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

TL;DR

This paper constructs and analyzes the partition function of the $D_{N+1}$ topological string within the KP hierarchy framework, linking it to singularity theory and Landau-Ginzburg models.

Contribution

It introduces a new formulation of the $D_{N+1}$ topological string partition function using $GL( olinebreak} ext{(} olinebreak ext{infinity)}$ elements, connecting it to singularity theory and dispersionless limits.

Findings

Partition function formulated via $GL( ext{infinity)}$ elements.

Dispersionless limit matches the topological Landau-Ginzburg model of type $D_{N+1}$.

Connection established between topological string, singularity theory, and integrable hierarchies.

Abstract

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 $D_{N+1}$.