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Higher form symmetries and orbifolds of two-dimensional Yang-Mills theory

Leonardo Santilli, Richard J. Szabo

Abstract

We undertake a detailed study of the gaugings of two-dimensional Yang-Mills theory by its intrinsic charge conjugation 0-form and centre 1-form global symmetries, elucidating their higher algebraic and geometric structures, as well as the meaning of dual lower form symmetries. Our derivations of orbifold gauge theories make use of a combination of standard continuum path integral methods, networks of topological defects, and techniques from higher gauge theory. We provide a unified description of higher and lower form gauge fields for a $p$-form symmetry in the geometric setting of $p$-gerbes, and derive reverse orbifolds by the dual $(-1)$-form symmetries. We identify those orbifolds in which charge conjugation symmetry is spontaneously broken, and relate the breaking to mixed anomalies involving $(-1)$-form symmetries. We extend these considerations to gaugings by the non-invertible 1-form symmetries of two-dimensional Yang-Mills theory by introducing a notion of generalized $θ$-angle.

Higher form symmetries and orbifolds of two-dimensional Yang-Mills theory

Abstract

We undertake a detailed study of the gaugings of two-dimensional Yang-Mills theory by its intrinsic charge conjugation 0-form and centre 1-form global symmetries, elucidating their higher algebraic and geometric structures, as well as the meaning of dual lower form symmetries. Our derivations of orbifold gauge theories make use of a combination of standard continuum path integral methods, networks of topological defects, and techniques from higher gauge theory. We provide a unified description of higher and lower form gauge fields for a -form symmetry in the geometric setting of -gerbes, and derive reverse orbifolds by the dual -form symmetries. We identify those orbifolds in which charge conjugation symmetry is spontaneously broken, and relate the breaking to mixed anomalies involving -form symmetries. We extend these considerations to gaugings by the non-invertible 1-form symmetries of two-dimensional Yang-Mills theory by introducing a notion of generalized -angle.
Paper Structure (35 sections, 13 theorems, 178 equations, 3 figures, 1 table)

This paper contains 35 sections, 13 theorems, 178 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Gauge theory preliminaries
  3. Geometry of gauge fields in two dimensions
  4. Representation theory
  5. Yang--Mills theory in two dimensions
  6. Orbifolds of two-dimensional Yang--Mills theory
  7. The orbifold gauge theories
  8. The path integral coupled to a background field
  9. Derivation in higher gauge theory
  10. Gauging the 1-form symmetry
  11. Comparison with combinatorial quantization
  12. Gauging the dual (-1)-form symmetry
  13. Line operators in two-dimensional Yang--Mills theory
  14. Charge conjugation symmetry orbifolds
  15. Spontaneous breaking of charge conjugation symmetry
  16. ...and 20 more sections

Key Result

Proposition 3.1

The partition function of the orbifold $\mathrm{SU}( N)/\mathop{\mathrm{\mathbb{Z}}}\nolimits_k$ pure Yang--Mills theory on $\Sigma$ is given by

Figures (3)

  • Figure 1: (a) Schematic depiction of a $G$-connection $\widetilde{A}$ on $\mathbb{P}^1$. (b) Schematic depiction of an $\mathrm{SU}( N)$-connection $\widetilde{A}^{\,\circ}$ on $\mathbb{P}^1 \setminus \left\{ \wp \right\}$ together with its monodromy $u$.
  • Figure 2: Inserting a topological line operator $L_{\mathcal{C}}$ wrapping a 1-cycle $L \subset \mathop{\mathrm{\mathbb{T}}}\nolimits^2$ is equivalent to cutting $\mathop{\mathrm{\mathbb{T}}}\nolimits^2$ open along $L$ and then closing it with a pair of cross-caps. The black boundaries in the picture are glued together.
  • Figure 3: The mixed anomaly between $\widehat{\mathsf{C}}^{ (0)}$, where $\mathsf{C}^{ (0)}$ is generated by the topological line defects $L_\eta$, and $\mathop{\mathrm{\mathbb{Z}}}\nolimits_N^{ (1)}$ generated by the topological point defects $\wp_\beta$.

Theorems & Definitions (15)

  • Proposition 3.1
  • Lemma 3.26
  • Lemma 3.29
  • Definition 3.30
  • Proposition 4.1
  • Proposition 4.2
  • Definition 5.1
  • Proposition 5.3
  • Proposition 5.5
  • Proposition 5.6
  • ...and 5 more