UC Berkeley

Starting from type IIB string theory on an $ADE$ singularity, the $\cN=(2,0)$ little string arises when one takes the string coupling $g_s$ to 0. We compactify the little string on the cylinder with punctures, which we fully characterize, for any simple Lie algebra $\fg$. Geometrically, these punctures are codimension two defects that are D5 branes wrapping 2-cycles of the singularity. Equivalently, the defects are specified by a certain set of weights of $^L \fg$, the Langlands dual of $\fg$.

As a first application of our formalism, we show that at low energies, the defects have a description as a $\fg$-type quiver gauge theory. We compute its partition function, and prove that it is equal to a conformal block of $\fg$-type $q$-deformed Toda theory on the cylinder, in the Coulomb gas formalism.

After taking the string scale limit $m_s\rightarrow\infty$, the little string becomes a $(2,0)$ superconformal field theory (SCFT). As a second application, we study how this limit affects the codimension two defects of the SCFT:

we show that the Coulomb branch of a given defect flows to a nilpotent orbit of $\fg$, and that all nilpotent orbits of $\fg$ arise in this way. We give a physical realization of the Bala--Carter labels that classify nilpotent orbits of simple Lie algebras, and we interpret our results in the context of $\fg$-type Toda. Finally, after compactifying our setup on a torus $T^2$, we make contact with the description of surface defects of 4d $\mathcal{N}=4$ Super Yang-Mills theory due to Gukov and Witten \cite{Gukov:2006jk}.