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Little String Origin of Surface Defects

Nathan Haouzi, Christian Schmid

Abstract

We derive the codimension-two defects of 4d $\mathcal{N} = 4$ Super Yang-Mills (SYM) theory from the (2, 0) little string. The origin of the little string is type IIB theory compactified on an ADE singularity. The defects are D-branes wrapping the 2-cycles of the singularity. We use this construction to make contact with the description of SYM defects due to Gukov and Witten [arXiv:hep-th/0612073]. Furthermore, we derive from a geometric perspective the complete nilpotent orbit classification of codimension-two defects, and the connection to ADE-type Toda CFT. The only data needed to specify the defects is a set of weights of the algebra obeying certain constraints, which we give explicitly. We highlight the differences between the defect classification in the little string theory and its (2, 0) CFT limit.

Little String Origin of Surface Defects

Abstract

We derive the codimension-two defects of 4d Super Yang-Mills (SYM) theory from the (2, 0) little string. The origin of the little string is type IIB theory compactified on an ADE singularity. The defects are D-branes wrapping the 2-cycles of the singularity. We use this construction to make contact with the description of SYM defects due to Gukov and Witten [arXiv:hep-th/0612073]. Furthermore, we derive from a geometric perspective the complete nilpotent orbit classification of codimension-two defects, and the connection to ADE-type Toda CFT. The only data needed to specify the defects is a set of weights of the algebra obeying certain constraints, which we give explicitly. We highlight the differences between the defect classification in the little string theory and its (2, 0) CFT limit.

Paper Structure

This paper contains 31 sections, 92 equations, 18 figures, 2 tables.

Table of Contents

  1. Introduction
  2. SYM Surface Defects From Little String Theory
  3. Gukov--Witten Surface Defects of $\mathcal{N}=4$ SYM
  4. Little String on a Riemann surface and D5 Branes
  5. Little String Theory origin of SYM surface defects and S-duality
  6. From Brane Defects to Parabolic Subalgebra Classification
  7. Mathematics Preliminaries
  8. Parabolic Subalgebras from Weight Data
  9. Parabolic subalgebras from Higgs field data
  10. Surface defects and Nilpotent Orbits
  11. A short review
  12. Nilpotent orbits from Levi subalgebras
  13. Surface Defect classification and $\mathcal{W}(\mathfrak{g})$-algebras
  14. Levi subalgebras from level-1 null states of Toda CFT
  15. Seiberg--Witten curves from $\mathcal{W}(\mathfrak{g})$-algebras
  16. ...and 16 more sections

Figures (18)

  • Figure 1: The quiver theory describing a full puncture for $\mathfrak{g}=A_3$
  • Figure 2: One starts with the $(1,1)$ little string theory on $T^2\times\mathbb{R}^2\times\mathcal{C}$. After doing two T-dualities in the torus directions, we get the $(1,1)$ little string theory on the T-dual torus; in the low energy limit, the pair of $(1,1)$ theories gives an S-dual pair of $\mathcal{N}=4$ SYM theories. D3 branes at a point on $T^2$ map to D4 branes in either $(1,1)$ theory, while D5 branes wrapping $T^2$ map to another set of D4 branes.
  • Figure 3: From the set of weights ${\cal W}_{\cal S}$, we read off the parabolic subalgebra $\mathfrak{p}_\varnothing$ of $A_3$ (in this case, the choice of weights is unique up to global $\mathbb{Z}_2$ action on the set). Reinterpreting each weight as a sum of "minus a fundamental weight and simple roots," we obtain the 2d quiver gauge theory shown on the right. The white arrow implies we take the CFT limit.
  • Figure 4: From the two sets of weights ${\cal W}_{\cal S}$, we read off the parabolic subalgebra $\mathfrak{p}_{\{\alpha_3,\alpha_4\}}$ of $D_4$. Reinterpreting each weight as a sum of "minus a fundamental weight and simple roots," we obtain two different 2d quiver gauge theories shown on the right. The white arrows imply we take the CFT limit.
  • Figure 5: Two sets of weights ${\cal W}_{\cal S}$ which spell out the same quiver, but denote two different defects; we see it is really the weights, and not quivers, that define a defect. This is clear in the CFT limit, where two distinct parabolic subalgebras are distinguished.
  • ...and 13 more figures

Theorems & Definitions (11)

  • Example 2.1
  • Example 3.1
  • Example 3.2
  • Example 3.3
  • Example 3.4: $A_3$ example
  • Example 3.5: $D_4$ example
  • Example 3.6
  • Example 4.1: $A_3$
  • Example 4.2
  • Example 5.1
  • ...and 1 more