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Generalized symmetries and 2-groups via electromagnetic duality in AdS/CFT

Oliver DeWolfe, Kenneth Higginbotham

TL;DR

The paper shows that electromagnetic duality in AdS/CFT braids together gauging of generalized global symmetries and the emergence of higher-group structures in the dual field theory. By analyzing the 1-form $A_ u$ and its magnetic dual 2-form $B_{ mu u}$ in AdS$_5$, it demonstrates how regular versus alternate boundary conditions encode global versus dynamical (gauged) $U(1)$ symmetries and identify the dualized 2-form current with the field strength of the gauged $U(1)$. In a concrete example with a mixed 't Hooft anomaly, the gravity dual of a 2-group symmetry is realized via a Chern–Simons theory, and after dualizing a gauge field to a 2-form one obtains a modified field strength that matches the 2-group structure; holographic renormalization confirms the results. The framework extends naturally to general dimensions and to $n$-group symmetries, highlighting how bulk dualities provide a unifying picture for generalized symmetries in holography.

Abstract

We discuss how electromagnetically dualizing a 1-form to a 2-form in AdS$_5$ exchanges regular and alternate boundary conditions, and thus gauges the originally global $U(1)$ symmetry in the dual field theory. The generalized symmetry current dual to the 2-form in the bulk is identified as the dual field strength of the gauged $U(1)$, and the associated double-trace operator with a logarithmically running coupling is just the gauged $U(1)$ Maxwell action. Applying this dualization to an AdS Maxwell-Chern-Simons theory dual to a global $U(1) \times U(1)$ model with an 't Hooft anomaly results in a theory with a modified field strength that holographically realizes a 2-group symmetry. We explicitly carry out the holographic renormalization to verify this, and discuss the generalization to other rank fields in other dimensions.

Generalized symmetries and 2-groups via electromagnetic duality in AdS/CFT

TL;DR

The paper shows that electromagnetic duality in AdS/CFT braids together gauging of generalized global symmetries and the emergence of higher-group structures in the dual field theory. By analyzing the 1-form and its magnetic dual 2-form in AdS, it demonstrates how regular versus alternate boundary conditions encode global versus dynamical (gauged) symmetries and identify the dualized 2-form current with the field strength of the gauged . In a concrete example with a mixed 't Hooft anomaly, the gravity dual of a 2-group symmetry is realized via a Chern–Simons theory, and after dualizing a gauge field to a 2-form one obtains a modified field strength that matches the 2-group structure; holographic renormalization confirms the results. The framework extends naturally to general dimensions and to -group symmetries, highlighting how bulk dualities provide a unifying picture for generalized symmetries in holography.

Abstract

We discuss how electromagnetically dualizing a 1-form to a 2-form in AdS exchanges regular and alternate boundary conditions, and thus gauges the originally global symmetry in the dual field theory. The generalized symmetry current dual to the 2-form in the bulk is identified as the dual field strength of the gauged , and the associated double-trace operator with a logarithmically running coupling is just the gauged Maxwell action. Applying this dualization to an AdS Maxwell-Chern-Simons theory dual to a global model with an 't Hooft anomaly results in a theory with a modified field strength that holographically realizes a 2-group symmetry. We explicitly carry out the holographic renormalization to verify this, and discuss the generalization to other rank fields in other dimensions.

Paper Structure

This paper contains 12 sections, 93 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Electromagnetic duality, boundary conditions and gauging global symmetries
  3. Review of holographic renormalization
  4. The one-form $A_\mu$ in AdS$_5$ and its dual global $U(1)$ symmetry
  5. The two-form $B_{\mu\nu}$ in AdS$_5$ gauges the global $U(1)$ dual to $A_\mu$
  6. Electromagnetic duality and generalized symmetries in general dimension
  7. Gauging 't Hooft anomalies and 2-group symmetries
  8. The $U(1) \times U(1)$ model
  9. Gravity dual of the global $U(1) \times U(1)$ is a Chern-Simons theory
  10. Gravity dual of 2-group structure from electromagnetic duality
  11. $n$-group symmetries in AdS/CFT in general dimension
  12. Conclusions

Figures (2)

  • Figure 1: A flowchart depicting the relationships between electromagnetic duality, boundary conditions and gauging described in this section.
  • Figure 2: A diagram of how electromagnetic duality on a bulk Chern-Simons theory leads to a theory with 2-group symmetry. In this case the blue arrows would represent alternate boundary conditions for the $B$ and $C$ fields, but still regular boundary conditions for $A$.