The Algebraic Structure of BRST Operators and their Applications

Abstract

This dissertation has two related but distinct parts. In the first part of the thesis, we construct a new sequence of generators of the BRST complex and reformulate the BRST differential so that it acts on elements of the complex much like the Maurer-Cartan differential acts on left-invariant forms. Thus our BRST differential is formally analogous to the differential defined on the BRST formulation of the Chevalley-Eilenberg cochain complex of a Lie algebra. Moreover, for an important class of physical theories, we show that in fact the differential is a Chevalley-Eilenberg differential. As one of the applications of our formalism, we show that the BRST differential provides a mechanism which permits us to extend a nonintegrable system of vector fields on a manifold to an integrable system on an extended manifold.

In the second part of the thesis, we isolate a new concept which we call the chain extension of a $D$-algebra. We demonstrate that this idea is central to to a number of applications to algebra and physics. Chain extensions may be regarded as generalizations of ordinary algebraic extensions of Lie algebras. Applications of our theory provide a new constructive approach to BRST theories which only contains three terms; in particular, this provides a new point of view concerning consistent deformations and constrained Hamiltonian systems. Finally, we show that a similar development provides a method by which Lie algebra deformations may be encoded into the structure maps of an sh-Lie algebra with three terms.