Workshop, Alexander Buryak: Pairs of compatible Poisson brackets and cohomological field theories

Speaker
Alexander Buryak, HSE, Moscow

Abstract
Many integrable systems (of evolutionary PDEs with one spatial variable) are bihamiltonian, i.e., can be reconstructed from
a pair of compatible Poisson brackets. The most famous example of such integrable system is the KdV hierarchy. By a result
of Dubrovin and Novikov, a pair of compatible Poisson brackets of hydrodynamic type is equivalent to a differential-geometric
structure: a pair of compatible flat metrics. By a result of Dubrovin, an important class of such pairs is given by the
Dubrovin–Frobenius (DF) manifolds: there is an explicit formula for such a pair using a potential of a DF manifold.
A cohomological field theory (CohFT) can be considered as a genus expansion of a DF manifold. In a recent paper with P. Rossi,
starting from a semisimple CohFT, we proved an explicit formula for a pair of compatible Poisson brackets, which is a dispersive
deformation of Dubrovin’s pair of Poisson brackets. In this way, one obtains all dispersive deformations of Dubrovin’s pairs will all
central invariants equal to the same constant.