from jack polynomials to minimal model spectra in this note a deep connection between free field realizations of conformal field theories and symmetric polynomials is presented we give a brief introduction into the necessary prerequisites of both free field realizations and symmetric polynomials in particular jack symmetric polynomials then we combine these two fields to classify the irreducible representations of the minimal model vertex operator algebras as an illuminating example of the power of these methods while these results on the representation theory of the minimal models are all known this note exploits the full power of jack polynomials to present significant simplifications of the original proofs in the literature

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

the kazhdan lusztig conjecture for w algebras the main result in this paper is the character formula for arbitrary irreducible highest weight modules of w algebras the key ingredient is the functor provided by quantum hamiltonian reduction which constructs the w algebras from affine kac moody km algebras and in a similar fashion w modules from km modules assuming certain properties of this functor the w characters are subsequently derived from the kazhdan lusztig conjecture for km algebras the result can be formulated in terms of a double coset of the weyl group of the km algebra the hasse diagrams give the embedding diagrams of the verma modules and the kazhdan lusztig polynomials give the multiplicities in the characters

modular invariant representations of infinite dimensional lie algebras and superalgebras abstract in this paper we launch a program to describe and classify modular invariant representations of infinite dimensional lie algebras and superalgebras we prove a character formula for a large class of highest weight representations l of a kac moody algebra unk with a symmetrizable cartan matrix generalizing the weyl kac character formula kac v g 1974 funct anal appl 8 68 70 in the case of an affine unk this class includes modular invariant representations of arbitrary rational level m t u where t unk z and u unk n are relatively prime and m g g u g is the dual coxeter number we write the characters of these representations in terms of theta functions and calculate their asymptotics generalizing the results of kac and peterson kac v g peterson d h 1984 adv math 53 125 264 and of kac and wakimoto kac v g wakimoto m 1988 adv math 70 156 234 for the u 1 integrable case we work out in detail the case unk a1 1 in particular classifying all its modular invariant representations furthermore we show that the modular invariant representations of the virasoro algebra vir are precisely the minimal series of belavin et al belavin a a polyakov a m zamolodchikov a b 1984 nucl phys b 241 333 380 using the character formulas of feigin and fuchs feigin b l fuchs d b 1984 lect notes math 1060 230 245 we show that tensoring the basic representation and modular invariant representations of a1 1 produces all modular invariant representations of vir generalizing the results of goddard et al goddard p kent a olive d 1986 commun math phys 103 105 119 and of kac and wakimoto kac v g wakimoto m 1986 lect notes phys 261 345 371 in the unitary case we study the general branching functions as well all these results are generalized to the kac moody superalgebras introduced by kac kac v g 1978 adv math 30 85 136 and to n 1 super virasoro algebras we work out in detail the case of the superalgebra b 0 1 1 showing in particular that restricting to its even part produces again all modular invariant representations of vir these results lead to general conjectures about asymptotic behavior of positive energy representations and classification of modular invariant representations

quantum reduction for affine superalgebras we extend the homological method of quantization of generalized drinfeld sokolov reductions to affine superalgebras this leads in particular to a unified representation theory of superconformal algebras

pattern search algorithms for bound constrained minimization we present a convergence theory for pattern search methods for solving bound constrained nonlinear programs the analysis relies on the abstract structure of pattern search methods and an understanding of how the pattern interacts with the bound constraints this analysis makes it possible to develop pattern search methods for bound constrained problems while only slightly restricting the flexibility present in pattern search methods for unconstrained problems we prove global convergence despite the fact that pattern search methods do not have explicit information concerning the gradient and its projection onto the feasible region and consequently are unable to enforce explicitly a notion of sufficient feasible decrease

argyres seiberg duality and the higgs branch we demonstrate the agreement between the higgs branches of two mathcal n 2 theories proposed by argyres and seiberg to be s dual namely the su 3 gauge theory with six quarks and the su 2 gauge theory with one pair of quarks coupled to the superconformal theory with e 6 flavor symmetry in mathematical terms we demonstrate the equivalence between a hyperkahler quotient of a linear space and another hyperkahler quotient involving the minimal nilpotent orbit of e 6 modulo the identification of the twistor lines

on quantum gauge theories in two dimensions two dimensional quantum yang mills theory is studied from three points of view i by standard physical methods ii by relating it to the largek limit of three dimensional chern simons theory and two dimensional conformal field theory iii by relating its weak coupling limit to the theory of reidemeister ray singer torsion the results obtained from the three points of view agree and give formulas for the volumes of the moduli spaces of representations of fundamental groups of two dimensional surfaces

kac wakimoto characters and universal mock theta functions in recent work bringmann and ono answer a question of kac and show that character formulas for sl r 1 1 modules due to kac and wakimoto are holomorphic parts of nonholomorphic modular functions here we confirm a speculation of ono that these characters are up to a simple q series the universal mock theta functions g 2 w q and g 3 w q of gordon and mcintosh using recent work of bringmann ono kang zwegers and gordon mcintosh we show that g 2 w q and g 3 w q are up to classical theta functions and products the characters of kac and wakimoto as a consequence we include a dictionary that gives a character formula for every classical mock theta function of ramanujan as well as subsequent natural generalizations

quantized enveloping algebras for borcherds superalgebras the rst author was supported in part by nsf grant dms 9300523 the second author was supported in part by the nondirected research fund korea research foundation 1996 the third author was supported in part by a faculty research grant from st lawrence university

on the convergence of pattern search algorithms we introduce an abstract definition of pattern search methods for solving nonlinear unconstrained optimization problems our definition unifies an important collection of optimization methods that neither compute nor explicitly approximate derivatives we exploit our characterization of pattern search methods to establish a global convergence theory that does not enforce a notion of sufficient decrease our analysis is possible because the iterates of a pattern search method lie on a scaled translated integer lattice this allows us to relax the classical requirements on the acceptance of the step at the expense of stronger conditions on the form of the step and still guarantee global convergence

pattern search methods for linearly constrained minimization we extend pattern search methods to linearly constrained minimization we develop a general class of feasible point pattern search algorithms and prove global convergence to a karush kuhn tucker point as in the case of unconstrained minimization pattern search methods for linearly constrained problems accomplish this without explicit recourse to the gradient or the directional derivative of the objective key to the analysis of the algorithms is the way in which the local search patterns conform to the geometry of the boundary of the feasible region

s duality for the large n 4 superconformal algebra we prove some conjectures about vertex algebras which emerge in gauge theory constructions associated to the geometric langlands program in particular we present the conjectural kernel vertex algebra for the s t 2 s duality transformation in su 2 gauge theory we find a surprising coincidence which gives a powerful hint about the nature of the corresponding duality wall concretely we determine the branching rules for the small n 4 superconformal algebra at central charge 9 as well as for the generic large n 4 superconformal algebra at central charge 6 moreover we obtain the affine vertex superalgebra of mathfrak osp 1 2 and the n 1 superconformal algebra times a free fermion as quantum hamiltonian reductions of the large n 4 superconformal algebras at c 6

on the superconformal index of argyres douglas theories we conjecture a closed form expression for the schur limit of the superconformal index of two infinite series of argyres douglas ad superconformal field theories scfts the a1 a2n 3 and the a1 d2n theories while these scfts can be realized at special points on the coulomb branch of certain n 2 gauge theories their superconformal r symmetries are emergent and hence their indices cannot be evaluated by localization instead we construct the a1 a2n 3 and a1 d2n indices by using a relation to two dimensional qdeformed yang mills theory and data from the class s construction our results generalize the indices derived from the torus partition functions of the two dimensional chiral algebras associated with the a1 a3 and a1 d4 scfts as checks of our conjectures we study the consistency of our results with an s duality recently discussed by us in collaboration with giacomelli and papageorgakis we reproduce known higgs branch relations we check consistency with a series of renormalization group flows and we verify that the small s 1 limits of our indices reproduce expected cardy like behavior we will discuss the s 1

a brief review of the 2d 4d correspondences an elementary introduction to the 2d 4d correspondences is given after quickly reviewing the 2d q deformed yang mills theory and the liouville theory we will introduce 4d theories obtained by coupling trifundamentals to su 2 gauge fields we will then cleraly see that the supersymmetric partition function of these theories on s 3 times s 1 and on s 4 is given respectively by the q deformed yang mills theory and the liouville theory after giving a short discussion on how this correspondence may be understood from the viewpoint of the 6d mathcal n 2 0 theory we conclude the review by enumerating future directions most of the technical points will be referred to more detailed review articles

superalgebras symplectic bosons and the sugawara construction new representations of affine lie algebras are constructed using symplectic bosons of the sort that occur naturally in the brst treatment of fermionic string theories these representations are shown to have analogous properties to the current algebra representations in terms of free fermion fields though they do not act in a positive space in particular the condition for the sugawara construction of the virasoro algebra to equal the free one is the existence of a superalgebra with a quadratic casimir operator paralleling the symmetric space theorem for fermionic field constructions both results are seen to be particular cases of a more general super symmetric space theorem which arises from considering an affinisation of the superalgebras these algebras are realised in terms of free fermions and symplectic bosons and lead to a super sugawara construction of the virasoro algebra the conditions for this to equal a virasoro algebra obtained from the free fields are provided by the super symmetric space theorem

affine lie algebras and product sum identities abstract we provide four different decompositions for a special infinite product first studied by i schur our product sum decomposition gives two more correspondences between subsets of partitions with parts congruent to 1 or 5 modulo 6 our method uses a special vertex representation of affine lie algebra c 3 1 and admissible representations of a 1 1 as well as quintuple product identity

gauge theories and macdonald polynomials we study the mathcal n 2 four dimensional superconformal index in various interesting limits such that only states annihilated by more than one supercharge contribute extrapolating from the su 2 generalized quivers which have a lagrangian description we conjecture explicit formulae for all a type quivers of class mathcal s which in general do not have one we test our proposals against several expected dualities the index can always be interpreted as a correlator in a two dimensional topological theory which we identify in each limit as a certain deformation of two dimensional yang mills theory the structure constants of the topological algebra are diagonal in the basis of macdonald polynomials of the holonomies

vertex algebras for s duality we define new deformable families of vertex operator algebras mathfrak a mathfrak g psi sigma associated to a large set of s duality operations in four dimensional supersymmetric gauge theory they are defined as algebras of protected operators for two dimensional supersymmetric junctions which interpolate between a dirichlet boundary condition and its s duality image the mathfrak a mathfrak g psi sigma vertex operator algebras are equipped with two mathfrak g affine vertex subalgebras whose levels are related by the s duality operation they compose accordingly under a natural convolution operation and can be used to define an action of the s duality operations on a certain space of vertex operator algebras equipped with a mathfrak g affine vertex subalgebra we give a self contained definition of the s duality action on that space of vertex operator algebras the space of conformal blocks in the derived sense i e chiral homology for mathfrak a mathfrak g psi sigma is expected to play an important role in a broad generalization of the quantum geometric langlands program namely we expect the s duality action on vertex operator algebras to extend to an action on the corresponding spaces of conformal blocks this action should coincide with and generalize the usual quantum geometric langlands correspondence the strategy we use to define the mathfrak a mathfrak g psi sigma vertex operator algebras is of broader applicability and leads to many new results and conjectures about deformable families of vertex operator algebras

graded lie superalgebras and the superdimension formula in this paper we investigate the structure of graded lie superalgebras l a a l a where is a countable abelian semigroup and a is a countable abelian group with a coloring map satisfying a certain finiteness condition given a denominator identity for the graded lie superalgebra l we derive a superdimension formula for the homogeneous subspaces l a a a which enables us to study the structure of graded lie superalgebras in a unified way we discuss the applications of our superdimension formula to free lie superalgebras generalized kac moody superalgebras and monstrous lie superalgebras in particular the product identities for normalized formal power series are interpreted as the denominator identities for free lie superalgebras we also give a characterization of replicable functions in terms of product identities and determine the root multiplicities of monstrous lie superalgebras

quantum reduction and representation theory of superconformal algebras abstract we study the structure and representations of a family of vertex algebras obtained from affine superalgebras by quantum reduction as an application we obtain in a unified way free field realizations and determinant formulas for all superconformal algebras

some zero one laws for additive functionals of markov processes let xt px be anm symmetric markov process with a strictly positive transition density consider the additive functionalat 0tf xs wheref e 0 is a universally measurable function on the state spacee among others we prove thatpx at 0 already impliespx at 0 the latter is also equivalent topx at 0 for quasi everyx e and allt 0 and to the analytic condition smallint f n fdm infty for a sequence of finely open borel setsfn such thate fn is polar in the special cases of brownian motion and bessel process these results were obtained earlier by h j engelbert w schmidt x x xue and the authors

hecke type double sums appell lerch sums and mock theta functions i by introducing a dual notion between partial theta functions and appell lerch sums we find and prove a formula which expresses hecke type double sums in terms of appell lerch sums not only does our formula prove classical hecke type sum identities such as those in work of kac and peterson on affine lie algebras and hecke modular forms but once we have the hecke type forms for ramanujan s mock theta functions our formula gives straightforward proofs of many of the classical mock theta identities indeed we obtain a new proof of the mock theta conjectures our formula also applies to positive level string functions associated with admissible representations of the affine lie algebra a 1 1 of kac and wakimoto

the formulae of kontsevich and verlinde from the perspective of the drinfeld double a two dimensional gauge theory is canonically associated to every drinfeld double for particular doubles the theory turns out to be e g the ordinary yang mills theory the g g gauged wznw model or the poisson model that underlies the kontsevich quantization formula we calculate the arbitrary genus partition function of the latter the result is the q deformation of the ordinary yang mills partition function in the sense that the series over the weights is replaced by the same series over the q weights for q equal to a root of unity the series acquires the affine weyl symmetry and its truncation to the alcove coincides with the verlinde formula

on infinite root systems we define in an axiomatic fashion the concept of a set of root data that generalizes the usual concept of root system of a kac moody lie algebra we study these objects from a purely formal and geometrical point of view as well as in relation to their associated lie algebras this leads to a coherent theory of root systems bases subroot systems lie algebras defined by root data and subalgebras

fock representations and brst cohomology insl 2 current algebra we investigate the structure of the fock modules overa 1 1 introduced by wakimoto we show that irreducible highest weight modules arise as degree zero cohomology groups in a brst like complex of fock modules chiral primary fields are constructed as brst invariant operators acting on fock modules as a result we obtain a free field representation of correlation functions of thesu 2 wzw model on the plane and on the torus we also consider representations of fractional level arising in polyakov s 2d quantum gravity finally we give a geometrical borel weil like interpretation of the wakimoto construction

voas and rank two instanton scfts we analyze the mathcal n 2 superconformal field theories that arise when a pair of d3 branes probe an f theory singularity from the perspective of the associated vertex operator algebra we identify these vertex operator algebras for all cases we find that they have a completely uniform description parameterized by the dual coxeter number of the corresponding global symmetry group we further present free field realizations for these algebras in the style of recent work by three of the authors these realizations transparently reflect the algebraic structure of the higgs branches of these theories we find fourth order linear modular differential equations for the vacuum characters schur indices of these theories which are again uniform across the full family of theories and parameterized by the dual coxeter number we comment briefly on expectations for the still higher rank cases

level rank duality in nonunitary coset theories it is shown that the conformal field theories su n k su n 1 su n k 1 and su m l su m 1 su m 1 1 are equivalent where k n n m l m m n for a pair m n of relatively prime positive integers

elliptic modular double and 4d partition functions we consider 4d supersymmetric special unitary gamma quiver gauge theories on compact manifolds which are t 2 fibrations over s 2 we show that their partition functions are correlators of vertex

infinite chiral symmetry in four dimensions we describe a new correspondence between four dimensional conformal field theories with extended supersymmetry and two dimensional chiral algebras the meromorphic correlators of the chiral algebra compute correlators in a protected sector of the four dimensional theory infinite chiral symmetry has far reaching consequences for the spectral data correlation functions and central charges of any four dimensional theory with mathcal n 2 superconformal symmetry

equivariant verlinde algebra from superconformal index and argyres seiberg duality in this paper we show the equivalence between two seemingly distinct 2d tqfts one comes from the coulomb branch index of the class ss theory t g on l k 1 s 1 the other is the lglg equivariant verlinde formula or equivalently partition function of lgclgc complex chern simons theory on s 1 we first derive this equivalence using the m theory geometry and show that the gauge groups appearing on the two sides are naturally g and its langlands dual lglg when g is not simply connected we provide a recipe of computing the index of t g as summation over the indices of t g with non trivial background t hooft fluxes where g is the universal cover of g then we check explicitly this relation between the coulomb index and the equivariant verlinde formula for g su 2 or so 3 in the end as an application of this newly found relation we consider the more general case where g is su n or psu n and show that equivariant verlinde algebra can be derived using field theory via generalized argyres seiberg duality we also attach a mathematica notebook that can be used to compute the su 3 equivariant verlinde coefficients

two dimensional lattice gauge theory based on a quantum group in this article we analyse a two dimensional lattice gauge theory based on a quantum group the algebra generated by gauge fields is the lattice algebra introduced recently by a yu alekseev h grosse and v schomerus in 1 we define and study wilson loops this theory is quasi topological as in the classical case which allows us to compute the correlation functions of this theory on an arbitrary surface

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

hyperkahler and quaternionic kahler geometry a quaternion hermitian manifold of dimension at least 12 with closed fundamental 4 form is shown to be quaternionic kahler a similar result is proved for 8 manifolds hyperkahler metrics are constructed on the fundamental quaternionic line bundle with the zero section removed of a quaternionic kahler manifold indefinite if the scalar curvature is negative this construction is compatible with the quaternionic kahler and hyperkahier quotient constructions and allows quaternionic kahler geometry to be subsumed into the theory of hyperkahler manifolds it is shown that the hyperkahler metrics that arise admit a certain type of su 2 action possess functions which are kahler potentials for each of the complex structures simultaneously and determine quaternionic kahler structures via a variant of the moment map construction quaternionic kahler metrics are also constructed on the fundamental quaternionic line bundle and a twistor space analogy leads to a construction of hyperkahler metrics with circle actions on complex line bundles over kahler einstein complex contact manifolds nilpotent orbits in a complex semi simple lie algebra with the hyperkahler metrics defined by kronheimer are shown to give rise to quaternionic kahler metrics and various examples of these metrics are identified it is shown that any quaternionic kahler manifold with positive scalar curvature and sufficiently large isometry group may be embedded in one of these manifolds the twistor space structure of the projectivised nilpotent orbits is studied

vertex operators of admissible modules ofuq c 1 n using our recent bosonic realization ofuq sp2n we construct explicitly the vertex operators for the level 1 2 modules ofuq sp2n using bosonic fields our method contains a detailed analysis of all theq intertwining relations

massive ode im correspondence and nonlinear integral equations for a_r 1 type modified affine toda field equations the massive ordinary differential equation integrable model ode im correspondence is a relation between the linear problem associated with modified affine toda field equations and 2d massive integrable models we study the massive ode im correspondence for the type modified affine toda field equations based on the system satisfied by the solutions of the linear problem we derive the bethe ansatz equations and determine the asymptotic behavior of the q functions for large values of the spectral parameter we derive the non linear integral equations for the q functions from the bethe ansatz equations we compute the effective central charge in the uv limit which is identified by one of the non unitary wa r minimal models when the solution has trivial monodromy around the origin of the complex plane

mathcal w algebras and surface operators in mathcal n 2 gauge theories a general class of algebras can be constructed from the affine sl n algebra by quantum drinfeld sokolov reduction and are classified by partitions of n surface operators in an su n 4d gauge theory are also classified by partitions of n we argue that instanton partition functions of gauge theories in the presence of a surface operator can also be computed from the corresponding algebra we test this proposal by analysing the polyakov bershadsky algebra obtaining results that are in agreement with the known partition functions for su 3 gauge theories with a so called simple surface operator as a byproduct our proposal implies relations between and algebras

analysis of generalized pattern searches this paper contains a new convergence analysis for the lewis and torczon generalized pattern search gps class of methods for unconstrained and linearly constrained optimization this analysis is motivated by a desire to understand the successful behavior of the algorithm under hypotheses that are satisfied by many practical problems specifically even if the objective function is discontinuous or extended valued the methods find a limit point with some minimizing properties simple examples show that the strength of the optimality conditions at a limit point depends not only on the algorithm but also on the directions it uses and on the smoothness of the objective at the limit point in question the contribution of this paper is to provide a simple convergence analysis that supplies detail about the relation of optimality conditions to objective smoothness properties and to the defining directions for the algorithm and it gives previous results as corollaries

argyres douglas theories s 1 reductions and topological symmetries in a recent paper we proposed closed form expressions for the superconformal indices of the a1 a2n 3 and a1 d2n argyres douglas ad superconformal field theories scfts in the schur limit following up on our results we turn our attention to the small s 1 regime of these indices as expected on general grounds our study reproduces the s 3 partition functions of the resulting dimensionally reduced theories however we show that in all cases with the exception of the reduction of the a1 d4 scft certain imaginary partners of real mass terms are turned on in the corresponding mirror theories we interpret these deformations as r symmetry mixing with the topological symmetries of the direct s 1 reductions moreover we argue that these shifts occur in any of our theories whose fourdimensional n 2 superconformal u 1 r symmetry does not obey an su 2 quantization condition we then use our r symmetry map to find the four dimensional ancestors of certain three dimensional operators somewhat surprisingly this picture turns out to imply that the scaling dimensions of many of the chiral operators of the four dimensional theory are encoded in accidental symmetries of the three dimensional theory we also comment on the implications of our work on the space of general n 2 scfts

parameter estimation of coupled water and energy balance models based on stationary constraints of surface states 1 we use a conditional averaging approach to estimate the parameters of a land surface water and energy balance model and then use the estimated parameters to partition net radiation into latent sensible and ground heat fluxes and precipitation into evapotranspiration and drainage plus runoff through conditional averaging of the modeled fluxes with respect to soil moisture and temperature we write an objective function that approximates the temperature and moisture dependent errors of the modeled fluxes in terms of atmospheric forcing e g precipitation and radiation surface states moisture s and temperature ts and model parameters the novelty of the approach is that the error term is estimated without comparison to measured fluxes instead it is inferred from the deviation of the conditionally averaged tendency terms expectation and expectation from zero since each of these terms equals zero in stationary systems but diverges from zero in the presence of misspecified parameters minimization of the approximated error yields parameters for model applications this strategy was previously studied for simple water balance models using soil moisture conditional averaging here we extend the idea to include energy balance fluxes and surface temperature conditioning the method is tested at two ameriflux sites vaira ranch california and kendall grassland arizona the estimated fluxes using only observed forcing and state variables are in reasonable agreement with field measurements because this method is based on conditional averages it can be applied to situations with subsampled or missing data that is continuous integration in time is not required

characters and fusion rules for w algebras via quantized drinfeld sokolov reduction using the cohomological approach tow algebras we calculate characters and fusion coefficients for their representations obtained from modular invariant representations of affine algebras by the quantized drinfeld sokolov reduction

quantitative comparison of canopy conductance models using a bayesian approach 1 a quantitative model comparison methodology based on deviance information criterion a bayesian measure of the trade off between model complexity and goodness of fit is developed and demonstrated by comparing semiempirical transpiration models this methodology accounts for parameter and prediction uncertainties associated with such models and facilitates objective selection of the simplest model out of available alternatives which does not significantly compromise the ability to accurately model observations we use this methodology to compare various jarvis canopy conductance model configurations embedded within a larger transpiration model against canopy transpiration measured by sap flux the results indicate that descriptions of the dependence of stomatal conductance on vapor pressure deficit photosynthetic radiation and temperature as well as the gradual variation in canopy conductance through the season are essential in the transpiration model use of soil moisture was moderately significant but only when used with a hyperbolic vapor pressure deficit relationship subtle differences in model quality could be clearly associated with small structural changes through the use of this methodology the results also indicate that increments in model complexity are not always accompanied by improvements in model quality and that such improvements are conditional on model structure possible application of this methodology to compare complex semiempirical models of natural systems in general is also discussed

punctures and p spin curves from matrix models this article investigates the intersection numbers of the moduli space of p spin curves with the help of matrix models the explicit integral representations that are derived for the generating functions of these intersection numbers exhibit p stokes domains labelled by a spin component l taking values l 1 0 1 2 p 2 earlier studies concerned integer values of p but the present formalism allows one to extend our study to half integer or negative values of p which turn out to describe new types of punctures or marked points on the riemann surface they fall into two classes ramond l 1 absent for positive integer p and neveu schwarz l ne 1 the intersection numbers of both types are computed from the integral representation of the n point correlation functions in a large n scaling limit we also consider a supersymmetric extension of the random matrix formalism to show that it leads naturally to an additional logarithmic potential open boundaries on the surface or admixtures of r and ns punctures may be handled by this extension

ramanujan s lost notebook and the virasoro algebra by using the theory of vertex operator algebras we gave a new proof of the famous ramanujan s modulus 5 modular equation from his lost notebook p 139 in r furthermore we obtained an infinite list of q identities for all odd moduli thus we generalized the result of ramanujan

hyperkahler metrics and supersymmetry we describe two constructions of hyperkahler manifolds one based on a legendre transform and one on a sympletic quotient these constructions arose in the context of supersymmetric nonlinear models but can be described entirely geometrically in this general setting we attempt to clarify the relation between supersymmetry and aspects of modern differential geometry along the way reviewing many basic and well known ideas in the hope of making them accessible to a new audience

n point and higher genus osp 1 2 fusion we study affine osp 1 2 fusion the fusion in osp 1 2 conformal field theory for example higher point and higher genus fusion is discussed the fusion multiplicities are characterized as discretized volumes of certain convex polytopes and are written explicitly as multiple sums measuring those volumes we extend recent methods developed to treat affine su 2 fusion they are based on the concept of generalized berenstein zelevinsky triangles and virtual couplings higher point tensor products of finite dimensional irreducible osp 1 2 representations are also considered the associated multiplicities are computed and written as multiple sums

relaxed highest weight modules i rank 1 cases relaxed highest weight modules play a central role in the study of many important vertex operator super algebras and their associated logarithmic conformal field theories including the admissible level affine models indeed their structure and their super characters together form the crucial input data for the standard module formalism that describes the modular transformations and grothendieck fusion rules of such theories in this article character formulae are proved for relaxed highest weight modules over the simple admissible level affine vertex operator superalgebras associated to mathfrak s mathfrak l 2 and mathfrak osp 1 vert 2 moreover the structures of these modules are specified completely this proves several conjectural statements in the literature for mathfrak s mathfrak l 2 at arbitrary admissible levels and for mathfrak osp 1 vert 2 at level frac 5 4 for other admissible levels the mathfrak osp 1 vert 2 results are believed to be new

representation theory of the affine lie superalgebra at fractional level n 2 noncritical strings are closely related to the wess zumino novikov witten model and there is much hope to further probe the former by using the algebraic apparatus provided by the latter an important ingredient is the precise knowledge of the representation theory at fractional level in this paper the embedding diagrams of singular vectors appearing in verma modules for fractional values of the level p and q coprime are derived analytically the nilpotency of the fermionic generators in hslc requires the introduction of a nontrivial generalisation of the mff construction to relate singular vectors among themselves the diagrams reveal a striking similarity with the degenerate representations of the n 2 superconformal algebra

conformal field theories via hamiltonian reduction constraining thesl 3 wzw model we construct a reduced theory which is invariant with respect to the new chiral algebraw 3 2 this symmetry is generated by the stress energy tensor two bosonic currents with spins 3 2 and theu 1 current we conjecture a kac formula that describes the highly reducible representation for this algebra we also discuss the quantum hamiltonian reduction for the general type of constraints that leads to the new extended conformal algebras

bosonic realizations ofuq c 1 n abstract we construct explicitly the quantum symplectic affine algebrauq sp 2n using bosonic fields the fock space decomposes into irreducible modules of level 1 2 quantizing the feingold frenkel construction forq 1

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

logarithmic conformal field theory beyond an introduction this article aims to review a selection of central topics and examples in logarithmic conformal field theory it begins with the remarkable observation of cardy that the horizontal crossing probability of critical percolation may be computed analytically within the formalism of boundary conformal field theory cardy s derivation relies on certain implicit assumptions which are shown to lead inexorably to indecomposable modules and logarithmic singularities in correlators for this a short introduction to the fusion algorithm of nahm gaberdiel and kausch is provided while the percolation logarithmic conformal field theory is still not completely understood there are several examples for which the formalism familiar from rational conformal field theory including bulk partition functions correlation functions modular transformations fusion rules and the verlinde formula has been successfully generalized this is illustrated for three examples the singlet model related to the triplet model symplectic fermions and the fermionic bc ghost system the fractional level wess zumino witten model based on at related to the bosonic ghost system and the wess zumino witten model for the lie supergroup related to at and 1 the bershadsky polyakov algebra and the feigin semikhatov algebras these examples have been chosen because they represent the most accessible and most useful members of the three best understood families of logarithmic conformal field theories the logarithmic minimal models the fractional level wess zumino witten models and the wess zumino witten models on lie supergroups excluding in this review the emphasis lies on the representation theory of the underlying chiral algebra and the modular data pertaining to the characters of the representations each of the archetypal logarithmic conformal field theories is studied here by first determining its irreducible spectrum which turns out to be continuous as well as a selection of natural reducible but indecomposable modules this is followed by a detailed description of how to obtain character formulae for each irreducible a derivation of the action of the modular group on the characters and an application of the verlinde formula to compute the grothendieck fusion rules in each case the genuine fusion rules are known so comparisons can be made and favourable conclusions drawn in addition each example admits an infinite set of simple currents hence extended symmetry algebras may be constructed and a series of bulk modular invariants computed the spectrum of such an extended theory is typically discrete and this is how the triplet model arises for example moreover simple current technology admits a derivation of the extended algebra fusion rules from those of its continuous parent theory finally each example is concluded by a brief description of the computation of some bulk correlators a discussion of the structure of the bulk state space and remarks concerning more advanced developments and generalizations the final part gives a very short account of the theory of staggered modules the simplest class of representations that are responsible for the logarithmic singularities that distinguish logarithmic theories from their rational cousins these modules are discussed in a generality suitable to encompass all the examples met in this review and some of the very basic structure theory is proven then the important quantities known as logarithmic couplings are reviewed for virasoro staggered modules and their role as fundamentally important parameters akin to the three point constants of rational conformal field theory is discussed an appendix is also provided in order to introduce some of the necessary but perhaps unfamiliar language of homological algebra

weyl kac type character formula for admissible representations of borcherds kac moody lie superalgebras we prove that borcherds kac moody lie superalgebras yield weyl kac type character formulas for some specific types of admissible representations our results generalize the results of kac and wakimoto proc natl acad sci usa 85 4956 4960 1988 by extending the weyl kac character formula for integrable irreducible highest weight modules over kac moody algebras to the case of admissible irreducible highest weight modules over borcherds kac moody lie superalgebras