The algebraic structure of cohomological field theory

Abstract The algebraic foundation of cohomological field theory is presented. It is shown that these theories are based upon realizations of an algebra which contains operators for both BRST and vector supersymmetry. Through a localization of this algebra, we construct a theory of cohomological gravity in arbitrary dimensions. The observables in the theory are polynomial, but generally non-local operators, and have a natural interpretation in terms of a universal bundle for gravity. As such, their correlation functions correspond to cohomology classes on moduli spaces of Riemannian connections. In this uniformization approach, different moduli spaces are obtained by introducing curvature si…

A field theoretic realization of a universal bundle for gravity

Abstract Based upon a local vector supersymmetry algebra, we discuss the general structure of the quantum action for topological gravity theories in arbitrary dimensions. The precise form of the action depends on the particular dimension, and also on the moduli space of interest. We describe the general features by examining a theory of topological gravity in two dimensions, with a moduli space specified by vanishing curvature two-form. It is shown that these topological gravity models together with their observables provide a field theoretic realization of a universal bundle for gravity.

Vector supersymmetry in the universal bundle

Abstract We present a vector supersymmetry for Witten-type topological gauge theories, and examine its algebra in terms of a superconnection formalism. When covariant constraints on the supercurvature are chosen, a correspondence is established with the universal bundle construction of Atiyah and Singer. The vector supersymmetry represents a certain shift operator in the curvature of the universal bundle, and can be used to generate the hierarchy of observables in these theories. This formalism should lead to the construction of vector supergravity theories, and perhaps to the gravitational analogue of the universal bundle.

Topological field theory

On the universal bundle for gravity

Abstract We construct a supergravity type theory based on a superspace whose odd directions consist of a vector, together with a scalar representing a topological BRST shift symmetry. As such, the resulting theory is a theory of topological gravity. The gravitino is interpreted as a ghost field for this shift symmetry and plays the usual role of gauge field for local supersymmetry. Our construction is within the bundle of frames approach to superspace where covariant torsion constraints are analyzed, and we find that the resulting theory contains additional fields which are not present in existing theories of topological gravity. In particular, a minimal solution exists which contains a BRS…

Equivariance in topological gravity

Abstract We present models of topological gravity for a variety of moduli space conditions. In four dimensions, we construct a model for self-dual gravity characterized by the moduli condition R + μν =0, and in two dimensions we treat the case of constant scalar curvature. Details are also given for both flat and Yang-Mills type moduli conditions in arbitrary dimensions. All models are based on the same fundamental multiplet which conveniently affords the construction of a complete hierarchy of observables. This approach is founded on a symmetry algebra which includes a local vector supersymmetry, in addition to a global BRST-like symmetry which is equivariant with respect to Lorentz transf…

A star product in lattice gauge theory

Abstract We consider a variant of the cup product of simplicial cochains and its applications in discrete formulations of non-abelian gauge theory. The standard geometrical ingredients in the continuum theory all have natural analogues on a simplicial complex when this star product is used to translate the wedge product of differential forms. Although the star product is non-associative, it is graded-commutative, and the coboundary operator acts as a deviation on the star algebra. As such, it is reminiscent of the star product considered in some approaches to closed string field theory, and we discuss applications to the three dimensional non-abelian Chern-Simons theory.