Gauging, Equivariantization, Orbifolds, and Condensation

Gauging, equivariantization, orbifolding, and condensation are related categorical constructions, but they are not synonyms. Equivariantization forms objects with coherent group-equivariant structure. Physical gauging of a finite symmetry first requires an anomaly-free crossed extension and then equivariantization. Condensation forms modules over a connected separable algebra object, with locality selecting deconfined objects. If the anomaly, separability, or locality hypothesis fails, the advertised gauged or modular category does not follow.

Required background. Fusion Categories, Module Categories, and Bimodule Defects supplies algebra objects and modules. Noninvertible Symmetries, Fusion, and Junction Data supplies coherence. Categorical Symmetries, Higher Representations, and Charges supplies actions. Constructions from Gauging, Duality, and Condensation supplies the physical constructions. Helpful background. Gauging Continuous and Finite Symmetries states the path-integral operation, while Duality Operations: Gauging, Quotients, and Orbifolds tracks global data across duality frames.

Four constructions and their hypotheses

Let a finite group $G$ act on a fusion category $\mathcal C$ by tensor autoequivalences $T_g$ with coherent multiplication maps. The equivariantization $\mathcal C^G$ has objects $(X,u)$, where

$$u_g:T_g(X)\xrightarrow{\cong}X, \qquad u_{gh}\circ\mu_{g,h}=u_g\circ T_g(u_h).$$

Simple equivariant objects are organized by $G$-orbits of simples of $\mathcal C$ together with irreducible projective representations of their stabilizers; the projective cocycle comes from the action coherence. One has $\operatorname{FPdim}(\mathcal C^G)=|G|\operatorname{FPdim}(\mathcal C)$.

For a braided category describing a topological phase, physical gauging needs more. Symmetry fluxes live in a $G$-crossed braided extension $\mathcal C_G^\times=\bigoplus_g\mathcal C_g$. Obstructions to constructing its associators and crossed braiding are the categorical form of the anomaly. When those obstructions vanish, gauging is $(\mathcal C_G^\times)^G$. Etingof, Nikshych, and Ostrik identify the extension problem with maps into the Brauer–Picard classifying space and display the successive obstruction data in Etingof, Nikshych, and Ostrik 2010, §§7–8, printed pp. 34–51 (PDF).

Condensation starts instead with an algebra object $(A,\mu,\eta)$ in a braided category. For a bosonic bulk condensation, $A$ is normally required to be connected, commutative, and separable. Right $A$-modules describe wall excitations; local modules, whose action is compatible with double braiding, describe deconfined bulk excitations. Kong derives

$$\mathcal D\simeq\mathcal C_A^{\mathrm{loc}}$$

under these hypotheses in Kong 2014, §§2.3–2.5, printed pp. 6–16 (PDF). Separability makes module formation exact in the semisimple setting; commutativity and locality are what preserve a braided deconfined sector.

Orbifold completion is a defect-bicategory construction: it adjoins suitable separable Frobenius algebra objects as new phases and bimodules as walls. Carqueville and Runkel prove idempotence and describe the equivariant and orbifold completions in Carqueville and Runkel 2016, §§4–5, printed pp. 25–36 (PDF). It is a theorem for the specified pivotal bicategory, not a universal analytic construction of every QFT orbifold.

Gauging a finite example

Begin with the trivial $2+1$-dimensional phase and an anomaly-free $G=\mathbb Z_2$ zero-form symmetry. The crossed extension has two simple flux sectors, so its total Frobenius–Perron dimension is $2$. Equivariantization resolves each flux by a character of its stabilizer. The result has four one-dimensional simple objects, conventionally $\mathbf1,e,m,\epsilon$, with

$$e^2=m^2=\epsilon^2=\mathbf1, \qquad e\otimes m=\epsilon.$$

Its total dimension is $4=|G|^2$. This is the quantum double category of $\mathbb Z_2$, the categorical data of the toric-code topological order. The electric object records a gauge charge, the magnetic object a flux, and their mutual braiding is nontrivial. A dual one-form symmetry emerges after gauging. The exact physical interpretation and construction return to Constructions from Gauging, Duality, and Condensation.

The dimension check distinguishes ordinary equivariantization from full physical gauging: applying equivariantization only to the neutral category multiplies dimension by $2$, whereas adjoining flux sectors and then equivariantizing multiplies the original dimension by $4$.

Obstructions and stop rules

If $A$ is not separable, its module category can be nonsemisimple even when $\mathcal C$ is semisimple, invalidating the finite fusion formulas. If $A$ is not commutative or a module is not local, the surviving excitations need not form the claimed braided bulk category. If the $G$-crossed associator obstruction is nonzero, no choice of equivariant objects repairs the missing flux theory.

The adversarial fixture condenses a nonseparable algebra and then discards nonlocal modules as though semisimplicity and modularity were automatic. The safe conclusion is only that an algebra object and its module category have been formed. A consistent gauged phase requires the missing separability, locality, anomaly cancellation, and physical realization.

De-equivariantization is a conditional inverse, not a formal cancellation symbol. It requires a Tannakian subcategory equivalent to $\operatorname{Rep}(G)$, embedded with the appropriate central structure. If the symmetric subcategory is super-Tannakian or if the embedding is not central, the output and tangential requirements change. Likewise, recovering the original category after gauging requires retaining the emergent dual symmetry and its action. Forgetting that structure destroys invertibility of the operation.

Exercises

Why does the four-simple-object $\mathbb Z_2$ gauged category have total dimension $4$ rather than $2$?

Solution

The anomaly-free crossed extension first adds the two group-graded flux sectors, multiplying total dimension by $|G|=2$. Equivariantization then adds charge labels, multiplying it by another factor of $2$. Thus the final total dimension is $|G|^2=4$.

References

• Carqueville, Nils, and Ingo Runkel. “Orbifold Completion of Defect Bicategories.” Quantum Topology 7 (2016): 203–279. DOI; Open PDF.
• Etingof, Pavel, Dmitri Nikshych, and Victor Ostrik. “Fusion Categories and Homotopy Theory.” Quantum Topology 1 (2010): 209–273. DOI; Open PDF.
• Kong, Liang. “Anyon Condensation and Tensor Categories.” Nuclear Physics B 886 (2014): 436–482; erratum and addendum incorporated in the 2021 version. DOI; Open PDF.