Supersymmetry and cohomology of graph complexes

Abstract

This is preprint HAL-00429963 (2009). I describe a combinatorial construction of the cohomology classes in compactified moduli spaces of curves $\widehat{Z}_{I}\in H^{*}(\bar{\mathcal{M}}_{g,n})$ starting from the following data: an odd derivation $I$, whose square is non-zero in general, $I^{2}\neq 0$, acting on a $\mathbb{Z}/2\mathbb{Z}$-graded associative algebra with odd scalar product. The constructed cocycles were first described in the theorem 2 in the author's paper "Noncommmutative Batalin-Vilkovisky geometry and Matrix integrals". Comptes Rendus Mathematique, 348, pp. 359-362, arXiv:0912.5484 , preprint HAL-00102085 (09/2006). By the theorem 3 from loc.cit. the family of the cohomology classes obtained in the case of the algebra $Q(N)$ and the derivation $I=\left[\Lambda,\cdot\right]$ coincided with the generating function of products of $\psi-$classes. This was the first nontrivial computation of categorical Gromov-Witten invariants of higher genus. The result matched with the mirror symmetry prediction, i.e. with the classical (non-categorical) Gromov-Witten descendent invariants of a point for all genus. As a byproduct of that computation a new combinatorial formula for products of $\psi$-classes $\psi_{i}=c_{1}(T_{p_{i}}^{*})$ in the cohomology $H^{*}(\bar{\mathcal{M}}_{g,n})$ is written out.