On Some Algebraic Structures Arising in String Theory

TL;DR

This paper simplifies the proof that the homology of topological chiral algebras has a BV-algebra structure and extends these results to non-chiral cases, using a characterization involving an odd second order derivation.

Contribution

The authors provide a simplified proof of BV-algebra structures in topological chiral algebra homology and generalize the results to non-chiral cases using a new characterization theorem.

Findings

Homology of topological chiral algebras admits a BV-algebra structure.

A simple proof of the BV-algebra structure is provided.

Results are extended to non-chiral algebra cases.

Abstract

Lian and Zuckerman proved that the homology of a topological chiral algebra can be equipped with the structure of a BV-algebra; \ie one can introduce a multiplication, an odd bracket, and an odd operator $\Delta$ having the same properties as the corresponding operations in Batalin-Vilkovisky quantization procedure. We give a simple proof of their results and discuss a generalization of these results to the non chiral case. To simplify our proofs we use the following theorem giving a characterization of a BV-algebra in terms of multiplication and an operator $\Delta$: {\em If $A$ is a supercommutative, associative algebra and $\Delta$ is an odd second order derivation on $A$ satisfying $\Delta^2=0$, one can provide $A$ with the structure of a BV-algebra.}