Non-invertible symmetries in finite-group gauge theory

TL;DR

This work develops a comprehensive lattice framework for both invertible and non-invertible symmetries in finite group gauge theories across dimensions, focusing on 0-form domain walls and their boundary manifestations. It constructs and analyzes three wall families—automorphism, diagonal, and magnetic—and a twisted class implementing electric-magnetic duality, deriving their fusion algebras and actions on gapped boundaries, Wilson lines, and magnetic defects. A central result is that non-invertible domain walls can mix electric and magnetic sectors, exemplified by non-factorized EM duality walls that exhibit Fibonacci-like fusion in specific models such as D=3 with G= D_4, and by a general EM-duality extension to higher dimensions. The paper further introduces Cheshire strings as higher codimensional diagonal defects with universal fusion rules, and provides explicit Lagrangian BF-type realizations in concrete models, underscoring the interplay between symmetry, topology, and boundary physics in topological gauge theories.

Abstract

We investigate the invertible and non-invertible symmetries of topological finite-group gauge theories in general spacetime dimensions, where the gauge group can be abelian or non-abelian. We focus in particular on the 0-form symmetry. The gapped domain walls that generate these symmetries are specified by boundary conditions for the gauge fields on either side of the wall. We investigate the fusion rules of these symmetries and their action on other topological defects including the Wilson lines, magnetic fluxes, and gapped boundaries. We illustrate these constructions with various novel examples, including non-invertible electric-magnetic duality symmetry in 3+1d $\mathbb{Z}_2$ gauge theory, and non-invertible analogs of electric-magnetic duality symmetry in non-abelian finite-group gauge theories. In particular, we discover topological domain walls that obey Fibonacci fusion rules in 2+1d gauge theory with dihedral gauge group of order 8. We also generalize the Cheshire string defect to analogous defects of general codimensions and gauge groups and show that they form a closed fusion algebra.

Figures (19)

• Figure 2: Example of valid flat gauge field configuration in a local region of $\mathcal{D}_H(\Sigma)$. Note that the holonomies of $(v_1,v_3,v_4,v_1)$ and $(v_2,v_3,v_4,v_2)$ are trivial, but the holonomy of $(v_1,v_3,v_2,v_4,v_1)$ is not trivial in general.
• Figure 3: Example of equivalent gauge field configuration for the same local region. They are related by a gauge transformation with parameter given by $(k_L,k_R)\in H$ on $v_3$ and $1\in H$ on $v_1,v_2,v_4$.
• Figure 4: Local region of $\Sigma$ in the presence of $\mathcal{D}_{H,\alpha}(\Sigma)\times\mathcal{D}_{H^\prime,\alpha^\prime}(\Sigma)$. We use a cellular decomposition of $\Sigma\times[0,1]$ obtained from two copies of a given triangulation of $\Sigma$ by joining equivalent vertices of the two copies.
• Figure 5: Local region of the boundary $\partial\mathcal{M}$ in the presence of $\mathcal{D}_{H,\alpha}(\partial\mathcal{M})\times\mathcal{B}_{K,\beta}(\partial\mathcal{M})$. We use a cellular decomposition of $\partial\mathcal{M}\times[0,1]$ obtained from two copies of a given triangulation of $\partial\mathcal{M}$ by joining equivalent vertices of the two copies.
• Figure 6: Derivation of the automorphism fusion rule \ref{['eq:fusionautomorphism']}. From the gauge transformation with image $\vec{h}(v_i)=(\phi\cdot g_i^{-1},g_i^{-1})\in G^{(\phi)}$ for all $v_i$ in $\mathcal{D}_{G^{(\phi)}}$ we go from a generic gauge field configuration to temporal gauge in the second figure. By the flatness condition, the group elements on the right and left of each domain wall are equal. Gauge transformations with $\vec{h}(v_i)=(\phi\cdot \phi^\prime\cdot h,\phi^\prime \cdot h)\in G^{(\phi)}$ and $\vec{h}(v_i^\prime)=(\phi^\prime\cdot h,h)\in G^{(\phi^\prime)}$ preserve the temporal gauge and make the gauge transformations of $\mathcal{D}_{G^{(\phi\circ\phi^\prime)}}$.