Dual Little Strings and their Partition Functions

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 $X_{N,M}\sim X_{N^{\mathrm{\prime }},M^{\mathrm{\prime }}}$ for $NM=N^{\mathrm{\prime }}{M}^{\mathrm{\prime }}$ and $\text{gcd}(N,M)=\text{gcd}(N^{\mathrm{\prime }},M^{\mathrm{\prime }})$.