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]). After that the question about the Hamiltonian nature of such hydrodynamic type systems arose. 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 examples of the WDVV equations which are equivalent to the hydrodynamic type systems with local homogeneous first-order Dubrovin-Novikov type Hamiltonian structures, and those which are equivalent to the hydrodynamic type systems without such structures. 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 has been solved by O.I. Mokhov and the author. The results of O.I. Bogoyavlenskij and A.P. Reynolds [3] for the three-component nondiagonalizable hydrodynamic type systems are essentially used for the solution. The classification will be presented. The work is supported by Russian Science Foundation under grant 16-11-10260.