BRST cohomology for certain reducible topological symmetries

In this paper, two different definitions of the BRST complex are connected. We obtain the BRST complex of topological quantum field theories (leading to equivariant cohomology) from the standard definition of the classical BRST complex (leading to Lie algebra cohomology) provided that we include ghosts for ghosts. Hereby, we use a finite dimensional model with a semi-direct product action ofH DiffM on a configuration spaceM, whereH is a compact Lie group representing the gauge symmetry in this model.     

BRST cohomology in classical mechanics

The classical (non-quantum) cohomology of the Becchi-Rouet-Stora-Tyutin (BRST) symmetry in phase space is defined and worked out. No group action for the gauge transformations is assumed. The results cover, therefore, the general case of an open algebra and are valid off-shell. Each cohomology class contains all BRST invariant functions with fixed ghost number (an integer) which differ from each other by a BRST variation. If the ghost number is negative there is only the trivial class whose elements are equivalent to zero. If the ghost number is positive or zero there is a bijective correspondence between the BRST classes and those of the exterior derivative along the gauge orbits. These gauge orbits lie in the phase space surface on which the gauge generators are constrained to vanish. The BRST invariant functions of ghost numberp are then related to closedp-forms along the orbits. The addition of a BRST variation corresponds to the addition of an exact form. Some comments about the quantum case are included. The physical meaning of the classes with ghost number greater than zero is not discussed.Chercheur qualifié du Fonds National de la Recherche Scientifique (Belgium)     

Algebraic renormalization of twisted N=2 supersymmetry with Z=2 central extension

We study the renormalizability of (massive) topological QCD based on the algebraic BRST technique by adopting a noncovariant Landau type gauge and making use of the full topological superalgebra. The most general local counter terms are determined and it is shown that in the presence of central charges the BRST cohomology remains trivial. By imposing an additional set of stability constraints it is proven that the matter action of topological QCD is perturbatively finite.     

BRST analysis of physical states for 2D gravity coupled toc≦1 matter

We consider 2D gravity coupled toc1 conformal matter in the conformal gauge. The Liouville system is represented by a free scalar field, ^L , with background charge such that the BRST operator imposing reparametrization invariance is nilpotent. We compute the cohomology of this BRST charge on the product of the Fock space of ^L with those of the ghosts and one other free scalar field, ^M representing the matter system. From this calculation the physical states of the full theory are determined. For thec<1 case the further projection from the Fock space of ^M to the irreducible representation, using Felder's resolution, reproduces the results of Lian and Zuckerman.Supported by the NSF Grant # PHY-88-04561Supported in part by the Department of Energy Contract # DE-FG03-84ER-40168 and by the USC Faculty Research and Innovation Fund     

Topological Field Theory Interpretation of String Topology

The string bracket introduced by Chas and Sullivan is reinterpreted from the point of view of topological field theories in the Batalin–Vilkovisky or BRST formalisms. Namely, topological action functionals for gauge fields (generalizing Chern–Simons and BF theories) are considered together with generalized Wilson loops. The latter generate a (Poisson or Gerstenhaber) algebra of functionals with values in the S¹-equivariant cohomology of the loop space of the manifold on which the theory is defined. It is proved that, in the case of GL(n,) with standard representation, the (Poisson or BV) bracket of two generalized Wilson loops applied to two cycles is the same as the generalized Wilson loop applied to the string bracket of the cycles. Generalizations to other groups are briefly described.     

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.Address after September 1992     

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.In [7], we have shown that the cohomology of a topological conformal field theory carries the structure of a batalin-Vilkovisky algebra (actually, two commuting such structures, corresponding to the two chiral sectors of the theory). In the second part of this paper, we describe the analogous algebraic structure on the equivariant cohomology of a topological conformal field theory: we call this structure a gravity algebra. This algebraic structure is a certain generalization of a Lie algebra, and is distinguished by the fact that it has an infinite sequence of independent operations {a ₁, ...,a _k },k2, satisfying quadratic relations generalizing the Jacobi rule. (The operad underlying the category of gravity algebras has been studied independently by Ginzburg-Kapranov [9].)The author is grateful to M. Bershadsky, E. Frenkel, M. Kapranov, G. Moore, R. Plesser and G. Zuckerman for the many ways in which they helped in the writing of this paper; also to the Department of Mathematics at Yale University for its hospitality while part of this paper was written.The author is partially supported by a fellowship of the Sloan Foundation and a research grant of the NSF.     

Kodaira-Spencer theory of gravity and exact results for quantum string amplitudes

We develop techniques to compute higher loop string amplitudes for twistedN=2 theories with=3 (i.e. the critical case). An important ingredient is the discovery of an anomaly at every genus in decoupling of BRST trivial states, captured to all orders by a master anomaly equation. In a particular realization of theN=2 theories, the resulting string field theory is equivalent to a topological theory in six dimensions, the Kodaira-Spencer theory, which may be viewed as the closed string analog of the Chern-Simons theory. Using the mirror map this leads to computation of the number of holomorphic curves of higher genus curves in Calabi-Yau manifolds. It is shown that topological amplitudes can also be reinterpreted as computing corrections to superpotential terms appearing in the effective 4d theory resulting from compactification of standard 10d superstrings on the correspondingN=2 theory. Relations withc=1 strings are also pointed out.This article was processed by the author using the Springer-Verlag TEX CoMaPhy macro package 1991.     

On moduli spaces of graded bundles

This is a review of the main topological properties of moduli spaces of graded bundles. These spaces consist of such elements of the cohomology H ¹(M, GL(n, E)), where E»M is a vector bundle, which are identical when restricted to ⁰ M=M × . Some explicit formulae in the case when M = P ^m and M is a torus are quoted. Applications to instantons and supergravity are discussed.     

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 -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.     

Refined Chiral Slavnov–Taylor Identities: Renormalization and Local Physics

We study the quantization of chiral QED with one family of massless fermions and the Stueckelberg field in order to give mass to the Abelian gauge field in a BRST-invariant way. We show that an extended Slavnov–Taylor (ST) identity can be introduced and fulfilled to all orders in perturbation theory by a suitable choice of the local actionlike counterterms, order by order in the loopwise expansion. This ST identity incorporates the Adler–Bardeen anomaly and involves the introduction of a doublet (K, c), where K is an external source of dimension 0 and c is the ghost field. By a purely algebraic argument we show that the introduction of the source K trivializes the cohomology of the extended linearized classical ST operator S ₀ in the Fadeev–Popov (FP) charge + 1 sector.We discuss the physical content of the extended ST identity and prove that the cohomology classes associated with S ₀ are modified with respect to the ones of the classical BRST differential s in the FP neutral sector (physical observables). This provides a counterexample showing that the introduction of a doublet can modify the cohomology of the model, as a consequence of the fact that the counting operator for the doublet (K, c) does not commute with S ₀ .We explicitly check that the physical states defined by s are no more physical states of the full quantized theory by showing that the subspace of the physical states corresponding to s is not left-invariant under the application of the S matrix, as a consequence of the extended ST identity.     

Observables in topological Yang-Mills theorieswith extended shift supersymmetry

We present a complete classification, at the classical level, of the observables of topological Yang-Mills theories with an extended shift supersymmetry of N generators, in any space-time dimension. The observables are defined as the Yang-Mills BRST cohomology classes of shift supersymmetry invariants. These cohomology classes turn out to be solutions of an N-extension of Witten's equivariant cohomology. This work generalizes results known in the case of shift supersymmetry with a single generator. Received: 8 March 2005, Published online: 21 October 2005 Supported in part by the Conselho Nacional de Desenvolvimento Científico e Tecnológico CNPq, Brazil     

The coupling of Chern–Simons theory to topological gravity

We couple Chern–Simons gauge theory to 3-dimensional topological gravity with the aim of investigating its quantum topological invariance. We derive the relevant BRST rules and Batalin–Vilkovisky action. Standard BRST transformations of the gauge field are modified by terms involving both its anti-field and the super-ghost of topological gravity. Beyond the obvious couplings to the metric and the gravitino, the BV action includes hitherto neglected couplings to the super-ghost. We use this result to determine the topological anomalies of certain higher ghost deformations of SU(N)$\mathit{SU}(N)$ Chern–Simons theory, introduced years ago by Witten. In the context of topological strings these anomalies, which generalize the familiar framing anomaly, are expected to be cancelled by couplings of the closed string sector. We show that such couplings are obtained by dressing the closed string field with topological gravity observables.     

Basic and equivariant cohomology in balanced topological field theory

We present a detailed algebraic study of the N=2 cohomological set-up describing the balanced topological field theory of Dijkgraaf and Moore. We emphasize the role of N=2 topological supersymmetry and internal symmetry by a systematic use of superfield techniques and of an covariant formalism. We provide a definition of N=2 basic and equivariant cohomology, generalizing Dijkgraaf’s and Moore’s, and of N=2 connection. For a general manifold with a group action, we show that: (i) the N=2 basic cohomology is isomorphic to the tensor product of the ordinary N=1 basic cohomology and a universal group theoretic factor; (ii) the affine spaces of N=2 and N=1 connections are isomorphic.     

Explicit form and convergence of 1-differential formal deformations of the poisson Lie algebra

Introducing the notion of an admissible graded Lie subalgebra A of the Nijenhui-Richardson algebra A(V) of the vector space V, it is shown that each cohomology class of a subcomplex C _A of the Chevalley-Eilenberg complex (C ⁰ M), extends in a cononical way as a graded cohomology class of weight — 1 of A. Applying this when V is the space N of smooth functions of a smooth manifold M, shows that the de Rham cohomology of M is induced by the graded cohomology of weight — 1 of the Schouten graded Lie algebra of M. This allows us to construct explicitly all 1-differential, nc formal deformations of the Poisson bracket of a symplectic manifold. The construction also applies for an arbitrary Poisson manifold but leads to only part of these deformations when the structure degenerates, as shown by an example.     

Geometry of the Aharonov–Bohm Effect

We show that the connection responsible for any Abelian or non-Abelian Aharonov–Bohm effect with n parallel “magnetic” flux lines in ℝ³, lies in a trivial G-principal bundle P→M, i.e. P is isomorphic to the product M×G, where G is any path connected topological group; in particular a connected Lie group. We also show that two other bundles are involved: the universal covering space , where path integrals are computed, and the associated bundle P× _G ℂ ^m →M, where the wave function and its covariant derivative are sections.     

BRST Cohomology and Phase Space Reduction in Deformation Quantization

In this article we consider quantum phase space reduction when zero is a regular value of the momentum map. By analogy with the classical case we define the BRST cohomology in the framework of deformation quantization. We compute the quantum BRST cohomology in terms of a "quantum" Chevalley-Eilenberg cohomology of the Lie algebra on the constraint surface. To prove this result, we construct an explicit chain homotopy, both in the classical and quantum case, which is constructed out of a prolongation of functions on the constraint surface. We have observed the phenomenon that the quantum BRST cohomology cannot always be used for quantum reduction, because generally its zero part is no longer a deformation of the space of all smooth functions on the reduced phase space. But in case the group action is "sufficiently nice", e.g. proper (which is the case for all compact Lie group actions), it is shown for a strongly invariant star product that the BRST procedure always induces a star product on the reduced phase space in a rather explicit and natural way. Simple examples and counterexamples are discussed.     

Krichever-Novikov formulation of topological conformal field theory

The Krichever-Novikov (KN) global operator formalism is applied to construct a topological conformal field theory on a compact Riemann surface from an N=2 super-conformal field theory. The topological version of the KN algebra is derived and the BRST charge is shown to be genus-dependent in this formulation. This leads to an interesting cohomology structure for the physical subspace of the Hilbert space.