  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 $\mathcal M_{g,n}.$
