Geometry and Integrability in Mathematical Physics GIMP'06

May 15 - 19, 2006, Moscow, Russia

General

Announcement

Poster

Participants

Practical details

Schedule

Photos

Serguei Barannikov

DMA ENS Paris, France

Modular operads and Batalin-Vilkovisky geometry

Modular operads were introduced by Getzler and Kapranov as the higher genus analogs of cyclic operads. The basic idea is to replace the trees, playing the central role in the theory of cyclic operads, by graphs. In particular, the Kontsevich graph complexes arise naturally as the modular analog of cobar transformation. The calculation of homology of graph complexes is a very complicated combinatorial problem. The examples here are the homology of chord diagrams, which encode the Vassiliev invariants, or the homology of moduli spaces of Riemann surfaces, whose stable limit modulo torsion is calculated by the Mumford conjecture proven recently. I shall explain that the modular operads are intimately related with noncommutative generalisation of Batalin-Vilkovisky geometry. The classical limit of the latter is the noncommutative symplectic geometry described by Kontsevich in connection with cyclic operads.

I shall show, in particular, that the algebras over the Feynman transform of a twisted modular operad P are in one-to-one correspondence with solutions to quantum master equation of Batalin-Vilkovisky geometry on the affine P-manifolds. As an application a construction of cohomology classes in the Deligne-Mumford moduli spaces will be given.