Homotopy G-algebras and moduli space operad

TL;DR

This paper explores the deep connections between moduli spaces of algebraic curves, operads, and homotopy G-algebra structures, revealing new algebraic structures arising from geometric and topological contexts.

Contribution

It demonstrates that the space of an operad with multiplication forms a homotopy Gerstenhaber algebra and that decorated moduli spaces naturally act on the de Rham complex of Kähler manifolds.

Findings

The space of an operad with multiplication is a homotopy Gerstenhaber algebra.

The singular cochain complex naturally forms an operad.

Decorated moduli spaces act on the de Rham complex of Kähler manifolds.

Abstract

This paper emphasizes the ubiquitous role of moduli spaces of algebraic curves in associative algebra and algebraic topology. The main results are: (1) the space of an operad with multiplication is a homotopy Gerstenhaber (i.e., homotopy graded Poisson) algebra; (2) the singular cochain complex is naturally an operad; (3) the operad of decorated moduli spaces acts naturally on the de Rham complex $\Omega^\bullet X$ of a K\"{a}hler manifold $X$, thereby yielding the most general type of homotopy G-algebra structure on $\Omega^\bullet X$.