Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Gaillard-Zumino non-invertible symmetries

Fabio Apruzzi, Luca Martucci

TL;DR

The paper demonstrates that classical Gaillard–Zumino duality symmetries in broad four-dimensional gauge theories survive quantum mechanically as non-invertible zero-form symmetries when restricted to rational subgroups ${ m G}_{f Q}={ m G}igcap { m Sp}(2n,f Q)$. It constructs explicit topological defects ${ m D}_{ m S}$ realizing these symmetries by fusing invertible and non-invertible components, and analyzes their action on line operators, revealing non-invertible transformations that can map genuine lines to non-genuine ones. The authors develop generating interfaces ${ m W}_{ m S}$ across generating pieces ${ m A,C,oldsymbol{ abla}}$ and illustrate the framework with axion–Maxwell, ${ au}$–Maxwell, and perturbative Calabi–Yau models, including detailed fusion and condensation phenomena. They also discuss gauging integral subgroups and the implications for symmetry breaking in quantum gravity contexts, suggesting connections to Swampland ideas and SymTFT formalisms. Overall, the work provides a concrete, programmable mechanism by which classical dualities yield a rich spectrum of non-invertible quantum symmetries in non-gravitational and gravitationally coupled settings, with several explicit examples and future directions for minimal defects, anomalies, and moduli-space fixed points.

Abstract

We uncover an infinite class of novel zero-form non-invertible symmetries in a broad family of four-dimensional models, studied years ago by Gaillard and Zumino (GZ), which includes several extended supergravities as particular subcases. The GZ models consist of abelian gauge fields coupled to a neutral sector, typically including a set of scalars, whose equations of motion are classically invariant under a continuous group $\mathscr{G}$ acting on the electric and magnetic field strengths via symplectic transformations. The standard lore holds that, at the quantum level, these symmetries are broken to an integral subgroup $\mathscr{G}_\mathbb{Z}$. We show that, in fact, a much larger subgroup $\mathscr{G}_\mathbb{Q}$ survives, albeit through non-invertible topological defects. We explicitly construct these defects and compute some of their fusion rules. As illustrative examples, we consider the axion-dilaton-Maxwell model and the bosonic sector of a class of $\mathcal{N}=2$ supergravities of the kind that appear in type II Calabi-Yau compactifications. Finally, we comment on how (part of) these non-invertible zero-form symmetries can be broken by gauging the $\mathscr{G}_\mathbb{Z}$ subgroup of invertible symmetries.

Gaillard-Zumino non-invertible symmetries

TL;DR

The paper demonstrates that classical Gaillard–Zumino duality symmetries in broad four-dimensional gauge theories survive quantum mechanically as non-invertible zero-form symmetries when restricted to rational subgroups . It constructs explicit topological defects realizing these symmetries by fusing invertible and non-invertible components, and analyzes their action on line operators, revealing non-invertible transformations that can map genuine lines to non-genuine ones. The authors develop generating interfaces across generating pieces and illustrate the framework with axion–Maxwell, –Maxwell, and perturbative Calabi–Yau models, including detailed fusion and condensation phenomena. They also discuss gauging integral subgroups and the implications for symmetry breaking in quantum gravity contexts, suggesting connections to Swampland ideas and SymTFT formalisms. Overall, the work provides a concrete, programmable mechanism by which classical dualities yield a rich spectrum of non-invertible quantum symmetries in non-gravitational and gravitationally coupled settings, with several explicit examples and future directions for minimal defects, anomalies, and moduli-space fixed points.

Abstract

We uncover an infinite class of novel zero-form non-invertible symmetries in a broad family of four-dimensional models, studied years ago by Gaillard and Zumino (GZ), which includes several extended supergravities as particular subcases. The GZ models consist of abelian gauge fields coupled to a neutral sector, typically including a set of scalars, whose equations of motion are classically invariant under a continuous group acting on the electric and magnetic field strengths via symplectic transformations. The standard lore holds that, at the quantum level, these symmetries are broken to an integral subgroup . We show that, in fact, a much larger subgroup survives, albeit through non-invertible topological defects. We explicitly construct these defects and compute some of their fusion rules. As illustrative examples, we consider the axion-dilaton-Maxwell model and the bosonic sector of a class of supergravities of the kind that appear in type II Calabi-Yau compactifications. Finally, we comment on how (part of) these non-invertible zero-form symmetries can be broken by gauging the subgroup of invertible symmetries.
Paper Structure (39 sections, 295 equations, 3 figures)

This paper contains 39 sections, 295 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Gaillard-Zumino models
  3. Classical continuous GZ symmetries
  4. GZ symmetry breaking by flux quantization
  5. GZ topological defects: general strategy
  6. Construction of ${\cal W}_{\cal S}$ interfaces
  7. ${\rm Sp}(2n,\mathbb{Z})$ generating interfaces.
  8. A interfaces
  9. C interfaces from TQFTs
  10. Minimal C interfaces
  11. Condensation defects
  12. Action on line defects
  13. Examples with one gauge field
  14. Warm-up: the axion-Maxwell model
  15. $\tau$-Maxwell model
  16. ...and 24 more sections

Figures (3)

  • Figure 1: Construction of a $\mathcal{D}_{{\cal S}}$ defect by fusing a ${\cal U}_{\cal S}$ defect and a ${\cal W}_{\cal S}$ interface.
  • Figure 2: Action of the defect $\mathcal{D}_{{\cal S}}$ on the line operator $L_{\mathbb{L}}$.
  • Figure 3: Action of the defect $\mathcal{D}_{{\cal S}}$ on the line operator $L_{\mathbb{L}}$.