ИСТИНА

The associativity equations or the Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations (see, e.g. [1]) of the two-dimensional topological quantum field theory can be represented as integrable nondiagonalizable systems of hydrodynamic type [2] (see also [3]). For any evolution flow, the reduction on the stationary set of its integral is a canonical Hamiltonian system according to Mokhov‘s fundamental principle [4, 5]. O.I. Mokhov and the author constructed the reduction on the stationary set of an integral of the hydrodynamic type system which is equivalent to the WDVV equations in the case of three primary fields and proved the Liouville integrability of the reduction. The author also obtained such reduction of the hydrodynamic type systems which are equivalent to the WDVV equations in the case of four primary fields. The work is supported by Russian Science Foundation under grant 18-11-00316. References [1] Dubrovin B.A. Geometry of 2D topological field theories. Preprint SISSA-89/94/FM, SISSA, Trieste, Italy, 1994; Lecture Notes in Math. 1996. V.1620. P. 120–348; arXiv: hep-th/9407018 (1994). [2] Mokhov O.I. Symplectic and Poisson geometry on loop spaces of manifolds and nonlinear equations. Topics in Topology and Mathematical Physics. Ed. S.P. Novikov. Amer. Math. Soc., Providence, RI. 1995. P. 121–151; arXiv: hep-th/9503076 (1995). [3] O.I. Mokhov. Symplectic and Poisson structures on loop spaces of smooth manifolds, and integrable systems. Russian Mathematical Surveys. 1998 V. 53:3. P. 515–622. [4] O.I. Mokhov. The Hamiltonian property of an evolutionary flow on the set of stationary points of its integral Russian Mathematical Surveys. 1984. V. 39:4. P. 133–134. [5] O.I. Mokhov. On the Hamiltonian property of an arbitrary evolution system on the set of stationary points of its integral. Mathematics of the USSR-Izvestiya. 1988. V. 31:3. P. 657–664.