arithmetic properties of mirror map and quantum coupling we study some arithmetic properties of the mirror maps and the quantum yukawa couplings for some 1 parameter deformations of calabi yau manifolds first we use the schwarzian differential equation which we derived previously to characterize the mirror map in each case for algebraic k3 surfaces we solve the equation in terms of thej function by deriving explicit modular relations we prove that some k3 mirror maps are algebraic over the genus zero function fieldq j this leads to a uniform proof that those mirror maps have integral fourier coefficients regarding the maps as riemann mappings we prove that they are genus zero functions by virtue of the conway norton conjecture proved by borcherds using frenkel lepowsky meurman s moonshine module we find that these maps are actually the reciprocals of the thompson series for certain conjugacy classes in the griess fischer group this also gives as an immediate consequence a second proof that those mirror maps are integral we thus conjecture a surprising connection between k3 mirror maps and the thompson series for threefolds we construct a formal nonlinear ode for the quantum coupling reduced modp under the mirror hypothesis and an integrality assumption we derive modp congurences for the fourier coefficients for the quintics we deduce at least for 5 d that the degreed instanton numbersnd are divisible by 53 a fact first conjectured by clemens

holonomic functions of several complex variables and singularities of anisotropic ising n fold integrals focusing on examples associated with holonomic functions we try to bring new ideas on how to look at phase transitions for which the critical manifolds are not points but curves depending on a spectral variable or even fill higher dimensional submanifolds lattice statistical mechanics often provides a natural holonomic framework to perform singularity analysis with several complex variables that would in the most general mathematical framework be too complex or simply could not be defined in a learn by example approach considering several picard fuchs systems of two variables above calabi yau odes associated with double hypergeometric series we show that d finite holonomic functions are actually a good framework for finding properly the singular manifolds the singular manifolds are found to be genus zero curves we then analyze the singular algebraic varieties of quite important holonomic functions of lattice statistical mechanics the n fold integrals n corresponding to the n particle decomposition of the magnetic susceptibility of the anisotropic square ising model in this anisotropic case we revisit a set of so called nickelian singularities that turns out to be a two parameter family of elliptic curves we then find the first set of non nickelian singularities for 3 and 4 that also turns out to be rational or elliptic curves we underline the fact that these singular curves depend on the anisotropy of the ising model or equivalently that they depend on the spectral parameter of the model this has important consequences on the physical nature of the anisotropic n s which appear to be highly composite objects we address from a birational viewpoint the emergence of families of elliptic curves and that of calabi yau manifolds on such problems we also address the question of singularities of non holonomic functions with a discussion on the accumulation of these singular curves for the non holonomic anisotropic full susceptibility this article is part of lattice models and integrability a special issue of journal of physics a mathematical and theoretical in honour of f y wu s 80th birthday

a differential operator for symmetric functions and the combinatorics of multiplying transpositions by means of irreducible characters for the symmetric group formulas have previously been given for the number of ways of writing permutations in a given conjugacy class as products of transpositions these formulas are alternating sums of binomial coefficients and powers of integers combinatorial proofs are obtained in this paper by analyzing the action of a partial differential operator for symmetric functions

double vogan diagrams and semisimple symmetric spaces a vogan diagram is a set of involution and painting on a dynkin diagram it selects a real form or equivalently an involution from a complex simple lie algebra we introduce the double vogan diagram which is two sets of vogan diagrams superimposed on an affine dynkin diagram they correspond to pairs of commuting involutions on complex simple lie algebras and therefore provide an independent classification of the simple locally symmetric pairs

algorithm of the finite lattice method for high temperature expansion of the ising model in three dimensions we propose an algorithm of the finite lattice method to generate the high temperature series for the ising model in three dimensions it enables us to extend the series for the free energy of the simple cubic lattice from the previous series of 26th order to 46th order in the inverse temperature the obtained series give the estimate of the critical exponent for the specific heat in high precision

on the bilinear structure associated to bezoutians abstract this paper is partly a survey of known results on quadratic forms that are hard to find in the literature our main focus is a twisted form of a construction due to bezout this skew bezoutian is a symplectic resp quadratic space associated to a pair of reciprocal or skew reciprocal coprime polynomials of same degree the isometry group of this space turns out to contain a certain associated hypergeometric group using the skew bezoutian we construct explicit isometries of bilinear spaces with given invariants such as the characteristic polynomial or jordan form and in the quadratic case the spinor norm

on globally nilpotent differential equations abstract in a previous work of the authors a middle convolution operation on the category of fuchsian differential systems was introduced in this note we show that the middle convolution of fuchsian systems preserves the property of global nilpotence this leads to a globally nilpotent fuchsian system of rank two which does not belong to the known classes of globally nilpotent rank two systems

25th order high temperature expansion results for three dimensional ising like systems on the simple cubic lattice 25th order high temperature series are computed for a general nearest neighbor three dimensional ising model with arbitrary potential on the simple cubic lattice in particular we consider three improved potentials characterized by suppressed leading scaling corrections critical exponents are extracted from high temperature series specialized to improved potentials obtaining gamma 1 2373 2 nu 0 63012 16 alpha 0 1096 5 eta 0 03639 15 beta 0 32653 10 delta 4 7893 8 moreover biased analyses of the 25th order series of the standard ising model provide the estimate delta 0 52 3 for the exponent associated with the leading scaling corrections by the same technique we study the small magnetization expansion of the helmholtz free energy the results are then applied to the construction of parametric representations of the critical equation of state using a systematic approach based on a global stationarity condition accurate estimates of several universal amplitude ratios are also presented

a quick proof on the equivalence classes of extended vogan diagrams abstract an extended vogan diagram is an extended dynkin diagram together with a diagram involution such that the vertices fixed by the involution are colored white or black every extended vogan diagram represents an almost compact real form of the affine kac moody lie algebra two extended diagrams are said to be equivalent if they represent isomorphic real forms the equivalence classes of extended vogan diagrams have earlier been classified by the authors in this paper we present a much shorter and instructive argument

lattice green functions and calabi yau differential equations by making the connection between four dimensional lattice green functions lgfs and picard fuchs ordinary differential equations of calabi yau manifolds we have given explicit forms for the coefficients of the four dimensional lgfs on the simple cubic and body centred cubic lattices in terms of finite sums of products of binomial coefficients and have shown that the corresponding four dimensional face centred cubic lgf satisfies a fourth order ode of degree 7 which we have found but not solved the fact that the picard fuchs equations that appear here are similar to those that appear in string theory perhaps makes the first connection between these two apparently unrelated topics

generalized hypergeometric functions and rational curves on calabi yau complete intersections in toric varieties we formulate general conjectures about the relationship between the a model connection on the cohomology of ad dimensional calabi yau complete intersectionv ofr hypersurfacesv1 vr in a toric varietyp and the system of differential operators annihilating the special generalized hypergeometric series 0 constructed from the fan using this generalized hypergeometric series we propose conjectural mirrorsv ofv and the canonicalq coordinates on the moduli spaces of calabi yau manifolds

belyi functions for hyperbolic hypergeometric to heun transformations abstract a complete classification of belyi functions for transforming certain hypergeometric equations to heun equations is given the considered hypergeometric equations have the local exponent differences 1 k 1 l 1 m that satisfy k l m n and the hyperbolic condition 1 k 1 l 1 m 1 there are 366 galois orbits of belyi functions giving the considered non parametric hypergeometric to heun pull back transformations their maximal degree is 60 which is well beyond reach of standard computational methods to obtain these belyi functions we developed two efficient algorithms that exploit the implied pull back transformations

iterated elliptic and hypergeometric integrals for feynman diagrams we calculate 3 loop master integrals for heavy quark correlators and the 3 loop quantum chromodynamics corrections to the parameter they obey non factorizing differential equations of second order with more than three singularities which cannot be factorized in mellin n space either the solution of the homogeneous equations is possible in terms of 2f1 gaus hypergeometric functions at rational argument in some cases integrals of this type can be mapped to complete elliptic integrals at rational argument this class of functions appears to be the next one arising in the calculation of more complicated feynman integrals following the harmonic polylogarithms generalized polylogarithms cyclotomic harmonic polylogarithms square root valued iterated integrals and combinations thereof which appear in simpler cases the inhomogeneous solution of the corresponding differential equations can be given in terms of iterative integrals where the new innermost letter itself is not an iterative integral a new class of iterative integrals is introduced containing letters in which multiple definite integrals appear as factors for the elliptic case we also derive the solution in terms of integrals over modular functions and also modular forms using q product and series representations implied by jacobi s i functions and dedekind s function the corresponding representations can be traced back to polynomials out of lambert eisenstein series having representations also as elliptic polylogarithms a q factorial 1 k logarithms and polylogarithms of q and their q integrals due to the specific form of the physical variable x q for different processes different representations do usually appear numerical results are also presented

shimura curves of genus at most two we enumerate all shimura curves x d 0 r of genus at most two there are exactly 858 such curves up to equivalence

admissible positive systems of affine non twisted kac moody lie algebras abstract we introduce admissible positive systems and hermitian real forms of affine non twisted kac moody lie algebras and show that a real form has admissible positive system if and only if it is hermitian we use the vogan diagrams to classify the hermitian real forms and show that their symmetric spaces carry complex structures

ising model susceptibility the fuchsian differential equation for χ 4 and its factorization properties we give the fuchsian linear differential equation satisfied by x 4 the four particle contribution to the susceptibility of the isotropic square lattice ising model this fuchsian differential equation is deduced from a series expansion method introduced in two previous papers and is applied with some symmetries and tricks specific to x 4 the corresponding order ten linear differentia operator exhibits a large set of factorization properties among these factorizations one is highly remarkable it corresponds to the fact that the two particle contribution x 2 is actually a solution of this order ten linear differential operator this result together with a similar one for the order seven differential operator corresponding to the three particle contribution x 3 leads us to a conjecture on the structure of all the n particle contributions x n

equivalence classes of vogan diagrams abstract a vogan diagram is a dynkin diagram with an involution and the vertices fixed by the involution may be painted they represent real simple lie algebras and two diagrams are said to be equivalent if they represent the same lie algebra in this article we classify the equivalence classes of all vogan diagrams in doing so we find that the underlying dynkin diagrams have certain properties in graph painting we show that this combinatorial property provides an easy classification for most of the simply laced dynkin diagrams

differential algebra on lattice green functions and calabi yau operators we revisit miscellaneous linear differential operators mostly associated with lattice green functions in arbitrary dimensions but also calabi yau operators and order 7 operators corresponding to exceptional differential galois groups we show that these irreducible operators are not only globally nilpotent but are such that they are homomorphic to their formal adjoints considering these operators or sometimes equivalent operators we show that they are also such that either their symmetric square or their exterior square have a rational solution this is a general result an irreducible linear differential operator homomorphic to its formal adjoint is necessarily such that either its symmetric square or its exterior square has a rational solution and this situation corresponds to the occurrence of a special differential galois group we thus define the notion of being special geometry for a linear differential operator if it is irreducible globally nilpotent and such that it is homomorphic to its formal adjoint since many derived from geometry n fold integrals periods occurring in physics are seen to be diagonals of rational functions we address several examples of minimal order operators annihilating diagonals of rational functions and remark that they also seem to be systematically associated with irreducible factors homomorphic to their adjoint

modular solutions to equations of generalized halphen type solutions to a class of differential systems that generalize the halphen system are determined in terms of automorphic functions whose groups are commensurable with the modular group these functions uniformize riemann surfaces of genus zero and have q series with integral coefficients rational maps between these functions are derived implying subgroup relations between their automorphism groups as well as symmetrization maps between the associated differential systems

automata and the susceptibility of the square lattice ising model modulo powers of primes we study the full susceptibility of the ising model modulo powers of primes we find exact functional equations for the full susceptibility modulo these primes revisiting some lesser known results on discrete finite automata we show that these results can be seen as a consequence of the fact that modulo 2 r one cannot distinguish the full susceptibility from some simple diagonals of rational functions which reduce to algebraic functions modulo 2 r and consequently satisfy exact functional equations modulo 2 r we sketch a possible physical interpretation of these functional equations modulo 2 r as reductions of a master functional equation corresponding to infinite order symmetries such as the isogenies of elliptic curves one relevant example is the landen transformation which can be seen as an exact generator of the ising model renormalization group we underline the importance of studying a new class of functions corresponding to ratios of diagonals of rational functions they reduce to algebraic functions modulo powers of primes and they may have solutions with natural boundaries

a classification of involutive automorphisms of an affine kac moody lie algebra abstract in this thesis we classify the conjugacy classes of involutions in aut g where g is an affine kac moody lie algebra we distinguish between two kinds of involutions those which preserve the conjugacy class of a borel subalgebra and those which don t we give a complete and non redundant list of representatives of involutions of the first kind and we compute their fixed points sets we prove that any involution of the first kind has a conjugate which leaves invariant the components of the gauss decomposition g n h n we also give a complete list of representatives of the conjugacy classes of involutions of the second kind

lattice green functions the seven dimensional face centred cubic lattice we present a recursive method to generate the expansion of the lattice green function of the d dimensional face centred cubic fcc lattice we produce a long series for d 7 then we show and recall that in order to obtain the linear differential equation annihilating such a long power series the most economic way amounts to producing the non minimal order differential equations we use the method to obtain the minimal order linear differential equation of the lattice green function of the seven dimensional fcc lattice we give some properties of this irreducible order eleven differential equation we show that the differential galois group of the corresponding operator is included in this order eleven operator is non trivially homomorphic to its adjoint and we give a decomposition of this order eleven operator in terms of four order one self adjoint operators and one order seven self adjoint operator furthermore using the landau conditions on the integral we forward the regular singularities of the differential equation of the d dimensional lattice and show that they are all rational numbers we evaluate the return probability in random walks in the seven dimensional fcc lattice we show that the return probability in the d dimensional fcc lattice decreases as d 2 as the dimension d goes to infinity

the modularity of k3 surfaces with non symplectic group actions we consider complex k3 surfaces with a non symplectic group acting trivially on the algebraic cycles vorontsov and kond classified those k3 surfaces with transcendental lattice of minimal rank the purpose of this note is to study the galois representations associated to these k3 surfaces the rank of the transcendental lattices is even and varies from 2 to 20 excluding 8 and 14 we show that these k3 surfaces are dominated by fermat surfaces and hence they are all of cm type we will establish the modularity of the galois representations associated to them also we discuss mirror symmetry for these k3 surfaces in the sense of dolgachev and show that a mirror k3 surface exists with one exception

on the integrality of the taylor coefficients of mirror maps we continue our study begun in on the integrality of the taylor coefficients of mirror maps duke math j to appear of the fine integrality properties of the taylor coefficients of the series q z z exp g z f z where f z and g z log z f z are specific solutions of certain hypergeometric differential equations with maximal unipotent monodromy at z 0 more precisely we address the question of finding the largest integer v such that the taylor coefficients of z 1q z 1 v are still integers in particular we determine the dwork kontsevich sequence un n 1 where un is the largest integer such that qn z n is a series with integer coefficients where qn z exp gn z fn z fn z m 0 nm z m and gn z m 1 hnm hm nm z m with hn denoting the n th harmonic number conditional on the conjecture that there are no prime number p and integer n such that the p adic valuation of hn 1 is strictly greater than 3

counting cycles in permutations by group characters with an application to a topological problem the character theory of the symmetric group is used to derive properties of the number of permutations with k cycles which are expressible as the product of a full cycle with an element of an arbitrary but fixed conjugacy class for the conjugacy class of fixed point free involutions this problem has application to the analysis of singularities in surfaces 1 introduction for nonnegative integers k and n and a partition p of n let ef denote the number of permutations w on n symbols such that it has exactly k cycles and such that it can be expressed as a product of an arbitrary but fixed cycle of length n and a permutation in the conjugacy class indexed by p the purpose of this paper is to derive the generating function for these numbers and to obtain some of their properties the method makes direct use of combinatorial and algebraic properties of the group algebra of the symmetric group a special case of this problem is of particular interest let e p n denote the number e when p indexes the conjugacy class of permutations on pn symbols with n cycles of length p the matter of calculating ek2 n arose in connection with work by harris and morrison 4 on singularities in surfaces it has also occurred indirectly in the work of gross 2 on graph embeddings harer and zagier 3 have shown by an independent method that the sequence e k2 n for k n 1 satisfies a three term linear recurrence equation with coefficients which are polynomials in n to fix ideas note that for n 2 the permissible permutations are

vogan diagrams of real forms of affine kac moody lie algebras abstract a vogan diagram is actually a dynkin diagram with some additional structure this paper develops theory of vogan diagrams for almost compact real forms of indecomposable nontwisted affine kac moody lie algebras here the equivalence classes of vogan diagrams are in one one correspondence with the isomorphism classes of almost compact real forms

the diagonal ising susceptibility we use the recently derived form factor expansions of the diagonal two point correlation function of the square ising model to study the susceptibility for a magnetic field applied only to one diagonal of the square lattice for the isotropic ising model we exactly evaluate the one and two particle contributions 1 d and 2 d of the corresponding susceptibility and obtain linear differential equations for the three and four particle contributions as well as the five particle contribution 5 d but only modulo a given prime we use these exact linear differential equations to show that not only the russian doll structure but also the direct sum structure on the linear differential operators for the n particle contributions n d are quite directly inherited from the direct sum structure on the form factors f n we show that the nth particle contributions n d have their singularities at roots of unity these singularities become dense on the unit circle sinh 2ev ktsinh 2eh kt 1 as n

singularities of n fold integrals of the ising class and the theory of elliptic curves we introduce some multiple integrals that are expected to have the same singularities as the singularities of the n particle contributions n to the susceptibility of the square lattice ising model we find the fuchsian linear differential equation satisfied by these multiple integrals for n 1 2 3 4 and only modulo some primes for n 5 and 6 thus providing a large set of possible new singularities of n we discuss the singularity structure for these multiple integrals by solving the landau conditions we find that the singularities of the associated odes identify up to n 6 with the leading pinch landau singularities the second remarkable obtained feature is that the singularities of the odes associated with the multiple integrals reduce to the singularities of the odes associated with a finite number of one dimensional integrals among the singularities found we underline the fact that the quadratic polynomial condition 1 3w 4w2 0 that occurs in the linear differential equation of 3 actually corresponds to a remarkable property of selected elliptic curves namely the occurrence of complex multiplication the interpretation of complex multiplication for elliptic curves as complex fixed points of the selected generators of the renormalization group namely isogenies of elliptic curves is sketched most of the other singularities occurring in our multiple integrals are not related to complex multiplication situations suggesting an interpretation in terms of motivic mathematical structures beyond the theory of elliptic curves

admissible positive systems of affine kac moody lie algebras the twisted cases abstract we introduce admissible positive systems and hermitian real forms of affine twisted kac moody lie algebras and show that a real form has admissible positive system if and only if it is hermitian we use the vogan diagrams to classify the hermitian real forms and show that their symmetric spaces carry complex structures the affine non twisted kac moody lie algebras have been treated in an earlier work and this article deals with the twisted cases

on rigid tuples in linear groups of odd dimension abstract we construct examples of linearly rigid tuples which lead to regular galois realizations over q for linear and unitary groups gl2m 1 q and u2m 1 q2 where q is odd and m q 1 resp m q 1 the notion of linear rigidity was introduced by strambach and volklein j reine angew math to appear as an analogue of katz physically rigid systems see n m katz rigid local systems annals of mathematics studies vol 139 princeton univ press princeton nj 1996

high order fuchsian equations for the square lattice ising model χ 6 this paper deals with the six particle contribution to the magnetic susceptibility of the square lattice ising model we have generated modulo a prime series coefficients for the length of the series is sufficient to produce the corresponding fuchsian linear differential equation modulo a prime we obtain the fuchsian linear differential equation that annihilates the depleted series the factorization of the corresponding differential operator is performed using a method of factorization modulo a prime introduced in a previous paper the depleted differential operator is shown to have a structure similar to the corresponding operator for it splits into factors of smaller orders with the left most factor of order 6 being equivalent to the symmetric fifth power of the linear differential operator corresponding to the elliptic integral e the right most factor has a direct sum structure and using series calculated modulo several primes all the factors in the direct sum have been reconstructed in exact arithmetics

globally nilpotent differential operators and the square ising model we recall various multiple integrals with one parameter related to the isotropic square ising model and corresponding respectively to the n particle contributions of the magnetic susceptibility to the lattice form factors to the two point correlation functions and to their extensions the univariate analytic functions defined by these integrals are holonomic and even g functions they satisfy fuchsian linear differential equations with polynomial coefficients and have some arithmetic properties we recall the explicit forms found in previous work of these fuchsian equations as well as their russian doll and direct sum structures these differential operators are selected fuchsian linear differential operators and their remarkable properties have a deep geometrical origin they are all globally nilpotent or sometimes even have zero p curvature we also display miscellaneous examples of globally nilpotent operators emerging from enumerative combinatorics problems for which no integral representation is yet known focusing on the factorized parts of all these operators we find out that the global nilpotence of the factors resp p curvature nullity corresponds to a set of selected structures of algebraic geometry elliptic curves modular curves curves of genus five six and even a remarkable weight 1 modular form emerging in the three particle contribution 3 of the magnetic susceptibility of the square ising model noticeably this associated weight 1 modular form is also seen in the factors of the differential operator for another n fold integral of the ising class 3 h for the staircase polygons counting and in ap ry s study of 3 g functions naturally occur as solutions of globally nilpotent operators in the case where we do not have g functions but hamburger functions one irregular singularity at 0 or that correspond to the confluence of singularities in the scaling limit the p curvature is also found to verify new structures associated with simple deformations of the nilpotent property

lattice green function at 0 for the 4d hypercubic lattice the generating function for recurrent polya walks on the four dimensional hypercubic lattice is expressed as a kampe de feriet function various properties of the associated walks are enumerated

on lame operators which are pull backs of hypergeometric ones we give a method that would allow one to calculate the number of lame operators 5n n e n with prescribed finite monodromy and do the calculation for the case n 1 we find a bound for the degree over q of the field of definition of the coefficients of a lame operator with prescribed finite monodromy and give examples of lame operators with finite monodromy finally we study lame operators with infinite monodromy and generic second order differential operators which are pull backs of hypergeometric ones under algebraic maps consider an ordinary differential operator 0 1 l a x an x where as x e k x and k is a number field over q what one can say about its solutions ly 0 and monodromy looking only at the coefficients as x in this paper we give some partial answers to this question for the special case of a lame operator one of the questions we will approach is the following for every n e n how many lame operators yn are there whose monodromy group is a dihedral group of order 2n n e n we will answer in the case n 1 2 14 2 15 the method we use for the proof works for general n for example the case n 2 can be found in co before summarizing our work we give a brief review of the known results since the work of schwarz on hypergeometric equations and the classification of the relative finite monodromy groups no basic improvements have been made recently the challenge of the grothendieck conjecture has stimulated some progress see ka ka1 ka2 ho be he dw1 we recall that this conjecture asserts that if in 0 1 almost all the reductions mod p prime ideal in the ring of integers of k have a complete set of solutions which is equivalent to l having zero p curvature for almost all p then equation 0 1 has only algebraic solutions on the other hand some improvements in the sense of klein and schwarz can be found in ba dw sing ba dw but we are still a long way from achieving the same level of understanding for general operators that we have attained in the hypergeometric case in received by the editors january 2 1991 1991 mathematics subject classification primary 34b30 secondary 14e20 the author was a visiting fellow at princeton university supported by c n r italy 1995 american mathematical society

monodromy of picard fuchs differential equations for calabi yau threefolds abstract in this paper we are concerned with the monodromy of picardfuchs differential equations associated with one parameter families of calabiyau threefolds our results show that in the hypergeometric cases the matrix representations of monodromy relative to the frobenius bases can be expressed in terms of the geometric invariants of the underlying calabi yau threefolds this phenomenon is also verified numerically for other families of calabi yau threefolds in the paper furthermore we discover that under a suitable change of bases the monodromy groups are contained in certain congruence subgroups of sp 4 z of finite index and whose levels are related to the geometric invariants of the calabi yau threefolds

the ising model and special geometries we show that the globally nilpotent g operators corresponding to the factors of the linear differential operators annihilating the multifold integrals n of the magnetic susceptibility of the ising model n 6 are homomorphic to their adjoint this property of being self adjoint up to operator homomorphisms is equivalent to the feature of their symmetric squares or their exterior squares having rational solutions the differential galois groups are in the special orthogonal or symplectic groups this self adjoint up to operator equivalence property means that the factor operators that we already know to be derived from geometry are special globally nilpotent operators they correspond to special geometries beyond the small order factor operators occurring in the linear differential operators associated with 5 and 6 and in particular those associated with modular forms we focus on the quite large order 12 and order 23 operators we show that the order 12 operator has an exterior square which annihilates a rational solution then its differential galois group is in the symplectic group the order 23 operator is shown to factorize into an order 2 operator and an order 21 operator the symmetric square of this order 21 operator has a rational solution its differential galois group is thus in the orthogonal group

landau singularities and singularities of holonomic integrals of the ising class we consider families of multiple and simple integrals of the ising class and the linear ordinary differential equations with polynomial coefficients they are solutions of we compare the full set of singularities given by the roots of the head polynomial of these linear odes and the subset of singularities occurring in the integrals with the singularities obtained from the landau conditions for these ising class integrals we show that the landau conditions can be worked out either to give the singularities of the corresponding linear differential equation or the singularities occurring in the integral the singular behaviour of these integrals is obtained in the self dual variable w s 2 1 s2 with s sinh 2k where k j kt is the usual ising model coupling constant switching to the variable s we show that the singularities of the analytic continuation of series expansions of these integrals actually break the kramers wannier duality we revisit the singular behaviour zenine et al 2005 j phys a math gen 38 9439 74 of the third contribution to the magnetic susceptibility of ising model 3 at the points 1 3w 4w2 0 and show that 3 s is not singular at the corresponding points inside the unit circle s 1 while its analytical continuation in the variable s is actually singular at the corresponding points 2 s s2 0 outside the unit circle s 1

integrals of the ising class from an experimental mathematical perspective we analyze ising class integrals our experimental results involvedextreme precision multidimensional quadrature on intricate integrands thus highly parallel computation was required

the perimeter generating functions of three choice imperfect and one punctured staircase polygons we consider the isotropic perimeter generating functions of three choice imperfect and one punctured staircase polygons whose 8th order linear fuchsian odes are previously known we derive simple relationships between the three generating functions and show that all three generating functions are joint solutions of a common 12th order fuchsian linear ode we find that the 8th order differential operators can each be rewritten as a direct sum of a direct product with operators no larger than 3rd order we give closed form expressions for all the solutions of these operators in terms of 2 f 1 hypergeometric functions with rational and algebraic arguments the solutions of these linear differential operators can in fact be expressed in terms of two modular forms since these 2 f 1 hypergeometric functions can be expressed with two rational or algebraic pullbacks

diagonal ising susceptibility elliptic integrals modular forms and calabi yau equations we give the exact expressions of the partial susceptibilities 3 d and 4 d for the diagonal susceptibility of the ising model in terms of modular forms and calabi yau odes and more specifically 3f2 1 3 2 3 3 2 1 1 z and 4f3 1 2 1 2 1 2 1 2 1 1 1 z hypergeometric functions by solving the connection problems we analytically compute the behavior at all finite singular points for 3 d and 4 d we also give new results for 5 d we see in particular the emergence of a remarkable order 6 operator which is such that its symmetric square has a rational solution these new exact results indicate that the linear differential operators occurring in the n fold integrals of the ising model are not only derived from geometry globally nilpotent but actually correspond to special geometry homomorphic to their formal adjoint this raises the question of seeing if these special geometry ising operators are special ones reducing in fact systematically to selected k balanced q 1fq hypergeometric functions or correspond to the more general solutions of calabi yau equations

mirror symmetry and rational curves on quintic threefolds a guide for mathematicians we give a mathematical account of a recent string theory calcula tion which predicts the number of rational curves on the generic quintic three fold our account involves the interpretation of yukawa couplings in terms of variations of hodge structure a new q expansion principle for functions on the moduli space of calabi yau manifolds and the mirror symmetry phe nomenon recently observed by string theorists department of mathematics duke university durham north carolina 27706 e mail address drm math duke edu this content downloaded from 157 55 39 224 on wed 14 dec 2016 04 59 36 utc all use subject to http about jstor org terms

calabi yau threefolds fibred by mirror quartic k3 surfaces abstract we study threefolds fibred by mirror quartic k3 surfaces we begin by showing that any family of such k3 surfaces is completely determined by a map from the base of the family to the moduli space of mirror quartic k3 surfaces this is then used to give a complete explicit description of all calabi yau threefolds fibred by mirror quartic k3 surfaces we conclude by studying the properties of such calabi yau threefolds including their hodge numbers and deformation theory

square lattice ising model susceptibility connection matrices and singular behaviour of χ 3 and χ 4 we present a simple but efficient way to calculate connection matrices between sets of independent local solutions defined at two neighbouring singular points of fuchsian differential equations of quite large orders such as those found for the third and fourth contribution 3 and 4 to the magnetic susceptibility of the square lattice ising model we deduce all the critical behaviours of the solutions 3 and 4 as well as the asymptotic behaviour of the coefficients in the corresponding series expansions we confirm that the newly found quadratic singularities of the fuchsian ode associated with 3 are not singularities of the particular solution 3 itself we use the previous connection matrices to get the exact expressions of all the monodromy matrices of the fuchsian differential equation for 3 and 4 expressed in the same basis of solutions these monodromy matrices are the generators of the differential galois group of the fuchsian differential equations for 3 and 4 whose analysis is just sketched here as far as the physics implications of the solutions are concerned we find challenging qualitative differences when comparing the corrections to scaling for the full susceptibility at high temperature respectively low temperature and the first two terms 1 and 3 respectively 2 and 4

extended vogan diagrams an extended vogan diagram is an extended dynkin diagram with a diagram involution such that the vertices fixed by the involution can be painted or unpainted every extended vogan diagram represents an almost compact real form of some affine kac moody lie algebra two diagrams may represent isomorphic algebras and in this case we say that the diagrams are equivalent in this paper we classify the equivalence classes of extended vogan diagrams and provide a complete list of all diagrams within each class it gives a combinatorial classification of the isomorphic classes of almost compact real forms of the affine kac moody lie algebras

ising n fold integrals as diagonals of rational functions and integrality of series expansions we show that the n fold integrals n of the magnetic susceptibility of the ising model as well as various other n fold integrals of the ising class or n fold integrals from enumerative combinatorics like lattice green functions correspond to a distinguished class of functions generalizing algebraic functions they are actually diagonals of rational functions as a consequence the power series expansions of the analytic at x 0 solutions of these linear differential equations derived from geometry are globally bounded which means that after just one rescaling of the expansion variable they can be cast into series expansions with integer coefficients we also give several results showing that the unique analytical solution of calabi yau odes and more generally picard fuchs linear odes with solutions of maximal weights are always diagonals of rational functions besides in a more enumerative combinatorics context generating functions whose coefficients are expressed in terms of nested sums of products of binomial terms can also be shown to be diagonals of rational functions we finally address the question of the relations between the notion of integrality series with integer coefficients or more generally globally bounded series and the modularity of odes

selected non holonomic functions in lattice statistical mechanics and enumerative combinatorics we recall that the full susceptibility series of the ising model modulo powers of the prime reduce to algebraic functions we also recall the nonlinear polynomial differential equation obtained by tutte for the generating function of the q coloured rooted triangulations by vertices which is known to have algebraic solutions for all the numbers of the form the holonomic status of being unclear we focus on the analysis of the case showing that the corresponding series is quite certainly non holonomic along the line of a previous work on the susceptibility of the ising model we consider this series modulo the first eight primes and show that this probably non holonomic function reduces modulo these primes to algebraic functions we conjecture that this probably non holonomic function reduces to algebraic functions modulo almost every prime or power of prime numbers this raises the question of whether such remarkable non holonomic functions can be seen as a ratio of diagonals of rational functions or even algebraic functions of diagonals of rational functions

square lattice ising model tilde chi 5 ode in exact arithmetic we obtain in exact arithmetic the order 24 linear differential operator l24 and the right hand side e 5 of the inhomogeneous equation l24 5 e 5 where is a linear combination of n particle contributions to the susceptibility of the square lattice ising model in bostan et al 2009 j phys a math theor 42 275209 the operator l24 modulo a prime was shown to factorize into here we prove that no further factorization of the order 12 operator is possible we use the exact ode to obtain the behaviour of at the ferromagnetic critical point and to obtain a limited number of analytic continuations of beyond the principal disc defined by its high temperature series contrary to a speculation in boukraa et al 2008 j phys a math theor 41 455202 we find that is singular at w 1 2 on an infinite number of branches

experimental mathematics on the magnetic susceptibility of the square lattice ising model we calculate very long low and high temperature series for the susceptibility of the square lattice ising model as well as very long series for the five particle contribution 5 and six particle contribution 6 these calculations have been made possible by the use of highly optimized polynomial time modular algorithms and a total of more than 150 000 cpu hours on computer clusters the series for low and high temperature regimes 5 and 6 are now extended to 2000 terms in addition for 5 10 000 terms of the series are calculated modulo a single prime and have been used to find the linear ode satisfied by 5 modulo a prime a diff pade analysis of the 2000 terms series for 5 and 6 confirms to a very high degree of confidence previous conjectures about the location and strength of the singularities of the n particle components of the susceptibility up to a small set of additional singularities the exponents at all the singularities of the fuchsian linear ode of 5 and the as yet unknown ode of 6 are given they are all rational numbers we find the presence of singularities at w 1 2 for the linear ode of 5 and w2 1 8 for the ode of 6 which are not singularities of the physical 5 and 6 that is to say the series solutions of the ode s which are analytic at w 0 furthermore analysis of the long series for 5 and 6 combined with the corresponding long series for the full susceptibility yields previously conjectured singularities in some n n 7 the exponents at all these singularities are also seen to be rational numbers we also present a mechanism of resummation of the logarithmic singularities of the n leading to the known power law critical behaviour occurring in the full and perform a power spectrum analysis giving strong arguments in favour of the existence of a natural boundary for the full susceptibility

counting semiregular permutations which are products of a full cycle and an involution character theoretic methods and the group algebra of the symmetric group are used to derive properties of the number of permutations with only p cycles for an arbitrary but fixed p which are expressible as the product of a full cycle and a fixed point free involution this problem has application to single face embeddings of p regular graphs on surfaces of given genus

invariants of real forms of affine kac moody lie algebras abstract the theory of vogan diagrams which are dynkin diagrams with an overlay of certain additional information allows one to give a rapid classification of finite dimensional real semisimple lie algebras and to make use of this classification in practice this paper develops a corresponding theory of vogan diagrams for almost compact real forms of indecomposable nontwisted affine kac moody lie algebras in this case also the equivalence classes of vogan diagrams correspond to the isomorphism classes of almost compact real forms although the real forms of such algebras had already been classified the theory of vogan diagrams introduces invariants for such algebras and makes it possible to locate a given real form within the classification

monodromy for the hypergeometric function nfn 1 la fonction hypergeometrique le groupe hypergeometrique la forme hermitienne invariante le cas imprimitif theorie de galois differentielle fonctions hypergeometriques algebriques

the fuchsian differential equation of the square lattice ising model χ 3 susceptibility using an expansion method in the variables xi that appear in n 1 dimensional integrals representing the n particle contribution to the ising square lattice model susceptibility we generate a long series of coefficients for the three particle contribution 3 using an n4 polynomial time algorithm we give the fuchsian differential equation of order 7 for 3 that reproduces all the terms of our long series an analysis of the properties of this fuchsian differential equation is performed

infinite flag varieties and conjugacy theorems abstract we study the orbit of a highest weight vector in an integrable highest weight module of the group g associated to a kac moody algebra unk a we obtain applications to the geometric structure of the associated flag varieties and to the algebraic structure of unk a in particular we prove conjugacy theorems for cartan and borel subalgebras of unk a so that the cartan matrix a is an invariant of unk a

canonical decomposition of irreducible linear differential operators with symplectic or orthogonal differential galois groups we first revisit an order six linear differential operator already introduced in a previous paper having a solution which is a diagonal of a rational function of three variables this linear differential operator is such that its exterior square has a rational solution indicating that it has a selected differential galois group and is actually homomorphic to its adjoint we obtain the two corresponding intertwiners giving this homomorphism to the adjoint we show that these intertwiners are also homomorphic to their adjoint and have a simple decomposition already underlined in a previous paper in terms of order two self adjoint operators from these results we deduce a new form of decomposition of operators for this selected order six linear differential operator in terms of three order two self adjoint operators we generalize this decomposition to decomposition in terms of three self adjoint operators of arbitrary orders provided the three orders have the same parity we then generalize the previous decomposition to decompositions in terms of an arbitrary number of self adjoint operators of the same parity order this yields an infinite family of linear differential operators homomorphic to their adjoint and thus with a selected differential galois group we show that the equivalence of such operators with selected differential galois groups is compatible with these canonical decompositions the rational solutions of the symmetric or exterior squares of these selected operators are noticeably seen to depend only on the rightmost self adjoint operator in the decomposition these results and tools are applied on operators of large orders for instance it is seen that a large set of quite massive operators associated with reflexive 4 polytopes defining calabi yau three folds obtained recently by lairez correspond to a particular form of the decomposition detailed in this paper all the results of this paper can be seen as providing an algebraic characterization of linear differential operators with selected symplectic or orthogonal differential galois groups

some explicit solutions of the additive deligne simpson problem and their applications abstract in this paper we construct three infinite series and two extra triples e8 and e 8 of complex matrices b c and a b c of special spectral types associated to simpson s classification in amer math soc proc 1 1992 157 and magyar et al classification in adv math 141 1999 97 this enables us to construct fuchsian systems of differential equations which generalize the hypergeometric equation of gauss riemann in a sense they are the closest relatives of the famous equation because their triples of spectral flags have finitely many orbits for the diagonal action of the general linear group in the space of solutions in all the cases except for e8 we also explicitly construct scalar products such that a b and c are self adjoint with respect to them in the context of fuchsian systems these scalar products become monodromy invariant complex symmetric bilinear forms in the spaces of solutions when the eigenvalues of a b and c are real the matrices and the scalar products become real as well we find inequalities on the eigenvalues of a b and c which make the scalar products positive definite as proved by klyachko spectra of three hermitian or real symmetric matrices b c and a b c form a polyhedral convex cone in the space of triple spectra he also gave a recursive algorithm to generate inequalities describing the cone the inequalities we obtain describe non recursively some faces of the klyachko cone

an on line library of extended high temperature expansions of basic observables for the spin s ising models on two and three dimensional lattices we present an on line library of unprecedented extension for high temperature expansions of basic observables in the ising models of general spin s with nearest neighbor interactions we have tabulated through order 25 the series for the nearest neighbor correlation function the susceptibility and the second correlation moment in two dimensions on the square lattice and in three dimensions on the simple cubic and the body centered cubic lattices the expansion of the second field derivative of the susceptibility is also tabulated through 23 for the same lattices we have thus added several terms from four up to thirteen to the series already published for spin s 1 2 1 3 2 2 5 2 3 7 2 4 5

exact evaluation of the green functions for the anisotropic face centred and simple cubic lattices the analytic properties of the lattice green functions and are investigated where w1 w2 are complex variables and 1 2 are real parameters in the interval 0 in particular simple and direct methods are developed which enable one to evaluate g1 1 w1 and g2 2 w2 in terms of products of two complete elliptic integrals of the first kind kampe de feriet series are also used to derive new transformation formulae which give connections between g1 1 w1 and g2 2 w2

satake diagrams of affine kac moody algebras satake diagrams of affine kac moody algebras untwisted and twisted are obtained from their dynkin diagrams these diagrams give a classification of restricted root systems associated with these algebras in the case of simple lie algebras these root systems and satake diagrams correspond to symmetric spaces which have recently found many physical applications in quantum integrable systems quantum transport problems random matrix theories etc we hope these types of root systems may have similar applications in theoretical physics in future and may correspond to symmetric spaces analogue of affine kac moody algebras if they exist

a differential equation with monodromy group 2 j2 abstract we construct a sixth order differential equation having the central extension of c 2 by the hall janko group j 2 as monodromy group moreover it arises from an iterated application of tensor products and convolution operations from a first order differential equation

the ising model from elliptic curves to modular forms and calabi yau equations we show that almost all the linear differential operators factors obtained in the analysis of the n particle contributions s of the susceptibility of the ising model for n 6 are linear differential operators associated with elliptic curves beyond the simplest differential operators factors which are homomorphic to symmetric powers of the second order operator associated with the complete elliptic integral e the second and third order differential operators z2 f2 f3 can actually be interpreted as modular forms of the elliptic curve of the ising model a last order 4 globally nilpotent linear differential operator is not reducible to this elliptic curve modular form scheme this operator is shown to actually correspond to a natural generalization of this elliptic curve modular form scheme with the emergence of a calabi yau equation corresponding to a selected 4f3 hypergeometric function this hypergeometric function can also be seen as a hadamard product of the complete elliptic integral k with a remarkably simple algebraic pull back square root extension the corresponding calabi yau fourth order differential operator having a symplectic differential galois group the mirror maps and higher order schwarzian odes associated with this calabi yau ode present all the nice physical and mathematical ingredients we had with elliptic curves and modular forms in particular an exact isogenies representation of the generators of the renormalization group extending the modular group to a symmetry group