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. 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 sum runs over $G$-bundles $P$. The new background $\widehat A$ couples to a residual or topological symmetry. For a finite group the functional integral becomes a weighted finite sum; for a continuous group one must add kinetic terms or specify the fixed-point gauging operation and its counterterms.

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.

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 $Z_{g,h}$ is the partition function with $g$ and $h$ holonomies around the two cycles. 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. The phase $\epsilon(g,h)$ represents an allowed discrete-torsion choice when its cocycle condition is satisfied. The operator-algebra construction and modularly consistent twisted sectors are developed in Dijkgraaf, Vafa, Verlinde, and Verlinde 1989, pp. 485–526.

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 $G=\mathbb Z_n$, gauging often produces a dual quantum symmetry $\widehat G=\operatorname{Hom}(G,U(1))$. Under suitable anomaly and normalization conditions, gauging $\widehat G$ can recover the original theory. The recovery can include an invertible topological factor or depend on spacetime dimension, 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, changing a gauge group from $\widetilde G$ to $\widetilde G/K$ amounts to gauging an appropriate one-form symmetry only after the allowed line operators and discrete theta term are specified. The local Lie algebra and perturbative Feynman rules do not determine this operation; see Gaiotto, Kapustin, Seiberg, and Willett 2015, §§3–4 for the background-field formulation of higher-form symmetry and gauging.

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 QFT with a $U(1)$ current and background $A$, define schematically

$$S:\quad Z[A]mapsto \int\mathcal Da\,Z[a] \exp\!\left(\frac{i}{2\pi}\int a\wedge dA\right),$$

which gauges the $U(1)$ and couples the new topological current to $A$. A $T$ operation adds a quantized background Chern–Simons contact term. Their composition generates new duality frames and obeys group relations only up to spin, framing, and invertible topological factors. This $SL(2,\mathbb Z)$ action is constructed in Witten 2003, §§2–3.

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. Let $m\mathcal O$ break $G$ to a subgroup $H$. Then

$$(\mathcal T/G)+m\mathcal O \quad\text{and}\quad (\mathcal T+m\mathcal O)/H$$

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 paths explicitly and compare their endpoints, including residual symmetries and decoupled sectors. 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 and endpoint comparison after it

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

• 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. arXiv:1412.5148.
• 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. arXiv:hep-th/0307041.