Supersymmetry and cohomology of graph complexes

Abstract

I describe a combinatorial construction of the cohomology classes in compactified moduli spaces of curves Z^ I ∈ H ∗ (M¯ g,n ) starting from the following data: Z/ 2 Z-graded finite-dimensional associative algebra equipped with odd scalar product and an odd compatible derivation I, whose square is nonzero in general, I 2 ≠ 0. As a byproduct I obtain a new combinatorial formula for products of ψ-classes, ψi=c1(Tpi∗), in the cohomology H ∗ (M¯ g,n ).