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2-Group Symmetries in Class S

Lakshya Bhardwaj

TL;DR

This work identifies and computes 2-group symmetries in 4d $\mathcal{N}=2$ Class S theories arising from compactifications of 6d $\mathcal{N}=(2,0)$ theories on punctured Riemann surfaces with twist lines. It develops a framework where 1-form symmetries (line defects), 0-form flavor symmetries (flavor Wilson lines), and their mixing are encoded by a trio of abelian groups linked by exact sequences and a Postnikov class $[\Theta]=\mathrm{Bock}[w_2]$, with the non-closure relation $\delta B_2 + B_1^* w_3 = 0$ governing the mixing. The authors provide a general method to extract these data from the geometric surface data of Class S theories and illustrate it with a detailed SO$(4n+2)$-based example, where a nontrivial 2-group appears for certain polarizations and is corroborated by a gauge-theory analysis. The results illuminate how higher-form symmetries and their interplay with flavor symmetries are encoded in the geometry of compactifications, offering a concrete non-Lagrangian handle on generalized symmetries in Class S theories.

Abstract

2-group symmetries are generalized symmetries that arise when 1-form and 0-form symmetries mix with each other. We uncover the existence of a class of 2-group symmetries in general 4d N=2 theories of Class S that can be constructed by compactifying 6d N=(2,0) SCFTs on Riemann surfaces carrying arbitrary regular punctures and outer-automorphism twist lines. The 2-group structure can be captured in terms of equivalence classes of line defects plus flavor Wilson lines, which can be thought of as accounting for screening of line defects while keeping track of flavor charges. We describe a method for computing these equivalence classes for a general Class S theory using the data on the Riemman surface used for compactifying its parent 6d N=(2,0) theory.

2-Group Symmetries in Class S

TL;DR

This work identifies and computes 2-group symmetries in 4d Class S theories arising from compactifications of 6d theories on punctured Riemann surfaces with twist lines. It develops a framework where 1-form symmetries (line defects), 0-form flavor symmetries (flavor Wilson lines), and their mixing are encoded by a trio of abelian groups linked by exact sequences and a Postnikov class , with the non-closure relation governing the mixing. The authors provide a general method to extract these data from the geometric surface data of Class S theories and illustrate it with a detailed SO-based example, where a nontrivial 2-group appears for certain polarizations and is corroborated by a gauge-theory analysis. The results illuminate how higher-form symmetries and their interplay with flavor symmetries are encoded in the geometry of compactifications, offering a concrete non-Lagrangian handle on generalized symmetries in Class S theories.

Abstract

2-group symmetries are generalized symmetries that arise when 1-form and 0-form symmetries mix with each other. We uncover the existence of a class of 2-group symmetries in general 4d N=2 theories of Class S that can be constructed by compactifying 6d N=(2,0) SCFTs on Riemann surfaces carrying arbitrary regular punctures and outer-automorphism twist lines. The 2-group structure can be captured in terms of equivalence classes of line defects plus flavor Wilson lines, which can be thought of as accounting for screening of line defects while keeping track of flavor charges. We describe a method for computing these equivalence classes for a general Class S theory using the data on the Riemman surface used for compactifying its parent 6d N=(2,0) theory.

Paper Structure

This paper contains 16 sections, 78 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Generalities
  3. 1-Form Symmetries
  4. Global Form of 0-Form Flavor Symmetry Group
  5. 2-Group Symmetries
  6. Gauge Theories
  7. Class S
  8. $6d$ $\mathcal{N}=(2,0)$ Theories and Their Surface Defects
  9. 1-Form Symmetries
  10. Global Form of 0-Form Flavor Symmetry Group
  11. 2-Group Symmetries
  12. An Illustrative Example
  13. 1-Form Symmetry
  14. Global Form of 0-Form Flavor Symmetry Group
  15. 2-Group Symmetry
  16. ...and 1 more sections

Figures (6)

  • Figure 1: Consider two line defects $L_1$ and $L_2$ such that there exists a junction local operator $O_{21}\neq0$ between them. Then, we can construct a configuration as shown on the left side of figure with time flowing horizontally. For charge conservation, the charges $q(L_1)$ and $q(L_2)$ of $L_1$ and $L_2$ must match.
  • Figure 2: An outer-automorphism twist line (shown in black) acts on the element of $\widehat{Z}_\mathcal{G}$ carried by a K1-chain (shown in red and blue).
  • Figure 3: A $6d$ surface defect compactified on a K2-chain (shown in red) whose boundary is a K1-cycle (shown in blue) leads to a $4d$ local operator (shown in red) living at the end of a line defect corresponding to the K1-cycle (shown in blue). If the K2-chain passes over a puncture $\mathcal{P}_i$, then the local operator is charged under a representation $\mathcal{R}_i$ of $\mathcal{P}_i$. See (\ref{['R4']}).
  • Figure 4: Taking OPE of two local operators living at the ends of two line defects that are inverse of each other leads to a genuine local operator. The flavor center charges are added in this process.
  • Figure 5: Taking OPE of a genuine local operator with a local operator living at the end of a line defect leads to another local operator living at the end of that line defect. The flavor center charges are added in this process.
  • ...and 1 more figures