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## Supersymmetry and cohomology of graph complexes.

Letters in Mathematical Physics. 2019. Vol. 109. No. 3. P. 699-724.

https://arxiv.org/abs/1803.11549

I describe a combinatorial construction of the cohomology classes in compactified moduli spaces of curves ZˆI∈H∗(barM_g,n) starting from the following data: an odd derivation I, whose square is non-zero in general, I2≠0, acting on a ℤ/2ℤ-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=[Λ,⋅] coincided with the generating function of products of ψ−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 ψ-classes ψi=c1(T∗pi) in the cohomology H∗(barM_g,n) is written out.