Abstract
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 \(H^{*}\left( \bar{\mathcal {M}}_{g,n}\right) \).
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Notes
We work with finite-dimensional algebras, however it is straightforward to adapt our construction to infinite dimensional situations, for example of dg-associative algebra A with \(\dim _{k}H^{*}(A)<\infty \), provided that the propagator \(g^{-1}\) with the corresponding properties is defined.
References
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Acknowledgements
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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Preprint hal-00429963 (2009).
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Barannikov, S. Supersymmetry and cohomology of graph complexes. Lett Math Phys 109, 699–724 (2019). https://doi.org/10.1007/s11005-018-1123-7
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DOI: https://doi.org/10.1007/s11005-018-1123-7