THE HIERARCHY OF ASSOCIATIVITY EQUATIONS FOR n=3 CASE WITH AN METRIC ƞ11≠0

This paper describes the hierarchy for N = 2 and n=3 case with an metric ƞ11≠0 when V0 = 0 of associativity equations. The equation of associativity arose from the 2D topological field theory. 2D topological field theory represent the matter sector of topological string theory. These theories covariant before coupling to gravity due to the presence of a nilpotent symmetry and are therefore often referred to as cohomological field theories. We give a description of nonlinear partial differential equations of associativity in 2D topological field theories as integrable nondiagonalizable weakly nonlinear homogeneous system of hydrodynamic type. The article discusses nonlinear equations of the third order for a function f = f(x,t)) of two independent variables x, t. In this work we consider the associativity equation for n=3 case with an a metric 0 11   . The solution of some cases of hierarchy when N = 2 and V0 = 0 equations of associativity illustrated

LIOUVILLE INTEGRABLE REDUCTIONS OF THE ASSOCIATIVITY EQUATIONS ON THE SET OF STATIONARY POINTS OF AN INTEGRAL IN THE CASE OF THREE PRIMARY FIELDS

In this work, in the case of three primary fields, a reduction of the associativity equations (the Witten–Dijkgraaf–Verlinde–Verlinde system, see (Witten, 1990, Dijkgraaf et al., 1991, Dubrovin, 1994) with antidiagonal matrix ηij on the set of stationary points of a nondegenerate integral quadratic with respect to the first-order partial derivatives is constructed in an explicit form and its Liouville integrability is proved. In Mokhov’s paper (Mokhov, 1995, Mokhov, 1998), these associativity equations were presented in the form of an integrable nondiagonalizable system of hydrodynamic type. In the papers (Ferapontov, Mokhov, 1996, Ferapontov et al., 1997, Mokhov, 1998), a bi-Hamiltonian representation for these equations and a nondegenerate integral quadratic with respect to the first-order partial derivatives were found. Using Mokhov’s construction on canonical Hamiltonian property of an arbitrary evolutionary system on the set of stationary points of its nondegenerate integral of the papers (Mokhov, 1984, Mokhov, 1987), we construct explicitly the reduction for the integral quadratic with respect to the first-order partial derivatives, found explicitly the Hamiltonian of the corresponding canonical Hamiltonian system. We also found three functionally-independent integrals in involution with respect to the canonical Poisson bracket on the phase space for the constructed reduction of the associativity equations and thus proved the Liouville integrability of this reduction. This work is supported by the Russian Science Foundation under grant No. 18-11-00316.