Residual, Quotient, and Emergent Dual Symmetries

Gauging a subgroup changes symmetry in two distinct ways. First, a parent transformation survives only if it preserves the subgroup and every choice used to gauge it. Because the gauged subgroup now acts as redundancy, the inherited action factors through a quotient and must then be made faithful. Second, finite gauging creates twisted or topological sectors on which a new dual symmetry can act. This page derives both mechanisms for ordinary zero-form symmetries, with the dual-group example restricted to finite Abelian gauging in two dimensions.

Required background. Gauging Continuous and Finite Symmetries supplies the gaugeability conditions, groupoid sum, invariant-state projection, and twisted sectors whose symmetries are tracked here.

Helpful background. Quantum Implementations, Projective Actions, and Central Extensions supplies exact-sequence and extension language used to distinguish a quotient from a split product.

Which parent transformations descend

Assume first that spacetime is closed. Let a theory $\mathcal T$ have global symmetry group $G$, and gauge a subgroup $H\subset G$. Write $\mathcal D$ for the complete gauging data: the action or topological weight, allowed bundles, measure, observable spectrum, and boundary or asymptotic conditions.

A transformation $g\in G$ sends an $H$-bundle and its fields to data with structure group $gHg^{-1}$. It can therefore act within the same gauged theory only if it normalizes $H$. That condition is not enough: $g$ must also preserve $\mathcal D$ up to an equivalence of the gauged theory. Define the data-preserving normalizer

$$N_G(H;\mathcal D) = \left\{ g\in G\ \middle|\ gHg^{-1}=H,\quad g\mathbin{\cdot}\mathcal D\simeq\mathcal D \right\}.$$

Every element of $H$ lies in this normalizer and acts on the gauged fields by a gauge transformation. Consequently, two elements of $N_G(H;\mathcal D)$ that differ by $H$ induce the same map on gauge orbits.

There is one further coherence requirement. Choose the equivalences $g\mathbin{\cdot}\mathcal D\simeq\mathcal D$ so that their induced maps compose on physical gauge orbits; composition may differ by an $H$-gauge transformation, which is trivial there. Under this hypothesis the parent action descends to a homomorphism

$$\bar\rho: \frac{N_G(H;\mathcal D)}{H} \longrightarrow \operatorname{Aut}\!\left(\mathfrak D_{\mathcal T/H}\right),$$

where $\mathfrak D_{\mathcal T/H}$ denotes the physical states, operators, sectors, and correlation functions of the gauged theory. The actual inherited symmetry is its faithful image:

$$\begin{aligned} G_{\mathrm{res}} &=\operatorname{im}\bar\rho, \\ G_{\mathrm{res}} &\cong \frac{N_G(H;\mathcal D)/H}{\ker\bar\rho}. \end{aligned}$$

If coherent choices do not exist, the displayed quotient remains only a candidate. If chosen representatives compose as $\widetilde g_1\widetilde g_2 =\eta(g_1,g_2)\widetilde{g_1g_2}$ with a nontrivial physical transformation $\eta(g_1,g_2)$, they instead define an extended or projective action; failure of any consistent composition law is an obstruction. The later discussion shows where additional sector symmetries can enter such a structure.

This derivation explains each quotient. Dividing by $H$ removes transformations that have become gauge redundancy; dividing by $\ker\bar\rho$ removes any further elements acting trivially on all physical data. If the chosen gauge group acts nonfaithfully before gauging, first separate its faithful image from the inert kernel. Gauging the nonfaithful presentation can retain extra topological data and is not captured by silently replacing one group name with another. In the simplest product case $H=H_{\mathrm{eff}}\times K$, with finite $K$ acting trivially, gauging $H$ also sums flat $K$-bundles and leaves a decoupled finite gauge sector; gauging only $H_{\mathrm{eff}}$ omits that sector.

Gaiotto, Kapustin, Seiberg, and Willett make the physical mechanism explicit: after gauging, the original symmetry defects are gauge transformations on physical operators, while twisted-sector operators become genuine observables Gaiotto et al. 2015, § 2, p. 7, arXiv PDF.

Normality, choices, and faithful action

Three increasingly restrictive tests determine the inherited symmetry.

Subgroup test. If $H\triangleleft G$, every element of $G$ normalizes $H$, so the first candidate is $G/H$. If $H$ is not normal, only $N_G(H)/H$ can even act on the same set of gauge orbits. A transformation outside the normalizer maps the construction to a different gauging; it may define an interface or duality, but not an internal symmetry of $\mathcal T/H$.

Gauging-data test. A discrete theta term, restricted bundle sum, boundary condition, or regulator can fail to be invariant under part of the normalizer. The relevant group is then $N_G(H;\mathcal D)$, not the normalizer of the abstract subgroup alone.

Faithfulness test. An element that preserves every construction choice can still act trivially on the physical operator algebra and all sectors. It belongs to $\ker\bar\rho$ and must be removed from the faithful symmetry group.

These tests also show why the result need not be a direct product. Even when a quotient group and another symmetry are both present, their combined action can be a semidirect product, a non-split extension, or an anomaly-modified structure. A product symbol is justified only after the action, kernel, and splitting have been checked.

At a boundary, the first paragraph requires revision: gauge transformations that do not vanish there can carry physical charges. Whether they remain redundancies depends on the boundary conditions and boundary degrees of freedom; Proper and Improper Gauge Transformations develops that distinction.

The scalar quotient changes charge normalization

Return to the charge-one complex scalar with the symmetry-breaking coupling set to zero. Its exact connected symmetry is $G=U(1)$. Gauge the subgroup $H=\mathbb Z_N$ generated by a phase $e^{2\pi i/N}$. All of $U(1)$ normalizes $H$, and for invariant gauging data the inherited candidate is

$$U(1)/\mathbb Z_N.$$

This quotient is abstractly isomorphic to $U(1)$ through

$$\begin{aligned} U(1)/\mathbb Z_N&\longrightarrow U(1), \\ [e^{i\alpha}]&\longmapsto e^{iN\alpha}. \end{aligned}$$

Calling the symmetry “unchanged” would hide its physical normalization. The field $\phi$ is not a local gauge-invariant operator after the $\mathbb Z_N$ gauging, whereas

$$\phi^N\longmapsto e^{iN\alpha}\phi^N$$

is local and has unit charge when the quotient angle is $\beta=N\alpha$. Thus the abstract group can be the same while its faithful action and minimal local charge are different.

For the fixed interaction $h\phi^N+h^*(\phi^\dagger)^N$, the residual subgroup of the original $U(1)$ is already $\mathbb Z_N$. If other $\mathbb Z_N$-invariant interactions remove every compatible reflection, gauging this exact parent group leaves no inherited ordinary quotient symmetry. If a reflection survives, the parent group is instead dihedral and its quotient can leave a $\mathbb Z_2$ candidate. This is why the complete action, not one displayed term, enters $\mathcal D$.

Finite gauging creates a dual symmetry

Now specialize to an anomaly-free finite Abelian group $A$ gauged in two dimensions. Its character group is

$$\widehat A=\operatorname{Hom}(A,U(1)).$$

The notation keeps $A$ and $\widehat A$ distinct. They can be abstractly isomorphic, as for $\mathbb Z_N$, but there is generally no canonical isomorphism.

For the $\mathbb Z_N$ scalar sectors of the previous page, let $\mathcal H_k$ have spatial holonomy $k$. The new generator acts by the sector label:

$$\widehat R^{\,m}|k,\psi\rangle = e^{2\pi i mk/N}|k,\psi\rangle, \qquad m,k\in\mathbb Z_N.$$

This is not the original $R^\ell$ insertion. The temporal average over $R^\ell$ projects states within each $\mathcal H_k$ onto gauge invariants; $\widehat R^{\,m}$ instead assigns a charge to the twisted sector itself. Its characters obey

$$\frac{1}{N} \sum_{m=0}^{N-1} e^{2\pi i mk/N} = \delta_{k,0}.$$

The twisted sectors created by gauging therefore carry a new $\widehat{\mathbb Z_N}$ symmetry even when no inherited ordinary symmetry remains. Bhardwaj and Tachikawa define this character symmetry and its sector action at Bhardwaj and Tachikawa 2018, § 2.1, pp. 4–5, arXiv PDF.

The dimensional restriction matters. Gauging a finite Abelian ordinary symmetry in $d$ dimensions produces a dual $(d-2)$-form symmetry, so it is another ordinary zero-form symmetry only for $d=2$. Gaiotto, Kapustin, Seiberg, and Willett state this degree shift and the dual-gauging relation at Gaiotto et al. 2015, § 3, p. 14, arXiv PDF.

Double gauging is finite Fourier inversion

The reversal mechanism is transparent on a closed connected oriented two-dimensional spacetime $M$. Take the untwisted gauging with the canonical groupoid weight. A flat $A$ background determines $a\in H^1(M,A)$, and a background for the dual symmetry determines $b\in H^1(M,\widehat A)$. Their cup-product pairing gives a phase $e^{2\pi i\langle b,a\rangle_M}$. Because every flat Abelian bundle has automorphism group $H^0(M,A)$,

$$Z_{\mathcal T/A}[M;b] = \frac{1}{|H^0(M,A)|} \sum_{a\in H^1(M,A)} e^{2\pi i\langle b,a\rangle_M} Z_{\mathcal T}[M;a].$$

Gauging $\widehat A$ performs the inverse finite transform. To retain a background $a'$ for the double-dual symmetry $\widehat{\widehat A}\cong A$, insert the inverse pairing with $a'$ during the second sum:

$$Z_{(\mathcal T/A)/\widehat A}[M;a'] = \frac{1}{|H^0(M,\widehat A)|} \sum_{b\in H^1(M,\widehat A)} e^{-2\pi i\langle b,a'\rangle_M} Z_{\mathcal T/A}[M;b].$$

Substituting the first transform, perfect character pairing gives

$$\begin{aligned} &\sum_{b\in H^1(M,\widehat A)} e^{2\pi i\langle b,a-a'\rangle_M} \\ &\qquad = |H^1(M,\widehat A)|\,\delta_{a,a'}. \end{aligned}$$

For a genus-$g$ surface, $\chi(M)=2-2g$ and $|H^0(M,A)|=|H^0(M,\widehat A)|=|A|$ and $|H^1(M,\widehat A)|=|A|^{2g}$. Therefore

$$Z_{(\mathcal T/A)/\widehat A}[M;a'] = |A|^{-\chi(M)}Z_{\mathcal T}[M;a'].$$

Thus character orthogonality recovers the original background-dependent theory, while the canonical groupoid normalization leaves the displayed invertible Euler factor. A convention that compensates this local factor makes double gauging literally equal to $\mathcal T$.

The formulas above are untwisted. If the first gauging includes $e^{iS_{\mathrm{top}}[a]}$, that weight multiplies the forward kernel. Recovering $\mathcal T$ then requires the corresponding inverse or conjugate choice in the second operation; otherwise double gauging returns a topologically modified theory. In every case, both gaugings must be anomaly-free and must include compatible global sectors.

Bhardwaj and Tachikawa present the same finite-Fourier argument up to overall proportionality at Bhardwaj and Tachikawa 2018, § 2.1, pp. 4–5, eqs. (2.1)–(2.4), arXiv PDF.

Residual and dual pieces need not split

After gauging $A\subset G$, two sources of symmetry may coexist:

• the faithful inherited image of $N_G(A;\mathcal D)/A$;
• the dual symmetry carried by new topological or twisted sectors.

It is unsafe to write their total symmetry as $(G/A)\times\widehat A$ without further work. The inherited group may act nontrivially on $\widehat A$; a topological weight can modify the sector action; and mixed anomaly data can organize the two pieces into a nontrivial extension or obstruct part of the candidate action. These are questions about the complete operator and sector structure, not just group orders. A bounded two-dimensional analysis of how subgroup gauging and anomaly data reorganize the result appears in Bhardwaj and Tachikawa 2018, § 5.3, pp. 40–43, arXiv PDF.

For non-Abelian finite $A$, the set of one-dimensional characters is generally too small to encode the post-gauging symmetry. In two dimensions, Wilson lines form representation-category and potentially non-invertible symmetry data. That categorical structure is outside this page; the finite-Abelian calculation above is not a classification of the non-Abelian case. A bounded entry point is Constructions from Gauging, Duality, and Condensation.

An anomaly of $A$ can obstruct gauging altogether. A mixed anomaly between $A$ and a candidate residual symmetry can instead require a nontrivial extension, an inflow system, or the loss of part of that symmetry. The operational anomaly criterion is developed in What Is an Anomaly?.

A practical determination procedure

For a proposed gauging, determine the post-gauging symmetry in this order:

1. Specify the parent action on states, operators, sectors, and boundaries, and identify its kernel.
2. Specify whether the subgroup $H$ or its faithful image is being gauged, track any inert-kernel topological sector, and verify gaugeability.
3. Include all action, bundle, measure, topological, and boundary choices in $\mathcal D$.
4. Compute $N_G(H;\mathcal D)$ and quotient by transformations in $H$ that have become gauge redundancy.
5. Compute the kernel on the physical data of $\mathcal T/H$ and take the faithful image.
6. Inventory the new twisted or topological sectors and test whether a dual symmetry acts on them.
7. Check dimension, anomalies, extension data, and boundaries before combining inherited and dual pieces.

This order separates three questions that are often conflated: what survives from the parent, what has become redundancy, and what appears only because new sectors were added.

Common pitfalls

Quotienting by $H$ without checking normality. If $H$ is not normal, $G/H$ is not the inherited group. Begin with the data-preserving normalizer.

Stopping at the abstract quotient. The quotient may have an additional kernel on all physical data. A physical symmetry group is the faithful image, not merely a convenient presentation.

Calling $U(1)/\mathbb Z_N$ unchanged. It is abstractly $U(1)$, but the local gauge-invariant operators have a rescaled faithful charge normalization.

Confusing projection with the dual action. The original temporal insertion projects within a twisted Hilbert space. The dual character acts on the twisted-sector label.

Calling the dual symmetry ordinary in every dimension. For finite zero-form gauging it is a $(d-2)$-form symmetry. It is ordinary only in two dimensions.

Assuming a direct product. Residual and dual pieces can mix through an action, extension, topological term, or anomaly. Demonstrate splitness before writing a product.

Calling gauging emergence an RG effect. The dual symmetry appears because the theory is redefined by a bundle sum and new sectors. This is different from accidental symmetry at an infrared fixed point.

Check your understanding

Assume all finite subgroups below are gaugeable and spacetime is closed. Ignore additional restrictions from $\mathcal D$ and any further faithful kernel: compute only the abstract normalizer-quotient candidates.

1. In $S_3$, gauge $H=\langle(12)\rangle$. Find the inherited quotient candidate.
2. Repeat for $H=A_3$.
3. For $\mathbb Z_N$ gauging in two dimensions, show directly that averaging the dual characters removes every nontrivial twisted-sector label.

Solution

For $H=\langle(12)\rangle$, a normalizing element must preserve the transposition $(12)$ under conjugation. Its normalizer in $S_3$ is $H$ itself, so $N_{S_3}(H)/H$ is trivial.

The alternating subgroup $A_3$ is normal in $S_3$. Therefore its normalizer is all of $S_3$, and

$$S_3/A_3\cong\mathbb Z_2.$$

Finally, on sector $k$ the average of dual characters is

$$\frac{1}{N} \sum_{m=0}^{N-1} e^{2\pi i mk/N}.$$

It equals $1$ for $k=0$ and is a finite geometric series with value $0$ for $k\ne0\pmod N$. A second gauging therefore selects the untwisted background sector, subject to the normalization and anomaly qualifications stated above.

What to carry forward

Gauge Fields, Redundancy, and Observable Content develops the local and global data of ordinary gauge fields. Gauging a Higher-Form Symmetry develops the higher-dimensional dual symmetry just signposted. Full duality claims belong to Duality Claims, Dictionaries, Regimes, and Evidence, while RG-emergent symmetry belongs to Universality Classes and Scaling Functions.

References

• Bhardwaj, Lakshya, and Yuji Tachikawa. “On Finite Symmetries and Their Gauging in Two Dimensions.” Journal of High Energy Physics 03 (2018): 189. DOI. Open PDF
• Gaiotto, Davide, Anton Kapustin, Nathan Seiberg, and Brian Willett. “Generalized Global Symmetries.” Journal of High Energy Physics 02 (2015): 172. DOI. Open PDF