A complex of ribbon quivers and $\mathcal{M}_{g,m}$

Abstract

For any integer $d\in \mathbb{Z}$ we introduce a complex $\mathsf{ORGC}_{d}^{(g,m)}$ spanned by genus $g$ ribbon quivers with $m$ marked boundaries and prove that its cohomology computes (up to a degree shift) the compactly supported cohomology of the moduli space $\mathcal{M}_{g,m}$ of genus $g$ algebraic curves with $m$ marked points. We show that the totality of complexes $$ \mathsf{orgc}_{d}= \prod_{g\geq 1} \mathsf{ORGC}_{d}^{(g,1)}{\simeq} \prod_{g\geq 1} H_c^{\bullet-1+2g(d-1)}(\mathcal{M}_{g,1}) $$ has a natural dg Lie algebra structure which controls the deformation theory of the dg properad $\mathcal{P}re\mathcal{CY}_d$ governing a certain class of (possibly, infinite-dimensional) degree $d$ pre-Calabi-Yau algebras. This result implies, in particular, that for $d\leq 2$ the zero-th cohomology group of the derivation complex $\mathrm{Der}(\mathcal{P}re\mathcal{CY}_d)$ is one-dimensional (i.e. $\mathcal{P}re\mathcal{CY}_{d\leq 2}$ has no homotopy non-trivial automorphisms except rescalings), while for $d=2$ the cohomology group $H^1(\mathrm{Der} (\mathcal{P}re\mathcal{CY}_2))$ contains a subspace isomorphic to the Grothendieck-Teichm\"uller Lie algebra.