Dual little strings from F-theory and flop transitions

Abstract

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 ( N M 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 N M/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.

Dual little strings and their partition functions

“…Using a general building block W fαg fβg to compute Z N;M ðω; ϵ 1;2 Þ, we explicitly confirm Eq. (1.1) for gcdðN; MÞ ¼ 1, thus verifying the duality proposed in [12] explicitly at the level of the partition function. It is pertinent to mention here that the worldsheet theory of little strings realized as M2-branes suspended between the M5-branes is an N ¼ ð2; 0Þ supersymmetric two dimensional theory.…”

“…Given the web diagram, Z N;M can be computed efficiently using the (refined) topological vertex formalism: to this end, a preferred direction in the web needs to be chosen, which from the perspective of LSTs determines the parameters that play the role of coupling constants. Concretely, different choices of the preferred direction lead to different (but equivalent) power series expansions of Z N;M in terms of either h, v, or m. In this paper, 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: indeed, it was argued in [12] that X N;M and X N 0 ;M 0 are related to each other through a series of SLð2; ZÞ symmetry and flop transformations if…”

“…where the duality map ðh; v; mÞ ↦ ðh 0 ; v 0 ; m 0 Þ was proposed in [12] (see also Appendix C for an example), taking into account the consistency conditions on both sides. The relation (2.1) was tested in [12] (based on a conjecture put forward in [35]) at a particular region in the Kähler moduli space of X N;M (with h 36,37].…”

“…Moreover, using for example the refined topological vertex formalism, the topological string partition function Z N;M 1 on X N;M can be computed explicitly [11]. By analyzing the web diagrams associated with X N;M , it was shown in [12] that X N;M ∼ X N 0 ;M 0 for MN ¼ M 0 N 0 and gcdðM; NÞ ¼ k ¼ gcdðM 0 ; N 0 Þ; i.e., the two Calabi-Yau threefolds lie in the same extended Kähler moduli space and can be related by flop transitions. The partition function of the two dual theories should be the same and so it was expected that the topological string partition function of X N;M and X N 0 ;M 0 should be the same, i.e., Z N;M ðω; ϵ 1;2 Þ ¼ Z N 0 ;M 0 ðω 0 ; ϵ 1;2 Þ ð 1:1Þ…”

Little strings on D n orbifolds

“…As a byproduct, our result also verifies the 5d/6d dualities involving D n singularities, discussed in [26][27][28]. Similar studies on T-duality of LSTs were already made in [25,[29][30][31] for the (2, 0) LSTs and the (1, 0) LSTs found from NS5-branes on A n singularities.…”