BV-Structure of the Cohomology of Nilpotent Subalgebras and the Geometry of (W-) Strings

TL;DR

This paper constructs a BV-algebra from the cohomology of nilpotent subalgebras in simple Lie algebras, conjecturing its relation to physical operators in noncritical W-strings, and verifies this in two specific cases.

Contribution

It introduces a new BV-algebra structure on the cohomology related to nilpotent subalgebras and connects it to the algebra of physical operators in noncritical W-strings.

Findings

Constructed a BV-algebra from the cohomology of regular functions on G.

Conjectured the algebra describes all physical operators in noncritical W-strings.

Verified the conjecture for the Virasoro and W_3 string cases.

Abstract

Given a simple, simply laced, complex Lie algebra $\bfg$ corresponding to the Lie group $G$, let $\bfnp$ be the subalgebra generated by the positive roots. In this paper we construct a BV-algebra $\fA[\bfg]$ whose underlying graded commutative algebra is given by the cohomology, with respect to $\bfnp$, of the algebra of regular functions on $G$ with values in $\mywedge (\bfnp\backslash\bfg)$. We conjecture that $\fA[\bfg]$ describes the algebra of {\it all} physical (i.e., BRST invariant) operators of the noncritical $\cW[\bfg]$ string. The conjecture is verified in the two explicitly known cases, $\bfg=\sltw$ (the Virasoro string) and $\bfg=\slth$ (the $\cW_3$ string).