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6d $\mathcal{N}{=}(1,0)$ theories on $S^1/T^2$ and class S theories: part II

Kantaro Ohmori, Hiroyuki Shimizu, Yuji Tachikawa, Kazuya Yonekura

TL;DR

The paper develops a systematic framework for the T^2 compactification of a broad class of 6d N=(1,0) theories that can Higgs to N=(2,0) theories, revealing that the resulting 4d N=2 theories at the Coulomb origin are governed by coupled conformal-matter sectors linked by infrared-free and conformal gauge factors. By reducing first on S^1 to 5d, and then on another S^1 to 4d, the authors express the 4d theories as two interacting SCFTs linked by an infrared-free gauging, with the entire structure organized by class-S data and puncture collisions; when IR-free factors are absent, the 4d theory remains an SCFT with SL(2,Z) duality and enhanced flavor symmetry. The work provides explicit constructions and checks for A-type conformal matters and extends to general types, including D and E, offering detailed class-S descriptions, M-theory interpretations, and anomaly-based constraints that govern the existence and structure of IR-free sectors. It also discusses special cases without IR-free gauge groups and offers predictions for T-duality relations of little strings, as well as constraints on 6d matter content from 4d anomaly considerations. Overall, the results deepen the link between 6d Higgsable theories, their 5d and 4d reductions, and the class-S framework, and furnish concrete tools for constructing and analyzing the resulting 4d SCFTs and their duality properties.

Abstract

We study the $T^2$ compactification of a class of 6d $\mathcal{N}{=}(1,0)$ theories that is Higgsable to $\mathcal{N}{=}(2,0)$ theories. We show that the resulting 4d $\mathcal{N}{=}2$ theory at the origin of the Coulomb branch and the parameter space is generically given by two superconformal matter sectors coupled by an infrared-free gauge multiplet and another conformal gauge multiplet. Our analysis utilizes the 5d theories obtained by putting the same class of 6d theories on $S^1$. Our class includes, among others, the 6d theories describing multiple M5 branes on an ALE singularity, and we analyze them in detail. The resulting 4d theory has manifestly both the $\mathrm{SL}(2,\mathbb{Z})$ and the full flavor symmetry. We also discuss in detail the special cases of 6d theories where the infrared-free gauge multiplet is absent. In an appendix, we give a field-theoretical argument for an F-theoretic constraint that forbids a particular 6d anomaly-free matter content, as an application of our analysis.

6d $\mathcal{N}{=}(1,0)$ theories on $S^1/T^2$ and class S theories: part II

TL;DR

The paper develops a systematic framework for the T^2 compactification of a broad class of 6d N=(1,0) theories that can Higgs to N=(2,0) theories, revealing that the resulting 4d N=2 theories at the Coulomb origin are governed by coupled conformal-matter sectors linked by infrared-free and conformal gauge factors. By reducing first on S^1 to 5d, and then on another S^1 to 4d, the authors express the 4d theories as two interacting SCFTs linked by an infrared-free gauging, with the entire structure organized by class-S data and puncture collisions; when IR-free factors are absent, the 4d theory remains an SCFT with SL(2,Z) duality and enhanced flavor symmetry. The work provides explicit constructions and checks for A-type conformal matters and extends to general types, including D and E, offering detailed class-S descriptions, M-theory interpretations, and anomaly-based constraints that govern the existence and structure of IR-free sectors. It also discusses special cases without IR-free gauge groups and offers predictions for T-duality relations of little strings, as well as constraints on 6d matter content from 4d anomaly considerations. Overall, the results deepen the link between 6d Higgsable theories, their 5d and 4d reductions, and the class-S framework, and furnish concrete tools for constructing and analyzing the resulting 4d SCFTs and their duality properties.

Abstract

We study the compactification of a class of 6d theories that is Higgsable to theories. We show that the resulting 4d theory at the origin of the Coulomb branch and the parameter space is generically given by two superconformal matter sectors coupled by an infrared-free gauge multiplet and another conformal gauge multiplet. Our analysis utilizes the 5d theories obtained by putting the same class of 6d theories on . Our class includes, among others, the 6d theories describing multiple M5 branes on an ALE singularity, and we analyze them in detail. The resulting 4d theory has manifestly both the and the full flavor symmetry. We also discuss in detail the special cases of 6d theories where the infrared-free gauge multiplet is absent. In an appendix, we give a field-theoretical argument for an F-theoretic constraint that forbids a particular 6d anomaly-free matter content, as an application of our analysis.

Paper Structure

This paper contains 58 sections, 115 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction and summary
  2. Conventions:
  3. The 6d theory:
  4. The 5d theory:
  5. The 4d theory:
  6. Organization:
  7. Two M5-branes probing an ALE singularity on $T^2$
  8. Two M5-branes on $\mathbb{C}^2/\mathbb{Z}_2$
  9. Two M5-branes on other ALE singularities
  10. General structure of theories on $(\mathbf{-2})$-curves
  11. A brief review of structure of 6d SCFTs
  12. Non-Higgsable component and nonrenormalization
  13. $S^1$ compactification to five dimensions
  14. Tensor branch effective action.
  15. Mass deformation of 5d SCFT and 5d quiver.
  16. ...and 43 more sections

Figures (3)

  • Figure 1: $S^1$ reductions of the theory ${\cal T}^\text{6d}_N\{\mathfrak{g},\mathfrak{g}\}$.
  • Figure 2: $S^1$ reductions of the theory ${\cal T}^\text{6d}_\mathfrak{g}\{\mathrm{SU}(N)\}$.
  • Figure 3: $S^1$ reductions of two T-dual theories ${\cal T}^\text{A}_{\mathfrak{g},N}$ and ${\cal T}^\text{B}_{\mathfrak{g},N}$.