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On Lagrangians of Non-abelian Dijkgraaf-Witten Theories

Yuan Xue, Eric Y. Yang

Abstract

Dijkgraaf-Witten theories have a wide range of applications in topological phases of matter and the study of generalized global symmetries. We develop a method to construct BF-type Lagrangians for Dijkgraaf-Witten theories with non-abelian gauge group by gauging $H^{(0)}$ symmetries from a BF-Lagrangian of an abelian Dijkgraaf-Witten theory. When $H$ nontrivially permutes the operators of the original theory, the Lagrangian of the $H$-gauged theory is constructed with cohomologies with local coefficients. We analyze the structure of the Lagrangians and their gauge transformations with homotopy theory. We also construct the operator spectrum and verify the Lagrangians by matching elementary linking invariants.

On Lagrangians of Non-abelian Dijkgraaf-Witten Theories

Abstract

Dijkgraaf-Witten theories have a wide range of applications in topological phases of matter and the study of generalized global symmetries. We develop a method to construct BF-type Lagrangians for Dijkgraaf-Witten theories with non-abelian gauge group by gauging symmetries from a BF-Lagrangian of an abelian Dijkgraaf-Witten theory. When nontrivially permutes the operators of the original theory, the Lagrangian of the -gauged theory is constructed with cohomologies with local coefficients. We analyze the structure of the Lagrangians and their gauge transformations with homotopy theory. We also construct the operator spectrum and verify the Lagrangians by matching elementary linking invariants.

Paper Structure

This paper contains 13 sections, 88 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Lagrangian Constructions and Gauge Transformations
  3. BF Theory Formulation of Discrete Gauge Theories
  4. Charge Conjugation Gauged Lagrangian
  5. $\mathbb{D}_4$ Lagrangian
  6. Anomaly-Free Conditions
  7. Linking and Higher Gauging Condensation Defect Dressing
  8. $\mathbb{D}_{k}$ for $k$-odd
  9. $\mathbb{D}_{k}$ for $k$-even
  10. Equivalent Formulations of Untwisted $\mathbb{D}_{4}$ Gauge Theories
  11. Conclusion
  12. Homotopy Theory Basics
  13. Linking Invariant Calculations

Figures (2)

  • Figure 1: A triangulation of $S^1$.
  • Figure 2: An on-shell gauge transformation of $e^{i\oint_{S^1}a_1}$ on a loop with nontrivial $c_1$ profile. $0_i, 0_j, 0_k$ means that there are no data associated to the sites $i,j,k$.