The algebraic structure of cohomological field theory

Abstract

The algebraic foundation of cohomological eld 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. A s such, their correlation functions correspond to cohomology classes on moduli spaces of Riemannian connections. In this uniformization approach, di erent moduli spaces are obtained by i n troducing curvature singularities on codimension two submanifolds via a puncture operator. This puncture operator is constructed from a naturally occurring di erential form of co-degree two in the theory, and we are led to speculate on connections between this continuum quantum eld theory, and the discrete Regge calculus.

Symmetries and observables for BF theories in superspace

“…This is a symmetry of the quantum action (4,10) and satisfies the anticommutation relations (26) with the BRST-operator (7,9). The analysis done on the renormalization, finiteness and anomalies of ordinary BF-theories using vector supersymmetry can thus be applied directly to the supersymmetric BF-theories.…”

“…The anticommutation relations of the operators δ,δ and s,s are Together with the BRST-operators δ andδ the operators s ands can be combined to form a generator of N = 2 supersymmetry algebra [10,17,9]. The vector supersymmetries can be formulated also for the non-supersymmetric BF-theories.…”

“…One might argue though that because the theories are topological their physical quantities are not dependent on the metric of the manifold. In any case, the vector supersymmetry has been established as a common feature of many topological theories [9] and a useful tool, not only in the study of renormalization and related topics but also in finding new observables.…”

“…The BRST transformations of the fields can be derived from the superspace curvature twoform using a method similar to that of [9,8,13,14,15,16] for Donaldson theory and Witten type topological theories. But because of the N = 2 superspace with two anticommuting coordinates of opposite ghost numbers we can extend this method to include also the anti-BRST symmetry.…”

“…By formulating the theory in superspace a large set of observables, including previously unknown ones, can be derived form the superspace curvature. In [9] a vector-like supersymmetry similar to that found in ordinary BF-models and Chern-Simons theory [10,11] was constructed for the SBF-models. In particular, the hierarchy of observables constructed from the supercurvature can be derived from one initial observable with the help of the vector supersymmetry.…”