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Six-dimensional D_N theory and four-dimensional SO-USp quivers

Yuji Tachikawa

TL;DR

The paper extends Gaiotto's construction of 4d $\mathcal{N}=2$ SCFTs from the 6d $\mathrm{A}_{N-1}$ theory to the 6d $\mathrm{D}_{N}$ theory with orientifolds, realizing $\mathrm{SO}$–$\mathrm{USp}$ quivers as compactifications on punctured G-curves. It introduces tails labeled by embeddings of $\mathrm{SU}(2)$ into $\mathrm{SO}(2N)$ or $\mathrm{USp}(2N-2)$, with flavor symmetries given by the corresponding commutants, and constructs new isolated theories such as $T_{\mathrm{SO}(2N)}$ with $SO(2N)^3$ flavor symmetry. The work also uncovers dualities, including an $E_7$-flavored SCFT arising from puncture collisions, and demonstrates a nontrivial equivalence between the $A_3$ and $D_3$ six-dimensional theories in this setup. Overall, it broadens the Gaiotto program to the $D_N$ class and suggests rich connections to six-dimensional defects and potential holographic realizations.

Abstract

We realize four-dimensional N=2 superconformal quiver gauge theories with alternating SO and USp gauge groups as compactifications of the six-dimensional D_N theory with defects. The construction can be used to analyze infinitely strongly-coupled limits and S-dualities of such gauge theories, resulting in a new class of isolated four-dimensional N=2 superconformal field theories with SO(2N)^3 flavor symmetry.

Six-dimensional D_N theory and four-dimensional SO-USp quivers

TL;DR

The paper extends Gaiotto's construction of 4d SCFTs from the 6d theory to the 6d theory with orientifolds, realizing – quivers as compactifications on punctured G-curves. It introduces tails labeled by embeddings of into or , with flavor symmetries given by the corresponding commutants, and constructs new isolated theories such as with flavor symmetry. The work also uncovers dualities, including an -flavored SCFT arising from puncture collisions, and demonstrates a nontrivial equivalence between the and six-dimensional theories in this setup. Overall, it broadens the Gaiotto program to the class and suggests rich connections to six-dimensional defects and potential holographic realizations.

Abstract

We realize four-dimensional N=2 superconformal quiver gauge theories with alternating SO and USp gauge groups as compactifications of the six-dimensional D_N theory with defects. The construction can be used to analyze infinitely strongly-coupled limits and S-dualities of such gauge theories, resulting in a new class of isolated four-dimensional N=2 superconformal field theories with SO(2N)^3 flavor symmetry.

Paper Structure

This paper contains 14 sections, 72 equations, 12 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Review: 6d $A_{N-1}$ theory and $\mathrm{SU}(N)$ quivers
  3. Superconformal quivers, G-curve and Young tableaux
  4. Punctures and associated flavor symmetries
  5. SCFT with $\mathrm{SU}(N)^3$ flavor symmetry
  6. 6d $D_{N}$ theory and $\mathrm{SO}$--$\mathrm{USp}$ quivers
  7. Preliminary comments on $\mathrm{SO}$ and $\mathrm{USp}$ gauge groups
  8. From type IIA brane configuration to the G-curve
  9. SCFT with $\mathrm{SO}(2N)^3$ flavor symmetry
  10. Tails, tableaux and flavor symmetries
  11. Duality with SCFT with $E_7$ flavor symmetry
  12. Discussion
  13. $\mathrm{SO}(4)$--$\mathrm{USp}(2)$ quivers
  14. Curves for general $\mathrm{SO}$--$\mathrm{USp}$ quivers

Figures (12)

  • Figure 1: Brane configuration involving "$\mathrm{SU}(1)$" part. Vertical lines stand for NS5 branes extended along $x^{0,1,2,3,4,5}$, and horizontal lines are D4-branes extended along $x^{0,1,2,3,6}$ where the extent $x^6$ is bounded by two NS5-branes. It would correspond to a quiver with gauge groups $\cdots\times\mathrm{SU}(3)\times\mathrm{SU}(2)\times \text{"$\mathrm{SU}(1)$"}$.
  • Figure 2: Construction of $T_{\mathrm{SU}(4)}$. A circle or a box with $n$ inside is an $\mathrm{SU}(n)$ gauge group or flavor symmetry, respectively. A line connecting two objects is a bifundamental hypermultiplet. The symbol $\subset$ means that the subgroup of the flavor symmetry indicated couples to the corresponding gauge field. The triangle with three $\mathrm{SU}(4)$ flavor symmetry attached stands for the $T_{\mathrm{SU}(4)}$ theory.
  • Figure 3: On the left: an example of $\mathrm{SO}$--$\mathrm{USp}$ quiver gauge theory. A circle or a box stands for a gauge group or a flavor symmetry, respectively. A gray object with $n$ inside is an $\mathrm{SO}(n)$ group, a black object with $n$ inside is a $\mathrm{USp}(n)$ group. On the right: the brane configuration realizing the quiver. The vertical lines stand for NS5-branes, the horizontal lines D4-branes suspended between them, and $\otimes$ D6 branes. The dotted line represents the O4-plane. The color distinguishes two types of O4-planes.
  • Figure 4: Examples of $\mathrm{SO}$--$\mathrm{USp}$ quivers and their G-curves. Simple punctures are marked by $\times$. There are two types of full punctures. Each of the punctures labeled by $\odot$ or $\star$ has one $\mathrm{SO}(2N)$ or $\mathrm{USp}(2N-2)$ flavor symmetry, respectively.
  • Figure 5: Construction of $T_{\mathrm{SO}(6)}$. The triangle with three $\mathrm{SO}(6)$ flavor symmetries attached stands for the $T_{\mathrm{SO}(6)}$. The symbol $\subset$ between the $\mathrm{SO}(6)$ flavor symmetry and the $\mathrm{SO}(5)$ gauge symmetry signifies that $\mathrm{SO}(5)\subset \mathrm{SO}(6)$ is gauged.
  • ...and 7 more figures