The ABCDEFG of Little Strings

TL;DR

This paper explores the geometric and algebraic structures of codimension-two defects in little string theory, revealing their relation to Lie algebras, Toda theories, and nilpotent orbits, thus unifying various aspects of string and gauge theories.

Contribution

It provides a unified description of defects in little string theory using Lie algebraic data and links instanton partition functions to Toda conformal blocks.

Findings

Defects are labeled by weights of the Langlands dual algebra.

Instanton partition functions match Toda conformal blocks.

Defects' Coulomb branch flows to nilpotent orbits.

Abstract

Starting from type IIB string theory on an $ADE$ singularity, the $(2,0)$ little string arises when one takes the string coupling $g_s$ to 0. In this setup, we give a unified description of the codimension-two defects of the little string, labeled by a simple Lie algebra ${\mathfrak{g}}$. Geometrically, these are D5 branes wrapping 2-cycles of the singularity, subject to a certain folding operation when the algebra is non simply-laced. Equivalently, the defects are specified by a certain set of weights of $^L {\mathfrak{g}}$, the Langlands dual of ${\mathfrak{g}}$. As a first application, we show that the instanton partition function of the ${\mathfrak{g}}$-type quiver gauge theory on the defect is equal to a 3-point conformal block of the ${\mathfrak{g}}$-type deformed Toda theory in the Coulomb gas formalism. As a second application, we argue that in the $(2,0)$ CFT limit, the Coulomb branch of the defects flows to a nilpotent orbit of ${\mathfrak{g}}$.