From gravity to string topology

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