For any integer d Z we introduce a complex ORGC₃^ (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 M₆, ₌ of genus g algebraic curves with m marked points. We show that the totality of complexes orgc₃= ₆ ₁ ORGC₃^ (g, 1) ₆ ₁ Hc^-1+2g (d-1) (M₆, ₁) has a natural dg Lie algebra structure which controls the deformation theory of the dg properad PreCYd 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 (PreCYd) is one-dimensional (i. e. PreCY₃ 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-Teichm\"uller Lie algebra.
Sergei Merkulov (Mon,) studied this question.