# Topological $\\sigma$-Models and Large-$N$ Matrix Integral Source: [Topological $\\sigma$-Models and Large-$N$ Matrix Integral]() Collection index: [llms.txt]() Data updated: 2025-10-24T20:35:27.373Z Authors: T. Eguchi, K. Hori, S.-K. Yang Published: 1995-03-03T06:45:26.000Z ## 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. ## Recorded benchmark results A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage. ## Original sources and implementation [Original paper]() [PDF]()