The Topological CP^1 Model and the Large-N Matrix Integral

Abstract

We discuss the topological $C{P}^1$ model which consists of the holomorphic maps from Riemann surfaces onto $C{P}^1$. We construct a large-$N$ matrix model which reproduces precisely the partition function of the $C{P}^1$ model at all genera of Riemann surfaces. The action of our matrix model has the form $\mathrm{Tr}V(M)=-2\mathrm{Tr}M(\log M-1)+2\sum t_{n,P}\mathrm{Tr}{M}^n(\log M-c_n)+\sum 1/n\cdot t_{n-1,Q}\mathrm{Tr}{M}^n(c_n={\sum }_1^{n}1/j)$ where $M$ is an $N\times N$ hermitian matrix and $t_{n,P}(t_{n,Q}),(n=0,1,2,\cdots )$ are the coupling constants of the $n$-th descendant of the puncture (Kähler) operator.

13 pages, LaTeX, UT-684