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,X-M at a generic point in the Kahler 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 us to compute the topological string partition function of X-N,X-M as a series expansion in different Kahler parameters. Using this result, we give further explicit proof for a duality found previously in the literature, which relates X-N,X-M similar to X-N',X-M' for NM = N'M' and gcd(N,M) = gcd(N',M').