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Comments on the twisted punctures of $A_\text{even}$ class S theory

Yuji Tachikawa, Yifan Wang, Gabi Zafrir

TL;DR

The paper shows that the USp flavor symmetry of a full twisted puncture in A_{even} class S theories has a global anomaly governed by $\pi_4(\mathrm{USp})=\mathbb{Z}_2$, and explores this in the SCFT $R_{2,2N}$. By three complementary viewpoints—class S puncture data, 5d twisted compactifications, and IIA orientifold constructions—it demonstrates how the anomaly manifests and constrains how $R_{2,2N}$ can couple to bulk theories. It identifies $R_{2,2N}$ as a natural UV completion of an infrared-free $\mathrm{SO}(2N+1)$ gauge theory with $2N$ flavors, where the $USp(4N)$ symmetry carries the anomaly, and shows the Seiberg-Witten curves agree across different realizations with the bulk scale identified with a mass parameter. These results illuminate how global anomalies propagate from bulk to boundary in class S constructions and connect non-Lagrangian theories to familiar gauge-theoretic limits.

Abstract

We point out that the $\text{USp}$ symmetry associated to a full twisted puncture of a class S theory of type $A_\text{even}$ has the global anomaly associated to $π_4(\text{USp})=\mathbb{Z}_2$. We discuss manifestations of this fact in the context of the superconformal field theory $R_{2,2N}$ introduced by Chacaltana, Distler and Trimm. For example, we find that this theory can be thought of as a natural ultraviolet completion of an infrared-free $\text{SO}(2N+1)$ gauge theory with $2N$ flavors, whose $\text{USp}(4N)$ symmetry clearly has the global anomaly.

Comments on the twisted punctures of $A_\text{even}$ class S theory

TL;DR

The paper shows that the USp flavor symmetry of a full twisted puncture in A_{even} class S theories has a global anomaly governed by , and explores this in the SCFT . By three complementary viewpoints—class S puncture data, 5d twisted compactifications, and IIA orientifold constructions—it demonstrates how the anomaly manifests and constrains how can couple to bulk theories. It identifies as a natural UV completion of an infrared-free gauge theory with flavors, where the symmetry carries the anomaly, and shows the Seiberg-Witten curves agree across different realizations with the bulk scale identified with a mass parameter. These results illuminate how global anomalies propagate from bulk to boundary in class S constructions and connect non-Lagrangian theories to familiar gauge-theoretic limits.

Abstract

We point out that the symmetry associated to a full twisted puncture of a class S theory of type has the global anomaly associated to . We discuss manifestations of this fact in the context of the superconformal field theory introduced by Chacaltana, Distler and Trimm. For example, we find that this theory can be thought of as a natural ultraviolet completion of an infrared-free gauge theory with flavors, whose symmetry clearly has the global anomaly.

Paper Structure

This paper contains 10 sections, 27 equations, 5 figures.

Table of Contents

  1. Introduction and summary
  2. Class S constructions
  3. Review of $R_{2,2N}$
  4. Class S theories of type $A_{2N}$ with a twisted irregular puncture
  5. Class I
  6. Class II
  7. $R_{2,2N}$ from twisted irregular punctures
  8. Half-hypermultiplet from irregular twisted puncture
  9. 5d constructions
  10. IIA constructions

Figures (5)

  • Figure 1: The three-punctured sphere for the theory $R_{2,2N}$.
  • Figure 2: The sphere with one twisted regular puncture and one twisted irregular puncture that engineers a Class S SCFT of type $A_{2N}$ with ${\mathbb Z}_2$ twist.
  • Figure 3: (a) The brane web for a 5d SCFT with global symmetry $\mathrm{SU}(4N)\times \mathrm{U}(1)^2$. (b) The same web after moving some of the $7$-branes. This corresponds to a mass deformation of the 5d SCFT. As can be seen from the web, this mass deformation sends the 5d SCFT to an $\mathrm{SU}(2N+1)$ gauge theory with $4N$ flavors in the fundamental representation.
  • Figure 4: (a) The brane web for a 5d SCFT with global symmetry $\mathrm{SU}(4N+2)\times \mathrm{SU}(2)^2$. (b) The same web after moving some of the $7$-branes. This corresponds to a mass deformation of the 5d SCFT. As can be seen from the web, this mass deformation sends this 5d SCFT to the 5d SCFT in figure \ref{['WebTw']} (a).
  • Figure 5: (a) The IIA brane setup to engineer $\mathrm{SO}(2N+1)$ gauge theories with $n+n'$ flavors. This requires the use of two half-NS5-branes. (b) When we collapse the two half-NS5-branes, we obtain the IIA reduction for the $R_{2,2N}$ theory.