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Non-Invertible Global Symmetries and Completeness of the Spectrum

Ben Heidenreich, Jacob McNamara, Miguel Montero, Matthew Reece, Tom Rudelius, Irene Valenzuela

TL;DR

The paper demonstrates that for compact gauge theories, completeness of the charged spectrum is equivalent to the absence of topological (potentially non-invertible) 1-form electric symmetries, expanding the classic invertible-1-form story to include non-invertible topological operators. It shows that, for connected, finite, or disconnected compact groups, removing all topological Gukov-Witten/Z(G) operators guarantees a complete spectrum, and similarly for twist vortices when all GW operators are endable. The work further analyzes how these relations adapt under Higgsing and the addition of Chern-Simons/BF couplings, revealing higher-group structures that mix electric and magnetic symmetries and refining the completeness criterion. It discusses noncompact cases where the correspondence can break down, and explores Swampland implications, punctuating potential phenomenological consequences of twist strings in cosmology. Overall, the absence of topological operators emerges as a unifying principle linking symmetry, spectrum completeness, and quantum gravity constraints across a broad class of gauge theories.

Abstract

It is widely believed that consistent theories of quantum gravity satisfy two basic kinematic constraints: they are free from any global symmetry, and they contain a complete spectrum of gauge charges. For compact, abelian gauge groups, completeness follows from the absence of a 1-form global symmetry. However, this correspondence breaks down for more general gauge groups, where the breaking of the 1-form symmetry is insufficient to guarantee a complete spectrum. We show that the correspondence may be restored by broadening our notion of symmetry to include non-invertible topological operators, and prove that their absence is sufficient to guarantee a complete spectrum for any compact, possibly disconnected gauge group. In addition, we prove an analogous statement regarding the completeness of twist vortices: codimension-2 objects defined by a discrete holonomy around their worldvolume, such as cosmic strings in four dimensions. We discuss how this correspondence is modified in various, more general contexts, including non-compact gauge groups, Higgsing of gauge theories, and the addition of Chern-Simons terms. Finally, we discuss the implications of our results for the Swampland program, as well as the phenomenological implications of the existence of twist strings.

Non-Invertible Global Symmetries and Completeness of the Spectrum

TL;DR

The paper demonstrates that for compact gauge theories, completeness of the charged spectrum is equivalent to the absence of topological (potentially non-invertible) 1-form electric symmetries, expanding the classic invertible-1-form story to include non-invertible topological operators. It shows that, for connected, finite, or disconnected compact groups, removing all topological Gukov-Witten/Z(G) operators guarantees a complete spectrum, and similarly for twist vortices when all GW operators are endable. The work further analyzes how these relations adapt under Higgsing and the addition of Chern-Simons/BF couplings, revealing higher-group structures that mix electric and magnetic symmetries and refining the completeness criterion. It discusses noncompact cases where the correspondence can break down, and explores Swampland implications, punctuating potential phenomenological consequences of twist strings in cosmology. Overall, the absence of topological operators emerges as a unifying principle linking symmetry, spectrum completeness, and quantum gravity constraints across a broad class of gauge theories.

Abstract

It is widely believed that consistent theories of quantum gravity satisfy two basic kinematic constraints: they are free from any global symmetry, and they contain a complete spectrum of gauge charges. For compact, abelian gauge groups, completeness follows from the absence of a 1-form global symmetry. However, this correspondence breaks down for more general gauge groups, where the breaking of the 1-form symmetry is insufficient to guarantee a complete spectrum. We show that the correspondence may be restored by broadening our notion of symmetry to include non-invertible topological operators, and prove that their absence is sufficient to guarantee a complete spectrum for any compact, possibly disconnected gauge group. In addition, we prove an analogous statement regarding the completeness of twist vortices: codimension-2 objects defined by a discrete holonomy around their worldvolume, such as cosmic strings in four dimensions. We discuss how this correspondence is modified in various, more general contexts, including non-compact gauge groups, Higgsing of gauge theories, and the addition of Chern-Simons terms. Finally, we discuss the implications of our results for the Swampland program, as well as the phenomenological implications of the existence of twist strings.

Paper Structure

This paper contains 56 sections, 6 theorems, 90 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Topological Operators in Gauge Theories
  3. Higher-form Symmetries and Topological Operators
  4. Wilson Operators and Gukov-Witten Operators
  5. Topological Operators in Pure Gauge Theories
  6. Topological Gukov-Witten Operators and Endable Wilson Lines
  7. Topological Wilson Lines and Endable Gukov-Witten Operators
  8. 't Hooft Operators and Magnetic Global Symmetries
  9. Compact, Connected Gauge Groups
  10. U1 Gauge Theory
  11. SUNZK Gauge Theory
  12. G2 Gauge Theory
  13. General Story for Compact, Connected Groups
  14. Electric Completeness
  15. Magnetic Completeness
  16. ...and 41 more sections

Key Result

Lemma 1

Let $G$ be a compact Lie group, and $\rho$ a (complex) representation of $G$. Then, $\chi_{\rho}([g]) = \chi_\rho(1)$ if and only if $\rho(g)$ is the identity for all $g \in [g]$.

Figures (3)

  • Figure 1: Endable operators and topological operators. If a dimension $p$ charged operator $V(\mathcal{C}^{(p)})$ can end, a topological operator $T_{a}(S^{d-p-1})$ surrounding it may be shrunk to a point, yielding a factor of $B_V(a)$ (left), or it may be unlinked, yielding a trivial factor $B_1(a)$. This means that if $T_a$ does not satisfy $B_V(a)=B_1(a)$, then it cannot be topological whenever $V(\mathcal{C}^{(p)})$ is endable.
  • Figure 2: Cross-section of a particle worldline going around a vortex; the time direction is shown, and the $(d - 3)$ spatial directions along which the vortex extends are suppressed. The fact that the particle winds around the vortex means that the corresponding Wilson line and Gukov-Witten operators are linked, in the Euclidean picture. The linking property \ref{['eq:linking']} of topological and charged operators encodes the fact that the correlator on the left and right pictures differ by a multiplication by the character of the representation, which is the Aharonov-Bohm factor.
  • Figure 3: A Gukov-Witten operator for a conjugacy class $[g]$ in the identity component of $G$ (represented by the fuzzy line in the picture) may end on an "improperly quantized" 't Hooft operator, depicted here by an open dot. To have a Gukov-Witten operator end, we specify a boundary condition for the gauge field on a hemisphere of the ending locus (depicted by dotted lines) by demanding that the holonomy becomes trivial in a smooth way. By Lemma \ref{['ambrose_singer']}, this is only possible for $g$ in the identity component.

Theorems & Definitions (12)

  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • ...and 2 more