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 $$\bar{H}_\cdot ^{S^1}(LM)$$ .