From gravity to string topology

TL;DR

This paper develops a new algebraic framework called the string topology properad, which acts on loop space homology of manifolds, unifying and extending several known geometric and algebraic structures in string topology.

Contribution

It introduces the cohomology properad of ribbon graphs, showing its non-triviality and canonical action on loop space homology, connecting gravity, involutive Lie bialgebras, and moduli space cohomology.

Findings

The cohomology properad acts on the reduced equivariant homology of loop spaces.

It contains a morphism from the gravity properad related to moduli spaces of curves.

It reproduces and unifies known structures like the Chas-Sullivan construction and Getzler's gravity operad.

Abstract

The chain gravity properad introduced earlier by the author acts on the cyclic Hochschild of any cyclic $A_\infty$ algebra equipped with a scalar product of degree $-d$. In particular, it acts on the cyclic Hochschild complex of any Poincare duality algebra of degree $d$, and that action factors through a quotient dg properad $ST_{3-d}$ of ribbon graphs which is in focus of this paper. We show that its cohomology properad $H^\bullet(ST_{3-d})$ is highly non-trivial and that it acts canonically on the reduced equivariant homology $\bar{H}_\bullet^{S^1}(LM)$ of the loop space $LM$ of any simply connected $d$-dimensional closed manifold $M$. By its very construction, the string topology properad $H^\bullet(ST_{3-d})$ comes equipped with a morphism from the gravity properad which is fully determined by the compactly supported cohomology of the moduli spaces $M_{g,n}$ of stable algebraic curves of genus $g$ with marked points. This result gives rise to new universal operations in string topology as well as reproduces in a unified way several known constructions: we show that (i) $H^\bullet(ST_{3-d})$ is also a properad under the properad of involutive Lie bialgebras in degree $3-d$ whose induced action on $\bar{H}_\bullet^{S^1}(LM)$ agrees precisely with the famous purely geometric construction of M. Chas and D. Sullivan, (ii) $H^\bullet(ST_{3-d})$ is a properad under the properad of homotopy involutive Lie bialgebras in degree $2-d$; (iii) E. Getzler's gravity operad injects into $H^\bullet(ST_{3-d})$ implying a purely algebraic counterpart of the geometric construction of C. Westerland establishing an action of the gravity operad on $\bar{H}_\bullet^{S^1}(LM)$.