general β jacobi corners process and the gaussian free field we prove that the two dimensional gaussian free field describes the asymptotics of global fluctuations of a multilevel extension of the general jacobi random matrix ensembles our approach is based on the connection of the jacobi ensembles to a degeneration of the macdonald processes that parallels the degeneration of the macdonald polynomials to the heckman opdam hypergeometric functions of type a we also discuss the limit 2015 wiley periodicals inc

twisted modules and co invariants for commutative vertex algebras of jet schemes abstract let z a k be an affine scheme over c and j z its jet scheme it is well known that c j z the coordinate ring of j z has the structure of a commutative vertex algebra this paper develops the orbifold theory for c j z a finite order linear automorphism g of z acts by vertex algebra automorphisms on c j z we show that c j g z where j g z is the scheme of g twisted jets has the structure of a g twisted c j z module we consider spaces of orbifold coinvariants valued in the modules c j g z on orbicurves y g with y a smooth projective curve and g a finite group and show that these are isomorphic to c z g

bps jumping loci are automorphic we show that bps jumping loci loci in the moduli space of string compactifications where the number of bps states jumps in an upper semi continuous manner naturally appear as fourier coefficients of vector space valued automorphic forms for the case of t2 compactification the jumping loci are governed by a modular form studied by hirzebruch and zagier while the jumping loci in k3 compactification appear in a story developed by oda and kudla millson in arithmetic geometry we also comment on some curious related automorphy in the physics of black hole attractors and flux vacua

on classification of modular tensor categories we classify all unitary modular tensor categories umtcs of rank 4 there are a total of 35 umtcs of rank 4 up to ribbon tensor equivalence since the distinction between the modular s matrix s and s has both topological and physical significance so in our convention there are a total of 70 umtcs of rank 4 in particular there are two trivial umtcs with s 1 each such umtc can be obtained from 10 non trivial prime umtcs by direct product and some symmetry operations explicit data of the 10 non trivial prime umtcs are given in sect 5 relevance of umtcs to topological quantum computation and various conjectures are given in sect 6

congruence subgroups and generalized frobenius schur indicators we introduce generalized frobenius schur indicators for pivotal categories in a spherical fusion category mathcal c an equivariant indicator of an object in mathcal c is defined as a functional on the grothendieck algebra of the quantum double z mathcal c via generalized frobenius schur indicators the set of all equivariant indicators admits a natural action of the modular group using the properties of equivariant indicators we prove a congruence subgroup theorem for modular categories as a consequence all modular representations of a modular category have finite images and they satisfy a conjecture of eholzer in addition we obtain two formulae for the generalized indicators one of them a generalization of bantay s second indicator formula for a rational conformal field theory this formula implies a conjecture of pradisi sagnotti stanev as well as a conjecture of borisov halpern schweigert

c 2 cofiniteness of the 2 cycle permutation orbifold models of minimal virasoro vertex operator algebras in this article we give a sufficient and necessary condition for the c 2 cofiniteness of widetilde v v otimes v sigma for a c 2 cofinite vertex operator algebra v and the 2 cycle permutation of v otimes v as an application we show that the 2 cycle permutation orbifold model of the simple virasoro vertex operator algebra l c 0 of minimal central charge c is c 2 cofinite

height fluctuations in the honeycomb dimer model we study a model of random surfaces arising in the dimer model on the honeycomb lattice for a fixed wire frame boundary condition as the lattice spacing 0 cohn kenyon and propp 3 showed the almost sure convergence of a random surface to a non random limit shape 0 in 12 okounkov and the author showed how to parametrize the limit shapes in terms of analytic functions in particular constructing a natural conformal structure on them we show here that when 0 has no facets for a family of boundary conditions approximating the wire frame the large scale surface fluctuations height fluctuations about 0 converge as 0 to a gaussian free field for the above conformal structure we also show that the local statistics of the fluctuations near a given point x are as conjectured in 3 given by the unique ergodic gibbs measure on plane configurations whose slope is the slope of the tangent plane of 0 at x

on the computation of local components of a newform the problem let f be a cuspidal newform for 1 n with weight k 2 and character e there are well established methods for computing such forms using modular symbols see ste07 let f be the corresponding automorphic representation of the adele group gl2 aq

galois representations and torsion in the coherent cohomology of hilbert modular varieties let f be a totally real number eld and let p be a prime unramied in f we prove the existence of galois pseudo representations attached to modp m hecke eigenclasses of paritious weight occurring in the coherent cohomology of hilbert modular varieties forf of level prime to p

central limit theorems for biorthogonal ensembles and asymptotics of recurrence coefficients we study fluctuations of linear statistics corresponding to smooth functions for certain biorthogonal ensembles we study those biorthogonal ensembles for which the underlying biorthogonal family s

scale invariance of the png droplet and the airy process we establish that the static height fluctuations of a particular growth model the png droplet converges upon proper rescaling to a limit process which we call the airy process a y the airy process is stationary it has continuous sample paths its single time fixed y distribution is the tracy widom distribution of the largest eigenvalue of a gue random matrix and the airy process has a slow decay of correlations as y 2 roughly the airy process describes the last line of dyson s brownian motion model for random matrices our construction uses a multi layer version of the png model which can be analyzed through fermionic techniques specializing our result to a fixed value of y one reobtains the celebrated result of baik deift and johansson on the length of the longest increasing subsequence of a random permutation

a periodic hexagon tiling model and non hermitian orthogonal polynomials we study a one parameter family of probability measures on lozenge tilings of large regular hexagons that interpolates between the uniform measure on all possible tilings and a particular fully frozen tiling the description of the asymptotic behavior can be separated into two regimes the low and the high temperature regime our main results are the computations of the disordered regions in both regimes and the limiting densities of the different lozenges there for low temperatures the disordered region consists of two disjoint ellipses in the high temperature regime the two ellipses merge into a single simply connected region at the transition from the low to the high temperature a tacnode appears the key to our asymptotic study is a recent approach introduced by duits and kuijlaars providing a double integral representation for the correlation kernel one of the factors in the integrand is the christoffel darboux kernel associated to polynomials that satisfy non hermitian orthogonality relations with respect to a complex valued weight on a contour in the complex plane we compute the asymptotic behavior of these orthogonal polynomials and the christoffel darboux kernel by means of a riemann hilbert analysis after substituting the resulting asymptotic formulas into the double integral we prove our main results by classical steepest descent arguments

the pearcey process the extended airy kernel describes the space time correlation functions for the airy process which is the limiting process for a polynuclear growth model the airy functions themselves are given by integrals in which the exponents have a cubic singularity arising from the coalescence of two saddle points in an asymptotic analysis pearcey functions are given by integrals in which the exponents have a quartic singularity arising from the coalescence of three saddle points a corresponding pearcey kernel appears in a random matrix model and a brownian motion model for a fixed time this paper derives an extended pearcey kernel by scaling the brownian motion model at several times and a system of partial differential equations whose solution determines associated distribution functions we expect there to be a limiting nonstationary process consisting of infinitely many paths which we call the pearcey process whose space time correlation functions are expressible in terms of this extended kernel

elliptic genera of singular varieties the notions of orbifold elliptic genus and elliptic genus of singular varieties are introduced and the relation between them is studied the elliptic genus of singular varieties is given in terms of a resolution of singularities and extends the elliptic genus of calabi yau hypersurfaces in fano gorenstein toric varieties introduced earlier the orbifold elliptic genus is given in terms of the fixed point sets of the action we show that the generating function for the orbifold elliptic genus ellorb x 6n p for symmetric groups 6n acting on n fold products coincides with the one proposed by r dijkgraaf g moore e verlinde and h verlinde the two notions of elliptic genera are conjectured to coincide

heisenberg algebra and hilbert schemes of points on projective surfaces the purpose of this paper is to throw a bridge between two seemingly unrelated subjects one is the hilbert scheme of points on projective surfaces which has been intensively studied by various people see e g i br es g61 g62 the other is the infinite dimensional heisenberg algebra which is closely related to affine lie algebras see e g k we shall construct a representation of the heisenberg algebra on the homology group of the hilbert scheme in other words the homology group will become a fock space the basic idea is to introduce certain correspondences in the product of the hilbert scheme then they define operators on the homology group by a well known procedure they give generators of the heisenberg algebra and the only thing we must check is that they satisfy the defining relation here we remark that the components of the hilbert scheme are parameterized by numbers of points and our representation will be constructed on the direct sum of homology groups of all components our correspondences live in the product of the different components thus it is quite essential to study all components together our construction has the same spirit as the author s construction nal na4 of representations of affine lie algebras on homology groups of moduli spaces of instantons 1 on ale spaces which are minimal resolutions of simple singularities certain correspondences called hecke correspondences were used to define operators these twist instantons along curves irreducible components of the exceptional set while ours twist ideals around points in fact the hilbert scheme of points can be considered as the moduli space of rank 1 vector bundles or more precisely torsion free sheaves our construction should be considered as a first step to extend nal na4 to general 4 manifolds the same program was also proposed by ginzburg kapranov and vasserot gkv

elliptic genera of symmetric products and second quantized strings in this note we prove an identity that equates the elliptic genus partition function of a supersymmetric sigma model on the n fold symmetric product m n sn of a manifold m to the partition function of a second quantized string theory on the space m s 1 the generating function of these elliptic genera is shown to be almost an automorphic form for o 3 2 z in the context of d brane dynamics this result gives a precise computation of the free energy of a gas of d strings inside a higher dimensional brane

random skew plane partitions and the pearcey process we study random skew 3d partitions weighted by qvol and specifically the q 1 asymptotics of local correlations near various points of the limit shape we obtain sine kernel asymptotics for correlations in the bulk of the disordered region airy kernel asymptotics near a general point of the frozen boundary and a pearcey kernel asymptotics near a cusp of the frozen boundary

on serre s conjecture for mod l galois representations over totally real fields in 1987 serre conjectured that any mod two dimensional irre ducible odd representation of the absolute galois group of the rationals came from a modular form in a precise way we present a generalisation of this conjecture to 2 dimensional representations of the absolute galois group of a totally real eld where is unramied the hard work is in formulating an analogue of the weight part of serre s conjecture serre furthermore asked whether his conjecture could be rephrased in terms of a mod langlands philosophy using ideas of emerton and vign eras we formulate a mod local global principle for the group d where d is a quaternion algebra over a totally real eld split above and at 0 or 1 innite places and show how it implies the conjecture

s n orbifolds and string interactions we study interacting features of s n orbifold cfts concentrating on characters associated with s n orbifold primaries we first formulate a novel procedure for evaluating them through tracing the result is a polynomial formula which we show gives results equivalent to those found by bantay from this we deduce a hierarchy of commuting hamiltonians featuring locality in the induced space and nonlinear string type interactions

siegel modular forms of genus 2 with the simplest divisor we prove that there exist exactly eight siegel modular forms with respect to the congruence subgroups of hecke type of the paramodular groups of genus 2 vanishing precisely along the diagonal of the siegel upper half plane this is a solution of a question formulated during the conference black holes black rings and modular forms ens paris august 2007 these modular forms generalize the classical igusa form and the forms constructed by gritsenko and nikulin in 1998

gaussian free field in an interlacing particle system with two jump rates we study the fluctuations of a random surface in a stochastic growth model on a system of interlacing particles placed on a two dimensional lattice there are two different types of particles one with a low jump rate and the other with a high jump rate in the large time limit the random surface has a deterministic shape due to the different jump rates the limit shape and the domain on which it is defined are not smooth the main result is that the fluctuations of the random surface are governed by the gaussian free field 2012 wiley periodicals inc

2 permutations of lattice vertex operator algebras higher rank abstract this paper is a continuation of our investigation on 2 permutation of lattice vertex operator algebras in particular the quantum dimensions of irreducible modules and the fusion rules are determined if the rank of the lattice is one these results have been obtained previously in 27

elliptic genera of toric varieties and applications to mirror symmetry the paper contains a proof that elliptic genus of a calabi yau manifold is a jacobi form finds in which dimensions the elliptic genus is determined by the hodge numbers and shows that elliptic genera of a calabi yau hypersurface in a toric variety and its mirror coincide up to sign the proof of the mirror property is based on the extension of elliptic genus to calabi yau hypersurfaces in toric varieties with gorenstein singularities

on the central charge of a factorizable hopf algebra abstract for a semisimple factorizable hopf algebra over a field of characteristic zero we show that the value that an integral takes on the inverse drinfel d element differs from the value that it takes on the drinfel d element itself by at most a fourth root of unity this can be reformulated by saying that the central charge of the hopf algebra is an integer if the dimension of the hopf algebra is odd we show that these two values differ by at most a sign which can be reformulated by saying that the central charge is even we give a precise condition on the dimension that determines whether the plus sign or the minus sign occurs to formulate our results we use the language of modular data

algebraic cycles on a product of two hilbert modular surfaces in this paper we prove the tate conjecture for a product of two hilbert modular surfaces for non cm submotives

genus two partition functions and renyi entropies of large c conformal field theories we compute genus two partition functions in two dimensional conformal field theories at large central charge focusing on surfaces that give the third renyi entropy of two intervals we compute this for generalized free theories and for symmetric orbifolds and compare it to the result in pure gravity we find a new phase transition if the theory contains a light operator of dimension 0 19 this means in particular that unlike the second rnyi entropy the third one is no longer universal

the kernel of the modular representation and the galois action in rcft it is shown that for the modular representations associated to rational conformal field theories the kernel is a congruence subgroup whose level equals the order of the dehn twist an explicit algebraic characterization of the kernel is given it is also shown that the order of the dehn twist is bounded by a function of the number of primary fields allowing for a systematic enumeration of the modular representations coming from rcfts restrictions on the spectrum of the dehn twist and arithmetic properties of modular matrix elements are presented

siegel automorphic form corrections of some lorentzian kac moody lie algebras we find automorphic form corrections which are generalized lorentzian kac moody superalgebras without odd real simple roots for two elliptic lorentzian kac moody algebras of rank 3 with a lattice weyl vector and calculate multiplicities of their simple and arbitrary roots these kac moody algebras are defined by hyperbolic symmetrized generalized cartan matrices inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i of rank 3 both these algebras have elliptic type i e their weyl groups have fundamental polyhedra of finite volume in corresponding hyperbolic spaces and have a lattice weyl vector the correcting automorphic forms are siegel modular forms the form corresponding to g 1 is the classical siegel cusp form of weight 5 which is the product of ten even theta constants in particular we find an infinite product formula for this modular form

a remark on the c 2 cofiniteness condition on vertex algebras we show that a finitely strongly generated non negatively graded vertex algebra is c 2 cofinite if and only if it is lisse in the sense of beilinson et al preprint this shows that the c 2 cofiniteness is indeed a natural finiteness condition

the hodge elliptic genus spinning bps states and black holes we perform a refined count of bps states in the compactification of m theory on k3 times t 2 keeping track of the information provided by both the su 2 l and su 2 r angular momenta in the so 4 little group mathematically this four variable counting function may be expressed via the motivic donaldson thomas counts of k3 times t 2 simultaneously refining katz klemm and pandharipande s motivic stable pairs counts on k3 and oberdieck pandharipande s gromov witten counts on k3 times t 2 this provides the first full answer for motivic curve counts of a compact calabi yau threefold along the way we develop a hodge elliptic genus for calabi yau manifolds a new counting function for bps states that interpolates between the hodge polynomial and the elliptic genus of a calabi yau

second quantized frobenius algebras we show that given a frobenius algebra there is a unique notion of its second quantization which is the sum over all symmetric group quotients of nth tensor powers where the quotients are given by symmetric group twisted frobenius algebras to this end we consider the setting of frobenius algebras given by functors from geometric categories whose objects are endowed with geometric group actions and prove structural results which in turn yield a constructive realization in the case of nth tensor powers and the natural permutation action we also show that naturally graded symmetric group twisted frobenius algebras have a unique algebra structure already determined by their underlying additive data together with a choice of super grading furthermore we discuss several notions of discrete torsion and show that indeed a non trivial discrete torsion leads to a non trivial super structure on the second quantization

fc sets and twisters the basics of orbifold deconstruction we present a detailed account of the properties of text twister s and their generalizations text fc set s which are essential ingredients of the orbifold deconstruction procedure aimed at recognizing whether a given conformal model may be obtained as an orbifold of another one and if so to identify the twist group and the original model the close analogy with the character theory of finite groups is discussed and its origin explained

level spacing of random matrices in an external source in an earlier work we had considered a gaussian ensemble of random matrices in the presence of a given external matrix source the measure is no longer unitary invariant and the usual techniques based on orthogonal polynomials or on the coulomb gas representation are not available nevertheless the n point correlation functions are still given in terms of the determinant of a kernel known through an explicit integral representation this kernel is no longer symmetric though and is not readily accessible to standard methods in particular finding the level spacing probability is always a delicate problem in fredholm theory and we have to reconsider the problem within our model we find a new class of universality for the level spacing distribution when the spectrum of the source is ajusted to produce a vanishing gap in the density of the state the problem is solved through coupled non linear differential equations which turn out to form a hamiltonian system as a result we find that the level spacing probability p s behaves like exp c s 8 over 3 for large spacing s this is consistent with the asymptotic behavior exp c s 2 beta 2 whenever the density of state behaves near the edge as rho lambda sim lambda beta

modular data for the extended haagerup subfactor we compute the modular data that is the s and t matrices for the centre of the extended haagerup subfactor bmps12 the full structure i e the associativity data also known as 6 j symbols or f matrices still appears to be inaccessible nevertheless starting with just the number of simple objects and their dimensions obtained by a combinatorial argument in mw14 we find that it is surprisingly easy to leverage knowledge of the representation theory of sl 2 mathbb z into a complete description of the modular data we also investigate the possible character vectors associated with this modular data this is the published version of arxiv 1606 07165

freely generated vertex algebras and non linear lie conformal algebras we introduce the notion of a non linear lie conformal superalgebra and prove a pbw theorem for its universal enveloping vertex algebra we also show that conversely any graded freely generated vertex algebra is the universal enveloping algebra of a unique up to isomorphism non linear lie conformal superalgebra this correspondence will be applied in the subsequent work to the problem of classification of finitely generated simple graded vertex algebras

large n limit of gaussian random matrices with external source part i we continue the study of the hermitian random matrix ensemble with external source open image in new window where a has two distinct eigenvalues a of equal multiplicity this model exhibits a phase transition for the value a 1 since the eigenvalues of m accumulate on two intervals for a 1 and on one interval for 0 1 was treated in part i where it was proved that local eigenvalue correlations have the universal limiting behavior which is known for unitarily invariant random matrices that is limiting eigenvalue correlations are expressed in terms of the sine kernel in the bulk of the spectrum and in terms of the airy kernel at the edge in this paper we establish the same results for the case 0 a 1 as in part i we apply the deift zhou steepest descent analysis to a 3 3 matrix riemann hilbert problem due to the different structure of an underlying riemann surface the analysis includes an additional step involving a global opening of lenses which is a new phenomenon in the steepest descent analysis of riemann hilbert problems

dominos and the gaussian free field we define a scaling limit of the height function on the domino tiling model dimer model on simply connected regions in z 2 and show that it is the massless free field a gaussian process with independent coefficients when expanded in the eigenbasis of the laplacian

limit shapes and the complex burgers equation in this paper we study surfaces in r 3 that arise as limit shapes in random surface models related to planar dimers these limit shapes are surface tension minimizers that is they minimize a functional of the form h dx dy among all lipschitz functions h taking given values on the boundary of the domain the surface tension has singularities and is not strictly convex which leads to formation of facets and edges in the limit shapes we find a change of variables that reduces the euler lagrange equation for the variational problem to the complex inviscid burgers equation complex hopf equation the equation can thus be solved in terms of an arbitrary holomorphic function which is somewhat similar in spirit to weierstrass parametrization of minimal surfaces we further show that for a natural dense set of boundary conditions the holomorphic function in question is in fact algebraic the tools of algebraic geometry can thus be brought in to study the minimizers and especially the formation of their singularities this is illustrated by several explicitly computed examples

on the hilbert eigenvariety at exotic and cm classical weight 1 points let f be a totally real number field and let f be a classical cuspidal p regular hilbert modular eigenform over f of parallel weight 1 let x be the point on the p adic hilbert eigenvariety mathcal e corresponding to an ordinary p stabilization of f we show that if the p adic schanuel conjecture is true then mathcal e is smooth at x if f has cm if we additionally assume that f mathbb q is galois we show that the weight map is etale at x if f has either cm or exotic projective image which is the case for almost all cuspidal hilbert modular eigenforms of parallel weight 1 we prove these results by showing that the completed local ring of the eigenvariety at x is isomorphic to a universal nearly ordinary galois deformation ring

galois symmetry induced by hecke relations in rational conformal field theory and associated modular tensor categories hecke operators relate characters of rational conformal field theories rcfts with different central charges and extend the previously studied galois symmetry of modular representations and fusion algebras we show that the conductor n of a rcft and the quadratic residues mod n play an important role in the computation and classification of galois permutations we establish a field correspondence in different theories through the picture of effective central charge which combines galois inner automorphisms and the structure of simple currents we then make a first attempt to extend hecke operators to the full data of modular tensor categories the galois symmetry encountered in the modular data transforms the fusion and the braiding matrices as well and yields isomorphic structures in theories related by hecke operators

mckay correspondence for elliptic genera we establish a correspondence between orbifold and singular elliptic genera of a global quotient while the former is defined in terms of the fixed point set of the action the latter is defined in terms of the resolution of singularities as a byproduct the second quantization formula of dijkgraaf moore verlinde and verlinde is extended to arbitrary kawamata log terminal pairs

orbifold genera product formulas and power operations we generalize the definition of orbifold elliptic genus and introduce orbifold genera of chromatic level h using h tuples rather than pairs of commuting elements we show that our genera are in fact orbifold invariants and we prove integrality results for them if the genus arises from an h8 map into the morava lubin tate theory eh then we give a formula expressing the orbifold genus of the symmetric powers of a stably almost complex manifold m in terms of the genus of m itself our formula is the p typical analogue of the dijkgraaf moore verlinde verlinde formula for the orbifold elliptic genus r dijkgraaf et al elliptic genera of symmetric products and second quantized strings comm math phys 185 1 1997 197 209 it depends only on h and not on the genus

higher level appell functions modular transformations and characters we study modular transformation properties of a class of indefinite theta series involved in characters of infinite dimensional lie superalgebras the level l appell functions open image in new window satisfy open quasiperiodicity relations with additive theta function terms emerging in translating by the period generalizing the well known interpretation of theta functions as sections of line bundles the open image in new window function enters the construction of a section of a rank l 1 bundle open image in new window we evaluate modular transformations of the open image in new window functions and construct the action of an sl 2 subgroup that leaves the section of open image in new window constructed from open image in new window invariant

fluctuations of particle systems determined by schur generating functions abstract we develop a new toolbox for the analysis of the global behavior of stochastic discrete particle systems we introduce and study the notion of the schur generating function of a random discrete configuration our main result provides a central limit theorem clt for such a configuration given certain conditions on the schur generating function as applications of this approach we prove clt s for several probabilistic models coming from asymptotic representation theory and statistical physics including random lozenge and domino tilings non intersecting random walks decompositions of tensor products of representations of unitary groups

on simplicity of vacuum modules we find necessary and sufficient conditions of irreducibility of vacuum modules over affine lie algebras and superalgebras from this we derive conditions of simplicity of minimal w algebras moreover in the case of the virasoro and neveu schwarz algebras we obtain explicit formulas for the vacuum determinants

on galois representations associated to hilbert modular forms in this paper we prove that to any hilbert cuspidal eigenform one may attach a compatible system of galois representations this result extends the analogous results of deligne and deligne serre for elliptic modular forms the principal work on this conjecture was carried out by carayol and taylor but their results left one case remaining which we complete in this paper we also investigate the compatibility of our results with the local langlands correspondence and prove that whenever the local component of the automorphic representation is not special then the results coincide

generalized moonshine ii borcherds products the goal of this paper is to construct infinite dimensional lie algebras by using infinite product identities and to use these lie algebras to reduce the generalized moonshine conjecture to a pair of hypotheses about group actions on vertex algebras and lie algebras we expect the lie algebras that we construct to manifest as algebras of physical states in an orbifold conformal field theory yet to be fully constructed with symmetries given by the monster simple group we introduce vector valued modular functions attached to families of modular functions of different levels and we prove infinite product identities for a distinguished class of automorphic functions on a product of two half planes we recast this result using the borcherds harvey moore singular theta lift and show that the vector valued functions attached to completely replicable modular functions with integer coefficients lift to automorphic functions with infinite product expansions at all cusps for each element of the monster simple group we construct an infinite dimensional lie algebra such that its denominator formula is an infinite product expansion of the automorphic function arising from that element s mckay thompson series these lie algebras have the unusual property that their simple roots and all root multiplicities are known we show that under certain hypotheses characters of groups acting on these lie algebras form functions on the upper half plane that are either constant or invariant under a genus zero congruence group

universal singularity at the closure of a gap in a random matrix theory we consider a hamiltonian h h 0 v in which h 0 is a given non random hermitian matrix and v is an n times n hermitian random matrix with a gaussian probability distribution we had shown before that dyson s universality of the short range correlations between energy levels holds at generic points of the spectrum independently of h 0 we consider here the case in which the spectrum of h 0 is such that there is a gap in the average density of eigenvalues of h which is thus split into two pieces when the spectrum of h 0 is tuned so that the gap closes a new class of universality appears for the energy correlations in the vicinity of this singular point

on the elliptic genus of generalised kummer varieties borisov and libgober 2 recently proved a conjecture of dijkgraaf moore verlinde and verlinde see 6 on the elliptic genus of a hilbert scheme of points on a surface we show how their result can be used together with our work on complex genera of generalised kummer varieties 17 to deduce the following formula conjectured by kawai and yoshioka 15 on the elliptic genus of a generalised kummer variety a n of dimension 2 n 1 here is the weak jacobi form of weight 1 and index and v n is the hecke operator sending jacobi forms of index r to jacobi forms of index nr see 7