Dual little strings from F-theory and flop transitions

A particular two-parameter class of little string theories can be described by M parallel M5-branes probing a transverse affine A _{N - 1} singularity. We previously discussed the duality between the theories labelled by ( N, M) and ( M, N). In this work, we propose that these two are in fact only part of a larger web of dual theories. We provide evidence that the theories labelled by ( N, M) and (NM/k,k) are dual to each other, where k = gcd( N, M). To argue for this duality, we use a geometric realization of these little string theories in terms of F-theory compactifications on toric, non-compact Calabi-Yau threefolds X _{N, M} which have a double elliptic fibration structure. We show explicitly for a number of examples that X _{NM/ k, k} is part of the extended moduli space of X _{N, M} , i.e. the two are related through symmetry transformations and flop transitions. By working out the full duality map, we provide a simple check at the level of the free energy of little string theories.