Triality in little string theories

Abstract

We study a class of eight-supercharge little string theories (LSTs) on the world-volume of N M5-branes with transverse space S 1 × (C 2 /Z M ). These M-brane configurations compactified on a circle are dual to M D5-branes intersecting N NS5-branes on T 2 × R 7,1 as well as to F-theory compactified on a toric Calabi-Yau threefold X N,M . We argue that the Kähler cone of X N,M admits three regions associated with weakly coupled quiver gauge theories of gauge groups [U (N )] M , [U (M )] N and [U ( N M k )] k where k = gcd(N, M ). These provide low-energy descriptions of different LSTs. The duality between the first two gauge theories is well known and is a consequence of the S-duality between D5-and NS5-branes or the T-duality of the LSTs. The triality involving the third gauge theory is new and we demonstrate it using several examples. We also discuss implications of this triality for the W-algebras associated with the Alday-Gaiotto-Tachikawa dual theories.

Dual little strings and their partition functions

“…As we will discuss in [42], this turns out indeed the case. The parameters m i can be expressed in terms of coupling constants of a quiver gauge theory related by U-dualities to the usual quiver theory on the compactified M5-branes dual to X N;M .…”

“…The dual gauge theory and the role of these partitions labeling the building block will be discussed in detail in [42]. However, as any given summand in Eq.…”

Dihedral symmetries of gauge theories from dual Calabi-Yau threefolds

“…where f NS i 1 ,...,i N ,k,n ∈ Z. In [23,21,25] different duality transformations have been discussed, which involve flop transformations [33,34] of various curves of X N,1 , SL(2, Z) transformations as well as cutting and re-gluing of the web diagram. While these duality transformations were shown in [21] to leave Z N,1 (and thus also F N,1 ) invariant, they generically act in a rather nontrivial fashion on the web diagram Fig.…”

Beyond triality: dual quiver gauge theories and little string theories

“…It is expected that the UV completions of (at least some of) these gauge theories are LSTs. The relation among the three low-energy theories with gauge groups (1.2) was dubbed triality in [32], reflecting the fact that all three are engineered by the same Calabi-Yau threefold. It is important to realise that the mapping of the various gauge theory parameters is highly non-trivial among the members of the triality: indeed, the coupling constants, mass parameters and other Coulomb branch parameters are mixed in a highly non-trivial fashion when going from one theory to another.Combining the triality discussed in [32] with the observations of [31] that the Calabi-Yau manifold X N,M itself can be dualised to other manifolds in the sense of eq.…”

“…The relation among the three low-energy theories with gauge groups (1.2) was dubbed triality in [32], reflecting the fact that all three are engineered by the same Calabi-Yau threefold. It is important to realise that the mapping of the various gauge theory parameters is highly non-trivial among the members of the triality: indeed, the coupling constants, mass parameters and other Coulomb branch parameters are mixed in a highly non-trivial fashion when going from one theory to another.Combining the triality discussed in [32] with the observations of [31] that the Calabi-Yau manifold X N,M itself can be dualised to other manifolds in the sense of eq. (1.1) suggests an even more elaborate picture at the level of the low-energy theories: indeed it implies that a (circular) quiver gauge theory with N gauge nodes of type U (M ) (that is engineered by X N,M for arbitrary (N, M )) is dual to a whole web of other quiver gauge theories that have N gauge nodes of type U (M ) such that N M = N M and gcd(N, M ) = gcd(N , M ).…”