COMPATIBLE POISSON STRUCTURES OF HYDRODYNAMIC TYPE AND THE EQUATIONS OF ASSOCIATIVITY IN TWO-DIMENSIONAL TOPOLOGICAL FIELD THEORY

We study the problem of classification or efficient description of compatible Poisson structures of hydrodynamic type, i.e. compatible local first-order homogeneous Poisson brackets in field theory. It is proved that all two-component compatible Poisson structures of hydrodynamic type are explicitly described by solutions of a homogeneous four-component system of hydrodynamic type, and an integrable two-component reduction of this system is found. It turns out that this reduction is connected to the equations of associativity in two-dimensional topological field theory and deformations of two special Frobenius algebras. An analogous connection with the equations of associativity takes place for an arbitrary number of components and the corresponding general theory of compatible Poisson structures of hydrodynamic type, generated by some special solutions of the equations of associativity, and the theory of compatible deformations of two Frobenius algebras are constructed.