Category of SET orders

TL;DR

This work formulates a representation principle for physical systems with symmetry by identifying SET orders with the category $\mathrm{Fun}(\Sigma {\cal T}, {\cal X})$, where $\cal T$ encodes a fusion $n$-category symmetry and $\cal X$ encodes topological order data. It develops a comprehensive higher-categorical framework (including Karoubi condensation, centers $Z_k$, and the notion of boundaries) to capture defects, charges, symmetry breaking, gauging, and SymTFT, and introduces a detailed categorical algorithm for generalized gauging that is reversible via Morita equivalence. The paper defines the charge category as a relative center $Z(\phi)$ and shows how gauging corresponds to Morita-equivalent modifications, with explicit data for ungauging. Through numerous examples—ordinary and twisted gauging, DW theories, fermionic cases, and partial gauging—it demonstrates how the SET order category encodes both the symmetry and anomaly structure and their physical consequences. This unified, algebraic approach provides a powerful toolkit for analyzing symmetry enrichment in topological phases and their higher-dimensional generalizations, with clear links to holographic bulk theories (SymTFT).

Abstract

We propose the representation principle to study physical systems with a given symmetry. In the context of symmetry enriched topological orders, we give the appropriate representation category, the category of SET orders, which include SPT orders and symmetry breaking orders as special cases. For fusion n-category symmetries, we show that the category of SET orders encodes almost all information about the interplay between symmetry and topological orders, in a natural and canonical way. These information include defects and boundaries of SET orders, symmetry charges, explicit and spontaneous symmetry breaking, stacking of SET orders, gauging of generalized symmetry, as well as quantum currents (SymTFT or symmetry TO). We also provide a detailed categorical algorithm to compute the generalized gauging. In particular, we proved that gauging is always reversible, as a special type of Morita-equivalence. The explicit data for ungauging, the inverse to gauging, is given.

Figures (11)

• Figure 1: Center depicted as bulk.
• Figure 2: Zipping and unzipping.
• Figure 3: Graphical representation for generalized gauging, in terms of ${\cal T}$-module ${\cal K}$ v.s. in terms of algebra $A\in {\cal T}$ where ${\cal K} ={\newline}_{A} { {\cal T} }.$
• Figure 4: Gauging is reversible.
• Figure 5: Gauging a subsymmetry as partial gauging

Theorems & Definitions (109)

• Definition 1.1
• Remark 1.2
• Definition 2.1
• Remark 2.2
• Proposition 2.3
• Definition 2.4: $E_k$-multi-fusion $n$-category
• Remark 2.5
• Proposition 2.6
• proof
• Remark 2.7