A geometrical description of local and global anomalies

The general topological framework for testing the possible occurrence of anomalies in gauge theories can be constructed in terms of the theory of group actions on line bundles through the introduction of a suitable group cohomology. In this Letter, we generalize this construction in such a way that it can be applied to a larger class of theories, allowing for a noncontractible configuration space and a nonconnected ‘gauge’ group. This construction find applications to the problem of the lifts of principal group actions. As a physical application, we compare the mechanisms of the anomalies cancelation in gauge and string theories, through a geometrical splitting of local and global anomalies.     

Local BRST Cohomology and Covariance

The paper provides a framework for a systematic analysis of the local BRST cohomology in a large class of gauge theories. The approach is based on the cohomology of s+d in the jet space of fields and antifields, s and d being the BRST operator and exterior derivative respectively. It relates the BRST cohomology to an underlying gauge covariant algebra and reduces its computation to a compactly formulated problem involving only suitably defined generalized connections and tensor fields. The latter are shown to provide the building blocks of physically relevant quantities such as gauge invariant actions, Noether currents and gauge anomalies, as well as of the equations of motion. Received: 25 July 1996 / Accepted: 23 April 1997     

Cohomological analysis of bosonic D-strings and 2d sigma models coupled to abelian gauge fields

We analyze completely the BRST cohomology on local functionals for two-dimensional sigma models coupled to abelian world-sheet gauge fields, including effective bosonic D-string models described by Born-Infeld actions. In particular we prove that the rigid symmetries of such models are exhausted by the solutions to generalized Killing vector equations which we have presented recently, and provide all the consistent first order deformations and candidate gauge anomalies of the models under study. For appropriate target space geometries we find nontrivial deformations both of the abelian gauge transformations and of the world-sheet diffeomorphisms, and antifield-dependent candidate anomalies for both types of symmetries separately, as well as mixed ones.     

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 model for equivariant cohomology and representatives for the equivariant thom class

In this paper the BRST formalism for topological field theories is studied in a mathematical setting. The BRST operator is obtained as a member of a one parameter family of operators connecting the Weil model and the Cartan model for equivariant cohomology. Furthermore, the BRST operator is identified as the sum of an equivariant derivation and its Fourier transform. Using this, the Mathai-Quillen representative for the Thom class of associated vector bundles is obtained as the Fourier transform of a simple BRST closed element.Supported by the SV FOM/SMC Mathematical Physics, The Netherlands     

Some remarks on BRS transformations,anomalies and the cohomology of the Lie algebra of the group of gauge transformations

We show that ghosts in gauge theories can be interpreted as Maurer-Cartan forms in the infinite dimensional group ? of gauge transformations. We examine the cohomology of the Lie algebra of ? and identify the coboundary operator with the BRS operator. We describe the anomalous terms encountered in the renormalization of gauge theories (triangle anomalies) as elements of these cohomology groups.     

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.     

Existence,uniqueness and cohomology of the classical BRST charge with ghosts of ghosts

A complete canonical formulation of the BRST theory of systems with redundant gauge symmetries is presented. These systems includep-form gauge fields, the superparticle, and the superstring. We first define the Koszul-Tate differential and explicitly show how the introduction of the momenta conjugate to the ghosts of ghosts makes it acyclic. The global existence of the BRST generator is then demonstrated, and the BRST charge is proved to be unique up to canonical transformations in the extended phase space, which includes the ghosts. Finally, the BRST cohomology in classical mechanics is investigated and shown to be equal to the cohomology of the exterior derivative along the gauge orbits, as in the irreducible case. This is done by re-expressing the exterior algebra along the gauge orbits as a free differential algebra containing generators of higher degree, which are identified with the ghosts of ghosts. The quantum cohomology is not dealt with.Aspirant du Fonds National de la Recherche Scientifique (Belgium)Chercheur qualifié au Fonds National de la Recherche Scientifique (Belgium)     

TOPOLOGY OF CURRENTS,EFFECTIVE LAGRANGIANS AND ANOMALIES—SOME RECENT PROGRESS IN TOPOLOGICAL INVESTIGATIONS ON GAUGE THEORIES

We briefly review some recent progress i.n topological investigations on quantum field theories with gauge symmetries.The subjects include fractional charges, topological currents,chiral effective Lagrangians and tt:cir gauge invariances,gauge anomalies in different dimensions etc, We explain tree generalization of the Chern-Simons secondary classes, the cohomologies of gauge groups realized upon these generalized classes and their cornbinatory forms which satisfy the generalized transgression formulas, and make use of them in the cohomological analyses on the structure of anomalies in different dimensionswhich represent Topological obstructions to the existence of consistent quantized gauge field theories.     

A Poincaré lemma for sigma models of AKSZ type

For a sigma model of AKSZ type, we show that the local BRST cohomology is isomorphic to the cohomology of the target space differential when restricted to coordinate neighborhoods both in the base and in the target. An analogous result is shown to hold for the cohomology in the space of functional multivectors. Applications of these latter cohomology classes in the context of the inverse problem of the calculus of variation for general gauge systems are also discussed.     

Quaternionic Kähler Detour Complexes and $${\mathcal{N} = 2}$$ Supersymmetric Black Holes

We study a class of supersymmetric spinning particle models derived from the radial quantization of stationary, spherically symmetric black holes of four dimensional \({{\mathcal N} = 2}\) supergravities. By virtue of the c-map, these spinning particles move in quaternionic Kähler manifolds. Their spinning degrees of freedom describe mini-superspace-reduced supergravity fermions. We quantize these models using BRST detour complex technology. The construction of a nilpotent BRST charge is achieved by using local (worldline) supersymmetry ghosts to generate special holonomy transformations. (An interesting byproduct of the construction is a novel Dirac operator on the superghost extended Hilbert space.) The resulting quantized models are gauge invariant field theories with fields equaling sections of special quaternionic vector bundles. They underly and generalize the quaternionic version of Dolbeault cohomology discovered by Baston. In fact, Baston’s complex is related to the BPS sector of the models we write down. Our results rely on a calculus of operators on quaternionic Kähler manifolds that follows from BRST machinery, and although directly motivated by black hole physics, can be broadly applied to any model relying on quaternionic geometry.     

The topological meaning of non-abelian anomalies

We show how the non-abelian anomaly for gauge fields coupled to Weyl fermions in 2n dimensions is related to the non-trivial topology of gauge orbit space. The form of the anomaly and its normalization are shown to follow from a familiar index theorem for a certain (2n + 2)-dimensional Dirac operator. We are thus able to recover and give topological meaning to a variety of results concerning anomalies in 4- and higher-dimensional theories.     

Characteristic cohomology ofp-form gauge theories

The characteristic cohomologyH ^k _char(d) for an arbitrary set of freep-form gauge fields is explicitly worked out in all form degreesk < n — 1, wheren is the spacetime dimension. It is shown that this cohomology is finite-dimensional and completely generated by the forms dual to the field strengths. The gauge invariant characteristic cohomology is also computed. The results are extended to interactingp-form gauge theories with gauge invariant interactions. Implications for the BRST cohomology are mentioned.     

Duality theorems in BRST cohomology

In classical mechanics, there is no duality theorem relating the BRST cohomologies at positive and negative ghost numbers since these generically fail to be isomorphic. It is shown in this paper, however, that a duality theorem for the BRST operator cohomology can be established in quantum mechanics. Furthermore, when the hermicity properties of the quantum BRST formalism—which are in general just formal—turn out to be actually well defined, this duality theorem also holds for the state cohomology, as a consequence of the non degenerate pairing between subspaces at positive and negative ghost numbers defined by the BRST scalar product. In the case of gauge systems quantized in the Schrödinger representation with compact gauge orbits, the duality theorem contains ordinary Poincaré duality for a compact manifold. In the Fock representation, the duality theorem sheds a new light on existing decoupling theorems. The comparison with the classical situation is also briefly discussed.     

Notes on Certain (0,2) Correlation Functions

In this paper we shall describe some correlation function computations in perturbative heterotic strings that, for example, in certain circumstances can lend themselves to a heterotic generalization of quantum cohomology calculations. Ordinary quantum chiral rings reflect worldsheet instanton corrections to correlation functions involving products of elements of Dolbeault cohomology groups on the target space. The heterotic generalization described here involves computing worldsheet instanton corrections to correlation functions defined by products of elements of sheaf cohomology groups. One must not only compactify moduli spaces of rational curves, but also extend a sheaf (determined by the gauge bundle) over the compactification, and linear sigma models provide natural mechanisms for doing both. Euler classes of obstruction bundles generalize to this language in an interesting way.     

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.     

Modular Localization and the Foundational Origin of Integrability

The main aim of this work is to relate integrability in QFT with a complete particle interpretation directly to the principle of causal localization, circumventing the standard method of finding sufficiently many conservation laws. Its precise conceptual-mathematical formulation as “modular localization” within the setting of local operator algebras also suggests novel ways of looking at general (non-integrable) QFTs which are not based on quantizing classical field theories. Conformal QFT, which is known to admit no particle interpretation, suggest the presence of a “partial” integrability, referred to as “conformal integrability”. This manifests itself in a “braid-permutation” group structure which contains in particular informations about the anomalous dimensional spectrum. For chiral conformal models this reduces to the braid group as it is represented in Hecke- or Birman-Wenzl-algebras associated to chiral models. Another application of modular localization mentioned in this work is an alternative to the BRST formulation of gauge theories in terms of stringlike vectorpotentials within a Hilbert space setting.     

On equivalence of Floer's and quantum cohomology

We show that the Floer cohomology and quantum cohomology rings of the almost Kähler manifoldM, both defined over the Novikov ring of the loop space M, are isomorphic. We do it using a BRST trivial deformation of the topological A-model. The relevant aspect of noncompactness of the moduli of pseudoholomorphic instantons is discussed. It is shown nonperturbatively that any BRST trivial deformation of A model which does not change the dimensions of BRST cohomology does not change the topological correlation functions either.