Modular operads and Batalin-Vilkovisky geometry

TL;DR

This paper explores the noncommutative Batalin-Vilkovisky geometry linked to modular operads, establishing connections with quantum master equations and constructing characteristic classes related to moduli space homology.

Contribution

It introduces a novel noncommutative BV geometry framework associated with modular operads and relates algebraic structures to solutions of quantum master equations.

Findings

Algebras over the Feynman transform correspond to solutions of quantum master equations.

Constructs characteristic classes in the homology of Deligne-Mumford moduli spaces.

Links noncommutative symplectic geometry with modular operad structures.

Abstract

This is a copy of the article published in IMRN (2007). I describe the noncommutative Batalin-Vilkovisky geometry associated naturally with arbitrary modular operad. The classical limit of this geometry is the noncommutative symplectic geometry of the corresponding tree-level cyclic operad. I show, in particular, that the algebras over the Feynman transform of a twisted modular operad P are in one-to-one correspondence with solutions to quantum master equation of Batalin-Vilkovisky geometry on the affine P-manifolds. As an application I give a construction of characteristic classes with values in the homology of the quotient of Deligne-Mumford moduli spaces. These classes are associated naturally with solutions to the quantum master equation on affine S[t]-manifolds, where S[t] is the twisted modular Det-operad constructed from symmetric groups, which generalizes the cyclic operad of associative algebras.