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

TL;DR

This paper constructs a gravity properad structure on the cohomology of moduli spaces of algebraic curves, revealing deep algebraic relations and connections to graph complexes, expanding understanding of moduli space symmetries.

Contribution

It introduces the gravity properad structure on moduli space cohomology and links it to quasi-Lie bialgebra relations and Kontsevich's graph complex.

Findings

The collection of cohomology groups forms a properad called the gravity properad.

Generators satisfy relations of the degree shifted quasi-Lie bialgebra.

The complex contains infinitely many cohomology classes from Kontsevich's odd graph complex.

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).