New Perspectives on the BRST-algebraic Structure of String Theory
Bong H. Lian, Gregg J. Zuckerman
TL;DR
The paper shows that BRST cohomology in string theory forms a coboundary Gerstenhaber algebra with a dot product and a descent-derived bracket, identities holding up to homotopy off-shell. In the c=1 model, it gives a precise realization of this structure, identifying an exact sequence $0\to H(-)\to H\to {\cal A}\to 0$ and an isomorphism $H(+)\cong {\cal A}^*$, while connecting the bracket to the BV anti-bracket via a $b_0$–$\Delta$ correspondence. It extends these results to left-right moving sectors, topological chiral algebras, and general topological CFTs, and discusses deformations, modules, and closed-string field theory through the lens of Gerstenhaber algebra theory. The work links BRST algebraic structures to broader mathematical frameworks (Hochschild cohomology, Schouten/BDV formalisms) and provides concrete computational descriptions (ground ring, generators, dualities) that illuminate symmetry and deformation phenomena in string backgrounds. Overall, it establishes a unified algebraic perspective on BRST cohomology that unifies physical descent relations with Gerstenhaber and BV formalisms, with concrete realizations in the c=1 model and generalizations to topological backgrounds.
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.