Triality in Little String Theories

We study a class of eight-supercharge little string theories (LSTs) on the world-volume of $N$ M5-branes with transverse space $$\mathbb{S}$^{1}\times ($\mathbb{C}$^2/$\mathbb{Z}$_M)$. These M-brane configurations compactified on a circle are dual to $M$ D5-branes intersecting $N$ NS5-branes on ${\mathbb{T}}^2\times {\mathbb{R}}^{7,1}$ as well as to F-theory compactified on a toric Calabi-Yau threefold $X_{N,M}$. We argue that the K"ahler 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 $[U(\frac{NM}{k}){]}^k$ 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.