On Lagrangians of Non-abelian Dijkgraaf-Witten Theories

TL;DR

This paper develops a method to construct BF-type Lagrangians for non-abelian Dijkgraaf-Witten theories by gauging symmetries and analyzing their structure with homotopy theory, relevant for topological phases.

Contribution

It introduces a novel approach to derive non-abelian Dijkgraaf-Witten Lagrangians using symmetry gauging and cohomology with local coefficients.

Findings

Constructed BF-type Lagrangians for non-abelian Dijkgraaf-Witten theories.

Analyzed gauge transformations using homotopy theory.

Matched operator spectra with linking invariants.

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.