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.