TOPOLOGICAL σ MODELS AND LARGE N MATRIX INTEGRAL

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 one-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 a 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 Kähler class). The form of the superpotential indicates the close connection between CP 1 and N=2 supersymmetric sine-Gordon theory which was noted sometime ago by several authors. We also discuss possible generalizations of our construction to other manifolds and present an LG formulation of the topological CP 2 model.