string center of mass operator and its effect on brst cohomology we consider the theory of bosonic closed strings on the flat background 25 1 we show how the brst complex can be extended to a complex where the string center of mass operator x 0 is well defined we investigate the cohomology of the extended complex we demonstrate that this cohomology has a number of interesting features unlike in the standard brst cohomology there is no doubling of physical states in the extended complex the cohomology of the extended complex is more physical in a number of aspects related to the zero momentum states in particular we show that the ghost number one zero momentum cohomology states are in one to one correspondence with the generators of the global symmetries of the backgroundi e the poincare algebra

compatibility of local and global langlands correspondences we prove the compatibility of local and global langlands correspondences for gln which was proved up to semisimplification in ht more precisely for the ndimensional l adic representation rl of the galois group of an imaginary cm field l attached to a conjugate self dual regular algebraic cuspidal automorphic representation of gln al which is square integrable at some finite place we show that frobenius semisimplification of the restriction of rl to the decomposition group of a place v of l not dividing l corresponds to v by the local langlands correspondence if v is square integrable for some finite place v 6 l we deduce that rl is irreducible we also obtain conditional results in the case v l

convergence analysis of a high order nystrom integral equation method for surface scattering problems in this paper we present a convergence analysis for the nystrom method proposed in j comput phys 169 1 80 110 2001 for the solution of the combined boundary integral equation formulations of sound soft acoustic scattering problems in three dimensional space this fast and efficient scheme combines fft techniques and a polar change of variables that cancels out the kernel singularity we establish the stability of the algorithms in the l 2 norm and we derive convergence estimates in both the l 2 and l infty norms in particular our analysis establishes theoretically the previously observed super algebraic convergence of the method in cases in which the right hand side is smooth

hidden sl n symmetry in conformal field theories we prove that an irreducible representation of the virasoro algebra can be extracted from an irreducible representation space of thesl 2 current algebra by putting a constraint on the latter using the becchi rouet stora tyutin formalism thus there is asl 2 symmetry in the virasoro algebra but it is gauged and hidden this construction of the virasoro algebra is the quantum analogue of the hamiltonian reduction we then are naturally lead to consider a constrainedsl 2 wess zumino witten model this system is also related to quantum field theory of coadjoint orbit of the virasoro group based on this result we present a canonical derivation of thesl 2 current algebra in polyakov s theory of two dimensional gravity it is a manifestation of thesl 2 symmetry in conformal field theory hidden by the quantum hamiltonian reduction we also discuss the quantum hamiltonian reduction of thesl 2 current algebra and its relation to thew n algebra of zamolodchikov this makes it possible to define a natural generalization of the geometric action for thew n algebra despite its non lie algebraic nature

a natural construction of borcherds fake baby monster lie algebra we use a z 2 orbifold of the vertex operator algebra associated to the niemeier lattice with root lattice a 8 3 and the no ghost theorem of string theory to construct a generalized kac moody algebra borcherds theory of automorphic products allows us to determine the simple roots and identify the algebra with the fake baby monster lie algebra

the cohomology algebra of the semi infinite weil complex in 1993 lian zuckerman constructed two cohomology operations on the brst complex of a conformal vertex algebra with central charge 26 they gave explicit generators and relations for the cohomology algebra equipped with these operations in the case of the c 1 model in this paper we describe another such example namely the semi infinite weil complex of the virasoro algebra the semi infinite weil complex of a tame mathbb z graded lie algebra was defined in 1991 by feigin frenkel and they computed the linear structure of its cohomology in the case of the virasoro algebra we build on this result by giving an explicit generator for each non zero cohomology class and describing all algebraic relations in the sense of lian zuckerman among these generators

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

brst cohomology of the super virasoro algebras we study the superextension of the semi infinite cohomology theory of the virasoro algebra in particular we examine the brst complex with coefficients in the fock space of the rns superstring we prove a theorem of vanishing cohomology and establish the unitary equivalence between a positive definite transversal space a physical subspace and the zeroth cohomology group the cohomology of a subcomplex is identified as the covariant equivalent of the well known gso subspace an exceptional case to the vanishing theorem is discussed

galois representations arising from some compact shimura varieties our aim is to establish some new cases of the global langlands correspondence for glm along the way we obtain a new result on the description of the cohomology of some compact shimura varieties let f be a cm eld with complex conjugation c and

the brst cohomology of the nsr string vanishing and no ghost theorems we investigate the brst cohomology of the nsr string we prove vanishing theorems for the full and relative sub complexes generalizing the work of frenkel garland and zuckerman for the bosonic string using these results we give simple proofs of the no ghost theorems for both sectors

the fontaine mazur conjecture for gl _2 introduction 641 1 breuil mezard conjecture and the p adic local langlands 644 1 1 the breuil mezard conjecture 644 1 2 review of colmez s functor 647 1 3 hilbert samuel multiplicities 650 1 4 representations and pseudo representations 653 1 5 gl2 qp representations mod p 657 1 6 local patching and multiplicities 661 1 7 from pseudo representations to representations 669 2 modularity via the breuil mezard conjecture 675 2 1 quaternionic forms 675 2 2 global patching and multiplicities 677 2 3 the breuil mezard conjecture 686 addendum to ki 1 687 references 689

a family of calabi yau varieties and potential automorphy we prove potential modularity theorems for l adic representations of any dimension from these results we deduce the sato tate conjecture for all elliptic curves with nonintegral j invariant defined over a totally real field

semi infinite cohomology and string theory abstract we develop the theory of semi infinite cohomology of graded lie algebras first introduced by feigin we show that the relative semi infinite cohomology has a structure analogous to that of the de rham cohomology in kahler geometry we prove a vanishing theorem for a special class of modules and we apply our results to the case of the virasoro algebra and the fock module in this case the zero cohomology is identified as the physical subspace of the fock module and the no ghost theorem follows we reveal the profound relation of semi infinite cohomology theory to the gauge invariant free string theory constructed by banks and peskin we then indicate the connection between gauge invariant interacting string theories and the geometric realizations of the infinite dimensional lie algebras

brst cohomologies for non critical strings the brst cohomologies of non critical massive string models are analysed in detail in the range of dimensions 1 d 26 the classical theorems on vanishing of dolbeaut type and relative cohomologies are proved the no ghost theorem is proved in the same range of dimensions it is also shown that the brst complex of the massive string model is a resolution of the non critical nambu goto string in all spacetime dimensions 1 d 25

the structure of verma modules over the n equals 2 superconformal algebra we classify degeneration patterns of verma modules over the n equals 2 superconformal algebra in two dimensions explicit formulae are given for singular vectors that generate maximal submodules in each of the degenerate cases the mappings between verma modules defined by these singular vectors are embeddings in particular their compositions never vanish as a by product we also obtain general formulae for n equals 2 subsingular vectors

the construction of the sh lie algebra of closed bosonic strings we present a construction of the closed string algebra in terms of gaussian processes and crossed products also we give a purely functional analytical prove of the sh lie structure

the sato tate conjecture for hilbert modular forms we prove the sato tate conjecture for hilbert modular forms more precisely we prove the natural generalisation of the sato tate conjecture for regular algebraic cuspidal automorphic representations of gl2 af f a totally real field which are not of cm type the argument is based on the potential automorphy techniques developed by taylor et al but makes use of automorphy lifting theorems over ramified fields together with a topological argument with local deformation rings in particular we give a new proof of the conjecture for modular forms which does not make use of potential automorphy theorems for non ordinary n dimensional galois representations

on the density of modular points in universal deformation spaces based on comparison theorems for hecke algebras and universal deformation rings with strong restrictions at the critical prime l as provided by the results of wiles taylor diamond et al we prove under rather general conditions that the corresponding universal deformation spaces with no restrictions at l can be identified with certain hecke algebras of l adic modular forms as conjectured by gouvea thus generalizing previous work of gouvea and mazur along the way we show that the universal deformation spaces we consider are complete intersections flat over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i l of relative dimension three in which the modular points form a zariski dense subset furthermore the fibers above inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i l of these spaces are generically smooth

a generalized kac moody algebra of rank 14 abstract we construct a vertex algebra of central charge 26 from a lattice orbifold vertex operator algebra of central charge 12 the brst cohomology group of this vertex algebra is a new generalized kac moody algebra of rank 14 we determine its root space multiplicities and a set of simple roots

corps de nombres peu ramifies et formes automorphes autoduales let s be a finite set of primes p in s and q s a maximal algebraic extension of q unramified outside s and infinity assume that s 2 we show that the natural maps gal q p bar q p gal q s q are injective much of the paper is devoted to the problem of constructing selfdual automorphic cuspidal representations of gl 2n a q with prescribed properties at all places that we study via the twisted trace formula of j arthur the techniques we develop shed also some lights on the orthogonal symplectic alternative for selfdual representations of gl 2n

the breuil mezard conjecture when l neq p let l and p be primes let f qp be a finite extension with absolute galois group gf let f be a finite field of characteristic l and let w gf gln f be a continuous representation let r be the universal framed deformation ring for if l d p then the breuil mezard conjecture as recently formulated by emerton and gee relates the mod l reduction of certain cycles in r to the mod l reduction of certain representations of gln of we state an analogue of the breuil mezard conjecture when l p and we prove it whenever l 2 using automorphy lifting theorems we give a local proof when l is quasibanal for f and is tamely ramified we also analyze the reduction modulo l of the types defined by schneider and zink

on the modularity of 2 adic potentially semi stable deformation rings using p adic local langlands correspondence for text gl 2 mathbb q 2 and an ordinary r mathbb t theorem we prove that the support of patched modules for quaternionic forms meet every irreducible component of the potentially semi stable deformation ring this gives a new proof of the breuil mezard conjecture for 2 dimensional representations of the absolute galois group of mathbb q 2 which is new in the case overline r a twist of an extension of the trivial character by itself as a consequence a local restriction in the proof of fontaine mazur conjecture in pask nas algebra number theory 10 6 1301 1358 2016 is removed

a fast high order algorithm for the solution of surface scattering problems basic implementation tests and applications we present a new algorithm for the numerical solution of problems of acoustic scattering by surfaces in three dimensional space this algorithm evaluates scattered fields through fast high order solution of the corresponding boundary integral equation the high order accuracy of our solver is achieved through use of partitions of unity together with analytical resolution of kernel singularities the acceleration in turn results from use of a novel approach which based on high order two face equivalent source approximations reduces the evaluation of far interactions to evaluation of 3 d fast fourier transforms ffts this approach is faster and substantially more accurate and it runs on dramatically smaller memories than other fft and k space methods the present algorithm computes one matrix vector multiplication in o n6 5logn to o n4 3logn operations where n is the number of surface discretization points the latter estimate applies to smooth surfaces for which our high order algorithm provides accurate solutions with small values of n the former more favorable count is valid for highly complex surfaces requiring significant amounts of subwavelength sampling further our approach exhibits super algebraic convergence it can be applied to smooth and nonsmooth scatterers and it does not suffer from accuracy breakdowns of any kind in this paper we introduce the main algorithmic components in our approach and we demonstrate its performance with a variety of numerical results in particular we show that the present algorithm can evaluate accurately in a personal computer scattering from bodies of acoustical sizes of several hundreds

on the relevance of some constructions of highest weight modules over super kac moody algebras in connection with a paper by wakimoto we correct and improve results of wakimoto on the ir reducibility of his construction ofa 1 1 highest weight modules hwm for a very large class of super kac moody algebras we argue that such a hwm is most relevant when it is isomorphic to a proper factor module of the corresponding reducible verma module with the same highest weight in the same situation we present a general procedure to check the reducibility of the hwm in consideration

a high order algorithm for obstacle scattering in three dimensions in this work we describe implement and analyse in detail a high order fully discrete spectral algorithm for solving the helmholtz equation exterior to a bounded sound soft sound hard or absorbing obstacle in three space dimensions with dirichlet neumann or robin impedance boundary conditions our algorithm may be thought of as a discrete glerkin method but it is also equivalent to a nystrom method after a simple transformation we test our algorithm with extensive computational experiments on a variety of three dimensional smooth and non smooth obstacles with conical singularities our tests include the computation of scattered and far fields induced by incident plane waves our method is shown to be very accurate for scattering from surfaces which are globally parameterised by spherical coordinates and tests show that it performs very much better than several of the well established fast algorithms for obstacle sattering on a range of such surfaces even some which are non smooth further we prove superalgebraic convergence of the scattered and far fields obtained using our algorithm in the case of smooth scatterers

serre s modularity conjecture ii we provide proofs of theorems 4 1 and 5 1 of khare and wintenberger invent math doi 10 1007 s00222 009 0205 7 2009

β γ systems and the deformations of the brst operator we describe the relation between simple logarithmic cfts associated with closed and open strings and their infinite metric limits corresponding to the systems this relation is studied on the level of the brst complex we show that the consideration of metric as a perturbation leads to a certain deformation of the algebraic operations of the lian zuckerman type on the vertex algebra associated with the systems the maurer cartan equations corresponding to this deformed structure in the quasi classical approximation lead to the nonlinear field equations as an explicit example we demonstrate that using this construction yang mills equations can be derived this gives rise to a nontrivial relation between the courant dorfman algebroid and homotopy algebras emerging from the gauge theory we also discuss a possible algebraic approach to the study of beta functions in sigma models

multiplicites modulaires et representations de rm gl sb 2 mathbf z sb p et de rm gal overline mathbf q sb p mathbf q sb p en ell p appendice par guy henniart sur l unicite des types pour rm gl sb 2

the ricci curvature of diff s1 sl 2 r previous calculations of the ricci curvature for the manifold diff diff s1 s1 are extended to diff s1 sl 2r these manifolds are distinguished by being coadjoint orbits of diff s1 which admit compatible symphectic and complex structures making them kahler manifolds

quasiclassical lian zuckerman homotopy algebras courant algebroids and gauge theory we define a quasiclassical limit of the lian zuckerman homotopy bv algebra quasiclassical lz algebra on the subcomplex corresponding to light modes i e the elements of zero conformal weight of the semi infinite brst cohomology complex of the virasoro algebra associated with vertex operator algebra voa with a formal parameter we also construct a certain deformation of the brst differential parametrized by a constant two component tensor such that it leads to the deformation of the a subalgebra of the quasiclassical lz algebra altogether this gives a functor the category of voa with a formal parameter to the category of a algebras the associated generalized maurer cartan equation gives the analogue of the yang mills equation for a wide class of voas applying this construction to an example of voa generated by systems we find a remarkable relation between the courant algebroid and the homotopy algebra of the yang mills theory

modularity of 2 adic barsotti tate representations we prove a modularity lifting theorem for two dimensional 2 adic potentially barsotti tate representations this proves hypothesis h of khare wintenberger and completes the proof of serre s conjecture the main new ingredient is a classification of connected finite flat group schemes over rings of integers of finite extensions of 2

a collocation method for weakly singular integral equations with super algebraic convergence rate we consider biperiodic weakly singular integral equations of the second kind such as they arise in boundary integral equation methods these are solved numerically using a collocation scheme based on trigonometric polynomials the singularity is removed by a local change to polar coordinates the resulting operators have smooth kernels and are discretized using the tensor product composite trapezoidal rule we prove stability and convergence of the scheme achieving algebraic convergence of any order under appropriate regularity assumptions the method can be applied to typical boundary value problems such as potential and scattering problems both for bounded obstacles and for periodic surfaces we present numerical results demonstrating that the expected convergence rates can be observed in practice

monstrous moonshine and monstrous lie superalgebras we prove conway and norton s moonshine conjectures for the infinite dimensional representation of the monster simple group constructed by frenkel lepowsky and meurman to do this we use the no ghost theorem from string theory to construct a family of generalized kac moody superalgebras of rank 2 which are closely related to the monster and several of the other sporadic simple groups the denominator formulas of these superalgebras imply relations between the thompson functions of elements of the monster i e the traces of elements of the monster on frenkel lepowsky and meurman s representation which are the replication formulas conjectured by conway and norton these replication formulas are strong enough to verify that the thompson functions have most of the moonshine properties conjectured by conway and norton and in particular they are modular functions of genus 0 we also construct a second family of kac moody superalgebras related to elements of conway s sporadic simple group col these superalgebras have even rank between 2 and 26 for example two of the lie algebras we get have ranks 26 and 18 and one of the superalgebras has rank 10 the denominator formulas of these algebras give some new infinite product identities in the same way that the denominator formulas of the a n n e kac moody algebras give the macdonald identities

locality in free string field theory free string field operators are constructed for the open bosonic string in the light cone gauge in any dimension these are naturally localized by the center of mass coordinate relative to this localization they are shown to have a causal commutator provided there are no tachyons for the critical string in d 26 the result still holds if the tachyon is suppressed we also show a causal commutator relative to the string light cone

topological representations of the quantum groupuq sl2 we define a topological action of the quantum groupuq sl2 on a space of homology cycles with twisted coefficients on the configuration space of the punctured disc this action commutes with the monodromy action of the braid groupoid which is given by ther matrix ofuq sl2

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

kazhdan lusztig conjecture for generalized kac moody algebras ii proof of the conjecture generalized kac moody algebras were introduced by borcherds in the study of conway and norton s moonshine conjectures for the monster sporadic simple group in this paper we prove the kazhdan lusztig conjecture for generalized kac moody algebras under a certain mild condition by using a generalization to the case of generalized kac moody algebras of jantzen s character sum formula our main formula generalizes the celebrated result for the case of kac moody algebras and describes the characters of irreducible highest weight modules over generalized kac moody algebras in terms of the extended kazhdan lusztig polynomials

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

potentially semi stable deformation rings let k qp be a finite extension and gk gal k k the galois group of an alge braic closure k let f be a finite field of characteristic p and vf a finite dimensional f vector space equipped with a continuous action of gkthe study of the defor mation theory of galois representations was initiated by mazur ma who showed that if vf has no non trivial endomorphisms then it admits a universal deformation ring ryw after the work of wiles wi and the conjectures of fontaine mazur fm it became clear that for arithmetic applications it was important to understand cer tain quotients of ry corresponding to deformations satisfying certain conditions for example wiles uses deformations which arise from finite flat group schemes and the corresponding quotient of ryw was constructed by ramakrishna ra suppose that l k is a finite extension and let a b be integers it seems to be a kind of folklore conjecture that there should be a quotient of ry whose points in finite extensions of w f l p correspond to deformations of vf which become semi stable over l and have hodge tate weights in the interval a b this is closely related to the final conjecture of fo 3 special cases of this when vf is 2 dimensional are conjectured in the papers of fontaine mazur fm p 191 breuil conrad diamond taylor bcdt conj 1 1 1 and breuil m zard bm 2 2 2 4 the purpose of this paper is to prove such a result

representations of kac moody algebras by step algebras representations of a kac moody algebra associated to a semisimple lie algebra g which decompose to a direct sum of finite dimensional representations of the subalgebra g are constructed using a step algebra method the cases g su 2 su 3 are considered in detail

the riemann hilbert problem approach to a representation of the virasoro group ii the development of a riemann hilbert rh problem approach to the representation of the virasoro group introduced in our earlier paper w li and b y hou j math phys 30 1198 1989 is continued the rouche theorem is employed to prove that there exists a set of scalar functions in the complex plane in terms of which the structure of the virasoro group can be decided the general form of the rh problem will also be presented and the uniqueness of its solution will be proven for future use a standard form of a matrix fredholm equation equivalent to our rh problem will be deduced this may help to establish in a future paper the existence of a solution of the rh problem

new perspectives on the brst algebraic structure of string theory motivated by the descent equation in string theory we give a new interpretation for the action of the symmetry charges on the brst cohomology in terms of what we callthe gerstenhaber bracket this bracket is compatible with the graded commutative product in cohomology and hence gives rise to a new class of examples of what mathematicians call agerstenhaber algebra the latter structure was first discussed in the context of hochschild cohomology theory 11 off shell in the chiral brst complex all the identities of a gerstenhaber algebra hold up to homotopy applying our theory to thec 1 model we give a precise conceptual description of the brst gerstenhaber algebra of this model we are led to a direct connection between the bracket structure here and the anti bracket formalism in bv theory 29 we then discuss the bracket in string backgrounds with both the left and the right movers we suggest that the homotopy lie algebra arising from our gerstenhaber bracket is closely related to the hla recently constructed by witten zwiebach finally we show that our constructions generalize to any topological conformal field theory

note on spin modules associated to z graded lie superalgebras the aim of this note is to clarify what can be a correct analog of modules of semi infinite wedges over the virasoro algebra considered by feigin and fuchs invariant skew symmetric differential operators on the line and verma modules over the virasoro algebra funct anal appl 16 114 1982 for a certain class of lie superalgebras

combinatorial bases of basic modules for affine lie algebras cn 1 lepowsky and wilson initiated the approach to combinatorial rogers ramanujan type identities via vertex operator constructions of standard i e integrable highest weight representations of affine kac moody lie algebras meurman and primc developed further this approach for sl 2 c by using vertex operator algebras and verma modules in this paper we use the same method to construct combinatorial bases of basic modules for affine lie algebras of type cn 1 and as a consequence we obtain a series of rogers ramanujan type identities a major new insight is a combinatorial parametrization of leading terms of defining relations for level one standard modules for affine lie algebra of type cn 1

even galois representations and the fontaine mazur conjecture we prove under mild hypotheses that there are no irreducible two dimensional ordinary even galois representations of mathrm gal overline mathbf q mathbf q with distinct hodge tate weights this is in accordance with the fontaine mazur conjecture if k q is an imaginary quadratic field we also prove again under certain hypotheses that mathrm gal overline mathbf q k does not admit irreducible two dimensional ordinary galois representations of non parallel weight

quantum group as semi infinite cohomology we obtain the quantum group slq 2 as semi infinite cohomology of the virasoro algebra with values in a tensor product of two braided vertex operator algebras with complementary central charges c bar c 26 each braided voa is constructed from the free fock space realization of the virasoro algebra with an additional q deformed harmonic oscillator degree of freedom the braided voa structure arises from the theory of local systems over configuration spaces and it yields an associative algebra structure on the cohomology we explicitly provide the four cohomology classes that serve as the generators of slq 2 and verify their relations we also discuss the possible extensions of our construction and its connection to the liouville model and minimal string theory

the c algebra of bosonic strings we give a rigorous definition of witten sc string algebra to this end we present a new construction ofc algebras associated to special geometric situations kahler foliations and generalize this later construction to the string case through this we get a natural geometrical interpretation of the string of semi infinite forms as well as the fermionic algebra structure using the non commutative geometric concepts for investigating the string algebra we get a natural fredholm module representation of dimension 26

quantum dynamics on super groups constraints and the brst supergroup the smallest supergroup b k containing among its generators those of a constraint group k the brst charge and the corresponding ghost and antighost operators is constructed the supergroup b k can be enlarged to include the ordinary dynamical variables of the unconstrained physical system in a group approach to quantization b k generalizes the ordinary u 1 phase invariance of wave functions in particular this mechanism reproduces the essential features of the brst quantization when k is diff s1 the critical values of string theory are those for which the space of polarized functions on the enlarged brst supergroup is not irreducible

semi infinite homology and 2d gravity i in 33 we studied the constraint problem for two dimensional quantum gravity in the conformal gauge in this gauge we proposed an ansatz for the gravitational sector using this ansatz we established a striking connection between the matrix models and continuum 2d gravity we also announced several results on semi infinite homology of the virasoro algebra with coefficients in a suitable class of positive energy modules in this article we will provide details of the proof of the announced results

string quantization on group manifolds and the holomorphic geometry of diffs1 s1 the recent results by bowick and rajeev on the relation of the geometry of diffs1 s1 and string quantization in d are extended to a string moving on a group manifold a new derivation of the curvature formula 26 12m3 1 6m n m for the canonical holomorphic line bundle over diffs1 s1 is given which clarifies the relation of that bundle with the complex line bundles over infinite dimensional grassmannians studied by pressley and segal

generalized classical brst cohomology and reduction of poisson manifolds in this paper we formulate a generalization of the classical brst construction which applies to the case of the reduction of a poisson manifold by a submanifold in the case of symplectic reduction our procedure generalizes the usual classical brst construction which only applies to symplectic reduction of a symplectic manifold by a coisotropic submanifold i e the case of reducible first class constraints in particular our procedure yields a method to deal with second class constraints we construct the brst complex and compute its cohomology brst cohomology vanishes for negative dimension and is isomorphic as a poisson algebra to the algebra of smooth functions on the reduced poisson manifold in zero dimension we then show that in the general case of reduction of poisson manifolds brst cohomology cannot be identified with the cohomology of vertical differential forms

semi infinite induction and wakimoto modules the purpose of this paper is to suggest the construction and study properties of semi infinite induction which relates to semi infinite cohomology the same way induction relates to homology and coinduction to cohomology we prove a version of the shapiro lemma showing that the semi infinite cohomology of a module is isomorphic to that of the semi infinitely induced module a practical outcome of our construction is a simple construction of the wakimoto modules highest weight modules used in double sided bgg resolutions of irreducible modules

a class of non standard modules for affine lie algebras on etudie la structure des modules de verma pour des algebres de lie affines quand le poids le plus haut est orthogonal aux racines imaginaires on considere l algebre de lie a 1 1 sl 2 c 1

geometric realization of conformal field theory on riemann surfaces conformal field theory on a family of riemann surfaces is formulated we derive equations of motion of vacua which are parametrized by moduli of riemann surfaces and show that these vacua are characterized uniquely by these equations our theory has a deep connection with sato s theory of kp equations

numerical solution of the helmholtz equation in 2d and 3d using a high order nystrom discretization we show how to solve time harmonic scattering problems by means of a high order nystrom discretization of the boundary integral equations of wave scattering in 2d and 3d the novel aspect of our new method is its use of local corrections to the discretized kernel in the vicinity of the kernel singularity enhanced by local corrections the new algorithm has the simplicity and speed advantages of the traditional nystrom method but also enjoys the advantages of high order convergence for controlling solution error we explain the practical details of implementing a scattering code based on a high order nystrom discretization and demonstrate by numerical example that a scattering code based on this algorithm can achieve high order convergence to the correct answer we also demonstrate its performance advantages over a high order galerkin code

on a theorem of jordan the theorem of jordan which i want to discuss here dates from 1872 it is an elementary result on finite groups of permutations i shall first present its translations in number theory and topology 1 statements 1 1 number theory let f n m 0 amx m be a polynomial of degree n with coefficients in z if p is prime let np f be the number of zeros of f in fp z pz theorem 1 assume i n 2 ii f is irreducible in q x then a there are infinitely many p s with np f 0 b the set p0 f of p s with np f 0 has a density c0 c0 f which is 0 recall that a subset p of the set of primes has density c if lim x number of p p with p x x c where x is as usual the number of primes x moreover theorem 2 with the notation of theorem 1 one has c0 f 1 n with strict inequality if n is not a power of a prime example let f x 1 one has p p0 f if and only if p 1 mod 4 this set is well known to have density 1 2 we shall see more interesting examples in 5 1 2 topology let s1 be a circle let f t s be a finite covering of a topological space s assume i f has degree n i e every fiber of f has n elements with n 2 ii t is arcwise connected and not empty theorem 3 there exists a continuous map s1 s which cannot be lifted to the covering t i e there does not exist any continuous map s1 t such that f received by the editors march 1 2003 2000 mathematics subject classification primary 06 xx 11 xx 11f11 this text first appeared in math medley 29 2002 3 18 the writing was done with the help of heng huat chan c 2002 singapore mathematical society reprinted with permission

string theory and loop space index theorems we study index theorems for the dirac ramond operator on a compact riemannian manifold the existence of a group action on the loop space makes possible the definition of a character valued index which we calculate by using a two dimensional sigma model withn 1 2 supersymmetry we compute the euler characteristic the hirzebruch signature and the dirac ramond genus of loop space we compare our results to the calculations made by using the atiyah singer character valued index theorem

derivations and central extensions of finitely generated graded lie algebras abstract some general results concerning derivations of finitely generated lie algebras are established these are employed in order to determine the derivations and central extensions of kac moody lie algebras

chiral equivariant cohomology iii this is the third of a series of papers on a new equivariant cohomology that takes values in a vertex algebra and contains and generalizes the classical equivariant cohomology of a manifold with a lie group action a la h cartan in this paper we compute this cohomology for spheres and show that for any simple connected group g there is a sphere with infinitely many actions of g which have distinct chiral equivariant cohomology but identical classical equivariant cohomology unlike the classical case the description of the chiral equivariant cohomology of spheres requires a substantial amount of new structural theory which we fully develop in this paper this includes a quasi conformal structure equivariant homotopy invariance and the values of this cohomology on homogeneous spaces these results rely on crucial features of the underlying vertex algebra valued complex that have no classical analogues

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

twisted strong macdonald theorems and adjoint orbits the strong macdonald theorems state that for l reductive and s an odd variable the cohomology algebras h l z z n and h l z s are freely generated and describe the cohomological s and z degrees of the generators the resulting identity for the z weighted euler characteristic is equivalent to macdonald s constant term identity for a finite root system we calculate h mathfrak p z n mathfrak p and h mathfrak p s for mathfrak p a standard parahoric in a twisted loop algebra giving strong macdonald theorems that take into account both a parabolic component and a possible diagram automorphism twist in particular we show that h mathfrak p z n mathfrak p contains a parabolic subalgebra of the coinvariant algebra of the fixed point subgroup of the weyl group of l and thus is no longer free we also prove a strong macdonald theorem for h mathfrak b s mathfrak n and h mathfrak b z n mathfrak n when mathfrak b and mathfrak n are iwahori and nilpotent subalgebras respectively of a twisted loop algebra for each strong macdonald theorem proved taking z weighted euler characteristics gives an identity equivalent to macdonald s constant term identity for the corresponding affine root system as part of the proof we study the regular adjoint orbits for the adjoint action of the twisted arc group associated to l proving an analogue of the kostant slice theorem

characters of irreducible representations of the virasoro algebra on obtient les caracteres des modules irreductibles non integrables de plus haut poids l h 0 on determine les caracteres des familles l t 25 et l t 26 duales de l h 1 et de l h 0

how to regularize singular vectors and kill the dynamical weyl group abstract let g be a simple lie algebra and let m be the verma module over g with highest weight for a finite dimensional g module u we introduce a notion of a regularizing operator acting in u which makes the meromorphic family of intertwining operators m m u holomorphic and conjugates the dynamical weyl group operators aw end u to constant operators we establish fundamental properties of regularizing operators including uniqueness and prove the existence of a regularizing operator in the case g sl 3

new weyl groups for a 1 1 and characters of singular highest weight modules we consider singular verma modules overa1 1 i e verma modules for which the central charge is equal to minus the dual coxeter number we calculate the characters of certain factor modules of these verma modules in one class of cases we are able to prove that these factor modules are actually the irreducible highest modules for those highest weights we introduce new weyl groups which are infinitely generated abelian groups and are proper subgroups or isomorphic between themselves using these weyl groups we can rewrite the character formulae obtained in the paper in the form of the classical weyl character formula for the finite dimensional irreducible representations of semisimple lie algebras respectively weyl kac character formula for the integrable highest weight modules over affine kac moody algebras so that the new weyl groups play the role of the usual weyl group respectively affine weyl group

brst cohomology and highest weight vectors i we initiate a program to study certain recent problems in non compact coset cft by the brst approach we derive a reduction formula for the brst cohomology by making use of a twisting by highest weight modules as illustrations we apply the formula to the bosonic string model and a rank one non compact coset model dpl our formula provides a completely new approach to non compact coset construction