Dispersionless Toda hierarchy and two-dimensional string theory

TL;DR

This paper reveals how the dispersionless Toda hierarchy underpins a Landau-Ginzburg approach to two-dimensional string theory, linking integrable systems with tachyon dynamics through Riemann-Hilbert problems.

Contribution

It establishes a novel connection between the dispersionless Toda hierarchy and 2D string theory, translating Landau-Ginzburg elements into the Lax formalism and analyzing tachyon correlation functions.

Findings

Tachyon dynamics are encoded in a special solution of the hierarchy.

Recursion relations for tachyon correlators are derived.

A wedge algebra structure is identified behind the Riemann-Hilbert problem.

Abstract

The dispersionless Toda hierarchy turns out to lie in the heart of a recently proposed Landau-Ginzburg formulation of two-dimensional string theory at self-dual compactification radius. The dynamics of massless tachyons with discrete momenta is shown to be encoded into the structure of a special solution of this integrable hierarchy. This solution is obtained by solving a Riemann-Hilbert problem. Equivalence to the tachyon dynamics is proven by deriving recursion relations of tachyon correlation functions in the machinery of the dispersionless Toda hierarchy. Fundamental ingredients of the Landau-Ginzburg formulation, such as Landau-Ginzburg potentials and tachyon Landau-Ginzburg fields, are translated into the language of the Lax formalism. Furthermore, a wedge algebra is pointed out to exist behind the Riemann-Hilbert problem, and speculations on its possible role as generators of ``extra'' states and fields are presented.