On Gauging Finite Symmetries by Higher Gauging Condensation Defects

TL;DR

This work develops an EFT-style Lagrangian procedure for gauging finite 0-form symmetries in untwisted Dijkgraaf-Witten theories using higher gauging condensation defects, and analyzes its range of validity. It constructs effective actions for untwisted DW theories with non-Abelian gauges, notably the Heisenberg group $H_3(\mathbb{Z}_p)$, and shows that braiding and fusion data from Hopf links align with discrete gauge theory expectations, with a detailed 2+1D lattice regularization to fix sign issues in linking invariants. The paper also clarifies the role of these actions as symTFTs, discusses their relation to higher group global symmetries, and proves no-go results for realizing nontrivial higher groups via generalized Type-I actions. It further distinguishes between Type-I and Type-II actions, providing concrete on-shell matches and off-shell inconsistencies for Type-II, and outlines a general framework for understanding finite-symmetry gauging in DW theories across dimensions. The results contribute a practical, EFT-style toolkit for probing generalized and higher symmetries in topological phases and their boundary/categorical structures, while highlighting important limitations and open questions for future work.

Abstract

Based on the work by C{ó}rdova-Costa-Hsin (arXiv:2412.16681), we propose an EFT-style, Lagrangian procedure to gauge finite 0-form symmetries in untwisted Dijkgraaf-Witten gauge theories on closed oriented manifolds using higher gauging condensation defects and point out its limitations. Using this proposal, we construct effective actions of untwisted Dijkgraaf-Witten theories with Heisenberg gauge group over $\mathbb{Z}_p$ and show that the braiding data from Hopf link and the fusion rules match with the expected discrete gauge theories. We also study the symTFT implications of these effective Lagrangians and clarify their relations with higher group global symmetries.

Figures (5)

• Figure 1: Example of holonomies in the presence of a domain wall $\mathcal{D}_H$. The holonomy $(v_1,v_3,v_2,v_4,v_1)$ piercing the wall is nontrivial, while other holonomies remain trivial.
• Figure 2: An equivalent configuration where we performed a defect worldvolume gauge transformation $(k_L, k_R)\in H$ at site $v_3$.
• Figure 3: Two equivalent notions of a symmetry transformation on a line operator by a symmetry defect. We associate $g\in G$ to the symmetry defect $U_g$ and denote the line as $L$. The action of the symmetry defect on the line is denoted as $L \mapsto {}^g\!L$.
• Figure 4: An illustration of a symTFT sandwich. The operator $D_2^c$ restricted to the topological boundary $\mathcal{B}^{\text{Sym} }_{\mathcal{C} }$ produces a Hilbert space twisted by the symmetry defect $D_2^c$ upon interval compactification.
• Figure 5: A lattice configuration with a nontrivial 2-cochain coupled to the red link.

Theorems & Definitions (6)

• Theorem C.1
• Theorem C.2
• Theorem D.1
• Theorem D.2
• Theorem D.3
• Theorem D.4