Topological Landau-Ginzburg Formulation and Integrable Structure of 2d String Theory

TL;DR

This paper develops a topological Landau-Ginzburg model for 2D string theory at the self-dual radius, linking it to integrable hierarchies and matrix models, and identifying the structure of tachyon operators.

Contribution

It introduces a novel Landau-Ginzburg formulation for 2D string theory, connecting it with integrable systems and providing explicit calculations of correlators and operator identifications.

Findings

Superpotential matches matrix model results.

Identifies tachyons as primary and descendant fields.

Establishes link between Landau-Ginzburg and Toda lattice hierarchy.

Abstract

We construct a topological Landau-Ginzburg formulation of the two-dimensional string at the self-dual radius. The model is an analytic continuation of the $A_{k+1}$ minimal model to $k=-3$. We compute the superpotential and calculate tachyon correlators in the Landau-Ginzburg framework. The results are in complete agreement with matrix model calculations. We identify the momentum one tachyon as the puncture operator, non-negative momentum tachyons as primary fields, and negative momentum ones as descendants. The model thus has an infinite number of primary fields, and the topological metric vanishes on the small phase space when restricted to these. We find a parity invariant multi-contact algebra with irreducible contact terms of arbitrarily large number of fields. The formulation of this Landau-Ginzburg description in terms of period integrals coincides with the genus zero $W_{1+\infty}$ identities of two-dimensional string theory. We study the underlying Toda lattice integrable hierarchy in the Lax formulation and find that the Landau-Ginzburg superpotential coincides with a derivative of the Baker-Akhiezer wave function in the dispersionless limit. This establishes a connection between the topological and integrable structures. Guided by this connection we derive relations formally analogous to the string equation.