Gravity prop and moduli spaces $\mathcal{M}_{g,n}$

TL;DR

This work constructs a gravity properad GRav from the compactly supported cohomology of moduli spaces $H_c^{\bullet}(\mathcal{M}_{g,m+n})$, embedding Getzler’s gravity operad as a genus-zero subcollection and linking the structure to Costello’s nodal disk moduli and Willwacher’s twisting. It develops the twisted ribbon-graph framework ${\mathsf{tw}{\mathcal{R}}{\mathcal G}ra_d}$, proves its degree-zero cohomology recovers $GRav$ (Theorem A), and embeds quasi-Lie bialgebra data via a nontrivial map $j: {\mathcal L}ieb_{-1,0}\to GRav$ (Theorem B). The paper then studies the deformation theory of these maps, showing the total deformation complex $\mathsf{Def}(q\mathcal{L}ieb_{-1,0}\to GRav)$ contains infinitely many cohomology classes described by twisted ribbon graphs, many arising from Kontsevich’s graph complexes (Theorem C). Overall, the results illuminate how moduli-space cohomology controls and is controlled by a gravity-type operadic/properadic structure, with deep ties to graph complexes and quantum-algebraic deformations. These insights offer a unifying perspective on the cohomology of moduli spaces, operadic twists, and the algebraic structures governing stringy and topological field-theoretic phenomena.

Abstract

Let $\mathcal{M}_{g,n}$ be the moduli space of algebraic curves of genus $g$ with $m+n$ marked points decomposed into the disjoint union of two sets of cardinalities $m$ and $n$, and $H_c^{\bullet}(\mathcal{M}_{m+n})$ its compactly supported cohomology group. We prove that the collection of $S$-bimodules $\left\{H_c^{\bullet-m}(\mathcal{M}_{g,m+n})\right\}$ has the structure of a properad (called the gravity properad) such that it contains the (degree shifted) E. Getzler's gravity operad as the sub-collection $\{H_c^{\bullet-1}(\mathcal{M}_{0,1+n})\}_{n\geq 2}$. Moreover, we prove that the generators of the 1-dimensional cohomology groups $H_c^{\bullet-1}(\mathcal{M}_{0,1+2})$, $H_c^{\bullet-2}(\mathcal{M}_{0,2+1})$ and $H_c^{\bullet-3}(\mathcal{M}_{0,3+0})$ satisfy with respect to this properadic structure the relations of the (degree shifted) quasi-Lie bialgebra, a fact making the totality of cohomology groups $ \prod_{g,m,n} H_c^{\bullet}(\mathcal{M}_{g,m+n})\otimes_{S_m^{op}\times S_n} (sgn_m\otimes Id_n)$ into a complex with the differential fully determined by the just mentioned three cohomology classes . It is proven that this complex contains infinitely many cohomology classes, all coming from M. Kontsevich's odd graph complex. The gravity prop structure is established with the help of T. Willwacher's twisting endofunctor (in the category of properads under the operad of Lie algebras) and K. Costello's theory of moduli spaces of nodal disks with marked boundaries and internal marked points (such that each disk contains at most one internal marked point).

Theorems & Definitions (2)

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