Mark Rakowski

SUPERFIELD FORMULATION OF CHERN-SIMONS SUPERSYMMETRY

We discuss an extra supersymmetry present in the covariantly quantized Chern-Simons action within the superfield formalism. By introducing scalar superfields we show how the component transformations are naturally reproduced from the superfield transformation. When the superspace is extended to include an additional odd coordinate for the BRST symmetry, the entire theory is described by a single odd scalar superfield. The implications of this supersymmetry for the renormalized theory are also discussed.

Vector supersymmetry in topological field theory

Abstract We consider a type of supersymmetry which is present in the Chern-Simons and BF-type topological field theories in terms of a superconnection formalism. The combined global supersymmetry and BRST symmetry are realized in this superspace when non-covariant constraints on the supercurvature are chosen. This construction extends naturally within the bundle of frames approach to superspace, and we present a formalism which may lead to the construction of vector supergravity theories and their coupling to the topological field theories considered here.

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.

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 transformations. Different moduli spaces are accommodated through the choice of additional multiplets, and we explicitly construct these together with associated invariant actions.

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 BRST invariant scalar field. Vector supergravity presents a field theoretic model of a universal bundle for gravity, and we show how the vector supersymmetry operator can be used to generate a hierachy of observables.

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.

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.

DISCRETE QUANTUM FIELD THEORIES AND THE INTERSECTION FORM

It is shown that the standard mod-p valued intersection form can be used to define Boltzmann weights of subdivision invariant lattice models with gauge group Zp. In particular, we discuss a four-dimensional model which is based on the assignment of field variables to the two-simplices of the simplicial complex. The action is taken to be the intersection form defined on the second cohomology group of the complex, with coefficients in Zp. Subdivision invariance of the theory follows when the coupling constant is quantized and the field configurations are restricted to those satisfying a mod-p flatness condition. We present an explicit computation of the partition function for the manifold ±CP2, demonstrating non-triviality.

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 singularities on codimension two submanifolds via a puncture operator. This puncture operator is constructed from a naturally occuring differential form of co-degree two in the theory, and we are led to speculate on connections between this continuum quantum field theory, and the discrete Regge calculus.

Observable hierarchies from vector supersymmetry in cohomological field theories

Abstract We examine the underlying algebraic structure of cohomological field theories. The pattern of observable hierarchies is shown to be universally derived from the vector supersymmetry algebra. As such, vector supersymmetry plays a fundamental role in all topological field theories of this type. Within this algebraic approach, we present observable hierarchies for topological gravity, gauge theory and sigma models, together with their coupling.