algebraic k theory of group rings and the cyclotomic trace map we prove that the farrell jones assembly map for connective alge braic k theory is rationally injective under mild homological niteness condi tions on the group and assuming that a weak version of the leopoldt schneider conjecture holds for cyclotomic elds this generalizes a result of bokstedt hsiang and madsen and leads to a concrete description of a large direct summand of kn z g z q in terms of group homology in many cases the number theoretic conjectures are true so we obtain rational injectivity results about assembly maps in particular for whitehead groups under niteness assumptions on the group only the proof uses the cyclotomic trace map to topological cyclic homology bokstedt hsiang madsen s functor c and new general injectivity results about the assembly maps for topological hochschild homology and c

the bloch okounkov correlation functions of negative levels abstract bloch and okounkov introduced an n point correlation function on the fermionic fock space and found a closed formula in terms of theta functions this function affords several distinguished interpretations and in particular can be formulated as correlation functions on irreducible gl modules of level one these correlation functions have been generalized for irreducible integrable modules of gl and its classical lie subalgebras of positive levels by the authors in this paper we extend further these results and compute the correlation functions as well as the q dimensions for modules of gl and its classical subalgebras at negative levels

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

a q deformation of wakimoto modules primary fields and screening operators theq vertex operators of frenkel and reshetikhin are studied by means of aq deformation of the wakimoto module for the quantum affine algebrau q widehat mathfrak s mathfrak l 2 at an arbitrary levelk 0 2 a fock module version of theq deformed primary field of spinj is introduced as well as the screening operators which anti commute with the action ofu q widehat mathfrak s mathfrak l 2 up to a total difference of a field a proof of the intertwining property is given for theq vertex operators corresponding to the primary fields of spinj 1 2z 0 a sample calculation of the correlation function is also given

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

the geometry of tensor calculus i this paper defines and proves the correctness of the appropriate string diagrams for various kinds of monoidal categories with duals mathematics subject classifications 1991 18d10 52b11 53a45 57m25 68q10 82b23

equivariant operads string topology and tate cohomology from an operad fancyscript c with an action of a group g we construct new operads using the homotopy fixed point and orbit spectra these new operads are shown to be equivalent when the generalized g tate cohomology of fancyscript c is trivial applying this theory to the little disk operad fancyscript c 2 which is an s 1 operad we obtain variations on getzler s gravity operad which we show governs the chas sullivan string bracket

superpotential algebras and manifolds we study a special class of calabi yau algebras in the sense of ginzburg those arising as the fundamental group algebras of acyclic manifolds motivated partly by the usefulness of superpotential descriptions in motivic donaldson thomas theory we investigate the question of whether these algebras admit superpotential presentations we establish that the fundamental group algebras of a wide class of acyclic manifolds including all hyperbolic manifolds do not admit such descriptions disproving a conjecture of ginzburg regarding them we also describe a class of manifolds that do admit such descriptions and discuss a little their motivic donaldson thomas theory finally some links with topological field theory are described

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

the free loop space equivariant cohomology algebra of some formal spaces let mathbb k be a field of characteristic p 0 and s1 the unit circle we construct a model for the negative cylic homology of a commutative cochain algebra with two stages sullivan minimal model using the notion of shc formality introduced in bitjong and thomas topology 41 85 106 the main result of bitjong and el haouari math ann 338 347 354 and techniques of vigue poirrier j pure appl algebra 91 347 354 we compute the s1 equivariant cohomology algebras of the free loop spaces of the infinite complex projective space mathbb cp infty and the odd spheres s2q 1

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

batalin vilkovisky algebras and two dimensional topological field theories by a batalin vilkovisky algebra we mean a graded commutative algebraa together with an operator a a 1 such that 2 0 and a a is a graded derivation ofa for alla a in this article we show that there is a natural structure of a batalin vilkovisky algebra on the cohomology of a topological conformal field theory in two dimensions we make use of a technique from algebraic topology the theory of operads

reduction of jacobi manifolds a reduction procedure for jacobi manifolds is described in the algebraic setting of jacobi algebras as applications reduction by arbitrary submanifolds distributions and the reduction of jacobi manifolds with symmetry are discussed this generalized reduction procedure extends the well known reduction procedures for symplectic poisson contact and co symplectic structures

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

affine kac moody algebras and semi infinite flag manifolds we study representations of affine kac moody algebras from a geometric point of view it is shown that wakimoto modules introduced in 18 which are important in conformal field theory correspond to certain sheaves on a semi infinite flag manifold with support on its schhubert cells this manifold is equipped with a remarkable semi infinite structure which is discussed in particular the semi infinite homology of this manifold is computed the cousin grothendieck resolution of an invertible sheaf on a semi infinite flag manifold gives a two sided resolution of an irreducible representation of an affine algebras consisting of wakimoto modules this is just the brst complex as a byproduct we compute the homology of an algebra of currents on the real line with values in a nilpotent lie algebra

a homotopy theoretic realization of string topology let m be a closed oriented manifold of dimension d let lm be the space of smooth loops in m in 2 chas and sullivan defined a product on the homology h lm of degree d they then investigated other structure that this product induces including a batalin vilkovisky structure and a lie algebra structure on the s1 equivariant homology h s 1 lm these algebraic structures as well as others came under the general heading of the string topology of m in this paper we will describe a realization of the chas sullivan loop product in terms of a ring spectrum structure on the thom spectrum of a certain virtual bundle over the loop space we also show that an operad action on the homology of the loop space discovered by voronov has a homotopy theoretic realization on the level of thom spectra this is the cactus operad defined in 6 which is equivalent to operad of framed disks in this operad action realizes the chas sullivan bv structure on h lm we then describe a cosimplicial model of this ring spectrum and by applying the singular cochain functor to this cosimplicial spectrum we show that this ring structure can be interpreted as the cup product in the hochschild cohomology hh c m c m

on the negative cyclic homology of shc algebras let mathbb k be a field of characteristic p geq 0 and s1 the unit circle we prove that the shc structure on a cochain algebra a da induces an associative product on the negative cyclic homology hc a when the cochain algebra a da is the algebra of normalized cochains of the simply connected topological space x with coefficients in mathbb k then hc a is isomorphic as a graded algebra to h s 1 lx mathbb k the s1 equivariant cohomology algebra of lx the free loop space of x we use the notion of shc formality introduced in topology 41 85 106 2002 to compute the s1 equivariant cohomology algebras of the free loop space of the complex projective space mathbb c p n when n 1 0 p and of the even spheres s2n when p 2

on the newtonian limit of general relativity we establish rigorous results about the newtonian limit of general relativity by applying to it the theory of different time scales for non linear partial differential equations as developed in 4 1 8 roughly speaking we obtain a priori estimates for solutions to the einstein s equations an intermediate but fundamental step to show that given a newtonian solution there exist continuous one parameter families of solutions to the full einstein s equations the parameter being the inverse of the speed of light which for a finite amount of time are close to the newtonian solution these one parameter families are chosen via aninitialization procedure applied to the initial data for the general relativistic solutions this procedure allows one to choose the initial data in such a way as to obtain a relativistic solution close to the newtonian solution in any a priori given sobolev norm in some intuitive sense these relativistic solutions by being close to the newtonian one have little extra radiation content although actually this should be so only in the case of the characteristic initial data formulation along future directed light cones our results are local in the sense that they do not include the treatment of asymptotic regions global results are admittedly very important in particular they would say how differentiable the solutions are with respect to the parameter but their treatment would involve the handling of tools even more technical than the ones used here on the other hand this local theory is all that is needed for most problems of practical numerical computation

hochschild homology criteria for trivial algebra structures we prove two similar results by quite different methods the first one deals with augmented artinian algebras over a field we characterize the trivial algebra structure on the augmentation ideal in terms of the maximality of the dimensions of the hochschild homology or cyclic homology groups for the second result let x be a 1 connected finite cw complex we characterize the trivial algebra structure on the cohomology algebra of x with coefficients in a fixed field in terms of the maximality of the betti numbers of the free loop space

the homotopy theory of cyclic sets the aim of this note is to show that the homotopy theory of the cyclic sets of connes 3 is equivalent to that of s0 2 spaces i e spaces with a circle action and hence to that of spaces over k z 2

a chern character in cyclic homology we show that inner derivations act trivially on the cyclic cohomol ogy of the normalized cyclic complex w cl 3s c where i2 is a differential graded algebra this is then used to establish the fact that the map introduced in gj defines a chern character in k theory j m bismut recently extended the classical theory of chern characters to the equivariant theory of the free loop space lx he explicitly constructed an s invariant equivariantly closed form a e v on lx whose restriction to x the space of constant loops is ch f tr expv2 closely related to the equivariant theory of the loop space of x is a connes cyclic theory of the differential graded algebra q of differential forms on x more precisely chen s theory of iterated integrals allows us to view cyclic cycles of forms on x as equivariantly closed forms on the loop space lx this is a generalization of the point of view that a one form co on x can be viewed as a function on lx by y fy this suggests the possibility of interpreting the classical theory of chern char acters in the context of cyclic homology i e of extending ch e off the fixed point set x of the circle action on lx by way of a cyclic cycle over q in gjp getzler jones and petrack gave an explicit formula for a cyclic cy cle y e e which via iterated integrals maps to the bismut chern character il e over lx a priori their construction of y e e appears to depend on the choice of idempotent e describing the vector bundle e but in fact it is shown that y e e depends only on the isomorphism class of e in kq x and hence defines the desired chern character this is essentially proved by establishing the triviality of certain group actions on the cyclic homology of a differential graded algebra q these actions are natural generalizations of the usual adjoint action ad g given by conjugation by an invertible element g and its infinitesimal analogue ad given by inner derivation to connes cyclic bar complex of q 1 cyclic homology in this section we review the definition of the cyclic homology of a dga according to c kassel by a mixed complex we mean a triplet fe d b where g7 is a graded algebra and where d w ff x and b w w x are derivations of degrees 1 and 1 respectively and subject to the rela

mirror symmetry in two steps a i b we suggest an interpretation of mirror symmetry for toric varieties via an equivalence of two conformal field theories the first theory is the twisted sigma model of a toric variety in the infinite volume limit the a model the second theory is an intermediate model which we call the i model the equivalence between the a model and the i model is achieved by realizing the former as a deformation of a linear sigma model with a complex torus as the target and then applying to it a version of the t duality on the other hand the i model is closely related to the twisted landau ginzburg model the b model that is mirror dual to the a model thus the mirror symmetry is realized in two steps via the i model in particular we obtain a natural interpretation of the superpotential of the landau ginzburg model as the sum of terms corresponding to the components of a divisor in the toric variety we also relate the cohomology of the supercharges of the i model to the chiral de rham complex and the quantum cohomology of the underlying toric variety

n 2 structures in string theories the brst cohomology of any topological conformal field theory admits the structure of a batalin vilkovisky algebra and string theories are no exception loosely speaking we say that two topological conformal field theories are cohomologically equivalent if their brst cohomologies are isomorphic as batalin vilkovisky algebras in this paper we argue that any string theory regardless of the matter background is cohomologically equivalent to some twisted n 2 superconformal field theory we discuss three string theories in detail the bosonic string the nsr string and the w3 string in each case the way the cohomological equivalence is constructed can be understood as coupling the topological conformal field theory to topological gravity these results lend further supporting evidence to the conjecture that any topological conformal field theory is cohomologically equivalent to some topologically twisted n 2 superconformal field theory we end the paper with some speculative comments on massey products i

decouplings for surfaces in r4 abstract we prove a sharp decoupling for nondegenerate surfaces in r 4 this puts the progress in 1 on the lindelof hypothesis into a more general perspective

chiral equivariant cohomology of a point a first look the chiral equivariant cohomology contains and generalizes the classical equivariant cohomology of a manifold m with an action of a compact lie group g for any simple g there exist compact manifolds with the same classical equivariant cohomology which can be distinguished by this invariant when m is a point this cohomology is an interesting conformal vertex algebra whose structure is still mysterious in this paper we scratch the surface of this object in the case g su 2

bv structure on hochschild cohomology of the group ring of quaternion group of order eight in characteristic two abstract let k be an algebraically closed field of characteristic two and let q 8 be the quaternion group of order 8 we determine the gerstenhaber lie algebra structure and the batalin vilkovisky structure on the hochschild cohomology ring of the group algebra k q 8

quantization of the lie bialgebra of string topology let m be a smooth simply connected closed oriented manifold and lm the free loop space of m using a poincare duality model for m we show that the reduced equivariant homology of lm has the structure of a lie bialgebra and we construct a hopf algebra which quantizes the lie bialgebra

two constructions of affine lie algebra representations and boson fermion correspondence in quantum field theory we establish an isomorphism between the vertex and spinor representations of affine lie algebras for types dl 1 and dl 1 2 we also study decomposition of spinor representations using the infinite family of casimir operators and prove that they are either irreducible or have two irreducible components we show that the vertex and spinor constructions of the representations can be reformulated in the language of two dimensional quantum field theory in this physical context the two constructions yield the generalized sine gordon and thirring models respectively already in renormalized form the isomorphism of representations implies an equivalence of these two models which is known in quantum field theory as the boson fermion correspondence

entropy and collapsing of compact complex surfaces we study the problem of existence of mathcal f structures in the sense of cheeger and gromov but not necessarily polarized on compact complex surfaces we give a complete classification of compact complex surfaces of kahler type admitting mathcal f structures in the non kahler case we give a complete classification modulo the gap in the classification of surfaces of class vii we then use these results to study the minimal entropy problem for compact complex surfaces we prove that modulo the gap in the classification of surfaces of class vii all compact complex surfaces of kodaira dimension less than or equal to 1 have minimal entropy zero and such a surface admits a smooth metric g with zero topological entropy if and only if it is mathbb c p 2 a ruled surface of genus 0 or 1 a hopf surface a complex torus a kodaira surface a hyperelliptic surface or a kodaira surface modulo a finite group the key result we use to prove this is a new topological obstruction to the existence of metrics with vanishing topological entropy finally we show that these results fit perfectly into wall s study of geometric structures on compact complex surfaces for instance we show that the minimal entropy problem can be solved for a minimal compact kahler surface s of kodaira dimension infty 0 or 1 if and only if s admits a geometric structure modelled on mathbb c p 2 s 2 times s 2 s 2 times mathbb e 2 or mathbb e 4

grassmannian and string theory the infinite dimensional grassmannian manifold contains moduli spaces of riemann surfaces of all genera this well known fact leads to a conjecture that non perturbative string theory can be formulated in terms of the grassmannian we present new facts supporting this hypothesis in particular it is shown that grassmannians can be considered as generalized moduli spaces this statement permits us to define corresponding string amplitudes at least formally one can conjecture that it is possible to explain the relation between non perturbative and perturbative string theory by means of localization theorems for equivariant cohomology this conjecture is based on the characterization of moduli spaces relevant to string theory as sets consisting of points with large stabilizers in certain groups acting on the grassmannian we describe an involution on the grassmannian that could be related to s duality in string theory

on integer solutions of parsell vinogradov systems we prove a sharp upper bound on the number of integer solutions of the parsell vinogradov system in every dimension d ge 2

equivariant cohomology syzygies and orbit structure let x be a nice space with an action of a torus t we consider the atiyah bredon sequence of equivariant cohomology modules arising from the filtration of x by orbit dimension we show that a front piece of this sequence is exact if and only if the h bt module h t x is a certain syzygy moreover we express the cohomology of that sequence as an ext module involving a suitably defined equivariant homology of x one consequence is that the gkm method for computing equivariant cohomology applies to a poincare duality space if and only if the equivariant poincare pairing is perfect

the proof of the l 2 decoupling conjecture we prove the l 2 decoupling conjecture for compact hypersurfaces with positive denite second fundamental form and also for the cone this has a wide range of important consequences one of them is the validity of the discrete restriction conjecture which implies the full range of expected l p strichartz estimates for both the rational and up to n losses the irrational torus another one is an improvement in the range for the discrete restriction theorem for lattice points on the sphere various applications to additive combinatorics incidence geometry and number theory are also discussed our argument relies on the interplay between linear and multilinear restriction theory

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

hochschild homology of structured algebras abstract we give a general method for constructing explicit and natural operations on the hochschild complex of algebras over any prop with a multiplication we think of such algebras as a algebras with extra structure as applications we obtain an integral version of the costello kontsevich soibelman moduli space action on the hochschild complex of open tcfts the tradler zeinalian and kaufmann actions of sullivan diagrams on the hochschild complex of strict frobenius algebras and give applications to string topology in characteristic zero our main tool is a generalization of the hochschild complex

open closed homotopy algebra in mathematical physics in this paper we discuss various aspects of open closed homotopy algebras ochas presented in our previous paper inspired by zwiebach s open closed string field theory but that first paper concentrated on the mathematical aspects here we show how an ocha is obtained by extracting the tree part of zwiebach s quantum open closed string field theory we clarify the explicit relation of an ocha with kontsevich s deformation quantization and with the b models of homological mirror symmetry an explicit form of the minimal model for an ocha is given as well as its relation to the perturbative expansion of open closed string field theory we show that our open closed homotopy algebra gives us a general scheme for deformation of open string structures a algebras by closed strings l algebras

hypercommutative operad as a homotopy quotient of bv we give an explicit formula for a quasi isomorphism between the operads hycomm the homology of the moduli space of stable genus 0 curves and bv the homotopy quotient of batalin vilkovisky operad by the bv operator in other words we derive an equivalence of hycomm algebras and bv algebras enhanced with a homotopy that trivializes the bv operator

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

from zwiebach invariants to getzler relation we introduce the notion of zwiebach invariants that generalize gromov witten invariants and homotopical algebra structures we outline the induction procedure that induces the structure of zwiebach invariants on the sub bicomplex that gives the structure of gromov witten invariants on sub bicomplex with zero differentials we propose to treat hodge dgbv with 1 12 axiom as the simplest set of zwiebach invariants and explicitly prove that it induces wdvv and getzler equations in genera 0 and 1 respectively

two dimensional topological gravity and equivariant cohomology the analogy between topological string theory and equivariant cohomology for differentiable actions of the circle group on manifolds has been widely remarked on one of our aims in this paper is to make this analogy precise we show that topological string theory is the derived functor of semi relative cohomology just as equivariant cohomology is the derived functor of basic cohomology that homological algebra finds a place in the study of topological string theory should not surprise the reader granted that topological string theory is the conformal field theorist s algebraic topology

decouplings for three dimensional surfaces in mathbb r 6 r 6 we obtain the sharp l p decoupling for three dimensional nondegenerate surfaces in mathbb r 6 this can be thought of as a generalization of bourgain and demeter s result which is the sharp l p decoupling for two dimensional nondegenerate surfaces in mathbb r 4

a multiplication in cyclic homology we define a multiplication on the cyclic homology of a commutative cocommutative bialgebra h with superproduct in the case when h is a field of characteristic zero the cyclic homology becomes a polynomial algebra in one generator the loday quillen multiplication is trivial in that case in this paper we define an associative multiplication on the cyclic homology of certain commutative algebras h the definition is based on the cartesian product of sets instead of the shuffle product as in the case of lq and ga when h contains q our product is compatible with the hodge decomposition of hc h so it has the form hc n h hc j m h hc n m h the existence of this multiplication in the case h k is classical since it is just given by the tensor product of complex line bundle however its significance came to our attention in joint work with john klein as a consequence of our study of fiberwise morse theory on circle bundles ik and in other joint works with r penner and r hain by doing morse theory on circles john klein and i realized that there was a rationally nontrivial associative multiplication on the cyclic homology of the integral group ring on any cyclic group and that it had important consequences for pseudoisotopy theory this will be explained in a later paper in this paper we show that this multiplication in cyclic homotopy is defined first in the case when the algebra is h km k m where m is the underlying additive group of a ring then in the case when h is a hopf algebra with superproduct in the sense of gaucher ga and finally in the more general case of any algebra example 12 4 in fact we do not need additive inverses in m nor do we need an antipode for h so we can take m to be a monoid and h to be a bialgebra when h k i e m 1 the cyclic homology becomes a divided power algebra which is of course isomorphic to a polynomial algebra in one generator if k is a field of characteristic zero we assume throughout the paper that k is a commutative ring with 1 and that all tensor products are taken over k this includes those tensor products which appear in the definition of cyclic homology we also use connes original definition that a cyclic object in a category c is a covariant functor c where is connes cyclic category since this is the same as a contravariant functor c an outline of the paper is as follows in the first section we give a topological definition of the multiplication in hc km the construction is so elementary that received by the editors april 14 1994 1991 mathematics subject classification primary 18g60 secondary 16w30 this research was supported by nsf grant no dms 90 02512 c 1999 american mathematical society

the fixed point theorem in equivariant cohomology in this paper we study the s equivariant de rham cohomology of infinite dimensional s manifolds our main example is the free loop space lx where x is a finite dimensional manifold with the circle acting by rotating loops we construct a new form of equivariant cohomology h which agrees with the usual periodic equivariant cohomology in finite dimensions and we prove a suitable analogue of the classical fixed point theorem which is valid for loop spaces lx this gives a cohomological framework for studying differential forms on loop spaces and we apply these methods to various questions which arise from the work of witten 16 atiyah 2 and bismut 5 in particular we show following atiyah in 2 that the a polynomial of x arises as an equivariant characteristic class in the theory h of the normal bundle to x considered as the space of constant loops in lx

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

minimal models of quantum homotopy lie algebras via the bv formalism using the batalin vilkovisky formalism of mathematical physics an explicit construction for the minimal model of a quantum l algebra is given as a formal super integral the approach taken herein to these formal integrals is axiomatic and they can be approached using perturbation theory to obtain combinatorial formulae as shown in the appendix additionally there exists a canonical differential graded lie algebra morphism mapping formal functions on homology to formal functions on the whole space an inverse l algebra morphism to this differential graded lie algebra morphism is constructed as a formal super integral using the batalin vilkovisky formalism of mathematical physics an explicit construction for the minimal model of a quantum l algebra is given as a formal super integral the approach taken herein to these formal integrals is axiomatic and they can be approached using perturbation theory to obtain combinatorial formulae as shown in the appendix additionally there exists a canonical differential graded lie algebra morphism mapping formal functions on homology to formal functions on the whole space an inverse l algebra morphism to this differential graded lie algebra morphism is constructed as a formal super integral

wakimoto modules opers and the center at the critical level abstract wakimoto modules are representations of affine kac moody algebras in fock modules over infinite dimensional heisenberg algebras in this paper we present the construction of the wakimoto modules from the point of view of the vertex algebra theory we then use wakimoto modules to identify the center of the completed universal enveloping algebra of an affine kac moody algebra at the critical level with the algebra of functions on the space of opers for the langlands dual group on the punctured disc giving another proof of the theorem of b feigin and the author

the frobenius properad is koszul we show koszulness of the prop governing involutive lie bialgebras and also of the props governing non unital and unital counital frobenius algebras solving a long standing problem this gives us minimal models for their deformation complexes and for deformation complexes of their algebras which are discussed in detail using an operad of graph complexes we prove with the help of an earlier result of one of the authors that there is a highly non trivial action of the grothendieck teichm uller group grt 1 on completed versions of the minimal models of the properads governing lie bialgebras and involutive lie bialgebras by automorphisms as a corollary one obtains a large class of universal deformations of any involutive lie bialgebra and any frobenius algebra parameterized by elements of the grothendieck teichm uller lie algebra we also prove that for any given homotopy involutive lie bialgebra structure in a vector space there is an associated homotopy batalin vilkovisky algebra structure on the associated chevalley eilenberg complex

the cyclic homology of crossed product algebras in their article 9 on cyclic homology feigin and tsygan have given a spectral sequence for the cyclic homology of a crossed product algebra generalizing burghelea s calculation 4 of the cyclic homology of a group algebra for an analogous spectral sequence for the hochschild homology of a crossed product algebra see brylinski 2 3 in this article we give a new derivation of this spectral sequence and generalize it to negative and periodic cyclic homology hc a and hp a the method of proof is itself of interest since it involves a natural generalization of the notion of a cyclic module in which the condition that the morphism n n is cyclic of order n 1 is relaxed to the condition that it be invertible we call this category the paracyclic category given a paracyclic module p we can define a chain complex c p with differentials b and b which respectively lower and raise degree the condition that the module p is paracyclic translates to the condition on c p that 1 bb bb is invertible our main result is to show that there is an analogue of the eilenberg zilber theorem for bi paracyclic modules it is then easy to obtain a new expression for the cyclic homology of a crossed product algebra which leads immediately to the spectral sequence of feigin and tsygan if m is a module over a commutative ring k we will denote by m k the iterated tensor product defined by m 0 k and m k 1 m k m if m and n are graded modules we will denote by m n their graded tensor product we would like to thank j block j l brylinski and c kassel for a number of interesting discussions on the results presented here

einstein s equations with asymptotically stable constraint propagation we introduce a proposal to modify einstein s equations by embedding them in a larger symmetric hyperbolic system the additional dynamical variables of the modified system are essentially first integrals of the original constraints the extended system of equations reproduces the usual dynamics on the constraint surface of general relativity and therefore naturally includes the solutions to einstein gravity the main feature of this extended system is that at least for a linearized version of it the constraint surface is an attractor of the time evolution this feature suggests that this system may be a useful alternative to einstein s equations when obtaining numerical solutions to full nonlinear gravity

bosonization on higher genus riemann surfaces we prove the equivalence between certain fermionic and bosonic theories in two spacetime dimensions the theories have fields of arbitrary spin on compact surfaces with any number of handles global considerations require that we add new topological terms to the bosonic action the proof that our prescription is correct relies on methods of complex algebraic geometry

decoupling for moment manifolds associated to arkhipov chubarikov karatsuba systems abstract we prove l p l p decoupling inequalities for a class of moment manifolds these inequalities imply optimal mean value estimates for multidimensional weyl sums of the kind considered by arkhipov chubarikov and karatsuba and by parsell in our proofs we take a new point of view on the bourgain demeter guth induction on scales argument this point of view substantially simplifies even the proof of l 2 l p decoupling for the moment curve

cyclic homology and equivariant homology the purpose of this paper is to explore the relationship between the cyclic homology and cohomology theories of connes 9 11 see also loday and quillen 20 and if equivariant homology and cohomology theories here ii is the circle group the most general results involve the definitions of the cyclic homology of cyclic chain complexes and the notions of cyclic and cocyclic spaces so precise statements will be postponed until w 3 in this introduction we explain some of the formal similarities between the cyclic theory and the equivariant theory and give two examples where the general results apply let a be an associative algebra over a commutative ring k then one can form the cyclic homology hc a and cohomology hc a of a these groups have periodicity operators

boundary quantum knizhnik zamolodchikov equations and bethe vectors solutions to boundary quantum knizhnik zamolodchikov equations are constructed as bilateral sums involving off shell bethe vectors in case the reflection matrix is diagonal and only the 2 dimensional representation of mathcal u q widehat mathfrak sl 2 is involved we also consider their rational and classical degenerations

the unstable integral homology of the mapping class groups of a surface with boundary we construct a graph complex calculating the integral homology of the bordered mapping class groups we compute the homology of the bordered mapping class groups of the surfaces s 1 1 s 1 2 and s 2 1 using the circle action on this graph complex we build a double complex and a spectral sequence converging to the homology of the unbordered mapping class groups we compute the homology of the punctured mapping class groups associated to the surfaces s 1 1 and s 2 1 finally we use miller s operad to get the first kudo araki and browder operations on our graph complex we also consider an unstable version of the higher kudo araki dyer lashoff operations

on the topological hochschild homology of bu i the purpose of this paper and its sequel is to determine the homotopy groups of the spectrum thh here p is an odd prime is the adams summand of p local connective k theory see for example 25 and thh is the topological hochschild homology construction introduced by bokstedt in 3 in the present paper we will determine the mod p homotopy groups of thh and also the integral homotopy groups of thh l where l denotes the periodic adams summand in the sequel we will investigate the integral homotopy groups of thh using our present results as a starting point the thh construction appears to be of basic importance in algebraic k theory because it combines two useful properties it can be used to construct good approximations to the algebraic k theory functor and it is very accessible to calculation we shall review what is known about the first property in a moment the second property was demonstrated by bokstedt s calculation in his paper 4 of the homotopy groups of thh hz p and thh hz here hz p and hz denote the evident eilenberg mac lane spectra it is natural to ask about thh r for other popular ring spectra r and our work is a first step in this direction we pay special attention to the connective case because this is the case which is likely to be relevant in applications see subsection 1 4 below the calculation which we present in this paper is a homotopy theoretic one which uses the adams spectral sequence hereafter abbreviated ass this calculation has several interesting features in particular it is a pleasing example of an ass calculation in which although there are infinitely many differentials it is still possible to get the complete answer here is a summary of the contents of the paper in section 2 we review the facts we need to know about ordinary hochschild homology in section 3 we do the same

homotopy algebras inspired by classical open closed string field theory we define a homotopy algebra associated to classical open closed strings we call it an open closed homotopy algebra ocha it is inspired by zwiebach s open closed string field theory and also is related to the situation of kontsevich s deformation quantization we show that it is actually a homotopy invariant notion for instance the minimal model theorem holds also we show that our open closed homotopy algebra gives us a general scheme for deformation of open string structures a algebras by closed strings l 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

well posed forms of the 3 1 conformally decomposed einstein equations we show that well posed conformally decomposed formulations of the 3 1 einstein equations can be obtained by densitizing the lapse and by combining the constraints with the evolution equations we compute the characteristics structure and verify the constraint propagation of these new well posed formulations in these formulations the trace of the extrinsic curvature and the determinant of the 3 metric are singled out from the rest of the dynamical variables but are evolved as part of the well posed evolution system the only free functions are the lapse density and the shift vector we find that there is a 3 parameter freedom in formulating these equations in a well posed manner and that part of the parameter space found consists of formulations with causal characteristics namely characteristics that lie only within the lightcone in particular there is a 1 parameter family of systems whose characteristics are either normal to the slicing or lie along the lightcone of the evolving metric

on the relation between open and closed topological strings we discuss the relation between open and closed string correlators using topological string theories as a toy model we propose that one can reconstruct closed string correlators from the open ones by considering the hochschild cohomology of the category of d branes we compute the hochschild cohomology of the category of d branes in topological landau ginzburg models and partially verify the conjecture in this case

the second gelfand dickey bracket as a bracket on a poisson lie grassmannian we introduce a poisson structure on a grassmannian grk v on which the poisson lie group gl v acts in a poisson lie way we discuss the analytic complications connected with the infinite dimensional casev c and show that an open subset of grk v with this poisson structure is isomorphic to the gelfand dickey manifold of differential operators of orderk with the second gelfand dickey bracket in fact we introduce as a consequence a poisson lie action of an enormous group on the gelfand dickey manifold generalizing on the semiclassical level the sugavara inclusion

equivariant localization and completion in cyclic homology and derived loop spaces abstract we prove an equivariant localization theorem over an algebraically closed field of characteristic zero for smooth quotient stacks by reductive groups x g in the setting of derived loop spaces as well as hochschild homology and its cyclic variants we show that the derived loop spaces of the stack x g and its classical z fixed point stack 0 x z g z become equivalent after completion along a semisimple parameter z g g implying the analogous statement for hochschild and cyclic homology of the dg category of perfect complexes perf x g we then prove an analogue of the atiyah segal completion theorem in the setting of periodic cyclic homology where the completion of the periodic cyclic homology of perf x g at the identity e g g is identified with a 2 periodic version of the derived de rham cohomology of x g together these results identify the completed periodic cyclic homology of a stack x g over a parameter z g g with the 2 periodic derived de rham cohomology of its z fixed points

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