On Symmetry Properties of Frobenius Manifolds and Related Lie-Algebraic Structures

The aim of this paper is to develop an algebraically feasible approach to solutions of the oriented associativity equations. Our approach was based on a modification of the Adler–Kostant–Symes integrability scheme and applied to the co-adjoint orbits of the diffeomorphism loop group of the circle. A new two-parametric hierarchy of commuting to each other Monge type Hamiltonian vector fields is constructed. This hierarchy, jointly with a specially constructed reciprocal transformation, produces a Frobenius manifold potential function in terms of solutions of these Monge type Hamiltonian systems.

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.

ON THE HAMILTONIAN REDUCTION OF THE ASSOCIATIVITY EQUATIONS IN THE CASE OF FOUR PRIMARY FIELDS

The associativity equations arose in the papers of Witten (Witten, 1990) and Dijkgraaf, Verlinde, (Dijkgraaf et al., 1991) on two-dimensional topological field theories and subsequently they became to play a key role in many other important domains of mathematics and mathematical physics: in quantum cohomology, Gromov–Witten invariants, enumerative geometry, theory of submanifolds and so on. In Mokhov’s papers (Mokhov, 1984), (Mokhov, 1987) a general fundamental principle stating a canonical Hamiltonian property for the restriction of an arbitrary flow on the set of stationary points of its nondegenerate integral was proposed and proved. In this paper the Hamiltonians of the reductions of the associativity equations with antidiagonal matrix ηij in the case of four primary fields according to Mokhov`s construction is found in an explicit form.