Topological String, Matrix Integral, and Singularity Theory

We study the relation between topological string theory and singularity theory using the partition function of $A_{N-1}$ topological string defined by matrix integral of Kontsevich type. Genus expansion of the free energy is considered, and the genus $g=0$ contribution is shown to be described by a special solution of $N$-reduced dispersionless KP system. We show a universal correspondences between the time variables of dispersionless KP hierarchy and the flat coordinates associated with versal deformations of simple singularities of type $A$. We also study the behavior of topological matter theory on the sphere in a topological gravity background, to clarify the role of the topological string in the singularity theory. Finally we make some comment on gravitational phase transition.