weakly holomorphic modular forms and rank two hyperbolic kac moody algebras in this paper we compute basis elements of certain spaces of weight 0 weakly holomorphic modular forms and consider the integrality of fourier coefficients of the modular forms we use the results to construct automorphic correction of the rank 2 hyperbolic kac moody algebras h a a 4 5 6 through hilbert modular forms explicitly given by borcherds lifts of the weakly holomorphic modular forms we also compute asymptotics of the fourier coefficients as they are related to root multiplicities of the rank 2 hyperbolic kac moody algebras this work is a continuation of an earlier work of the first and second authors where automorphic correction was constructed for h a a 3 11 66

modularity of relatively rational vertex algebras and fusion rules of principal affine w algebras we study modularity of the characters of a vertex super algebra equipped with a family of conformal structures along the way we introduce the notions of rationality and cofiniteness relative to such a family we apply the results to determine modular transformations of trace functions on admissible modules over affine kac moody algebras and via brst reduction trace functions on minimal series representations of principal affine w algebras

some hecke rogers type identities abstract recently a new partition function associated with ramanujan s third order mock theta function q was discovered and subsequently its overpartition analogue was introduced in this paper we prove an intriguing identity arising from the study of that analogue which involves a double series of hecke rogers type in addition two further identities will be given our proofs heavily rely on formulas from the work of liu 12

on the complete classification of unitary n 2 minimal superconformal field theories aiming at a complete classification of unitary n 2 minimal models where the assumption of space time supersymmetry has been dropped it is shown that each modular invariant candidate partition function of such a theory is indeed the partition function of a fully fledged unitary n 2 minimal model subject to the assumptions that orbifolding is a physical process and that the space time supersymmetric mathcal a mathcal d mathcal e models are physical a family of models constructed via orbifoldings of either the diagonal model or of the space time supersymmetric exceptional models then demonstrates that there exists a unitary n 2 minimal model for every one of the allowed partition functions in the list obtained from gannon s work gannon in nucl phys b 491 659 688 1997

the w n minimal model classification we first rigourously establish for any n 2 that the toroidal modular invariant partition functions for the not necessarily unitary w n p q minimal models biject onto a well defined subset of those of the su n su n wess zumino witten theories at level p n q n this permits considerable simplifications to the proof of the cappelli itzykson zuber classification of virasoro minimal models more important we obtain from this the complete classification of all modular invariants for the w 3 p q minimal models all should be realised by rational conformal field theories previously only those for the unitary models i e w 3 p p 1 were classified for all n our correspondence yields for free an extensive list of w n p q modular invariants the w 3 modular invariants like the virasoro minimal models all factorise into su 3 modular invariants but this fails in general for larger n we also classify the su 3 su 3 modular invariants and find there a new infinite series of exceptionals

simple wzw currents a complete classification of simple currents of wzw theories is obtained the proof is based on an analysis of the quantum dimensions of the primary fields simple currents are precisely the primaries with unit quantum dimension for wzw theories explicit formulae for the quantum dimensions can be derived so that an identification of the fields with unit quantum dimension is possible

spin and wedge representations of infinite dimensional lie algebras and groups abstract we suggest a purely algebraic construction of the spin representation of an infinite dimensional orthogonal lie algebra sections 1 and 2 and a corresponding group section 4 from this we deduce a construction of all level one highest weight representations of orthogonal affine lie algebras in terms of creation and annihilation operators on an infinite dimensional grassmann algebra section 3 we also give a similar construction of the level one representations of the general linear affine lie algebra in an infinite dimensional wedge space along these lines we construct the corresponding representations of the universal central extension of the group sln k t t 1 in spaces of sections of line bundles over infinite dimensional homogeneous spaces section 5

lie group weight multiplicities from conformal field theory dominant weight multiplicities of simple lie groups are expressed in terms of the modular matrices of wess zumino witten conformal field theories and related objects symmetries of the modular matrices give rise to new relations among multiplicities at least for some lie groups these new relations are strong enough to completely fix all multiplicities

teichmuller groupoids and monodromy in conformal field theory we study monodromy representations of the teichmuller groupoid for the moduli space of pointed compact riemann surfaces of any genus with first order infinitesimal structure to calculate these representations using arithmetic schottky mumford uniformization theory we construct a real orbifold in the moduli space consisting of fusing and simple moves which gives tangential base points for a certain vector bundle on the moduli space with projectively flat connection we show that the monodromy of each fusing move can be expressed as a connection matrix and give the relations to the monodromy of simple moves furthermore we describe the monodromy representation associated with tsuchiya ueno yamada s conformal field theory and show that this representation can be expressed as the monodromy of the wess zumino witten model

ruijsenaars commuting difference operators and invariant subspace spanned by theta functions we study a family of mutually commutative difference operators introduced by ruijsenaars the conjugations of these operators with an appropriate function give the hamiltonians of some relativistic quantum systems these operators can be regarded as elliptic analogs of the macdonald operators and their coefficients consist of the jacobi theta functions we show that these operators act on the space of meromorphic functions on the cartan subalgebra of affine lie algebras and that the space spanned by characters of a fixed positive level is invariant under the action of these operators

c 2 rational toroidal conformal field theories via the gauss product we find a concise relation between the moduli of a rational narain lattice and the corresponding momentum lattices of left and right chiral algebras via the gauss product as a byproduct we find an identity which counts the cardinality of a certain double coset space defined for isometries between the discriminant forms of rank two lattices

a chebyshev collocation algorithm for the solution of advection diffusion equations an original method to precondition chebyshev collocated advection diffusion operators is proposed the preconditioner based on an upwind finite difference approximation is tested on one and two dimensional problems the preconditioner behaves efficiently for advection dominated cases and produces rates of convergence independent of the number of nodes finally an ad hoc designed multi domain strategy is introduced

principally specialized characters of sl m 1 modules in this paper we calculate a series of principally specialized characters of the sl m 1 modules of level 1 in particular we show that the principally specialized characters of the basic modules l 0 are expressed as an infinite product in addition we deduce the specialized character formula of quasiparticle type

a commutant of βγ system associated to the highest weight module v4 of sl 2 c analogous to the commutant in the theory of associative algebras one can construct a new subalgebra of a vertex algebra known as the vertex algebra commutant in this paper for the highest weight module v4 of lie algebra sl 2 c we describe a commutant of system s v4 by giving a finite set of generators and operator product expansion relations among generators

quantum r matrix for g2 and a solvable 175 vertex model the quantum r matrix for cartan s exceptional simple lie algebra g2 in its seven dimensional minimal representation is presented the r matrix is determined through the irreducible decomposition of the tensor modules over the quantised universal enveloping algebra uq g2 this defines a new solvable seven state 175 vertex model on the planar square lattice whose boltzmann weights satisfy the yang baxter equation for the equivalent face model the local state probability is obtained in terms of a relative of a level 1 string function related to the affine lie algebra pair b3 1 contains implies g2 1

mathematics underlying the f theory heterotic string duality in eight dimensions we give an analytic description of the moduli space of classical vacua for f theory compactified on elliptic k3 surfaces on open tubular regions near the two type ii boundary divisors the structure of these open sets is related to the total spaces of certain holomorphic theta fibrations over the corresponding boundary divisors as the two type ii divisors can be naturally identified as moduli spaces of elliptic curves and flat g bundles with g e8 e8 or spin 32 one is led to an analytic isomorphism between these open domains and regions of the moduli spaces of heterotic string theory compactified on the two torus corresponding to large volumes of the torus this provides a proof for the classical version of f theory heterotic string duality in eight dimensions a description of the type ii boundary points in terms of elliptic stable k3 surfaces is also given

ward identities and characters of kac moody algebras the authors derive partial differential equations for characters of kac moody algebras by using ward identities for energy momentum tensors and currents on a torus and the null vectors of current algebras the solutions of the partial differential equations for affine algebras a 1 l d 1 l e 1 l l 6 7 8 and b 1 l at level one are given

conformal embeddings rank level duality and exceptional modular invariants we compute the branching rules of the conformal embeddingsso 4nk 1 sp 2n k sp 2k n andso rq 1 so r q so q r forrq even using this we prove that the affine algebrassp 2n k andsp 2k n have the sames matrix and modular invariants as a second application we show how the triality ofso 8 leads to an exceptional modular invariant forsu 2 at level 16 and for allso q 4 at level 8

three manifold invariants from chern simons field theory with arbitrary semi simple gauge groups invariants for framed links in s 3 obtained from chern simons gauge field theory based on an arbitrary gauge group semi simple have been used to construct a three manifold invariant this is a generalization of a similar construction developed earlier for su 2 chern simons theory the procedure exploits a theorem of lickorish and wallace and also those of kirby fenn and rourke which relate three manifolds to surgeries on framed unoriented links the invariant is an appropriate linear combination of framed link invariants which does not change under kirby calculus this combination does not see the relative orientation of the component knots the invariant is related to the partition function of chern simons theory this thus provides an efficient method of evaluating the partition function for these field theories as some examples explicit computations of these manifold invariants for a few three manifolds have been done

on fusion algebra of chiral models we discuss some algebraic setting of chiral models in terms of the statistical dimensions of their fields in particular the conformal dimensions and the central charge of the chiral models are calculated from their braid matrices furthermore at level k 2 we present the characteristic polynomials of their fusion matrices in a factored form

demazure characters and wzw fusion rules the demazure character formula is applied to the verlinde formula for the fusion rules of wess zumino witten wzw models we follow littelmann s derivation of a generalized littlewood richardson rule that computes tensor products for simple lie algebras a combinatorial rule for wzw fusions does not result however only a modified version of the littlewood richardson rule is obtained that computes an old upper bound on the fusion coefficients of ar wzw models we argue that this is because the characters of simple lie algebras appear in this treatment instead of the corresponding affine characters the bruhat order on the affine weyl group must be implicated in any combinatorial rule for wzw fusions the bruhat order on subgroups of this group such as the finite weyl group does not suffice

property t for kac moody groups over rings abstract let r be a finitely generated commutative ring with 1 let a be an indecomposable 2 spherical generalized cartan matrix of size at least 2 and m m a the largest absolute value of a non diagonal entry of a we prove that there exists an integer n n a such that the kac moody group g a r has property t whenever r has no proper ideals of index less than n and all positive integers less than or equal to m are invertible in r

quantumr matrices related to the spin representations ofbn anddn we present the explicit form of the trigonometricr matrices related to the spin representations of the simple lie algebrasx n b n d n we conjecture that one dimensional configuration sums of the corresponding vertex models in the face formulation are the string functions ofx 1 modules

semi infinite weil complex and the virasoro algebra we define a semi infinite analogue of the weil algebra associated an infinite dimensional lie algebra it can be used for the definition of semi infinite characteristic classes by analogy with the chern weil construction the second term of a spectral sequence of this weil complex consists of the semi infinite cohomology of the lie algebra with coefficients in its adjoint semi infinite symmetric powers we compute this cohomology for the virasoro algebra this is just the brst cohomology of the bosonic system with central charge 26 we give a complete description of the fock representations of this bosonic system as modules over the virasoro algebra using friedan martinec shenker bosonization we derive a combinatorial identity from this result

decomposition rules for conformal pairs associated to symmetric spaces and abelian subalgebras of z2 graded lie algebras we give uniform formulas for the branching rules of level 1 modules over orthogonal affine lie algebras for all conformal pairs associated to symmetric spaces we also provide a combinatorial interpretation of these formulas in terms of certain abelian subalgebras of simple lie algebras

chebyshev 3 d spectral and 2 d pseudospectral solvers for the helmholtz equation two chebyshev solvers are presented for the linear helmholtz equation the first algorithm is a 3 d direct spectral solver based on a diagonalization technique whilst the second performs an iterative pseudospectral 2 d calculation with finite difference preconditioning both techniques handle general nonhomogeneous boundary conditions computing times and accuracies of the two methods are compared

quantum mckay correspondence and global dimensions for fusion and module categories associated with lie groups abstract global dimensions for fusion categories a k g defined by a pair g k where g is a lie group and k a positive integer are expressed in terms of lie quantum superfactorial functions the global dimension is defined as the square sum of quantum dimensions of simple objects for the category of integrable modules over an affine lie algebra at some level the same quantities can also be defined from the theory of quantum groups at roots of unity or from conformal field theory wzw models similar results are also presented for those associated module categories that can be obtained via conformal embeddings they are quantum subgroups of a particular kind as a side result we express the classical or quantum weyl denominator of simple lie groups in terms of classical or quantum factorials calculated for the exponents of the group some calculations use the correspondence existing between periodic quivers for simply laced lie groups and fusion rules for module categories associated with a k su 2

and as vertex operator extensions of dual affine algebras we discover a realisation of the affine lie superalgebra and of the exceptional affine superalgebra as vertex operator extensions of two algebras with dual levels and an auxiliary level 1 algebra the duality relation between the levels is we construct the representation of on a sum of tensor products of and modules and decompose it into a direct sum over the spectral flow orbit this decomposition gives rise to character identities which we also derive the extension of the construction to is traced to the properties of embeddings into and their relation with the dual pairs conversely we show how the representations are constructed from representations

virasoro character identities and artin l functions some identities between unitary minimal virasoro characters at levels m 3 4 5 are shown to arise as a consequence of relations between artin l functions of different quadratic fields the definitions and concepts of number theory necessary to present the theta function identities which can be derived from these relations are introduced a new infinite family of identities between virasoro characters at level 3 and level m 4a 2 for a odd and 1 4a 2 a 2 p where p is prime is obtained as a by product

characters for from a novel thermodynamic bethe ansatz motivated by the recent development on the exact thermodynamics of one dimensional quantum systems we propose quasiparticle like formulae for characters the case is re examined first the novel formulation yields a direct connection to the fractional statistics in the short range interacting model and provides a clear description of the spinon character formula generalizing the observation we find formulae for which can be proved by the durfee rectangle formula

meromorphic c 24 conformal field theories modular invariant conformal field theories with just one primary field and central charge c 24 are considered it has been shown previously that if the chiral algebra of such a theory contains spin 1 currents it is either the leech lattice cft or it contains a kac moody sub algebra with total central charge 24 in this paper all meromorphic modular invariant combinations of the allowed kac moody combinations are obtained the result suggests the existence of 71 meromorphic c 24 theories including the 41 that were already known

coulomb gas formulation of su 2 branes and chiral blocks we construct boundary states in su 2 k wznw models using the bosonized wakimoto free field representation and study their properties we introduce a fock space representation of ishibashi states which are coherent states of bosons with zero mode momenta boundary coulomb gas charges summed over certain lattices according to fock space resolution of su 2 k the virasoro invariance of the coherent states leads to families of boundary states including the b type d branes found by maldacena moore and seiberg as well as the a type corresponding to trivial current gluing conditions we then use the coulomb gas technique to compute exact correlation functions of wznw primary fields on the disk topology with a and b type cardy states on the boundary we check that the obtained chiral blocks for a branes are solutions of the knizhnik zamolodchikov equations

adjoint affine fusion and tadpoles we study affine fusion with the adjoint representation for simple lie algebras elementary and universal formulas determine the decomposition of a tensor product of an integrable highest weight representation with the adjoint representation using the refined affine depth rule we prove that equally striking results apply to adjoint affine fusion for diagonal fusion a coefficient equals the number of nonzero dynkin labels of the relevant affine highest weight minus 1 a nice lattice polytope interpretation follows and allows the straightforward calculation of the genus 1 1 point adjoint verlinde dimension the adjoint affine fusion tadpole explicit formulas piecewise polynomial in the level are written for the adjoint tadpoles of all classical lie algebras we show that off diagonal adjoint affine fusion is obtained from the corresponding tensor product by simply dropping non dominant representations

nonabelian orbifolds and the boson fermion correspondence for a positive integerl divisible by 8 there is a bosonic holomorphic vertex operator algebra voa v gamma l associated to the spin lattice l for a broad class of finite groupsg of automorphisms of v gamma l we prove the existence and uniqueness of irreducibleg twisted v gamma l modules and establish the modular invariance of the partition functionsz g h for commuting elements ing in particular for any finite group there are infinitely many holomorphic voas admittingg for which these properties hold the proof is facilitated by a boson fermion correspondence which gives a voa isomorphism between v gamma l and a certain fermionic construction and which extends work of frenkel and others

free field construction for the abf models in regime ii the wakimoto construction for the quantum affine algebra u q widehat mathfrak s mathfrak l 2 admits a reduction to the q deformed parafermion algebras we interpret the latter theory as a free field realization of the andrews baxter forrester models in regime ii we give multi particle form factors of some local operators on the lattice and compute their scaling limit where the models are described by a massive field theory with mathbb z k symmetric minimal scattering matrices

a spectral element method for fluid dynamics laminar flow in a channel expansion a spectral element method that combines the generality of the finite element method with the accuracy of spectral techniques is proposed for the numerical solution of the incompressible navier stokes equations in the spectral element discretization the computational domain is broken into a series of elements and the velocity in each element is represented as a high order lagrangian interpolant through chebyshev collocation points the hyperbolic piece of the governing equations is then treated with an explicit collocation scheme while the pressure and viscous contributions are treated implicitly with a projection operator derived from a variational principle the implementation of the technique is demonstrated on a one dimensional inflow outflow advection diffusion equation and the method is then applied to laminar two dimensional separated flow in a channel expansion comparisons are made with experiment and previous numerical work

a simulation based evolutionary approach to lna circuit design optimization abstract in this paper we propose a simulation based evolutionary approach for designing low noise amplifier lna integrated circuits ics based on a genetic algorithm ga the levenberg marquardt lm method and a circuit simulator the simulation based evolutionary approach is developed for design optimization of lna circuits for a given lna circuit the simulation based evolutionary approach simultaneously optimizes the electrical specifications such as s 11 s 12 s 21 s 22 k factor the noise figure and the input third order intercept point in the process between simulation and optimization first of all the necessary parameters of the lna circuit for circuit simulation are loaded by solving a set of nonlinear ordinary differential equations the circuit simulator will then be performed for the circuit simulation and specification evaluation once the specification meets the aforementioned seven constraints we output the optimized parameters otherwise we activate the ga for the global optimization in the meanwhile the lm method searches the local optima according to the results of the ga we then call circuit simulator to compute and evaluate newer results until the specification is matched in numerical experiment 10 parameters of the lna circuit including device configuration and biasing condition are optimized with respect to the constraints the design of lna circuit is with 0 18 m metal oxide silicon filed effect transistors benchmark results also computationally confirm the robustness and efficiency of the proposed method this simulation based evolutionary approach in general can be applied to optimal design of other analog and radio frequency circuits we believe that this systematical approach will help ic design optimization and benefit computer aided design of wireless communication system on a chip

the semi infinite cohomology of affine lie algebras we study the semi infinite or brst cohomology of affine lie algebras in detail this cohomology is relevant in the brst approach to gauged wznw models our main result is to prove necessary and sufficient conditions on ghost numbers and weights for non trivial elements in the cohomology in particular we prove the existence of an infinite sequence of elements in the cohomology for non zero ghost numbers this will imply that the brst approach to the topological wznw model admits many more states than a conventional coset construction this conclusion also applies to some non topological models

boundary conditions in chebyshev and legendre methods we discuss two different ways of treating non dirichlet boundary conditions in chebyshev and legendre collocation methods for second order differential problems an error analysis is provided the effect of preconditioning the corresponding spectral operators by finite difference matrices is also investigated

cartan preserving automorphisms of untwisted and twisted kac moody algebras the group n defined as the normalizer of the cartan subalgebra in the group of all inner and outer automorphisms of affine kac moody km algebras is a natural extension of the weyl group and is shown to play a fundamental role in the structure of these algebras both untwisted and twisted it is a discrete galilean group which incorporates the affine and weyl group structure of the km algebra and the space time structure of the string it links the virasoro and km algebras in a nontrivial way and plays a key role in the vertex construction moreover the elements of its covering group can be used to uniquely parametrize and untwist the pseudotwisted km algebras

representations of affine kac moody algebras and the affine scalar product simple procedures are given for finding all the dominant weights in a highest weight representation of an affine algebra for finding the weyl orbit of an arbitrary weight and for determining whether or not any given weight is in any given representation a simple definition of congruency is given that applies to all affine algebras the standard indefinite scalar product is generalized the generalization is used in the procedures

some characters of kac and wakimoto and nonholomorphic modular functions answering a question of kac we relate the character formulas for certain sl m 1 modules to automorphic forms we show that these q series are the holomorphic parts of nonholomorphic modular functions

iwahori hecke algebras for p adic loop groups this paper is a continuation of braverman and kazhdan ann math 2 174 3 1603 1642 2011 in which the first two authors have introduced the spherical hecke algebra and the satake isomorphism for an untwisted affine kac moody group over a non archimedian local field in this paper we develop the theory of the iwahori hecke algebra associated to these same groups the resulting algebra is shown to be closely related to cherednik s double affine hecke algebra furthermore using these results we give an explicit description of the affine satake isomorphism generalizing macdonald s formula for the spherical function in the finite dimensional case the results of this paper have been previously announced in braverman and kazhdan european congress of mathematics european mathematical society zurich 2014

chaotic eigenfunctions in phase space we study individual eigenstates of quantized area preserving maps on the 2 torus which are classically chaotic in order to analyze their semiclassical behavior we use the bargmann husimi representations for quantum states as well as their stellar parametrization which encodes states through a minimal set of points in phase space the constellation of zeros of the husimi density we rigorously prove that a semiclassical uniform distribution of husimi densities on the torus entails a similar equidistribution for the corresponding constellations we deduce from this property a universal behavior for the phase patterns of chaotic bargmann eigenfunctions which is reminiscent of the wkb approximation for eigenstates of integrable systems though in a weaker sense in order to obtain more precise information on chaotic eigenconstellations we then model their properties by ensembles of random states generalizing former results on the 2 sphere to the torus geometry this approach yields statistical predictions for the constellations which fit quite well the chaotic data we finally observe that specific dynamical information e g the presence of high peaks like scars in husimi densities can be recovered from the knowledge of a few long wavelength fourier coefficients which therefore appear as valuable order parameters at the level of individual chaotic eigenfunctions