Discovering T-Dualities of Little String Theories

We describe a general method for deducing T-dualities of little string theories, which are dualities between these theories that arise when they are compactified on circle. The method works for both untwisted and twisted circle compactifications of little string theories and is based on surface geometries associated to these circle compactifications. The surface geometries describe information about Calabi-Yau threefolds on which M-theory can be compactified to construct the corresponding circle compactified little string theories. Using this method, we deduce at least one T-dual, and in some cases multiple T-duals, for untwisted and twisted circle compactifications of most of the little string theories that can be described on their tensor branches in terms of a 6d supersymmetric gauge theory with a simple non-abelian gauge group, which are also known as rank-0 little string theories. This includes little string theories carrying N=(1,1) and N=(1,0) supersymmetries. For many, but not all, circle compactifications of N=(1,1) little string theories, we have T-dualities that realize Langlands dualities between affine Lie algebras. Along the way, we find another discrete theta angle distinct from 0 and $\pi$ for an E-string node.