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

TL;DR

This paper explores the topological Landau-Ginzburg theory with a rational potential within the dispersionless KP hierarchy, revealing connections to Toda hierarchies, flat solutions, and topological recursion relations.

Contribution

It introduces a comprehensive framework linking rational potential LG theories to the dispersionless KP hierarchy and constructs explicit solutions and recursion relations.

Findings

Hierarchy includes dispersionless Toda limits

Constructed explicit flat solutions and gravitational descendants

Derived residue formula for 3-point functions and topological recursion

Abstract

Based on the dispersionless KP (dKP) theory, we give a comprehensive study of the topological Landau-Ginzburg (LG) theory characterized by a rational potential. Writing the dKP hierarchy in a general form, we find that the hierarchy naturally includes the dispersionless (continuous) limit of Toda hierarchy and its generalizations having 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.