M. Kontsevich [in: The Gelfand Seminars, 1990–1992, Birkhäuser 173–187 (1993; Zbl 0821.58018)] introduces non-commutative analogues of symplectic geometry to produce classes in the homology of various graph complexes. The idea, formalized subsequently, is to associate a Lie algebra of Hamiltonian vector fields to any cyclic operad \(P\), a structure that models invariant multlinear operations on a vector space equipped with a non-degenerate inner product. The homology classes are determined by structure constants of strongly homotopy algebras over a cyclic dual operad of \(P\).
In the paper under review, the author extends the construction of these homology classes in the context of modular operads, a higher genus generalization of the notion of a cyclic operad. For this aim, he proves that the algebras over a Feynman transform of a twisted modular operad, which generalize the strongly homotopy algebras associated to cyclic operads, are equivalent to solutions of a quantum master equation of Batalin-Vilkovisky type.
As an application, the author addresses the construction of classes in the homology of Deligne-Mumford moduli spaces with unordered marked points.