Classes on compactifications of the moduli space of curves through solutions to the quantum master equation

TL;DR

This paper introduces a new construction using quantum A-infinity algebras to generate classes in the compactified moduli space of curves, extending Kontsevich's approach from the open moduli space.

Contribution

It extends Kontsevich's construction by incorporating quantum A-infinity algebras, providing a method to produce classes in the compactified moduli space of curves.

Findings

Constructs classes in the compactification of the moduli space of curves.

Extends Kontsevich's classes from the open to the compactified moduli space.

Provides explicit deformation theory for quantum A-infinity algebras.

Abstract

In this paper we describe a construction which produces classes in a compactification of the moduli space of curves. This construction extends a construction of Kontsevich which produces classes in the open moduli space from the initial data of a cyclic A-infinity algebra. The initial data for our construction is what we call a `quantum A-infinity algebra', which arises as a type of deformation of a cyclic A-infinity algebra. The deformation theory for these structures is described explicitly. We construct a family of examples of quantum A-infinity algebras which extend a family of cyclic A-infinity algebras, introduced by Kontsevich, which are known to produce all the Miller-Morita-Mumford classes using his construction.