ИСТИНА

In the case of three primary fields, the associativity equations or the Witten-Dijkgraaf-Verlinde-Verlinde (WDVV) equations of the two-dimensional topological quantum field theory can be represented as integrable nondiagonalizable systems of hydrodynamic type (O.I. Mokhov, [1]). O.I. Mokhov and E.V. Ferapontov [2] have shown that the Hamiltonian geometry of these systems essentially depends on the metric of the associativity equations. Namely, there are the WDVV equations, which are equivalent to the hydrodynamic type systems with local homogeneous first-order Hamiltonian structures of the Dubrovin-Novikov type, and the WDVV equations, which are equivalent to the hydrodynamic type systems without such structures. O.I. Mokhov and the author have completely solved the classification problem of existence of a local first-order Hamiltonian structure for the associativity equations in the representation of hydrodynamic type system in the case of three primary fields. Methods of the problem solving use recent results of O.I. Bogoyavlenskij and A.P. Reynolds [3] for the three-component nondiagonalizable hydrodynamic type systems. The results of the classification will be presented. The work is supported by RSF under grant 16-11-10260.