The ABCDEFG of Little Strings

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 ${\mathfrak{g}}$.

Figures (12)

• Figure 1: The action of the outer automorphism group $A$ on the simply-laced Lie algberas. In the case of $D_4$, the outer automorphism can be either $\mathbb{Z}_2$ (resulting in $B_3$) or $\mathbb{Z}_3$ (resulting in $G_2$).
• Figure 2: Sphere with 3 full $A_n$ punctures: 5d theory $T^{5d}$ and 3d theory $G^{3d}$ resulting from $\mathcal{W}_{\mathcal{S}}$.
• Figure 3: Sphere with 3 full $D_n$ punctures: 5d theory $T^{5d}$ and 3d theory $G^{3d}$ resulting from $\mathcal{W}_{\mathcal{S}}$.
• Figure 4: Sphere with 3 full $E_6$ punctures: 5d theory $T^{5d}$ and 3d theory $G^{3d}$ resulting from $\mathcal{W}_{\mathcal{S}}$.
• Figure 5: Sphere with 3 full $E_7$ punctures: 5d theory $T^{5d}$ and 3d theory $G^{3d}$ resulting from $\mathcal{W}_{\mathcal{S}}$.

Theorems & Definitions (5)

• Example 2.1: Polarized and Unpolarized Defects
• Example 4.1: $F_4$ example 1
• Example 4.2: $F_4$ example 2
• Example 4.3: $F_4$ example 3
• Example 4.4: $F_4$ example 4