

Seminar of the Department of Geometry and Topology "Geometry, Topology and Mathematical Physics", Steklov Mathematical Institute of RAS
August 3, 2005, Moscow, Lomonosov Moscow State University, Steklov Mathematical Institute of RAS






Kontsevich and Batalin–Vilkovisky classes in the ribbon graph complex
A. Yu. Lazarev^{} ^{} University of Bristol, Department of Mathematics

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Abstract:
The ribbon graph complex computes the cohomology of moduli spaces of complex algebraic curves of all genera and with any number of marked points.
An Ainfinity algebra with an even invariant scalar product determines a cycle in the graph complex. Similarly, a differential contractible algebra with an odd invariant scalar product determines a cocycle in the graph complex. These constructions were originally sketched by Kontsevich over a decade ago but until now they have not been properly utilized.
Using homologucal algebra and a finite dimensional analogue of the Batalin–Vilkovisky formalism in quantum field theory we give a conceptual reformulation of these constructions. We also construct infinite series of explicit nontrivial examples of these classes and compute their pairings in terms of asymptotic expansions of certain finitedimensional integrals.

