Topological $\sigma$-Models and Large-$N$ Matrix Integral

TL;DR

This paper explores the topological $CP^1$ model through matrix integrals, revealing its integrable structure, computing intersection numbers, and establishing a Landau-Ginzburg description linked to the Toda hierarchy and supersymmetric theories.

Contribution

It introduces a matrix model for the $CP^1$ topological model, connects it to the Toda hierarchy, and develops a Landau-Ginzburg framework, extending to $CP^2$ and relating to supersymmetric theories.

Findings

Matrix model reproduces sum over holomorphic maps.

Intersection numbers match matrix model predictions.

Landau-Ginzburg superpotential links to Toda hierarchy and supersymmetric sine-Gordon.

Abstract

In this paper we describe in some detail the representation of the topological $CP^1$ model in terms of a matrix integral which we have introduced in a previous article. We first discuss the integrable structure of the $CP^1$ model and show that it is governed by an extension of the 1-dimensional Toda hierarchy. We then introduce a matrix model which reproduces the sum over holomorphic maps from arbitrary Riemann surfaces onto $CP^1$. We compute intersection numbers on the moduli space of curves using geometrical method and show that the results agree with those predicted by the matrix model. We also develop a Landau-Ginzburg (LG) description of the $CP^1$ model using a superpotential $e^X+e^{t_{0,Q}}e^{-X}$ given by the Lax operator of the Toda hierarchy ($X$ is the LG field and $t_{0,Q}$ is the coupling constant of the K\"ahler class). The form of the superpotential indicates the close connection between $CP^1$ and $N=2$ supersymmetric sine-Gordon theory which was noted some time ago by several authors. We also discuss possible generalizations of our construction to other manifolds and present a LG formulation of the topological $CP^2$ model.