A complex of ribbon quivers and $\mathcal{M}_{g,m}$
Sergei Merkulov
TL;DR
The work develops a ribbon-graph–based framework to model the compactly supported cohomology of moduli spaces $\mathcal{M}_{g,m}$ via ribbon quivers $\mathsf{ORGC}_{d}^{(g,m)}$, establishing a core isomorphism $H^{\bullet}(\mathsf{RGC}_{d}) \cong H^{\bullet}(\mathsf{ORGC}_{d+1})$ and a dg Lie algebra structure on the one-boundary sectors that reflects moduli-space cohomology. It then connects these combinatorial models to the deformation theory of degree-$d$ pre-Calabi–Yau algebras by constructing a dg properad $\mathcal{P}re\mathcal{CY}_{d}$ and a quasi-isomorphism from $\mathfrak{orgc}_{d}$ to $\mathrm{Der}(\mathcal{P}re\mathcal{CY}^{3}_{d})$, yielding an explicit computation of the derivation cohomology in terms of $H^{\bullet}_{c}(\mathcal{M}_{g,1})$ plus a rescaling class. For $d\le 2$ the results imply rigidity (no nontrivial automorphisms up to rescaling), while for $d=2$ a Grothendieck–Teichmüller-type subspace appears in $H^{1}$. The framework leads to applications including a proof of non-Koszulness of the balanced infinitesimal bialgebra properad $\mathsf{BIB}_{d}$ and a perspective on the deformation theory of a broad class of noncommutative Calabi–Yau-type structures, connecting graph complexes to deep arithmetic and geometric invariants of moduli spaces.
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-Teichmüller Lie algebra.