Topological Landau-Ginzburg theory with a rational potential and the dispersionless KP hierarchy

Abstract

Based on the dispersionless KP (dKP) theory, we study a topological Landau-Ginzburg (LG) theory characterized by a rational potential, Writing the dKP hierarchy in a general form treating all the primaries in an equal basis, we find that the hierarchy naturally includes the dispersionless (continuous) limit of Toda hierarchy and its generalizations having a finite number of primaries, Several flat solutions of the topological LG theory are obtained in this formulation, and are identified with those discussed by Dubrovin. We explicitly construct gravitational descendants for all the primary fields. Giving a residue formula for the 3-point functions of the fields, we show that these 3-point functions satisfy the topological recursion relation. The string equation is obtained as the generalized hodograph solutions of the dKP hierarchy, which show that all the gravitational effects to the constitutive equations (2-point functions) can be renormalized into the coupling constants in the small phase space.

Logarithmic Deformations of the Rational Superpotential/Landau-Ginzburg Construction of Solutions of the WDVV Equations

“…Another possible generalization, already alluded to above, is to replace the polynomial part of λ by an arbitrary rational function, generalizing the construction of [1,2]. The Frobenius manifold structure on the space of such rational functions has been much studied and these results can be generalized to include logarithmic terms.…”

“…where F (0) , F (1) , F (2) are independent of the parameters k i . F (0) is the prepotential for the C N /A N orbit space with λ + as the Landau-Ginzburg potential, and as such is a polynomial in the flat coordinates {t 1 , .…”

“…. , N where the coordinates {t i } are a set of flat coordinates for the metric η defined by (2). Geometrically, a solution defines a multiplication • : T M × T M → T M of vector fields on the parameter space M , i.e.…”

Dispersionful analog of the Whitham hierarchy

“…For the same class of Lax functions the more correct approach is given in [25], where the logarithmic terms are constructed using contour integrals. However, the generalization of such approaches to the whole Whitham hierarchy makes problems.…”

Recursion operator and dispersionless rational Lax representation

“…Here we consider a dispersionless Lax equation on the Poisson algebra Λ with a rational Lax function. Such equations one can find in context of topological field theories (see [4,5]). …”