cal G} m_1} \\ .\n\\llabel{101220a}\n\\end{equation}\n${\\cal G}$ is the gravitational constant, $R_{}$ is the radius of the planet, and\n$ k_2 $ is the second Love number for potential (pertaining to a perfectly fluid body).\nWe furthermore assumed that the obliquity of the planet to the orbital plane is zero ($\\varepsilon = 0^\\circ$), that is, $\\vv{L}$ and $\\vv{G}_1$ are aligned.\nTherefore, the angle between the two orbital planes, i.e., the mutual inclination $I$, satisfies the \nrelation\n\\vskip-.5cm\n\\begin{equation}\nH^2 = (L + G_1)^2 + G_2^2 +{2 (L + G_1) G_2}\\cos I \\ .\n\\llabel{120329d}\n\\end{equation}\n\n\\subsection{Conservative motion}\n\nBecause we are interested in the secular behavior of the system, we averaged the equations of motion over the mean anomalies of both orbits.\nIn the invariable plane, the averaged potential, quadrupole-level for the spin \\citep[e.g.][]{Correia_Laskar_2010}, octupole-level for the orbits \\citep[e.g.][]\nFord_etal_2000,Laskar_Boue_2010}, and with general relativity corrections \\citep[e.g.][]{Touma_etal_2009} is given by\n\n\\vskip-.6cm\n\\begin{eqnarray}\nU & =& - \\frac{C_0}{(1-e_1^2)^{1\/2}} - \\frac{C_1}{(1-e_1^2)^{3\/2}} \\cr\\noalign{\\medskip} & &\n- C_2 \\frac{(1 + \\frac{3}{2} e_1^2) (1-\\frac{3}{2} \\sin^2 I)}{(1-e_2^2)^{3\/2}\n- C_2 \\frac{ \\frac{15}{4} e_1^2 \\sin^2 I}{(1-e_2^2)^{3\/2}} \\cos 2 \\omega_1 \\cr\\noalign{\\medskip} & &\n+ C_3 \\frac{ {\\cal A} }{(1-e_2^2)^{5\/2}} e_1 e_2 \\cos \\varphi \\cr\\noalign{\\medskip} & &\n+ C_3 \\frac{\\frac{5}{2} (1-e_1^2) \\cos I \\sin^2 I}{(1-e_2^2)^{5\/2}} e_1 e_2 \\sin \\omega_1 \\sin \\omega_2\n\\ , \\llabel{090514a}\n\\end{eqnarray}\nwhere \n\\begin{equation}\nC_0 = \\frac{3 \\beta_1 {\\cal G}^2 (m_0 + m_1)^2}{a_1^2 c^2} \\ , \\quad\nC_1 = \\frac{{\\cal G} m_0 m_1 J_2 R_{}^2}{2 a_1^3} \\llabel{110816a} \\ ,\n\\end{equation}\n\\begin{equation}\nC_2 = \\frac{{\\cal G} \\beta_1 m_2 a_1^2}{4 a_2^3 } \\ , \\quad\nC_3 = \\frac{15 {\\cal G} \\beta_1 m_2 a_1^3}{16 a_2^4} \\frac{(m_0-m_1)}{m_0 + m_1} \\llabel{110816c} \\ ,\n\\end{equation}\n\\begin{equation}\n{\\cal A}= 1 + \\frac{3}{4} e_1^2 - \\frac{5}{4} {\\cal B} \\sin^2 I \\llabel{120402a} \\ ,\n\\end{equation}\n\\begin{equation}\n{\\cal B} = 1+ \\frac{5}{2} e_1^2 - \\frac{7}{2} e_1^2 \\cos 2 \\omega_1 \\llabel{120329a} \\ ,\n\\end{equation}\nand $\\varphi$ is the angle between the directions of the periastrons:\n\\begin{equation}\n\\cos \\varphi = - \\cos \\omega_1 \\cos \\omega_2 - \\sin \\omega_1 \\sin \\omega_2 \\cos I \\llabel{120329a} \\ .\n\\end{equation}\n$a_i$ is the semi-major axis (that can also be expressed using the mean motion $n_i $),\n$e_i$ is the eccentricity, and $\\omega_i $ is the argument of the periastron.\nWe also have $ \\beta_1 = m_0 m_1 \/ (m_0 + m_1) $, $ \\beta_2 = (m_0 + m_1) m_2 \/ (m_0 + m_1 + m_2) $,\n$ G_i = \\beta_i n_i a_i^2 (1-e_i^2)^{1\/2}$, and\n$ L = \\xi m_1 R_{}^2 \\omega_{} $, where $ \\xi $ is the normalized moment of inertia.\n\nThe contributions to the orbits are easily obtained using the Lagrange planetary equations \\citep[e.g.][]{Murray_Dermott_1999}:\n\\begin{equation}\n\\dot e_i = \\frac{\\sqrt{1-e_i^2}}{\\beta_i n_i a_i^2 e_i} \\frac{\\partial U}{\\partial \\omega_i} \\ , \\quad\n\\dot \\omega_i = - \\frac{\\sqrt{1-e_i^2}}{\\beta_i n_i a_i^2 e_i} \\frac{\\partial U}{\\partial e_i} \\llabel{110816d} \\ .\n\\end{equation}\nIn addition, since the variations in $e_1$, $e_2$ and $I$ are related by the conservation of the total angular momentum (Eq.\\,\\ref{120329d}), we have \n\\begin{eqnarray}\n\\frac{\\partial \\cos I}{\\partial G_1} &= -\\left[ \\FFrac{1}{G_2} + \\FFrac{1}{L + G_1} \\cos I \\right] \\ , \\cr\\noalign{\\medskip}\n\\frac{\\partial \\cos I}{\\partial G_2} &= -\\left[ \\FFrac{1}{L + G_1} + \\FFrac{1}{G_2} \\cos I \\right] \\ . \n\\end{eqnarray}\nAs we assumed that $L \/ G_1 \\sim (R_{} \/ a_1)^2 \\ll 1$, if we neglect first-order terms in $L\/G_1$, \nwe simply have ($i \\ne j = 1, 2$):\n\n\\begin{equation}\n\\frac{\\partial \\cos I}{\\partial e_i} = \\frac{G_i \\, e_i}{1-e_i^2} \\frac{\\partial \\cos I}{\\partial G_i} \\approx \\frac{e_i}{1-e_i^2} \\left( \\frac{G_i}{G_j} + \\cos I \\right) \\ . \\llabel{120329e}\n\\end{equation}\n\n\n\nThus,\n\\begin{eqnarray}\n\\dot e_1 & = & \\nu_{21} \\frac{ \\frac{5}{2} (1-e_1^2)^{1\/2} \\sin^2 I}{(1-e_2^2)^{3\/2}} e_1 \\sin 2 \\omega_1 \\cr\\noalign{\\medskip} \n&-& \\nu_{31} \\frac{ {\\cal A} (1-e_1^2)^{1\/2}}{(1-e_2^2)^{5\/2}} e_2 \\sin \\varphi_1 \\cr\\noalign{\\medskip}\n&-& \\nu_{31} \\frac{ \\frac{35}{4} e_1^2 \\sin 2 \\omega_1 \\sin^2 I}{(1-e_1^2)^{-1\/2} (1-e_2^2)^{5\/2}} e_2 \\cos \\varphi \\cr\\noalign{\\medskip}\n&+& \\nu_{31} \\frac{\\frac{5}{2} (1-e_1^2)^{3\/2} \\cos I \\sin^2 I}{(1-e_2^2)^{5\/2}} e_2 \\cos \\omega_1 \\sin \\omega_2 \n\\ , \\llabel{110816h}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\dot e_2 & = & \\nu_{32} \\frac{ {\\cal A} }{(1-e_2^2)^{2}} e_1 \\sin \\varphi_2 \\cr\\noalign{\\medskip}\n&+& \\nu_{32} \\frac{\\frac{5}{2} (1-e_1^2) \\cos I \\sin^2 I}{(1-e_2^2)^{2}} e_1 \\sin \\omega_1 \\cos \\omega_2 \\ , \\llabel{110816i} \n\\end{eqnarray}\nand\n\n\\begin{eqnarray}\n\\dot \\omega_1 &=& \\frac{\\nu_0}{ (1-e_1^2)} + \\frac{\\nu_1 \\, x_{}^2}{(1-e_1^2)^2} \\cr\\noalign{\\medskip} \n&+& \\nu_{21} \\frac{2 (1-e_1^2) + \\frac{5}{2} (e_1^2 - \\sin^2 I) (1 - \\cos 2 \\omega_1)}{(1-e_1^2)^{1\/2} (1-e_2^2)^{3\/2}} \\cr\\noalign{\\medskip} \n&+& \\nu_{22} \\frac{(1 + \\frac{3}{2} e_1^2 - \\frac{5}{2} e_1^2 \\cos 2 \\omega_1 ) \\cos I}{(1-e_2^2)^{2}} \\cr\\noalign{\\medskip} \n&-& \\nu_{31} \\frac{ {\\cal A} + \\frac{3}{2} e_1^2 - \\frac{5}{4} (5 -7 \\cos 2 \\omega_1 )e_1^2 \\sin^2 I}{e_1 (1-e_1^2)^{-1\/2} (1-e_2^2)^{5\/2}} e_2 \\cos \\varphi \\cr\\noalign{\\medskip} \n&-& \\nu_{31} \\frac{ \\frac{5}{2} {\\cal B} e_1^2 \\cos^2 I}{e_1 (1-e_1^2)^{1\/2} (1-e_2^2)^{5\/2}} e_2 \\cos \\varphi \\cr\\noalign{\\medskip}\n&+& \\nu_{31} \\frac{{\\cal A} e_1^2 + \\frac{5}{2} (1-e_1^2) (2 e_1^2 - \\sin^2 I)}{e_1 (1-e_1^2)^{1\/2} (1-e_2^2)^{5\/2}} e_2 \\cos I \\sin \\omega_1 \\sin \\omega_2 \\cr\\noalign{\\medskip}\n&-& \\nu_{32} \\frac{\\frac{5}{2} {\\cal B} \\cos I}{(1-e_2^2)^{3}} e_1 e_2 \\cos \\varphi \\cr\\noalign{\\medskip} \n&+& \\nu_{32} \\frac{{\\cal A} - \\frac{5}{2} (1-e_1^2) (1 - 3 \\cos^2 I)}{(1-e_2^2)^{3}} e_1 e_2 \\sin \\omega_1 \\sin \\omega_2\n \\ , \\llabel{110819a1}\n\\end{eqnarray}\n\\begin{eqnarray}\n\\dot \\omega_2 &=& \\nu_{21} \\frac{(1 + \\frac{3}{2} e_1^2 - \\frac{5}{2} e_1^2 \\cos 2 \\omega_1) \\cos I}{(1-e_1^2)^{1\/2} (1-e_2^2)^{3\/2}} \\cr\\noalign{\\medskip} \n&+& \\nu_{22} \\frac{1 + \\frac{3}{2} e_1^2 + (1-\\frac{5}{2} \\sin^2 I) (1 + \\frac{3}{2} e_1^2 - \\frac{5}{2} e_1^2 \\cos 2 \\omega_1)}{(1-e_2^2)^{2}} \\cr\\noalign{\\medskip} \n&-& \\nu_{31} \\frac{ \\frac{5}{2} {\\cal B} \\cos I}{(1-e_1^2)^{1\/2}(1-e_2^2)^{5\/2}} e_1 e_2 \\cos \\varphi \\cr\\noalign{\\medskip}\n&+& \\nu_{31} \\frac{ {\\cal A} - \\frac{5}{2} (1-e_1^2) (1 - 3 \\cos^2 I)}{(1-e_1^2)^{1\/2}(1-e_2^2)^{5\/2}} e_1 e_2 \\sin \\omega_1 \\sin \\omega_2 \\cr\\noalign{\\medskip}\n&-& \\nu_{32} \\frac{ {\\cal A} (1+4e_2^2) + \\frac{5}{2} {\\cal B} e_2^2 \\cos^2 I}{e_2 (1-e_2^2)^{3}} e_1 \\cos \\varphi \\cr\\noalign{\\medskip}\n&+& \\nu_{32} \\frac{{\\cal A} e_2^2 + 5 (1-e_1^2) (e_2^2 - \\frac{1}{2}(1+ 7e_2^2) \\sin^2 I)}{e_2 (1-e_2^2)^{3}} \\quad \\times \\cr\\noalign{\\medskip}\n&& \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad \\quad\n e_1 \\cos I \\sin \\omega_1 \\sin \\omega_2 \n\\ , \\llabel{110819a2} \n\\end{eqnarray}\nwhere $ x_{} = \\omega_{} \/ n_1 $, the constant frequencies\n\\begin{equation}\n\\nu_0 = 3 n_1 \\pfe{n_1 a_1}{c}{2} \\ , \\llabel{110817f}\n\\end{equation}\n\\begin{equation}\n\\nu_1 = n_1 \\frac{k_2}{2} \\frac{m_0 + m_1}{m_1} \\pfe{R_{}}{a_1}{5} \\ , \\llabel{110817a}\n\\end{equation}\n\\begin{equation}\n\\nu_{21} = n_1 \\frac{3}{4} \\frac{m_2}{m_0 + m_1} \\pfe{a_1}{a_2}{3} \\ , \\llabel{110817b}\n\\end{equation}\n\\begin{equation}\n\\nu_{22} = n_2 \\frac{3}{4} \\frac{m_0 m_1}{(m_0 +m_1)^2} \\pfe{a_1}{a_2}{2} \\ , \\llabel{110817c}\n\\end{equation}\n\\begin{equation}\n\\nu_{31} = n_1 \\frac{15}{16} \\frac{m_2}{m_0+m_1} \\frac{m_0-m_1}{m_0+m_1} \\pfe{a_1}{a_2}{4} \\ , \\llabel{110817d}\n\\end{equation}\n\\begin{equation}\n\\nu_{32} = n_2 \\frac{15}{16} \\frac{m_0 m_1}{(m_0+m_1)^2} \\frac{m_0-m_1}{m_0+m_1} \\pfe{a_1}{a_2}{3} \\ , \\llabel{110817e}\n\\end{equation}\nand\n\\begin{equation}\n\\sin \\varphi_1 = - \\frac{\\partial (\\cos \\varphi)}{\\partial \\omega_1} = - \\sin \\omega_1 \\cos \\omega_2 + \\cos \\omega_1 \\sin \\omega_2 \\cos I \\llabel{120329a1} \\ ,\n\\end{equation}\n\\begin{equation}\n\\sin \\varphi_2 = \\frac{\\partial (\\cos \\varphi)}{\\partial \\omega_2} = \\cos \\omega_1 \\sin \\omega_2 - \\sin \\omega_1 \\cos \\omega_2 \\cos I \\llabel{120329a2} \\ .\n\\end{equation}\nWhen $ I = 0^\\circ $, we have $ \\varphi = \\varphi_1 = \\varphi_2 = \\omega_2 - \\omega_1 $.\nNote also that the longitude of the node does not appear in the equations of motion (Eqs.\\,\\ref{110816h}$-$\\ref{110819a2}) because we used the invariable plane as the reference plane (Eq.\\,\\ref{090514a}), and thus $\\Delta \\Omega = 180^\\circ$.\n\n\n\\begin{table*}\n\\begin{center}\n\\caption{Hierarchical two-planet non-resonant systems with $0.1 < a_1 < 0.5 $~AU and \n$ a_1 \/ a_2 < 0.1 $.\n \\llabel{table1}}\n\\begin{tabular}{lcccccccccccc}\n\\hline\n\\hline\nStar & Age& $m_0$ & $m_1$ & $m_2$ & $a_1$ & $e_1$ & $a_2$ & $e_2$ & $\\varphi^* $ & $R_{} $ & $I_\\mathrm{max}$ & Ref. \\\\\n(name) & (Gyr) & ($M_\\odot$) & ($M_{J}$) & ($M_{J}$) & (AU) & & (AU) & & (deg) & ($R_\\mathrm{Jup}$) & (deg) & \\\\\n\\hline\nHD\\,190360 & 7.8 & 0.96 & .057 & 1.50 & 0.128 & 0.01 & 3.92 & 0.36 & 219. & 0.54 & $\\sim 40^\\star$ & [1] \\\\\nHD\\,38529 & 3.3 & 1.48 & 1.13 & 17.6 & 0.131 & 0.25 & 3.70 & 0.36 & 269. & 1.13 & $\\sim 40^\\star$ & [2] \\\\\nHD\\,11964 & 9.6 & 1.12 & .079 & 0.62 & 0.229 & 0.30 & 3.16 & 0.04 & 307. & 0.60 & $\\sim 40$ & [3,4] \\\\\nHD\\,147018 & 6.4 & 0.93 & 2.12 & 6.56 & 0.239 & 0.47 & 1.92 & 0.13 & 251. & 1.25 & $\\sim 40$ & [5] \\\\\nHD\\,168443 & 9.8 & 0.99 & 7.66 & 17.2 & 0.293 & 0.53 & 2.84 & 0.21 & 252. & 1.51 & $\\sim 50$ & [6] \\\\\nHD\\,74156 & 3.7 & 1.24 & 1.88 & 8.03 & 0.294 & 0.64 & 3.40 & 0.43 & 266. & 1.23 & $\\sim 20$ & [7,8,9] \\\\\nHD\\,163607 & 8.6 & 1.09 & 0.77 & 2.29 & 0.360 & 0.73 & 2.42 & 0.12 & 186. & 1.05 & $\\sim 30$ & [10] \\\\\n\\hline\n\\end{tabular}\n\\end{center}\nNotes: All masses $m_i$ correspond to minimum values ($I_i = 90^\\circ$), except for HD\\,38529, which has $I_i = 48^\\circ$; $\\varphi^* = \\omega_2^* -\\omega_1^* $; $R_{} $ was estimated using Eq.(\\ref{120529b}); The maximal inclination $I_\\mathrm{max}$ was estimated using $k_2 \\Delta t = 100 $\\,s, $(^\\star)$ for HD\\,190360 and HD\\,38529 starting with $a_1 = 0.2$~AU and $e_1 = 0.25$.\n\nReferences: \n[1] \\citet{Vogt_etal_2005}; \n[2] \\citet{Benedict_etal_2010}; \n[3] \\citet{Butler_etal_2006}; [4] \\citet{Wright_etal_2009}; \n[5] \\citet{Segransan_etal_2010}; \n[6] \\citet{Pilyavsky_etal_2011};\n[7] \\citet{Naef_etal_2004}; \n[8] \\citet{Bean_etal_2008};\n[9] \\citet{Meschiari_etal_2011}; \n[10] \\citet{Giguere_etal_2012}.\n\\end{table*}\n\n\n\\subsection{Tidal effects}\n\n\\llabel{TidalEffects}\n\nIn our model, we additionally considered tidal dissipation raised by the central star on the inner planet.\nThe dissipation of the mechanical energy of tides in the planet's interior is responsible for a time delay $\\Delta t_{}$ between the initial perturbation and the maximal deformation. \nBecause the rheology of planets is poorly known, the exact dependence of $\\Delta t_{}$ on the tidal frequency is unknown.\nSeveral models exist \\citep[for a review see][]{Correia_etal_2003,Efroimsky_Williams_2009}, but for simplicity we adopted a model with constant $\\Delta t_{}$, \nwhich can be made linear \\citep{Singer_1968,Mignard_1979}.\nThe contributions to the equations of motion are given by \\citep[e.g.][]{Correia_2009, Correia_etal_2011}\n\\begin{equation}\n\\frac{\\dot \\omega_{}}{n_1} = - K_{}\n\\left( f_1(e_1) x_{} - f_2(e_1) \\right) \\ , \\llabel{090515a}\n\\end{equation}\n\\begin{equation}\n\\frac{\\dot a_1}{a_1} = 2 K'_{} \\,\n\\left( f_2(e_1) x - f_3(e_1) \\right) \\ , \\llabel{090515b}\n\\end{equation}\n\\begin{equation}\\dot e_1 = 9 K'_{} \\left( \\frac{11}{18} f_4(e_1) x - f_5(e_1) \\right) e_1 \\ ,\n\\llabel{090515c}\n\\end{equation}\n\\begin{equation}\n\\dot I = - \\frac{K'_{} }{2} \\frac{f_1(e_1)}{(1-e_1^2)^{1\/2}} x_{} \\sin \\varepsilon \\, = 0 \\ ,\n\\llabel{120529a}\n\\end{equation}\nwhere\n\\begin{equation}\nK_{} = n_1 \\frac{3 k_2}{\\xi Q} \\frac{m_0 \\beta_1}{m_1^2} \\pfe{R_{}}{a_1}{3} \\ ,\n\\end{equation}\n\\begin{equation}\nK'_{} = \\frac{K_{}}{1\/\\xi}\n\\frac{m_1}{\\beta_1} \\pfe{R_{}}{a_1}{2} \\llabel{090514m} \\ , \n\\end{equation}\n\\begin{equation} Q_{}^{-1} \\equiv n_1 \\Delta t_{} \\ , \\llabel{120704a} \\end{equation}\nand\n\\begin{equation} f_1(e) = \\frac{1 + 3e^2 + 3e^4\/8}{(1-e^2)^{9\/2}} \\ , \\end{equation} \n\\begin{equation} f_2(e) = \\frac{1 + 15e^2\/2 + 45e^4\/8 + 5e^6\/16}{(1-e^2)^{6}} \\ , \\end{equation}\n\\begin{equation} f_3(e) = \\frac{1 + 31e^2\/2 + 255e^4\/8 + 185e^6\/16 + 25e^8\/64}{(1-e^2)^{15\/2}} \\ , \\end{equation}\n\\begin{equation} f_4(e) = \\frac{1 + 3e^2\/2 + e^4\/8}{(1-e^2)^5} \\ , \\end{equation}\n\\begin{equation} f_5(e) = \\frac{1 + 15e^2\/4 + 15e^4\/8 + 5e^6\/64}{(1-e^2)^{13\/2}} \\ . \\end{equation}\n\nWe neglected the effect of tides over the argument of the periastron, as well as the flattening of the central star. Their effect is only to add a small supplementary frequency to $\\dot \\omega_1$, similar to the contributions from the general relativity\n\\citep[for a complete model see][]{Correia_etal_2011}.\nExpression (\\ref{120529a}) for the inclination is zero, because we assumed the obliquity of the planet to be zero ($\\varepsilon = 0^\\circ)$.\n\nUnder the effect of tides alone,\nthe equilibrium rotation rate, obtained when $ \\dot \\omega_{} = 0 $, is attained for (Eq.\\,\\ref{090515a})\n\\begin{equation}\n\\frac{\\omega_{}}{n_1} = f(e_1) = \\frac{f_2(e_1)}{f_1(e_1)} = 1 + 6 e_1^2 + {\\cal O}(e_1^4)\n \\ . \\llabel{090520a}\n\\end{equation}\nUsually, $ K'_{} \\ll K_{} $, so tidal effects modify the rotation rate much faster than the orbit. It is thus tempting to replace the equilibrium rotation in expressions (\\ref{090515b}) and (\\ref{090515c}).\nWith this simplification, one always obtains negative contributions for $ \\dot a_1 $ and $ \\dot e_1 $ \\citep{Correia_2009}, \n\\begin{equation}\n\\frac{\\dot a_1}{a_1} = - 7 K_{}' \\, f_6(e_1) e_1^2 \\, < 0\n\\ , \\llabel{090522a}\n\\end{equation}\n\\begin{equation}\n\\dot e_1 = - \\frac{7}{2} K_{}' f_6(e_1) (1-e_1^2) e_1 \\, < 0 \n\\ , \\llabel{090522b}\n\\end{equation}\nwith \n\\begin{equation}\nf_6 (e) = \\frac{1 + \\frac{45}{14} e^2 + 8e^4 + \\frac{685}{224} e^6 + \\frac{255}{448} e^8 + \\frac{25}{1792}e^{10}}{(1-e^2)^{15\/2} (1 + 3e^2 + 3e^4\/8)} \\ . \\llabel{090527a}\n\\end{equation}\nThus, the semi-major axis and the eccentricity can only decrease until the orbit of the planet becomes circular.\nHowever, planet-planet interactions can produce eccentricity\noscillations with a period shorter than, or comparable to, the damping timescale of the\nspin. In that case, expression (\\ref{090520a}) is not satisfied\nand multi-planetary systems may show non-intuitive eccentricity evolutions,\nsuch as eccentricity pumping of the inner orbit \\citep{Correia_etal_2012}.\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=8.5cm]{fig1.pdf}\n\\caption{Radius versus the mass of the planet. We plot all known close-in planets in the range $ 0.1 < a_1 < 0.5 $~AU, for which the radius was determined by the transiting method. We observe that the radius decreases with the mass in a relatively regular way. \\llabel{fig1}}\n\\end{figure}\n\n\\begin{figure*}[ht!]\n\\includegraphics[width=18cm]{fig2.pdf}\n\\caption{Possible secular trajectories for the HD\\,74156 system seen in the $(\\omega_1,I)$ plane ({\\it left}), and in the $(\\omega_1, e_1)$ plane ({\\it right}). We show the trajectories using the quadrupolar approximation ({\\it top}), corresponding to the level curves of constant energy, and the octupolar approximation ({\\it bottom}). The dashed black curves in the $(\\omega_1,I)$ plane correspond to the separatrix between the circulation and libration zones of $ \\omega_1$. The colors are preserved in all pictures, each one corresponding to a given value of the total angular momentum of the system (determined by different values of the initial mutual inclination). All trajectories are compatible to the present knowledge for this system (Table\\,\\ref{tabD}). \\llabel{fig2}}\n\\end{figure*}\n\n\\begin{table}\n\\begin{center}\n\\caption{Stability analysis of the HD\\,74156 system for different sets ($\\omega_1, I$) of initial conditions (Fig. \\ref{fig2}).\nChaotic diffusion is present whenever $D > 10^{-6}$ (in bold). \n \\llabel{tabD}}\n\\begin{tabular}{lccrr}\n\\hline\n\\hline\ntrajectory & $\\omega_1$ & $I_0$ & $2 \\pi\/g$ & $\\log D$ \\\\\n(color) & (deg) & (deg) & (kyr) & \\\\\n\\hline\n\\multicolumn{5}{c}{\\it circulation} \\\\\n\\hline\nred & 90 & 10 & 13.657 & -9.41 \\\\\nblue & 90 & 20 & 15.065 & -9.12 \\\\\ngreen & 90 & 30 & 18.554 & -7.66 \\\\\nbrown & 90 & 38 & 20.877 & {\\bf -4.78} \\\\\ngray & 30 & 60 & 16.992 & {\\bf -5.90} \\\\\norange & 30 & 70 & 17.773 & {\\bf -5.33} \\\\\npurple & 30 & 80 & 20.289 & {\\bf -1.60} \\\\\n\\hline\n\\multicolumn{5}{c}{\\it libration} \\\\\n\\hline\npink & 90 & 42 & 15.482 & -9.08 \\\\\nmangenta & 90 & 50 & 9.918 & -7.20 \\\\\ncyan & 90 & 60 & 7.703 & -8.50 \\\\\nyellow & 90 & 70 & 6.804 & -9.99 \\\\\nblack & 90 & 83 & 6.330 & -7.60 \\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\begin{figure}[ht!]\n\\centering\n\\includegraphics[width=8.5cm]{fig3.pdf}\n\\caption{Possible secular trajectories for the HD\\,74156 system seen in the $(\\omega_1, e_1)$ plane for two different values of the initial mutual inclination, $I_0 = 40^\\circ$ ({\\it top}), and $I_0 = 50^\\circ$ ({\\it bottom}). We show the trajectories using the quadrupolar approximation, corresponding to the level curves of constant energy. Each one corresponds to an initial value of the argument of the periastron $ \\omega_1$, ranging from $0^\\circ$ ({\\it black}) to $90^\\circ$ ({\\it red}) with a step of $10^\\circ$. The dashed black curves corresponds to the observed eccentricity of the planet, that is, it gives the initial condition for $ \\omega_1$. All trajectories are compatible with the current knowledge for this system (Table\\,\\ref{table1}). \\llabel{fig3}}\n\\end{figure}\n\n\\begin{figure}[ht!]\n\\includegraphics[width=8.5cm]{fig4.pdf}\n\\caption{Possible secular trajectories for the HD\\,74156 system seen in the $(\\varphi, e_1)$ plane for $I=0^\\circ$ ({\\it top}) and in the $(\\omega_1, e_1)$ plane for $I=35^\\circ$ ({\\it bottom}).\nAll trajectories are compatible with the current knowledge for this system (Table\\,\\ref{table1}).\nWe show the trajectories using the octupolar approximation (blue) and direct numerical simulations (red). In (a) the blue path corresponds to the level curves of constant energy for coplanar orbits. The dot marks the present position of the planet. In (b) we additionally show the trajectories using the quadrupolar approximation (green), which corresponds to the level curves of constant energy. \\llabel{fig4}}\n\\end{figure}\n\n\n\\section{Application to exoplanets}\n\n\nIn the following sections we apply the model described in Section~\\ref{TheModel} to different configurations of hierarchical two-planet systems.\nTo observe the damping effect of the mutual inclination, the spin of the planet must be fully damped, but not its orbit, that is, $ K'_{} \\ll K_{} $ (see Appendix~\\ref{InclinationDamping}).\nIn addition, the damping timescale of the spin should be of the same order as the period of the eccentricity oscillations, $ K_{} \\sim g $ (Eq.\\,\\ref{110902b}).\nThis is valid for gaseous planets roughly within $0.1 < a_1 < 0.5 $~AU, which we call ``moderate close-in planets''.\nIn Table~\\ref{table1} we list all hierarchical systems known to date whose inner orbit satisfies the above condition.\nWe focus our analysis on the HD\\,74156 system, but all the main results are easily extended to the remaining planetary systems.\n\n\n\\subsection{Radius of close-in exoplanets}\n\nAccording to expression (\\ref{110902b}), the $\\nu_1$ frequency (Eq.\\,\\ref{110817a}) is a key parameter for the observation of the eccentricity pumping of the inner orbit and consequent damping of the inclination (by means of $ g_x $).\nThe minimum masses and the semi-major axis are relatively well determined from the observations, so the largest incertitudes in $\\nu_1$ come from the Love number $k_2$, and particularly from the radius of the planet, which appears as a power of 5.\nTherefore, a correct estimate of the planetary radius is necessary to observe some effect on the inclination.\n\nSince the radius of the planet is correlated with its mass, one solution is to adopt a constant value for the density, $\\rho_{}$, and then compute the radius simply as $R_{}^3 = 3 m_1 \/ (4 \\pi \\rho_{})$.\nHowever, by applying this strategy to the two largest planets of the solar system, Jupiter and Saturn, we immediately see that it can give very distinct results.\nThe density of a planet depends on many factors, such as the age of the system, the initial composition of the accretion disk, or where the planet formed in the disk.\nAny theoretical estimation of the radius is then subject to large incertitude, and only direct observations can give reliable values.\n\n\nWe used an empirical expression based on the previously observed radius of close-in planets in the range $ 0.1 < a_1 < 0.5 $~AU.\nWe found ten planets in this range\\footnotemark[\\value{footnote}] whose radius were determined by the transiting method (Fig.\\,\\ref{fig1}).\nWe observe that the radius decreases with the mass in a relatively regular way, therefore we performed a linear regression of the observational data:\n\\begin{equation}\nR_{} \/ R_\\mathrm{Jup} = 0.46 \\, \\log_{} \\left( m_{} \/ M_\\mathrm{Jup} \\right) + 1.10\n\\ . \\llabel{120529b}\n\\end{equation}\nThis expression also agrees well with the solar system data, giving $1.1 \\, R_\\mathrm{Jup}$ for Jupiter and $ 1.0 \\, R_\\mathrm{Sat}$ for Saturn.\n\n\n\n\n\\subsection{Initial conditions uncertainty}\n\n\\llabel{uncertainty}\n\nAssuming that the observational values of the minimum masses, semi-major axis, and eccentricities of the planetary systems listed in Table~\\ref{table1} are relatively well determined, we can use them as a starting point to study these systems.\n\nA striking observation is that the eccentricity of at least one of the planets can be very high.\nBecause hierarchical systems exhibit high eccentricity values, the mutual inclinations can also be very high \\citep[e.g.][]{Chatterjee_etal_2008}.\nHowever, currently their mutual inclinations are unknown, not only because we are unable to determine the inclination with respect to the plane of the sky, $I_i$ (and hence the true masses), but also because we are unable to determine the longitude of the nodes in the plane of the sky, $\\Omega_i$:\n\\begin{equation}\n\\cos I = \\cos I_1 \\cos I_2 + \\sin I_1 \\sin I_2 \\cos (\\Omega_2 - \\Omega_1) \\ .\n\\llabel{120627a}\n\\end{equation}\nThe only partial exception is the system HD\\,38529, for which $I_2 \\approx 48^\\circ$, estimated using astrometric measurements from the {\\it Hubble Space Telescope} \\citep{Benedict_etal_2010}.\nThe planetary masses $m_1$ and $m_2$ given in Table~\\ref{table1} correspond to the minimum masses (assuming $I_1 = I_2 = 90^\\circ $), except for HD\\,38529, where the masses were estimated using $I_1 = I_2 = 48^\\circ $.\n\nAccording to expression (\\ref{120329d}), without knowing the mutual inclination of these systems it is impossible to determine the total angular momentum $H$, which is a constant of the motion.\nEven if we assume that the system is coplanar and prograde ($\\cos I = 1$), the total angular momentum is undetermined because the true planetary masses appearing in the expressions of $G_1$ and $G_2$ are also unknown (except for HD\\,38529).\nAs a consequence, the present dynamics of these systems can be considerably different, depending on the true $H$ value.\n\nMoreover, although the argument of the periastron of these planets is known in the plane of the sky (angle $\\varphi^* = \\omega_1^* - \\omega_2^*$ in Table\\,\\ref{table1}), the arguments of the periastron in the invariable plane of the system ($ \\omega_1$ and $ \\omega_2$) are also unknown because they depend on $I_i$ and $\\Omega_i$ \\citep[e.g.][]{Giuppone_etal_2012}:\n\\begin{equation}\n\\cos (\\omega_1 - \\omega_1^*) = \\frac{\\cos I_2 - \\cos I_1 \\cos I}{\\sin I_1 \\sin I} \\ ,\n\\llabel{120629a}\n\\end{equation}\nwhere $ \\omega_i^*$ is the argument of the periastron in the plane of the sky.\nFor $\\omega_2$ we have an identical relation, the only difference is that $\\omega_1$ is measured from the ascending node, while $\\omega_2$ is measured from the descending node.\n\nTherefore, apart from the semi-major axes, eccentricities and minimum masses, currently there are few constraints on the remaining orbital parameters of hierarchical systems.\nBecause we are only concerned with the tidal evolution of the mutual inclination, we adopted the masses listed in Table\\,\\ref{table1} for the numerical simulations.\nThe only free parameters are then the mutual inclination and the arguments of the periastron, whose uncertainty are related to the lack of knowledge of the longitude of the node in the plane of the sky $(\\Omega_2-\\Omega_1)$.\n\n\nUsing the quadrupolar approximation for gravitational interactions, the potential energy (Eq.\\,\\ref{090514a}) is independent of the $ \\omega_2$, and therefore $e_2$ is constant (Eq.\\,\\ref{110816d}).\nThe dynamics of the system is then fully described by the couples $(I, \\omega_1)$ or $(e_1, \\omega_1)$.\nAdopting the minimum masses (Table\\,\\ref{table1}), we show in Figure~\\ref{fig2}(a,b) some possibilities for the HD\\,74156 system for different mutual inclinations (corresponding to different $H$ values).\nFor nearly coplanar systems ($I < 20^\\circ $), only small variations are observed for $e_1$ and $I$.\nHowever, as the inclination increases, the dynamics of the system is considerably perturbed by the presence of Lidov-Kozai cycles \\citep{Lidov_1961,Lidov_1962,Kozai_1962}.\nIn this regime, we can observe significant exchanges between the inclination and the eccentricity of the inner orbit. \nIn some cases the eccentricity can reach values much higher than today, and thus enhance the tidal dissipation.\n\n\\begin{figure*}[ht!]\n\\includegraphics[width=18cm]{fig5.pdf}\n\\caption{Long-term evolution of the HD\\,74156 system for different tidal models with $I_0 = 40^\\circ$: only dissipation in the orbit is considered (a), only dissipation in the spin is considered (b), full model (c). We plot the mutual inclination $I$ ({\\it top}), the eccentricities $e_1$ (blue) and $e_2$ (green) ({\\it middle}), and the semi-major axis $a_1$ ({\\it bottom}). \\llabel{fig5}}\n\\end{figure*}\n\n\\begin{figure*}[ht!]\n\\includegraphics[width=18cm]{fig6.pdf}\n\\caption{Long-term evolution of the HD\\,74156 system with $I_0 = 40^\\circ$ for different values of the argument of the periastron, $ \\omega_{1} = 30^\\circ$ (a), $50^\\circ$ (b), and $ 70^\\circ$ (c). We plot the mutual inclination $I$ ({\\it top}), the eccentricities $e_1$ (blue) and $e_2$ (green) ({\\it middle}), and the semi-major axis $a_1$ ({\\it bottom}). \\llabel{fig6}}\n\\end{figure*}\n\n\\begin{figure*}[ht!]\n\\includegraphics[width=18cm]{fig7.pdf}\n\\caption{Long-term evolution of the HD\\,74156 system with $I_0 = 50^\\circ$ for different values of the argument of the periastron, $ \\omega_{1} = 30^\\circ$ (a), $60^\\circ$ (b), and $ 90^\\circ$ (c). As a function of time, we plot the mutual inclination $I$ ({\\it top}) and the eccentricities $e_1$ (blue) and $e_2$ (green) ({\\it middle}). As a function of $ \\omega_1$, we plot again the mutual inclination, where the color of each dot becomes darker with time ({\\it bottom}). \\llabel{fig7}}\n\\end{figure*}\n\n\nFigure~\\ref{fig2}(a,b) is different from the standard Lidov-Kozai diagrams that show level curves of the quadrupole Hamiltonian at fixed values $G_1 \\cos I$, because for HD\\,74156 the initial eccentricity of the inner planet is already fixed at $e_1 = 0.64$ (Table\\,\\ref{table1}). \nInstead, we followed the procedure in \\citet{Giuppone_etal_2012} and show the trajectories for different values of the initial mutual inclination $I$, i.e., we varied the total angular momentum $H$. This explains why librating orbits in the Lidov-Kozai regime do not encircle a Lidov-Kozai equilibrium point (which occurs at the current eccentricity value). \n\nThe impact of the initial uncertainty on the $ \\omega_1$ value is shown in Figure~\\ref{fig3}. \nDepending on this value, the observed eccentricity can correspond to a maximum or to a minimum for an identical total angular momentum $H$ (Fig.\\,\\ref{fig3}a). \nMoreover, two trajectories may be in circulation or in libration (Fig.\\,\\ref{fig3}b).\nIt is therefore very important to completely explore the phase space of the initial conditions $(I, \\omega_1)$ to capture the global dynamics of hierarchical two-planet systems.\n\n\n\n\n\\subsection{Octupole contribution}\n\nSo far, we restricted our analysis to the quadrupolar gravitational interactions, because they are mainly responsible for the inclination variations.\nHowever, for the hierarchical systems listed in Table\\,\\ref{table1}, the range of semi-major ratios is $ 0.03 < a_1 \/ a_2 < 0.1 $, meaning that octupolar interactions cannot be neglected.\nIndeed, although the impact of octupolar terms on the eccentricity variations is weaker than that of the quadrupole terms, octupolar interactions are strong enough to produce secular drifts when combined with tidal effects \\citep{Correia_etal_2012}.\n\nIn the planar prograde case ($\\cos I = 1$), the potential energy (Eq.\\,\\ref{090514a}) only depends on $\\varphi = \\omega_2 - \\omega_1 $, and therefore $I$ is constant (Eq.\\,\\ref{120329e}).\nThe dynamics of the system is then fully described by the couples $(\\varphi, e_1)$ or $(\\varphi, e_2)$.\nAdopting the minimum masses, the angle $\\varphi^*$ in the plane of the sky listed in Table\\,\\ref{table1} corresponds to the angle $(\\omega_2 - \\omega_1 + 180^\\circ)$ in the invariant plane of the system, because for $I_1 = I_2 = 90^\\circ$ we have $ \\omega_1 = \\omega_1^* + 90^\\circ $ and $ \\omega_2 = \\omega_2^* - 90^\\circ $ (Eq.\\,\\ref{120629a}).\nIn Figure~\\ref{fig4}(a) we show the expected eccentricity variations for the \nHD\\,74156 system in this unique situation for which the system is fully characterized.\n\n\nWith increasing mutual inclination, we are left with four free parameters ($e_1, \\omega_1, e_2, \\omega_2$), and it becomes impossible to capture the dynamics of the systems in a two-dimensional plot.\nNevertheless, we can perform numerical simulations of the equations of motion (Eqs.\\,\\ref{110816h}$-$\\ref{110819a2}) to understand how the octupolar terms modify the quadrupolar approximation (Fig.\\,\\ref{fig2}).\nWe observe that the main effect is to add some diffusion around the quadrupolar trajectories.\nThe diffusion is more pronounced for orbits in circulation around the separatrix ($35^\\circ < I < 145^\\circ$).\nThe stability of the orbits can be measured with a frequency analysis \\citep{Laskar_1990,Laskar_1993PD}.\nWe determined the precession frequency $g$ and $g'$ of the argument of the pericenter $\\omega_1$ over two consecutive time intervals of length $ T = 5 $\\,Myr.\nIn Table\\,\\ref{tabD} we compute the difference $ D = | g - g' | \/ g $, which is a measure of the chaotic diffusion of the trajectories \\citep{Correia_etal_2005, Couetdic_etal_2010}.\nIt should be close to zero for a regular solution, and values with $D > 10^{-6}$ correspond to chaotic motion.\nThis is observed for all trajectories in circulation close to the separatrix.\n\n\n\nIn Figure~\\ref{fig4}(b) we plot simultaneously the eccentricity evolution obtained with the two approximations for $ I_0 = 35^\\circ $, and compare it with direct numerical simulations.\nWe conclude that 1) the quadrupolar approximation correctly describes the average dynamics in inclined hierarchical systems; 2) the octupolar approximation is essential to derive a more realistic behavior and obtain results similar to direct numerical simulations.\n\n\n\n\n\\section{Tidal evolution}\n\n\n\n\n\nWe now include the effect of tides described in section~\\ref{TidalEffects} to the conservative equations of the motion (Eqs.\\,\\ref{110816h}$-$\\ref{110819a2}), and perform some numerical simulations.\nIn all simulations we adopt for the innermost planet $ \\xi_{} = 1\/5 $, $ k_2 = 1\/2 $, and a dissipation time lag $ \\Delta t_{} = 200 $\\,s.\nFor HD\\,74156 this dissipation is equivalent to $ k_2 \/ Q_{} \\approx 1.4 \\times 10^{-4}$ (Eq.\\,\\ref{120704a}), which is comparable to the value $ k_2 \/ Q_{} = (2.3 \\pm 0.7) \\times 10^{-4}$ estimated for Saturn \\citep[][]{Lainey_etal_2012}.\nIn addition, we always set $ \\omega_2 = 180^\\circ $ and $ 2 \\pi \/ \\omega_{} = 50 $\\,day. \nThe impact of the initial $ \\omega_2$ value can be obtained by adjusting a different value for $ \\omega_1$.\nSimilarly, the initial rotation rate is not a critical initial parameter, since tidal effects quickly bring the rotation near to the equilibrium value (Eq.\\,\\ref{090520a}). \n\n\n\n\n\\subsection{Effect of the spin}\n\nIn Figure~\\ref{fig5} we show some examples for the evolution of the HD\\,74156 planetary system, in three different situations.\nThe radius of the inner planet is estimated to be $ R_{} = 1.23 \\, R_{Jup} $ (Eq.\\,\\ref{120529b}), and we initially assume $ I_0 = 40^\\circ $ and $ \\omega_1 = 0^\\circ $.\n\n\nIn a first experiment, we only consider tidal effects on the orbit (Eqs.\\,\\ref{090522a},\\,\\ref{090522b}), as it is often done in previous studies.\nThat is, since the rotation of close-in planets evolves very fast, we assume that the spin is locked in its equilibrium position (Eq.\\,\\ref{090520a}).\nWe observe that the eccentricity and the semi-major axis of the inner orbit slowly decrease, while the mutual inclination and the eccentricity of the outer orbit only oscillate around a constant mean value (Fig.\\,\\ref{fig5}a).\nTherefore, we conclude that in this case the only effect of tides is to circularize the inner orbit in a timescale longer than the age of the system, a well-know result in the literature \\citep[e.g.][]{Correia_Laskar_2010B}.\n\nIn a second experiment, we neglect the effects on the orbit and we only consider the effect on the rotation (Eq.\\,\\ref{090515a}). \nThis situation corresponds to the opposite of the previous one, and it is not realistic, but it allows to highlight the importance of not neglecting the rotation rate evolution.\nIndeed, although there is no direct dissipative contribution to the eccentricity or to the inclination, we observe that these two parameters undergo significant variations (Fig.\\,\\ref{fig5}b), the eccentricity of the inner planet rising almost up to 1.\nThe pumping effect on the eccentricity due to the spin excitation was reported in the planar case by \\citet{Correia_etal_2012}, for which only octupolar terms are important.\nIn the non-planar case, the pumping effect it is even more pronounced, since quadrupole order terms additionally contribute.\nIn addition, because the angular momentum is mainly exchanged between the inner planet eccentricity and the inclination, while the first increases, the second decreases.\nIn Appendix~\\ref{InclinationDamping} we provide the full explanation for this effect in the frame of the quadrupolar approximation.\nWe also observe that the eccentricity of the outer planet is slightly damped during this process, because of the octupole order interactions \\citep{Correia_etal_2012}.\n\nFinally, since orbital and spin evolution cannot be dissociated, we integrate the full set of equations for the tidal evolution (Eqs.\\,\\ref{090515a}$-$\\ref{120529a}) (Fig.\\,\\ref{fig5}c).\nWe observe that the initial behavior of the system is identical to the situation without dissipation on the orbit (Fig.\\,\\ref{fig5}b), with a significant damping of the mutual inclination.\nHowever, as the eccentricity increases, the inner planet comes closer to the star at periastron, and tidal effects on the orbit become stronger.\nAs a consequence, the semi-major axis decreases and the damping effect on the eccentricity (Eq.\\,\\ref{090515c}) overrides the pumping drift (Eq.\\,\\ref{110902b}).\nAt this point, the inclination damping is less efficient, and it ceases when the pumping drift is completely gone.\nThe system ultimately evolves into a circular orbit as usual, but in a considerable much shorter timescale and it is left with a final lower mutual inclination ($\\sim 15^\\circ$).\n\n\n\n\\subsection{Effect of the argument of periastron}\n\nIn Section~\\ref{uncertainty} we have seen that the initial choice of the argument of the periastron of the inner planet, $ \\omega_1$, plays an important role on how the present eccentricity is changing (Fig.\\,\\ref{fig3}).\nSimilarly, it also changes the initial trend of the inclination: for increasing eccentricity $e_1$ the inclination decreases, and vice-versa.\nIn the example from previous section (Fig.\\,\\ref{fig5}c), we used $ \\omega_1 = 0^\\circ$, that is, we assumed that the observed value of the eccentricity ($e_1 = 0.64$) is a minimum, and the inclination $I = 40^\\circ$ a maximum (Fig.\\,\\ref{fig3}a).\n\nIn order to test the impact of the initial argument of the periastron in the HD\\,74156 system, in Figure~\\ref{fig6} we plot its evolution for different initial $ \\omega_1$ values.\nWe observe similar behavior as before for all situations, the only significant difference being the evolution timescale.\nUntil $ \\omega_1 < 50^\\circ $ this timescale increases, because the present eccentricity is no longer a minimum value.\nAs a consequence, the average value of the eccentricity oscillations is shifted down, and tidal dissipation is less effective, since at the periastron the inner planet is farther from the star.\n\n\nIn the quadrupolar approximation, the behavior described above should be maintained up to $ \\omega_1 < 90^\\circ $, for which the observed eccentricity is a maximum and the inclination a minimum (Fig.\\,\\ref{fig3}a).\nHowever, the fact that the initial inclination increases when the eccentricity decreases has a strong implication when including octupolar terms (Fig.\\,\\ref{fig2}): for high inclinations the trajectories are closer to the separatrix, which results in a higher oscillation of the eccentricity.\nThus, for $ \\omega_1 > 60^\\circ$ we observe that the evolution timescale is reduced again, since the inner orbit eccentricity is allowed to reach much higher values than those predicted by the quadrupolar approximation.\n\n\n\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{fig6bis.pdf}\n\\caption{Circularization time ($e_1 < 0.01$) for the HD\\,74156 system with $I_0 = 40^\\circ $ (red) and $I_0 = 50^\\circ $ (blue) using different values of the initial argument of the pericenter. \nFor $I_0 = 40^\\circ $ the timescale decreases around the Lidov-Kozai equilibria, because the system is the circulation regime.\nFor $I_0 = 50^\\circ $, the system is in libration, but it is most likely destroyed after it crosses the separatrix.\n \\llabel{fig6bis}}\n\\end{figure}\n\n\n\nUp to now, we have been considering an initial mutual inclination $I_0 = 40^\\circ$.\nFor $I_0\\la 40^\\circ$, there is only one dynamical regime for the HD\\,74256 planetary system, consisting of trajectories in circulation around the Lidov-Kozai equilibira (Fig.\\,\\ref{fig3}a).\nHowever, for higher values of the initial inclination we can also observe the libration regime (Fig.\\,\\ref{fig3}b). \nIn Figure~\\ref{fig7} we show some numerical simulations with initial $I_0 = 50^\\circ$ using different values for the argument of the periastron.\n\nFor $ \\omega_1 = 30^\\circ$ the inner planet is still in circulation (Fig.\\,\\ref{fig7}a), so we observe identical behavior for the eccentricity and inclination as in the case with $I_0 = 40^\\circ$ (Fig.\\,\\ref{fig6}).\nAt the bottom of Figure~\\ref{fig7} we plot the inclination as a function of the argument of the periastron ($I, \\omega_1$).\nWe plot a dot each $10^5$\\,yr and its color becomes darker with time.\nThere we can clearly see that the planet is always in circulation, and that the amplitude of the oscillations is damped with time.\n\nIn the remaining two situations shown in Figure~\\ref{fig7}, the inner planet is in libration around the Lidov-Kozai equilibrium located at $ \\omega_1 = 90^\\circ$.\nThe effect of tides is to slowly increase the amplitude of both the eccentricity and inclination.\nAs a consequence, the orbit of the planet will cross the separatrix of the libration zone and start to circulate as in the previous case.\n\nFor $ \\omega_1 = 60^\\circ$ (Fig.\\,\\ref{fig7}b) the planet already starts close to the separatrix, so the initial oscillations are higher and it takes only about 75\\,Myr to cross it.\nFor $ \\omega_1 = 90^\\circ$ (Fig.\\,\\ref{fig7}c) the planet is placed close to the Lidov-Kozai equilibrium, so it takes much longer to reach the separatrix.\nIn both situations, just after the transition of dynamical regime, the eccentricity reaches a very high value close to unity.\nTherefore, tidal effects with the central star become very strong and the final evolution is rapid: the semi-major axis decreases, and the inner orbit becomes circular.\nHowever, in a more realistic simulation where we integrate the full equations of motion and take into account the bodies dimensions, the planet most likely collides with the star.\nIn both situations the system is either destroyed, or its configuration completely modified from the initial situation.\n\nIn order to better understand the variation of the evolution timescale with the initial choice of the argument of the pericenter, in Figure~\\ref{fig6bis} we plot the circularization time ($ e_1 < 0.01 $) as a function of $ \\omega_1$.\nThe circularization time is more or less equivalent to the inclination damping time, since the final stages of the evolution are very fast.\nFor $I_0 = 40^\\circ $ the timescale decreases around the Lidov-Kozai equilibria, because the system is the circulation regime.\nFor $I_0 = 50^\\circ $, the system is in libration, but it is most likely destroyed after it crosses the separatrix.\n\n\n\\citet{Giuppone_etal_2012} also studied the evolution of planets inside the circulation zone of Lidov-Kozai equilibriums.\nThey performed some numerical simulations using the quadrupolar approximation and damping of the inner orbit eccentricity due to the presence of a primordial disk.\nThey concluded that the planet stays in libration and migrates into the Lidov-Kozai equilibrium position, which is exactly the contrary that we observed here.\nSince the eccentricity of the inner orbit is also damped in our model, these results appear somehow contradictory.\nTherefore, we performed one simulation where the eccentricity is damped, but the semi-major axis is held constant.\nIn this unrealistic situation, one observe that the planet migrates into the equilibrium like in \\citet{Giuppone_etal_2012}.\nAs a consequence, it seems that there is no inconsistency between the two models, but it becomes clear that the semi-major axis evolution plays an important role in destabilizing the Lidov-Kozai equilibria.\nIt appears that it cannot be neglected in future studies on the migration of the initial orbits as in \\citet{Giuppone_etal_2012}.\n\n\n\\subsection{Constraints for the mutual inclination}\n\n\\llabel{constraints}\n\nIn previous sections, we saw that in mutually inclined hierarchical two-planet systems there is a significant increase in the eccentricity of the inner planet's orbit.\nAs a result, the tidal dissipation is enhanced when the planet is at the periastron, and the system evolves faster into an equilibrium configuration.\n\nIn Figure~\\ref{fig8} we show the evolution of the HD\\,74156 system for three different values of the initial inclination $I_0 = 15^\\circ$, $30^\\circ$ and $45^\\circ$.\nAs expected, when we increase the mutual inclination, the evolution timescale decreases.\nFor $ I_0 = 15^\\circ $ the eccentricity and the semi-major take more than 10~Gyr to be completely damped, while for $I_0 = 45^\\circ$ the system is fully evolved only after 100\\,Myr.\nMoreover, for $I_0 = 15^\\circ$ there is almost no effect on the mutual inclination, we only observe some amplitude damping when the eccentricity is decreased to low values, because gravitational perturbations no longer force the inclination.\nOn the contrary, for $I_0 = 30^\\circ$, the pumping effect on the eccentricity is already present, and hence we observe a significant reduction of the final mutual inclination.\n\n\\begin{figure*}[ht!]\n\\includegraphics[width=18cm]{fig8.pdf}\n\\caption{Long-term evolution of the HD\\,74156 system for different values of the initial inclination, $I_0 = 15^\\circ$ (a), $30^\\circ$ (b), and $ 45^\\circ$ (c). We plot the mutual inclination $I$ ({\\it top}), the eccentricities $e_1$ (blue) and $e_2$ (green) ({\\it middle}), and the semi-major axis $a_1$ ({\\it bottom}). \\llabel{fig8}}\n\\end{figure*}\n\nIn Figure~\\ref{fig9} we show the same kind of evolution as before, but for an initial retrograde orbit with initial inclinations $I_0 = 165^\\circ$, $150^\\circ$ and $135^\\circ$.\nIn this case, the evolution of the system does not differ much from the prograde situation, the only significant difference is that the inclination is damped to high values close to $180^\\circ$ (coplanar system with a retrograde orbit).\nFor $I_0 = 165^\\circ$ the inclination is more or less conserved and the eccentricity is damped over 10~Gyr, while for lower values of the initial inclination, the inclination is damped and the system evolves in much shorter timescales.\n\n\\begin{figure*}[ht!]\n\\includegraphics[width=18cm]{fig9.pdf}\n\\caption{Long-term evolution of the HD\\,74156 system for different values of the initial inclination, $I_0 = 165^\\circ$ (a), $150^\\circ$ (b), and $135^\\circ$ (c). We plot the mutual inclination $I$ ({\\it top}), the eccentricities $e_1$ (blue) and $e_2$ (green) ({\\it middle}), and the semi-major axis $a_1$ ({\\it bottom}). \\llabel{fig9}}\n\\end{figure*}\n\nFrom the observation of Figures~\\ref{fig8} and~\\ref{fig9} we then conclude that mutual inclinations closer to $90^\\circ$ speed up the final evolution of the system. \nSince most hierarchical systems listed in Table\\,\\ref{table1} (except HD\\,190360 and HD\\,38529) still present substantial values for the inner-orbit's eccentricity, we then expect that their mutual inclinations are not extremely high.\nFor all those systems we run several numerical simulations starting with the present initial conditions from Table\\,\\ref{table1}, adopting $ k_2 \\Delta t = 100 $\\,s, and different initial values for $I$ and $ \\omega_1$. \nAll trajectories that circularize the inner-orbit in less than $\\sim 10$\\,Gyr\ncan then be ruled out, while those not showing significant modifications can be retained as possible representations of the real system.\nTherefore, we are able to set some constraints for the maximal mutual inclination of each system, whose limitations are listed in Table\\,\\ref{table1}.\nIn Figure~\\ref{fig10} we show the outcome of these simulations for the HD\\,74156 system, which corresponds to a summary of the more detailed evolutions shown in previous Figures.\nOrbits with $20^\\circ < I < 150^\\circ$ circularize the system in less than 10~Gyr, so they can be discarded.\n\nWhen we run the same kind of simulations for the HD\\,38529, we observe that the eccentricity is damped very quickly, even for coplanar orbits.\nOne possibility is that we overestimated the dissipation. \nHowever, even if we adopt $ k_2 \\Delta t = 10 $\\,s, that is, one order of magnitude lower than for the remaining planets, the system still circularizes in a timescale shorter than the age of the system.\nAnother possibility is to suppose that\nthe inner planet semi-major axis was higher in the past.\nThis hypothesis can also be extended to the HD\\,190360 system, for which the inner orbit is already circularized, but it may have had a higher eccentricity value in the past.\nIn order to test this scenario,\nfor all planetary systems in Table\\,\\ref{table1}, we run several numerical simulations for different initial values for $I$ and $ \\omega_1$, keeping $ k_2 \\Delta t = 100 $\\,s, but adopting $a_1 = 0.2 $~AU and $ e_1 = 0.25 $ as initial values, instead of the current values.\nIn Figure~\\ref{fig12} we plot an example for the HD\\,190360 system.\nBy modifying the initial conditions, we are able to reproduce the present observations.\nNote that the HD\\,190360 system is older than the HD\\,38529 one (Table\\,\\ref{table1}), so both systems may have undergone an identical evolution, but they are observed at different stages. \nThe initial semi-major axis could have been higher, providing that the inner orbit eccentricity was also higher (for instance $a_1 = 0.25 $~AU and $ e_1 = 0.4 $).\n\n\\begin{figure}[ht!]\n\\includegraphics[width=8.5cm]{fig10.pdf}\n\\caption{Circularization time ($e_1 < 0.01$) for the HD\\,74156 system using different values of the initial mutual inclination and argument of the pericenter. Since the estimated age of the system is\nseveral Gyr (Table\\,\\ref{table1}), we can rule out mutual inclinations within $ 20^\\circ < I < 150^\\circ$. \\llabel{fig10}}\n\\end{figure}\n\n\n\n\\begin{figure}[ht!]\n\\includegraphics[width=8.5cm]{fig12.pdf}\n\\caption{Possible past evolution of the HD\\,190360 system with $I_0 = 43^\\circ$, $a_1 = 0.2 $~AU, and $e_1 = 0.25$. We plot the mutual inclination $I$ ({\\it top}), the eccentricities $e_1$ (blue) and $e_2$ (green) ({\\it middle}), and the semi-major axis $a_1$ ({\\it bottom}). \\llabel{fig12}}\n\\end{figure}\n\n\n\n\n\\subsection{Effect of the dissipation rate}\n\n\\begin{figure}[ht!]\n\\includegraphics[width=8.5cm]{fig13.pdf}\n\\caption{Possible past evolution of the HD\\,74156 system with $I_0 = 50^\\circ$, using different dissipation rates. We plot the mutual inclination $I$ for $k_2 \\Delta t = 100$\\,s ({\\it top}), $k_2 \\Delta t = 10$\\,s ({\\it middle}), and $k_2 \\Delta t = 1$\\,s ({\\it bottom}). \\llabel{figN}}\n\\end{figure}\n\nIn the former sections we have been using $k_2 \\Delta t = 100 $\\,s, or, in terms of $Q$-factor, $ k_2 \/ Q_{} \\approx 1.4 \\times 10^{-4}$ (Eq.\\,\\ref{120704a}), which is similar to the present value measured for Saturn $ k_2 \/ Q_{} = (2.3 \\pm 0.7) \\times 10^{-4}$ \\citep[][]{Lainey_etal_2012}.\nWe adopted this value mainly for a better comparison with the previous paper on the planar case \\citep{Correia_etal_2012}.\nOther works on the tidal evolution of hot-Jupiters also adopted similar values for $\\Delta t$ \\citep[e.g.][]{Fabrycky_Tremaine_2007, Correia_etal_2011}.\n\nHowever, the $Q$-factor of planets is unknown, and may vary by some orders of magnitude \\citep[e.g.][]{Goldreich_Soter_1966}.\nIndeed, the value measured for Jupiter appears to differ by a factor of ten, $ k_2 \/ Q_{} = (1.1 \\pm 0.2) \\times 10^{-5}$ \\citep[][]{Lainey_etal_2009}.\nOn the other hand, statistical studies on the observed eccentricity distribution of hot-Jupiters give $k_2 \/ Q_{} \\sim 10^{-6}$ \\citep[e.g.][]{Jackson_etal_2008a, Hansen_2010}.\nPart of the problem is that the nature of tidal dissipation in these planets is still poorly understood. \nIn addition, the $Q$-factor is frequency dependent (i.e., model dependent), and therefore sometimes it is difficult to translate from one system to another.\nIn order to test the robustness of the inclination damping, here we test our model with lower dissipation rates (higher $Q$ values).\n\nAccording to the tidal equations (Eqs.\\,\\ref{090515a}$-$\\ref{120529a}), the evolution timescales are linearly proportional to $Q_{}$, so higher $Q$-values delay the final evolution of a system.\nIn Figure~\\ref{figN} we plot the evolution of the HD\\,74156 system for three different dissipation values, $k_2 \\Delta t_{} = 1,\\, 10, \\, 100$\\,s (which is equivalent to $k_2\/Q \\approx 10^{-6}, \\, 10^{-5}, \\, 10^{-4}$, respectively), starting with $I_0 = 50^\\circ$.\nWe observe that, although the evolution timescale is longer for higher $Q_{}$ values (as expected), the inclination damping is still present.\n\nIn section~\\ref{constraints} we saw that when we increase the initial mutual inclination $I_0$, the evolution timescale decreases very fast.\nFor instance, in the case of the HD\\,74156 system, for $I_0 = 40^\\circ$ the inner orbit becomes fully damped after 1\\,Gyr, for $I_0 = 50^\\circ$ it takes about 100\\,Myr, and for $I_0 = 60^\\circ$ only 10\\,Myr.\nTherefore, for different dissipation rates we can still put constraints on the mutual inclination of hierarchical systems, the only consequence is that as we increase $Q_{}$, the maximal mutual inclination that one can expect to observe also increases.\n\n\n\n\n\\section{Conclusion}\n\n\nMany two-planet systems have been reported in hierarchical configurations.\nFor most of these systems the mutual inclinations are unknown. However, since the orbital eccentricities are typically high, we may expect that the formation mechanism that increased the eccentricities also increased the mutual inclinations.\nVery often the innermost planet in these systems is close enough to the star to undergo tidal dissipation, which can pose constraints on the final evolution.\n\nHere we have studied a particular subgroup of hierarchical planetary systems in which the inner planet's semi-major axis $0.1 < a_1 < 0.5$~AU (we called these ``moderate close-in planets'').\nThis range is important to ensure that the spin of the inner planet is fully evolved, but not its orbit.\nUsing an averaged secular model that takes into account gravitational interactions\nup to octupole order, we showed that for many initial conditions the mutual inclination is damped to relatively low values ($ I \\sim 15^\\circ $ for HD\\,74156) on timescales shorter than the age of the system (less than one Gyr).\n\nWithout planetary perturbations and for zero obliquity there is no effect from tides on the evolution of the inclination (Eq.\\,\\ref{120529a}).\nThe inclination damping is thus not a direct consequence of tidal effects on the orbits.\nThe key element is a inner planet's eccentricity oscillation at a secular timescale similar to the synchronization time of its spin.\nIndeed, the rotation (and thus the flattening (Eq.\\,\\ref{101220a})) of the planet is driven by its eccentricity variations (Eq.\\,\\ref{090515a}). \nIn response to these excitations, the rotation is phase-shifted (Eq.\\,\\ref{110920a}) and the lag \ntends to pump the eccentricity (Eq.\\,\\ref{110902b}). \nBecause the total angular momentum must be conserved, the increase in the inner orbit's eccentricity is accompanied by a subsequent reduction of the mutual inclination.\nWhen the eccentricity pumping ceases, the inclination damping also stops.\n\nFor high mutual inclination values, quadrupolar gravitational exchanges with the eccentricity are more efficient, and so is the eccentricity pumping.\nAs a consequence, the inner orbit's eccentricity reaches higher values, tidal effects are enhanced at the periastron, and the system evolves on shorter timescales.\nA strong inclination damping is then often followed by a fast circularization of the inner orbit.\nSince most of the observed hierarchical systems still present substantial values of the inner-orbit's eccentricity after several Gyr, we expect that they cannot have very high mutual inclinations.\nIn particular, we are able to set constraints on the highest mutual inclination in these systems.\n\nThe evolution timescale also depends on the argument of the periastron in the invariant plane of the system, which is unknown at present for most systems.\nIndeed, the uncertainty on this parameter is much higher than the uncertainty on the dissipation time lag $\\Delta t$.\nIf $ \\omega_1 $ is in circulation, the evolution can be extremely fast (a few million years) for values close to $ \\omega_1 = 90^\\circ $ or $ \\omega_1 = 270^\\circ $.\nHowever, for high mutual inclinations $ \\omega_1 $ can be in Lidoz-Kozai libration for values close to $ \\omega_1 = 90^\\circ $ or $ \\omega_1 = 270^\\circ $.\nIn that case, the evolution timescale can be delayed to several Gyr, but the equilibrium is unstable and broken when the system crosses the separatrix with the circulation regime.\nAfter that, the inner planet is most likely lost.\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nTully-Fisher (TF) surveys at cosmological distances provide a direct\nway to track the evolution in fundamental scaling relations of disk\ngalaxies. Such surveys permit, for example, the measurement of the\ndisk size or luminosity evolution under the assumption that deviations\nfrom fiducial TF relations can be understood as photometric changes in\nconstant-mass potentials. Deviations from TF may also be produced by\ndynamical asymmetries due to instabilities, interactions, minor\nmergers, or slow and possibly lumpy mass-accretion. Semi-analytic\nmodels of galaxy formation are able to reproduce the basic slope and\nscatter in the TF relationship at $z=0$, yet most fail to recover the\ncorrect zero-point (e.g. Steinmetz \\& Navarro, 1999). We anticipate\nthat in the future, however, the scatter in the TF relation will be a\nmore useful diagnostic of how disks are assembled. The scatter about\nthe TF relation should constrain the modes by which disks form,\nparticularly if the nature of this scatter can be determined as a\nfunction of redshift. For example, are offsets from a fiducial TF\nrelation accompanied by an increase in disk asymmetry?\n\n\\section{Towards a Tully-Fisher Relation for Face-on Galaxies}\n\n\\begin{figure}\n\\plotfiddle{bershadym_fig1.eps}{2.25in}{-90}{60}{60}{-230}{280}\n\\vskip 0.15in\n\\caption{\\hsize 5.25in WIYN\/Densepak H$\\alpha$ velocity-field for PGC 46767 ({\\bf\nleft}), and the residuals from a single, inclined, rotating disk model\n({\\bf right}). The kinematic major and minor axes are drawn for\nreference. Note the {\\it low} amplitude, yet coherent structure of\nthe velocity residuals. Also note the large projected velocity for a\n(photometrically) apparently face-on disk.}\n\\vskip -0.15in\n\\end{figure}\n\nHistorically, TF surveys have targeted galaxies with photometric\ninclinations above 40-45$\\deg$ in order to minimize errors in the\ncorrected rotation velocities. HI kinematic estimates of inclination\nalso have been difficult to measure below $i\\sim40\\deg$ given the\nprecision of HI maps in the past, as well as the intrinsic problems\nassociated with flaring and warping of the outer parts of HI disks\n(e.g. Begeman, 1989). We have recently acquired integral-field echelle\nspectroscopy with the WIYN telescope's Densepak fiber-bundle of over a\ndozen, nearby disk galaxies ($0.020.7$ (where apparent size is a weak\nfunction of redshift). Only modest gains ($-1.25$ in the exponent) are\nmade from space.\n\nA zero-point for this telescope-diameter--time--redshift relation\ncan be set from the Keck observations of rotation curves at $z\\sim1$,\ntaken in several hours with a 1 arcsec wide slit (Vogt et al. 1997). A\nfactor of 3 higher spatial resolution would just resolve galaxies at\ncosmological distances at the equivalent of one disk scale-length for a\ntypical disk today. In the background limit, one would then require a\nthree times larger aperture at constant exposure time, i.e. a 30-m\ntelescope. Other gains can come from (i) improved instrument\nthroughput, and (ii) enhancing surface-brightness contrast by\nresolving bright, star-forming knots in distant galaxy disks via\nhigh-order adaptive optics (Koo, 1999). The latter is not applicable,\nhowever, for the study of stellar motions in disks, which are of\nparticular interest for understanding asymmetries in disk mass\ndistributions.\n\n\\begin{figure}\n\\plotfiddle{bershadym_fig3.eps}{1.85in}{-90}{60}{60}{-235}{270}\n\\vskip 0.1in\n\\caption{\\hsize 5.25in The velocity-field residuals ({\\bf left})\nbetween two rotating disks at inclinations of 10$\\deg$ and 20$\\deg$\nshows the maximum sensitivity to inclination lies at $\\theta\\sim\\pm45$\nfrom the kinematic major axis. For our characteristic instrumental\nprecision ($\\delta$V), observed residuals (V-V$_{model}$), and number\nof beams per galaxy (fibers, N), the {\\bf right} panel shows our\nestimated precision ($\\Delta$i) as a function of true inclination for\nderiving kinematic inclinations from WIYN\/Densepak H$\\alpha$ echelle\nspectroscopy for our nearby galaxy sample.}\n\\vskip -0.15in\n\\end{figure}\n\n\n\\acknowledgments\nWe are grateful to acknowledge our collaborators in these projects --\nL. S. Sparke, J. S. Gallagher, III, E. M. Wilcots, W. van Driel,\nC.J. Conselice, and D. Ragaigne; M. Haynes, R. Giovanelli, C. Mihos,\nand D. Koo. This research was supported by AST-96-18849 and\nAST-99-70780.\n\\vskip -0.3in\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}