Maurer-Cartan Elements and Cyclic Operads

TL;DR

This paper unifies various BV and homotopy BV structures through cyclic operads and Maurer-Cartan elements, introduces the cyclic operad of cyclically invariant operations, and connects these to moduli space homology and string topology.

Contribution

It introduces a new construction linking cyclic operads and Maurer-Cartan elements to BV structures, and computes the homology of cyclically invariant operations as the gravity operad.

Findings

Homology of cyclically invariant operations is the gravity operad.

Constructs cyclic brace operations inducing gravity relations.

Establishes connections between BV structures, moduli spaces, and string topology.

Abstract

First we argue that many BV and homotopy BV structures, including both familiar and new examples, arise from a common underlying construction. The input of this construction is a cyclic operad along with a cyclically invariant Maurer-Cartan element in an associated Lie algebra. Using this result we introduce and study the operad of cyclically invariant operations, with instances arising in cyclic cohomology and $S^1$ equivariant homology. We compute the homology of the cyclically invariant operations; the result being the homology operad of $\mathcal{M}_{0,n+1}$, the uncompactified moduli spaces of punctured Riemann spheres, which we call the gravity operad after Getzler. Motivated by the line of inquiry of Deligne's conjecture we construct `cyclic brace operations' inducing the gravity relations up-to-homotopy on the cochain level. Motivated by string topology, we show such a gravity-BV pair is related by a long exact sequence. Examples and implications are discussed in course.