Hierarchy of WDVV associativity equations for n = 3 case and N = 2 when V0 = 0

Abstract

We investigate solutions to a hierarchy for N = 2 case when V0 = 0 of WittenDijkgraaf-E.Verlinde-H.Verlinde (WDVV) equations. We give a description of nonlinear partial
differential equations of associativity in 2D topological field theories (for some special type solutions of the Witten-Dijkgraaf-E.Verlinde-H.Verlinde (WDVV) system) 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. The equations of associativity reduce to the nonlinear equations of the third order for a function f = f(x, t)) when prepotential F dependet of the metric η . In this work we consider the WDVV equation for n = 3 case with an antidiagonal metric η . The solution of some cases of hierarchy when N = 2 and V₀ = 0 equations of associativity illustrated. Lax pairs for the system of three equations, that contains the equation of associativity are written to find the hierarchy of associativity equation. Using the compatibility condition are found the relations between the matrices U, V₂, V₁ . The elements of matrix V₂ are found with the expression of z_ij and independent and dependent variables for the matrix V² . Also solving elements of matrix V1 expressed through y_ij and independent and dependent variables for the matrix V₁ . We accepted that elements of matrix V0 are zero. In
the physical setting the solutions of WDVV describe moduli space of topological conformal field theories. In the above variables the nonlinear equations of the third order for a function f = f(x,t)) we rewritten as a new system of three equations. It is found the relationship between the elements at, b_t , c_t and y_ijx of the matrices U_t, V_1x . It is found that only z11, z12, z13 are independent elements of V2 , and the other elements can be written in terms of them. Expressed are variables at , bt , ct of three equations are written with the help of matrix elements z12, z13, y12, y13 .