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.
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Alastair Hamilton, Noncommutative geometry and compactifications of the moduli space of curves . J. Noncommut. Geom. 4 (2010), no. 2, pp. 157–188DOI 10.4171/JNCG/52