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 \(\bar{H}_\cdot ^{S^1}(LM)\).
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Notes
For a nice introduction into the theory of props and properads, we refer to the paper [20] by B. Vallette.
When representing elements of operads and props as decorated graphs, we tacitly assume that all edges and legs are directed along the flow going from the bottom of the graph to the top. The action of the element \(\sum _{k=1}^{3} (123)^k\in {\mathbb {K}}[{\mathbb {S}}_3]\) on any element a of an \({\mathbb {S}}_3\)-module is denoted by \(\oint _{123} a\).
For a ribbon graph \(\Gamma \), we denote by \(V(\Gamma )\) its set of vertices, \(B(\Gamma )\) its set of boundaries and by \(E(\Gamma )\) its set of edges. The genus of \(\Gamma \) is defined by \(g= 1+\frac{1}{2}\left( \# E(\Gamma ) - \# V(\Gamma )- \# B(\Gamma )\right) \).
We work in this section in the category of possibly infinite-dimensional vector spaces V which are direct limits, \(\displaystyle V=\lim _{\longrightarrow } V_p\), of finite-dimensional ones. Their dual vector spaces are defined as projective limits, \(\displaystyle V^*=\lim _{\longleftarrow } V^*_p\).
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Merkulov, S.A. From gravity to string topology. Lett Math Phys 113, 62 (2023). https://doi.org/10.1007/s11005-023-01686-8
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DOI: https://doi.org/10.1007/s11005-023-01686-8