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 $\text{gcd}(N,M)=\text{gcd}(N',M')$.
Highlights
We study the topological string partition function of a class of toric, double elliptically fibered Calabi-Yau threefolds XN;M at a generic point in the Kähler moduli space
The study of little string theories (LSTs) has received renewed interest: first proposed two decades ago [1,2,3,4], during the last years, the construction of LSTs from various M-brane constructions and their dual geometric description in F-theory [7] has led to a better understanding of supersymmetric quantum theories in six dimensions
By analyzing the web diagrams associated with XN;M, it was shown in [12] that XN;M ∼ XN0;M0 for MN 1⁄4 M0N0 and gcdðM; NÞ 1⁄4 k 1⁄4 gcdðM0; N0Þ; i.e., the two Calabi-Yau threefolds lie in the same extended Kähler moduli space and can be related by flop transitions
Summary
The study of little string theories (LSTs) has received renewed interest: first proposed two decades ago [1,2,3,4] (see [5,6] for a review), during the last years, the construction of LSTs from various M-brane constructions and their dual geometric description in F-theory [7] has led to a better understanding of supersymmetric quantum theories in six dimensions. We shall use this fact to prove a duality (that was first proposed in [12]) at the level of the partition function for generic values of the Kähler parameters: it was argued in [12] that XN;M and XN0;M0 are related to each other through a series of SLð2; ZÞ symmetry and flop transformations if NM 1⁄4 N0M0 and gcdðN; MÞ 1⁄4 k 1⁄4 gcdðN0; M0Þ. We shall verify (3.1) explicitly at a generic point in the moduli space of Kähler parameters by performing a novel expansion of the lefthand side of this relation
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