Noncommutative geometry and compactifications of the moduli space of curves

TL;DR

This paper demonstrates that the homology of a natural compactification of the moduli space of curves can be fully described algebraically using a differential graded Lie algebra, linking noncommutative geometry with moduli space topology.

Contribution

It introduces a novel algebraic description of the homology of a compactified moduli space via a differential graded Lie algebra derived from noncommutative symplectic geometry.

Findings

Homology of the compactification is described by a differential graded Lie algebra.

A two-parameter family is constructed using a Lie cobracket on noncommutative 0-forms.

The structure corresponds to pinching simple closed curves on Riemann surfaces.

Abstract

In this paper we show that the homology of a certain natural compactification of the moduli space, introduced by Kontsevich in his study of Witten's conjectures, can be described completely algebraically as the homology of a certain differential graded Lie algebra. This two-parameter family is constructed by using a Lie cobracket on the space of noncommutative 0-forms, a structure which corresponds to pinching simple closed curves on a Riemann surface, to deform the noncommutative symplectic geometry described by Kontsevich in his subsequent papers.