The topological string associated with a simple singularity of type $D$

The partition function of $D_{N+1}$ topological string, the coupled system of topological gravity and $D_{N+1}$ topological minimal matter , is proposed in the framework of KP hierarchy. It is specified by the elements of $GL(\infty)$ which constitute the deformed family from the $A_{2N-1}$ topological string. Its dispersionless limit is investigated from the view of both dispersionless KP hierarchy and singularity theory. In particular the free energy restricted on the small phase space coincides with that for the topological Landau-Ginzburg model of type $D_{N+1}$.