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

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).