The Topological CP¹ Model and the Large-N Matrix Integral

We discuss the topological $CP^1$ model which consists of the holomorphic maps from Riemann surfaces onto $CP^1$. We construct a large-$N$ matrix model which reproduces precisely the partition function of the $CP^1$ model at all genera of Riemann surfaces. The action of our matrix model has the form ${\rm Tr}\, V(M)=-2{\rm Tr}\, M(\log M -1) +2\sum t_{n,P}{\rm Tr}\, M^n(\log M-c_n) +\sum 1/n\cdot t_{n-1,Q}{\rm Tr}\, M^n~(c_n=\sum_1^n 1/j )$ 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 (K\"ahler) operator.