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 $${\mathsf{orgc}}_d=\prod _{g\ge 1}{\mathsf{ORGC}}_d^{(g,1)}\simeq \prod _{g\ge 1}{H}_c^{\bullet -1+2g(d-1)}({\mathcal{M}}_{g,1})$$ has a natural dg Lie algebra structure which controls the deformation theory of the dg properad $\mathcal{P}re{\mathcal{C}\mathcal{Y}}_d$ governing a certain class of (possibly, infinite-dimensional) degree $d$ pre-Calabi-Yau algebras. This result implies, in particular, that for $d\le 2$ the zero-th cohomology group of the derivation complex $\mathrm{Der}(\mathcal{P}re{\mathcal{C}\mathcal{Y}}_d)$ is one-dimensional (i.e. $\mathcal{P}re{\mathcal{C}\mathcal{Y}}_{d\le 2}$ has no homotopy non-trivial automorphisms except rescalings), while for $d=2$ the cohomology group $H^1(\mathrm{Der}(\mathcal{P}re{\mathcal{C}\mathcal{Y}}_2))$ contains a subspace isomorphic to the Grothendieck-Teichmüller Lie algebra.

arXiv commentIn v2 the definition of the derivation complex given in Section 5 is rewritten in a more detailed and explicit way. Some misprints are corrected