Dual Little Strings and their Partition Functions

TL;DR

This paper computes the topological string partition functions for a class of toric Calabi-Yau threefolds related to little string theories, confirming a duality between different geometries with the same product of parameters.

Contribution

It provides explicit calculations of the partition functions for $X_{N,M}$ manifolds and proves a previously conjectured duality relating different geometries with the same product of parameters.

Findings

Explicit topological string partition functions for $X_{N,M}$.

Verification of the duality $X_{N,M} o X_{N',M'}$ with $NM=N'M'$.

Application of refined topological vertex formalism.

Abstract

We study the topological string partition function of a class of toric, double elliptically fibered Calabi-Yau threefolds $X_{N,M}$ at a generic point in the K\"ahler moduli space. These manifolds engineer little string theories in five dimensions or lower and are dual to stacks of M5-branes probing a transverse orbifold singularity. Using the refined topological vertex formalism, we explicitly calculate a generic building block which allows to compute the topological string partition function of $X_{N,M}$ as a series expansion in different K\"ahler parameters. Using this result we give further explicit proof for a duality found previously in the literature, which relates $X_{N,M}\sim X_{N',M'}$ for $NM=N'M'$ and $\text{gcd}(N,M)=\text{gcd}(N',M')$.