On the Caldararu and Willwacher conjectures

TL;DR

This paper proves two conjectures relating the cohomology of moduli stacks of curves to ribbon graph complexes, advancing understanding in algebraic geometry and topological field theories.

Contribution

It provides proofs for Willwacher's and Caldararu's conjectures connecting moduli space cohomology with ribbon graph complexes.

Findings

Proof of Willwacher's conjecture

Proof of Caldararu's conjecture on Bridgeland differential cohomology

Discussion of connections to string topology and weight zero cohomology

Abstract

In the present paper, we study a relation between the cohomology of moduli stacks of smooth and proper curves $\mathcal M_{g,n}$ and the cohomology of ribbon graph complexes. The main results of this work are proofs of T. Willwacher's conjecture and A. C$\check{\mathrm a}$ld$\check{\mathrm a}$raru's conjecture about the cohomology of the Bridgeland differential. We also discuss the relation to string topology and the Chan-Galatius-Payne theorem about the weight zero part of the compactly supported cohomology of $\mathcal M_{g,n}.$