From gravity to string topology

Abstract

The chain gravity properad introduced in Merkulov (Gravity prop and moduli spaces ${\mathcal {M}}_{g,n}$, 2021, http://arxiv.org/abs/2108.10644) acts on the cyclic Hochschild complex 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 ${\mathcal{S}\mathcal{T}}_{3-d}$ of ribbon graphs which is in focus of this paper. We show that its cohomology properad $H^\cdot ({\mathcal{S}\mathcal{T}}_{3-d})$ is highly non-trivial and that it acts canonically on the reduced equivariant homology $\bar{H}_\cdot ^{S^1}(LM)$ of the loop space of any simply connected d-dimensional closed manifold M. By its very construction, the string topology properad $H^\cdot ({\mathcal{S}\mathcal{T}}_{3-d})$ comes equipped with a morphism from the gravity properad ${\mathcal {G}} rav_{3-d}$ which is fully determined by the compactly supported cohomology of the moduli spaces ${\mathcal {M}}_{g,n}$ 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) $H^\cdot ({\mathcal{S}\mathcal{T}}_{3-d})$ is also a properad under the properad of involutive Lie bialgebras ${\mathcal {L}}{ ieb }^{\diamond }_{3-d}$ whose induced (via $H^\cdot ({\mathcal{S}\mathcal{T}}_{3-d})$) action on $\bar{H}_\cdot ^{S^1}(LM)$ agrees precisely with the famous purely geometric construction of Chas and Sullivan (String topology, ; in: The legacy of Niels Henrik Abel, Springer, Berlin 2004), (ii) $H^\cdot ({\mathcal{S}\mathcal{T}}_{3-d})$ is a properad under the properad of homotopy  involutive Lie bialgebras ${\mathcal {H}}{ olieb }^{\diamond }_{2-d}$ which controls (via $H^\cdot ({\mathcal{S}\mathcal{T}}_{3-d})$) four universal string topology operations introduced in Merkulov (Propof ribbon hypergraphs and strongly homotopy involutive Lie bialgebras, 2020, http://arxiv.org/abs/1812.04913), (iii) E. Getzler’s gravity operad injects  into $H^\cdot ({\mathcal{S}\mathcal{T}}_{3-d})$ implying a purely algebraic counterpart of the geometric construction of Westerland (Math Ann 340:97–142, 2008) establishing an action of the gravity operad on $\bar{H}_\cdot ^{S^1}(LM)$.