Duality Operations: Gauging, Quotients, and Orbifolds

Gauging a shared global symmetry can turn one duality into another, but only when the original equivalence is valid as a function of the full background field. The operation sums over bundles, projects states, adds twisted sectors, and creates residual or emergent symmetries. Applying it to a zero-background partition function is not enough to determine the resulting theory.

Required background. Duality claims and dictionaries supplies the source comparison, and gauging continuous and finite symmetries defines the operation. Helpful background. See residual and emergent symmetries and noninvertible constructions by gauging.

Gauging as a transformation of a background-dependent theory

Let $\mathcal T$ have a nonanomalous global symmetry $G$ with background connection or cocycle $A$. The ungauged theory is the functional $Z_{\mathcal T}[A]$, including its response under large gauge transformations. Schematically, gauging promotes $A$ to a dynamical field and sums over its topological sectors:

$$Z_{\mathcal T/G}[\widehat A] =\frac{1}{\operatorname{vol}\mathcal G} \sum_{[P]} \int_{A\in\operatorname{Conn}(P)}\!\mathcal DA\, Z_{\mathcal T}[A]\, e^{iS_{\mathrm{gauge}}[A]+iS_{\mathrm{top}}[A,\widehat A]}.$$

The formula suppresses gauge fixing and stabilizer factors. The sum runs over $G$-bundles $P$; for a finite group it is a weighted sum over flat bundles with the appropriate automorphism factors. A new background $\widehat A$ exists only when the output has the corresponding residual, dual, or topological symmetry. For a continuous group one must specify the kinetic or fixed-point gauging operation, its counterterms, and the RG endpoint.

Suppose a proposed duality gives

$$Z_A[A]=e^{iC[A]}Z_B[f(A)]$$

for every allowed background, where $C[A]$ is a declared local counterterm. If the symmetry map $f$ is compatible with bundles and the anomaly allows gauging, integrating this equality over $A$ produces a new duality. If equality was checked only at $A=0$, the conclusion does not follow: twisted sectors and topologically nontrivial bundles were never compared.

The typed operation-square map makes this failure direction explicit: the bundle sum is an operation edge, while agreement at $A=0$ is only an insufficient check.

Projection and twisted sectors

For a finite symmetry $G$ in two dimensions, the torus orbifold partition function takes the schematic form

$$Z_{\mathcal T/G} =\frac{1}{|G|}\sum_{\substack{g,h\in G\\gh=hg}} \epsilon(g,h)Z_{g,h}.$$

Here $g$ is the spatial twist and $h$ is the temporal insertion, so $Z_{g,h}$ is a trace over the $g$-twisted Hilbert space with $h$ inserted. The terms with $g=1$ implement projection onto invariant states in the untwisted sector. Terms with $g\neq1$ are twisted sectors required by locality and modular covariance. Discrete torsion is a class $[\alpha]\in H^2(G,U(1))$; the torus weight $\epsilon(g,h)$ is derived from a representative cocycle and is not an arbitrary phase. The operator-algebra construction is developed in Dijkgraaf, Vafa, Verlinde, and Verlinde 1989, pp. 485–526, while the discrete-torsion choice originates in Vafa 1986, §2.

Keeping only the invariant subspace is not gauging. It discards the twisted states and generally destroys modular invariance. Conversely, adding twisted sectors without projection double-counts gauge-related states.

For a two-dimensional finite-abelian orbifold, gauging produces the ordinary quantum symmetry $\widehat G=\operatorname{Hom}(G,U(1))$; under suitable anomaly and normalization conditions, gauging $\widehat G$ can recover the original theory. In general dimension, gauging a finite-abelian zero-form symmetry instead produces a dual $(d-2)$-form symmetry, with qualifications from anomalies and topological actions. Recovery can include a topological factor, so it must be checked rather than assumed.

Anomaly is the first gate

A symmetry with a nontrivial ’t Hooft anomaly cannot be gauged as an ordinary standalone $d$-dimensional symmetry. In background notation,

$$Z[A^g]=Z[A]e^{i\mathcal A[A,g]}.$$

If no local counterterm cancels $\mathcal A$, the integrand is not a function on gauge-equivalence classes. Options are to couple to a $(d+1)$-dimensional inflow theory, gauge an anomaly-free subgroup, add degrees of freedom with the opposite anomaly, or accept a relative theory.

Two dual descriptions can use different counterterm schemes. Before gauging, translate them so their background responses agree. A contact term that was harmless for separated-point correlators can become a dynamical Chern–Simons or Dijkgraaf–Witten term after the background is promoted.

Quotients and the faithful symmetry

If a subgroup $K$ of a nominal global symmetry acts trivially on all genuine operators, the faithful symmetry is $G/K$, possibly combined with a higher-form symmetry into a higher group. Gauging $G$ and gauging $G/K$ are different sums over bundles. The first may include redundant gauge fields or a decoupled topological sector.

Likewise, writing $\widetilde G/K$ requires $K\subset Z(\widetilde G)$ to act trivially on the dynamical matter. Changing from $\widetilde G$ to this quotient amounts to gauging the corresponding one-form symmetry only if that symmetry passes its anomaly gate and the allowed line lattice and discrete theta term are specified. The local Lie algebra and perturbative Feynman rules do not determine the result Gaiotto, Kapustin, Seiberg, and Willett 2015, §§3–4, arXiv PDF; different genuine-line choices can define distinct theories with the same local correlators Aharony, Seiberg, and Tachikawa 2013, §§1–2, arXiv v5 PDF.

A quotient notation should therefore answer:

• Which subgroup is being gauged, and in what degree?
• Which bundles and background fields are summed over?
• Which topological action weights them?
• Which operators survive, which become endpoints, and which twisted operators appear?
• What residual ordinary or generalized symmetry remains?

Generating dualities with S and T operations

For a three-dimensional CFT—or a QFT together with a declared gauged RG endpoint—with a $U(1)$ current and background $A$, choose the orientation convention

$$\begin{aligned} (SZ)[A] &=\int\mathcal Da\,Z[a] \exp\!\left(\frac{i}{2\pi}\int a\wedge dA\right),\\ (TZ)[A] &=Z[A]\exp\!\left(\frac{i}{4\pi}\int A\wedge dA\right). \end{aligned}$$

The $S$ operation gauges the $U(1)$ and couples the new topological current to $A$; $T$ shifts the background Chern–Simons contact term by one unit. With this unit $T$ requires spin data; on a purely bosonic nonspin theory the admissible subgroup can contain only $T^2$. These operations generally produce new CFTs rather than symmetries of the original one, and relations such as $S^2=-1$ and $(ST)^3=1$ can carry charge-conjugation, framing, gravitational, or invertible factors. Witten constructs the action and its qualifications in Witten 2003, §3, arXiv PDF.

This is a precise example of an operation acting on a theory with background data, not on a list of local operators. Quantization of Chern–Simons levels, spin-$c$ structure, and monopole operators is essential.

Noncommuting operations

Gauging and deformation need not commute, and one order may not even exist. Let $m\mathcal O$ explicitly break $G$ to a subgroup $H$. After $G$ is gauged, $\mathcal O$ is generally not gauge invariant, so

$$(\mathcal T/G)+m\mathcal O \quad\text{is undefined without a gauge-invariant spurion or Higgs completion}, \qquad (\mathcal T+m\mathcal O)/H \quad\text{can be well defined}.$$

Even when both paths are completed, they can have different bundles, topological sectors, and domain walls. Similarly, gauging before compactification can retain holonomies that become lower-dimensional scalars, whereas compactifying first and gauging only the zero-mode symmetry can omit them.

To claim that an operation transports a duality, form both composed maps explicitly. Matching endpoint theory cards is necessary but not sufficient: the induced maps of operators, backgrounds, extended sectors, and predeclared counterterms must agree as well. A commuting diagram is a result, not a formatting choice.

A practical operation record

Stage	Required data
Input	Complete theory, faithful symmetry, background-field functional, anomaly
Choice	Subgroup, bundles, kinetic/topological action, discrete torsion, spin structure
Projection	Gauge-invariant local and extended operators
New sectors	Twists, monopoles, flux sectors, boundary degrees of freedom
Output symmetry	Residual, quotient, topological, higher-form, or noninvertible symmetry
Duality test	Background equality before integration; comparison of endpoint theories and composed operator/background maps afterward

Blank entries narrow the claim. In particular, an operation is not invertible merely because its notation has an inverse-looking symbol.

Common pitfalls

Gauging a symmetry known only at zero background. Nontrivial bundles and contact terms can spoil the transported relation.

Projecting without adding twisted sectors. That operation is not an orbifold path integral and generally fails locality or modular covariance.

Ignoring the anomaly after promoting the field. A background counterterm becomes part of the dynamical action and can change the theory decisively.

Exercises

For a two-dimensional theory with nonanomalous $\mathbb Z_2$ symmetry, label torus sectors by $Z_{a,b}$ with $a,b\in\{0,1\}$.

1. Write the orbifold partition function with trivial discrete torsion.
2. Identify the untwisted projection and the twisted sectors.
3. Explain what data a proposed duality must match before the same orbifold can be applied to both sides.

Solution

The answer is

$$Z_{\mathrm{orb}}=\frac12 (Z_{0,0}+Z_{0,1}+Z_{1,0}+Z_{1,1}).$$

The combination $(Z_{0,0}+Z_{0,1})/2$ projects the untwisted Hilbert space onto even states. The $a=1$ terms arise from the twisted Hilbert space, with $b$ inserting the group element in its trace. A duality must match all four background sectors, their modular transformations, and any allowed counterterm phase—not only $Z_{0,0}$—before summing them.

References

• Aharony, Ofer, Nathan Seiberg, and Yuji Tachikawa. “Reading between the Lines of Four-Dimensional Gauge Theories.” Journal of High Energy Physics 08 (2013): 115. DOI. Open PDF.
• Dijkgraaf, Robbert, Cumrun Vafa, Erik Verlinde, and Herman Verlinde. “The Operator Algebra of Orbifold Models.” Communications in Mathematical Physics 123 (1989): 485–526. doi:10.1007/BF01238812.
• Gaiotto, Davide, Anton Kapustin, Nathan Seiberg, and Brian Willett. “Generalized Global Symmetries.” Journal of High Energy Physics 02 (2015): 172. DOI. Open PDF.
• Vafa, Cumrun. “Modular Invariance and Discrete Torsion on Orbifolds.” Nuclear Physics B 273 (1986): 592–606. DOI.
• Witten, Edward. “$SL(2,\mathbb Z)$ Action on Three-Dimensional Conformal Field Theories with Abelian Symmetry.” In From Fields to Strings, vol. 2, 1173–1200. World Scientific, 2005. Open PDF.