Physics – High Energy Physics – High Energy Physics - Theory
Scientific paper
1995-03-03
Int.J.Mod.Phys. A10 (1995) 4203
Physics
High Energy Physics
High Energy Physics - Theory
25 pages, phyzzx, no figures
Scientific paper
10.1016/0920-5632(95)00632-X
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.
Eguchi Takami
Hori Kentaro
Yang Shan Kuo
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