On higher order Leibniz identities in TCFT

TL;DR

This paper extends the algebraic structure of topological conformal field theories by introducing nonlocal operators and higher order Leibniz identities, suggesting a homotopy algebra framework.

Contribution

It introduces a novel extension of local observables with nonlocal operators and constructs parameter-dependent operations satisfying homotopy Leibniz identities.

Findings

Construction of parameter-dependent operations via integrals over moduli spaces

Extension of algebraic structures in TCFT with higher order identities

Conjecture on a complete set of operations leading to homotopy Leibniz algebras

Abstract

We extend the algebra of local observables in topological conformal field theories by nonlocal operators. This allows to construct parameter-dependent operations realized via certain integrals over the compactified moduli spaces, satisfying analogues of the Leibniz and higher order Leibniz identities holding up to homotopy. We conjecture that one can construct a complete set of such operations which lead to a parameter-dependent version of Loday's homotopy Leibniz algebras.