Letters in Mathematical Physics

, Volume 109, Issue 3, pp 699–724 | Cite as

Supersymmetry and cohomology of graph complexes

  • Serguei BarannikovEmail author


I describe a combinatorial construction of the cohomology classes in compactified moduli spaces of curves \(\widehat{Z}_{I}\in H^{*}\left( \bar{\mathcal {M}}_{g,n}\right) \) starting from the following data: \(\mathbb {Z}/2\mathbb {Z}\)-graded finite-dimensional associative algebra equipped with odd scalar product and an odd compatible derivation I, whose square is nonzero in general, \(I^{2}\ne 0\). As a byproduct I obtain a new combinatorial formula for products of \(\psi \)-classes, \(\psi _{i}=c_{1}\left( T_{p_{i}}^{*}\right) \), in the cohomology \(H^{*}\left( \bar{\mathcal {M}}_{g,n}\right) \).


Mirror symmetry Graph complexes Gromov–Witten invariants Moduli spaces 

Mathematics Subject Classification

Primary 14J33 53D37 81Q30 Secondary 05A15 



The results of this paper were presented starting from 2006 at conferences in Cambridge, Berkeley, Grenoble, Miami, Vienna, Brno, Tokyo, Moscow, Boston. I am thankful to organizers of these conferences for their hospitality, for the opportunity to present these results to large audience and to the participants for interesting questions and comments. The author is partially supported by Laboratory of Mirror Symmetry NRU HSE, RF Government grant, Ag. No 14.641.31.0001.


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Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.UMR7586 CNRSInstitut de mathématiques de JussieuParisFrance
  2. 2.National Research University Higher School of EconomicsMoscowRussian Federation

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