Degeneration of Topological String partition functions and Mirror curves of the Calabi-Yau threefolds $X_{N,M}$

TL;DR

This paper investigates how degenerations of mirror curves in Calabi-Yau threefolds affect topological string partition functions, revealing factorization properties and methods to derive partition functions of related geometries.

Contribution

It demonstrates that mirror curve degenerations lead to partition function factorization and provides a way to obtain partition functions of simpler geometries from more complex ones.

Findings

Partition functions factorize during mirror curve degenerations.

Degeneration of mirror curves allows deriving partition functions of related geometries.

The method connects complex Calabi-Yau geometries through mirror curve degenerations.

Abstract

In this paper we study certain degenerations of the mirror curves, associated with Calabi-Yau threefolds $X_{N,M}$, and the effect of these degenerations on the topological string partition function of $X_{N,M}$. We show that when the mirror curve degenerates and become the union of the lower genus curves the corresponding partition function factorizes into pieces corresponding to the components of the degenerate mirror curve. Moreoever we show that using degeneration of a generalised mirror curve it is possible to obtain the partition function corresponding to $X_{N,M-1}$ from $X_{N,M}$.