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Geometric Categories for Continuous Gauging

Devon Stockall, Matthew Yu

Abstract

We develop a unified categorical framework for gauging both continuous and finite symmetries in arbitrary spacetime dimensions. Our construction applies to geometric categories i.e. categories internal to stacks. This generalizes the familiar setting of fusion categories, which describe finite group symmetries, to the case of Lie group symmetries. Within this framework, we obtain a functorial Symmetry Topological Field Theory together with its natural boundaries, allowing us to compute associated endomorphism categories and Drinfeld centers in a uniform way. For a given symmetry group $G$, our framework recovers the electric and magnetic higher-form symmetries expected in $G$-gauge theory. Moreover, it naturally encodes electric breaking symmetry in the presence of charged matter, reproducing known physical phenomena in a categorical setting.

Geometric Categories for Continuous Gauging

Abstract

We develop a unified categorical framework for gauging both continuous and finite symmetries in arbitrary spacetime dimensions. Our construction applies to geometric categories i.e. categories internal to stacks. This generalizes the familiar setting of fusion categories, which describe finite group symmetries, to the case of Lie group symmetries. Within this framework, we obtain a functorial Symmetry Topological Field Theory together with its natural boundaries, allowing us to compute associated endomorphism categories and Drinfeld centers in a uniform way. For a given symmetry group , our framework recovers the electric and magnetic higher-form symmetries expected in -gauge theory. Moreover, it naturally encodes electric breaking symmetry in the presence of charged matter, reproducing known physical phenomena in a categorical setting.

Paper Structure

This paper contains 32 sections, 37 theorems, 85 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. The External Perspective on Symmetries
  3. From cobordism to Morita categories
  4. From Morita categories to SymTFT
  5. From Morita categories to SymTFT
  6. The Gauging Prescription
  7. Outline
  8. Glossary
  9. Warm-up: (1+1)-Dimensional Gauging and Symmetries
  10. Symmetries of Gauge Theories
  11. Constructing the (2+1)d SymTFT
  12. Symmetries of Gauge Theories in (1+1)d
  13. Identifying the Electric Symmetry
  14. Identifying the Magnetic Symmetry
  15. Geometric Gadgets
  16. ...and 17 more sections

Key Result

Theorem 1

Let $G$ be an affine algebraic group over characteristic $0$ field $\mathds{K}$. Then there is a fully extended once-categorified $(n+1)$-dimensional TQFT associated to $\mathbf{nRep}(G)$, for all $n\geq 1$.The reader is directed to read warning:shiftindex before comparing thm:generalTQFT with the c

Figures (3)

  • Figure 1: Left: The (1+1)d empty theory is equipped with the trivial action of a group $G$. Right: The bulk-boundary system obtained from equivariantizing the $G$-action. The bulk contains the (2+1)d TQFT constructed from $\mathbf{Rep}(G)$. The physical boundary $\mathfrak{B}_{\mathrm{Phys}}$ contains only local degrees of freedom. The topological boundary $\mathfrak{B}_{\mathrm{Top}}$ contains only topological degrees of freedom, and corresponds to a $\mathbf{Rep}(G)$-module category. The TQFT is a once-categorified (1+1)d theory, and we denote this by not capping off the ends in the 3d bulk. A detailed physical account of this "non-compact" TQFT, in the sense that the objects are non-compact, is descibed in Antinucci:2024bcm.
  • Figure 2: $\mathbf{Rep}(H)$ determines a topological boundary for the bulk $\mathbf{Rep}(G)$ TQFT. The blue lines are objects in $\text{Z}(\mathbf{Rep}(G))$ which can end on the topological boundary. The end points carry a label $h_i\in \mathrm Z(H)$.
  • Figure 3: Sheaves on $S^1$: a skyscraper supported at $0$, and a sheaf supported on $[\frac{\pi}{2},\pi]$.

Theorems & Definitions (105)

  • Definition 1.1
  • Example 1.2
  • Example 1.3
  • Example 1.4
  • Claim 1.5
  • Theorem 1: \ref{['thm:generalTQFT']}
  • Proposition 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Theorem 2: \ref{['thm:MoritaEquivalence']}
  • ...and 95 more