A complex of ribbon quivers and M_g,m

2025/03/03 by Sergei Merkulov, Merkulov, Sergei
Mathematics · #Algebraic structures and combinatorial models #Advanced Combinatorial Mathematics #Homotopy and Cohomology in Algebraic Topology

paper · pdf · doi:10.48550/arxiv.2503.02020

Abstract

For any integer d∈ ℤ we introduce a complex 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 M_g,m of genus g algebraic curves with m marked points. We show that the totality of complexes orgc_d= ∏_{g≥ 1} ORGC_d^(g,1)≃ ∏_{g≥ 1} H_c^{\bullet-1+2g(d-1)}(M_g,1) has a natural dg Lie algebra structure which controls the deformation theory of the dg properad PreCY_d governing a certain class of (possibly, infinite-dimensional) degree d pre-Calabi-Yau algebras. This result implies, in particular, that for d≤ 2 the zero-th cohomology group of the derivation complex Der(PreCY_d) is one-dimensional (i.e. PreCY_{d≤ 2} has no homotopy non-trivial automorphisms except rescalings), while for d=2 the cohomology group H¹(Der (PreCY₂)) contains a subspace isomorphic to the Grothendieck-Teichmüller Lie algebra.

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