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. The Hamiltonian geometry of these systems essentially depends on the metric of the associativity equations (O.I. Mokhov and E.V. Ferapontov, [2]). 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. O.I. Mokhov and the author have obtained the classification of existence of a local first-order Hamiltonian structure for the hydrodynamic type systems which are equivalent to the WDVV equations in the case of three primary fields. 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 of this problem. The results of classification will be presented. The work is supported by Russian Science Foundation under grant 16-11- 10260. References [1] O.I. Mokhov. Symplectic and Poisson structures on loop spaces of smooth manifolds, and integrable systems. Russian Mathematical Surveys. 53(3) (1998), 515–622. [2] E.V. Ferapontov, O.I. Mokhov. The associativity equations in the twodimensional topological field theory as integrable Hamiltonian nondiagonalizable systems of hydrodynamic type. Functional Anal. Appl. 30(3) (1996), 195–203. arXiv: hep-th/9505180. [3] O.I. Bogoyavlenskij, A.P. Reynolds. Criteria for Existence of a Hamiltonian Structure. Regular and Chaotic Dynamics. 15(4–5) (2010), 431–439.