Higher Gauging & Condensation Defects

• Higher gauging is the procedure of gauging discrete p-form symmetries on submanifolds to yield topological condensation defects with non-invertible fusion rules.
• This approach employs both field-theoretic and lattice constructions to derive defects characterized by group cohomology and higher-categorical fusion structures.
• The framework imposes anomaly constraints and informs RG flows, enabling analysis of dualities and emergent TQFTs in quantum matter.

A higher gauging procedure generalizes the ordinary gauging of global symmetries by allowing discrete $p$-form symmetries to be gauged only on submanifolds of non-maximal codimension. The resulting topological defects—condensation defects—are supported on the locus where gauging occurs and often exhibit non-invertible fusion rules. These defects, and the associated categorical and field-theoretic structures, form the foundation for a unified treatment of non-invertible symmetries, dualities, and constraints in quantum field theory and topological phases.

1. Definition and Construction of Higher Gauging and Condensation Defects

Higher gauging is the operation of gauging a discrete $q$-form symmetry $G$ on a codimension-$p$ submanifold $\Sigma$ within a $D$-dimensional spacetime. The result is a topological defect of dimension $D-p$—the condensation defect—whose worldvolume supports a sum over insertions of symmetry defects of codimension $q+1$ (for $n$-form symmetry, these are typically Wilson operators or their higher-form analogs).

For a 3+1d QFT with a finite one-form symmetry $G\simeq \mathbb{Z}_n$, condensation defects of codimension one are labeled by discrete torsion classes $q$0. Explicitly, the condensation defect inserted along an oriented three-manifold $q$1 is given by

$q$2

where $q$3 is the symmetry surface operator along $q$4 and $q$5, with $q$6 and $q$7 the Bockstein (Choi et al., 2022). For higher codimension, the procedure generalizes: e.g., in (3+1)d, codimension-2 defects arise from gauging a 1-form symmetry on a surface, and in general, a $q$8-form symmetry can be $q$9-gauged on codimension-$G$0 submanifolds (Roumpedakis et al., 2022, Barkeshli et al., 2022, Bah et al., 4 Jun 2025).

In lattice models, higher gauging is realized by introducing appropriate gauge degrees of freedom, imposing local Gauss law constraints, and minimally coupling the original matter operators to the new gauge fields, with the condensation defect arising as a localized projector or interface Hamiltonian supported on the gauged region (Hofmeier et al., 12 May 2026, Ebisu et al., 2024).

2. Algebraic Structure and Fusion Rules

Condensation defects generally form a higher-categorical algebra, often a fusion 2-category, whose fusion product is non-invertible. Fusion of two parallel defects $G$1 wrapping the same submanifold $G$2 is defined by juxtaposition and collapse of the intermediate region, yielding

$G$3

where $G$4 is another condensation defect and $G$5 is the partition function of a lower-dimensional TQFT evaluated on $G$6 (e.g., a $G$7-dimensional topological phase) (Choi et al., 2022, Roumpedakis et al., 2022). For one-form symmetries in (3+1)d, the fusion coefficients include $G$8 (a 2+1d $G$9 gauge theory), $p$0 Chern-Simons theories, and invertible SPTs (Choi et al., 2022).

More generally, for abelian $p$1, fusion rules of condensation defects on a manifold $p$2 encode the structure of the group cohomology of $p$3:

• Fusion coefficients are partition functions of lower-dimensional TQFTs (not just numbers).
• Non-invertibility is manifest in relations such as $p$4 with $p$5 the partition function of a decoupled theory, e.g., $p$6 gauge theory or $p$7 BF theory (Bah et al., 4 Jun 2025, Roumpedakis et al., 2022, Lin et al., 2022).

For codimension-2 defects (e.g., twist strings in 3+1d), the algebraic structure generalizes to a higher fusion category, with the interplay between fusion, associativity, and TQFT partition functions governed by nontrivial relations involving group cohomology and Postnikov classes (Barkeshli et al., 2022, Lin et al., 2022).

3. Duality, Triality, and Non-invertible Topological Defects

Duality and triality defects are particular types of codimension-one defects produced by higher gauging on half of spacetime (or half of the system in a lattice construction). For a theory $p$8 with a finite one-form symmetry $p$9:

• If $\Sigma$0 is self-dual under gauging $\Sigma$1 (the S operation), gauging $\Sigma$2 on half of spacetime yields a duality defect $\Sigma$3, obeying fusion rules with non-invertible structure:

$\Sigma$4

where $\Sigma$5 is orientation-reversal and $\Sigma$6 is the trivial condensation defect (Choi et al., 2022).

• For triality defects, one requires additional symmetry (S and T), leading to more intricate fusion rules involving further non-invertible structures. These defects are realized in free Maxwell theory (at specific couplings), SO(N) Yang-Mills, and super Yang-Mills theories (Choi et al., 2022).
• In all cases, the fusion algebra is non-invertible, with TQFTs as structure coefficients.

Codimension-2 non-invertible condensation defects arise from higher gauging on surfaces, leading to string-like or membrane-like excitations (e.g., "twist strings" or "Cheshire strings" in (3+1)D toric code, with fusion governed by $\Sigma$7, where $\Sigma$8 is the Euler characteristic) (Barkeshli et al., 2022).

4. Categorical and Representation-Theoretic Frameworks

The structures induced by higher gauging and condensation defects are naturally organized within higher fusion categories or multifusion 2-categories. In (2+1)d topological order:

• The condensation completion $\Sigma$9 of a fusion category $D$0 produces a fusion 2-category whose objects are separable algebra objects (string defects), 1-morphisms are bimodules (point defects/DW junctions), and 2-morphisms are module maps (Yue et al., 2024).
• Horizontal fusion of string defects in $D$1 implements higher gauging of one-form symmetries, with explicit decomposition into simples via Morita equivalence.
• For finite-group gauge theories, the fusion 2-category of defects is realized as $D$2, with objects corresponding to module categories for $D$3 and fusion rules encoding condensation and duality of defects (Bartsch et al., 2022).

In general, gauging of finite higher groups creates topological surface (and higher) defects organized as fusion 2-categories, classified (up to Morita equivalence and cohomological data) by categorical extensions and Postnikov invariants (Décoppet et al., 2022, Bartsch et al., 2022).

The requirement of Karoubi completeness (splitting of idempotents) in the defect category ensures that all possible condensation defects—at each codimension—are properly captured, with fusion rules reflecting direct-sum decompositions and further condensation relations (Bah et al., 4 Jun 2025).

5. Field-Theoretic and Lattice Realizations

Higher gauging and condensation defects are not just formal algebraic objects but admit concrete field-theoretic and lattice constructions:

• In field theory, condensation defects correspond to worldvolume insertions in the path integral that sum over background $D$4-form gauge fields with appropriate discrete torsion (encoded in group or higher-group cohomology) (Bah et al., 4 Jun 2025, Xue et al., 27 Dec 2025).
• In lattice models, these are implemented as patchwise gauging interfaces or projectors onto gauge-invariant subspaces in a local region, with movement operators giving topological invariance of defect position (and serving as generators of non-invertible dualities) (Hofmeier et al., 12 May 2026, Lyons et al., 2024, Ebisu et al., 2024).
• As a key example, gauging diagonal 1-form symmetries in stacks of 2+1d topological orders produces the X-Cube and related fracton models as string-membrane-nets, with condensation defects classifying immobile fractons, planons, and lineons (Gorantla et al., 19 May 2025).

Notably, higher gauging of spatially modulated symmetries (e.g., subsystem or dipole symmetries) produces condensation defects whose non-invertible fusion rules encode the structure of emergent excitations and dual symmetries (including lineon and fracton sectors) (Ebisu et al., 2024).

6. Constraints from Anomalies and RG Flows

The presence of non-invertible condensation defects has dynamical consequences through higher-form ’t Hooft anomalies:

• Non-invertible symmetries generated by condensation defects often imply that a trivial symmetric gapped phase (with only invertible topological sectors) is forbidden. That is, a flow to such a phase would require matching the UV non-invertible symmetry to an invertible (SPT) phase, often impossible due to group cohomology obstructions (Choi et al., 2022, Bah et al., 4 Jun 2025).
• Direct cohomological analysis reveals that, for many symmetry types and torsion classes (e.g., even $D$5 for $D$6 one-form symmetries), the obstruction class ruling out a trivial IR exists, requiring either symmetry breaking or a non-trivial topological order in the IR.
• Mixed anomalies engender higher-group symmetry structures upon partial gauging, with the fusion rules and categorical structures of condensation defects encoding the nontrivial constraints of the anomaly (Oishi et al., 3 Apr 2026, Arbalestrier et al., 18 Feb 2025).

7. Generalizations and Implications

Higher gauging and condensation defects unify and generalize the algebraic and geometric manipulation of symmetries, dualities, and interfaces in quantum field theory:

• All forms of gauging (0-form, 1-form, higher-form) and condensation are special cases of this framework, with extensions to non-invertible and categorical symmetries.
• The same machinery underpins the construction of gapped boundaries, symmetry-enriched phases, and the full web of dualities accessible through tensor network and categorical methods (Williamson et al., 2017, Yue et al., 2024).
• Non-invertible condensation defects and their fusion categories provide a rigorous approach for the classification of modular invariants, mapping class group actions, and the analysis of strongly interacting phases with emergent higher or categorical symmetries.

These approaches enable explicit calculations of fusion, braiding, symmetry fractionalization, and their constraints both in continuum and lattice frameworks across dimensions, laying a unified foundation for the study of generalized symmetries, anomalies, and topological orders in modern quantum matter and quantum field theory (Choi et al., 2022, Barkeshli et al., 2022, Bah et al., 4 Jun 2025, Hofmeier et al., 12 May 2026).