Homotopy Relations for Topological VOA

TL;DR

This paper explores a parameter-dependent homotopy algebra related to topological VOAs, interpreting multilinear operations via integrals over polytopes, and proves the pentagon relation up to homotopy with a construction for higher operations.

Contribution

It introduces a parameter-dependent homotopy algebra for topological VOAs, providing integral interpretations and explicit proofs of key relations.

Findings

Proved the pentagon relation up to homotopy.

Provided a construction for higher multilinear operations.

Interpreted algebraic operations as integrals over polytopes.

Abstract

We consider a parameter-dependent version of the homotopy associative part of the Lian-Zuckerman homotopy algebra and provide the interpretation of multilinear operations of this algebra in terms of integrals over certain polytopes. We explicitly prove the pentagon relation up to homotopy and propose a construction of higher operations.