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+{"text":"\\section{Introduction}\n\nThe nucleation of Abrikosov vortices \\cite{Abrikosov57} in the\nmixed state of type-II superconductors with periodic artificial\npinning centers attracted a great attention since 1970's. Recent\nprogress in the fabrication of nanostructures provides the\npossibility to realize superconducting thin films containing\nartificial defects as pinning sites with well-defined size,\ngeometry and spatial arrangement \\cite{Mosh98,Schuller00}. Vortex\npinning was extensively explored by many groups to develop a\nfundamental understanding of flux dynamics and for its relevance\nin applications which require enhancements of the critical current\ndensity. Thus, several types of artificial pinning centers, such\nas square, rectangular or triangular arrays, have been introduced\nin a controlled way in the superconducting films. In particular,\nthe use of regular array of pinning centers such as antidots\n\\cite{Mosh98,Fiory78,Lykov93,Castellanos97} or magnetic dots\n\\cite{Schuller00,Schuller97,Mosh99,Schuller08} brings to new\ncommensurability effects, which give additional insight into the\npinning properties of vortices. The most notable phenomenon for\nthese studies is the so-called matching effect which occurs when\nthe vortex lattice is commensurate with the periodic pinning\narray. This situation occurs, in particular, at fractional or\ninteger values of the so-called first matching field\n$H_1=\\Phi_0\/a_{0}^{2}$, i.e., when the applied field $H$\ncorresponds to one flux quantum, $\\Phi_0=h\/2e$, per unit cell\narea, $a_{0}^{2}$, of the pinning array. Here $a_{0}$ is the\nlattice constant of the pinning arrangement. As a result, at the\nmatching field, the critical current density, $J_c$, is\ndrastically enhanced \\cite{Schuller00,Fiory78,Silhanek05} and\nmoreover, as a consequence of the Little-Parks effect\n\\cite{Little-Parks}, the upper critical magnetic field is\nincreased at the matching values. Recently antidot arrangements\nwith a big variety of symmetries have been investigated. Matching\neffects have been reported in perforated Nb thin films for\nantidots lattices with short range order \\cite{Ziemann}, or\nquasiperiodic fivefold Penrose structures \\cite{Kemmler06}.\nMoreover asymmetric pinning arrays have been suggested as\nsuperconducting rectifiers \\cite{Mosh05}.\\\\\nIf the artificial structure of defects is created by lithographic\ntechnique, the matching fields are usually in the range of a few\noersteds. For this reason, matching effects are observed in a very\nnarrow temperature region, close to the critical temperature\n$T_c$, for a reduced value $t=T\/T_c\\geq0.95$. In order to both\nincrease the matching field and decrease the temperature where the\neffect is present, the period of the pinning structure should be\nreduced to less than 100 nm. This gives, in fact, the possibility\nto increase $H_1$ up to 1 tesla or even higher. A reasonable\nmethod to achieve this goal is to use self-assembled substrates,\nsuch as, for example, $Al_{2}O_{3}$ templates with characteristic\nfeatures in the nanometric scale \\cite{Mosh07}. The pore diameter\nin $Al_{2}O_{3}$ substrates could easily be varied in the range\n25-200 nm with porosity (i.e. interpore spacing) around 50\\%, and\nthis gives the possibility to achieve matching fields of thousands\nof oersteds \\cite{Mosh07}. To prepare $Al_{2}O_{3}$ substrates\nbulk Al \\cite{Prischepa-Cryogenics94}, Al foils \\cite{Welp-PRB},\nand deposited thick Al films were used \\cite{Mosh07}.\\\\\nVery recently, another very promising material for self-assembled\nsubstrates and an optimum candidate for the Nb growth was\nproposed, namely, porous silicon (PS) \\cite{Trezza-JAP08}. PS is\nconstituted by a network of pores immersed in a nanocrystalline\nmatrix \\cite{Pavesi} and it is a material which offers a\nconsiderable technological interest in different fields, as for\ninstance micro and optoelectronics \\cite{Collins} and gas sensing\n\\cite{Cheraga,Lysenko}. The diameter of pores, ${\\O}$, in PS can\neasily be varied from 200 nm down to 5 nm by using substrates with\nappropriate doping (n or p) and different regimes of anodization.\nThe porosity, in fact, can be varied in the range 30-90\\% by\nadjusting parameters such as the acid solution, the anodizing\ncurrent density and the illumination of the substrate during the\nanodization. The regularity of the pores arrangement, however, is\nof the order of 10\\% lower than the one observed in $Al_{2}O_{3}$\ntemplates obtained by electrochemical oxidation \\cite{Piraux}. It\nhas been demonstrated \\cite{Trezza-JAP08} that thin Nb films\ndeposited on PS substrates can inherit their structure. The\nresulting samples then consist of porous Nb thin films with in\nplane geometrical dimensions, $a_{0}$ and ${\\O}$, comparable with\nthe superconducting coherence length, $\\xi(T)$. In these samples,\nmatching fields of the order of 1 Tesla\nwere experimentally observed \\cite{Trezza-JAP08}.\\\\\nAim of this work is to deepen the study of the matching effect in\nsuperconducting Nb thin films deposited on PS. Superconducting\nproperties  were investigated by transport measurements in the\npresence of magnetic fields applied perpendicularly to the samples\nsurface, down to t = 0.52. As a consequence of the high density of\nthe pore network, the (H,T) phase diagram presents a deviation\nfrom the classic linear dependence. This effect appears at the\nmatching field $H_{1}\\approx$ 1 Tesla, a value larger than those\ntypical of periodic pinning arrays obtained both by lithographic\ntechniques and by using another kind of self-organized templates.\nMoreover a new effect related to the commensurability between the\nvortex lattice and the underlaying pinning structure was found. It\nconsists in the appearance of pronounced structures in the\nderivative of the $R(H)$ curves, $dR$\/$dH$, which can be observed\nin correspondence of the first matching field and its fractional\nvalues.\n\n\\section{Fabrication}\n\nPorous layers were fabricated by electrochemical anodic etching of\nn-type, 0.01 $\\Omega$cm, monocrystalline silicon wafers. The\nelectrochemical dissolution was performed in 48\\% water solution\nof HF, applying a current density of 20 mA\/cm$^{2}$. The\nanodization time was chosen in the range of 0.5 - 4 min in order\nto get porous layers with a thickness ranging from 0.5 to 4\n$\\mu$m. The pores extend on a surface of about 1 cm$^{2}$. The\nintegral porosity was estimated by gravimetry to be of about 50\\%\n\\cite{Lazarouk}. The resulting porous substrates have ${\\O}$=10 nm\nand $a_{0}$ = 40 nm. For this lattice, if the formula\n$H_1=\\Phi_0\/a_{0}^{2}$ for the square lattice is used, the\nexpected first matching field is $H_{1}$ = 1.3 Tesla.\\\\\nNb thin films were grown on top of the porous Si substrates in a\nUHV dc diode magnetron sputtering system with a base pressure in\nthe low $10^{-8}$ mbar regime and sputtering Argon pressure of\n$3.5\\times10^{-3}$ mbar. In order to reduce the possible\ncontamination of the porous templates, the substrates were heated\nat $120^{\\circ}$C for one hour in the UHV chamber. The deposition\nwas then realized at room temperature after the cool off of the\nsubstrates. Films were deposited at typical rates of 0.33 nm\/s,\ncontrolled by a quartz crystal monitor calibrated by low-angle\nreflectivity measurements. Since the effect of the periodic\ntemplate would be reduced when the film thickness, $d_{Nb}$,\nexceeds the pore diameter, ${\\O}$, \\cite{Trezza-JAP08} the Nb\nthickness was chosen to be 8.5 nm for the sample analyzed in this\npaper. A reference Nb thin film of the same thickness was grown on\na non-porous Si substrate in the same deposition run.\n\n\\section{Experimental results and discussion}\n\nThe superconducting properties were resistively measured in a\n$^{4}$He cryostat using a standard dc four-probe technique on\nunstructured samples. The critical temperature was defined at the\nmidpoint of the $R(T)$ transition curves. The value of the\ntransition temperatures of the film grown on the porous substrate\nand of the reference sample in the absence of the magnetic field\nwere $T_c$ = 3.83 K and $T_c$ = 4.53 K, respectively. The critical\ntemperature depression in the case of the porous sample is\nconsistent with what already reported in literature for  films\ngrown both on $Al_{2}O_{3}$ \\cite{Mosh07} and on PS\n\\cite{Trezza-JAP08}. The first step for the characterization of\nthe behavior of the porous Nb sample in the presence of\nperpendicular magnetic field is the determination of its ($H$,$T$)\nphase diagram. The temperature dependence of the perpendicular\nupper critical field, $H_{c2\\bot}$, was obtained performing\nresistance vs. field, $R(H)$, measurements at fixed values of the\ntemperature with a temperature stability of 1 mK.\n$H_{c2\\bot}$ was defined at the midpoint of each of the $R(H)$ curves.\\\\\nIn Fig. \\ref{Fig.1} the ($H$,$T$) phase diagrams of the Nb thin\nfilms are shown. In general, the perpendicular upper critical\nfield of superconducting films of thickness $\\textit{d}$ obeys a\nlinear temperature dependence, $H_{c2\\bot}(T)$ =\n($\\Phi_{0}$\/2$\\pi\\xi_{0\\parallel}^{2}$)(1-$T$\/$T_{c}$)\n\\cite{Tinkham}. $\\xi_{0\\parallel}$ is the Ginzburg-Landau\ncoherence length parallel to the sample surface at $T$ = 0. The\ntemperature dependence of $\\xi_{\\parallel}$ is\n$\\xi_{\\parallel}(T)$ = $\\xi_{0\\parallel}$\/$\\sqrt{1-T\/T_{c}}$.\nAnother superconducting parameter to be taken into account is the\nmagnetic field penetration depth, $\\lambda$, whose temperature\ndependence is $\\lambda(T)$ = $\\lambda_{0}$\/$\\sqrt{1-T\/T_{c}}$,\nwhere $\\lambda_{0}$ is the penetration depth at $T$ =0.\n\n\\begin{figure}\n\\includegraphics[width=8cm]{fig1.eps} \\caption{Left scale: Perpendicular upper\ncritical field $H_{c2\\bot}$ vs. temperature of the Nb thin film\nwith $d_{Nb}$ = 8.5 nm grown on (a) porous template and (b)\nnon-porous reference substrate. The linear fits to the data close\nto $T_{c}$ are also shown. Right scale: $dH_{c2\\bot}^{2}$\/$dT^{2}$\nversus temperature. The inset shows the comparison between the\nsecond derivatives as functions of the reduced temperature of two\nsamples, grown on the porous template (full circles) and on the\nnon-porous template (open circles). (Color online).} \\label{Fig.1}\n\\end{figure}\n\nThe $H_{c2\\bot}(T)$ curve obtained for the Nb film deposited on\nporous Si template, reported in Fig. \\ref{Fig.1}(a), presents some\npeculiarities, which indicate that the superconducting properties\nare influenced by the introduction of the porous array. In fact,\nif the $H_{c2\\bot}$ second derivative versus the temperature is\nplotted we can see that it changes its sign from positive to\nnegative at $H \\approx$ 1.16 Tesla. This field value is very close\nto the nominal first matching field that we expect for the porous\nSi template, $H_{1}\\approx$ 1.30 Tesla, assuming a square porous\narray. This change in concavity was already reported in a previous\nstudy on the same kind of samples, and it was ascribed to the\nformation of a commensurate vortex structure \\cite{Trezza-JAP08}.\nFrom the measured value of $H_{1}$ it follows that the period of\nthe porous template is $a$ = 42 nm. In the following we will\nidentify $a_{0}$ $\\equiv$ 42 nm. In Fig. \\ref{Fig.1}(b) is\nreported the $H_{c2\\bot}(T)$ curve for the Nb reference film of\nthe same thickness deposited on the non-porous template. As\nexpected the $H_{c2\\bot}(T)$ behavior is linear over the all\ntemperature range and the $H_{c2\\bot}$ second derivative versus\ntemperature does not present any peculiarity except for a shallow\npeak near $T_c$. In the inset of Fig. \\ref{Fig.1}(b), for sake of\ncomparison, the $dH_{c2\\bot}^{2}$\/$dT^{2}$ versus the reduced\ntemperature is reported for both the Nb films, in order to point\nout the difference in their magnitude. A fit to the data close to\n$T_{c}$ with the expression for $H_{c2\\bot}(T)$ reported above,\nyields a value of the Ginzburg-Landau coherence length at $T$ = 0,\n$\\xi_{0\\parallel}$ = 9.1 nm and $\\xi_{0\\parallel}$ = 9.5 nm,\nresulting in a superconducting coherence length $\\xi_{S}$ = 5.8 nm\nand $\\xi_{S}$ = 6.0 nm, for the Nb porous sample and the Nb\nreference film, respectively. The values of $\\xi_{0\\parallel}$ are\nsignificantly smaller than the BCS coherence length of Nb,\n$\\xi_{0}$ = 39 nm \\cite{Buckel}, indicating that our films are in\ndirty limit regime with an electron mean free path of $\\textit{l}$\n= 1.38 $\\xi_{0\\parallel}^{2}$ \/ $\\xi_{0}$ $\\approx$ 3 nm\n\\cite{Schmidt}. Since the film dimensions in the $xy$ plane are\nlarger than $\\xi_{\\parallel}(T)$, the expression for\n$H_{c2\\bot}(T)$, reported above, is verified in the whole\ntemperature range. The Ginzburg-Landau parameter, $\\kappa$ =\n$\\lambda$(0)\/$\\xi_{0\\parallel}$, can be estimated using the\nexpression $\\kappa$ = 0.72$\\lambda_{L}$\/$\\textit{l}$ = 9.6, where\n$\\lambda_{L}$ = 39 nm is the London penetration depth of Nb\n\\cite{Buckel}. Ratios of $\\xi_{0\\parallel}$\/$\\textit{a} \\approx$\n0.2 and $\\lambda$(0)\/$\\textit{a} \\approx$ 2.1, measured for\n$a_{0}$ = 42 nm, are larger than in previous works\n\\cite{Welp-PRB,Mosh06} on perforated Nb samples, and indicate that\nwe are in presence of individual vortex pinning \\cite{Brandt}.\nMoreover, the pore diameter, ${\\O}$, in our PS template is\ncomparable with the vortex core dimension at $T$=0, ${\\O} \\approx\n\\xi_{0\\parallel}$. This means that the saturation number, $n_{S}$\n= $\\frac{\\O}{2\\xi_{S}(T)}$, defined as the maximum number of\nvortices that fits into a pore with diameter ${\\O}$, is less or\nequal to 1, so that each pore can trap only one fluxon\n\\cite{Mkrtchyan}. Subsequently multiquanta vortex lattice\n\\cite{Mosh98} cannot be observed in our system.\\\\\nNow we move to a more careful inspection of the $R(H)$ curves of\nthe Nb porous film. This will lead to the observation of a\npeculiar behavior of these transitions, whose analysis represents\nthe main subject of this work. In Figs. \\ref{Fig.2}(a) and 2(b)\n$R(H)$ curves obtained for two different values of the\ntemperature, $T$ = 3.490 K and $T$ = 3.531 K, respectively, are\npresented.\n\n\\begin{figure}\n\\includegraphics[width=8cm]{fig2.eps} \\caption{Left scale: $R(H)$ measurement\nat (a) $T$ = 3.490 K and (b) $T$ = 3.531 K. Right scale: $dR$\/$dH$\nversus the applied magnetic field. In both panels the arrow\nindicates the field where the bump is present. (Color online)}\n\\label{Fig.2}\n\\end{figure}\n\n\\noindent At first glance both the curves are rather smooth and do\nnot present any structures or enlargements due, for example, to\nsample inhomogeneities. However if the dependence of the first\nderivative $dR$\/$dH$ versus the applied magnetic field is\nanalyzed, some distinct features can be observed. In particular in\nboth the curves a small local maximum is present at specific\nvalues of the magnetic field. Let's focus on the position where\nthe bumps, as indicated by an arrow in Fig. \\ref{Fig.2}, start to\ndevelop. The bumps in the first derivative reflects the presence\nof a small dip in the corresponding magnetic field dependence of\nthe resistance $R(H)$ at the same value of $H$. This effect was\nascribed to a pinning enhancement when the period of the vortex\nstructure is commensurate with the period of the antidots\n\\cite{Patel}. The bumps in the $dR$\/$dH$ appear indeed in our\ncurves at values of the magnetic fields $H_{n}$ when the magnetic\nflux threading each unit cell is equal to the flux quantum,\n$\\Phi_0$, or to fractional values of $\\Phi_0$. In Fig.\n\\ref{Fig.2}(a), where the $R(H)$ measurement at $T$ = 3.490 K is\nshown, the peculiarity in $dR$\/$dH$ is, in fact, observed at\n$H_{bump}\\approx$ 0.126 Tesla. The period of the vortex lattice at\nthis field value is $a'$ = 128 nm, i.e. about three times the\ninterpore spacing of this analyzed sample, $a_{0}$ = 42 nm.\nConsequently this field value corresponds to one-ninth of the\nmatching field $H_{1}$\/9 $\\approx$ 0.129 Tesla. Similarly, in Fig.\n\\ref{Fig.2}(b) where the $R(H)$ measurement at $T$ = 3.531 K is\nshown, the bump in $dR$\/$dH$ develops at $H_{bump}\\approx$ 0.065\nTesla. The period of the vortex array at this field is then $a''$\n= 178 nm, which is about four times the interpore spacing of this\nsample. Consequently this field value corresponds to one-sixteenth\nof the matching field $H_{1}$\/16 $\\approx$ 0.072 Tesla. An\nadditional bump structure is present at $H\\approx$ 0.2 Tesla.\nHowever, this field value does not correspond to any commensurate\nvortex configuration (see discussion below) and does not survive\nrepeating the measurement in the same temperature range. Many\n$R(H)$ measurements at different temperatures have been performed\nand the behavior of all the corresponding $dR$\/$dH$ curves has\nbeen analyzed. A selection of these curves is reported in Fig.\n\\ref{Fig.3}. Some of them have been obtained by sweeping the field\nupward and downward and no hysteresis has been detected.\n\n\\begin{figure}\n\\includegraphics[width=8.5cm]{fig3.eps} \\caption{First derivatives, $dR$\/$dH$, as a\nfunction of the applied magnetic field at different temperatures.\nThe arrows indicate the field where the bump is present for each\ntemperature. (Color online)} \\label{Fig.3}\n\\end{figure}\n\nFor instance, the curves at $T$ = 2.551 K and $T$ = 3.304 K\npresent a bump at $H_{bump}$ = $H_{1}$ and $H_{bump}$ = $H_{1}$\/4,\nrespectively. By comparison a curve with no bump, measured at\ntemperature $T$ = 2.805 K, is also shown. In all curves the fields\nat which the bumps are observed are related to the first matching\nfield through the relation: $H$ = $H_{1}$\/$n^{2}$ with $n$ =\n1,...,4. The temperatures at which bumps are observed, the\ncorresponding fields and their values normalized to $H_{1}$, the\n$\\xi_{S}$ values, the vortex-vortex distances, $a$, and their\nvalues normalized to $a_{0}$, are summarized in Table \\ref{table}.\n\n\n\\begin{table}\n\\caption{Temperatures at which the bumps are observed,\ncorresponding fields and their values normalized to $H_{1}$,\n$\\xi_{S}$ values at that temperature, vortex-vortex distances,\n$a_{k\/l}$, and their values normalized to $a_{0}$ = 42 nm.}\n\\label{table}\n\\begin{center}\n\\begin{tabular}{cccccc}\n$T$(K)  & $H_{bump}$(T) & $\\frac{H_{bump}}{H_{1}}$ & $\\xi_{S}$(nm) & $a_{k\/l}$(nm) & $\\frac{a_{k\/l}}{a_{0}}$ \\\\\n\\hline\n2.551 & 1.160  & 1 & 10.02  & 42.0  & 1.00  \\\\\n\n3.304  & 0.275  & 1\/4 & 15.66  & 87.0  &  2.07  \\\\\n\n3.490  & 0.126  &  1\/9 & 19.44  & 128.0  & 3.05  \\\\\n\n3.531  & 0.065  & 1\/16 & 20.76  & 178.0  & 4.24  \\\\\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nWe argue that the presence of the observed bumps in the $dR$\/$dH$\ncurves can be related to different vortex lattice arrangements\nmade possible by the lattice of holes. The specific vortex lattice\nconfigurations occurring at the first matching field and at its\nfractional values are shown in Fig. \\ref{Fig.4}.\n\n\\begin{figure}\n\\includegraphics[width=7cm]{fig4.eps} \\caption{Vortex lattice configurations\noccurring at the first matching field and its fractional values.\nIncreasing the temperature the vortices diameter and their\nreciprocal distance increase, as reported in Table \\ref{table}.\nBlue circles represent the holes, pink ones represent the\nvortices. (Color online)} \\label{Fig.4}\n\\end{figure}\n\n\\noindent In the case of $H_{bump}$\/$H_{1}$ = 1 a commensurate\nsquare vortex configuration is formed, where each pore is occupied\nby a fluxon and the side of this square array is just $a_{0}$ = 42\nnm. Increasing the temperature the vortices diameter\n($\\approx$2$\\xi_{S}$) and their reciprocal distance increase, as\nreported in Table \\ref{table}. When $H_{bump}$\/$H_{1}$ = 1\/4, 1\/9\nand 1\/16 a square vortex lattice is again obtained with $a$ = 87\nnm, 128 nm and 178 nm, respectively. This means that the pores act\nas an ordered template of strong pinning centers, which is able to\npreserve the long range positional order of the flux lattice also\nat low fields value, i.e. at higher vortex spacing. As already\npointed out the optimization of the vortex structures leads to the\nformation of larger square flux lattices with respect to the\nunderlying artificial pinning array with the lattice constant $a$\nexactly equal to $na_{0}$. The vortices tend to be placed as far\nfrom each other as possible due to the repulsive interaction\nbetween them and at the same time they want to follow the imposed\nsquare potential induced by the antidots. This constraint gives\n$a$ = $a_{0} \\sqrt{l^{2}+k^{2}}$, where $l$ and $k$ are integer\nnumbers. Therefore, we should expect the fractional matching\nfields at $H$ = $H_{k\/l}$ = $\\Phi_0\/a^{2}$ =\n$\\Phi_0\/[a_{0}^{2}(l^{2}+k^{2})]$ = $H_{1}\/(l^{2}+k^{2})$\n\\cite{Mosh95}. We observed bumps at fractional matching fields\n$H_{0\/2}$, $H_{0\/3}$ and $H_{0\/4}$. The other bumps expected from\nthe equation above at fractional fields $H_{k\/l}$ with $k \\neq$ 0\nhave not been observed. All the fields values at which the bumps\nin the $dR$\/$dH$ appear are shown as points of coordinates\n($H_{bump}$,$T$) in Fig. \\ref{Fig.5}.\n\n\\begin{figure}\n\\includegraphics[width=8.8cm]{fig5.eps} \\caption{The points of coordinates\n($H_{bump}$,$T$) identify the values of the fields and\ntemperatures at which bumps have been observed in the $dR$\/$dH$\ncurves at fixed temperatures. The solid lines correspond to the\ndifferent matching field orders achieved with the interpore\nspacing $a_{0}$ = 42 nm, while the dotted lines are obtained\nforasmuch as the regularity of the pore distance is achieved\nwithin the 10 percents of the average distance. (Color online)}\n\\label{Fig.5}\n\\end{figure}\n\n\\noindent In this figure the solid lines correspond to the\nmatching fields of different order, as calculated assuming an\ninterpore spacing $a_{0}$ = 42 nm, through the formula\n$H_1=\\Phi_0\/a_{0}^{2}$.  The dotted lines are obtained considering\na deviation from the corresponding mean interpore distance of the\norder of 10\\% \\cite{Trezza-JAP08}. It is worth noticing that all\nthe data fall into the range theoretically estimated, suggesting\nthat the observed peculiarities in the $R(H)$ curves can be indeed\nascribed to commensurability effect between the porous structure\nof the Nb film and the vortex lattice. The distribution of the\nexperimental points is consistent with the observation that a\ncertain temperature dependence of the matching effect can be found\nfor the case of short-range ordered templates \\cite{Ziemann}. We\nwould also point out that the effect is observable in our sample\nonly up to $H$ = $H_{1}$, due to the very high value of the first\nmatching field. The second matching field in fact is $H_{2}$ =\n2$H_{1}$ = 2.32 Tesla. From a linear extrapolation of the\n$H_{c2\\bot}$ curve, it follows that in order to see at this field\na bump in the $dR$\/$dH$ we should measure a $R(H)$ curve at $T$ =\n1.73 K, temperature which cannot be reached in our experimental\nsetup. All the field values reported above have been calculated\nassuming a square lattice. The measured field values do not match\nwith the ones calculated if a triangular array for the pores is\nconsidered. In fact, at $T$ = 3.490 K (see Fig. \\ref{Fig.2}(a))\nthe structure in the $dR$\/$dH$ curve for a triangular lattice\nwould have been observed at a field $2\/\\sqrt{3}$ times higher than\n$H_{0\/3}$ = 0.126 Tesla, where no peculiar feature has been\ndetected. This supports our assumption of considering a square\nlattice of holes in our system.\n\n\\section{Conclusions}\n\nMatching effects have been reported for Nb thin film grown on\nporous silicon. Due to the extremely reduced values of the\ninterpore distance the effect is present at fields values higher\nthan 1 Tesla and down to reduced temperatures as low as t $\\simeq$\n0.52. The commensurability manifests both in the ($H$,$T$) phase\ndiagram and in the $R(H)$ transitions. The latter in particular\nreveal the formation of fractional matching states. As it was\nargued in many works the vortex configuration at fractional\nmatching fields are characterized by striking domain structure and\nassociated grain boundaries \\cite{Field02,Mosh03}. The presence of\nmultiple degenerate states with domain formation at the fractional\nfield, directly observed with scanning Hall probe microscopy\n\\cite{Field02}, seems to be high probable in our films. The\nreduced regularity of our templates, in fact, could be compensated\nby the formation of domain walls of different complexity. The\nparticular domain configuration is of course a matter of energy\nbalance between the cost in energy for the wall formation and the\nenergy gain due to the vortex pinning.\n\n\n\n\n\n\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:1}\n\nBosonic sigma models with the metric action functional \npossess rigid symmetries induced by isometries\nof their target space. Such rigid symmetries may be gauged\nby the minimal coupling to the gauge fields of the isometry\ngroup. The gauged action is then invariant under arbitrary \nlocal gauge transformations. The minimal coupling does not work, \nhowever, for the topological Wess-Zumino term in the action \nfunctional of the sigma model, if such is present. In particular,\nit was shown in \\cite{JJMO,HS} for the two-dimensional sigma model \nwith the Wess-Zumino term corresponding to a closed $3$-form $H$ \non the target space that the gauging of rigid symmetries \nrequires satisfying certain conditions. Such conditions assure\nthe absence of local gauge anomalies and guarantee the existence\nof a gauging procedure that results in an action functional invariant \nunder infinitesimal local gauge transformations.  \nThe infinitesimal gauge invariance of the gauged action implies \nits invariance under all ``small'' local gauge transformations, \ni.e. the ones that are homotopic to unity. As was observed in\n\\cite{GSW}, it is possible, however, that the gauged action\nexhibits global gauge anomalies that lead to its non-invariance \nunder some ``large'' local gauge transformations non-homotopic \nto unity. The phenomenon was analyzed in detail for sigma models \non closed worldsheets in \\cite{GSW} and on worldsheets\nwith boundaries and defects in \\cite{GSW1}. In the case of Wess-Zumino-Witten \n(WZW) models of conformal field theory with Lie group $G=\\tilde G\/Z$ as the\ntarget, where $\\tilde G$ is the universal covering group of $G$ and $Z$ \nis a subgroup of the center $\\tilde Z$ of $\\tilde G$, \nwith the Wess-Zumino term corresponding to the bi-invariant \nclosed 3-form $H_k=\\frac{k}{12\\pi}\\,{\\rm tr}\\hspace{0.03cm}(g^{-1}dg)^3$, \nthe local gauge anomalies are absent for a restricted class\nof rigid symmetries. These include the symmetries induced \nby the adjoint action $g\\mapsto hgh^{-1}$ on $G$ for \n$h\\in\\tilde G\/\\tilde Z$, or by its twisted versions \n$g\\mapsto hg\\hspace{0.03cm}\\omega(h)^{-1}$, for \n$h\\in\\tilde G\/Z^\\omega$, where $\\omega$ is an automorphism\nof $\\tilde G$ and $Z^\\omega=\\{z\\in\\tilde Z\\,|\\,z\\hspace{0.02cm}\n\\omega(z)^{-1}\\in Z\\}$\nis the subgroup of elements in $\\tilde Z$ that acts trivially. \nIn these cases, the global gauge anomalies may occur for the target \ngroups $G$ that are not simply connected (corresponding to the so called \nnon-diagonal WZW models). They are detected by a cohomology class \n$\\varphi\\in H^2(\\tilde G\/Z^\\omega\\times G,U(1))$ that can be easily \ncomputed. Class $\\varphi$ is invariant under the \naction of $\\gamma\\in\\tilde G\/Z^\\omega$ on $\\tilde G\/\\tilde Z^\\omega\n\\times G$ given by \n\\begin{equation}\n(h,g)\\ \\mapsto\\ (\\gamma h\\gamma^{-1},\\gamma g\\hspace{0.04cm}\n\\omega(\\gamma)^{-1})\\,.\n\\label{ac}\n\\end{equation} \nThe simplest case when the anomaly class is nontrivial corresponds \nto $G=SU(3)\/\\mathbb{Z}_3$ at level $k=1$ or to $G=SU(4)\/\\mathbb{Z}_4$ \nat level $k=2$, both with $\\omega={I\\hspace{-0.04cm}d}$. Some other \ncases with global gauge anomalies for $\\omega={I\\hspace{-0.04cm}d}$ \nwere cited in \\cite{GSW}. In Sec.\\,\\ref{sec:3} of the present paper, \nwe obtain the full list of connected compact simple target groups $G$ \nfor which the WZW model with the gauged (twisted) adjoint action of \n$\\tilde G\/\\tilde Z^\\omega$ exhibits global gauge anomalies. In the twisted \ncase, we consider only outer automorphisms $\\omega$ since for \ninner automorphisms the twisted adjoint action may be \nreduced to the untwisted one by conjugating it with a right \ntranslation on $G$ which is a rigid symmetry of the WZW theory.\nThe classes of outer automorphisms of $\\tilde G$ modulo inner automorphisms\nare generated by automorphisms of the Lie algebra $\\mathfrak{g}$ that\npreserve the set of simple roots inducing a symmetry of the Dynkin \ndiagram of $\\mathfrak{g}$.\nGlobal gauge anomalies occur only for (non-simply connected) \ngroups $G$ with Lie algebras $\\mathfrak{g}=A_r,\\,D_r,\\,\\mathfrak{e}_6$ \nin the Cartan classification of simple Lie algebras\\footnote{We consider \nthe compact real forms $\\mathfrak{g}$ of complex simple Lie algebras that \nare in one-to-one correspondence with their complexifications \n$\\mathfrak{g}^{\\mathbb C}$.}.  \n\\vskip 0.1cm\n\nGauged WZW models serve to construct coset $G\/H$ models \\cite{GKO,Godd} \nof the two-dimensional conformal field theory \\cite{BRS,GK0,GK,KPSY}. \nIn such models, \none restricts the gauging to the (possibly twisted) adjoint action on \nthe target group of the subgroup \n$\\Gamma=\\tilde H\/(Z^\\omega\\cap\\tilde H)\\subset\n\\tilde G\/Z^\\omega$, where $\\tilde H$ a closed \nconnected subgroup of $\\tilde G$ (simply-connected or not). Global gauge \nanomalies are now detected by the pullback cohomology class \nin $H^2(\\Gamma\\times G,U(1))$. \nSecs.\\,\\ref{sec:4} and \\ref{sec:5} are devoted to finding out when the latter \nis nontrivial for groups $\\,G\\,$ as before and for a wide class of \nsubgroups $\\tilde H\\subset\\tilde G$ (the nontriviality of the pullback \nclass depends only on the subgroup $\\tilde H$ modulo conjugation \nby elements of $\\tilde G$ and it may occur only if the original anomaly class \n$\\varphi$ is nontrivial, hence for Lie algebras $\\mathfrak{g}$ \nenumerated above). Closed connected subgroups $\\tilde H\\subset\\tilde G$\nare in one-to-one correspondence to Lie subalgebras $\\mathfrak{h}\\subset\n\\mathfrak{g}$. We obtain the complete list of cases with global \ngauge anomalies for subgroups $\\tilde H$ with the Lie algebra $\\mathfrak{h}$ \nwhich is a semisimple regular subalgebra of $\\mathfrak{g}$ (i.e. such that \nthe roots of $\\mathfrak{h}$ form a  subset of roots of $\\mathfrak{g}$). \nThe complete classification (modulo conjugation) of regular subalgebras \nof simple Lie algebras was obtained in the classical work \\cite{Dynkin}\nof Dynkin. The complete classification of all semisimple subalgebras \nof simple Lie algebras is not known explicitly, except for low ranks \nand may be complicated. We give the complete list of non-regular \nsemisimple subalgebras $\\mathfrak{h}$ of $\\mathfrak{g}=\\mathfrak{e}_6$\ncorresponding to subgroups $\\tilde H\\subset\\tilde G$ that lead to global \ngauge anomalies. For $\\mathfrak{g}=A_r$ \nand $\\mathfrak{g}=D_r$, we limit ourselves to few examples of anomalous \nsubgroups $\\tilde H\\subset\\tilde G$ for which $\\mathfrak{h}$ is a non-regular \nsemisimple subalgebra of $\\mathfrak{g}$.\n\\vskip 0.1cm\n\nAs discussed in \\cite{GSW} for the untwisted case, the presence of global \ngauge anomalies of the type studied here renders the $G\/H$ coset models \ninconsistent on the quantum level (barring accidental degeneracies \nof the affine characters). Hence the importance of the classification \nof the anomalous cases.  \n\n\n\\section{No-anomaly condition}\n\\label{sec:2}\n\nThe WZ contribution to the action of the WZW model corresponding \nto the closed $3$-form $H_k$ on a connected compact simple Lie group $G=\\tilde G\/Z$ with $Z\\subset\\tilde Z$ may be defined (modulo $2\\pi$) whenever \nthe periods of $H_k$ (i.e.\nits integrals over closed $3$-cycles) belong to $2\\pi\\mathbb Z$. \nFor the standard \nnormalization of the invariant negative-definite quadratic \nform ${\\rm tr}$ on the Lie algebra \n$\\mathfrak{g}$ in which long roots (viewed as elements of \n$\\ii\\mathfrak t_{\\mathfrak g}$, where $\\mathfrak t_{\\mathfrak g}$ is the\nCartan subalgebra of $\\mathfrak g$) have length squared $2$, this happens \nfor levels $k\\in K_G\\subset\\mathbb Z$. If $G=\\tilde G$ then\n$K_G=\\mathbb Z$ whereas $K_G$ may be a proper subset \nof $\\mathbb Z$ if $G=\\tilde G\/Z$ with $Z$ nontrivial (i.e. $\\not=\\{1\\}$). \nSets $K_G$ of admissible levels are explicitly known \n\\cite{FGK,Gawedzki}. Besides, for $G=SO(2r)\/{\\mathbb Z_2}$ with $r$ even \n(where $K_G=\\mathbb{Z}$ when $4|r$ and $K_G\n=2\\mathbb{Z}$ if $4\\hspace{-0.2cm}\\not{\\hspace{-0.05cm}|}r$), there are \ntwo different consistent choices of the WZ term of the action. \nThe details of the construction of the WZ contribution \n$\\exp\\big[iS^{W\\hspace{-0.05cm}Z}_\\Sigma(g)\\big]$ to the Feynman \namplitude of the sigma-model field $g:\\Sigma\\to G$ defined on a closed \noriented worldsheet $\\Sigma$, discussed e.g. in \\cite{Gtop,GR}, will not \ninterest us here beyond the fact that the result is invariant under \nthe composition of fields $g$ with the left or right action of (fixed) \nelements of group $G$. The action functional with the (twisted) adjoint \nsymmetry of the WZW model gauged is a functional of field $g$ and of \ngauge-field $A$, a $\\mathfrak{g}$-valued $1$-form on $\\Sigma$. \nIt has the form\n\\begin{equation}\nS^{W\\hspace{-0.05cm}Z}_\\Sigma(g,A)\\,=\\,S^{W\\hspace{-0.05cm}Z}_\\Sigma(g)\\,\n+\\,\\frac{_k}{^{4\\pi}}\\int{\\rm tr}\\,\\big((g^{-1}dg)\\hspace{0.02cm}\\omega(A)\n+(dg)g^{-1}\\hspace{-0.05cm}A+g^{-1}\\hspace{-0.05cm}\nAg\\hspace{0.05cm}\\omega(A)\\big)\n\\label{gauged_action}\n\\end{equation}\n(for the untwisted case, $\\omega=I\\hspace{-0.03cm}d$).\nThe local gauge transformations $h:\\Sigma\\to\\tilde G\/Z^\\omega$ act \non the sigma model and gauge fields by\n\\begin{equation}\n{}^h\\hspace{-0.05cm}g\\,=\\,hg\\hspace{0.04cm}\\omega(h)^{-1}\\,,\n\\qquad{}^h\\hspace{-0.06cm}A\\,\n=\\,hA\\hspace{0.01cm}h^{-1}+hdh^{-1}\\,.\n\\end{equation}\nNote that $Z^\\omega=\\tilde Z$ for $\\omega=I\\hspace{-0.03cm}d$.\nIt is easy to show that the invariance of the gauged Feynman amplitudes \nunder such transformations: \n\\begin{equation}\n\\exp\\big[iS^{W\\hspace{-0.05cm}Z}_\\Sigma({}^h\\hspace{-0.05cm}g,\n{}^h\\hspace{-0.06cm}A)\n\\big]\\,=\\,\\exp\\big[iS^{W\\hspace{-0.05cm}Z}_\\Sigma(g,A)\\big]\n\\label{gaugeinv}\n\\end{equation}\nis equivalent to the identity\n\\begin{eqnarray}\n\\frac{\\exp\\big[iS^{W\\hspace{-0.05cm}Z}_\\Sigma({}^h\\hspace{-0.05cm}g)\\big]}\n{\\exp\\big[iS^{W\\hspace{-0.05cm}Z}_\\Sigma(g)\\,\n+\\,\\frac{_{\\ii k}}{^{4\\pi}}\\int_\\Sigma\n{\\rm tr}\\,\\big(g^{-1}dg\\hspace{0.05cm}\\omega(h^{-1}dh)+(dg)g^{-1}h^{-1}dh\n+g^{-1}(h^{-1}dh)g\\hspace{0.05cm}\\omega(h^{-1}dh)\\big)\\big]}\\,\n=\\,1\\,,\\  \n\\label{eval}\n\\end{eqnarray}\nsee Appendix \\ref{app:1}.\nThe ratio on the left hand side belongs always to $U(1)$. It coincides \nwith the evaluation of the anomaly class $\\varphi\\in \nH^2(\\tilde G\/Z^\\omega\\times G,U(1))$ on the 2-cocycle that is the image of the \nfundamental class of $\\Sigma$ under the map $(h,g):\\Sigma\\to\\tilde G\/\nZ^\\omega\\times G$.\n\nA simple analysis \\cite{GSW}\nof the structure of cohomology group $H^2(\\tilde G\/Z^\\omega\\times G,U(1))$ \nbased on the K\\\"unneth Theorem shows that class $\\varphi$ is trivial \nif and only if\nidentity (\\ref{eval}) holds for $\\Sigma=S^1\\times S^1$ and \n\\begin{equation}\nh({\\rm e}^{\\ii\\sigma_1},{\\rm e}^{\\ii\\sigma_2})={\\rm e}^{\\ii\\sigma_1\\tilde M}\\,,\n\\qquad g({\\rm e}^{\\ii\\sigma_1},\n{\\rm e}^{\\ii\\sigma_2})={\\rm e}^{\\ii\\sigma_2M}\n\\label{hg}\n\\end{equation}\nwhere $\\tilde M,M\\in\\ii\\mathfrak t_{\\mathfrak g}$ and are such that, \nin terms of the exponential map with values in $\\tilde G$,\n\\begin{equation} \n\\tilde z\\equiv e^{2\\ii\\pi\\tilde M}\\in Z^\\omega\\quad\\ {\\rm and}\n\\quad\\ z\\equiv e^{2\\ii\\pi M}\\in Z\\,.\n\\label{tzz}\n\\end{equation}\nBoth $\\tilde M$ and $M$ have to belong to the coweight lattice \n$P^\\vee(\\mathfrak{g})\\subset\\ii\\mathfrak{t}_{\\mathfrak g}$ dual to the\nweight lattice of $\\mathfrak{g}$ and composed of \n$M\\in\\ii\\mathfrak{t}_{\\mathfrak g}$ s.t. $\\exp[2\\ii\\pi M]\\in\\tilde Z$.\nFor $(h,g)$ given by Eqs.\\,(\\ref{hg}), the left hand side of Eq.\\,(\\ref{eval})\nis easily computable giving rise to the identity\n\\begin{equation}\nc_{\\tilde z\\omega(\\tilde z)^{-1},z}\\,\\exp\\big[-2\\ii\\pi k\\,\n{\\rm tr}(M\\omega(\\tilde M))\\big]\\,=\\,1\n\\label{idtytw}\n\\end{equation}\nwhich holds for all $\\tilde M,M\\in P^\\vee(\\mathfrak{g})$ \nas above if and only if there\nare no global gauge anomalies for the WZW model with gauged (twisted) adjoint \naction of $\\tilde G\/\\tilde Z^\\omega$ on the target group $G$. \nIn Eq.\\,(\\ref{idtytw}),\n\\begin{equation}\nZ^2\\ni(z,z')\\,\\mapsto\\,c_{z, z'}\\in U(1) \n\\label{bihom}\n\\end{equation}\nis a $k$-dependent bihomomorphism in $Hom(Z\\otimes Z,U(1))$\nwhose explicit form may be extracted from Appendix 2 of \\cite{FGK}.\nFor cyclic $Z\\equiv\\mathbb Z_p$ generated by $z_0=e^{2\\ii\\pi\\theta}$\nfor $\\theta\\in P^\\vee(\\mathfrak{g})$,\n\\begin{equation}\nc_{z_0^m,z_0^n}=\\exp[-\\ii\\pi k\\hspace{0.02cm}mn\\,{\\rm tr}(\\theta^2)].\n\\label{cc}\n\\end{equation}\nFor the only case with non-cyclic $Z$, we shall explicit $c_{z,z'}$\nin Sec.\\,\\ref{sec:D_reven_tw}. In the untwisted case with \n$\\omega=I\\hspace{-0.03cm}d$, \ncondition (\\ref{idtytw}) reduces to the requirement that\n\\begin{equation}\n\\exp\\big[-2\\ii\\pi k\\,{\\rm tr}(M\\tilde M)\\big]\\,=\\,1\\,.\n\\label{idty}\n\\end{equation}\nIf we gauge only the adjoint action of $\\tilde H\/(\\tilde Z^\\omega\n\\cap\\tilde H)$ \nthen there are no global gauge anomalies if and only if identity (\\ref{idtytw}) \nholds under the additional restriction that, as an element of $\\tilde G$, \n$\\,\\exp[2\\ii\\pi\\tilde M]\\in\\tilde H$. \n\\vskip 0.1cm\n\nIt is enough to check the above conditions for $\\tilde M,M$ in different\nclasses modulo the coroot lattice $Q^\\vee(\\mathfrak{g})$ (composed\nof $\\tilde M\\in\\ii\\mathfrak{t}_{\\mathfrak g}$ s.t. $\\exp[2\\ii\\pi\\tilde M]=1$\nin $\\tilde G$) since ${\\rm tr}\\,\\tilde M M\\in{\\mathbb Z}$ if \n$\\tilde M\\in P^\\vee(\\mathfrak{g})$ and $M\\in Q^\\vee(\\mathfrak{g})$ \nor {\\it vice versa}. In particular, if $Z=\\{1\\}$, i.e. if $G$ is simply\nconnected, then conditions (\\ref{idtytw}) and (\\ref{idty}) are \nalways satisfied so that there are no global gauge anomalies in that case. \nIn the sequel, we shall describe for each Lie algebra $\\mathfrak{g}$ \nthe center $\\tilde Z$ \nof the corresponding simply connected group $\\tilde G$ in terms \nof coweights of $\\mathfrak{g}$. Then choosing a Lie \nsubalgebra $\\mathfrak{h}\\subset\\mathfrak{g}$, we shall restrict elements \n$\\tilde{M}$ by requiring that $e^{2\\ii \\pi \\tilde{M}}\\in \\tilde{H}$. \nNote that $e^{2\\ii\\pi\\tilde M}\\in\\tilde H\\,$ \nif and only if $\\,e^{2\\ii\\pi\\tilde M}\\in g\\tilde H g^{-1}\\,$ \nfor $\\,g\\in\\tilde G$ and $\\,e^{2\\ii\\pi\\tilde M}\\in\\tilde Z$. \nHence the no-anomaly conditions coincide for conjugate subgroups \n$\\tilde H\\subset\\tilde G$. Thus it is \nenough to consider one Lie subalgebra $\\,\\mathfrak h\\subset\\mathfrak g\\,$ \nin each class of subalgebras related by inner automorphisms of $\\mathfrak g$. \nWe may also require that the Cartan subalgebra $\\mathfrak t_{\\mathfrak h}$\nof $\\mathfrak h$ be contained in the Cartan subalgebra \n$\\mathfrak t_{\\mathfrak g}$ of $\\mathfrak g$. Then \n$e^{2\\ii \\pi \\tilde{M}}\\in \\tilde{H}$ if and only if there is \n$q^\\vee\\in Q^\\vee(\\mathfrak g)$ such that $\\tilde M+q^\\vee\\in\\ii\\mathfrak \nt_{\\mathfrak h}$. This is the condition that we shall impose \non $\\tilde M$. \n\\vskip 0.1cm\n\nThe no-anomaly conditions for Lie subalgebras $\\mathfrak h\\subset\\mathfrak g$\nrelated by outer automorphisms $\\omega'$ of $\\mathfrak g$ are also related.\nIndeed, it is easy to see that the expression on the right hand side\nof Eq.\\,(\\ref{eval}) for gauge transformation $h$ and fields $g$ coincides \nwith the similar expression for gauge transformation $\\omega'(h)$ and \nfield $\\omega'(g)$ if in the latter case subgroup $Z\\subset\\tilde Z$ is \nreplaced by $\\omega'(Z)$ and the twist $\\omega$ by \n$\\omega'\\omega\\hspace{0.04cm}\\omega'^{-1}$. The only exception is \nthe case of $G=SO(2r)\/\\mathbb{Z}_2$ for even $r$ and odd $k$ where \none may also have to interchange the two different consistent choices \nof the theory, see Sec.\\,\\ref{sec:D_reven_tw}.\n\\vskip 0.2cm\n\n{\\bf Summarizing:} the necessary and sufficient condition for \nthe absence of global gauge anomalies requires that Eq.\\,(\\ref{idtytw})\nholds for all $\\tilde M,M\\in P^\\vee(\\mathfrak{g})$ such that\n\\begin{equation} \n\\tilde z\\equiv e^{2\\ii\\pi\\tilde M}\\in Z^\\omega\\cap\\tilde H\\quad\\ {\\rm and}\n\\quad\\ z\\equiv e^{2\\ii\\pi M}\\in Z\\,.\n\\label{tzzH}\n\\end{equation}\nIn the untwisted case, this reduces to the condition\\footnote{In the conformal\nfiled theory terminology \\cite{SchellYank}, condition\n(\\ref{coset_anomaly}) means that the monodromy charge $Q_J(\\tilde J)$ for the \nsimple currents $\\tilde J$ and $J$ corresponding to the central elements \n$\\tilde z$ and $z$ has to vanish modulo 1.}\n\\begin{equation}\\label{coset_anomaly}k~\\text{tr} \n(M \\tilde{M}) \\in \\mathbb{Z}\\ \\ \\text{for all}\\ \\ \n\\tilde M,M \\in P^\\vee(\\mathfrak{g})\\ \\ \ns.t.\\ \\ \\tilde z\\in\\tilde H, \n\\ \\ z\\in Z\\,. \n\\end{equation}\nThe no-anomaly conditions for subgroups $\\tilde H\\subset\\tilde G$\ncorresponding to Lie subalgebras $\\mathfrak h\\subset\\mathfrak g$ related \nby inner (outer) automorphisms of $\\mathfrak g$ coincide (are simply\nrelated).\n\n\n\\section{Cases with $\\mathfrak{h}=\\mathfrak{g}$}\n\\label{sec:3}\n\nAs the first step, we shall consider the cases with \n$\\mathfrak h=\\mathfrak g$ for all simple \nalgebras $\\mathfrak g$ according to the Cartan classification, and for \narbitrary nontrivial subgroups $Z\\subset\\tilde Z$. \nIf there are no global gauge anomalies in that case, then the anomalies are \nabsent \nalso for other $\\mathfrak h\\subset\\mathfrak g$. In other words,\nupon restricting $\\mathfrak h$ to a smaller subalgebra, the anomalies may \nonly disappear. In this way, a lot of trivial cases can be already treated\nwithout specifying the subalgebra $\\mathfrak{h}$. We shall then consider \nin the next section the classification of subalgebras $\\mathfrak{h}\\subset\n\\mathfrak{g}$ up to conjugation only for the remaining cases: those \nwith possible anomalies. \n\n\n\n\\subsection{Case $A_r = \\mathfrak{su}(r+1)$, $r\\geq1$}\n\nLie algebra $\\mathfrak{g} = A_r $, corresponding to group \n$\\tilde G=SU(r+1)$, is composed of traceless anti-hermitian\nmatrices of size $r+1$. Its Cartan subalgebra $\\mathfrak{t}_{\\mathfrak g}$ \nmay be taken \nas the subalgebra of diagonal traceless matrices with imaginary entries. \nWe define $e_i\\in\\ii\\mathfrak{t}_{\\mathfrak g},\\ i = 1, \\ldots, r+1$, \\,as \na diagonal matrix with the $j$'s diagonal entry equal to $\\delta_{ij}$, \nso that tr$(e_i e_j) \n= \\delta_{ij}$. Roots (viewed as elements of $\\ii\\mathfrak{t}_{\\mathfrak g}$) \nand coroots of $\\mathfrak{su}(r+1)$ have then the form\n$e_i - e_j$ for $i \\neq j$ and the standard choice of simple roots is \n$\\alpha_i = e_i-e_{i+1}, \\,i = 1 \\ldots r$. The center $\\tilde{Z} \\cong \n\\mathbb{Z}_{r+1}$ may be generated by $z = e^{2i\\pi \\theta}$ with \n$\\theta = \\lambda^\\vee_r = (1\/(r+1)) \\sum_{i=1}^{r+1}e_i-e_{r+1}$ where\n$\\lambda^\\vee_i$ denotes the $i$-th simple coweight satisfying\n$\\,{\\rm tr}(\\lambda_i^\\vee\\alpha_j)=\\delta_{ij}$. \nSubgroups $Z$ of $\\tilde{Z}$ are of the form $Z \\cong \\mathbb{Z}_p$ with $p | \n(r+1)$, and may be generated by $z^q= e^{2i\\pi q \\theta} $ \nfor $r+1=pq$. The admissible levels\nfor the WZW model based on group $G=\\tilde G\/\\mathbb Z_p$ are: \n\\begin{equation}\\label{Consistency_Ar}\n\\begin{array}{ll}\n k \\in 2\\mathbb{Z} & \\text{if } p \\text{ even and } q \\text{ odd,}\\\\\n k \\in \\mathbb{Z}  & \\text{otherwise,}\n\\end{array}\n\\end{equation}\nsee \\cite{FGK,Gawedzki}. If we now represent $M$ and $\\tilde{M}$ \nin the Euclidian space spanned by vectors $e_i$,\n\\begin{eqnarray}\\label{M_Ar}\n&&M = aq \\theta = \\left( \\dfrac{a}{p}, \\ldots , \\dfrac{a}{p},- \\dfrac{ar}{p}\n\\right), \\hspace{2.1cm} a \\in \\mathbb{Z}\\,, \\\\\n\\label{tildeM_Ar}\n&&\\tilde{M} = \\tilde{a} \\theta = \\left( \\dfrac{\\tilde{a}}{r+1}, \\ldots ,\n\\dfrac{\\tilde{a}}{r+1},- \\dfrac{\\tilde{a}r}{r+1} \\right), \\qquad \\tilde{a} \\in\n\\mathbb{Z}\\,,\n\\end{eqnarray}\nthe condition for $M$ in (\\ref{tzzH}) is satisfied and\n$e^{2\\ii \\pi \\tilde{M}}\\in \\tilde{Z}$. \n\\subsubsection{Untwisted case}\n\n\nIf $\\omega=I\\hspace{-0.03cm}d$, the global gauge invariance\nfor $\\mathfrak{h}=\\mathfrak{g}$ is assured if \n\\begin{equation}\\label{Quantity_Ar}\n k~\\text{tr} (M \\tilde{M}) = k \\dfrac{r a\\tilde{a}}{p}\\,\\in\\mathbb Z\\,.\n\\end{equation}\nIn particular, $k\\in p\\mathbb{Z}$ is a sufficient condition\nfor the absence of global anomalies.\nRecall that $p$ divides $r+1$. This implies that $p$ and $r$ are relatively \nprime. Hence $k\\in p\\mathbb{Z}$ is also a necessary condition for the absence\nof the anomalies if there are no further restrictions on the values of \n$\\tilde a$, \\,i.e. if $\\mathfrak h=\\mathfrak g$. \nTaking into account restrictions (\\ref{Consistency_Ar}), this leads to the \nfirst result:\n\n\\begin{prop}\\label{prop_Ar_h=g}\nThe untwisted coset models corresponding to Lie algebra $\\mathfrak{g} \n= \\mathfrak{su}(r+1)$, subgroups $Z \\cong \\mathbb{Z}_p$, $r+1 = pq$, \nand arbitrary subalgebras $\\mathfrak{h}$ do not have global gauge anomalies \nif $k \\in p\\mathbb{Z}$. \nThe models with $\\mathfrak h=\\mathfrak g$ and with \n$k \\notin p\\mathbb{Z}$ for $p>1$ \nodd or $q$ even, or with $k \\in 2\\mathbb{Z}\\setminus p\\mathbb{Z}$ \nfor $p > 2$ even and $q$ odd \nare anomalous.\n\\end{prop}\n\n\\subsubsection{Twisted case}\n\nFor $r>1$, there is one nontrivial outer automorphism of \n$\\mathfrak{su}(r+1)$. It maps simple root \n$\\alpha_i$ to $\\alpha_{r+1-i}$ so that for $\\tilde M$ given by \nEq.\\,(\\ref{tildeM_Ar}),\n\\begin{equation}\n\\omega(\\tilde M)\n= \\omega(\\tilde{a} \\theta) = \\left( \\dfrac{\\tilde{a}r}{r+1},\n\\dfrac{-\\tilde{a}}{r+1}, \\ldots ,\\dfrac{-\\tilde{a}}{r+1} \\right), \\qquad \n\\tilde{a} \\in\\mathbb{Z}\\,.\n\\label{omegaM_Ar}\n\\end{equation}\nThe condition\n\\begin{equation}\ne^{2\\ii\\pi\\tilde M}\\,\\omega(e^{-2\\ii\\pi\\tilde M})=e^{4\\ii\\pi\\tilde a\\theta}\\,\\in\\,Z\n\\end{equation}\nreduces to the requirement\n\\begin{eqnarray}\nq|\\tilde a\\quad\\ {\\rm for}\\ \\quad q\\ \\quad {\\rm odd\\ \\quad and}\n\\quad\\ \\frac{q}{2}|\\tilde a\\quad\\ {\\rm for}\\ \\quad q\\ \\ {\\rm even}\\,.\n\\end{eqnarray}\nIt follows that $Z^\\omega\\cong\\mathbb Z_p$ for $\\,q\\,$ odd and $Z^\\omega\n\\cong\\mathbb Z_{2p}$ for $\\,q\\,$ even.\nFrom Eq.\\,(\\ref{cc}), we obtain\n\\begin{eqnarray}\nc_{\\tilde z\\omega(\\tilde z)^{-1},z}=\\exp\\hspace{-0.07cm}\n\\big[-2\\ii\\pi k\\frac{\\tilde a a r}{p}\\big]\n\\end{eqnarray} \nand from Eqs.\\,(\\ref{M_Ar}) and (\\ref{omegaM_Ar}),\n\\begin{eqnarray}\n\\exp[-2\\ii\\pi k\\,{\\rm tr}(M\\omega(\\tilde M))]\n=\\exp\\hspace{-0.07cm}\\big[-2\\ii\\pi k\\frac{a\\tilde a}{p}\\big]\n\\end{eqnarray} \nso that the no-anomaly condition (\\ref{idtytw}) reduces to the identity\n\\begin{equation}\n\\exp[-2\\ii\\pi k a\\tilde aq]=1\n\\end{equation}\nwhich always holds implying\n\n\\begin{prop}\\label{prop_Ar_h=g_tw}\nThe twisted coset models corresponding to Lie algebra $\\mathfrak{g} \n= \\mathfrak{su}(r+1)$, subgroups $Z \\cong \\mathbb{Z}_p$, $r+1 = pq$, \nand arbitrary subalgebras $\\mathfrak{h}$ do not have global gauge anomalies.\n\\end{prop}\n\n\n \n\n\n\\subsection{Case $B_r = \\mathfrak{so}(2r+1)$, $r \\geq 2$}\n\nLie algebra $\\mathfrak{g} = B_r $, corresponding to group\n$\\tilde G=Spin(2r+1)$, is composed of real antisymmetric\nmatrices of size $2r+1$. The Cartan algebra $\\mathfrak{t}_{\\mathfrak g}$ \nmay be taken as composed of $r$ blocks\n\n\\begin{equation}\n \\begin{pmatrix}\n0 & - t_i \\\\\n t_i  & 0 \n \\end{pmatrix}\n\\end{equation}\nplaced diagonally, with the last diagonal entry vanishing. Let \n$e_i\\in\\ii\\mathfrak{t}_{\\mathfrak g}$ denote the\nmatrix corresponding to $t_j = \\ii\\delta_{ij}$. With the normalization such \nthat\ntr$(e_ie_j) = \\delta_{ij}$, roots of $\\mathfrak{g}$ have the form $\\pm e_i \\pm\ne_j$ for $i \\neq j$ and $\\pm e_i$, and one may choose $\\alpha_i = e_i -e_{i+1}$\nfor $i = 1 \\ldots r-1$ and $\\alpha_r = e_r$ as the simple roots. The center\n$\\tilde{Z} \\cong \\mathbb{Z}_2$ is generated by $z = e^{2 \\ii \\pi \\theta}$ with\n$\\theta = \\lambda^\\vee_1 = e_1$, and the only nontrivial subgroup of the\ncenter is $Z =\n\\tilde{Z}$. If we describe $M$ and $\\tilde{M}$ in the Euclidian space spanned\nby vectors $e_i$, it is enough to take\n\\begin{equation}\n M = a \\theta = \\left( a, 0, \\ldots, 0 \\right), \\qquad \n \\tilde{M} = \\tilde{a} \\theta = \\left( \\tilde{a}, 0, \\ldots, 0 \\right), \\qquad\na,\\tilde{a} \\in \\mathbb{Z}\\,.\n\\end{equation}\nLie algebra $\\mathfrak{so}_{r+1}$ does not have nontrivial\nouter automorphisms. For $\\omega=I\\hspace{-0.03cm}d$, the global gauge \ninvariance is assured if\n\\begin{equation}\n k~\\text{tr} (M \\tilde{M}) = k a\\tilde{a} \\in \\mathbb{Z}\n\\end{equation}\nwhich is always the case leading to\n\\begin{prop}\nThe coset models corresponding to Lie algebra \n$g = \\mathfrak{so}(2r+1)$ and any subalgebra $\\mathfrak{h}$ do not have \nglobal gauge anomalies.\n\\end{prop}\n\n\n\\subsection{Case $C_r = \\mathfrak{sp}(2r)$, $r \\geq 3$}\n\nLie algebra $\\mathfrak{g} = C_r$, corresponding to group \n$\\tilde G=Sp(2r)$, is composed of antihermitian matrices $X$ of\nsize $2r$ such that $\\Omega X$ is symmetric, with $\\Omega$ built \nof $r$ blocks \n\n\\begin{equation}\n\\omega = \\begin{pmatrix}\n0 & -1 \\\\\n1 & 0 \n \\end{pmatrix}\n\\end{equation}\nplaced diagonally. The Cartan algebra $\\mathfrak{t}_{\\mathfrak g}$ may be taken \nas composed of r blocks $t_i \\omega$ placed diagonally. \nLet $e_i\\in\\ii\\mathfrak{t}_{\\mathfrak g}$ \ndenote the matrix corresponding to\n$t_j = \\ii\\delta_{ij}$. With the normalization tr$(e_ie_j) = 2 \\delta_{ij}$, \nroots of $\\mathfrak{g}$ have the form $(1\/2) (\\pm e_i \\pm e_j)$ for \n$i \\neq j$ and\n$\\pm e_i$. The simple roots may be chosen as $\\alpha_i = (1\/2) (e_i - e_{i+1})$\nfor $i = 1, \\ldots r-1$ and $\\alpha_r = e_r$. The center $\\tilde{Z} \\cong \n\\mathbb{Z}_2$ is generated by $z = e^{2 \\ii \\pi \\theta}$ with $\\theta =\n\\lambda^\\vee_r = (1\/2) \\sum_{i=1}^{r} e_i$, and its only nontrivial subgroup is\n$Z = \\tilde{Z}$. We then take $M$ and $\\tilde{M}$ in the\nEuclidian space spanned by vectors $e_i$ of the form \n\\begin{equation}\n M = a \\theta = \\left( \\dfrac{a}{2}, \\ldots, \\dfrac{a}{2} \\right) \\qquad \n \\tilde{M} = \\tilde{a} \\theta = \\left( \\dfrac{\\tilde{a}}{2},  \\ldots,\n\\dfrac{\\tilde{a}}{2} \\right) \\qquad a,\\tilde{a} \\in \\mathbb{Z}.\n\\end{equation}\n \n\\noindent Lie algebra $\\mathfrak{sp}(2r)$ does not have nontrivial\nouter automorphisms. For $\\omega=I\\hspace{-0.03cm}d$,\ntaking into account the normalization of $\\rm tr$, we obtain: \n\\begin{equation}\\label{Quantity_Cr}\n k~\\text{tr} (M \\tilde{M}) = k \\dfrac{a\\tilde{a}r}{2} ,\n\\end{equation}\nensuring the global gauge invariance if it is an integer. The admissible \nlevels $k$ are \n\\begin{eqnarray}\n&&k \\in \\mathbb{Z} \\qquad\\ \\text{\\,if } r \\text{ is even,}\\\\\n&&k \\in 2\\mathbb{Z} \\qquad \\text{if } r \\text{ is odd,}\n\\end{eqnarray}\nsee \\cite{FGK,Gawedzki}, so that the above condition is always satisfied\nleading to  \n\n\\begin{prop}\nThe coset models corresponding to Lie algebra \n$\\mathfrak{g} = \\mathfrak{sp}(2r)$ \nand any subalgebra $\\mathfrak{h}$ do not have global gauge anomalies.\n\\end{prop}\n\n\\subsection{Case $D_r = \\mathfrak{so}(2r)$, $r \\geq 4$}\n\nLie algebra $\\mathfrak{g} = D_r$, corresponding to group \n$\\tilde G=Spin(2r)$, is composed of real antisymmetric\nmatrices of size $2r$. The Cartan algebra $\\mathfrak{t}_{\\mathfrak g}$\nmay be taken as composed of $r$ blocks\n\\begin{equation}\n\\omega = \\begin{pmatrix}\n0 & -t_i \\\\\nt_i & 0 \n \\end{pmatrix}\n\\end{equation}\nplaced diagonally. Let us denote by $e_i\\in\\ii\\mathfrak{t}_{\\mathfrak g}$ \nthe matrix corresponding to $t_j =\\ii\\delta_{ij}$. \nWith the normalization tr$(e_ie_j) =  \\delta_{ij}$, roots of\n$\\mathfrak{g}$ have the form $\\pm e_i \\pm e_j$ for $i \\neq j$, and the simple\nroots may be chosen as $\\alpha_i = e_i - e_{i+1}$ for $i = 1 \\ldots r-1$ and\n$\\alpha_r = e_{r-1} + e_{r}$. \n\n\\paragraph{Case of $r$ odd.}\n\nIf $r$ is odd, the center $\\tilde{Z} \\cong \\mathbb{Z}_4$ is generated by $z =\ne^{2 \\ii \\pi \\theta}$ with $\\theta = \\lambda^\\vee_r = (1\/2) \\sum_{i=1}^{r} e_i$.\nThe possible nontrivial subgroups are \n$Z = \\tilde{Z}$ and $Z \\cong \\mathbb{Z}_2$, \ngenerated by $z^2$. In particular, $Spin(2r)\/\\mathbb{Z}_2=SO(2r)$. \nTaking the general form of $M$ and $\\tilde{M}$ in the Euclidian\nspace spanned by vectors $e_i$, \n\\begin{equation}\\label{MMtilde_Dr_odd}\n\\begin{array} {ll}\n M = a \\theta = \\left( \\dfrac{a}{2}, \\ldots, \\dfrac{a}{2} \\right), & a \\in\n\\mathbb{Z} \\text{ if } Z \\cong \\mathbb{Z}_4\\,, \\\\\n & a \\in 2\\mathbb{Z} \\text{ if } Z \\cong \\mathbb{Z}_2\\,, \\\\\n \\tilde{M} = \\tilde{a} \\theta = \\left( \\dfrac{\\tilde{a}}{2},  \\ldots,\n\\dfrac{\\tilde{a}}{2} \\right), & \\tilde{a} \\in \\mathbb{Z}\\,.\n\\end{array}\n\\end{equation}\nThe admissibility condition for the levels\nin the corresponding WZW models are \\cite{Gawedzki}:\n\\begin{eqnarray}\n\\label{comp_Dr_odd}\n&&k \\in 2\\mathbb{Z} \\text{ if } Z \\cong \\mathbb{Z}_4\\,, \\\\\n&&k \\in \\mathbb{Z}\\hspace{0.06cm}\\ \\text{ if } Z \\cong \\mathbb{Z}_2\\,.\n\\end{eqnarray}\n\n\n\\subsubsection{Untwisted case}\n\nIf $\\omega=I\\hspace{-0.03cm}d$ then the global gauge invariance\nis assured if the quantity \n\\begin{equation}\\label{Quantity_Dr_odd}\n k~\\text{tr} (M \\tilde{M}) = k \\dfrac{a\\tilde{a}r}{4} ,\n\\end{equation}\nis an integer. The latter holds for \n\\begin{eqnarray}\n&&k \\in 4\\mathbb{Z} \\text{ if } Z \\cong \\mathbb{Z}_4\\,, \\\\\n&&k \\in 2\\mathbb{Z} \\text{ if } Z \\cong \\mathbb{Z}_2\\,.\n\\end{eqnarray}\n\n\\noindent Comparing to to the admissibility conditions\n(\\ref{comp_Dr_odd}), we deduce the following \n\\begin{prop}\\label{prop_Drodd_h=g}\nThe untwisted coset models corresponding to Lie algebra $\\mathfrak{g} =\n\\mathfrak{so}(2r)$, $r$ odd, and any subalgebra $\\mathfrak{h}$ \ndo not have global gauge anomalies for\n\\begin{eqnarray}\n&&k \\in 4\\mathbb{Z} \\text{ if } Z \\cong \\mathbb{Z}_4 \\\\\n&&k \\in 2\\mathbb{Z} \\text{ if } Z \\cong \\mathbb{Z}_2.\n\\end{eqnarray}\nThe models with $\\mathfrak h=\\mathfrak g$ and $k \\in 2\\mathbb{Z}$\nwith odd $k\/2$ for $Z \\cong \\mathbb{Z}_4$\nor with $k$ odd for $Z \\cong \\mathbb{Z}_2$ are anomalous.\n\\end{prop}\n\n\\subsubsection{Twisted case}\nThere is only one nontrivial outer automorphism $\\omega$ of \n$\\mathfrak{so}(2r)$ with odd $\\,r$, \\,It exchanges the simple roots \n$\\alpha_{r-1}$ and $\\alpha_{r}$ and does not change the other ones. Thus, \ntaking $M$ and $\\tilde{M}$ given by \\eqref{MMtilde_Dr_odd}, we get\n\\begin{equation}\\label{omegaM_Dr_odd}\n  \\omega(\\tilde{M}) = \\tilde{a}\\,\\omega(\\lambda_r^\\vee) = \\tilde{a} \n\\lambda_{r-1}^\\vee = -\\tilde{a}\\lambda_r^\\vee +\\tilde a q^\\vee \n= - \\tilde M +\\tilde a q^\\vee\n\\end{equation}\nwhere $q^\\vee \\in Q^\\vee(D_r)$. The condition\n\\begin{equation}\ne^{2\\ii\\pi\\tilde M}\\,\\omega(e^{-2\\ii\\pi\\tilde M})=e^{4\\ii\\pi\n\\tilde a\\theta}\\,\\in\\,Z\n\\end{equation}\nis always satisfied whatever the subgroup $Z \\cong \\mathbb Z_4$ \nor $\\mathbb Z_2$ considered. From Eq.\\,(\\ref{cc}), we obtain\n\\begin{eqnarray}\nc_{\\tilde z\\omega(\\tilde z)^{-1},z}=\\exp\\hspace{-0.07cm}\\big[\n\\hspace{-0.05cm}-\\ii\\pi k\\frac{\\tilde a a r}{2}\\big]\n\\end{eqnarray} \nand from Eqs.\\,(\\ref{MMtilde_Dr_odd}) and (\\ref{omegaM_Dr_odd}),\n\\begin{eqnarray}\n\\exp\\hspace{-0.07cm}\\big[\\hspace{-0.05cm}-2\\ii\\pi k\n\\,{\\rm tr}(M\\omega(\\tilde M))\\big]\n=\\exp\\hspace{-0.07cm}\\big[\\hspace{-0.05cm}+\\ii\\pi k\\frac{a\\tilde a r}{2}\\big]\n\\end{eqnarray} \nso that the no-anomaly condition (\\ref{idtytw}) always holds implying\n\n\\begin{prop}\\label{prop_Drodd_h=g_tw}\nThe twisted coset models corresponding to Lie algebra $\\mathfrak{g} \n= \\mathfrak{so}(2r)$, $r$ odd, subgroups $Z \\cong \\mathbb{Z}_4$ or \n$\\mathbb Z_2$, and arbitrary subalgebras $\\mathfrak{h}$ do not have \nglobal gauge anomalies.\n\\end{prop}\n\n\n\\paragraph{Case of $r$ even.}\n\nIf $r$ is even, the center $\\tilde{Z} \\cong \n\\mathbb{Z}_2 \\times \\mathbb{Z}_2$ is\ngenerated by $z_1 = e^{2 \\ii \\pi \\theta_1}$ with $\\theta_1 = \n\\lambda^\\vee_r = (1\/2)\\sum_{i=1}^{r} e_i$\nand $z_2 = e^{2 \\ii \\pi \\theta_2}$ with $\\theta_2=\\lambda^\\vee_1 =\ne_1$. The possible nontrivial subgroups are given in Table \\ref{Subgroups_Dr}.\n\\begin{table}[htb]\n\\centering\n \\begin{tabular}{|c|c|c|}\n\\hline\n Subgroup $Z$ & Type & Generator(s) $z_i$ \\\\\n\\hline \\hline\n$\\tilde{Z}$ & $\\mathbb{Z}_2 \\times \\mathbb{Z}_2$ & $z_1,\\,z_2$ \\\\\n \\hline\n$Z_1 := \\mathbb{Z}_2 \\times \\lbrace 1 \\rbrace$ & $\\mathbb{Z}_2$ & $z_1$ \\\\\n\\hline \n$ Z_2 := \\lbrace 1 \\rbrace \\times \\mathbb{Z}_2 $ & $\\mathbb{Z}_2$  & $z_2$\n\\\\\n\\hline \n$ Z_{\\rm diag}$ & $\\mathbb{Z}_2$ & $z_1z_2$ \\\\\n\\hline\n \\end{tabular}\n\\caption{Subgroups of $\\tilde Z(Spin(2r)) \\cong \\mathbb{Z}_2 \\times\n\\mathbb{Z}_2$, $r$ even, and their generators. }\n\\label{Subgroups_Dr}\n\\end{table}\n\n\\noindent Here, $SO(2r)=Spin(2r)\/Z_2$. The general form \nof $M$ and $\\tilde{M}$ in the Euclidian space spanned\nby vectors $e_i$ is\n\\begin{equation}\\label{MMtilde_Dr_even}\n\\begin{array} {ll}\n M = a_1 \\theta_1 + a_2 \\theta_2 = \\left(\\dfrac{a_1}{2} + a_2, \\dfrac{a_1}{2},\n\\ldots, \\dfrac{a_1}{2} \\right), & a_1,a_2 \\in \\mathbb{Z} \\text{ if } Z =\n\\tilde{Z}, \\\\\n & a_1 \\in \\mathbb{Z}, a_2 = 0 \\text{ if } Z = Z_1, \\\\\n & a_1= 0 , a_2 \\in \\mathbb{Z} \\text{ if } Z = Z_2, \\\\\n & a_1=a_2 \\in \\mathbb{Z} \\text{ if } Z = Z_{\\rm diag}, \\\\\n \\tilde{M} = \\tilde{a}_1 \\theta_1 + \\tilde{a}_2 \\theta_2 =\n\\left(\\dfrac{\\tilde{a}_1}{2} + \\tilde{a}_2, \\dfrac{\\tilde{a}_1}{2},  \\ldots,\n\\dfrac{\\tilde{a}_1}{2} \\right), & \\tilde{a}_1, \\tilde{a}_2 \\in \\mathbb{Z}.\n\\end{array}\n\\end{equation}\nIn this case, the conditions for admissible levels of the WZW model are \n\\cite{Gawedzki}:\n\\begin{equation}\\label{Consistency_Dreven}\n\\begin{array}{lcl}\n k \\in \\mathbb{Z} & \\text{ if } & r\/2 \\text{ is even for any } Z, \\\\\n& & r\/2 \\text{ is odd for } Z = Z_2,\\\\\n k \\in 2\\mathbb{Z} & \\text{ if } & r\/2 \\text{ is odd and } Z = \\tilde{Z}, Z_1\n\\text{ or } Z_\\text{diag}. \\\\\n\\end{array}\n\\end{equation}\n\n\\subsubsection{Untwisted case}\n\nIf $\\omega=I\\hspace{-0.03cm}d$ then the global gauge invariance\nis assured if\n\\begin{equation}\\label{Quantity_Dr_even}\n k~\\text{tr} (M \\tilde{M}) = k \\left( \\dfrac{a_1 \\tilde{a}_1 r}{4} + \\dfrac{a_1\n\\tilde{a}_2}{2} + \\dfrac{a_2 \\tilde{a}_1}{2}+a_2\\tilde a_2 \\right) ,\n\\end{equation}\nis an integer. This holds\nfor $k \\in 2\\mathbb{Z}$, whatever the subgroup considered. Comparing  \nto the admissibility conditions (\\ref{Consistency_Dreven}), \nwe deduce the following \n\n\\begin{prop}\\label{prop_Dreven_h=g}\nThe untwisted coset models corresponding to Lie algebra $\\mathfrak{g} =\n\\mathfrak{so}(2r)$, $r$ even, and any subalgebra $\\mathfrak{h}$ \ndo not have global gauge anomalies if $k \\in 2\\mathbb{Z}$. The models\nwith $\\mathfrak h=\\mathfrak g$ and with $k$ odd for $r\/2$ even and any \nnontrivial $Z$, or with $k$ odd for \n$r\/2$ odd and $Z = Z_2$, are anomalous.\n\\end{prop}\n\n\\subsubsection{Twisted case}\n\\label{sec:D_reven_tw}\n\nFor $r>4$, there is only one nontrivial outer automorphism $\\omega$ \nof $\\mathfrak{so}(2r)$, which is the same as the one described in \nthe case of $r$ odd: it interchanges the simple roots $\\alpha_{r-1}$ \nand $\\alpha_r$. Thus, taking $M$ and \n$\\tilde{M}$ given by \\eqref{MMtilde_Dr_even}, we get\n\\begin{eqnarray}\\label{omegaM_Dr_even}\n\\omega (\\tilde{M})&=&\\tilde{a}_1\\omega(\\theta_1) + \\tilde{a}_2\n\\hspace{0.02cm}\\omega(\\theta_2)\\,=\\,\n\\tilde{a}_1 \\lambda_{r-1}^\\vee + \\tilde{a}_2\\lambda_1^\\vee \\cr\n&=& \\tilde{a}_1\\lambda_r^\\vee + (\\tilde{a}_1+\\tilde{a}_2)\\lambda_1^\\vee + \n\\tilde a_1q^\\vee\\,=\\,\n\\tilde{M} + \\tilde{a}_1\\theta_2 + \\tilde a_1q^\\vee\n\\end{eqnarray}\nwhere $q^\\vee \\in Q^\\vee(D_r)$. The condition\n\\begin{equation}\ne^{2\\ii\\pi\\tilde M}\\,\\omega(e^{-2\\ii\\pi\\tilde M})\n=e^{-2\\ii\\pi\\tilde a_1\\theta_2}\\,\\in\\,Z\n\\end{equation}\nis satisfied for arbitrary $\\tilde a_1$ if $Z = \\tilde Z$ or \n$Z_2$, and for $\\tilde a_1=0\\,{mod}\\,2$ if $Z = Z_1$ or $Z_{diag}$. \nFor $Z=\\tilde Z$, the expression\nfor  bihomomorphism (\\ref{bihom})\nextracted from \\cite{FGK} reads:\n\\begin{eqnarray}\nc_{z_1^{m_1}z_2^{m_2},z_1^{n_1}z_2^{n_2}}=\\Big(\\hspace{-0.1cm}\n\\pm \\exp\\hspace{-0.07cm}\\big[\\dfrac{\\ii\\pi k}{2}\\big]\n\\Big)^{m_1n_2-m_2n_1}\\,\n\\exp\\hspace{-0.07cm}\\big[\\hspace{-0.05cm}-\\frac{\\ii\\pi k}{2}(m_1n_1 \n\\dfrac{r}{2}+m_1n_2+m_2n_1+2m_2n_2)\\big]\n\\label{pmbihol}\n\\end{eqnarray}\nfor $m_i,n_i\\in\\mathbb Z$, with the sign $\\pm$ corresponding to the two choices\nof WZ action functional. For the cyclic subgroups of $\\tilde Z$,\nthe above expression reduces to the one given by Eq.\\,(\\ref{cc}). \nWe have:\n\\begin{equation}\n  c_{\\tilde z\\omega(\\tilde z)^{-1},z}=\\big(\\pm 1)^{a_1\\tilde\n    a_1}\\,\\exp[\\ii\\pi k( a_1 \\tilde a_1 +  a_2 \\tilde a_1)]\n\\end{equation}\n\n\\noindent and, from Eqs.\\,(\\ref{MMtilde_Dr_even}) and (\\ref{omegaM_Dr_even}),\n\\begin{eqnarray}\n\\exp[-2\\ii\\pi k\\,{\\rm tr}(M\\omega(\\tilde M))]\n=\\exp\\hspace{-0.07cm}\\big[\\hspace{-0.05cm}-\\ii\\pi k \n\\big( (\\dfrac{r}{2}+1) a_1 \\tilde a_1 \n+ a_1 \\tilde a_2 + a_2 \\tilde a_1\\big)\\big].\n\\end{eqnarray} \nHence the no-anomaly condition \\eqref{idtytw} requires that\n\\begin{equation}\n\\label{Dreven_tw_c}\n  (\\pm 1)^{a_1 \\tilde a_1}\\exp\\hspace{-0.07cm}\\big[\\hspace{-0.05cm}\n- \\ii \\pi k \\big(\\dfrac{r}{2} a_1\n    \\tilde a_1 + a_1 \\tilde a_2 \\big)\\big] = 1\n\\end{equation}\nConsidering each subgroup $Z$ and the corresponding values of\n$a_1,\\,a_2,\\,\\tilde a_1,$ and $\\tilde a_2$, and recalling the\nconditions \\eqref{Consistency_Dreven} for the admissible levels \nof the corresponding WZW model, we deduce the\n\n\\begin{prop}\\label{prop_Dreven_h=g_tw}\nThe twisted coset model corresponding to Lie algebra $\\mathfrak{g} \n= \\mathfrak{so}(2r)$, $r>4$ even and arbitrary subalgebra \ndo not have anomalies for $Z = \\tilde Z$ (+ theory), $Z_1$ and\n$Z_{diag}$ if $k$ is even, and for $Z = Z_2$ if $k \\in \\mathbb Z$. \nThe twisted models with $\\mathfrak h = \\mathfrak g$ for \n$Z=\\tilde Z$ (- theory) and $k$ even, and for $Z = \\tilde Z$, $Z_1$ or \n$Z_{diag}$ and $k$ odd, $r\/2>2$ even, are anomalous.\n\\end{prop}\n\nFor $r=4$, there are more nontrivial outer automorphisms, because\nthe symmetries of the diagram of $D_4$ form the permutation group $S_3$\n(the well known ``triality''). They belong to two conjugacy classes, \nthe one composed of cyclic permutations of order 2,\n\\begin{equation}\n\\omega_1:\\,\\alpha_3\\rightarrow\\alpha_4\\rightarrow\\alpha_3\\,,\n\\qquad \\omega_2:\\,\\alpha_1\\rightarrow\\alpha_3\\rightarrow\\alpha_1\\,,\n\\qquad \\omega_3:\\,\\alpha_1\\rightarrow\\alpha_4\\rightarrow\\alpha_1\\,,\n\\label{omegas}\n\\end{equation}\nand the one containing cyclic permutations of order 3,\n\\begin{equation}\n\\omega_4:\\,\\alpha_1 \\rightarrow \\alpha_4 \\rightarrow \n\\alpha_3\\rightarrow \\alpha_1\\,,\\qquad\\omega_4^{-1}:\\,\n\\alpha_1 \\rightarrow \\alpha_3 \\rightarrow \n\\alpha_4\\rightarrow \\alpha_1\\,.\n\\label{permutalpha}\n\\end{equation}\nThe no-anomaly conditions for twists $\\omega$ and \n$\\omega'\\omega\\hspace{0.04cm}\\omega'^{-1}$ in the same conjugacy class\nare related, as was discussed at the end of Sec.\\,\\ref{sec:2}:\nthey coincide if in the latter case subgroup $Z\\subset\\tilde Z$ is \nreplaced by $\\omega'(Z)$. The only exception is \nthe case $Z=\\tilde Z$ for odd $k$ where one has also to interchange  \nthe $\\pm$ theories if $\\omega'$ is cyclic of order 2. \nIt is straightforward to see that\n\\begin{equation}\n\\omega_4\\omega_1\\omega_4^{-1}=\\omega_2\\,,\\qquad\\omega_4^{-1}\\omega_1\\omega_4=\n\\omega_3\n\\end{equation}\nand\n\\begin{equation}\n\\omega_4(Z_1)=Z_{diag}\\,,\\qquad\\omega_4(Z_2)=Z_1\\,,\\qquad\\omega_4(Z_{diag})\n=Z_2\\,.\n\\label{permutZ}\n\\end{equation}\nThe results of Proposition \\ref{prop_Dreven_h=g_tw} still hold for $r=4$ \nand twist $\\omega_1$ and the ones for $r=4$ and twists $\\omega_2$ and \n$\\omega_3$ follow from the latter by using the above remark \n(or by a direct calculation) giving:  \n\n\\begin{prop}\\label{prop_D4_h=g_tw}\nThe twisted coset models corresponding to Lie algebra $\\mathfrak{g} \n= \\mathfrak{so}(8)$ with twist $\\omega_1$ and arbitrary subalgebra \ndo not have anomalies for $Z = \\tilde Z$ (+ theory), $Z_1$ and\n$Z_{diag}$ if $k$ is even, and for $Z = Z_2$ if $k \\in \\mathbb Z$. \nThe models with $\\mathfrak h = \\mathfrak g$ for $Z=\\tilde Z$ (- theory)\nand $k$ even, and for $Z = \\tilde Z$, $Z_1$ or $Z_{diag}$ and $k$ odd are \nanomalous. The results for twist $\\omega_2$ ($\\omega_3$) are as \nthe ones for twist $\\omega_1$ except for the permutation (\\ref{permutZ})\nof the subgroups $Z\\rightarrow\\omega_4(Z)$ ($Z\\rightarrow\\omega_4^{-1}(Z))$.\n\\end{prop}\n\n\n\\noindent For the cyclic outer automorphism $\\omega_4$ of order 3,\ntaking $M$ and $\\tilde{M}$ given \nby Eqs.\\,\\eqref{MMtilde_Dr_even}, we obtain:\n\\begin{equation}\\label{omega2M_Dr_even}\n  \\omega_4 (\\tilde{M}) =\n (\\tilde{a}_1 + \\tilde{a}_2) \\theta_1 + \\tilde{a}_1 \n\\theta_2 + \\tilde a_1 q^\\vee\n\\end{equation}\nwhere $q^\\vee \\in Q^\\vee(D_4)$. The condition\n\\begin{equation}\ne^{2\\ii\\pi\\tilde M}\\,\\omega_4(e^{-2\\ii\\pi\\tilde M})=\\exp[2\\ii\\pi(-\\tilde\n  a_2\\theta_1+ (\\tilde a_2 - \\tilde a_1) \\theta_2)]\\,\\in\\,Z\n\\end{equation}\nis satisfied for arbitrary $\\tilde a_1, \\tilde a_2$ if $Z = \\tilde Z$,\nand for $\\tilde a_1 =\\tilde a_2$, $\\tilde a_2 = 0$, $\\tilde a_1 = 0$,\nall $mod\\,2$, if $Z = Z_1,\\, Z_2$ or $Z_{diag}$ respectively. \nExpression (\\ref{pmbihol}) for the bihomomorphism gives here:\n\\begin{equation}\n   c_{\\tilde z\\hspace{0.02cm}\\omega_4(\\tilde z)^{-1},z}\n= (\\pm 1)^{-a_2 \\tilde a_2 + a_1\n     \\tilde a_1 - a_1 \\tilde a_2}\\exp[\\ii \\pi k(a_1\\tilde a_1+ a_2 \n\\tilde a_1 - a_2 \\tilde a_2)]\n\\end{equation}\nFrom Eqs.\\,(\\ref{MMtilde_Dr_even}) and (\\ref{omegaM_Dr_even}),\n\\begin{eqnarray}\n\\exp[-2\\ii\\pi k\\,{\\rm tr}(M\\omega_4(\\tilde M))]\n=\\exp\\hspace{-0.07cm}\\big[\\hspace{-0.05cm}-\\ii\\pi k \n\\big( a_1 \\tilde a_1 + a_2 \\tilde a_1 + a_2 \\tilde\n  a_2 \\big)\\big]\n\\end{eqnarray} \nso that the no-anomaly condition \\eqref{idtytw} becomes\n\\begin{equation}\n  (\\pm 1)^{-a_2 \\tilde a_2 - a_1\n     \\tilde a_2 + a_1 \\tilde a_1}= 1.\n\\end{equation}\nConsidering each subgroup $Z$ and the corresponding values of\n$a_1,\\,a_2,\\,\\tilde a_1,$ and $\\tilde a_2$, and recalling the\nadmissible values \\eqref{Consistency_Dreven} of the level, we deduce\n\n\\begin{prop}\\label{prop_D4_h=g_tw2}\nThe twisted coset models corresponding to Lie algebra $\\mathfrak{g} \n= \\mathfrak{so}(8)$, outer automorphism $\\omega_4$ and arbitrary\nsubalgebra do not have anomalies for $Z=\\tilde Z$ (+ theory) and\n$Z=Z_1,\\,Z_2$ or $Z_{diag}$. The models with $\\mathfrak h = \\mathfrak g$ \nand $Z = \\tilde Z$ (- theory) is anomalous.  \n\\end{prop}\n\n\\noindent The results for the twist $\\omega_4^{-1}$ may be deduced from\nthe above proposition if we observe that $\\omega_4^{-1}$ may be obtained\nfrom $\\omega_4$ by the conjugation by any cyclic outer automorphism\n$\\omega'$ of order 2. Hence the conditions for the absence or the presence\nof anomalies for the theory twisted by $\\omega_4^{-1}$ are as for\nthe ones for the twist $\\omega_4$ except for the exchange of the $\\pm$\ntheories for $Z=\\tilde Z$ and $k$ odd leading to\n\n\\begin{prop}\\label{prop_D4_h=g_tw3}\nThe twisted coset models corresponding to Lie algebra $\\mathfrak{g} \n= \\mathfrak{so}(8)$, outer automorphism $\\omega_4^{-1}$ and arbitrary\nsubalgebra do not have anomalies for $Z=\\tilde Z$ (($-$)${}^k$ theory) and\n$Z=Z_1,\\,Z_2$ or $Z_{diag}$. The models with $\\mathfrak h = \\mathfrak g$, \nand $Z = \\tilde Z$ (($-$)${}^{k+1}$ theory) is anomalous.  \n\\end{prop}\n\n\\noindent This may be confirmed by a direct calculation.\n\n\n\n\\subsection{Case $\\mathfrak{e}_6$}\n\nThe imaginary part $\\,\\ii\\mathfrak{t}_{\\mathfrak g}\\,$ of the \ncomplexification of the Cartan \nsubalgebra $\\mathfrak{t}_{\\mathfrak g}$ of $\\mathfrak{g} = \\mathfrak{e}_6$ \nmay be identified \nwith the subspace of $\\mathbb{R}^{7}$ orthogonal to the vector \n$(1, \\ldots, 1, 0)$, with the scalar product inherited from \n$\\mathbb{R}^{7}$. The simple roots may be\ntaken as $\\alpha_i = e_i - e_{i+1}$ for $i=1 \\ldots 5$ and $\\alpha_6 =\n(1\/2)(-e_1-e_2-e_3+e_4+e_5+e_6) + (1\/\\sqrt{2})e_7$, where $e_i$ are the vectors\nof the canonical basis of $\\mathbb{R}^{7}$. The center $\\tilde{Z} \\cong\n\\mathbb{Z}_3$ is generated by $z = e^{2 \\ii \\pi \\theta}$ with $\\theta =\n\\lambda^\\vee_5 = (1\/6)(e_1 +e_2+e_3+e_4+e_5-5e_6) + (1\/\\sqrt{2})e_7$. The only\nnontrivial subgroup is $Z = \\tilde{Z}$. The general form of $M$ and\n$\\tilde{M}$ in the Euclidian space spanned by vectors $e_i$ is\n\n\\begin{equation}\\label{MMtilde_e6}\n\\begin{array}{l}\n M = a \\theta = \\left( \\dfrac{a}{6},\\ldots ,\\dfrac{a}{6},\\dfrac{-5a}{6},\n\\dfrac{a}{\\sqrt{2}} \\right) \\qquad  a \\in \\mathbb{Z},\\\\\n \\tilde{M} = \\tilde{a} \\theta = \\left( \\dfrac{\\tilde{a}}{6},\\ldots\n,\\dfrac{\\tilde{a}}{6},\\dfrac{-5\\tilde{a}}{6}, \\dfrac{\\tilde{a}}{\\sqrt{2}}\n\\right) \\qquad \\tilde{a} \\in \\mathbb{Z}.\n\\end{array}\n\\end{equation}\n\n\\subsubsection{Untwisted case}\n\nIf $\\omega=I\\hspace{-0.03cm}d$ then the global gauge invariance\nis assured if\n\\begin{equation}\\label{Quantity_e6}\n k~\\text{tr} (M \\tilde{M}) = k \\dfrac{4a\\tilde{a}}{3} ,\n\\end{equation}\nis an integer. This holds\nfor $k \\in 3\\mathbb{Z}$. Since all integer levels $k \\in \\mathbb{Z}$ \nare admissible \\cite{FGK,Gawedzki}, \nwe deduce \n\n\\begin{prop}\nThe untwisted coset models corresponding to Lie algebra $\\mathfrak{g} =\n\\mathfrak{e}_6$ and arbitrary subalgebra $\\mathfrak{h}$ do not have global \ngauge anomalies if $k \\in 3\\mathbb{Z}$. The models $Z=\\mathbb Z_3$, \n$\\mathfrak h=\\mathfrak g$ and $k \\in\n\\mathbb{Z} \\setminus 3\\mathbb{Z}$ are anomalous.\n\\end{prop}\n\n\\subsubsection{Twisted case}\nThere is only one nontrivial outer automorphism $\\omega$ of \n$\\mathfrak{e}_6$, which exchanges the simple roots $\\alpha_1$ \nand $\\alpha_2$ with $\\alpha_5$ and $\\alpha_4$ and does not change \nthe other ones. Thus, taking $M$ and $\\tilde{M}$ given by \n\\eqref{MMtilde_e6}, we get\n\\begin{equation}\\label{omegaM_e6}\n  \\omega(\\tilde{M}) = \\tilde{a}\\hspace{0.02cm}\\omega(\\lambda_5^\\vee) \n= \\tilde{a} \\lambda_{1}^\\vee = -\\tilde{a}\\lambda_5^\\vee + \\tilde a q^\\vee \n= - \\tilde M + \\tilde aq^\\vee\n\\end{equation}\nwhere $q^\\vee \\in Q^\\vee(\\mathfrak e_6)$. The condition\n\\begin{equation}\ne^{2\\ii\\pi\\tilde M}\\,\\omega(e^{-2\\ii\\pi\\tilde M})=e^{4\\ii\\pi\\tilde a\\theta}\\,\\in\\,Z\n\\end{equation}\nis always satisfied for $Z = \\tilde{Z}$. From Eq.\\,(\\ref{cc}), we obtain\n\\begin{eqnarray}\nc_{\\tilde z\\omega(\\tilde z)^{-1},z}=\\exp[-2\\ii\\pi k\\frac{4 \\tilde a a}{3}]\n\\end{eqnarray} \nand from Eqs.\\,(\\ref{MMtilde_e6}) and (\\ref{omegaM_e6}),\n\\begin{eqnarray}\n\\exp[-2\\ii\\pi k\\,{\\rm tr}(M\\omega(\\tilde M))]\n=\\exp\\hspace{-0.07cm}\\big[\\hspace{-0.05cm}+2\\ii\\pi k\\frac{4 a\\tilde a }{3}\n\\big]\n\\end{eqnarray} \nso that the no-anomaly condition (\\ref{idtytw}) always holds implying\n\n\\begin{prop}\\label{prop_e6_h=g_tw}\nThe twisted coset models corresponding to Lie algebra $\\mathfrak{g} \n= \\mathfrak e_6$, subgroup $Z \\cong \\mathbb{Z}_3$ and arbitrary subalgebras \n$\\mathfrak{h}$ do not have global gauge anomalies.\n\\end{prop}\n\n\n\\subsection{Case $\\mathfrak{e}_7$}\n\nThe imaginary part $\\,\\ii\\mathfrak{t}_{\\mathfrak g}\\,$ of the \ncomplexification of the Cartan \nsubalgebra $\\mathfrak{t}_{\\mathfrak g}$ of $\\mathfrak{g} = \\mathfrak{e}_7$ \nmay be \nidentified with the\nsubspace of $\\mathbb{R}^{8}$ orthogonal to the vector $(1, \\ldots, 1)$ with the\nsimple roots $\\alpha_i = e_i - e_{i+1}$ for $i=1 \\ldots 6$ and $\\alpha_7 =\n(1\/2)(-e_1-e_2-e_3-e_4+e_5+e_6+e_7+e_8)$, where $e_i$ are the vectors of the\ncanonical basis of $\\mathbb{R}^{8}$. The center $\\tilde{Z} \\cong \\mathbb{Z}_2$\nis generated by $z = e^{2 \\ii \\pi \\theta}$ with $\\theta = \\lambda^\\vee_1 =\n(1\/4)(3,-1,\\ldots, -1, 3)$. The only nontrivial subgroup is $Z = \\tilde{Z}$.\nThe general form of $M$ and $\\tilde{M}$ in the Euclidian space generated\nby $e_i$ is \n\\begin{equation}\n\\begin{array}{l}\n M = a \\theta = \\left( \\dfrac{3a}{4},\\dfrac{-a}{4}, \\ldots, \\dfrac{-a}{4},\n\\dfrac{3a}{4} \\right) \\qquad  a \\in \\mathbb{Z},\\\\\n \\tilde{M} = \\tilde{a} \\theta = \\left( \\dfrac{3\\tilde{a}}{4},\n\\dfrac{-\\tilde{a}}{4}, \\ldots,\\dfrac{-\\tilde{a}}{4}, \\dfrac{3\\tilde{a}}{4}\n\\right) \\qquad \\tilde{a} \\in \\mathbb{Z}.\n\\end{array}\n\\end{equation}\nLie algebra $\\mathfrak{e}_7$ does not have nontrivial outer automorphisms\nso that we may take $\\omega=I\\hspace{-0.03cm}d$.\nThe global gauge invariance is then assured if the quantity\n\\begin{equation}\\label{Quantity_e7}\n k~\\text{tr} (M \\tilde{M}) = k \\dfrac{3a\\tilde{a}}{2} ,\n\\end{equation}\nis an integer. This holds\nfor $k \\in 2\\mathbb{Z}$. The condition for admissible levels \nalso requires in this case that $k \\in 2\\mathbb{Z}$ \\cite{FGK,Gawedzki} \nso that we deduce: \n\n\\begin{prop}\nThe coset models corresponding to Lie algebra $\\mathfrak{g} =\n\\mathfrak{e}_7$ and any subalgebra $\\mathfrak{h}$ do not have global\ngauge anomalies.\n\\end{prop}\n\n\n\n\\subsection{Case $\\mathfrak{g}_2$, $\\mathfrak{f}_4$ and $\\mathfrak{e}_8$}\n\n\nThe center of the simply connected groups  corresponding to Lie\nalgebras  $\\mathfrak{g} =\n\\mathfrak{g}_2, \\mathfrak{f}_4$ or $\\mathfrak{e}_8$ is trivial~: \n$\\tilde{Z} \\cong \\lbrace 1\\rbrace$ so that there are no nontrivial\nsubgroups $Z$ in that case and we infer:\n\n\\begin{prop}\nThe coset models corresponding to Lie algebras $\\mathfrak{g} =\n\\mathfrak{g}_2, \\mathfrak{f}_4$ or $\\mathfrak{e}_8$ and any subalgebra\n$\\mathfrak{h}$ do not have global gauge anomalies.\n\\end{prop}\n\n\n\\section{Regular subalgebras}\n\\label{sec:4}\n\nLooking back at the previous section, the global gauge anomalies of the coset\nmodels may appear only for $\\mathfrak{g} = A_r$, $D_r$ \nand $\\mathfrak{e}_6$ in the untwisted case, and only for $\\mathfrak{g}=D_r$\nwith even $r$ in the twisted case (note that these are all simply laced \nLie algebras). Now we have to specify the Lie subalgebra $\\mathfrak{h}$ of \na simple algebra $\\mathfrak g $ to see in which cases the anomalies \nsurvive the restriction of the symmetry group. The first class of \nsemisimple subalgebras that we shall consider are the regular ones, \nintroduced by Dynkin in \\cite{Dynkin}. A Lie subalgebra $\\mathfrak{h}$ of \nan algebra $\\mathfrak{g}$ is called regular if, for a choice of the Cartan \nsubalgebra $\\,t_{\\mathfrak g}\\subset\\mathfrak g\\,$\n(defined up to conjugation), it's complexification is of the form\n\\begin{equation}\n\\mathfrak h^{\\mathbb C}\\,=\\,\\mathfrak t_{\\mathfrak\nh}^{\\mathbb C}\\oplus\\Big(\\mathop{\\oplus}\\limits_{\\alpha\\in\n\\Delta_{\\mathfrak h}\\subset\\Delta_{\\mathfrak g}}\\mathbb C e_\\alpha\\Big)\n\\end{equation}\nwhere $\\mathfrak t_{\\mathfrak h}\\subset\\mathfrak t_{\\mathfrak g}$ is\na Cartan subalgebra of $\\mathfrak{h}$. Subalgebra $\\,\\mathfrak h\\,$ \nis semisimple if $\\,\\alpha\\in\\Delta_{\\mathfrak h}\\,$ implies that \n$\\,-\\alpha\\in\\Delta_{\\mathfrak h}\\,$ and if \n$\\,\\alpha\\in\\Delta_{\\mathfrak h}\\,$  span $\\,\\mathfrak t_{\\mathfrak h}^{\\mathbb C}$.\n$\\,\\Delta_{\\mathfrak h}\\,$ is then the set of roots of $\\,\\mathfrak h$. \n\n\\paragraph{Construction of regular subalgebras.}\nThere is a nice diagrammatic method to obtain all the regular \nsemisimple subalgebras \nof a given semisimple algebra (up to conjugation), \nproposed by Dynkin in \\cite{Dynkin} and summarized in \\cite{Lorente}. \nWe briefly describe it here:\n\n\\begin{enumerate}\n \\item Take the Dynkin diagram of the ambient algebra $\\mathfrak{g}$, \nand adjoin to it a node corresponding to the lowest root $\\delta=-\\phi$ \n(negative of the highest root $\\phi$) of $\\mathfrak{g}$, obtaining the \nextended Dynkin diagram of $\\mathfrak{g}$.\n\n\\item Remove arbitrarily one root from this diagram, in order to obtain \nat most $r+1$ different diagrams, which may split into orthogonal subdiagrams.\n\\item Reapply the firsts two steps to each connected subdiagram obtained\nabove, until no new diagram appears. This way one gets all the regular \nsubalgebras $\\mathfrak{h} \\subset \\mathfrak{g}$ of maximal rank.\n\\item Remove again an arbitrarily root from each diagram, and apply the full \nprocedure to each connected subdiagram obtained this way (including the\nlast step).\n\\end{enumerate}\nThe algorithm stops when no root can be removed, hence one will obtain \nall the regular subalgebras of $\\mathfrak{g}$.\n\n\\subsection{Regular semisimple subalgebras of $A_r$}\n\nThe semisimple regular subalgebras of $A_r$ are \ngiven in \\cite{Dynkin} (Chapter II, Table 9) and have the form: \n\n\\begin{equation}\\label{regular_subalg_Ar}\n \\mathfrak{h} = A_{r_1} \\oplus \\ldots \\oplus A_{r_m}, \\qquad r_1 + 1 + \\ldots +\nr_m + 1 \\leq r+1\n\\end{equation}\n\n\\noindent The embedding of $\\mathfrak{h}$ in $\\mathfrak{g}$ realizing\nthe ideals $A_{r_i}$ as diagonal blocks in the matrices of $A_{r}$\nis unique up to an inner automorphism of $A_r$.\nTaking $M$ and $\\tilde{M}$ as given in Eqs. \\eqref{M_Ar} and \\eqref{tildeM_Ar} \nwe must require that $\\tilde M+q^\\vee\\in\\ii\\mathfrak{t}_{\\mathfrak h}$, \nfor some $q^\\vee\\in Q^\\vee(A_r)$. Looking block by block, we obtain \nthe conditions \n\\begin{equation}\n\\label{condA}\n \\dfrac{\\tilde{a}(r_i+1)}{r+1} \\in \\mathbb{Z} \\qquad \\forall i = 1, \\ldots, m\n\\end{equation}\nand that\n\\begin{equation}\n\\dfrac{\\tilde{a}}{r+1}\\in\\mathbb Z\n\\end{equation}\nif the inequality in (\\ref{regular_subalg_Ar}) is strict. The latter condition\nimplies that (\\ref{Quantity_Ar}) holds eliminating possible global \ngauge anomalies. We may then limit ourselves to the case when the inequality \nin (\\ref{regular_subalg_Ar}) is saturated. this implies that \nFor $i = 1, \\ldots, m$, we may then rewrite conditions (\\ref{condA}) as \n\\begin{equation}\n \\tilde{a}(r_i+1) = q_i (r+1) \\qquad q_i \\in \\mathbb{Z}.\n\\label{lst}\n\\end{equation}\nIn what follows, we shall denote by, respectively, \n$u_1 \\wedge\\cdots\\wedge u_n$ and $u_1\\vee\\cdots\\vee u_n$ the greatest \ncommon divisor and the least common multiple of $u_1,\\dots,u_n$. \nDividing both sides of Eq.\\,(\\ref{lst}) by $(r+1) \\wedge (r_i+1)$, we get\n\\begin{equation}\n \\tilde{a}\\frac{r_i+1}{(r+1) \\wedge (r_i+1)}= q_i\\frac{r+1}{(r+1) \n\\wedge (r_i+1)}\n\\end{equation}\nso that $\\frac{r+1}{(r+1) \n\\wedge (r_i+1)}|\\frac{\\tilde{a}\\,(r_i+1)}{(r+1)\\wedge(r_i+1)}$. Using the \nfact that $\\frac{r+1}{(r+1)\\wedge (r_i+1)}$ and \n$\\frac{r_i+1}{(r+1)\\wedge(r_i+1)}$ are relatively prime, we infer \nthat $\\frac{r+1}{(r+1) \n\\wedge (r_i+1)}|\\tilde a$, i.e. that\n\\begin{equation}\n \\tilde{a} \\in \\dfrac{r+1}{(r+1) \\wedge (r_i+1)}\\mathbb{Z} \\qquad \\forall i\n= 1, \\ldots, m\n\\end{equation}\nwhich leads, according to Proposition \\ref{lcm} of Appendix \\ref{app:2}, \nto the condition\n\\begin{equation}\n \\tilde{a} \\in \\left(\\dfrac{r+1}{(r+1) \\wedge (r_1+1)} \\vee \\cdots \\vee\n\\dfrac{r+1}{(r+1) \\wedge (r_m+1)} \\right) \\mathbb{Z}\n\\end{equation}\nThis property can be reformulated, using Proposition\n\\ref{lcmfrak} of Appendix \\ref{app:2}, as\n\\begin{equation}\\label{Condition_regular_Ar}\n \\tilde{a} \\in \\left(\\dfrac{r+1}{(r+1) \\wedge (r_1+1) \\wedge \\cdots \\wedge\n(r_m+1)} \\right) \\mathbb{Z}\n\\end{equation}\nSince we assumed that $r_1 + 1 + \\ldots + r_m + 1 = r+1$,\ncondition (\\ref{Condition_regular_Ar}) may be simplified to \n\\begin{equation}\n \\tilde{a} \\in \\left(\\dfrac{r+1}{(r_1+1) \\wedge \\cdots \\wedge (r_m+1)} \\right)\n\\mathbb{Z}\n\\end{equation}\nIn order to guarantee that the quantity \\eqref{Quantity_Ar}\nis an integer for every $a$ and $\\tilde{a}$, ensuring the global \ngauge invariance, it is enough to compute it for $a = 1$ and\n\\begin{equation}\n \\tilde{a} = \\dfrac{r+1}{(r_1+1) \\wedge \\cdots \\wedge (r_m+1)}\\,.\n\\end{equation}\nDenoting $(r_1+1) \\wedge \\ldots \\wedge\n(r_m+1)=l$, and $r+1 = pq$, the quantity \\eqref{Quantity_Ar} becomes\n\\begin{equation}\n k~\\text{tr} (M \\tilde{M}) = k \\dfrac{r q}{{l}} = k r\\dfrac{q\/(q\\wedge\n{l})}{{l}\/(q\\wedge {l})}.\n\\label{lst1}\n\\end{equation}\nFinally, recalling that ${l} | (r+1)$ and, consequently, \n$\\frac{l}{q\\wedge {l}}$ and $r$ are relatively prime, \nwe infer that the right hand side of Eq.\\,(\\ref{lst1})\nis be an integer if and only if \n\\begin{equation}\n k \\in \\dfrac{{l}}{q \\wedge {l}} \\mathbb{Z}\\,.\n\\end{equation}\nTaking into account condition \\eqref{Consistency_Ar}  for admissible\nlevels, we are now able to state \n\\begin{prop}\nThe untwisted coset models built with Lie algebra $\\mathfrak{g} = A_r$, \nsubgroup $Z \\cong \\mathbb{Z}_p$ for $ (r+1)=pq$ and any regular subalgebra\n$\\mathfrak{h} = A_{r_1} \\oplus \\ldots \\oplus A_{r_m}$ do not have global \ngauge anomalies for \n\\begin{itemize}\n \\item $r_1+1 + \\ldots r_m+1 < r+1 \\qquad k \\in \n\\left\\lbrace\\begin{array}{l}\n2\\mathbb{Z} \\text{ if } p \\text{ even and } q \\text{ odd}\\\\ \n\\mathbb{Z} \\text{ otherwise}\\end{array}\\right.$\n\n\\item $r_1+1 + \\ldots r_m+1 = r+1 \\qquad k \\in \n\\left\\lbrace\\begin{array}{l}\n\\dfrac{{l}}{q \\wedge {l}} \\mathbb{Z} \\cap 2\\mathbb{Z} \\text{ if } p \\text{ even\nand } q \\text{ odd} \\\\\n\\dfrac{{l}}{q \\wedge {l}} \\mathbb{Z} \\text{ otherwise}\n\\end{array}\\right.$\n\\end{itemize}\nwhere ${l} = (r_1+1) \\wedge \\ldots \\wedge (r_m+1)$.\nThe other untwisted models with admissible levels are anomalous.\n\\end{prop}\n\n\\paragraph{Example 1:  $\\mathfrak g = A_4 = \\mathfrak{su}(5)$.}\n\nThe center $\\tilde Z \\cong \\mathbb Z_5$ of the corresponding group has \nonly one nontrivial subgroup, $Z = \\tilde Z \\cong \\mathbb Z_5$, so with \n$p=5$ odd and $q =1$ odd with the previous notations. The admissible \nlevels are $k \\in \\mathbb Z$, according \nto \\eqref{Consistency_Ar}. \nFollowing Proposition \\ref{prop_Ar_h=g}, the regular subalgebra \n$\\mathfrak h = \\mathfrak g$ leads to the condition $k \\in 5\\mathbb Z$ \nfor non-anomalous models. Then, applying the last proposition above, \nthe cases $\\mathfrak h = A_1,\\,A_1 \\oplus A_1\\equiv 2A_1,\\,A_2$ and $A_3$ \nleads to non-anomalous models for every $k \\in \\mathbb Z$, because \nhere we have \n$r_1+1 + \\ldots r_m+1 < r+1 = 5$. For $\\mathfrak h = A_2 \\oplus A_1$, \nwe have an equality. However, $l = (r_1+1) \\wedge (r_2+1) = 3 \\wedge 2 = 1$, \nso $l \/ (l \\wedge q) = 1$ and the model has no anomalies \nfor every $k \\in \\mathbb Z$. Consequently, the only anomalous models \ncorresponding to $\\mathfrak g = A_4$ and $\\mathfrak h$ regular are those \nwith $\\mathfrak h = \\mathfrak g$, $Z = \\tilde Z$ and $k \\in \\mathbb Z \n\\setminus 5\\mathbb Z$.\n\n\\paragraph{Example 2:  $\\mathfrak g = A_5 = \\mathfrak{su}(6)$.}\n\nHere the center $\\tilde Z \\cong \\mathbb Z_6$ has three nontrivial \nsubgroups : $Z \\cong \\mathbb Z_6, \\mathbb Z_3$ and $\\mathbb Z_2$ with \nthe respective admissible levels $k \\in 2 \\mathbb Z,\\,\\mathbb Z$ and \n$2 \\mathbb Z$. The models corresponding to the case \n$\\mathfrak h = \\mathfrak g$ will be non-anomalous for \n\\begin{equation}\n  k \\in \\left\\lbrace\n  \\begin{array}{ll}\n    6 \\mathbb Z & \\text{if } Z \\cong \\mathbb Z_6 \\\\\n    3 \\mathbb Z & \\text{if } Z \\cong \\mathbb Z_3 \\\\    \n    2 \\mathbb Z & \\text{if } Z \\cong \\mathbb Z_2.\n  \\end{array}\n\\right.\n\\end{equation}\nRegular subalgebras $\\mathfrak h = A_1,\\,2A_1,\\,A_2,\\,A_2 \\oplus A_1, \nA_3$ and $A_4$ correspond to the strict inequality for ranks in \nthe proposition above, so there will be no anomalies for these models with\n\\begin{equation}\n  k \\in \\left\\lbrace\n  \\begin{array}{ll}\n    2 \\mathbb Z & \\text{if } Z \\cong \\mathbb Z_6 \\text{ or } \\mathbb Z_2 \\\\\n    \\mathbb Z & \\text{if } Z \\cong \\mathbb Z_3. \\\\    \n  \\end{array}\n\\right.\n\\end{equation}\nComputation shows that $\\mathfrak h = 2A_2$ leads to non-anomalous \nmodels for the same $k$ as for $\\mathfrak h = \\mathfrak g$, and that \nthe models corresponding to $\\mathfrak h = A_3 \\oplus A_1$ and to $3A_1$ \nhave no anomalies for $k \\in 2 \\mathbb Z$ if \n$Z \\cong \\mathbb Z_6$ or $\\mathbb Z_2$ and for $k\\in\\mathbb Z$ \nif $Z\\cong\\mathbb Z_3$ . \nThus, the anomalous models corresponding to $\\mathfrak g = A_5$ have either \n$\\mathfrak h = \\mathfrak g$ or $\\mathfrak{h}=2A_2$, \nwhere $k \\in 2\\mathbb Z \\setminus 6 \\mathbb Z$ for $Z \\cong \\mathbb Z_6$ \nand $k \\in \\mathbb Z \\setminus 3 \\mathbb Z$ for $Z \\cong \\mathbb Z_3$. \n\n\\subsection{Regular semisimple subalgebras of $D_r$}\n\nThe semisimple regular subalgebras of $D_r$ are given \nin \\cite{Dynkin} (Chapter II, Table 9) and have the form:\n\n\\begin{equation}\\label{regular_subalg_Dr}\n \\mathfrak{h} = A_{r_1} \\oplus \\ldots \\oplus A_{r_m} \\oplus D_{s_1} \\oplus\n\\ldots \\oplus D_{s_n}\n\\end{equation}\nwhere $r_1 + 1 + \\ldots + r_m + 1 + s_1 + \\ldots s_n \\leq r$.\\footnote{To \ntake into account all the possible cases with this formula, we may need \nto consider $D_2$ instead of $2A_1$ and $D_3$ instead of $A_3$ \nto respect the inequality. See examples below.} The embedding of\n$D_{s_i}$ subalgebras realizes them as diagonal \nblocks in $D_r$. Instead of giving an explicit embedding of \nsubalgebras $A_{r_i}$, it is enough to see that\n$A_l$ is trivially embedded in $D_{l+1}$, by sending the $l$ simple roots\n$\\alpha_i^{A_l}$ of $A_l$ to the $l$ first simple roots $\\alpha_i^{D_{l+1}}$ of\n$D_{l+1}$. Then, the Serre construction allows us to reconstruct the full\nstructure of $A_l$, embedded in $D_{l+1}$, which is then easily embedded in\n$D_r$ as a diagonal block. The embedding of $\\mathfrak h$ into $\\mathfrak g$ \ndescribed above is unique, up to inner automorphisms of $\\mathfrak g$, except\nfor even $r$ if there are no $D_{s_i}$ and $r_1+1+\\ldots+r_m+1=r$ with \nall $r_i$ odd. In the latter case there is a second \nindependent embedding of $A_{r_1}\\oplus\\ldots\\oplus A_{r_m}$ into $D_r$ that\nsends the simple roots of $A_{r_m}$ to the last $r_m+1$ simple roots of \n$D_r$ omitting $\\alpha_{r-1}$. That embedding is related to the previous \none by the outer automorphism $\\omega$ of $D_r$ that permutes roots \n$\\alpha_{r_1}$ and $\\alpha_r$, but not by an inner automorphism. \nRecall that the coroot lattice $Q^\\vee(D_r)$ is composed of vectors \n\\begin{equation}\nq^\\vee=\\sum\\limits_{i=1}^rq^\\vee_ie_i\\quad{\\rm with}\\quad \nq^\\vee_i\\in\\mathbb Z\\quad{\\rm and}\\quad\\sum\\limits_{i=1}^rq^\\vee_i\\in2\\mathbb Z\\,.\n\\label{coroot_lat_Dr}\n\\end{equation} \n\n\n\\paragraph{Case of $r$ odd.}\nTaking $M$ and $\\tilde{M}$ as given in \\eqref{MMtilde_Dr_odd}, \nwe shall impose the condition $e^{2\\ii \\pi \\tilde{M}}\\in \\tilde{H}$. On the \nLie-algebra level, we have to show that for some \n$q^\\vee\\in Q^\\vee(\\mathfrak{g})$, $\\tilde M+q^\\vee$ belongs to\n$\\ii\\mathfrak{t}_{\\mathfrak{h}}$. Looking block by block, we infer that \n\\begin{equation}\n\\label{D1}\n\\frac{\\tilde{a}\\hspace{0.01cm}(r_i+1)}{2}\\in\\mathbb Z\\,, \\qquad i=1, \\dots, m\\,,\n\\end{equation}\nand that\n\\begin{equation}\n\\label{D2}\n\\frac{\\tilde{a}}{2}\\in\\mathbb Z\n\\end{equation}\nif $r_1 + 1 + \\ldots + r_m + 1 + s_1 +\\ldots s_n < r$. The condition \nthat the sum of components of vectors in $Q^\\vee(\\mathfrak{so}(2r))$ is even\nimposes the additional requirement that   \n\\begin{equation}\n\\label{add_odd}\n\\frac{\\tilde{a}\\hspace{0.01cm}r}{2}\\in2\\mathbb Z\\,,\n\\end{equation}\ni.e. $\\tilde a\\in 4\\mathbb Z$, in the absence of $D_{s_i}$ components \nin $\\mathfrak{h}$,  (in that case conditions (\\ref{D1}) and (\\ref{D2}) imply \nalready that $\\tilde a\\in2\\mathbb Z$).\nRe-examining the quantity \\eqref{Quantity_Dr_odd} which \nhas to be an integer with the above restrictions in mind \nand taking into account the conditions for admissible levels, \nwe deduce \n\n\\begin{prop}\nThe untwisted coset models built with Lie algebra $\\mathfrak{g} =\n\\mathfrak{so}(2r)$, $r$ odd, and a regular subalgebra  $\\mathfrak{h} = A_{r_1}\n\\oplus \\ldots \\oplus A_{r_m} \\oplus D_{s_1} \\oplus \\ldots \\oplus D_{s_n}$ \ndo not have global gauge anomalies for the following cases \n\\begin{itemize}\n\n\\item\\hspace{-0.1cm}$r_1 + 1 + \\ldots + r_m + 1 + s_1 + \\ldots s_n = r \\text{ with all }\nr_i \\text{ odd and }\\quad k \\in \\left\\lbrace\n\\begin{array}{l} \n4\\mathbb{Z} \\text{ if } Z \\cong \\mathbb{Z}_4 \\\\\n 2\\mathbb{Z} \\text{ if } Z \\cong \\mathbb{Z}_2\\,\n\\end{array}\\right.$\n\\item $\\hspace{-0.3cm}\\left.\\begin{array}{l}r_1 + 1 + \\ldots + r_m + 1 + s_1 + \\ldots s_n < r \n\\text{ or}\\\\ r_1 + 1 + \\ldots + r_m + 1 + s_1 + \\ldots s_n = r\\ and\\ some\\  \nr_{i}\\ even\\end{array}\\right\\rbrace\n\\ k \\in\n\\left\\lbrace\n\\begin{array}{l} \n2\\mathbb{Z} \\text{ if } Z \\cong \\mathbb{Z}_4 \\\\\n\\mathbb{Z}\\ \\hspace{0.04cm} \\text{ if } Z \\cong \\mathbb{Z}_2\n\\end{array}\\right.$\n\\end{itemize}\nThe other untwisted models with admissible levels \nare not globally gauge invariant.\n\\end{prop}\n\n\n\\noindent{\\bf Remark}\\ \\ \\ In particular, the global gauge anomalies present\nif $\\mathfrak h=\\mathfrak g$ for $Z=\\mathbb Z_4$ and $k\\in2\\mathbb Z$,\n$k\/2$ odd, or for $Z=\\mathbb Z_2$ and $k$ odd, disappear for $\\mathfrak h= \nA_{r_1}\\oplus \\ldots\\oplus A_{r_m}\\oplus D_{s_1}\\oplus\\ldots\\oplus D_{s_n}$\nif $r_1 + 1 + \\ldots + r_m + 1 + s_1 + \\ldots s_n < r$ or if\n$r_1 + 1 + \\ldots + r_m + 1 + s_1 + \\ldots s_n = r$ with some\n$r_i$ even. Note that if there no $D_{s_i}$ and $r_1 + 1 + \\ldots + r_m + 1=r$\nthen all $r_i$ cannot be odd. \n\n\n\\paragraph{Example: $\\mathfrak{g}=D_5=\\mathfrak{so}(10)$.} The admissible\nlevels are $k \\in 2\\mathbb Z$ for \n$Z = \\tilde Z \\cong \\mathbb Z_4$ and $k \\in \\mathbb Z$ for $Z \\cong \n\\mathbb Z_2$. According to Proposition \\ref{prop_Drodd_h=g}, there are \nno gauge anomalies in the case $\\mathfrak h = \\mathfrak g$ for \n\\begin{equation}\\label{ex_Drodd_1}\n  k \\in \\left\\lbrace\n    \\begin{array}{ll}\n      4 \\mathbb Z \\text{ if } Z \\cong \\mathbb Z_4\\\\\n      2 \\mathbb Z \\text{ if } Z \\cong \\mathbb Z_2.\\\\\n    \\end{array}\n\\right.\n\\end{equation}\n For regular subalgebra $\\mathfrak h = A_1, 2A_1\\cong D_2, \nA_2, A_3\\cong D_3, D_4$, \nthe inequality on the ranks is strict so there are no anomalies for  \n\\begin{equation}\\label{ex_Drodd_2}\n  k \\in \\left\\lbrace\n  \\begin{array}{ll}\n    2 \\mathbb Z & \\text{if } Z = \\mathbb Z_4 \\\\\n    \\mathbb Z & \\text{if } Z = \\mathbb Z_2. \\\\    \n  \\end{array}\n\\right.\n\\end{equation}\nIn the case $\\mathfrak h = A_4$ and $A_2\\oplus A_1$, the rank inequality \nis saturated and there is one $r_i$ even, so \\eqref{ex_Drodd_2} \nstill gives the no-anomaly condition for $k$.\n$D_5$ admits also $D_3\\oplus D_2\\cong A_3 \\oplus 2 A_1$, \n$A_1\\oplus D_3\\cong A_3 \\oplus A_1$, $A_2\\oplus D_2\\cong \nA_2 \\oplus 2 A_1$, $2D_2\\cong 4A_1$ and $A_1\\oplus D_2\\cong 3 A_1$,\nsee \\cite{Lorente} or the method described above,\nwhere only the left hand sides respect the inequality for ranks\nand should be used to extract the no-anomaly conditions.\nFor $A_2\\oplus D_2$, $2D_2$ and $A_1\\oplus D_2$ either the inequality\nfor ranks is saturated and there is an even $r_i$ or the inequality\nfor ranks is strict, hence there are no anomalies for levels\nsatisfying \\eqref{ex_Drodd_2}. Finally, for $D_3\\oplus D_2$ and $A_1\\oplus D_3$ \nthe rank inequality is saturated by there is no even $r_i$ and the\ngauge anomalies persist for \n$Z \\cong \\mathbb Z_4$ if $k \\in 2\\mathbb Z \\setminus 4 \\mathbb Z$ \nand for $Z \\cong \\mathbb Z_2$ if $k$ odd.\n\n\n\\paragraph{Case of $r$ even.}\n\nTaking $M$ and $\\tilde{M}$ as given in \\eqref{MMtilde_Dr_even} and \nfollowing the same reasoning as for the case of $r$ odd, we get \nthe same conditions: \n\\begin{equation}\n\\frac{\\tilde{a}_1(r_i+1)}{2}\\in\\mathbb Z\\,, \\qquad i=1,\\dots,m\\,,\n\\label{=}\n\\end{equation}\nand, if $r_1 + 1 + \\ldots + r_m + 1 + s_1 + \n\\ldots s_n < r$, \n\\begin{equation}\n\\frac{\\tilde{a}_1}{2}\\in\\mathbb Z\n\\label{<}\n\\end{equation}\nAdditionally, if there are no $D_{s_i}$ components in $\\mathfrak{h}$,\nthen\n\\begin{eqnarray}\n&&\\tilde a_1\\frac{r}{2}+\\tilde a_2\\in2\\mathbb Z\\hspace{1.6cm}\\text{for the }\n1^{\\rm st}\\text{ \\,embedding}\\,\\cr\n&&\\tilde a_1\\big(\\frac{r}{2}-1\\big)+\\tilde a_2\\in2\\mathbb Z\\qquad\\text{for \nthe }2^{\\rm nd}\\text{ embedding}\\,\n\\end{eqnarray}\n(the last two conditions differ only if all $r_i$ are odd and the\nrank inequality is saturated because in the other cases $\\tilde a_1$ has\nto be even). Examining the quantity \n\\eqref{Quantity_Dr_even} which has to be an integer with this information \nin mind and taking into account the admissibility conditions \nfor the levels, we deduce \n\n\\begin{prop}\nThe untwisted  coset models built with Lie algebra $\\mathfrak{g} \n=\\mathfrak{so}(2r)$, $r$ even, and a regular \nsubalgebra  $\\mathfrak{h} = A_{r_1}\n\\oplus \\ldots \\oplus A_{r_m} \\oplus D_{s_1} \\oplus \n\\ldots \\oplus D_{s_n}$ do not\nhave global gauge anomalies for the following cases \n\\begin{itemize}\n\\item$r_1 + 1 + \\ldots + r_m + 1 + s_1 + \\ldots s_n = r \\text{ with all }\nr_i \\text{ odd }  \\\\ \\hspace*{0.2cm}k \\in \\left\\lbrace\n\\begin{array}{l} \n2\\mathbb{Z} \\text{ for any } Z \\\\\n\\mathbb{Z}\\ \\,\\text{ if } r\/2 \\text { even, no } D_{s_i} \\text{ and }Z=Z_1\n\\ \\ \\ \n\\text{ for the } 1^{\\rm st}\\text{\\, embedding}\\\\\n\\mathbb{Z}\\ \\,\\text{ if } r\/2 \\text { even, no } D_{s_i} \\text{ and }Z=Z_{diag}\n\\text{ for the } 2^{\\rm nd}\\text{ embedding}\n\\end{array}\\right.$\n\\item\\hspace{-0.3cm} \n$\\left.\\begin{array}{l}r_1 + 1 + \\ldots + r_m + 1 + s_1 + \\ldots s_n < r \n\\text{ or}\\\\ r_1 + 1 + \\ldots + r_m + 1 + s_1 + \\ldots s_n = r\n\\text{ and some } r_{i}\\text{ even}\\end{array}\n\\right\\rbrace\\\\ \\hspace*{0.26cm}k \\in\n\\left\\lbrace\n\\begin{array}{l} \n2\\mathbb{Z} \\text{ if } Z = \\tilde{Z},\\, Z_1 \\text{ or } Z_\\text{diag} \\\\\n\\mathbb{Z}\\ \\,\\text{ if } Z = Z_2 \\\\\n\\mathbb Z\\ \\,\\text{ if } r\/2 \\text{ even, no } D_{s_i} \\text{ and any } Z\\\\\n\\end{array}\\right.$\n\\end{itemize}\nThe other untwisted models with admissible levels are not globally \ngauge invariant.\n\\end{prop}\n\n\n\n\\noindent{\\bf Remark}\\ \\ \\ In particular, the global gauge anomalies present\nif $\\mathfrak h=\\mathfrak g$ for $Z=Z_2$ and $k$ odd disappear \nfor $\\mathfrak h=A_{r_1}\\oplus \\ldots\\oplus A_{r_m}\n\\oplus D_{s_1}\\oplus\\ldots\\oplus D_{s_n}$\nif $\\,r_1 + 1 + \\ldots + r_m + 1 + s_1 + \\ldots s_n < r\\,$ or if\n$\\,r_1 + 1 + \\ldots + r_m + 1 + s_1 + \\ldots s_n = r\\,$ with some\n$r_i$ even.\n\n\\paragraph{Example: $\\mathfrak{g}=D_4=\\mathfrak{so}(8)$.} Here $r$ \nand $r\/2$ are both even, so all levels \n$k \\in \\mathbb Z$ are admissible for all $Z$ and there are no anomalies \nin the case $\\mathfrak h = \\mathfrak g$ for $k$ even according\nto Proposition \\ref{prop_Dreven_h=g}, whereas \nthe cases with $k$ odd are anomalous. The possible (proper, nontrivial) \nsubalgebras $\\mathfrak h$ are: $A_1$, $A_2$, $2A_1$, $A_3$ (the latter \ntwo with 2 inequivalent embeddings), $D_2$, $D_3$, $2D_2$ and $A_1\\oplus D_2$.\nNote that the two embeddings of $2A_1$ and that of $D_2$ are \nrelated by the outer automorphisms of $D_4$ and similarly for \nthe two embeddings of $A_3$ and the one of $D_3$. For regular subalgebra \n$\\mathfrak h = A_1$ or $A_2$, \nthe inequality on ranks is strict and there are no $D_{s_i}$ so there\nare no anomalies for $k\\in\\mathbb Z$ for all $Z$. For $D_2$ or $D_3$,\nthe rank inequality is still strict and there are no anomalies for $k$\neven and all $Z$ and for $k$ odd and $Z=Z_2$. For $A_1\\oplus D_2$ or \n$2D_2$, the rank inequality is saturated and there are no anomalies \nfor even $k$ and any $Z$. Finally, for $2A_1$ or $A_3$ the rank inequality is \nsaturated and there are no $D_{s_i}$ so there are no anomalies for\n$k$ even and any $Z$ and for $k$ odd and $Z=Z_1$ for the 1$^{\\rm st}$ embedding\nand $Z=Z_{diag}$ for the 2$^{\\rm nd}$ one. \n\\vskip 0.4cm\n\nRecall from Sec.\\,\\ref{sec:D_reven_tw} that the twisted coset\nmodels for $\\mathfrak g=\\mathfrak{so}(2r)=\\mathfrak h$ \nwith $r>4$ even have gauge \nanomalies for  $Z=\\tilde Z$ (- theory) if $k$ is even and for \n$Z=\\tilde Z,\\,Z_1$ or $Z_{diag}$ if $k$ is odd for $r\/2$ even. \nThese are the cases where the no-anomaly condition (\\ref{Dreven_tw_c}) \nmay be violated. The restriction $e^{2\\ii\\pi\\tilde M}\\in\\tilde H$\nfor $\\mathfrak h=A_{r_1}\\oplus \\ldots\\oplus A_{r_m}\n\\oplus D_{s_1}\\oplus\\ldots\\oplus D_{s_n}$\nif $\\,r_1 + 1 + \\ldots + r_m + 1 + s_1 + \\ldots s_n < r\\,$ or if\n$\\,r_1 + 1 + \\ldots + r_m + 1 + s_1 + \\ldots s_n = r\\,$ with some\n$r_i$ even imposes the condition $\\tilde a_1\\in 2\\mathbb Z$ removing \nthe anomalies in the case $Z=\\tilde Z$ (- theory) for $k$ even and, if,\nadditionally, there are no $D_{s_i}$ components in $\\mathfrak{h}$, \nalso for $Z\\not=Z_2$ and $k$ odd. If there are no $D_{s_i}$ and\n$r_1+1+\\ldots+r_m+1=r$ with all $r_i$ odd then for $k$ odd ($r\/2$ even)\nthe anomalies for $Z=\\tilde Z$ are removed for the $+$ theory in the\ncase of the 1$^{\\rm st}$ embedding and for the $-$ theory in the case of\nthe 2$^{\\rm nd}$ embedding, and for $Z=Z_1,\\ Z_{diag}$ in the case \nof both embeddings. We obtain this way \n\n\\begin{prop}\nThe twisted  coset models built with Lie algebra $\\mathfrak{g} \n=\\mathfrak{so}(2r)$, $r>4$ even, and a regular subalgebra  \n$\\mathfrak{h} = A_{r_1}\\oplus \\ldots \\oplus A_{r_m} \\oplus D_{s_1} \n\\oplus \\ldots \\oplus D_{s_n}$ do not have global gauge anomalies for \nthe following cases \n\\begin{itemize}\n\\item$r_1 + 1 + \\ldots + r_m + 1 + s_1 + \\ldots s_n = r \\text{ with all }\nr_i \\text{ odd }  \\\\ \\hspace*{0.2cm}k \\in \\left\\lbrace\n\\begin{array}{l} \n2\\mathbb{Z} \\text{ if } Z=\\tilde Z \\text{ (+ theory) or } Z=Z_1,\\,Z_{diag} \n\\\\\n\\mathbb{Z}\\ \\,\\text{ if } Z=Z_2\\\\\n\\mathbb{Z}\\ \\,\\text{ if } r\/2 \\text { even, no } D_{s_i} \\text{ and }Z=\\tilde Z\n\\text{ (+ theory) for the } 1^{\\rm st}\\text{\\, embedding}\\\\\n\\mathbb{Z}\\ \\,\\text{ if } r\/2 \\text { even, no } D_{s_i} \\text{ and }Z=\\tilde Z\n\\text{ \\,(- theory) \\hspace{0.07cm}for the } 2^{\\rm nd}\\text{ embedding}\\\\\n\\mathbb{Z}\\ \\,\\text{ if } r\/2 \\text{ even, no } D_{s_i} \\text{ and }\nZ=Z_1,\\,Z_{diag}\n\\end{array}\\right.$\n\\item\\hspace{-0.3cm} \n$\\left.\\begin{array}{l}r_1 + 1 + \\ldots + r_m + 1 + s_1 + \\ldots s_n < r \n\\text{ or}\\\\ r_1 + 1 + \\ldots + r_m + 1 + s_1 + \\ldots s_n = r\n\\text{ and some } r_{i}\\text{ even}\\end{array}\n\\right\\rbrace\\\\ \\hspace*{0.26cm}k \\in\n\\left\\lbrace\n\\begin{array}{l} \n2\\mathbb{Z} \\text{ if } Z = \\tilde{Z},\\, Z_1 \\text{ or } Z_\\text{diag} \\\\\n\\mathbb{Z}\\ \\,\\text{ if } Z = Z_2 \\\\\n\\mathbb Z\\ \\,\\text{ if } r\/2 \\text{ even, no } D_{s_i} \\text{ and } \nZ=\\tilde Z,\\,Z_1,\\,Z_{diag}\n\\end{array}\\right.$\n\\end{itemize}\nThe other twisted models with admissible levels are not globally \ngauge invariant.\n\\end{prop}\n\n\nThe above results also hold for the coset model with \n$\\mathfrak g=\\mathfrak{so}(8)$ with twist $\\omega_1$,\nsee (\\ref{omegas}). Hence, for $\\mathfrak h=A_1$ or $A_2$\nthere are no gauge anomalies. For $D_2$ or $D_3$ there are\nno anomalies if $k$ is even for any $Z$ and if $k$ is odd\nfor $Z=Z_2$. For $A_1\\oplus D_2$ or $2D_2$ there are no anomalies\nif $k$ is even for $Z=\\tilde Z$ (+ theory) or $Z=Z_1,\\,Z_2,\\,Z_{diag}$\nor if $k$ is odd and $Z=Z_2$. Finally, for $2A_1$ or $A_3$ there are no\nanomalies for $Z=\\tilde Z$ (+ theory for the 1$^{\\rm st}$ embedding, \n- theory for the 2$^{\\rm nd}$ one) and for $Z=Z_1,Z_2,Z_{diag}$. In accordance\nwith the discussion of Sec.\\,\\ref{sec:D_reven_tw}, we may obtain\nthe result for twist $\\omega_2$ from the one for $\\omega_1$\nby applying the permutation $Z\\rightarrow\\omega_4(Z)$ induced \nby the outer automorphism $\\omega_4$ on the cyclic subgroups \nof $\\tilde Z$, see Eqs.\\,(\\ref{permutZ}), and on the one\n$\\mathfrak h\\rightarrow\\omega_4(\\mathfrak h)$  \non subalgebras (modulo inner automorphisms) induced by the\naction (\\ref{permutalpha}) of $\\omega_4$ on simple roots:\n\\begin{equation}\n\\begin{array}{ll}\\label{permuth}\n&\\hspace*{-0.6cm}\\omega_4(A_1)=A_1,\\ \\,\\omega_4(A_2)=A_2,\\ \\,\n\\omega_4((2A_1)^{(1)})\n=(2A_1)^{(2)},\\ \\,\\omega_4((2A_1)^{(2)})=D_2,\\ \\,\n\\omega_4(A_3^{(1)})=A_3^{(2)},\\\\\n&\\hspace*{-0.6cm}\\omega_4(A_3^{(2)})=D_3,\\ \\,\\omega_4(D_2)=(2A_1)^{(1)},\\ \\,\n\\omega_4(D_3)=A_3^{(1)},\\ \\,\\omega_4(2D_2)=2D_2,\\ \\,\n\\omega_4(A_1\\oplus D_2)=A_1\\oplus D_2.\n\\end{array}\n\\end{equation}\nwhere the superscript $(i),\\ i=1,2$, labels the independent embeddings.\nSimilarly, the result for twist $\\omega_3$ from the one for $\\omega_1$\nby applying the inverse permutations $Z\\rightarrow\\omega_4^{-1}(Z)$\nand $\\mathfrak h\\rightarrow\\omega_4^{-1}(\\mathfrak h)$.\n\\,For twists $\\omega_4,\\omega_4^{-1}$, the the remaining\ngauge anomalies are lifted if $\\mathfrak h=A_1$ or $A_2$ \nimposing the restrictions $\\tilde a_1,\\tilde a_2\\in 2\\mathbb Z$\nresulting in \n\\begin{prop}\nThe twisted coset models built with Lie algebra $\\mathfrak{g} =\n\\mathfrak{so}(8)$ with twist $\\omega_4$ have global gauge anomalies \nfor regular subalgebras \n$\\mathfrak{h}=2A_1$,\\,$A_3$,\\,$D_2$,\\,$D_3$,\\,$2D_2$,\\,$A_1\\oplus D_2$ \nand $Z=\\tilde Z$ (- theory). The other cases of coset models with \nLie algebra $\\mathfrak{so}(8)$ and twist $\\omega_4$ are without anomalies.\n\\end{prop}\n\n\\noindent Similarly\n\n\n\\begin{prop}\nThe twisted coset models built with Lie algebra $\\mathfrak{g} =\n\\mathfrak{so}(8)$ with twist $\\omega_4^{-1}$ have global gauge anomalies \nfor regular subalgebras $\\mathfrak{h}=2A_1$,\\,$A_3$,\\,$D_2$,\\,$D_3$,\\,$2D_2$\nand $A_1\\oplus D_2$ and $Z=\\tilde Z$ (($-$)$^k$ theory). The other cases \nof coset models with Lie algebra $\\mathfrak{so}(8)$ and twist $\\omega_4^{-1}$ \nare without anomalies.\n\\end{prop}\n\n\n\n\\subsection{Regular semisimple subalgebras of $\\mathfrak e_{6}$}\n\nIn this case with fixed rank $r=6$, one can establish a complete list \nof regular semisimple subalgebras, up to conjugation, with an embedding, \nhowever, that is not explicit \\cite{Dynkin,Lorente}. We shall only need \nthe embedding of simple roots\nin the ambient algebra which is enough to reconstruct the full embedding \nusing the Serre construction. The element $M$ and $\\tilde{M}$ will be\ndescribed employing the explicit\nrealization of the coweight and coroot lattices of $\\mathfrak{e}_6$,\n\\begin{equation}\\label{Coweight_Lattice_e6}\n P^\\vee(\\mathfrak{e}_6) = \\left\\lbrace \\left( \\dfrac{a}{6} + q_1, \\ldots,\n\\dfrac{a}{6} + q_6, \\dfrac{b}{\\sqrt{2}} \\right) \\left| \n\\begin{array}{l}a,b, q_1,\n\\ldots, q_6 \\in \\mathbb{Z}\\\\ a + q_1 + \\ldots + q_6 = 0\\\\ a + b \\in 2\\mathbb{Z}\n\\end{array} \\right. \\right\\rbrace\n\\end{equation}\nand the coroot lattice $Q^\\vee(\\mathfrak{e}_6)$ is defined the same way but\nadding the condition $a \\in 3 \\mathbb{Z}$. We shall consider only the\nuntwisted coset models because the twisted ones are non-anomalous,\nsee Proposition \\ref{prop_e6_h=g_tw}.\nTaking $M$ and $\\tilde{M}$ in\n$P^\\vee(\\mathfrak{e}_6)$ with the corresponding coefficients, the quantity\n\\eqref{Quantity_e6} becomes\n\\begin{equation}\\label{Quantity_e6_bis}\n k~\\text{tr}(M\\tilde{M}) = k \\dfrac{a \\tilde{a}}{3} + m, \\qquad \\text{ with }\nm \\in \\mathbb{Z}\n\\end{equation}\nNow, specifying a subalgebra $\\mathfrak{h}\\subset \\mathfrak e_{6}$ \nand requiring that $e^{2\\ii \\pi\n\\tilde{M}} \\in \\tilde{Z}\\cap\\tilde{H}$, two possibilities arise: if one can\nshow that $\\tilde{a} \\in 3\\mathbb{Z}$ then the previous quantity is an integer\nfor\nevery $k \\in \\mathbb{Z}$ and all the corresponding coset models are globally\ngauge invariant. Otherwise, if there exist an element $\\tilde{M}$ such that\n$\\tilde{a} \\notin 3\\mathbb{Z}$, then we have to require $k \\in 3\\mathbb{Z}$ to\nhave a globally gauge invariant coset model, and the other coset models \nare anomalous. Before examining the anomaly problem for every regular\nsubalgebra of $\\mathfrak{e}_6$, one can make four remarks:\n\n\\begin{itemize}\n\\item if there are no anomalies for a given subalgebra $\\mathfrak{h}$ of\n$\\mathfrak{e}_6$ ($\\tilde{a} \\in 3\\mathbb{Z}$), then the regular subalgebras \nthat are smaller (and will be obtained from the Dynkin diagram of \n$\\mathfrak{h}$ by the procedure described above) lead also\nto the condition $\\tilde{a} \\in 3\\mathbb{Z}$, inheriting \nit from $\\mathfrak{h}$. In other words, the regular subalgebra with \nno anomalies protects the cases of its regular subalgebras. Consequently, \nwe will look only at the cases where the anomalies are present and treat \nthe problem by decreasing rank.\n\n\n\\item Among the regular subalgebras generated by the algorithm described\nat the beginning of Sec.\\,\\ref{sec:4},\nmany can still be mapped into each other by the conjugations \nthat normalize $\\,\\mathfrak t_{\\mathfrak e_6}\\,$ (and induce on it Weyl group \ntransformations) and, as a result, they lead to the same condition \nfor the absence of anomalies. We may then consider only one regular \nsubalgebra in each class of subalgebras related by Weyl group transformations.\nIn particular, there are Weyl group transformations that permute the simple \nroots $\\,\\alpha_i\\,$ and $\\,\\delta=-\\phi\\,$ according to the symmetries of the \nextended Dynkin diagrams (see, e.g., Appendix B of \\cite{Gawedzki1})\nand they permit to restrict the count of regular subalgebras. \n\n\\item The subalgebras related by the outer automorphism of $\\mathfrak{e}_6$ \nlead to the same no-anomaly condition, see the remark at the end of\nSec.\\,\\ref{sec:2}.\n\n\\item Since $e^{2\\ii \\pi\\tilde{M}} \\in\\tilde Z\\cap \\tilde{H}$ if and only if\n$\\tilde M\\in P^\\vee(\\mathfrak g)$ and $\\tilde{M}+q^\\vee\\in\\ii\n\\mathfrak{t}_{\\mathfrak h}\\subset\\ii\\mathfrak{t}_\\mathfrak{g}$ for some\n$q^\\vee\\in Q^\\vee(\\mathfrak{e}_6)$, it is enough to check the no-anomaly\ncondition (\\ref{coset_anomaly}) only for $\\tilde{M}\\in P^\\vee(\\mathfrak g)$\nperpendicular to the orthogonal complement $\\ii\\mathfrak{t}^\\bot_{\\mathfrak h}$\nof $\\ii \\mathfrak{t}_{\\mathfrak h}$ in $\\ii\\mathfrak{t}_{\\mathfrak g}$.\n\\end{itemize}\n\n \nWe now consider the regular semisimple subalgebras, beginning \nby those of rank 6 and then decreasing the rank. Subspace \n$\\ii\\mathfrak{t}_\\mathfrak{h}^\\bot$  (which is small for high\nranks) is computed for each subalgebra and we look at the consequences \nof the condition $\\tilde M\\perp \\ii\\mathfrak{t}_\\mathfrak{h}^\\bot$ \non $\\tilde{M}$. Upon using the protection \nproperty and the Weyl transformations described above, as well as\nthe outer automorphism of $\\mathfrak{e}_6$, only a few cases have \nto be treated. The explicit computation is given in Table \n\\ref{Computation_e6_regular}. The subalgebras of rank 6 are not represented \nbecause we have $\\mathfrak{t}_\\mathfrak{h}^\\bot = \\emptyset$, so there is \nno supplementary condition for $\\tilde{M}$ and there are always anomalies if \n$k \\notin 3\\mathbb{Z}$. Only subalgebras of rank 5 and 4 have \npotential anomalies, the ones of lower ranks being protected by \na possible inclusion into non-anomalous subalgebras. \n\\begin{table}[htb]\n\\centering\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n   $\\mathfrak{h}$ & simple roots of $\\mathfrak{h}$ &\nbasis of $\\ii\\mathfrak{t}_{\\mathfrak{h}}^\\bot$ & $\\tilde{M}$ \\\\\n\\hline \\hline\n $D_5$ &  $\\alpha_1, \\alpha_2, \\alpha_3, \\alpha_4,\n\\alpha_6$ & $\\left(1,1,1,1,1,\\text{-}5, 3\\sqrt{2} \\right)$ &\n$\\tilde{a} \\in 3\\mathbb{Z} $ \\\\\n\\hline\n $A_3 \\oplus 2A_1$ &  $\\alpha_1, \\alpha_2, \\alpha_3\n\\oplus \\delta \\oplus \\alpha_5$ & $\\left(1,1,1,1,\\text{-}2,\\text{-}2, 0\n\\right)$ & $\\tilde{a} \\in 3\\mathbb{Z} $ \\\\\n\\hline \n  $A_4 \\oplus A_1$ &  $\\alpha_1, \\alpha_2, \\alpha_3,\n\\alpha_4 \\oplus \\delta$ & $\\left(1,1,1,1,1,\\text{-}5, 0 \\right)$ &\n$\\tilde{a} \\in 3\\mathbb{Z} $\\\\\n\\hline\n  $A_5$ &  $\\alpha_1, \\alpha_2, \\alpha_3, \\alpha_4, \\alpha_5$\n& $\\left(0,0,0,0,0,0, 1 \\right)$ & $\\tilde{a}\\in 2\\mathbb Z$ \\\\\n\\hline \n $2A_2\\oplus A_1$ &  $\\alpha_1, \\alpha_2 \\oplus\n\\alpha_4,\\alpha_5 \\oplus \\alpha_6$ & $\\left(1,1,1,-1,-1,-1,3\\sqrt{2}\n\\right)$ & $\\tilde{a}\\in\\mathbb Z$ \\\\\n\\hline \\hline\n $2A_2$ & $\\alpha_1, \\alpha_2 \\oplus \\alpha_4,\n\\alpha_5$ & $\\left(1,1,1,\\text{-}1,\\text{-}1,\\text{-}1, 0 \\right)$ &\n$\\tilde{a}\\in2\\mathbb Z$\\\\\n & & $\\left(0,0,0,0,0,0,1 \\right)$ & \\\\\n\\hline\n \\end{tabular}\n\\caption{$\\ii\\mathfrak{t}_\\mathfrak{h}^\\bot$ for the regular\nsubalgebras of $\\mathfrak{e}_6$ of rank $5$ and $4$ and consequences \nfor $\\tilde a$; the simple roots $\\alpha_i$ of $\\mathfrak{e}_6$ and \nits lowest root $\\delta$ are used to generate the regular subalgebras\n\\cite{Lorente}.}\\label{Computation_e6_regular}\n\\end{table}\n\n\\noindent We are thus able to state\n\n\\begin{prop}\nThe untwisted coset models built with Lie algebra $\\mathfrak{g} \n= \\mathfrak{e}_6$ \nand any regular subalgebra $\\mathfrak{h}$ do not have global gauge anomalies \nfor every $k\\in\\mathbb{Z}$, except for the cases \n$\\mathfrak{h} = \\mathfrak{e}_6, A_5 \\oplus A_1, 3A_2$, \nof rank 6, $A_5, 2A_2\\oplus A_1$,\nof rank 5, and $2A_2$ of rank 4, where the only globally gauge\ninvariant models are those with $k \\in 3 \\mathbb{Z}$. \n\\end{prop}\n\n\\section{R-subalgebras and S-subalgebras}\n\\label{sec:5}\n\nThe regular subalgebras are not the only possible Lie subalgebras for a given\nambient Lie algebra. We can use them, however, to classify all the remaining \nones. Let $\\mathfrak{h}$ be a semisimple subalgebra of $\\mathfrak{g}$. \nLet $\\mathcal{R}(\\mathfrak{h})$ be a minimal regular\nsubalgebra of $\\mathfrak{g}$ containing $\\mathfrak h$ (up to conjugation). \nIf $\\mathcal{R}(\\mathfrak{h}) =\\mathfrak{g}$, then \n$\\mathfrak{h}$ is called an S-subalgebra. Otherwise, \nit is called an R-subalgebra. For the exceptional simple algebras, the \nclassification of R- and S-subalgebras has been achieved by Dynkin \nin \\cite{Dynkin}. \nThe case of other simple algebras was discussed in \\cite{Dynk2} with\nless explicit results. In this section, we first treat completely the \ncase of non-regular subalgebras of the exceptional Lie algebra \n$\\mathfrak{g} = \\mathfrak{e}_6$ which may have anomalies and then we consider \nsome examples of non-regular subalgebras of classical Lie algebras.\n\n\\paragraph{Dynkin index.} Consider a simple Lie subalgebra $\\mathfrak\nh \\subset \\mathfrak g$ of a semisimple Lie algebra $\\mathfrak g$ \nand the corresponding embedding $\\iota$. \\,The relation \n\\begin{equation}\n  {\\rm tr}_{\\mathfrak{g}}(\\iota(X))^2\\,=\\,j\\,{\\rm tr}_{\\mathfrak{h}}X^2 \\qquad {\\rm for}\\quad X \\in \\mathfrak h\\,\n\\end{equation}\n\\noindent where the invariant quadratic forms $\\,{\\rm tr}_{\\mathfrak g}\\,$ \nand $\\,{\\rm tr}_{\\mathfrak h}\\,$ have the normalizations described in\nthe beginning of Sec.\\,\\ref{sec:2}, \ndefines the scalar factor $\\,j\\,$ (independent of $X$), \n called Dynkin index, which\nis always an integer \\cite{Dynkin}. Moreover, $\\,j\\,$ is invariant under\ncomposition of $\\,\\iota\\,$ with inner (and outer) automorphisms of \n$\\mathfrak g$, so that it depends on the class of equivalent \nembeddings.\n\n\n\\subsection{Simple nonregular subalgebras of $\\mathfrak{e}_6$}\n\\label{sec:SimpleCase}\n\n\\paragraph{Subalgebras of rank 1}\n\nAccording to Dynkin, the subalgebra $\\mathfrak{h} = A_1$ can be embedded in\nseveral different ways in $\\mathfrak{e}_6$, as regular, R- and S-subalgebra and\nthe embedding $\\iota$ is fully characterized by the embedding \nof the simple coroot $\\alpha^\\vee$ of $A_1$. Recall the compatibility \ncondition for $\\tilde{M}$ in the anomaly problem\n\n\\begin{equation}\n e^{2\\ii \\pi \\tilde{M}} \\in \\tilde{H} \\cap \\tilde{Z} \\subseteq\n\\mathcal{Z}(\\tilde{H})\\,,\n\\end{equation}\nwhere  $\\mathcal{Z}(\\tilde{H})=\\{1,e^{2\\ii\\pi\\iota(\\lambda^\\vee)}\\}$ \nwith $\\lambda^\\vee =\\frac{1}{2} \\alpha^\\vee$ is the center of $\\tilde H$\nwhich is either trivial (if $1=e^{2\\ii\\pi\\iota(\\lambda^\\vee)}$ and \n$\\tilde H\\cong SO(3)$) or is isomorphic to $\\mathbb Z_2$\n(if $1\\not=e^{2\\ii\\pi\\iota(\\lambda^\\vee)}$ and $\\tilde H\\cong SU(2)$). \nLooking at the embedding of $\\lambda^\\vee$ in $\\mathfrak{e}_6$, three\npossibilities can occur \n\n\n\\begin{enumerate}\n \\item If $\\iota(\\lambda^\\vee) \\notin P^\\vee(\\mathfrak{e}_6)$ \nthen $\\tilde{Z}\n\\cap \\tilde{H} = \\lbrace 1 \\rbrace$ and $\\tilde{M}$ is a coroot of\n$\\mathfrak{e}_6$, so the quantity \\eqref{Quantity_e6} is always \nan integer and there are no anomalies for this model.\n\\item If $\\iota(\\lambda^\\vee) \\in Q^\\vee(\\mathfrak{e}_6)$ \nthen $\\tilde{M}$ is still\nonly a coroot of $\\mathfrak{e}_6$, and there are no anomalies too.\n\\item If $\\,\\iota(\\lambda^\\vee) \\in P^\\vee(\\mathfrak{e}_6)\\setminus \nQ^\\vee(\\mathfrak{e}_6)$ then anomalies are\npossible and we have to check that the quantity \\eqref{Quantity_e6}\nis an integer for $\\tilde{M} = \\iota(\\lambda^\\vee)$ looking at the\ncorresponding value for $\\tilde{a}$, see Eq.\\,(\\ref{Quantity_e6_bis}).\n\\end{enumerate}\nThe explicit embeddings are given in \\cite{Dynkin} (Chapter III, Table 18), and\nthe computation of the intersection with the roots of $\\mathfrak{e}_6$ is done\nin Table \\ref{Computation_e6_rank1} for each subalgebra of rank 1: the\npossibility 3 never occurs, so there are no anomalies for \nthe corresponding coset models for any $k \\in \\mathbb{Z}$.\n\n\\begin{table}[htb]\n\\centering \n\\begin{tabular}{|c|c|c|c|}\n\\hline\n$\\mathcal{R}(\\mathfrak{h})$ & Index & $\\iota(\\lambda^\\vee)$ & Compatibility\\\\\n\\hline \\hline\n$A_1$ & 1 & $ \\left(0,0,0,0,0,0,\\frac{1}{\\sqrt{2}}\\right)$ & $\\notin P^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline \\hline\n$2 A_1$ & 2 &$ \\left( \\tfrac{1}{2},0,0,0,0,\\tfrac{\\text{-}1}{2},\\tfrac{1}{\\sqrt{2}}\n\\right)$ & $\\notin P^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$3 A_1$ & 3 & $ \\left(\n\\tfrac{1}{4},\\tfrac{1}{4},\\tfrac{1}{4},\\tfrac{\\text{-}1}{4},\\tfrac{\\text{-}1}{4}\n,\\tfrac{\\text{-}1}{4},\\tfrac{3}{2\\sqrt{2}} \\right)$ & $\\notin\nP^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$ A_2$ & 4 &$ \\left( 0,0,0,0,0,0,\\sqrt{2} \\right)$ & $\\in Q^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$ A_2 \\oplus A_1$ & 5 & $ \\left(\n\\tfrac{1}{2},0,0,0,0,\\tfrac{\\text{-}1}{2},\\sqrt{2} \\right)$ & $\\notin\nP^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$ A_2 \\oplus 2A_1$ & 6 & $ \\left(\n\\tfrac{1}{2},\\tfrac{1}{2},0,0,\\tfrac{\\text{-}1}{2},\\tfrac{\\text{-}1}{2},\\sqrt{2}\n\\right)$ & $\\notin P^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$ 2A_2$ & 8 &$ \\left(1,0,0,0,0,\\text{-}1,\\sqrt{2}\\right)$ & $\\in Q^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$2 A_2 \\oplus A_1$ & 9 & $ \\left(\n\\tfrac{3}{4},\\tfrac{1}{4},\\tfrac{1}{4},\\tfrac{\\text{-}1}{4},\\tfrac{\\text{-}1}{4}\n,\\tfrac{\\text{-}3}{4},\\tfrac{5}{2\\sqrt{2}} \\right)$ & $\\notin\nP^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$ A_3 $ & 10 & $ \\left( \\tfrac{1}{2},0,0,0,0,\\tfrac{\\text{-}1}{2},\\tfrac{3}{\\sqrt{2}}\n\\right)$ & $\\notin P^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$ A_3 \\oplus A_1$ & 11 & $ \\left(\n\\tfrac{1}{2},\\tfrac{1}{2},0,0,\\tfrac{\\text{-}1}{2},\\tfrac{\\text{-}1}{2},\\tfrac{3\n}{\\sqrt{2}} \\right)$ & $\\notin P^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$ A_3 \\oplus 2A_1$ & 12 & $ \\left(\n\\tfrac{1}{2},\\tfrac{1}{2},\\tfrac{1}{2},\\tfrac{\\text{-}1}{2},\\tfrac{\\text{-}1}{2}\n,\\tfrac{\\text{-}1}{2},\\tfrac{3}{\\sqrt{2}} \\right)$ & $\\in Q^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$ A_4$ & 20 & $ \\left( 1,0,0,0,0,\\text{-}1,2\\sqrt{2} \\right)$ & $\\in Q^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$ A_4 \\oplus A_1$ & 21 & $\n\\left(1,\\tfrac{1}{2},0,0,\\tfrac{\\text{-}1}{2},\\text{-}1,2\\sqrt{2}\n\\right)$ & $\\notin P^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$ D_4$ & 28 & $ \\left(\n\\tfrac{1}{2},\\tfrac{1}{2},\\tfrac{1}{2},\\tfrac{\\text{-}1}{2},\\tfrac{\\text{-}1}{2}\n,\\tfrac{\\text{-}1}{2},\\tfrac{5}{\\sqrt{2}} \\right)$ & $\\in Q^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$ D_5 (a_1)$ & 30 & $\n\\left(1,\\tfrac{1}{2},0,0,\\tfrac{\\text{-}1}{2},\\text{-}1,\\tfrac{5}{\\sqrt{2}}\n\\right)$ & $\\notin P^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$ A_5$ & 35 & $ \\left(\n\\tfrac{3}{2},\\tfrac{1}{2},0,0,\\tfrac{\\text{-}1}{2},\\tfrac{\\text{-}3}{2},\\tfrac{5\n}{\\sqrt{2}}\\right)$ & $\\notin P^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$ A_5 \\oplus A_1$ & 36 & $ \\left(\n\\tfrac{3}{2},\\tfrac{1}{2},\\tfrac{1}{2},\\tfrac{\\text{-}1}{2},\\tfrac{\\text{-}1}{2}\n,\\tfrac{\\text{-}3}{2},\\tfrac{5}{\\sqrt{2}}\\right)$ & $\\in Q^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$D_5 $ & 60 & $ \\left(\n\\tfrac{3}{2},\\tfrac{1}{2},\\tfrac{1}{2},\\tfrac{\\text{-}1}{2},\\tfrac{\\text{-}1}{2}\n,\\tfrac{\\text{-}3}{2},\\tfrac{7}{\\sqrt{2}}\\right)$ & $\\in Q^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline \\hline\n$ \\mathfrak{e}_6 (a_1)$ & 84 &$\n\\left(2,1,0,0,\\text{-}1,\\text{-}2,4\\sqrt{2}\\right)$ & $\\in Q^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$\\mathfrak{e}_6 $ & 156 & $ \\left(\n\\tfrac{5}{2},\\tfrac{3}{2},\\tfrac{1}{2},\\tfrac{\\text{-}1}{2},\\tfrac{\\text{-}3}{2}\n,\\tfrac{\\text{-}5}{2},\\tfrac{11}{\\sqrt{2}}\\right)$ & $\\in Q^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline \n  \\end{tabular}\n\\caption{The embedding of element $\\lambda^\\vee$ for rank 1\nsubalgebras and its intersection with the lattices of\n$\\mathfrak{e}_6$.}\n\\label{Computation_e6_rank1}\n\\end{table}\n\n\n\\paragraph{Simple S-subalgebras of rank > 1}\n\nFollowing \\cite{Dynkin} (Chapter IV, Table 24), there exist four S-subalgebras\nof $\\mathfrak{e}_6$ of rank $>1$: $\\mathfrak{h} = A_2, \\mathfrak{g}_2, C_4$ \nand $\\mathfrak{f}_4$. For the cases $\\mathfrak{g}_2$ and $\\mathfrak{f}_4$, \nthe center\nof the corresponding group is $\\mathcal{Z}(\\tilde{H}) \\cong \\lbrace 1 \\rbrace$.\nThen $\\tilde{M}$ can be only a coroot of $\\mathfrak{e}_6$ and the quantity\n\\eqref{Quantity_e6} is always an integer. For the two remaining cases, the\nexplicit\nembedding is still given in \\cite{Dynkin}, and the strategy is the same as for\nrank one: we look how the generating element $\\iota(\\lambda^\\vee)$ of\n$\\mathcal{Z}(\\tilde{H})$ intersects with the lattices of $\\mathfrak{e}_6$ and\ncheck which possibility occurs among those listed in the case of rank one\n(except that we would also have to check that for the low multiples \nof $\\lambda^\\vee$ if $\\iota(\\lambda^\\vee)$ were\nnot in $Q^\\vee(\\mathfrak{g})$). The results are described in Table \n\\ref{Computation_e6_S} from which we infer that there are no gauge anomalies \nfor all simple S-subalgebras of $\\mathfrak e_6$.\n\n\\begin{table}[htb]\n \\centering\n\\begin{tabular}{|c|c|c|c|c|}\n  \\hline\n$\\mathfrak{h}$ & $\\mathcal{R}(\\mathfrak{h})$ & Index & $\\iota(\\lambda^\\vee)$ &\nCompatibility \\\\\n\\hline \\hline\n$A_2$ & $\\mathfrak{e}_6$ & 9 & $ \\left(\\tfrac{1}{2},\\tfrac{1}{2},\\tfrac{1}{2},\n\\tfrac{\\text{-}1}{2},\\tfrac{\\text{-}1}{2},\\tfrac{\\text{-}1}{2},\n\\tfrac{3}{\\sqrt{2}}\\right)$ & $\\in Q^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n$C_4$ & $\\mathfrak{e}_6$ & 1 & $ \\left(\n\\tfrac{1}{2},\\tfrac{1}{2},\\tfrac{1}{2},\\tfrac{\\text{-}1}{2},\\tfrac{\\text{-}1}{2}\n,\\tfrac{\\text{-}1}{2},\\tfrac{1}{\\sqrt{2}}\\right)$ &\n$\\in Q^\\vee(\\mathfrak{e}_6)$\\\\\n\\hline\n\\end{tabular}\n\\caption{The embedding of element $\\lambda^\\vee$ for simple S-subalgebras of\n$\\mathfrak{e}_6$ and its intersection with the lattices.}\n\\label{Computation_e6_S}\n\\end{table}\n\n\\paragraph{Simple R-subalgebras of rank > 1}\\label{SimpleR}\n\nWe only need to look at the R-subalgebras $\\mathfrak{h}$ with potential \nanomalies. Indeed, the subalgebra $\\mathcal{R}(\\mathfrak{h})$ is \nregular, so has been already treated. If $\\mathcal{R}(\\mathfrak{h})$ \ncorresponds to a model without anomalies, then it protects also \nthe R-subalgebra $\\mathfrak{h}$ included in it and there will be no anomalies \nfor the model built with $\\mathfrak{h}$. The list of the R-subalgebras of \n$\\mathfrak{e}_6$ is\ngiven in \\cite{Dynkin} (Chapter IV, Table 25), but without explicit embedding.\nThere remain five cases with potential anomalies: $\\mathfrak{h} = A_2$, with\n$\\mathcal{R}(\\mathfrak{h}) = A_5$,\\,$2A_2$,\\,$3A_2$, and  $\\mathfrak{h} = A_3$ \nor $C_3$ with $\\mathcal{R}(\\mathfrak{h}) = A_5$. If\n$\\mathcal{R}(\\mathfrak{h})$ is simple, then the embedding of $\\mathfrak{h}$ in\n$\\mathcal{R}(\\mathfrak{h})$ is given in \\cite{Lorente} (Table XIII), \nconsidering\n$\\mathfrak{h}$ as an S-subalgebra of $\\mathcal{R}(\\mathfrak{h})$. \n\n\n\\begin{table}[htb]\n\\centering\n\\begin{tabular}{|c|c|c|c|c|c|}\n  \\hline\n$\\mathfrak{h}$ & $\\mathcal{R}(\\mathfrak{h})$ & Index & $\\iota(\\lambda^\\vee)$ &\nCompatibility & $\\tilde{a}$ \\\\\n\\hline \\hline\n$A_2$ & $2A_2(\\iota_1)$ & 2 & $ \\left(\n\\tfrac{1}{3},\\tfrac{1}{3},\\tfrac{\\text{-}2}{3},\\tfrac{2}{3},\\tfrac{\\text{-}1}{3}\n,\\tfrac{\\text{-}1}{3},0\\right)$ &\n$\\notin P^\\vee(\\mathfrak{e}_6)$ &\\\\\n\\hline\n$A_2$  & $2A_2 (\\iota_2)$ & 2 & $ \\left(\n\\tfrac{1}{3},\\tfrac{1}{3},\\tfrac{\\text{-}2}{3},\\tfrac{1}{3},\\tfrac{1}{3},\\tfrac{\n\\text{-}2}{3},0\\right)$ &\n$\\in P^\\vee(\\mathfrak{e}_6)\\setminus Q^\\vee(\\mathfrak{e}_6)$ & 2\\\\\n\\hline\n$A_2$ & $3A_2(\\iota_1)$ & 3 & $ \\left(\n0,0,-1,1,0,0,0\\right)$ &\n$\\in Q^\\vee(\\mathfrak{e}_6)$ &\\\\\n\\hline\n$A_2$ & $3A_2(\\iota_2)$ & 3 & $ \\left(0,0,-1,\\tfrac{2}{3},\\tfrac{2}{3}\n,\\tfrac{\\text{-}1}{3},0\\right)$ &\n$\\notin P^\\vee(\\mathfrak{e}_6)$ & \\\\\n\\hline\n$A_2$ & $A_5$ & 5& $ \\left(\n\\tfrac{2}{3},\\tfrac{2}{3},\\tfrac{\\text{-}1}{3},\\tfrac{2}{3},\\tfrac{\\text\n{-}1}{3},\\tfrac{\\text{-}4}{3},0\\right)$ &\n$\\in P^\\vee(\\mathfrak{e}_6)\\setminus Q^\\vee(\\mathfrak{e}_6)$ & 4\\\\\n\\hline\n$A_3$ & $A_5$ & 2 & $ \\left(\n\\tfrac{1}{2},\\tfrac{1}{2},\\tfrac{\\text{-}1}{2},\\tfrac{1}{2},\\tfrac{\\text{-}1}{2}\n,\\tfrac{\\text{-}1}{2},0\\right)$ & $\\notin P^\\vee(\\mathfrak{e}_6)$ & \\\\\n\\hline\n$C_3$ & $A_5$ & 1 & $ \\left(\n\\tfrac{1}{2},\\tfrac{1}{2},\\tfrac{1}{2},\\tfrac{\\text{-}1}{2},\\tfrac{\\text{-}1}{2},\\tfrac{\\text{-}1}{2},0\\right)$ &\n$\\notin P^\\vee(\\mathfrak{e}_6)$ & \\\\\n\\hline\n\\end{tabular}\n\\caption{The embedding of element $\\lambda^\\vee$ for simple R-subalgebras of\n$\\mathfrak{e}_6$ and its intersection with the lattices. In case of potential\nanomalies, the explicit value of $\\tilde{a}$ that enters quantity\n\\eqref{Quantity_e6_bis} is given.}\n\\label{Computation_e6_R}\n\\end{table}\n\n\n\\noindent If $\\mathcal{R}(\\mathfrak{h})$ is only semisimple, the problem \nof the embedding is\ntreated in \\cite{Minchenko}, where several inequivalent embeddings of\n$\\mathfrak{h}$ in $\\mathfrak{e}_6$ appear. \nFor the $\\mathfrak{h} = A_2$ and $\\mathcal{R}(\\mathfrak{h}) = 3 A_2$, the two\ninequivalent embeddings are the following, denoting by $\\tilde\\alpha_1^\\vee$ \nand $\\tilde\\alpha_2^\\vee$ the simple coroots of $A_2$.\n\\begin{align}\n \\iota_1(\\tilde\\alpha_1^\\vee) = \\alpha_1^\\vee + \\alpha_5^\\vee \n+ \\delta^\\vee & \\qquad &\n\\iota_2(\\tilde\\alpha_1^\\vee) = \\alpha_1^\\vee + \\alpha_4^\\vee + \\delta^\\vee\\\\\n \\iota_1(\\tilde\\alpha_2^\\vee) = \\alpha_2^\\vee \n+ \\alpha_4^\\vee + \\alpha_6^\\vee & \\qquad &\n\\iota_2(\\tilde\\alpha_2^\\vee) = \\alpha_2^\\vee + \\alpha_5^\\vee + \\alpha_6^\\vee\n\\end{align}\nwhere we have exchanged $\\alpha^\\vee_4$ and $\\alpha^\\vee_5$. The other possible\nexchanges are equivalent to $\\iota_1$ or $\\iota_2$ \\cite{Minchenko}. For\n$\\mathcal{R}(\\mathfrak{h}) = 2A_2$, the two embeddings are given by\nsimilar formulas but with omission of $\\alpha^\\vee_6$ and $\\delta^\\vee$. \nAgain, in order to find $Z(\\tilde H)\\cap\\tilde Z$, we have to check how \nthe generating element $\\iota(\\lambda^\\vee)$ of \n$\\mathcal{Z}(\\tilde{H})$ intersects \nwith the lattices of $\\mathfrak{e}_6$. An explicit calculation is done in \nTable \\ref{Computation_e6_R}, and this time potential anomalies occur. \nThen, looking at the value of $\\tilde{a}$ for $\\tilde{M} =  \n\\iota(\\lambda^\\vee)$, we deduce an, eventually more restrictive, condition \non level $k$ required to avoid the anomalies (to exclude the anomalies in \nthe case of $A_3\\subset A_5$, we also have to observe that $\\iota(2\\lambda^\\vee)\n\\in Q^\\vee(\\mathfrak{e}_6)$). \n\\eject\n\n\\noindent This way, we obtain the general result for \nsimple nonregular subalgebras of $\\mathfrak{e}_6$\n\n\\begin{prop}\nThe untwisted coset models with $\\mathfrak{g} = \\mathfrak{e}_6$ and any\nsimple, nonregular subalgebra $\\mathfrak{h}$ do not have global  gauge\nanomalies for $k \\in \\mathbb{Z}$ except for the R-subalgebras \n$\\mathfrak{h} = A_2$ with $\\mathcal{R}(\\mathfrak{h}) = A_5$ and \n$\\mathfrak h=A_2$ with $\\mathcal{R}(\\mathfrak h)=2A_2$ embedded \nvia $\\iota_2$. For those subalgebras, the global gauge invariance requires \nthat $k \\in 3\\mathbb{Z}$.\n\\end{prop}\n\n \n\n\\subsection{Semisimple nonregular subalgebras of $\\mathfrak{e}_6$}\n\nLet $\\mathfrak h$ be a semisimple subalgebra of $\\mathfrak e_6$:\n\\begin{equation}\n \\mathfrak h = \\mathop{\\oplus}\\limits_{i=1}^{n} \\mathfrak h_i\n\\end{equation}\nwhere the $\\mathfrak h_i$ are simple, and the corresponding subgroups \nare denoted by $\\tilde H_i$. The case $n =1$ has been already treated \nabove, so we now deal with $n \\geq 2$. First, suppose that one of the \n$\\mathfrak h_i$ considered as a simple subalgebra leads to anomalies: \nthere exists $\\tilde M_i$ such that $e^{2 \\ii \\pi \\tilde{M}_i} \\in \n\\tilde H_i \\cap \\tilde Z$ which imposes $k \\in 3 \\mathbb Z$ to ensure \nthat the quantity $(\\ref{Quantity_e6})$ is integral. Then, taking \n$\\tilde M = \\tilde M_i$ but now embedded in $\\mathfrak h$,\nwe shall still have to impose $k \\in 3\\mathbb Z$ \nto have a globally gauge invariant model with semisimple Lie algebra \n$\\mathfrak h$. In other words, semisimple algebras composed of simple \nideals with at least one leading to anomalies are also anomalous. \nHowever, the inverse \nis not true: one can have a semisimple subalgebra corresponding to \nan anomalous model with all its simple ideals without any anomaly. For \nexample, the model with regular subalgebra $2A_2$ of \n$\\mathfrak e_6$ is anomalous for $k \\in \\mathbb{Z}\\setminus 3\\mathbb{Z}$ \nwhereas the one with $A_2$ (still regular) is globally gauge invariant \nfor every $k \\in \\mathbb{Z}$. Thus we need to check all the cases where \nall the simple ideals correspond to models without anomaly. To do that, \nwe need to consider the elements $\\sum_{i=1}^n \\alpha_i \\iota \n(\\lambda^\\vee_i)$ where $\\alpha_i \\in \\mathbb{Z}$ and $\\lambda_i$ are \nthe generating elements of the center of the $\\tilde H_i$, which have \nall been described above in the simple case (Tables \n\\ref{Computation_e6_rank1}, \\ref{Computation_e6_S} and \n\\ref{Computation_e6_R}), and $\\iota: \\mathfrak h \\rightarrow \n\\mathfrak e_6$ is the embedding. Comparing how these elements are \ncompatible with the coroot and coweight lattices of $\\mathfrak e_6$, \nthe anomaly problem is reduced to the three possibilities described in \nthe simple case \\ref{sec:SimpleCase}. \n\n\\paragraph{S-subalgebras}\n\nIn \\cite{Dynkin}\n(Chapter V, Table 39) one can find all the S-subalgebra of $\\mathfrak e_6$ \nand their including relations. It turns out that subalgebra\n $\\mathfrak h = \\mathfrak g_2 \\oplus A_2$ \n(with the explicit embedding given in \\cite{Dynkin}, Chapter V, Table 35) \nleads to an anomaly if $k \\in \\mathbb Z \\setminus 3 \\mathbb Z$, and that \nthe other semisimple nonsimple S-subalgebras of $\\mathfrak e_6$ are protected. \n\n\\begin{table}[ht]\n\\centering\n  \\begin{tabular}{|c|c|c|c|c|}\n  \\hline \n$\\mathfrak{h}$ & $\\mathcal{R}(\\mathfrak{h})$ & $\\left\\lbrace \n\\mathcal{R}(\\mathfrak{h}_i) \\right\\rbrace$ & Indices &\n No anomaly for \\\\\n\\hline \\hline\n\n$A_2 \\oplus A_1$ & $A_5 \\oplus A_1$ & $A_5, A_1$ & 2,1 &\n${k \\in 3\\mathbb{Z}}$  \\\\\n$A_3 \\oplus A_1$ & $A_5 \\oplus A_1$ & $A_5, A_1$ & 2,1 &$k \\in \\mathbb{Z}$  \\\\\n$C_3 \\oplus A_1$ & $A_5 \\oplus A_1$ & $A_5, A_1$ & 1,1 &$k \\in \\mathbb{Z}$  \\\\\n$A_1 \\oplus A_1$ & $A_5 \\oplus A_1$ & $A_5, A_1$ & 35,1&$k \\in \\mathbb{Z}$  \\\\\n$(A_2 (\\iota_1) \\oplus A_1) \\oplus A_1$ & $A_5 \\oplus A_1$ \n& $A_5, A_1$ & 2,3,1 &$k \\in\n\\mathbb{Z}$  \\\\\n$(A_2 (\\iota_2) \\oplus A_1) \\oplus A_1$ & $A_5 \\oplus A_1$ & $A_5, A_1$ \n& 2,3,1 &${k \\in\n3\\mathbb{Z}}$  \\\\\n$(2A_1) \\oplus A_1$ & $A_5 \\oplus A_1$ & $A_5, A_1$ \n& 8,3,1&$k \\in \\mathbb{Z}$  \\\\\n\\hline\n$A_1 \\oplus (2A_2)$ & $A_2 \\oplus (2A_2)$ \n& $A_2, 2A_2 $ & 4,1,1&${k \\in\n3\\mathbb{Z}}$  \\\\\n$A_2 \\oplus A_2 (\\iota_1) $ & $A_2 \\oplus (2A_2)$ \n& $A_2, 2A_2 $ &1,2 &$k \\in\n\\mathbb{Z}$  \\\\\n$A_2 \\oplus A_2 (\\iota_2)$ & $A_2 \\oplus (2A_2)$ \n& $A_2, 2A_2 $ & 1,2&${k \\in\n3\\mathbb{Z}}$  \\\\\n$A_1 \\oplus A_2 (\\iota_2)$ & $A_2 \\oplus (2A_2)$ \n& $A_2, 2A_2 $ & 4,2&${k \\in\n3\\mathbb{Z}}$  \\\\\n\\hline\n$A_1 \\oplus A_1 \\oplus A_2$ & $A_2 \\oplus A_2 \\oplus A_2$ \n& $A_2, A_2, A_2$ & 4,4,1&\n$k\\in \\mathbb{Z}$  \\\\\n\\hline\n$A_2(\\iota_2) \\oplus A_1$ & $A_5$ & $A_5 $ \n& 2,3&${k \\in 3\\mathbb{Z}}$  \\\\\n\\hline\n$  A_1 \\oplus A_2 (\\iota_2)$ & $A_1 \\oplus (2A_2)\n$ & $A_1, 2A_2$ & 1,2&${k \\in 3\\mathbb{Z}}$  \\\\\n\\hline\n$A_1 \\oplus A_1 \\oplus A_2$ & $A_1 \\oplus A_2 \\oplus A_2$ \n& $A_1, A_2, A_2$  & 1,4,1&$k \\in \\mathbb{Z}$  \\\\\n\\hline\n\\end{tabular}\\small\n\\caption{Semisimple nonsimple R-subalgebras of\n$\\mathfrak{e}_6$ with possible anomalies and the conditions\non $k$ required for their absence}\n\\label{Computation_e6_ssimple}\n\\end{table}\n\n\n\n\\paragraph{R-subalgebras}\nThe end of \\cite{Lorente} proposes a method to construct all the\nsemisimple R-subalgebras: the idea is to take the semisimple S-subalgebras \nof the semisimple regular subalgebras of $\\mathfrak e_6$, treating each \nsemisimple ideal independently. The semisimple S-subalgebras are described \nfor the classical \nalgebras up to rank 6 in \\cite{Lorente}, which is enough to construct all \nthe semisimple R-subalgebras of $\\mathfrak e_6$. However, we only need to \ntreat the R-subalgebras $\\mathfrak h$ where the regular \nsubalgebras $\\mathcal R(\\mathfrak h)$ lead to an anomaly problem, because\nthe other cases are protected against anomalies. \nThe computation is given in Table \\ref{Computation_e6_ssimple}, using \nthe fact that one ideal leads to an anomaly or computing the elements \nof the center as described before. Note that for the nonsimple S-subalgebra \n$A_2 \\oplus A_1 \\subset A_5$, $A_2$ is actually embedded in \n$A_2 \\oplus A_2$ \\cite{Lorente}, so the question of the two inequivalent \nembeddings $\\iota_1$ and $\\iota_2$ arises also here, as in \\ref{SimpleR}. \nWorking by decreasing rank, we have excluded some algebras from this Table \nsince they are protected by the ones of higher rank that do not have anomalies.\n\n\\eject\n\n\n\n\n\n\n\n\n\n\n\\noindent Putting all that together, we obtain the following result:\n\n\\begin{prop}\nThe untwisted coset models with $\\mathfrak{g} = \\mathfrak{e}_6$ and any\nnonregular nonsimple semisimple subalgebra \n$\\mathfrak{h}$ do not have global  gauge anomaly\nfor $k \\in \\mathbb{Z}$, except for the S-subalgebra $\\mathfrak{h} =\n\\mathfrak{g}_2 \\oplus A_2$ and the R-subalgebras appearing in \nTable \\ref{Computation_e6_ssimple} with the condition $k\\in 3\\mathbb Z$\nwhich exhibit global gauge anomaly for $k\\in\\mathbb Z\\setminus3\\mathbb Z$.\n\\end{prop}\n\n\\subsection{Examples of nonregular subalgebras of classical Lie algebras}\n\nThe semisimple nonregular subalgebra of classical algebra have been \nclassified explicitly in \\cite{Lorente} only up to rank 6. The general \nclassification proposed by Dynkin in \\cite{Dynk2} is less explicit and \ndoes not allow us to treat the anomaly problem in a general form as for \nregular subalgebras. Here we only give some example of classical algebras, \nbut the method is always the same once the explicit embedding of a \nsubalgebra is known : as for $\\mathfrak{e}_6$, we need to look how \nthe embedding of the generating element of the center of the considered \nsubalgebra is compatible with the coroot lattice of the ambient algebra.\n\n\\paragraph{Nonregular semisimple subalgebras of $A_4$.}\n\nThe coroot lattice of $A_4$ is given by\n\\begin{equation}\n  P^\\vee(A_4) = \\left\\lbrace \\left. \\left( \\dfrac{a}{5} + q_1, \\dots , \n\\dfrac{a}{5} + q_4, -\\dfrac{4a}{5} - q_1 - \\dots - q_4\\right) \\right| a, \nq_1, \\dots q_4 \\in \\mathbb Z \\right\\rbrace\n\\end{equation}\nand the coweight lattice $Q^\\vee(A_4)$ is given by the same formula but \nwith $a = 0$. According to \\cite{Lorente}, $A_4$ admits two S-subalgebras \nwhich are simple : $A_1$ and $B_2$. For $\\mathfrak h = A_1$, the embedding \nof the generating element $\\lambda^\\vee$ of the center of the corresponding \ngroup is given by\n\\begin{equation}\n  \\iota(\\lambda^\\vee) = (2,1,0,-1,-2) \\in Q^\\vee(A_4)\n\\end{equation}\nso the quantity $k$ tr$(M\\tilde M)$ will be integral for every $k \\in \n\\mathbb Z$ and there will be no anomaly for this model. For $\\mathfrak{h} \n= B_2$, one have\n\\begin{equation}\n   \\iota(\\lambda^\\vee) = (1,0,0,0,-1) \\in Q^\\vee(A_4)\n\\end{equation}\nwhich leads to the same conclusion. As we have seen in the regular case, \nall regular subalgebras of $A_4$ (except $A_4$) leads to non-anomalous \nmodels. We immediately conclude that all the R-subalgebra of $A_4$ are \nprotected by their regular $\\mathcal R(\\mathfrak h)$, so there is also \nno anomaly for these models. Finally, the only anomalous models corresponding \nto $\\mathfrak g = A_4$ and an arbitrary semisimple subalgebra are those with \n$\\mathfrak h = \\mathfrak g$, $Z = \\tilde Z \\cong Z_5$ and $k \\in \\mathbb Z \n\\setminus 5\\mathbb Z$. \n\n\\paragraph{S-subalgebras of $A_5$.}\n\nThe coroot lattice of $A_5$ is given by\n\\begin{equation}\n  P^\\vee(A_5) = \\left\\lbrace \\left. \\left( \\dfrac{a}{6} + q_1, \\dots , \n\\dfrac{a}{6} + q_5, -\\dfrac{5a}{6} - q_1 - \\dots - q_5\\right) \\right| a, \nq_1, \\dots q_5 \\in \\mathbb Z \\right\\rbrace\n\\end{equation}\nand the coweight lattice $Q^\\vee(A_5)$ is given by the same formula \nbut with $a = 0$. According to \\cite{Lorente}, $A_5$ admits six \nS-subalgebras : $A_1$, $A_2$, $A_3$, $C_3$, $A_1 \\oplus A_1$ and \n$A_2 \\oplus A_1$. For $\\mathfrak{h}=A_1$, one has   \n\\begin{equation}\n\\iota(\\lambda^\\vee) = \\left( \\dfrac{5}{2},\\dfrac{3}{2},\\dfrac{1}{2},\n-\\dfrac{1}{2},-\\dfrac{3}{2},-\\dfrac{5}{2} \\right),\n\\end{equation}\nsee Table VI of \\cite{Lorente}, whereas for $\\mathfrak h = A_2,\\,A_3$ \nand $C_3$, one has \n\\begin{eqnarray}\n\\iota(\\lambda^\\vee) = \\left( \\dfrac{2}{3},\\dfrac{2}{3},-\\dfrac{1}{3},\n\\dfrac{2}{3},-\\dfrac{1}{3},-\\dfrac{1}{3} \\right),\\ \n\\left( \\dfrac{1}{2},\\dfrac{1}{2},-\\dfrac{1}{2},\n\\dfrac{1}{2},-\\dfrac{1}{2},-\\dfrac{1}{2} \\right),\\ \n\\left( \\dfrac{1}{2},\\dfrac{1}{2},\\dfrac{1}{2},\n-\\dfrac{1}{2},-\\dfrac{1}{2},-\\dfrac{1}{2} \\right),\n\\end{eqnarray}\nrespectively, see the last 3 entries of Table 5 above.\nIn all 4 cases, $\\iota(\\lambda^\\vee)\\in P^\\vee(A_5) \\setminus Q^\\vee(A_5)$.\nTaking $\\iota(\\lambda^\\vee) = \\tilde M$ with $\\tilde a = 3,4,3,3$, respectively,\nand appropriate $\\tilde q_i$, and $M\\in P^\\vee(A_5)$ such that \n$e^{2\\ii\\pi M}\\in Z\\cong\\mathbb{Z}_p$, we obtain\n  \\begin{equation}\n    \\text{tr}(M\\tilde M) = \\dfrac{5 a \\tilde{a}}{p} + n\\,,\n  \\end{equation}\nwhere $n \\in \\mathbb Z$. There will be no anomaly for $k$ such that \n$k\\,{\\rm tr}(M\\tilde M) \\in \\mathbb Z$. \nFor $\\tilde a=3$, this imposes on $k$ the same \nrestrictions that the admissibility conditions (\\ref{Consistency_Ar}), \nso that the untwisted coset theories corresponding to the S-subalgebras \n$\\mathfrak{h}=A_1,A_3,C_3\\subset A_5$ do not have anomalies.\n\\,For the S-subalgebra $\\mathfrak{h}=A_2$, we obtain\nthe non-anomalous models with admissible levels for \n\\begin{equation}\n  k \\in \\left\\lbrace \n    \\begin{array}{ll}\n\\mathbb Z \\cap 2 \\mathbb Z = 2 \\mathbb Z &\\text{if } Z\\cong\\mathbb Z_2\\\\      \n3 \\mathbb Z   &\\text{if } Z\\cong\\mathbb Z_3\\\\\n3 \\mathbb Z \\cap 2 \\mathbb Z = 6 \\mathbb Z &\\text{if } Z\\cong\\mathbb Z_6      \n    \\end{array}\\right.\n\\end{equation}\nThe other untwisted models corresponding to the S-subalgebra \n$\\mathfrak h = A_2\\subset A_5$ and non-trivial subgroups $Z$ are anomalous.\n\\vskip 0.1cm\n\nThere are no conceptual or technical difficulties to obtain the no-anomaly \nconditions on $k$ for other subalgebras of $A_5$, and also for other \nclassical algebra $\\mathfrak g$, once the embeddings are known, but there \nis no general result so each case has to be treated separately. The previous \nexamples show that different anomaly conditions could appear according to \nthe subalgebra considered.\n\n\\section{Conclusions}\n\nWe have studied above the conditions for the absence of global gauge\nanomaly in the coset models of conformal field theory derived from\nWZW models with connected simple compact groups $G=\\tilde G\/Z$ as the targets  \nby gauging a subgroup of the rigid adjoint or twisted-adjoint symmetries\n$\\,G\\ni g\\mapsto hg\\hspace{0.03cm}\\omega(h)^{-1}\\in G$, \\,where $\\omega$ is \na, possible trivial, automorphism of $G$. \\,The full group of such symmetries \nis equal to $\\tilde G\/Z^\\omega$, where $Z^\\omega$ is the maximal subgroup \nof the center $\\tilde Z$ of the universal covering group $\\tilde G$ \nof $G$ for which the (twisted) adjoint action \nis well defined. We considered both the coset models where the full group \n$\\tilde G\/Z^\\omega$ was gauged and the ones where the gauging concerned only \na closed connected subgroup of $\\tilde G\/Z^\\omega$. Global gauge \nanomalies obstructing the invariance of the Feynman amplitudes of the theory \nunder ``large'' gauge transformations non-homotopic to unity may appear \nonly for non-simply connected groups $G$ corresponding to Lie algebras \n$\\mathfrak{g}$ of types $A_r,\\,D_r$ and $\\mathfrak{e}_6\\,$ (that are all\nsimply-laced). Using the results\n\\cite{Dynkin,Lorente,Minchenko} on the classification of semisimple \nLie subalgebras of simple Lie algebras, we obtained a complete \nlist of non-anomalous coset models (without boundaries) for groups $G$ \nwith the Lie algebra $A_r,\\,D_r$ or $\\mathfrak{e}_6$ if the gauged symmetry \nsubgroup $\\subset\\tilde G\/Z^\\omega$ corresponds to a regular Lie subalgebra \n$\\mathfrak{h}\\subset\\mathfrak{g}$ or, for $\\mathfrak{g}=\\mathfrak{e}_6$,\nto any semisimple Lie subalgebra. The global gauge anomalies that appear\nin the other coset model should render them inconsistent on the quantum\nlevel, as was argued in \\cite{GSW}. \n\\eject\n\n\n\\noindent{\\bf\\Large Appendices}\n\\vskip -0.5cm\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\ \\ \\ \\ Previous research has shown that using deep CNN models can be outstanding performance. Image classifiers or object detectors usually use VGG\\cite{vgg}, ResNet\\cite{resnet}, and other high FLOPs models as the backbone. In object detection, although the one-stage model led by YOLO\\cite{yolov1} is much faster than the two-stage model led by Faster-RCNN\\cite{fasterrcnn}, the backbone DarkNet used by YOLO is still a VGG-like or ResNet-like deep CNN model. YOLO has a computing overhead of at least \\textasciitilde20k MFLOPs. For edge computing devices, such high computing resource requirements cannot be carried. In the past, the number of model parameters was regarded as a primary goal of model lightweight. However, people soon discovered that the number of parameters is not positively correlated with the running speed. In recent years, more attention has been paid to the number of FLOPs. Although the model's actual running speed is still affected by the framework and OS, and implementation details, FLOPs are still recognized by the mainstream as the most theoretical metrics. In this paper, we use FLOPs as the guide to design a novel lightweight convolution method CSL-Module. We have theoretically proved that CSL-Module is 5 to 7 times faster than convolution-3x3. In our experiments, we have also shown that CSL-Module is faster and performs better than other lightweight convolution methods. Furthermore, CSL-Module was used to construct two highly efficient components, which were finally combined into a new lightweight object detector CSL-YOLO. Compared with other similar lightweight YOLOs, CSL-YOLO has reached the state-of-the-art level.\n\nThe rest of the paper is organized as follows: Section 2 briefly reviews the related techniques for lightweight model design, followed by the proposed CSL-Module, CSL-Bone, CSL-FPN, and CSL-YOLO in section 3, the tricks of CSL-YOLO in section 4, the experiments and discussion are presented in section 5, and finally, the conclusion in section 6.\n\n\\begin{figure*}\n  \\centering\n  \\bmvaHangBox{\\fbox{\\includegraphics[width=10cm]{_imgs\/cslyolo.png}}}\\\\\n  \\caption{Overall architecture of CSL-YOLO. the convolution-1x1 is weights-sharing.}\n  \\label{cslyolo}\n\\end{figure*}\n\n\\section{Related Work}\n\\ \\ \\ \\ In recent years, many lightweight CNN models have been proposed. In this section, we first review the lightweight convolution methods and backbone, and then we review the lightweight model of the one-stage object detector, especially the SSD\\cite{ssd}\\cite{retinanet} series and YOLO\\cite{yolov1} series.\n\n\\subsection{Lightweight Convolution Methods}\n\\ \\ \\ \\ The previous works have shown that convolution-3x3 is a powerful feature extraction method, but this method is still too expensive for edge computing. Especially in the deep CNN model, many convolution layers are used, which causes unbearable large FLOPs. Depthwise Separable Convolution (DSC) proposed by Google\\cite{xception}\\cite{mobilenet}. They uses a depthwise convolution to extract feature in space, and then use a pointwise convolution to extract feature in depth. This decoupling convolution method has shown in their experiments that it can approximate the performance of convolution-3x3 with fewer FLOPs. The backbone MobileNet\\cite{mobilenet} uses DSC extensively reduces FLOPs but maintains good accuracy. The successor backbone MobileNetv2\\cite{mobilenetv2} found that doing nonlinear transformation when the feature dimension is small, it will lose too much useful information. This problem is significant in lightweight models where the feature dimension is strictly controlled. Therefore, they use a pointwise convolution to expand the feature dimension to avoid the information loss caused by nonlinear transformation before the feature-maps passes through the DSC. They call this method Inverted Residual Block (IRB).\n\nIn addition to the two methods of decoupling in space and depth to reduce calculations, DSC and IRB, there are also some methods to divide the feature-maps into multiple groups to reduce calculations. ShuffleNet\\cite{shufflenet} divides the feature-maps into $G$ groups and passes them through DSC. The formula for FLOPs of convolution is shown as \\myeqref{convflops}. $H'$ denotes the height of output. $W'$ denotes the width of output. $C$ denotes the channel of input. $N$ denotes the channel of output. $K$ denotes the kernel size of convolution. According to the it, the feature-maps are grouped so that the input channel $C$ is reduced to $C\/G$, the output channel $N$ is reduced to $N\/G$, and the FLOPs only have the original $1\/G$. CSPNet\\cite{cspnet} divides the feature-maps into two halves. The first half of which is generated by convolution, and the other half is directly output after concatenating the first half. GhostNet\\cite{ghostnet} discusses this problem more systematically. The proposed Ghost Module uses half of the feature-maps to produce half of the output feature-maps with more expensive transformations, and then uses these feature-maps to generate more redundant feature-maps through cheap linear transformations, and finally concatenates the two halves. These lightweight methods performed well in their experiments.\n\n\\begin{equation}\nFLOPs_{conv}=H'*W'*C*K^2*N\n\\label{convflops}\n\\end{equation}\n\n\\subsection{Lightweight Object Detection Method}\n\\ \\ \\ \\ As generally recognized, object detectors can be classified into two-stage and one-stage. The two-stage model has a stage to generate ROI(Region of Interest), so the accuracy is usually higher, but the speed is also slower than the one-stage model. Although the one-stage model represented by YOLO focuses on the characteristics of real-time detection. However, most FLOPs of the one-stage model are still unacceptable for the tight computing resources of edge computing devices. Therefore, we will review the representative lightweight models in the YOLO and SSD series with extremely low FLOPs.\n\n\\subsubsection{SSD Series}\n\\ \\ \\ \\ SSD is an essential branch of the one-stage detectors, and many designs have led to later detectors. The multi-scale prediction they recommended has indirectly affected the proposal and contribute to the popularization of Feature Pyramid Network(FPN)\\cite{fpn}, and they effectively integrated the anchor box proposed by Faster-RCNN\\cite{fasterrcnn} to improve the performance significantly. We next introduce a few representative lightweight models in the SSD series. MobileNet-SSD\\cite{mobilenet} achieves quite good results with the simple combination of lightweight backbone MobileNet + SSD head. MobileNet-SSDLite further modified the SSD head to make the entire model more lightweight, and the accuracy has improved under the same FLOPs. MobileNetv2-SSDLite\\cite{mobilenetv2} replaced the backbone with a lighter MobileNetv2\\cite{mobilenetv2} and achieved fantastic speed and accuracy. PeleeNet\\cite{peleenet} constructed a DenseNet-like\\cite{densenet} lightweight backbone, and at the same time, reduced the output scale of the SSD head to reduce the calculation. They also got impressive results in the experiments.\n\n\\subsubsection{YOLO Series}\n\\ \\ \\ \\ As the one-stage model pioneer, YOLO discards the stage that generates the prior ROI and directly predicts the bounding box. This solution dramatically improves the speed of inference and achieves real-time on the GPU. But YOLO is still too large for embedding devices. YOLO has updated four official versions so far. Although there are corresponding tiny versions from YOLOv1 to YOLOv4\\cite{yolov1}\\cite{yolov2}\\cite{yolov3}\\cite{yolov4}, the official updated version focuses more on improving accuracy than speed. Therefore, Tiny-YOLOv1 to Tiny-YOLOv4\\cite{yolov1}\\cite{yolov2}\\cite{yolov3}\\cite{yolov4} always follow a similar compression strategy. They remove some convolutional layers or remove some multi-scale output layers in FPN. This strategy achieves a good compression ratio, but it also causes a large loss of accuracy. YOLO-LITE\\cite{yololite} follows a more aggressive reduction strategy. YOLO-LITE even only has 482 MFLOPs, but it also loses more considerable accuracy.\n\n\\begin{figure}\n\\centering\n\\begin{tabular}{cc}\n\\bmvaHangBox{\\fbox{\\includegraphics[width=5cm]{_imgs\/cslm.png}}}&\n\\bmvaHangBox{\\fbox{\\includegraphics[width=5cm]{_imgs\/cslm_types.png}}}\\\\\n(a)&(b)\n\\end{tabular}\n\\caption{(a) Overall architecture of CSL-Module. (b) The two variants of CSL-Module are with attention and downsample version.}\n\\label{cslm}\n\\end{figure}\n\n\\section{Approaches}\n\\ \\ \\ \\ In this section, we first introduce the Cross-Stage Lightweight (CSL) Module. CSL-Module generates feature-maps with fewer FLOPs, and then we build two lightweight components necessary for object detectors based on CSL-Module.\n\\subsection{CSL-Module}\n\\ \\ \\ \\ Previous research has shown that use less computation to generate redundant feature-maps can reduce the FLOPs considerably. CSPNet\\cite{cspnet} presents a cross-stage method for solving it, and GhostNet\\cite{ghostnet} systematically verifies the effectiveness of the cheap operation in this issue. However, the problem is that the main operation to generate valuable feature-maps is still too expensive for edge computing. We propose dividing the input feature-maps into two branches. The first branch generates the half redundant feature-maps by a cheap operation like GhostNet did; the other branch generates the other half necessary feature-maps by lightweight main operation, then concatenate the two outputs together. The overall architecture is shown in \\myfigref{cslm}. The hyperparameter $t$ represents the ratio of feature expanding. We set $t$ as 3 in CSL-Bone, and set $t$ as 2 in else. We insert the SE module\\cite{squeezenet} or adaptive average pooling when down-sampling or attention is needed after expansion block. Besides, we use Mish\\cite{mish} as the activation function, and in their experiments show that Mish performs better in deep CNN models than ReLU\\cite{relu} and Swish\\cite{swish}.\n\\subsubsection{Difference from Existing Methods}\n\\ \\ \\ \\ The proposed CSL-Module generates half redundant feature-maps by a cheap operation at the skip branch. On the main branch, it is different from CSP Module and Ghost Module. We propose a lightweight main operation to generate the other half necessary feature-maps. In this branch, we design an IRB-like expansion block, using the input feature-maps and output feature-maps of the skip branch to generate intermediate candidate feature-maps by depthwise convolution. One of the great advantages of this block is pointwise convolution-free, and We all know that depthwise convolution has far fewer FLOPs than pointwise convolution. It is different from IRB. IRB uses pointwise convolution to generate candidate feature-maps. The other advantages of this block is it thoroughly considered all currently available features, this can minimizes redundant calculations. Besides, because there is already skip branch, the main branch only needs to generate half of the feature-maps, reducing FLOPs significantly. In general, the proposed CSL-Module reduces FLOPs by cheap operation and cross-stage ideas. On the other hand, we especially lightweight design for the main branch. We replace the convolution layers in VGG-16\\cite{vgg} to verify the effectiveness of CSL-Module, and the new models are denoted as IRB-VGG-16, Ghost-VGG-16, and CSL-VGG-16, respectively. We evaluate them on CIFAR-10, the training setting and tricks are all of the same (e.g., flip, affine, mix-up\\cite{mixup}, and steps learning rate). From \\mytabref{cslmoncifar}, it can be seen that CSL-Module is faster than other advanced lightweight convolution methods, and CSL-Module can more approximate the performance of convolution-3x3. This experiment proves that CSL-Module is a very competitive lightweight convolution method.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline\n& MFLOPs & Acc.(\\%)\\\\\n\\hline\nVGG-16 & 299 & 92.6\\\\\nIRB-VGG-16 & 226 & 92.6\\\\\nGhost-VGG-16 & 169 & 90.2\\\\\n\\textbf{CSL-VGG-16} & \\textbf{128} & 92.0\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Comparison of CSL-Module with the other lightweight convolution methods on CIFAR-10.}\n\\label{cslmoncifar}\n\\end{table}\n\n\\subsubsection{Analysis on FLOPs}\n\\ \\ \\ \\ The analysis is made with the well-known formula \\myeqref{convflops}. We assume the input shape and output shape are equal for computational simplicity to compare the speed-up ratio of FLOPs between CSL-Module and convolution-3x3. CSL-Module has a hyperparameter $t$ as same as IRB\\cite{mobilenetv2}, representing the expansion ratio of the expansion block. When $t$=2, the speed-up ratio reaches 7.2 times; when $t$=3, it reaches 5.1 times. The whole process is shown in \\myeqref{analysissr}\n\\begin{equation}\n\\begin{aligned}\nFLOPs_{csl}&=(H'*W'*C*0.5N)+t(H'*W'*K^2*0.5N)\\\\\n            &+(H'*W'*C*K^2)\\\\\n            &+(H'*W'*(C+0.5tN)*K^2)\\\\\n            &+(H'*W'*(C+0.5tN)*0.5N)\\\\\nwhen\\ C=&N,\\ K=3:\\\\\n&sr=\\frac{FLOPs_{conv}}{FLOPs_{csl}}\\simeq\\frac{9}{1+0.25t}\n\\label{analysissr}\n\\end{aligned}\n\\end{equation}\n\\subsection{Building Lightweight Components}\n\\ \\ \\ \\ We propose two lightweight components CSL-Bone and CSL-FPN. These two components are necessary for object detectors. CSL-Bone extracts the features of input image with fewer FLOPs than other backbone models; CSL-FPN predicts the bounding boxes on more different scales efficiently.\n\n\\subsubsection{Lightweight Backbone}\n\\ \\ \\ \\ The proposed CSL-Bone consists of several CSL-Module groups. SE Module\\ \\cite{squeezenet} has integrated to first CSL-Module in a group to enhance the feature extraction capabilities of the entire group.\\ Also, we insert pooling layers for down-sampling at appropriate locations to obtain high-level semantic feature. Finally, CSL-Bone outputs three different scale feature-maps. The overall architecture is shown in \\mytabref{cslyolo}. We evaluate CSL-Bone, MobileNetv2, and GhostNet on CIFAR-10 and applied the same training setting too. It can be seen from \\mytabref{cslboneoncifar}. Although the CSL-Bone gets lower accuracy than MobileNetv2, but the FLOPs of CSL-Bone just 58.7\\% than MobileNetv2. On the other hand, CSL-Bone gets higher accuracy than GhostNet, but only slightly increased FLOPs.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline\n& MFLOPs & Acc.(\\%)\\\\\n\\hline\nMobileNetv2 & 75 & 91.3\\\\\nGhostNet & 40 & 89.3\\\\\n\\textbf{CSL-Bone} & 44 & 90.7\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Comparison of CSL-Bone with the other lightweight backbons on CIFAR-10.}\n\\label{cslboneoncifar}\n\\end{table}\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n& MFLOPs & $AP$ & $AP_{50}$\\\\\n\\hline\nVanilla-FPN (conv-3x3) & 416 & 19.0 & 35.8\\\\\nCSL-FPN ($R$=1) & 127 & 18.7 & 35.5\\\\\nCSL-FPN ($R$=2) & 198 & 18.8 & 35.8\\\\\nCSL-FPN ($R$=3) & 268 & 18.8 & \\textbf{37.2}\\\\\nCSL-FPN ($R$=4) & 339 & \\textbf{19.8} & 37.0\\\\\nCSL-FPN ($R$=5) & 409 & \\textbf{19.8} & \\textbf{37.2}\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{The performance of the proposed CSL-FPN with different $R$ on MS-COCO.}\n\\label{cslfpnonmscoco}\n\\end{table}\n\n\\begin{equation}\n\\begin{aligned}\n&Let\\ l\\ be\\ the\\ number\\ of\\ layers.\\\\ \n&Let\\ k\\ be\\ the\\ number\\ of\\ anchors\\ per\\ layer.\\\\\n&Let\\ B\\ be\\ bounding\\ boxes\\ of\\ img.\\\\\n&Let\\ A_i\\ be\\ anchors\\ of\\ layer_i,\\ 0\\leq i<l\\\\\n&S=[0,\\ \\frac{1}{2^{l-1}},\\ \\frac{1}{2^{l-2}},\\ \\frac{1}{2^{l-3}},...,\\ \\frac{1}{2^{l-l}}]\\\\\n&X_i=\\{S_i\\leq b<S_{i+1}|\\forall b \\in B\\}\\\\\n&C_i=Kmeans(X_i,\\ k)\\ with\\ IoU\\ dist\\ function.\\\\\n&A_i=\\{Center(C_i^0),Center(C_i^1),...,Center(C_i^k)\\}\n\\label{anchorconstraint}\n\\end{aligned}\n\\end{equation}\n\n\\subsubsection{Lightweight FPN}\n\\ \\ \\ \\ Previous studies have shown that large-scale feature-maps have more object details, such as edges, corners, or textures, while small feature-maps have comprehensive semantic understanding. Vanilla FPN\\cite{fpn} up-sampling the small feature-maps then fuse them with the large feature-maps. On the other hand, vanilla FPN output three scale feature-maps. It is helpful for the model to detect objects of different sizes. The proposed CSL-FPN first replaces all convolution-3x3 in FPN to CSL-Module. Second, in the expansion stage, a middle-scale layer is formed between two scale layers, and these middle-scale layers can enhance the model's ability to detect objects of different sizes. Third, in the repeat stage,  there are (k)th layer, (k-1)th, and (k+1)th layer for feature fusion at the same time, but each time only odd or even layers are used. For example, there are only the 2nd and 4th layers in the first fusion, and in the second fusion, there are the 1st, 3rd, and 5th layers. In other words, the proposed CSL-FPN has the same number of convolutions as vanilla-FPN but has more feature fusion. The overall architecture is shown in \\myfigref{cslfpn}. In our implementation of the proposed CSL-FPN, to make element-wise addition easier, we set the channels of the five output layers to be the same in the layer expansion stage. The repeat stage uses a hyperparameter $R$ to indicate that CSL-FPN has stacked several blocks in total. A larger $R$ can achieve higher AP, but FLOPs also increase. There is a trade-off between speed and performance. We test the best value of $R$ on MS-COCO based on 320x320 CSL-YOLO. \\mytabref{cslfpnonmscoco} shows the results. As $R$ gets larger and larger, $AP$ also improved from 18.6\\% to 19.8\\%, $AP_{50}$ improved from 35.5\\% to 37.2\\%, but MFLOPs also deteriorated from 127 to 409. After considering the trade-off, we decided to set $R$ to 3.\n\\begin{figure}\n  \\centering\n  \\bmvaHangBox{\\fbox{\\includegraphics[width=10cm]{_imgs\/cslfpn.png}}}\\\\\n  \\caption{Overall architecture of CSL-FPN. the layer expansion stage is used to expand the output scale and enhance the continuity between the scales. the repeat stage uses a hyperparameter $R$ to represent the number of times the block is stacked. the blue arrow represents downsampling, the orange arrow represents upsampling, and the white hollow circle represents identity mapping.}\n  \\label{cslfpn}\n\\end{figure}\n\n\\section{TRICKS OF CSL-YOLO}\n\\subsection{Anchors Constraint}\n\\ \\ \\ \\ The original YOLO series uses K-means with the IoU distance function to cluster the height and width of ground truth, then take the center point as an anchor box. These $k$ anchors generated from $k$ cluster and assign to the output layers of FPN based on their scale. When we expand the output layers from 3 to 5, $k$ also increased from 9 to 15. If we use the above method, then most of these anchors are small-scale due to many small objects in MS-COCO\\cite{mscoco}. There will be high-level output layers forced to use small-scale anchors. However, it is well known that high-level feature-maps are not conducive to detect small objects. Therefore, we add the scale limit such as \\myeqref{anchorconstraint} before K-means so that the distribution of anchors generated is more in line with each output layer's scale. We conduct experiments in \\mytabref{improvementac}, and we can see that the original 3 output layers has worsened after being expanded to 5 output layers. After adding our constraint method, it has improved a considerable AP.\n\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|}\n\\hline\n& $AP$ & $AP_{50}$\\\\\n\\hline\nOutput Layers x 3 w\/o AC & 17.9 & 36.1\\\\\nOutput Layers x 5 w\/o AC & 17.2 & 35.1\\\\\n\\textbf{Output Layers x 5 w\/ AC} & \\textbf{18.8} & \\textbf{37.2}\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Improvement by anchors constraint.}\n\\label{improvementac}\n\\end{table}\n\n\\subsection{Non-Exponential Prediction}\nYOLO series actually predicts the offset of $x$, $y$, $w$, $h$, as follows:\n\n\\begin{equation}\n\\begin{aligned}\n&w=w_{anchor}*\\mathrm{e}^{w_{pred}},\\\\\n&h=h_{anchor}*\\mathrm{e}^{h_{pred}},\n\\label{logprediction}\n\\end{aligned}\n\\end{equation}\nwhere $h_{pred}$ and $w_{pred}$ denotes the offset of the object height and width predicted by the model, $h_{anchor}$ and $w_{anchor}$ denotes the height and width of anchor. Although the log function can limit the prediction of the model range, but the sensitivity of the exponential function makes the width and height quite unstable. Therefore, we removed the log function and let the model directly predict the offset. Then, \\myeqref{nonogprediction} can be modified as:\n\\begin{equation}\n\\begin{aligned}\n&w=w_{anchor}+w_{pred},\\\\\n&h=h_{anchor}+h_{pred}.\n\\label{nonogprediction}\n\\end{aligned}\n\\end{equation}\nFrom \\mytabref{nonexppred}, clearly non-exponential prediction can improve 1\\textasciitilde2\\% AP even under various image sizes. As shown in \\myfigref{cslyolo}, all components are then integrated to construct CSL-YOLO.  During inference, the soft-nms\\cite{softnms} technique is adopted to penalize overlapping boxes.\n\\begin{table}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|}\n\\hline\n& Input Size & $AP$ & $AP_{50}$\\\\\n\\hline\nw\/ Exp & 224x224 & 15.0 & 31.1\\\\\nw\/ Exp & 320x320 & 18.8 & 37.2\\\\\nw\/ Exp & 416x416 & 22.0 & 41.9\\\\\nw\/ Exp & 512x512 & 25.1 & 45.7\\\\\n\\hline\nw\/o Exp & 224x224 & \\textbf{16.5} & \\textbf{32.2}\\\\\nw\/o Exp & 320x320 & \\textbf{21.4} & \\textbf{39.9}\\\\\nw\/o Exp & 416x416 & \\textbf{24.5} & \\textbf{44.0}\\\\\nw\/o Exp & 512x512 & \\textbf{26.3} & \\textbf{46.2}\\\\\n\\hline\n\\end{tabular}\n\\end{center}\n\\caption{Improvement by non-exponential prediction.}\n\\label{nonexppred}\n\\end{table}\n\n\n\n\\section{EXPERIMENTS}\n\\ \\ \\ \\ The proposed CSL-YOLO is not pre-trained on Imagenet\\cite{imagenet}. We follow the conclusion of \\cite{rethinking}, and train from scratch on MS-COCO. We applied the basic augmentation methods such as flip, rotate, and mix-up\\cite{mixup} instead of the fancy methods. We set the batch size as 16, and choosing Adam as the optimizer, The learning rate starts from 1e-3, after 50 epochs, it drops to 1e-4, after 40 epochs, it drops to 1e-5, and finally it is completed after another 20 periods. We train the model on two GTX 1080ti. The entire training process has \\textasciitilde800K iterations. Finally, the result of the prediction is shown in \\mytabref{cslyoloonmscoco}. Under the input scale of 416x416, the proposed CSL-YOLO uses 3.2M parameters and 1470 MFLOPs to obtain 42.8\\% $AP_{50}$, and Tiny-YOLOv4 uses 6.1M parameters and 3450 MFLOPs to obtain 40.2\\% $AP_{50}$. It can be said that CSL-YOLO consumes much less time(FLOPs) and space(parameters) than the advanced Tiny-YOLOv4, and can achieve impressive AP performance. Besides, at the input scale of 224x224, compared with the lightest YOLO-LITE, CSL-YOLO still achieves higher AP performance with lower FLOPs.\n\n\\begin{table}\n\\begin{center}\n\\setlength{\\tabcolsep}{0.7mm}{\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\\hline\nModel & Backbone & Input Size & MFLOPs & Params & $AP$ & $AP_{50}$ &  $AP_{75}$\\\\\n\\hline\nYOLO-LITE & DarkNet-LITE & 224x224 & 482 & - & - & 12.3 & -\\\\\nTiny-YOLOv3 & Tiny-DarkNet & 320x320 & 1650 & 8.7M & 14.0 & 29.1 & -\\\\\nTiny-YOLOv3 & Tiny-DarkNet & 416x416 & 2780 & 8.7M & 16.0 & 33.1 & -\\\\\nTiny-YOLOv4 & Tiny-CSPDarkNet & 416x416 & 3450 & 6.1M & 21.7 & 40.2 & -\\\\\n\\textbf{CSL-YOLO} & CSL-Bone & 224x224 & \\textbf{425} & \\textbf{3.2M} & \\textbf{16.5} & \\textbf{32.2} & \\textbf{15.0}\\\\\n\\textbf{CSL-YOLO} & CSL-Bone & 320x320 & \\textbf{869} & \\textbf{3.2M} & \\textbf{21.4} & \\textbf{39.9} & \\textbf{20.6}\\\\\n\\textbf{CSL-YOLO} & CSL-Bone & 416x416 & \\textbf{1470} & \\textbf{3.2M} & \\textbf{24.5} & \\textbf{44.0} & \\textbf{24.2}\\\\\n\\textbf{CSL-YOLO} & CSL-Bone & 512x512 & \\textbf{2223} & \\textbf{3.2M} & \\textbf{26.3} & \\textbf{46.2} & \\textbf{26.5}\\\\\n\\hline\nMobileNet-SSD & MobileNet & 300x300 & 1200 & 6.8M & 19.3 & - & -\\\\\nMobileNet-SSDLite & MobileNet & 320x320 & 1300 & - & 22.2 & - & -\\\\\nMobileNetv2-SSDLite & MobileNetv2 & 320x320 & 800 & 4.3M & 22.1 & - & -\\\\\nPeleeNet & PeleeNet & 304x304 & 1290 & 6.0M & 22.4 & 38.3 & 22.9\\\\\n\\hline\n\\end{tabular}}\n\\end{center}\n\\caption{Comparison of CSL-YOLO with the other state-of-the-art lightweight object detection system on MS-COCO.}\n\\label{cslyoloonmscoco}\n\\end{table}\n\n\\section{CONCLUSION}\n\\ \\ \\ \\ We proposed a new lightweight convolution method CSL-Module, in the experiment proved that it is competitive compared to other similar methods. We further proposed two components CSL-Bone and CSL-FPN, both of which achieve better performance with fewer FLOPs. We build a new lightweight object detector CSL-YOLO with these components. Besides, we proposed two tricks, a new anchor generation method and a non-exponential prediction to improve AP. Finally, we trained CSL-YOLO from scratch on MS-COCO. From \\mytabref{cslyoloonmscoco}, we can see that CSL-YOLO has better AP and fewer FLOPs and parameters than other lightweight YOLOs. In summary, we have three contributions:\n\\begin{enumerate}\n\\item We proposed a novel lightweight convolution method CSL-Module that costs fewer FLOPs to approximate the fitting ability of convolution-3x3.\n\\item We proposed two lightweight components for object detector, CSL-Bone and CSL-FPN which showed excellent results.\n\\item We effectively integrated these components, and two tricks to improve AP, and finally proposed a state-of-the-art level lightweight detector CSL-YOLO.\n\\end{enumerate}\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nRedshift is one of well-known effects of gravity, it plays an essential\nrole in relativistic astrophysics. Its description entered many textbooks.\nIn particles, it concerns propagation of light in the black hole background.\nThus main attention is focused on properties of light outside the\nevent horizon. Less  these properties inside the horizon are discussed.\nAlso, strange as it may seem, the question of the redshift of a photon\nmoving along the horizon dropped out from consideration almost completely.\nThere is general discussion of this issue in \\cite{lake} for the\nReissner-Nordstr\\\"{o}m - de Sitter metric but the relation between redshift or\nblueshift and the nature of the horizon was not revealed there.\n\nMeanwhile, there are several points that can serve as a motivation for such a\nconsideration.\n\n(i) Recently, important methodic issues were discussed in \\cite{au}, \\cit\n{kas} concerning properties of the world visible by an observer falling into\nthe Schwarzschild black hole and communicating by radio signals with another\nfalling one. In doing so, some incorrect statements were made in \\cite{au}\nabout \"ghosts\" of the first observer supposedly waiting for the second one\non the horizon. As was argued in \\cite{kas}, there are no such ghosts at all\nalthough the second observer does receive a signal from the first one\nemitted at the moment of crossing the horizon. This required detailed\ncalculation of the frequency shift for a photon propagating along the\nhorizon with the result that a finite redshift occurs in this case.\n\nIn the present work, we generalize these observations, demonstrate that such\na redshift is present for any horizon of a spherically symmetric nonextremal\nblack hole and find its value.\n\n(ii) If the metric contains an inner horizon (say, for the Reissner-Nordst\n\\\"{o}m black hole), the calculation of the frequency\nshift for a photon emitted along such a horizon is of special interest. We demonstrate that now,\ninstead of a redshift, a blueshift occurs.  In the limit when a photon\nis received near the bifurcation point, the blueshift becomes infinite. This\nestablishes the connection of the issue under discussion with the analogue\nof the Ba\\~{n}ados-Silk-West (BSW) effect \\cite{ban}. It consists in the\ninfinite growth of the energy of colliding particles in the centre of mass\nframe. Originally, it was found near the event horizon but, later on, it\nturned out that another similar effect is valid also near the inner horizon\n(see \\cite{inner} and references therein). From another hand, the issue\nunder discussion can be considered as an effect supplemental to a well-known\ninstability of the inner horizon \\cite{fn}, \\cite{ham}.\n\n(iii) In addition to the propagation along the nonextremal horizon, there is\nalso a question what happens in the extremal case. We argue that, by\ncontrast to two previous ones, now the frequency shift is absent.\n\n(iv) In papers \\cite{au}, \\cite{kas} the Kruskal-Szekerez (KS) coordinate\nsystem was used. We also exploit it. In addition, it is of interest to\ncompare the results using another powerful system - the Lemaitre one. We\nconstruct such a system for a whole class of metrics that includes the\nSchwarzchild one as a particular case.\n\nIn the present work, we restrict ourselves by the simplest case of radially\nmoving observers.\n\n\\section{Motion outside the nonextremal event horizon}\n\nWe consider the metri\n\\begin{equation}\nds^{2}=-fdt^{2}+\\frac{dr^{2}}{f}+r^{2}(d\\theta ^{2}+\\sin ^{2}\\theta d\\phi\n^{2})\\text{.}  \\label{met}\n\\end{equation}\n\nWe suppose that the metric has the event horizon at $r=r_{+}$, so \nf(r_{+})=0 $. (For simplicity, we assumed that $g_{00}g_{11}=-1$ but this\ncondition can be relaxed easily.) We consider now a nonextremal black hole.\nNear the horizon\n\\begin{equation}\nf\\approx \\kappa (r-r_{+})\\text{,}  \\label{kar}\n\\end{equation\nwhere $\\kappa =\\frac{f^{\\prime }(r_{+})}{2}$ is the surface gravity.\n\nLet an observer has the four-velocity $u^{\\mu }=\\frac{dx^{\\mu }}{d\\tau }$,\nwhere $\\tau $ is the proper time. We restrict ourselves to the radial motion\nof a massive particle (we call it \"observer\"). Then, the four-velocity \nu^{\\mu }=(\\dot{t}$, $\\dot{r})$. Here, a dot denotes derivative with respect to \n$\\tau $.\n\nThe geodesic equations of motion for such a particle read\n\\begin{equation}\nm\\dot{t}=\\frac{E}{f}\\text{,}  \\label{mt}\n\\end{equation\n\\begin{equation}\nm\\dot{r}=-Z\\text{, }Z=\\sqrt{E^{2}-fm^{2}}\\text{,}  \\label{mr}\n\\end{equation\n$E=-mu_t$ is the conserved energy  of a particle.\n\nFor a photon having the wave vector $k^{\\mu }$, the equations of motion ar\n\\begin{equation}\nk^{t}=\\frac{\\omega _{0}}{f}\\text{,}  \\label{yt}\n\\end{equation\n\\begin{equation}\nk^{\\phi }=\\frac{l}{g_{\\phi }}\\text{,}\n\\end{equation\n\\begin{equation}\nk^{r}=-Q\\text{, }Q=\\sqrt{\\omega _{0}^{2}-f\\frac{l^{2}}{r^{2}}}  \\label{r}\n\\end{equation\nwhere $\\omega _{0}=-k_{0}$ is the conserved frequency, $l=k_{\\phi }$ is the\nconserved angular momentum.\n\nLet a free falling observer emits or receives a photon. Its frequency measured\nby this observer is equal to \n\\begin{equation}\n\\omega =-k_{\\mu }u^{\\mu }.  \\label{om}\n\\end{equation}\n\nTaking into account (\\ref{mt}) - (\\ref{r}), we find after straightforward\ncalculation\n\\begin{equation}\n\\omega =\\frac{E\\omega _{0}-\\varepsilon ZQ}{mf}\\text{.}  \\label{wzq}\n\\end{equation}\n\nHere, $\\varepsilon =+1$ if both objects (the observer and the photon) move in\nthe same direction and $\\varepsilon =-1$ if they do this in opposite ones.\nIf $\\varepsilon =-1$, this corresponds to head-on collision between a\nmassive and a massless particles and this means that either an observer receives\na photon emitted from a smaller value of $r$ or a falling observer emits a\nphoton in the backward direction.\n\nNear the horizon, $f\\ll 1.$ Let $\\varepsilon =+1$ and the observer looks\nback. Near the horizon, he will see the frequenc\n\\begin{equation}\n\\omega \\approx \\frac{1}{2}(\\frac{El^{2}}{m\\omega _{0}r_{+}^{2}}+\\frac{\\omega\n_{0}m}{E})\\text{.}  \\label{n1}\n\\end{equation\nIf $\\varepsilon =-1$\n\\begin{equation}\n\\frac{\\omega }{\\omega _{0}}\\approx \\frac{2E}{mf_{em}}\\text{.}  \\label{n2}\n\\end{equation\nHere $f_{em}$ is the value of the metric function in the point of emission.\nLet the observer emits in his frame a photon having the frequency $\\omega $\nand travelling to infinity. Then, at infinity it will be received with the\nfrequency $\\omega _{0}$ having the order $f_{em}$. Eq. (\\ref{n2}) agrees\nwith the standard result for the Schwarzschild metric (see Sec. XII.102 of \n\\cite{LL}, especially eq. 102.10).\n\nIt is instructive to compare this to another situation usually discussed in\ntextbooks when the emitter is not free falling but is static. In the latter\ncase, the frequency at infinity $\\omega _{0}=\\omega \\sqrt{f}$ (see, e.g. eq.\n88.6 of \\cite{LL}) has the order $\\sqrt{f_{em}}\\,$. Obviously, the\ndifference can be attributed to the motion of an emitter (the Doppler effect\nmakes the redshift stronger).\n\nSome reservations are in order. Throughout the paper, we assume that the\ngeometric optics is a reasonable approximation for propagation of light\nwaves. As usual, this implies that the wavelength $\\lambda $ satisfies\nrelations $\\frac{\\lambda }{2\\pi }\\ll \\mathcal{L},$ $\\mathcal{R},$ where \n\\lambda $ is the wave length, $\\mathcal{L}$ is a typical scale\ncharacterizing a wave packet and $\\mathcal{R}^{-2}$ is the typical component\nof the Riemann tensor (see eq. 22.23c in \\cite{grav}). For the Schwarzschild\nmetric, this entails, in terms of the frequency, the condition $\\omega M\\gg\n1,M$ is the black hole mass. In the more general case of, say, the\nReissner-Nordstr\\\"{o}m metric, this gives $\\omega r_{+}\\gg 1$, where $r_{+}$\nis the radius of the horison.\n\nAlso, we assume that backreaction of the photon on the metric is negligible.\nThis implies that, in any case, its energy is much less than the ADM mass of\na black hole, so in natural geometric units $\\omega \\ll M$. Thus we have the\ndouble inequality $r_{+}^{-1}\\ll \\omega \\ll M$. As for the Reissner-Nordstr\\\"{o}m\n metric$\\ M\\leq r_{+}\\leq 2M$, the conditions of validity of this\napproximation are practically the same for the Reissner-Nordstr\\\"{o}m and\nSchwarzschild metrics.\n\n\\section{Photon emitted at the horizon}\n\nWe see from (\\ref{n2}) that, as the point of emission of a photon is\napproaching the horizon, the frequency measured at infinity becomes smaller\nand smaller. However, this formula does not describe what happens if the\nphoton is emitted exactly on the horizon. Then, $\\omega _{0}=0$ and $l=0$, \nf_{em}=0$ from the very beginning (see also below), the photon does not\nreach infinity at all and moves along the leg of the horizon. Here, original\ncoordinates (\\ref{met}) are not applicable since the metric becomes\ndegenerate on the horizon. This can be remedied with the use of standard KS\ncoordinates or its generalization.\n\nLet us introduce coordinates $U$ and $V$, wher\n\\begin{equation}\nU=-\\exp (-\\kappa u)\\text{, }V=\\exp (\\kappa v)\\text{,}  \\label{uv}\n\\end{equation\n\\begin{equation}\nu=t-r^{\\ast }\\text{, }v=t+r^{\\ast }\\text{.}  \\label{uU}\n\\end{equation\n\\begin{equation}\nr^{\\ast }=\\int^{r}\\frac{dr}{f}  \\label{tc}\n\\end{equation\nis the tortoise coordinate. It is seen from (\\ref{uv}) tha\n\\begin{equation}\nUV=-\\exp (2\\kappa r^{\\ast })\\text{.}  \\label{uvr}\n\\end{equation}\n\nThen the metric read\n\\begin{equation}\nds^{2}=FdUdV+r^{2}d\\omega ^{2}\\text{,}  \\label{mF}\n\\end{equation\nHere\n\\begin{equation}\nF=-f\\frac{du}{dU}\\frac{dv}{dV}\\text{.}  \\label{fg}\n\\end{equation\nFor the transformation (\\ref{uv})\n\\begin{equation}\nF=\\frac{f}{UV\\kappa ^{2}}\\text{,}\n\\end{equation\n$F\\neq 0$ on the horizon. It is clear from (\\ref{met}), (\\ref{uvr}) that \nF=F(r)$. For example, for the Schwarzschild metric\n\\begin{equation}\nF=-\\frac{4r_{+}^{3}}{r}e^{-\\frac{r}{r_{+}}}\\text{,}\n\\end{equation\nprovided the constant of integration is chosen in (\\ref{tc}) in such a way\nthat $r^{\\ast }=r+r_{+}\\ln \\frac{r-r_{+}}{r_{+}}$.\n\nWe consider the vicinity of the future horizon, on which $U=0$. Along this\nhorizon, $V$ takes finite values.\n\nNear the horizo\n\\begin{equation}\nr^{\\ast }\\approx \\frac{1}{2\\kappa }\\ln \\frac{r-r_{+}}{r_{+}}+const\\text{,}\n\\end{equation\n\\begin{equation}\nUV\\approx \\frac{(r-r_{+})}{r_{+}}C_{0}\\text{,}  \\label{uvc}\n\\end{equation\nwhere $C_{0}$ is a constant. In the Schwarzschild case, $C_{0}=e$. It is\ninstructive to rewrite equations of motion for massive particles outside the\nevent horizon (\\ref{mt}) - (\\ref{mr})\\ in terms of the KS coordinates and\ntake the horizon limit afterwards. For an observer moving inward they rea\n\\begin{equation}\nmu^{U}=\\frac{E+Z}{f}\\frac{dU}{du}\\text{,}  \\label{mu}\n\\end{equation\n\\begin{equation}\nmu^{V}=\\frac{E-Z}{f}\\frac{dV}{dv}\\text{.}  \\label{mV}\n\\end{equation\nTaking also into account (\\ref{uv}) and (17), we hav\n\\begin{equation}\nm\\dot{U}=-\\frac{(E+Z)}{FV\\kappa }\\text{,}  \\label{uu}\n\\end{equation\n\\begin{equation}\nm\\dot{V}=\\frac{(E-Z)}{UF\\kappa }.  \\label{vv}\n\\end{equation}\n\nIn a similar way, we have for a photon moving in the outward direction\n\\begin{equation}\nk^{U}=\\frac{\\kappa U}{f}(Q-\\omega _{0})=\\frac{Q-\\omega _{0}}{F\\kappa V}\\text\n,}  \\label{ku}\n\\end{equation\n\\begin{equation}\nk^{V}=\\frac{\\kappa V}{f}(\\omega _{0}+Q)=\\frac{\\omega _{0}+Q}{FU\\kappa }\\text\n.}\n\\end{equation}\n\nLet $\\lambda $ be the affine parameter. On the future horizon $U=0$, $k^{U}\n\\frac{dU}{d\\lambda }=0=k_{V}$ which agrees with (\\ref{ku}) if we put $\\omega\n_{0}=0=f=Q$ in the right hand side. Thus only $k^{V\\text{ }}$remains\nnonzero. It follows from the geodesic equations that on the horizo\n\\begin{equation}\n\\frac{dk^{V}}{d\\lambda }=-\\Gamma _{VV}^{V}\\left( k^{V}\\right) ^{2}\\text{.}\n\\end{equation\nHere the Christoffel symbol $\\Gamma _{VV}^{V}\\sim \\frac{\\partial F}{\\partial\nV}\\sim \\frac{\\partial F}{\\partial r}\\frac{\\partial r}{\\partial V}\\sim f=0$\non the horizon. Therefore, $k^{V\\text{ }}=const$ along the horizon\ngenerator. We have from (\\ref{om}), (\\ref{uu}) on the horizo\n\\begin{equation}\n\\omega =-\\frac{1}{2} Fk^{V}u^{U}\\text{,}  \\label{omf0}\n\\end{equation\nand we have\n\n\\begin{equation}\n\\omega = \\frac{E}{Vm\\kappa }k^{V}\\text{.}  \\label{omf}\n\\end{equation}\n\nLet observer 1 crosses the horizon at some $V=V_{1}$ and the same for observer 2 but\nlater, at $V=V_{2}$. It follows from (\\ref{uv}), (\\ref{uU}) that $V_{2}>V_{1}\n. Assuming that observers are identical in that they have the same values of \n$E$ and $m$, we obtain\n\n\\begin{equation}\n\\frac{\\omega _{2}}{\\omega _{1}}=\\frac{V_{1}}{V_{2}}<1\\text{.}  \\label{12}\n\\end{equation}\n\nThis agrees with eq. (A16) of \\cite{kas} obtained for the Schwarzschild\nmetric by another method.\n\nIt is also instructive to check that indeed $\\omega _{0}=0$. By definition, \n\\omega _{0}$ is a constant Killing frequency \n\\begin{equation}\n\\omega _{0}=-k_{\\mu }\\xi ^{\\mu }\\text{,}  \\label{fr}\n\\end{equation\nwhere $\\xi ^{\\mu }$ is the Killing vector. In the original coordinates (\\re\n{met}), \n\\begin{equation}\n\\xi ^{\\mu }=(1,0,0,0)\\text{, }\\xi _{\\mu }=(-f,0,0,0)\\text{.}\n\\end{equation\nPassing to KS coordinates, one obtain\n\\begin{equation}\n\\xi ^{U}=-\\kappa U\\text{, }\\xi ^{V}=\\kappa V\\text{.}\n\\end{equation\nThen, we see from (\\ref{fr}) that\n\n\\begin{equation}\n\\omega _{0}=-F(k^{V}\\xi ^{U}+k^{U}\\xi ^{V})=F\\kappa (k^{V}U-k^{U}V)\\text{.}\n\\end{equation\nOn the future horizon, $k^{U}=0$ and $U=0$, so we see that indeed $\\omega\n_{0}=0$.\n\nAlso, it is easy to check that for a photon propagating along the horizon \nl=0$. Indeed, if we write down the condition $k_{\\mu }k^{\\mu }=0$ on the\nfuture horizon, we obtain that $k_{\\phi }=0$. This agrees with previous\nobservations concerning the properties of trajectories on the horizon \\cit\n{circkerr}, \\cite{nh15}.\n\n\\section{Generalized Lemaitre frame}\n\n\\subsection{Form of metric}\n\nIt is instructive to reformulate the redshift value in the Lemaitre-like\ncoordinates $\\rho ,\\tau $. In contrast to the Kruskal ones, this frame is\nbased on free falling particles. The Lemaitre frame is well known for the\nSchwarzschild metric. Now, we suggest its generalization valid for the\nmetric (\\ref{met}).\n\nThe general theory of transformations that make the metric of a spherically\nsymmetric black hole regular, was developed in \\cite{f}. For our goals, it\nis sufficient to find a particular class of transformations that (i) makes the\nmetric regular on the horizon, (ii) generalizes the Lemaitre metric (in\nparticular, the metric should have $g_{\\tau \\tau }=-1$). We make the\ntransformatio\n\\begin{equation}\n\\rho =t+\\int \\frac{dr^{\\ast }}{\\sqrt{1-f}}\\text{,}  \\label{rof}\n\\end{equation\n\\begin{equation}\n\\tau =t+\\int dr^{\\ast }\\sqrt{1-f}  \\label{tauf}\n\\end{equation\nwhere $r^{\\ast }$ is given by (\\ref{tc}). Eqs. (\\ref{rof}), (\\ref{tauf}) are\ndirect generalization of eqs. 102.1 of \\cite{LL}. Then, it is easy to check\nthat\n\n\\begin{equation}\nds^{2}=-d\\tau ^{2}+(1-f)d\\rho ^{2}+r^{2}(\\rho ,\\tau )(d\\theta ^{2}+\\sin\n^{2}\\theta d\\phi ^{2})\\text{.}  \\label{metL}\n\\end{equation}\n\nOn the horizon, $f=0$, the metric coefficient is regular, $g_{\\rho \\rho }=1\n. In the particular case of the Schwarzschild metric, $f=1-\\frac{r_{+}}{r}$\nand we return to the standard formula for the Lemaitre metric, when $r$ is\nexpressed in terms of $\\rho $ and $\\tau $. The coordinates (\\ref{rof}), (\\re\n{tauf}) are suitable for the description of a black hole including both the\nouter R region and the contracting T$^{\\_}$ one \\cite{nov}. In a similar\nway, one can use the expanding version that would result in a changing sign at \n$\\tau $.\n\nNow, we want to pay attention to some nice properties of the metric (\\re\n{metL}). The proper distance between points 1 and 2 calculated for a given \n\\tau $ is equal to $l=\\int d\\rho \\sqrt{1-f}$. Requiring $d\\tau =0$ in (\\re\n{tauf}) and substituting $dt$ into (\\ref{rof}), we obtain from (\\ref{tc}), \n\\ref{rof}) tha\n\\begin{equation}\nl=r_{2}-r_{1}\\text{.}  \\label{dist}\n\\end{equation}\n\nIt is also instructive to calculate the velocity. Let, say, point 1 be fixed\nand let us focus on the velocity of free fall $v=\\frac{dl}{d\\tau }$ of\na particle with $E=m$, where $r_{2}\\equiv r$ changes depending on time.\nThen, it is easy to  find from (\\ref{mr}), (\\ref{dist}) tha\n\\begin{equation}\nv=-\\sqrt{1-f}\\text{.}\n\\end{equation\nTaking the derivative ones more, we obtain $\\frac{dv}{dr}=\\frac{1}{2\\sqrt{1-\n}}\\frac{df}{dr}$. On the horizon, this gives u\n\\begin{equation}\n\\left( \\frac{dv}{dr}\\right) _{H}=\\kappa \\text{,}  \\label{ka}\n\\end{equation\nwhere we took into account that for our metric the surface gravity $\\kappa \n\\frac{1}{2}\\left( \\frac{df}{dr}\\right) _{H}$. The subscript \"H\" means that the\ncorresponding quantity is calculated on the horizon. Eq. (\\ref{ka}) will be\nused below. It is worth noting that for the extremal horizon ($\\kappa =0$)\nwe have also $\\left( \\frac{dv}{dr}\\right) _{H}=0$.\n\n\\subsection{Redshift: from Kruskal coordinates to Lemaitre ones}\n\nThe above frame is especially useful for the presentation of the redshift \n\\ref{12}). On the horizon, $f=0$. Then, in its vicinity, we obtain from (\\re\n{uU}), (\\ref{rof}), (\\ref{tauf}) that on the horizo\n\\begin{equation}\nv=\\tau +C_{1}=\\rho +C_{2}\\text{,}\n\\end{equation\nwhere $C_{1,2}$ are constants. As a result, we obtain from (\\ref{12}) tha\n\\begin{equation}\n\\frac{\\omega _{2}}{\\omega _{1}}=\\exp (\\kappa (\\tau _{1}-\\tau _{2}))=\\exp\n(\\kappa (\\rho _{1}-\\rho _{2}))\\text{.}  \\label{omk}\n\\end{equation}\n\nThus the Lemaitre frame allows us to present the resulting redshift along\nthe horizon in a simple and intuitively clear picture -- the redshift grows\n(and, consequently, the emitter looks dimmer) exponentially with respect to\nLemaitre time that passes from emitting to observation.\n\nIn the last paragraph of Sec. II, we listed the general condition for the\ngeometrical optic to be valid. Now, we can express it in another way. Since\na physical wave packet has a finite length, parts of it will move away from\nthe black hole horizon even if its center is located exactly on the horizon.\nSince the equation of light geodesics in the generalized Lemaitre frame\nreads $dr\/d\\tau =1-\\sqrt{1-f}$ for outward propagation, the Lemaitre time needed to leave the\nvicinity of the horizon $r=r_{+}$ diverges as $\\left\\vert \\ln ({r\/r_{+}-1)\n\\right\\vert $. Suppose, the emitter radiates light with the wavelength \n\\lambda $. Since in any case the wave packets cannot be smaller than \n\\lambda $, we can roughly estimate initial scale as $r-r_{+}\\ \\sim \\lambda \n. Then, we find that after the Lemaitre time $\\tau \/r_{+}\\sim \\ln \nr_{+}\/\\lambda }\\sim \\ln {\\omega _{0}r_{+}}$ the wave packet will reach the\nscale of black hole horizon, the geometric optic approximation fails and, in\nparticular, Eq. (\\ref{omk}) evidently breaks down.\n\n\\section{Photon emitted at the inner horizon}\n\nLet us consider the situation similar to that considered above. An observer\nmoves beyond the event horizon $r_{+}$and approaches the inner horizon \nr_{-}<r_{+}$. When he crosses it, he emits a photon. Another observer who\nalso crosses the inner horizon later, receives this very photon. What can be\nsaid about its frequency?\n\n\\subsection{The coordinates and metric}\n\nThe metric between the outer and inner horizons represents so-called $T$\nregion \\cite{nov}. For the definiteness, we consider $T^{-}$ region that\ncorresponds to a black hole but similar formulas are valid for the $T^{+}$\nregions (white holes). Now, the metric can be formally obtained from (\\re\n{met}) if one takes into account that for $r_{\\_}<r<r_{+}$the metric\nfunction $f<0$, so spacelike and timelike coordinates mutually interchange.\nWe can write $f=-g$, \n\\begin{equation}\ny=t,T=-r\\text{.}  \\label{T}\n\\end{equation\nThen, the metric can be rewritten in the for\n\\begin{equation}\nds^{2}=-\\frac{dT^{2}}{g}+gdy^{2}+T^{2}d\\omega ^{2}\\text{.}\n\\end{equation}\n\nThe equations of motion for a massive particle have the for\n\\begin{equation}\nm\\dot{y}=\\frac{P}{g}\\text{,}\n\\end{equation\n\\begin{equation}\nm\\dot{T}=z\\equiv \\sqrt{P^{2}+m^{2}g}\\text{,}  \\label{z}\n\\end{equation\nwhere $P=mu_{y}$ is the conserved momentum.\n\nNow, the KS transformation somewhat changes and reads\n\n\\begin{equation}\nU=\\exp (-\\kappa _{-}u)\\text{,}  \\label{ut}\n\\end{equation\n\\begin{equation}\nV=\\exp (\\kappa _{-}v)\\text{,}  \\label{vt}\n\\end{equation\nwhere $r^{\\ast }$ is given by the formula\n\n\\begin{equation}\nr^{\\ast }=\\int \\frac{dr}{g}\\text{.}  \\label{tg}\n\\end{equation\nThe metric takes the form (\\ref{mF}) wit\n\\begin{equation}\nF=-\\frac{g}{UV\\kappa _{-}^{2}}\\text{.}\n\\end{equation\nwhere $\\kappa _{-}$ is the surface gravity associated with the inner horizon\nand $F\\neq 0$ is finite there.\n\nRepeating the calculations step by step, we arrive to the same formula (\\ref{12}\n\\begin{equation}\n\\frac{\\omega _{2}}{\\omega _{1}}=\\frac{V_{1}}{V_{2}}.  \\label{vin}\n\\end{equation\nNow, we would like to pay attention that according to (\\ref{uv}), $V$ is a\nmonotonically increasing function of $v$. It is seen from (\\ref{uU}), (\\re\n{tc}) that for a fixed $u$, $\\frac{\\partial v}{\\partial r}>0$. However, it\nis seen from (\\ref{T}) that event 2 that takes place after 1, has \nr_{2}<r_{1}$. As a result, $v_{2}<v_{1}$ and $V_{2}<V_{1}$. Therefore, \n\\omega _{2}>\\omega _{1}$ and now we have a blueshift. Thus this is related\nto the fact that $r$ and $t$ coordinates change their character in the\nregion under discussion.\n\nThe results (\\ref{omk}) and (\\ref{vin}) can be united in one formul\n\\begin{equation}\n\\frac{\\omega _{2}}{\\omega _{1}}=\\exp [\\left( \\frac{dv}{dr}\\right) _{H}(\\rho\n_{1}-\\rho _{2})]=\\exp [\\left( \\frac{dv}{dr}\\right) _{H}(\\tau _{1}-\\tau _{2})\n\\text{,}\n\\end{equation\nwhere $\\tau _{2}>\\tau _{1}$, $\\rho _{2}>\\rho _{1}$. For the outer horizon we\ncan use eq. (\\ref{ka}) that gives us (\\ref{omk}) and we have a redshift.\nFor the inner horizon, the counterpart of (\\ref{ka}) gives us $\\left( \\frac\ndv}{dr}\\right) _{H}=-\\kappa _{-}$, where now $\\kappa _{-}=\\frac{1}{2\n\\left\\vert \\frac{df}{dr}\\right\\vert _{H}$ is the surface gravity of the\ninner horizon (where $\\left( \\frac{df}{dr}\\right) _{H}<0$). As a result, we\nobtain here a blueshift.\n\nIn a similar way, the procedure under discussion gives the same result when\nan observer crosses the event horizon of a white hole moving outward from the \nT^{+}$ to $R$ region. Then, he will detect all photons propagating along\nthis horizon to be blueshifted. In particular, this holds for the Schwarzschild\nmetric. Analogously, an observer entering $T^{+}$ region from the inner $R$\none (say, like in the Reissner-Nordstr\\\"{o}m metric)\\ will see a redshift at\nthe inner horizon. In other words, in both situations (either black or white\nhole) an observer crossing a horizon from the $T$ to $R$ region will see\na blueshift, while from the $R$ to $T$ region he will see a redshift.\n\n\\subsection{Relation to other effects}\n\nIn the previous subsection we have shown that the blueshift at the inner\nhorizon (and, consequently, the energy absorbed by the observer) grows\nexponentially with the Lemaitre time between the moments of emission and\nobservation. Here we compare this interesting effect with others known in\nliterature.\n\nIf two particle collide, their energy $E_{c.m.}$ in the centre of mass frame\ncan be defined on the point of collision according t\n\\begin{equation}\nE_{c.m.}^{2}=-P_{\\mu }P^{\\mu }\\text{,}\n\\end{equation\n$P^{\\mu }=p_{1}^{\\mu }+p_{2}^{\\mu }$ being the total momentum of two\nparticles. If particle 1 is massive and particle 2 is massless, $p_{1}^{\\mu\n}=mu^{\\mu }$ and $p_{2}^{\\mu }=k^{\\mu }$, where we put the Planck constant\nto unity. As a result\n\\begin{equation}\nE_{c.m.}^{2}=m^{2}+2m\\omega \\text{.}\n\\end{equation}\n\nIn the example under discussion, if $V_{1}=O(1)$ and $V_{2}\\rightarrow 0$,\nthe frequency $\\omega _{2}\\rightarrow \\infty $ according to (\\ref{vin}).\nThen, $E_{c.m.}\\rightarrow \\infty $ as well and we encounter a counterpart\nof the BSW effect near the inner horizon. But $V=0$ on the future horizon \nU=0$ is nothing else than the bifurcation point \\cite{inner} (see also below\nfor more details). Thus the present results for the blueshift agree with the\nprevious ones in the limit when the bifurcation point is reached.\n\nThere is also another issue to which we can compare the present\nconsideration. As is well known, near the inner (Cauchy) horizon an\ninstability develops inside black  holes. This happens when a decaying flux\nof radiation coming from infinity crosses the event horizon and concentrates\nnear the inner one - see, e.g. Chapter 14.3.1 in \\cite{fn}. (For a modern\nreview of the subject see \\cite{ham}.) However, now we consider radiation\nwhich is not coming from infinity but is emitted by an observer who crosses\nthe inner horizon. The resulting energy flux from an emitter at the inner\nhorizon appears to be finite, though it is not restricted from above if \nV\\rightarrow 0$.\n\nThus as far as the radiation near the inner horizon is concerned, we have\nthree situations: (i) the analogue of the BSW effect (relevant near the\nbifurcation point), (ii) blueshift of a photon in the situation under\ndiscussion (relevant near any point of the inner horizon, the blueshift is\nin general finite), (iii) the instability of the inner horizon (infinite\nblueshift due to concentration of radiation along the horizon). Cases (i)\nand (ii) are closely related in the sense that in the limit when the point\nwhere a photon is absorbed approaches the bifurcation point, one obtains (i)\nfrom (ii). Meanwhile, in case (iii) the effect is unbounded and this \npoints to a potential pathology connected with the nature of the \\textit{inner}\n horizon.\n\n\\section{Special case: emission at the bifurcation point}\n\nIn the Sec. V, we discussed briefly such spacetimes that contain $T^{+}$\nregions (white holes). Then the intersection between the future and past\nhorizons forms the so-called bifurcation point (sphere, if the angle\nvariables are taken into account), where it is possible to pass from the\nwhite hole region to the black one. White holes and bifurcation points do\nnot arise in the situation when a black hole is formed due to gravitational\ncollapse and in this sense they are not feasible astrophysically. However, they\nare inevitably present in the full picture of an eternal black-white hole.\nTherefore, we consider such objects for theoretical reasons and for\ncompleteness. In particular, in Sec. V, we saw that accounting for the\nbifurcation point arises naturally in the connection between our problem and\nthe BSW effect. In doing so, it is a receiver that passes near the\nbifurcation point.\n\nIn the present Section, we consider another case, when it is an emitter that\npasses through this point at the moment of radiation. Consideration of the\nfrequency shift when a photon emitted from the bifurcation point is a\nseparate case that does not follow directly from the previous formulas. For\nthe Reissner-Nordstr\\\"{o}m-de Sitter metric, such a problem was considered\nin Sec. IV b of \\cite{lake}. It follows from the corresponding results that\ndifferent cases are possible here: $\\omega _{2}<\\omega _{1}$, $\\omega\n_{2}=\\omega _{1}$, $\\omega _{2}>\\omega _{1}.$ On the first glance, this\ndisagrees with our results described above since we obtained either redshift\n(for the event horizon of a black hole or inner horizon of a white hole) or\nblueshift (for the inner one in a black hole or event horizon of a white\nhole). Fortunately, this contradiction is illusory. Now, we will explain how\none can obtain the results for the bifurcation point from ours. To this end,\nwe compare (i) the generic situation and (ii) that with crossing the\nbifurcation point and trace how (ii) arises from (i) within the limiting\ntransition.\n\nFor our purposes, it is sufficient to discuss the simplest metric that\npossesses the bifurcation point, so we can imply it to be, say, the\nSchwarzschild one. We assume that the emitter 1 moves from the inner\nexpanding $T^{+}$ region (i.e. white hole) \\cite{nov}, crosses the past\nhorizon and enters the R region. Afterwards, it crosses the event horizon\nfalling into a black hole. Let, as before, the emitter and receiver have\nequal masses $m_{1}=m_{2}=m$. However, now we cannot put \\ $E_{1}=E_{2}.$\nThis is because a particle with $E=m$ would escape to infinity instead of\nfalling into a black hole. We remind the reader that up to now, in all our\nconsiderations an emitter and an observer are set to be at rest in infinity.\nHowever, the most gereral case can easily be obtained by adding\ncorresponding Lorentz boosts. In the present subsection we meet the\nsituation where this procedure is needed.\n\nTherefore, we must use more general formula based on (\\ref{omf}\n\\begin{equation}\n\\frac{\\omega _{2}}{\\omega _{1}}=\\frac{E_{2}}{E_{1}}\\frac{V_{1}}{V_{2}}\\text{\n}  \\label{gen}\n\\end{equation\nThe first factor can be interpreted as a Lorentz boost responsible for the\nDoppler effect. For $E_{1}=E_{2}$ we return to the case considered by us\nabove but now the first factor is not equal to one and plays now a crucial\nrole.\n\nIf, by assumption, particle 1 falls into a black hole, this means that it\nmust bounce from the potential barrier in the turning point $r=r_{0}$.\nAccording to equations of motion (\\ref{mr}), this means tha\n\\begin{equation}\nE=m\\sqrt{f(r_{0})}\\text{.}  \\label{ef}\n\\end{equation}\n\nIf $r_{0}\\rightarrow r_{+}$, $f(r_{0})\\rightarrow 0$, so $E\\rightarrow 0$ as\nwell. More precisely, it is seen from (\\ref{kar}), (\\ref{uvc}) tha\n\\begin{equation}\nE\\sim \\sqrt{r_{0}-r_{+}}\\sim \\sqrt{\\left\\vert U\\right\\vert V}\\text{.}\n\\end{equation\nAs a result\n\\begin{equation}\n\\frac{\\omega _{2}}{\\omega _{1}}\\sim \\frac{\\alpha E_{2}}{V_{2}}\\text{, \n\\alpha \\equiv \\sqrt{\\frac{\\left\\vert V\\right\\vert _{1}}{U_{1}}}\\text{.}\n\\end{equation}\n\nIn the limit when the trajectory of particle 1 passes closer and closer to\nthe bifurcaiton point $U=0=V$, $\\alpha $ remains finite. Using equations of\nmotion in the T region (see the previous section), it is easy to show that\nin the limit $V\\rightarrow 0$, $U\\rightarrow 0$, the component of the\nvelocity $u^{U}$ contains just this factor $\\alpha $.\n\nThus depending on relation between $\\alpha $ and $V_{2}$ one can obtain any\nresult for $\\omega _{2}$ (redshift, blueshift, the absence of the frequency\nshift). In this sense, the general formula (\\ref{gen}) reproduces both\n\"standard\" fall of the emitter in a black hole and the behavior of the\nemitter that passes through the bifurcation point.\n\n\\section{Extremal horizon}\n\nLet an observer crosses the (ultra) extremal horizon $r_{+}$. By definition,\nthis means that near it the metric function is\n\\begin{equation}\nf\\sim (r-r_{+})^{n}\n\\end{equation\nwhere $n=2$ in the extremal case and $n=3,4...$ in the ultraextremal one.\nThe difference with the nonextremal case consists in a different nature of\ntransformation making the metric regular. Let the two-dimensional part of\nthe  metric has the same form as in (\\ref{met}). The subsequent procedure\nis known  - see, e.g., \\cite{lib}, \\cite{bron} (Sec. 3.5.1). We use the\nsame coordinates $u$, $v$ and want to find appropriate coordinates $U,$ $V$\n\\begin{equation}\nV=V(v)\\text{, }U=U(u)\\text{.}\n\\end{equation}\n\nNow, we are interested in the situation with emission of a photon exactly\nalong the horizon.Then, near the horizon it follows for the tortoise\ncoordinate (\\ref{tc}) tha\n\\begin{equation}\nr-r_{+}\\sim \\left\\vert r^{\\ast }\\right\\vert ^{\\frac{1}{1-n}}\\text{,}\n\\end{equation\n\\begin{equation}\nf\\sim \\left\\vert r^{\\ast }\\right\\vert ^{\\frac{n}{1-n}}\\text{.}\n\\end{equation\nWe consider the metric near the future horizon where $v$ is finite, $r^{\\ast\n}\\rightarrow -\\infty $, $u=v-2r^{\\ast }\\rightarrow +\\infty $. We hav\n\\begin{equation}\nf\\sim u^{\\frac{n}{1-n}}\\text{.}  \\label{fu}\n\\end{equation}\n\nWe try a transformation that behaves like \n\\begin{equation}\nU\\sim u^{-\\frac{1}{n-1}}\\text{,}  \\label{uextr}\n\\end{equation\nso that $U\\rightarrow 0$. Then, it is easy to check that the metric has the\nform (\\ref{mF}) where $F\\neq 0$ is finite on the horizon. To find the\nfrequency, we must use the expression for $u^{U}$ (\\ref{mu}) in which now \n\\ref{uextr}) is valid, so $\\frac{dU}{du}\\sim u^{\\frac{n}{1-n}}$. \\ It is\nseen from (\\ref{fu}) that $u^{U}\\rightarrow const$ on the horizon and it does\nnot contain $V$. Taking into account that $k^{V}$ is a constant along the\nhorizon generator as before, we come to the conclusion that $V$ drops out and \n\\frac{\\omega _{2}}{\\omega _{1}}=const.$ We see that in the horizon limit the\nquantity $V$ does not enter the frequency. In this sense, $\\frac{\\omega _{2\n}{\\omega _{1}}$ does not change along the horizon, so redshift or blueshift\nis absent.\n\nIn a sense, it is quite natural. Indeed, the extremal horizon is the double\none. The inner and outer horizons merge. But for an inner horizon we had\na blueshift, for the outer one we had a redshift. Together, they mutually cancel\nand produce no effect.\n\nThe absence of the redshift or blueshift formally agrees with (\\ref{omk}) if\none puts $\\kappa =0$ there. However, for (ultra)extremal black holes the\nKruskal-like transformation looks very different, so we could not use eq. \n\\ref{omk}) directly. Therefore, it was not obvious in advance, whether or\nnot the redshift for the extremal horizon can be obtained as the extremal\nlimit of a nonextremal one. Now, we see that this is the case.\n\n\\section{Summary}\n\nThus we showed that for emission along the outer horizon redshift occurs and\nwe derived a simple formula that generalized the one previously found in\nliterature. We also showed that along the inner horizon blueshift occurs and\nfound its relation with the BSW effect. We also showed how the previously known\nresults for the emission at the bifurcation point are reproduced from a\ngeneral formula and lead to a diversity of situations (redshift, blueshift\nor the absence of frequency shift). For (ultra)extremal horizons the effect\nis absent.\n\nThese observations have a quite general character in agreement with the\nuniversality of black hole physics. We also generalized the Lemaitre frame\nand in this frame derived a simple and instructive formula for a redshift\nalong the horizon in terms of the Lemaitre time and the surface gravity.\n\n\\section*{Acknowledgements}\n\nThe work was supported by the Russian Government Program of Competitive\nGrowth of Kazan Federal University.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:level1}Introduction}\n\n\nHelicity density (hereafter simply denoted as helicity) is defined as the inner product of velocity and vorticity and is known to play a crucial role termed as the $\\alpha$ effect in the dynamo action in magnetohydrodynamics \\cite{moffattbook}. \nIn contrast, the role of helicity in neutral hydrodynamic turbulence is not clearly understood to-date. Studies on helicity can be divided into two categories, namely studies on the emergence of helicity and studies on the effects of helicity on the dynamics of turbulence. In the former, the rise of statistically significant helicity spectrum of homogeneous turbulence is never found in the absence of ad hoc initialization or forcing \\cite{cj1989}. Conversely, helicity is known to emerge in rotating inhomogeneous turbulence such as a convection zone in a rotating sphere \\cite{duarteetal2016,ssd2014} or a rotating inhomogeneous turbulence in which the rotation axis is parallel to the inhomogeneous direction \\cite{gl1999,rd2014,kapylaetal2017}. Therefore, the key in the emergence of helicity corresponds to the inhomogeneity of rotating turbulence. In the latter, most studies focus on homogeneous turbulence and effects on energy cascade. A few studies revealed that helicity does not crucially influence hydrodynamic flows in the context of the energy cascade.\nFor example, with the aid of the eddy-damped quasi-normalized Markovian (EDQNM) approximation, Andr\\'e and Lesieur \\cite{al1977} showed that helicity does not affect the energy cascade once the inertial range is established. Rogers and Moin \\cite{rm1987} numerically showed that the correlation between helicity and the dissipation rate of the turbulent kinetic energy is tenuous in homogeneous isotropic turbulence, homogeneous shear turbulence, and turbulent channel flow. Wallace \\textit{et al.} \\cite{wallaceetal1992} experimentally confirmed the correlation between helicity and the dissipation rate in a turbulent boundary-layer, a two-stream mixing-layer, and grid-flow turbulence. They concluded that there is a tenuous relationship between small dissipation rate and large helicity except in the shear flows.\n\nIn contrast, helicity is expected to be important in dynamics of mean flow generation in inhomogeneous turbulence. This point was first discussed by Yokoi and Yoshizawa \\cite{yy1993} in terms of the closure scheme for the Reynolds-averaged Navier--Stokes (RANS) formulation. They suggested that the spatial gradient of helicity coupled with the vortical motion of fluid affects the Reynolds stress (velocity--velocity correlation) and diminishes the turbulent momentum transfer.\nRecently, Yokoi and Brandenburg \\cite{yb2016} numerically revealed that the mean flow is generated in a system with both inhomogeneous helicity and system rotation. This phenomenon can be explained with a model expression for the Reynolds stress obtained by Yokoi and Yoshizawa \\cite{yy1993}.\nFlow generation in the context of the large-scale flow instability was also discussed by Frisch \\textit{et al.} \\cite{aka}, and it is termed as the anisotropic kinetic alpha (AKA) effect. However, Yokoi and Brandenbrug \\cite{yb2016} noted that the flow generation due to the inhomogeneous helicity is suitable for treating flows at high Reynolds number, such as astro\/geophysical flows, while the AKA effect is valid only for flows at low Reynolds number. Thus the model proposed by Yokoi and Yoshizawa \\cite{yy1993} involves general physics of fully developed inhomogeneous turbulence. However, the origin of the helicity effect on the Reynolds stress was not demonstrated based on the Reynolds-stress transport equation. In this sense, the manner in which helicity affects the Reynolds-stress evolution continues to be unclear. \n\nThe Reynolds stress is typically modeled by the eddy-viscosity representation, which is one of the simplest models for the Reynolds stress. The eddy-viscosity model represents the momentum transfer enhanced by turbulence, and the effective viscosity is augmented by turbulent motions. Pope \\cite{pope1975} obtained a nonlinear eddy-viscosity model for the Reynolds stress from the Reynolds-stress transport equation model of Launder \\textit{et al.} \\cite{lrr1975} by neglecting the diffusion effect. The fore-mentioned nonlinear eddy-viscosity models represented a considerable improvement relative to the conventional models. However, in some flows, the models continue to exhibit difficulties in terms of performance. A representative case in which the models do not work well is a swirling flow in a straight pipe \\cite{steenbergen,kito1991}. In the flow, the mean axial velocity exhibits a dent profile in the center axis region of the pipe, and the dent profile is significantly more persistent in the downstream region than those predicted by the eddy-viscosity type models. \n\nYokoi and Yoshizawa \\cite{yy1993} applied the turbulence model with inhomogeneous helicity effect on the Reynolds stress to a swirling pipe flow and successfully reproduced the sustainment of the dent mean velocity. Another description of the effect of helicity on turbulence was constructed by Yoshizawa \\textit{et al.} \\cite{yoshizawaswirl2011}. They introduced a timescale of helical motion into the model and obtained good results in a swirling pipe flow. The results suggest the importance of helicity effect in describing the properties of swirling flows.\nThis helicity effect is also discussed in the context of the sub-grid scale (SGS) modeling in relation to the over-estimation of dissipation rate in the use of eddy-viscosity-type SGS stress models \\cite{yy2017}.\nHowever, the terms obtained in Yokoi and Yoshizawa \\cite{yy1993} or Yoshizawa \\textit{et al.} \\cite{yoshizawaswirl2011} were not directly linked to the systematic modeling of Pope \\cite{pope1975}. This is because the mechanism by which helicity affects the Reynolds stress is not fully known, and thus the helicity effect in the Reynolds-stress evolution is not explicitly considered. In order to reveal the helicity effect on the Reynolds stress, we investigate the physical origin of the effect at the level of the Reynolds-stress transport equation.\n\nIn this study, we perform a numerical simulation of a rotating inhomogeneous turbulence driven by a helical external forcing.\nAlthough the mechanism of the helicity generation is important, this is not examined here. We impose the helicity by external forcing in the present study and focus on the effect of inhomogeneous helicity on the mean flow.\nThe flow configuration is similar to that used by Yokoi and Brandenburg \\cite{yb2016}. It has two homogeneous directions and one inhomogeneous direction, and the rotation axis is perpendicular to the inhomogeneous direction. In the configuration the mean flow is expected to emerge in the rotation-axis direction.\nThis flow configuration is similar to the low-latitude region of rotating sphere in which turbulence is radially inhomogeneous and its rotation axis is mostly perpendicular to the inhomogeneous direction \\cite{duarteetal2016,ssd2014}. We also conduct simulations in non-rotating and\/or non-helical forcing cases to identify the condition for the mean-flow generation. The helicity effect is tested in relation to the Reynolds-stress transport equation, and the origin of the mean-flow generation is explored.\n\nThe rest of this study is organized as follows. Section~\\ref{sec:level2} summarizes the relationship between the eddy-viscosity-type turbulence model and the transport equation for the Reynolds stress. The model for the Reynolds stress including the helicity effect derived by Yokoi and Yoshizawa \\cite{yy1993} is also presented. Section~\\ref{sec:level3} presents the numerical setup and the simulation results. We also discuss the origin of the helicity effect on the Reynolds stress. A comparison between our results and the model expression of the Reynolds stress with helicity is given in Sec.~\\ref{sec:level4}. The conclusions are discussed in Sec.~\\ref{sec:level5}.\n\n\\section{\\label{sec:level2}Model representations of the Reynolds stress and helicity effect}\n\nThe Navier--Stokes equation and the continuity equation for an incompressible fluid in a rotating system are given respectively as follows:\n\\begin{align}\n\\frac{\\partial u_i}{\\partial t} & =\n- \\frac{\\partial}{\\partial x_j} u_i u_j - \\frac{\\partial p}{\\partial x_i}\n+ \\nu \\frac{\\partial^2 u_i}{\\partial x_j \\partial x_j} + 2 \\epsilon_{ij\\ell} u_j \\Omega^F_\\ell + f_i, \n\\label{eq:2.1} \\\\\n\\frac{\\partial u_i}{\\partial x_i} & = 0, \n\\label{eq:2.2}\n\\end{align}\nwhere $u_i$ denotes the $i$-th component of the velocity, $p$ the pressure divided by the fluid density with centrifugal force included, $\\nu$ the kinematic viscosity, $\\Omega^F_i$ the angular velocity of the system, $f_i$ the external force, and $\\epsilon_{ij\\ell}$ the alternating tensor. We decompose a physical quantity $q [= (u_i, p, f_i)]$ into mean and fluctuation parts as follows:\n\\begin{align}\nq = Q + q', \\ \\ & Q = \\left< q \\right>,\n\\label{eq:2.3}\n\\end{align}\nwhere $\\left< \\cdot \\right>$ denotes an ensemble average. Substituting Eq.~(\\ref{eq:2.3}) into Eqs.~(\\ref{eq:2.1}) and (\\ref{eq:2.2}), we obtain the mean field equations,\n\\begin{align}\n\\frac{\\partial U_i}{\\partial t} & =\n- \\frac{\\partial}{\\partial x_j} \\left( U_i U_j + R_{ij} \\right) - \\frac{\\partial P}{\\partial x_i}\n+ \\nu \\frac{\\partial^2 U_i}{\\partial x_j \\partial x_j} + 2 \\epsilon_{ij\\ell} U_j \\Omega^F_\\ell + F_i, \n\\label{eq:2.4} \\\\\n\\frac{\\partial U_i}{\\partial x_i} & = 0,\n\\label{eq:2.5}\n\\end{align}\nwhere $R_{ij} (= \\left< u_i' u_j' \\right>)$ denotes the Reynolds stress. The only difference between Eqs.~(\\ref{eq:2.1}) and (\\ref{eq:2.4}) corresponds to the Reynolds stress. Thus, the Reynolds stress solely represents the effects of turbulent motion on the mean velocity. In order to close the system of Eqs.~(\\ref{eq:2.4}) and (\\ref{eq:2.5}), a model expression for the Reynolds stress is required.\n\n\\subsection{\\label{sec:level2a}Relationship between model and transport equation for the Reynolds stress}\n\nThe simplest model for the Reynolds stress is the eddy-viscosity model that is expressed as follows:\n\\begin{align}\nR_{ij} = \\frac{2}{3} K \\delta_{ij} - 2 \\nu_T S_{ij},\n\\label{eq:2.6}\n\\end{align}\nwhere $K (= \\left<u_i' u_i' \\right>\/2)$ denotes the turbulent kinetic energy, $\\nu_T$ the eddy viscosity, \\linebreak $S_{ij} [= \\left( \\partial U_i \/ \\partial x_j + \\partial U_j \/ \\partial x_i \\right) \/ 2]$ the strain rate of the mean velocity, and $\\delta_{ij}$ the Kronecker delta. The eddy-viscosity model is not just an empirical model but can be obtained from the fundamental equation, i.e., the Navier--Stokes equation. Specifically, the model expression for the Reynolds stress is closely related to the transport mechanism of the Reynolds stress. A systematic way to obtain the eddy-viscosity-type model from the Reynolds-stress transport equation may be summarized as follows \\cite{pope1975,yoshibook}. The exact transport equation for the Reynolds stress is expressed as follows:\n\\begin{align}\n\\frac{\\mathrm{D} R_{ij}}{\\mathrm{D} t} & =\nP_{ij} - \\varepsilon_{ij} + \\Phi_{ij} + \\Pi_{ij} \n + T_{ij} + D_{ij} + C_{ij} + F_{ij}, \n\\label{eq:2.7}\n\\end{align}\nwhere $\\mathrm{D}\/\\mathrm{D} t = \\partial \/ \\partial t + U_\\ell \\partial \/ \\partial x_\\ell$ denotes the Lagrange derivative. Here, $P_{ij}$ denotes the production rate, $\\varepsilon_{ij}$ the destruction rate, $\\Phi_{ij}$ the pressure--strain correlation, $\\Pi_{ij}$ the pressure diffusion, $T_{ij}$ the turbulent diffusion, and $D_{ij}$ the viscous diffusion, $C_{ij}$ the Coriolis effect, and $F_{ij}$ the external work. They are respectively defined as follows:\n\\begin{subequations}\n\\begin{align}\nP_{ij} & = - R_{i\\ell} \\frac{\\partial U_j}{\\partial x_\\ell} - R_{j\\ell} \\frac{\\partial U_i}{\\partial x_\\ell}, \n\\label{eq:2.8a} \\\\\n\\varepsilon_{ij} & = \\left< 2 \\nu s_{i\\ell} \\frac{\\partial u_j'}{\\partial x_\\ell}\n   + 2 \\nu s_{j\\ell} \\frac{\\partial u_i'}{\\partial x_\\ell} \\right>, \n\\label{eq:2.8b} \\\\\n\\Phi_{ij} & = 2 \\left< p' s_{ij} \\right>, \n\\label{eq:2.8c} \\\\\n\\Pi_{ij} & = - \\frac{\\partial}{\\partial x_j} \\left< p' u_i' \\right>\n   - \\frac{\\partial}{\\partial x_i} \\left< p' u_j' \\right>,\n\\label{eq:2.8d} \\\\\nT_{ij} & = - \\frac{\\partial}{\\partial x_\\ell} \\left< u_i' u_j' u_\\ell' \\right>, \n\\label{eq:2.8e} \\\\\nD_{ij} & = \\frac{\\partial}{\\partial x_\\ell} \\left< 2 \\nu s_{i\\ell} u_j' + 2 \\nu s_{j\\ell} u_i' \\right>, \n\\label{eq:2.8f} \\\\\nC_{ij} & = 2\\left( \\epsilon_{im \\ell} R_{jm} + \\epsilon_{jm \\ell} R_{im} \\right) \\Omega_\\ell^F ,\n\\label{eq:2.8g} \\\\\nF_{ij} & = \\left< u_i' f_j + u_j' f_i \\right>,\n\\label{eq:2.8h}\n\\end{align}\n\\end{subequations}\nwhere $s_{ij}  [= \\left( \\partial u_i \/ \\partial x_j + \\partial u_j \/ \\partial x_i \\right) \/ 2]$ denotes the strain rate of the velocity. Pope \\cite{pope1975} obtained a general expression of the Reynolds stress based on the following two assumptions. First, to the right-hand side of Eq.~(\\ref{eq:2.7}), the model by Launder \\textit{et al.} \\cite{lrr1975} (LRR model) is adopted; $\\varepsilon_{ij}$ and $\\Phi_{ij}$ are modeled as follows:\n\\begin{align}\n\\varepsilon_{ij} & = \\frac{2}{3} \\varepsilon \\delta_{ij}, \n\\label{eq:2.9} \\\\\n\\Phi_{ij} & = - C_{S1} \\frac{\\varepsilon}{K} B_{ij} + C_{R1} K S_{ij} \\nonumber \\\\\n& \\hspace{1.2em} + C_{R2} \\left[ B_{i\\ell} S_{\\ell j} + B_{j\\ell} S_{\\ell i} \\right]_D \n + C_{R3} \\left( B_{i\\ell} \\Omega_{\\ell j} + B_{j\\ell} \\Omega_{\\ell i} \\right) ,\n\\label{eq:2.10}\n\\end{align}\nwhere $\\varepsilon (= \\varepsilon_{ii}\/2)$ denotes the dissipation rate of the turbulent energy $K$, $B_{ij} [= R_{ij} -(2\/3) K \\delta_{ij}]$ the deviatoric part of the Reynolds stress, $\\Omega_{ij} = \\left( \\partial U_j \/ \\partial x_i - \\partial U_i \/ \\partial x_j \\right)\/2$, $\\left[ A_{ij} \\right]_D = A_{ij} - A_{\\ell \\ell} \\delta_{ij} \/ 3$, and $C_{S1}$, $C_{R1}$, $C_{R2}$, and $C_{R3}$ denote the model constants.\nThe term with $C_{S1}$ describes the `return to isotropy' model while the terms with $C_{R1}$, $C_{R2}$, and $C_{R3}$ correspond to the `isotropization of production' model \\cite{lrr1975}. Although there are more elaborate models for the pressure--strain correlation, such as Craft and Launder \\cite{tcl}, we focus on simple models proportional to $B_{ij}$. Second, quasi-homogeneity of the flow field is assumed, and the diffusion terms are neglected as $\\Pi_{ij} = T_{ij} = D_{ij} = 0$. In addition to the two assumptions, it is necessary to handle the time derivative term, $\\mathrm{D} R_{ij} \/ \\mathrm{D} t$. In the algebraic stress models, the weak-equilibrium assumption, $\\mathrm{D} (R_{ij}\/K)\/\\mathrm{D} t = 0$, is applied. Here this assumption is not used; we introduce an appropriate time derivative instead of the Lagrange derivative in order to satisfy the frame invariance of the turbulence equation in a rotating system \\cite{hamba2006,ariki2015}. When the upper convected time derivative, $\\mathfrak{D} A_{ij}\/\\mathfrak{D} t = \\mathrm{D} A_{ij}\/\\mathrm{D} t - A_{i\\ell} \\partial U_j \/ \\partial x_\\ell - A_{j\\ell} \\partial U_i \/ \\partial x_\\ell$, is adopted, Eq.~(\\ref{eq:2.7}) is expressed as follows:\n\\begin{align}\n\\frac{\\mathfrak{D} B_{ij}}{\\mathfrak{D} t} & =\n - C_{S1} \\frac{\\varepsilon}{K} B_{ij}\n- \\left( \\frac{4}{3} - C_{R1} \\right) K S_{ij} \\nonumber \\\\\n& \\hspace{1.2em}\n- \\left( 1 - C_{R2} \\right) \\left[ B_{i\\ell} S_{\\ell j} + B_{j\\ell} S_{\\ell i} \\right]_D \n- \\left( 1 - C_{R3} \\right) \\left( B_{i\\ell} \\Omega_{\\ell j}^* + B_{j\\ell} \\Omega_{\\ell i}^* \\right),\n\\label{eq:2.11}\n\\end{align}\nwhere $\\Omega_{ij}^* (= \\Omega_{ij} + \\epsilon_{ij\\ell} \\Omega_\\ell^F)$ denotes the mean absolute vorticity tensor. Here it is assumed that the external work does not affect the Reynolds stress directly. The model for $\\Phi_{ij}$ is extended to a rotating system. Thus, we replace $\\Omega_{ij}$ in Eq.~(\\ref{eq:2.10}) by $\\Omega_{ij}^*$. This frame invariant formulation is performed to ensure the consistency of the equations in a rotating frame. The effect of rotation may affect the transport equation for $\\varepsilon$ \\cite{bfr1985}, and this type of a modification is needed to predict turbulent flows under the solid body rotation with the Reynolds-stress models. However, this point is beyond the scope of the present study that focuses on the effects on the mean flow. The first term on the right-hand side of Eq.~(\\ref{eq:2.11}) represents the destruction of $B_{ij}$ or the relaxation to an isotropic state. The second term denotes the production of $B_{ij}$ by the isotropic part of turbulence, while the third and fourth terms  denote the production by the anisotropic part of turbulence. Equation~(\\ref{eq:2.11}) is re-expressed as follows:\n\\begin{align}\nB_{ij} & =\n- 2 \\frac{4-3C_{R1}}{6C_{S1}} \\frac{K^2}{\\varepsilon} S_{ij} \\nonumber \\\\\n& \\hspace{1.2em}\n- \\frac{1-C_{R2}}{C_{S1}} \\frac{K}{\\varepsilon} \\left[ B_{i\\ell} S_{\\ell j} + B_{j\\ell} S_{\\ell i} \\right]_D \n- \\frac{1-C_{R3}}{C_{S1}} \\frac{K}{\\varepsilon} \\left( B_{i\\ell} \\Omega_{\\ell j}^* + B_{j\\ell} \\Omega_{\\ell i}^* \\right) \n\\nonumber \\\\\n& \\hspace{1.2em}\n- \\frac{1}{C_{S1}} \\frac{K}{\\varepsilon} \\frac{\\mathfrak{D} B_{ij}}{\\mathfrak{D} t}.\n\\label{eq:2.12}\n\\end{align}\nSubstituting this expression iteratively into $B_{ij}$ on the right-hand side, we obtain the following:\n\\begin{align}\nB_{ij} & =\n- 2 C_\\nu \\frac{K^2}{\\varepsilon} S_{ij} \\nonumber \\\\\n& \\hspace{1.2em}\n+ C_{q1} \\frac{K^3}{\\varepsilon^2} \\left[ S_{i\\ell} S_{\\ell j} + S_{j\\ell} S_{\\ell i} \\right]_D \n+ C_{q2} \\frac{K^3}{\\varepsilon^2} \\left( S_{i\\ell} \\Omega_{\\ell j}^* + S_{j\\ell} \\Omega_{\\ell i}^* \\right)\n\\nonumber \\\\\n& \\hspace{1.2em}\n+ C_d \\frac{K}{\\varepsilon} \\frac{\\mathfrak{D}}{\\mathfrak{D} t} \\left( \\frac{K^2}{\\varepsilon} S_{ij} \\right) \n+ \\text{(higher order terms)} , \n\\label{eq:2.13}\n\\end{align}\nwhere $C_\\nu = (4-3C_{R1})\/(6C_{S1})$, $C_{q1} = 2 C_\\nu (1-C_{R2}) \/C_{S1}$, $C_{q2} = 2 C_\\nu (1-C_{R3})\/C_{S1}$, and $C_d = 2C_\\nu\/C_{S1}$. In contrast to the formulation obtained by Pope \\cite{pope1975}, the time derivative term is retained in the right-hand side of Eq.~(\\ref{eq:2.13}) as shown in Yoshizawa \\cite{yoshibook}. This corresponds to a more general formulation when compared with that obtained by Pope \\cite{pope1975} since the time derivative term does not always disappear. The first term of Eq.~(\\ref{eq:2.13}) represents the eddy-viscosity term which corresponds to the second term on the right-hand side of Eq.~(\\ref{eq:2.6}), and this term is derived from the isotropic part of the production term. This reflects the point that the eddy-viscosity model constitutes a good approximation when the turbulence is nearly isotropic, quasi-homogeneous, and steady. \n\n\n\\subsection{\\label{sec:level2b}The Reynolds-stress expression accompanied with the helicity effect}\n\nThe eddy-viscosity-type models provide good results for simple flows such as free shear layer flows and channel flows. However, they perform poorly for more complex flows. An example in which the usual eddy-viscosity models do not work well is a swirling flow in a straight pipe \\cite{steenbergen,kito1991}. In the swirling-flow experiments, it is observed that the mean axial velocity shows a dent in the center axis region, and this inhomogeneous velocity profile is very persistent to the well downstream region. However, this type of a dent profile that is imposed at the pipe inlet cannot be sustained and decays rapidly in the usual eddy-viscosity model simulation \\cite{ky1987,steenbergen}. This is because the eddy viscosity is so strong that it smears out any large-scale velocity gradient. Jakirli\\'c \\textit{et al.} \\cite{jht2000} pointed out that even with an elaborate explicit Reynolds-stress model such as Craft \\textit{et al.} \\cite{cls1996} or Shih \\textit{et al.} \\cite{shihetal1997} as well as the standard eddy-viscosity model, it is difficult to accurately reproduce the fore-mentioned rotational flows without performing a few modifications in the model constants. With the aid of the two-scale direct-interaction approximation (TSDIA) \\cite{tsdia} that is an analytical statistical theory of inhomogeneous turbulence, Yokoi and Yoshizawa \\cite{yy1993} suggested that eddy viscosity may be suppressed by symmetry breaking swirling motion. They analytically constructed a new turbulence model in which the helicity effect is incorporated. In the formulation, homogeneous isotropic non-mirror-symmetric turbulence is assumed as the basic field, and the effects of inhomogeneity, anisotropy, and system rotation are incorporated in a perturbational manner based on the Navier--Stokes equation. Brief descriptions of the formulation are given in Appendix~\\ref{sec:a}. According to the formulation, the deviatoric or traceless part of the Reynolds stress is expressed as follows:\n\\begin{align}\nB_{ij} = - 2 \\nu_T S_{ij}\n+ \\eta \\left[ \\frac{\\partial H}{\\partial x_j} \\Omega^*_i + \\frac{\\partial H}{\\partial x_i} \\Omega^*_j \\right]_D,\n\\label{eq:2.14}\n\\end{align}\nwhere $\\eta$ denotes the transport coefficient, $H (= \\left< u_i' \\omega_i' \\right>)$ the turbulent helicity, and $\\Omega^*_i (= \\epsilon_{ij\\ell} \\partial U_\\ell \/ \\partial x_j + 2 \\Omega^F_i)$ the mean absolute vorticity. In this study, we refer to the model of Eq.~(\\ref{eq:2.14}) as the helicity model. This model allowed the successful reproduction of the sustainment of the dent mean axial velocity in a swirling flow. The helicity model is similar to the AKA model \\cite{aka} in the sense that the AKA describes the effect of lack of parity invariance on the mean flow. The helicity model is developed for high-Reynolds number flows since the TSDIA corresponds to perturbational expansion from fully developed homogeneous turbulence, while the AKA is valid for low-Reynolds number flows \\cite{yb2016}. Hence, it is expected the helicity model can be applied to realistic high-Reynolds number turbulent flows.\n\nIt is interesting to note that as pointed out in \\cite{yy1993} and \\cite{yb2016}, the present model accounts for the mean flow generation from the no-mean-velocity initial condition. Even if system does not have the mean velocity gradient, Eq.~(\\ref{eq:2.14}) may include a non-zero value when both the helicity gradient and the system rotation exist. In such cases, the deviatoric part of the Reynolds stress is expressed as follows:\n\\begin{align}\nB_{ij} = 2 \\eta \\left[ \\frac{\\partial H}{\\partial x_j} \\Omega^F_i + \\frac{\\partial H}{\\partial x_i} \\Omega^F_j \\right]_D \n\\neq 0.\n\\label{eq:2.15}\n\\end{align}\nThis suggests that the mean flow is generated by this helicity effect when the inhomogeneous helicity is coupled with the rotation since the mean velocity equation is expressed as\n\\begin{align}\n\\frac{\\partial U_i}{\\partial t} = - \\frac{\\partial}{\\partial x_j} \n\\left[ \\eta \\left( \\frac{\\partial H}{\\partial x_j} 2 \\Omega^F_i \n+ \\frac{\\partial H}{\\partial x_i} 2 \\Omega^F_j - \\frac{\\partial H}{\\partial x_\\ell} 2 \\Omega^F_\\ell \\frac{2}{3} \\delta_{ij} \\right) \\right] - \\frac{\\partial P}{\\partial x_i}\n\\neq 0.\n\\label{eq:2.16}\n\\end{align}\nYokoi and Brandenburg \\cite{yb2016} performed direct numerical simulations (DNSs) of a rotating inhomogeneous turbulence with an imposed turbulent helicity. They commenced with a no-mean-velocity configuration and observed a mean-flow generation in a rotating turbulence. Additionally, they confirmed that in the early stage of the simulation in which the mean-velocity gradient is not significantly developed, the Reynolds stress is well correlated with the middle part of Eq.~(\\ref{eq:2.15}). It is not possible to predict this type of a flow generation phenomenon by using a conventional model of the Reynolds stress as given by Eq.~(\\ref{eq:2.13}) since each term contains the mean shear rate.\n\nThe results indicate that inhomogeneous helicity coupled with the vortical motion of fluid affects the Reynolds stress and reduces turbulent momentum transport represented by the eddy viscosity. The following points should be noted. The model representation of Eq.~(\\ref{eq:2.14}) was analytically obtained from the Navier--Stokes equation with the aid of TSDIA. However, the second term on the right-hand side of Eq.~(\\ref{eq:2.14}) is not obtained in a direct manner from the systematic construction of the model shown in Sec.~\\ref{sec:level2a}. This is because the turbulent helicity is not explicitly included in the Reynolds-stress transport equation given in Eq.~(\\ref{eq:2.11}) on which the model constitution is based. Yokoi and Brandenburg \\cite{yb2016} compared the profile of the Reynolds stress with that of Eq.~(\\ref{eq:2.14}) to determine a very good correlation between them. However, the origin of the helicity effect on the Reynolds-stress equation was not shown. As shown in Sec.~\\ref{sec:level2a}, the model expression of the Reynolds stress is related to  its transport mechanism. The effect of helicity corresponding to the second term on the right-hand side of Eq.~(\\ref{eq:2.14}) should exist on the right-hand side of Eq.~(\\ref{eq:2.7}) as well as the production term corresponding to the eddy-viscosity term. Hence, the physical origin of the second term of Eq.~(\\ref{eq:2.14}) is not clarified in the sense  of the Reynolds-stress evolution.\n\n\\section{\\label{sec:level3}Numerical simulations}\n\nIn order to investigate the mechanism of the mean-flow generation and its relationship to the turbulent helicity, we perform a series of numerical simulations of a rotating inhomogeneous turbulence driven by a helical external force. We examine the transport equation for the Reynolds stress to explore the manner in which the turbulent helicity affects the Reynolds-stress transport.\n\n\\subsection{\\label{sec:level3a}Governing equations and numerical setup}\n\nIn order to simulate a high-Reynolds-number turbulent flow, the large eddy simulation (LES) is adopted instead of the DNS. The governing equations of the LES in a rotating system are expressed as follows:\n\\begin{align}\n\\frac{\\partial \\overline{u}_i}{\\partial t} & =\n - \\frac{\\partial}{\\partial x_j} \\overline{u}_i \\overline{u}_j\n  - \\frac{\\partial \\overline{p}}{\\partial x_i}\n + \\frac{\\partial}{\\partial x_j} 2 \\nu_{sgs} \\overline{s}_{ij} \n+ 2 \\epsilon_{ij\\ell} \\overline{u}_j \\Omega_\\ell^F\n  + \\overline{f}_i , \n\\label{eq:3.1} \\\\ \n\\frac{\\partial \\overline{u}_i}{\\partial x_i} & = 0,\n\\label{eq:3.2}\n\\end{align}\nwhere the kinematic viscosity is neglected, and $\\overline{q}$ denotes the grid-scale (resolved) component of $q$. It should be noted that $\\overline{q}$ is different from the ensemble average, $\\left< q \\right>$, which is already introduced in Eq.~(\\ref{eq:2.3}). With respect to the model of the subgrid-scale (SGS) viscosity, $\\nu_{sgs}$, the Smagorinsky model \\cite{smagorinsky1963},\n\\begin{align}\n\\nu_{sgs} = \\left( C_S \\Delta \\right)^2 \\sqrt{ 2 \\overline{s}_{ij} \\overline{s}_{ij}}, \n\\label{eq:3.3}\n\\end{align}\nis applied with the Smagorinsky constant $C_S = 0.19$, which is the optimized value for homogeneous isotropic turbulence \\cite{yoshibook}, and $\\Delta = \\left( \\Delta x \\Delta y \\Delta z \\right)^{1\/3}$ where $\\Delta x_i$ denotes the grid size of the $i$-th direction.\n\n\\begin{figure}[b]\n\\centering\n\\includegraphics[scale=0.45]{fig1.eps}\n\\caption{Computational domain and schematic profiles of turbulent energy and helicity. $K^{GS} (= \\left< \\overline{u}_i' \\overline{u}_i' \\right>\/2)$ and $H^{GS} (= \\left< \\overline{u}_i' \\overline{\\omega}_i' \\right>)$ denote the turbulent energy and helicity of the grid scale motions, respectively. An external forcing is applied only around $y=0$ plane.}\n\\label{fig:1}\n\\end{figure}\n\nIn the simulation, the computational domain is a rectangular parallelepiped region as shown in Fig.~\\ref{fig:1}. An external force applied around the center plane at $y=0$ injects turbulent energy and helicity. In the calculation, the rotation axis is set perpendicular to the inhomogeneous direction of the turbulence to assess the helicity model Eq.~(\\ref{eq:2.14}). This set up is similar to that used by Yokoi and Brandenburg \\cite{yb2016}. The configuration corresponds to the low-latitude region of a rotating  spherical convection in which the inhomogeneous direction of helicity is mainly perpendicular to the rotation axis in a low-latitude region \\cite{duarteetal2016,ssd2014}. The objective involves elucidating the effect of inhomogeneous helicity on the mean flow in rotating turbulence and not clarifying the mechanism of helicity generation, and thus helicity is injected by an external forcing to achieve simplicity in contrast to the simulation in which helicity emerges spontaneously \\cite{gl1999,rd2014,kapylaetal2017}. The external force is defined by the vector potential $\\overline{\\psi}_i$ as follows:\n\\begin{align}\n\\overline{f}_i = C \\epsilon_{ij\\ell} \\frac{\\partial}{\\partial x_j} \\left[ g (y) \\overline{\\psi}_\\ell \\right],\n\\label{eq:3.4}\n\\end{align}\nwhere $g(y)$ denotes a weighting function introduced to confine the external force around the $y=0$ plane. The coefficient $C$ is determined to satisfy $\\left< \\overline{u}_i' \\overline{u}_i' \\right>_S (y=0)\/2=1$ at each time step, where $\\left< \\cdot \\right>_S$ denotes the $x$--$z$ plane average and $\\overline{q}'$ denotes the fluctuation of $\\overline{q}$ around $\\left< \\overline{q} \\right>_S$; \n\\begin{align}\n\\overline{q} = \\left< \\overline{q} \\right>_S + \\overline{q}'. \n\\label{eq:3.5}\n\\end{align}\nThe force is solenoidal, $\\partial \\overline{f}_i \/ \\partial x_i = 0$. With respect to the weighting function, $g(y) = \\mathrm{exp} \\left[ -y^2\/\\sigma^2 \\right]$ with $\\sigma = L_y\/32 = 0.393$ is applied, and this is a value comparable to the forcing scale $\\pi\/k_f$ where $k_f$ is given in the following [Eq.~(\\ref{eq:3.6a})]. The vector potential $\\overline{\\psi}_i$ obeys a stochastic process like the Ornstein--Uhlenbeck process \\cite{ouforcing}, and is determined from the power and helicity spectra of $\\overline{f}_i$, $E^{ex}(k)$ and $E_H^{ex}(k)$ given as follows:\n\\begin{subequations}\n\\begin{align}\nE^{ex} (k) & \\propto\n\\begin{cases}\nk^{-5\/3} & k=k_f, 10 \\le k_f \\le 14 \\\\\n0 & \\text{otherwise} ,\n\\end{cases} \n\\label{eq:3.6a} \\\\\nE_H^{ex} (k) & = 2 \\alpha k E^{ex} (k) ,\n\\label{eq:3.6b}\n\\end{align}\n\\end{subequations}\nwhere $\\alpha$ denotes the parameter that determines the intensity of helicity of the external force. The spectrum $E^{ex}(k)$ is selected corresponding to the typical inertial-range form of turbulence, and $E^{ex}_H (k)$ corresponds to the statistical property of inertial wave when $\\alpha = \\pm 1$ \\cite{moffatt1970}. The range of $\\alpha$ should be $-1 \\le \\alpha \\le 1$ since the helicity spectrum must satisfy $|E_H^{ex}(k)| \\le 2k E^{ex}(k)$ \\cite{al1977}; $\\alpha = 0$ corresponds to the non-helical case and $\\alpha = 1$ ($-1$) is the most positively (negatively) helical case. Details of forcing are given in Appendix~\\ref{sec:b}.\n\nThe size of the computational domain is $L_x \\times L_y \\times L_z = 2\\pi \\times 4\\pi \\times 2\\pi$ and the number of the grid point is $N_x \\times N_y \\times N_z = 128 \\times 256 \\times 128$. The periodic boundary conditions are used in all directions, we use the second-order finite-difference scheme in space, and the Adams--Bashforth method is used for time integral. A triply periodic box is used, and thus the pseudo-spectral scheme may be more appropriate for DNS with the linear viscosity term. However, with respect to the LES, a complex nonlinear form of the SGS viscosity decreases the numerical accuracy of the pseudo-spectral scheme. Moreover, we are going to apply the code to homogeneous turbulence with a non-uniform grid. Thus, we adopt finite-difference scheme. The pressure is directly solved in the wave number space by using FFT. Parameters of the simulation are shown in Table~\\ref{tb:1}; namely Run 1 is non-helical and non-rotating case, Run 2 is helical but non-rotating, Run 3 is rotating but non-helical, and Runs 4, 5, and 6 are helical and rotating. We observe the effect of helical forcing by comparing Runs 1 and 2 for non-rotating case, and Runs 3, 4, and 6 for the rotating case. We also observe the effect of the system rotation by comparing Runs 2, 5, and 6. In all the runs, the external force is applied in the wavenumber band $10 \\le k \\le 14$. With respect to the helical cases, $\\alpha = 0.5$ for Runs 2 and 4 and $\\alpha = 0.2$ for Run 5, which are not fully helical ones, are adopted since the relative helicity [$u_i \\omega_i\/(|u_i||\\omega_i|)$] in realistic turbulence is modulated from the maximally helical case of the inertial wave, $\\alpha = \\pm 1$, due to buoyancy and nonlinear interaction of turbulence \\cite{ranjan2017}. $L^{GS}_0$ denotes the characteristic length scale of the turbulence and $\\mathrm{Ro}^{GS}_0$ denotes the Rossby number respectively defined by\n\\begin{align}\nL^{GS}_0 = \\frac{(K^{GS}_0)^{3\/2}}{\\varepsilon^{SGS}_0} \\ , \\ \\ \n\\mathrm{Ro}^{GS}_0 = \\frac{{K^{GS}_0}^{1\/2}}{L^{GS}_0 2 \\Omega^F} ,\n\\label{eq:3.7}\n\\end{align}\nwhere $K^{GS} = \\left< \\overline{u}_i' \\overline{u}_i' \\right>\/2$, $K^{GS}_0 = K^{GS} (y=0)$, $\\varepsilon^{SGS} = 2\\left< \\nu_{sgs} \\overline{s}_{ij} \\overline{s}_{ij}' \\right>$, $\\varepsilon^{SGS}_0 = \\varepsilon^{SGS} (y=0)$, and $\\left< \\cdot \\right>$ denotes the average over the homogeneous plane and over time. The time average is taken over $20 \\le t \\le 30$ as mentioned below. In our calculation, the length scale of inhomogeneity of turbulence is estimated as $\\ell^\\nabla = 0.4$ for all runs, in which $\\ell^\\nabla$ is defined such that $K^{GS} (y= \\ell^\\nabla) = \\mathrm{e}^{-1} K^{GS}_0$. The validity of turbulence models requires the length scale of inhomogeneity of turbulence is much longer than the scale of energy containing eddy $L_0^{GS}$ \\cite{corrsin1974}. These two scales are comparable in the simulation. However, the fore-mentioned lack of scale separation is often observed in actual turbulence such as in an atmospheric boundary layer \\cite{stull1993}. It should be emphasized that the mean velocity is set to zero in the initial condition, and the plane average of the external force is also zero, $\\left< \\overline{f}_i \\right>_S = 0$, such that the external force does not directly excite the mean velocity.\n\n\\begin{table}[ht]\n\\centering\n\\caption{Calculation parameters.}\n\\begin{ruledtabular}\n\\begin{tabular}{ccccc}\nRun & $\\alpha$ & $\\Omega_x^F$ & $L^{GS}_0$ & $\\mathrm{Ro}^{GS}_0$ \\\\ \\hline\n1 & $0$ & $0$ & $0.506$ & $\\infty$ \\\\\n2 & $0.5$ & $0$ & $0.547$ & $\\infty$ \\\\\n3 & $0$ & $5$ & $0.542$ & $0.185$ \\\\\n4 & $0.2$ & $5$ & $0.550$ & $0.182$ \\\\\n5 & $0.5$ & $2$ & $0.544$ & $0.459$ \\\\\n6 & $0.5$ & $5$ & $0.602$ & $0.166$ \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\label{tb:1}\n\\end{table}\n\n\\subsection{\\label{sec:level3b}Numerical results}\n\n\\subsubsection{\\label{sec:level3b1}Mean-flow generation}\n\nFigure~\\ref{fig:2} shows the time evolution of the mean axial velocity, $\\left< \\overline{u}_x \\right>_S$, for Run 6. The mean flow is generated around $y=0$ as time elapses and is sustained in subsequent periods. This result is the same as that obtained by Yokoi and Brandenburg \\cite{yb2016} in which the positive mean velocity directed to the rotation axis was generated around the positively helical region.\nIn the simulation performed by Yokoi and Brandenburg \\cite{yb2016}, helicity is distributed as $H(y) \\propto \\sin (\\pi y\/y_0)$ (in the original study, the inhomogeneous direction is $z$), and thus the positive axial velocity emerges in $y>0$ and the negative axial velocity emerges in $y<0$. Conversely, in the present simulation, the positive helicity is driven only in a limited region around $y=0$. Hence, the positive axial mean velocity emerges only around $y=0$.\nIt should be noted again that the mean velocity cannot be directly generated from the external force since the direct contribution from the external force is excluded in the calculation. Hereafter, we take the time average over $20 \\le t \\le 30$ as well as the homogeneous plane average.\nThe mean axial velocity of each run is given in Fig.~\\ref{fig:3}. Evidently, the positive axial mean velocity emerges only for the cases with both helicity injection and system rotation, namely Runs 4, 5, and 6. The difference between Run 3 and Runs 4 and 6 only corresponds to the existence of the helicity injection, and thus the external force with $\\alpha = 0$ does not influence the induction of the axial mean velocity. This indicates that neither inhomogeneous helicity nor system rotation by themselves are sufficient to obtain the mean-flow generation.\nIt is interesting to note that the maximum values of the mean flows for Runs 4 and 5 are the same. This suggests that the product of the helicity and the angular velocity of system rotation determines the mean-flow generation. The mean flow profile is expected to be symmetric about $y=0$. The present result is slightly asymmetric due to the limitations of time or ensemble average. \n\n\\begin{figure}[htp]\n \\begin{tabular}{c}\n  \\begin{minipage}{0.49\\hsize}\n   \\centering\n   \\includegraphics[scale=0.74]{fig2.eps}\n   \\caption{Time evolution of the axial mean velocity for Run 6. The horizontal axis denotes the time, and the vertical axis denotes the inhomogeneous direction, $y$, and the color contour denotes the value of $\\left< \\overline{u}_x \\right>_S$.}\n  \\label{fig:2}\n  \\end{minipage}\n  \\begin{minipage}{0.04\\hsize}\n  \\end{minipage}\n  \\begin{minipage}{0.49\\hsize}\n   \\centering\n   \\includegraphics[scale=0.67]{fig3.eps}\n   \\caption{Mean axial velocity of each run.}\n   \\label{fig:3}\n  \\end{minipage}\n \\end{tabular}\n\\end{figure}\n\nWhen the turbulent field is statistically steady, the equation for the mean axial velocity is expressed as follows:\n\\begin{align}\n\\frac{\\partial U_x}{\\partial t} \n= -\\frac{\\partial R_{xy}}{\\partial y} = 0 ,\n\\label{eq:3.8}\n\\end{align}\nwhere $R_{ij}$ satisfies $R_{ij} = R_{ij}^{GS} - 2\\left<\\nu_{sgs} \\overline{s}_{ij} \\right>$ in the framework of the eddy-viscosity representation of the SGS stress, and $R_{ij}^{GS} = \\left< \\overline{u}_i' \\overline{u}_j' \\right>$ denotes the Reynolds stress of the grid scale.\nIt should be noted that $\\overline{u}_i'$ denotes the fluctuation of the GS velocity $\\overline{u}_i$ and is defined as in Eq.~(\\ref{eq:3.5}).\nEquation~(\\ref{eq:3.8}) gives the Reynolds stress constant in the $y$ direction. The turbulence is inactive at the upper and lower boundaries, and thus $y=\\pm L_y\/2$, $R_{xy}$ disappears at this point. Therefore, the solution of the mean velocity equation is $R_{xy} = 0$. The green line with squares in Fig.~\\ref{fig:4} shows the profile of $R_{xy}$ for Run 6. It is nearly equal to zero although a slight non-zero value is observed around $y=0$ because the time averaging is insufficient for the statistically steady state. Here, we consider the appropriateness of the eddy-viscosity model,\n\\begin{align}\nR_{xy} = - \\nu_T \\frac{\\partial U_x}{\\partial y} \\ , \\ \\ \n\\nu_T = C_\\nu \\frac{K^2}{\\varepsilon},\n\\label{eq:3.9}\n\\end{align}\nIn Fig.~\\ref{fig:4}, the profile of $R_{xy}$ estimated by Eq.~(\\ref{eq:3.9}) is also plotted in the red line with crosses. It should be noted that $\\nu_T$ is evaluated by using $K^{GS}$ and $\\varepsilon^{SGS}$ instead of $K$ and $\\varepsilon$ as $\\nu_T = C\\nu (K^{GS})^2\/\\varepsilon^{SGS}$ with $C_\\nu =0.09$. This clearly indicates excessively high non-zero values around $y=0$. Since $\\nu_T \\neq 0$ around $y=0$, the velocity gradient must vanish in order to satisfy $R_{xy} = 0$. Therefore, the eddy-viscosity model is unable to reproduce the present result in which the mean flow is sustained around $y=0$.\n\n\\begin{figure}[htbp]\n\\centering\n\\includegraphics[scale=0.72]{fig4.eps}\n\\caption{The Reynolds stress $R_{xy}$ for Run 6. The green line with squares denotes the directly evaluated value, $R_{xy} = R_{xy}^{GS} - 2 \\left< \\nu_{sgs} \\overline{s}_{xy} \\right>$, and the red line with crosses denotes the value estimated by the eddy-viscosity model that is given by Eq.~(\\ref{eq:3.9}) with $C_\\nu = 0.09$.}\n\\label{fig:4}\n\\end{figure}\n\nIn order to rectify the inadequacy of the eddy-viscosity model, let us assume the following generic expression for the model,\n\\begin{align}\nR_{xy} = - \\nu_T \\frac{\\partial U_x}{\\partial y} + N_{xy},\n\\label{eq:3.10}\n\\end{align}\nwhere $N_{xy}$ denotes an additional term. As shown in Fig.~\\ref{fig:4}, the eddy-viscosity term, $-\\nu_T \\partial U_x \/ \\partial y$, has a large positive gradient around $y=0$. In order to satisfy $R_{xy} = 0$, $N_{xy}$ must involve a large negative gradient around $y=0$ to counterbalance the eddy-viscosity term. We expect that the second term on the right-hand side of Eq.~(\\ref{eq:2.14}) is a good candidate for $N_{xy}$ because the mean flow is only sustained when both the helical force and the system rotation are present.\n\n\n\\subsubsection{\\label{sec:level3b2}Origin of the helicity effect}\n\nIn order to investigate the origin of the additional term $N_{xy}$, we examine the transport equation for the Reynolds stress. The transport equation for $R_{xy}^{GS}$ is expressed as follows:\n\\begin{align}\n\\frac{\\partial R_{xy}^{GS}}{\\partial t} =\nP_{xy}^{GS} + \\Phi_{xy}^{GS} + \\Pi_{xy}^{GS} + C_{xy}^{GS},\n\\label{eq:3.11}\n\\end{align}\nwhere only the terms that significantly contribute to the simulation for Run 6 are included. Here $P_{xy}^{GS}$ denotes the production, $\\Phi_{xy}^{GS}$ the pressure--strain correlation, $\\Pi_{xy}^{GS}$ the pressure diffusion, and $C_{xy}^{GS}$ the Coriolis effect. They are respectively defined as follows:\n\\begin{subequations}\n\\begin{align}\nP_{xy}^{GS} & = - \\frac{2}{3} K^{GS} \\frac{\\partial U_x}{\\partial y} \n- B_{yy}^{GS} \\frac{\\partial U_x}{\\partial y} - B_{xz}^{GS} \\frac{\\partial U_z}{\\partial y}, \n\\label{eq:3.12a} \\\\\n\\Phi_{xy}^{GS} &\n = 2 \\left< \\overline{p}' \\overline{s}_{xy}' \\right>, \n\\label{eq:3.12b} \\\\\n\\Pi_{xy}^{GS} &\n = -\\frac{\\partial}{\\partial y} \\left< \\overline{p}' \\overline{u}_x' \\right>, \n\\label{eq:3.12c} \\\\\nC_{xy}^{GS} & = 2 R_{xz}^{GS} \\Omega_x^F,\n\\label{eq:3.12d}\n\\end{align}\n\\end{subequations}\nwhere $B_{ij}^{GS} = R_{ij}^{GS} - (2\/3) K^{GS} \\delta_{ij}$. The budget of the transport equation for $R_{xy}^{GS}$ for Run 6 is shown in Fig.~\\ref{fig:5}. It should be noted that the balance of the above four terms are mostly the same for Runs 4 and 5 especially in the sense that the pressure--strain correlation and the pressure diffusion are predominant (figures are not shown here). The production term plotted in the red line with crosses exhibits a positive gradient around $y=0$. It should be noted that with respect to the production term $P_{xy}^{GS}$, the first term on the right-hand side of Eq.~(\\ref{eq:3.12a}) is dominant [detailed contribution from each term in Eq.~(\\ref{eq:3.12a}) is not shown here]. Thus, as shown in Sec.~\\ref{sec:level2a}, the production term corresponds to the eddy-viscosity term, and it also exhibits a positive gradient around $y=0$ in Fig.~\\ref{fig:4}. Based on the discussion in Sec.~\\ref{sec:level3b1}, any candidate of the term corresponding to $N_{xy}$ that accounts for the sustainment of the mean velocity should exhibit a negative gradient around $y=0$. In Fig.~\\ref{fig:5}, two candidates are observed, namely the pressure diffusion $\\Pi_{xy}^{GS}$ (the blue line with circles) and the Coriolis effect $C_{xy}^{GS}$ (the magenta line with triangles). If the Coriolis effect corresponds to the origin of $N_{xy}$, the mean flow would be sustained for Run 3 in which  system rotation exists as well as for Runs 4, 5, and 6. Therefore, we focus on the pressure diffusion term. However, this does not deny the importance of Coriolis force in the flow generation phenomenon. As shown in Fig. 5, the Coriolis effect also contributes to the Reynolds stress in the sense that it sustains the mean flow. Additionally, the effect of the Coriolis force appears not only in the Coriolis effect but also in the pressure through the Poisson equation as discussed in the following paragraph.\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[scale=0.72]{fig5.eps}\n\\caption{Budget of the transport equation for $R_{xy}^{GS}$ for Run 6. The $y$ coordinate is limited to the region at $-2\\le y \\le 2$ where a high mean velocity exists.}\n\\label{fig:5}\n\\end{figure}\n\nIn order to investigate the pressure diffusion [Eq.(\\ref{eq:3.12c})], we consider the Poisson equation for the pressure fluctuation,\n\\begin{align}\n\\nabla^2 \\overline{p}' & =\n - 2\\overline{s}_{ab}' S_{ab} + \\overline{\\omega}_a' \\Omega_a^*\n- \\overline{s}_{ab}' \\overline{s}_{ab}'\n+ \\frac{1}{2} \\overline{\\omega}_a' \\overline{\\omega}_a' \n+ \\frac{\\partial^2}{\\partial x_a \\partial x_b} \\left[ 2\\left( \\nu_{sgs} \\overline{s}_{ab} - \\left< \\nu_{sgs} \\overline{s}_{ab} \\right> \\right) \\right].\n\\label{eq:3.13}\n\\end{align}\nWe approximate the left-hand side as\n\\begin{align}\n\\nabla^2 \\overline{p}' = - \\frac{\\overline{p}'}{\\ell_p^2},\n\\label{eq:3.14}\n\\end{align}\nwhere $\\ell_p$ denotes the length scale associated with the pressure fluctuation. Thus, the pressure diffusion $\\Pi_{xy}^{GS}$ is estimated as follows:\n\\begin{align}\n\\Pi_{xy}^{GS}\/\\ell_p^2 & =\n\\frac{\\partial}{\\partial y} \\left[\n - 2 \\left< \\overline{u}_x' \\overline{s}_{ab}'\\right> S_{ab}\n + \\left< \\overline{u}_x' \\overline{\\omega}_a' \\right> \\Omega_a^* \n- \\left< \\overline{u}_x' \\overline{s}_{ab}' \\overline{s}_{ab}' \\right> \n+ \\frac{1}{2} \\left< \\overline{u}_x' \\overline{\\omega}_a' \\overline{\\omega}_a' \\right> \n+ \\frac{\\partial^2}{\\partial y^2} \\left( 2 \\left< \\overline{u}_x' \\nu_{sgs} \\overline{s}_{ab} \\right> \\right) \\right] ,\n\\label{eq:3.15}\n\\end{align}\nwhere $\\ell_p$ is approximated as a constant in space for simplicity. Figure~\\ref{fig:6} shows the pressure diffusion $\\Pi_{xy}^{GS}$ evaluated from Eq.~(\\ref{eq:3.15}) for Run 6. As shown in the figure, the second term related to the mean absolute vorticity is dominant. Thus, $\\Pi_{xy}^{GS}$ is approximated as follows:\n\\begin{align}\n\\Pi_{xy}^{GS} \/ \\ell_p^2\n= \\frac{\\partial}{\\partial y} \\left( 2 \\left< \\overline{u}_x' \\overline{\\omega}_x' \\right> \\Omega_x^F \\right)\n= \\frac{\\partial}{\\partial y} \\left( \\frac{2}{3} H^{GS} \\Omega_x^F \\right),\n\\label{eq:3.16}\n\\end{align}\nand this includes $|\\Omega_i| \\ll |2\\Omega_i^F|$ and $\\left<\\overline{u}_x' \\overline{\\omega}_x' \\right> = \\left<\\overline{u}_y' \\overline{\\omega}_y' \\right> = \\left<\\overline{u}_z' \\overline{\\omega}_z' \\right> = H^{GS} \/3$. This indicates that the helicity gradient and the system rotation may account for the pressure diffusion that contributes to the mean velocity sustainment. A model expression of the pressure diffusion that is similar to Eq.~(\\ref{eq:3.16}) is also analytically obtained with the aid of the TSDIA \\cite{tsdia}. A brief introduction of the theory and the detailed calculation are given in Appendix~\\ref{sec:a}. The result is\n\\begin{align}\n\\Pi_{ij} & = \\frac{1}{3} \\left[ \\frac{\\partial}{\\partial x_j} \\left( L^2 H 2\\Omega_i^F \\right)\n+ \\frac{\\partial}{\\partial x_i} \\left( L^2 H 2\\Omega_j^F \\right) \\right] \n+ \\text{(non-helical term)} + O(|u^{(00)}|^3), \n\\label{eq:3.17}\n\\end{align}\nwhere $L$ denotes the length scale related to the energy containing eddy and $u^{(00)}$ is the lowest-order velocity corresponding to homogeneous isotropic turbulence defined in Eq.~(\\ref{eq:a4}). This model expression for the pressure diffusion is in good agreement with  Eq.~(\\ref{eq:3.16}).\n\n\\begin{figure}[htp]\n\\centering\n\\includegraphics[scale=0.72]{fig6.eps}\n\\caption{Approximate evaluation of $\\Pi_{xy}^{GS}$ for Run 6.}\n\\label{fig:6}\n\\end{figure}\n\nIn Fig.~\\ref{fig:5}, the pressure--strain correlation $\\Phi_{ij}^{GS}$ (the green line with squares) also significantly contributes to the Reynolds-stress transport. One might consider that the pressure diffusion and the pressure--strain correlation cancel each other. However, the sum of these two terms (as denoted by the cyan line with diamonds in Fig.~\\ref{fig:5}) contributes to exhibit a negative gradient around $y=0$ and plays the same role as the pressure diffusion itself. This tendency is also theoretically demonstrated as follows. The model expression of the pressure--strain correlation $\\Phi_{ij}$ is obtained with the aid of the TSDIA \\cite{tsdia}, and it is possible to analytically examine the balance of the two terms. The analytical result of the pressure--strain correlation is as follows:\n\\begin{align}\n\\Phi_{ij} & = - \\frac{3}{10} \\left[ \\frac{\\partial}{\\partial x_j} \\left( L^2 H 2 \\Omega_i^F \\right)\n   + \\frac{\\partial}{\\partial x_i} \\left( L^2 H 2\\Omega_j^F \\right) \\right]_D \n+ \\text{(non-helical term)} + O(|u^{(00)}|^3).\n\\label{eq:3.18}\n\\end{align}\nAlthough the helicity effect of the pressure--strain correlation $\\Phi_{ij}$ has the sign opposite to that of the pressure diffusion $\\Pi_{ij}$, its magnitude is slightly smaller. Thus the sum of $\\Pi_{ij}$ and $\\Phi_{ij}$,\n\\begin{align}\n\\Pi_{ij} + \\Phi_{ij} & = \n\\frac{1}{30} \\left[ \\frac{\\partial}{\\partial x_j} \\left( L^2 H 2\\Omega_i^F \\right)\n+ \\frac{\\partial}{\\partial x_i} \\left( L^2 H 2\\Omega_j^F \\right) \\right] \n+ \\text{(non-helical term)} + O(|u^{(00)}|^3),\n\\label{eq:3.19}\n\\end{align}\ncontributes in the same manner as the pressure diffusion $\\Pi_{ij}$ and sustains the mean flow.\n\n\n\\section{\\label{sec:level4}Correspondence of the pressure diffusion to the helicity model}\n\nThe results indicated that the pressure diffusion plays an important role in the sustainment of the mean velocity  in inhomogeneous helical turbulence. This fact contradicts the assumption for the derivation of the model for the Reynolds stress as given in Sec.~\\ref{sec:level2a}. In the current construction, the flow is assumed to be quasi-homogeneous for the diffusion to be neglected. However, the effect of the pressure diffusion is required to improve the Reynolds-stress model for inhomogeneous helical turbulence. As shown in Sec.~\\ref{sec:level3b2}, the effect of helicity is explicitly incorporated in the pressure diffusion term for the Reynolds-stress transport equation. Here, we add the helicity effect that originates from the pressure diffusion term to the LRR model \\cite{lrr1975} as follows:\n\\begin{align}\n\\begin{split}\n& \\Phi_{ij} + \\left[ \\Pi_{ij} \\right]_D\n= \\Phi_{ij}^{LRR} + C_{PH} \\Gamma_{ij},\n\\label{eq:4.1}\n\\end{split}\n\\end{align}\nwhere $\\Phi_{ij}^{LRR}$ denotes the LRR model given by Eq.~(\\ref{eq:2.10}), $C_{PH}$ is a positive constant, and\n\\begin{align}\n\\Gamma_{ij} = \n \\left[ \\frac{\\partial}{\\partial x_j} \\left( \\frac{K^3}{\\varepsilon^2} H \\Omega_i^* \\right)\n+ \\frac{\\partial}{\\partial x_i} \\left( \\frac{K^3}{\\varepsilon^2} H \\Omega_j^* \\right) \\right]_D .\n\\label{eq:4.2}\n\\end{align}\nHere, the length scale that corresponds to $\\ell_p$ in Eq.~(\\ref{eq:3.14}) or $L$ in Eq.~(\\ref{eq:3.17}) is expressed in terms of $K$ and $\\varepsilon$. Thus, the Reynolds-stress equation is re-expressed as follows:\n\\begin{align}\n\\frac{\\mathfrak{D} B_{ij}}{\\mathfrak{D} t} & =\n- C_{S1} \\frac{\\varepsilon}{K} B_{ij}\n- \\left( \\frac{4}{3} - C_{R1} \\right) K S_{ij} \n+ C_{PH} \\Gamma_{ij} \\nonumber \\\\\n& \\hspace{1.2em}\n- \\left( 1 - C_{R2} \\right) \\left[ B_{i\\ell} S_{\\ell j} + B_{j\\ell} S_{\\ell i} \\right]_D \n- \\left( 1 - C_{R3} \\right) \\left( B_{i\\ell} \\Omega_{\\ell j}^* + B_{j\\ell} \\Omega_{\\ell i}^* \\right),\n\\label{eq:4.3}\n\\end{align}\nThe third term on the right-hand side denotes the only difference between Eqs.~(\\ref{eq:2.11}) and (\\ref{eq:4.3}). Thus, the model expression corresponding to Eq.~(\\ref{eq:2.13}) is given as follows:\n\\begin{align}\nB_{ij} & = - 2 C_\\nu \\frac{K^2}{\\varepsilon} S_{ij}\n+ C_\\gamma \\frac{K}{\\varepsilon} \\Gamma_{ij}\n+ \\cdots,\n\\end{align}\nwhere $C_\\gamma = C_{PH} \/ C_{S1}$. The second term is significantly similar to Eq.~(\\ref{eq:2.14}) obtained by Yokoi and Yoshizawa \\cite{yy1993}. Hence, the helicity model given by Eq.~(\\ref{eq:2.14}) can trace part of its origin to the pressure diffusion in inhomogeneous helical turbulence in a rotating system.\n\n\n\\section{\\label{sec:level5}Conclusions}\n\nThe mechanism of the mean-flow generation and its relationship to the turbulent helicity were investigated by using the numerical simulation of a rotating inhomogeneous turbulence. In the simulation, an external forcing was applied to inject turbulent energy and helicity and the rotation axis was perpendicular to the inhomogeneous direction. The initial mean velocity and the mean part of the external force were set to zero, and this implies that it is not possible to directly excite the mean flow by the external forcing. The results showed that the mean flow is generated and sustained only when both helical forcing and system rotation exist. The flow-generation phenomenon originates from both the turbulent helicity and the rotational motion of fluid. \n\nThe usual eddy-viscosity model is unable to reproduce the mean-flow generation observed in the simulation, and therefore an additional term is needed to explain the phenomenon. In order to explore candidates for the additional term, the budget of the Reynolds-stress transport equation was investigated. The results suggested that the pressure diffusion significantly influences the sustainment of the mean flow. The approximation to the Poisson equation for the pressure fluctuation was used to obtain an expression for the pressure diffusion in terms of the turbulent helicity and the angular velocity of the system rotation. The effect of helicity in relation to the pressure diffusion term was considered to obtain a model for the Reynolds stress, and the obtained model is considerably similar to the one obtained by Yokoi and Yoshizawa \\cite{yy1993}. The model implies that the inhomogeneity of helicity plays a crucial role in rotating turbulence such as the momentum transport due to turbulence in the low-latitude region of a rotating sphere \\cite{duarteetal2016,ssd2014}.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}