New Perspectives on the BRST-algebraic Structure of String Theory

TL;DR

This paper introduces a new interpretation of the BRST algebra in string theory using Gerstenhaber brackets, revealing a rich algebraic structure that connects to BV theory and extends to topological conformal field theories.

Contribution

It provides a novel Gerstenhaber algebra framework for BRST cohomology in string theory, including off-shell identities up to homotopy and applications to various models.

Findings

Gerstenhaber bracket compatible with cohomology product

Connection between BRST-Gerstenhaber algebra and BV anti-bracket

Generalization to topological conformal field theories

Abstract

Motivated by the descent equation in string theory, we give a new interpretation for the action of the symmetry charges on the BRST cohomology in terms of what we call {\em the Gerstenhaber bracket}. This bracket is compatible with the graded commutative product in cohomology, and hence gives rise to a new class of examples of what mathematicians call a {\em Gerstenhaber algebra}. The latter structure was first discussed in the context of Hochschild cohomology theory \cite{Gers1}. Off-shell in the (chiral) BRST complex, all the identities of a Gerstenhaber algebra hold up to homotopy. Applying our theory to the c=1 model, we give a precise conceptual description of the BRST-Gerstenhaber algebra of this model. We are led to a direct connection between the bracket structure here and the anti-bracket formalism in BV theory \cite{W2}. We then discuss the bracket in string backgrounds with both the left and the right movers. We suggest that the homotopy Lie algebra arising from our Gerstenhaber bracket is closely related to the HLA recently constructed by Witten-Zwiebach. Finally, we show that our constructions generalize to any topological conformal field theory.