Soliton surface associated with the equation of associativity for n=3 case with an metric

Abstract

The Witten–Dijkgraaf–Verlinde–Verlinde (WDVV) equations, also called the associativity equations, is a system of nonlinear partial differential equations for one function, depending on a finite number of variables. The WDVV equations were introduced a few decades ago in the context of two-dimensional topological field theories. The task of giving the associativity equations a geometric interpretation has two complementary aspects. On one side, can write these equations in a form that does not depend on the choice of the coordinates. On the other side, one must demand that the geometrical structure should be capable to select a class of affinely related coordinates. The coordinate selection rule is important in the geometrization of the associativity equations. In this paper, we consider the soliton surface of the associativity equation. The equation of associativity originated from 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. The surface is constructed using Sym-Tafel formula, which is a connection between classical manifold geometry and soliton theory. The Sym-Tafel formula reconstructs a surface from knowledge of its fundamental forms, combines integrable nonlinearities, and allows the application of soliton theory methods to geometric problems. The soliton surfaces approach is necessary in the construction of so-called integrable geometries. Any class of soliton surfaces is integrable. Geometric objects associated with the surfaces of the solitons can usually be identified with the solutions to the strings. Thus in this work soliton surfaces for the associativity equation for n=3 case with an metric h11 not equal to 0 are considered, and first and second fundamental forms of soliton surfaces are found for this case. In addition, we study an area of surfaces for the associativity equation for  case n=3 with an metric h11 not equal to 0.