Gauging Continuous and Finite Symmetries

Gauging an ordinary global symmetry turns its background gauge data into dynamical variables and identifies configurations related by gauge transformations. For a continuous group this introduces local connection fluctuations and, usually, a gauge-field action. For a finite group there is no infinitesimal gauge field; gauging instead sums over flat bundles, projects onto invariant states, and adds twisted sectors.

The operation is defined only for a gaugeable symmetry and only after its global form, allowed bundles, boundary conditions, measure, local or topological action, and observable content have been specified. This page develops that structural construction and one continuous and one finite scalar example. Detailed Faddeev–Popov or BRST machinery, Yang–Mills dynamics, orbifold modular consistency, lattice algorithms, and anomaly cancellation mechanisms are outside its scope.

Required background. Background Fields versus Dynamical Gauging supplies the distinction between a prescribed connection and a gauge-field integration variable. Vector, Principal, and Associated Bundles supplies the bundle, section, transition-function, and associated-bundle language used in the global sector sum.

Helpful background. What Is an Anomaly? supplies the background-field criterion for whether the proposed symmetry is gaugeable.

What data define a gauging

Start from a QFT $\mathcal T$ with ordinary global symmetry $G$ and background functional

$$Z_{\mathcal T}[M;P,a],$$

where $P\to M$ is a principal $G$-bundle and $a$ a connection. Gauging requires more than replacing ordinary derivatives by covariant ones. One must choose:

• the group $G$ with its global form, its action on all operators and sectors, and the kernel of that action, not only its Lie algebra;
• the allowed bundles, singular sectors, and boundary conditions;
• which gauge transformations are quotiented and which may act at a boundary;
• a measure on the resulting groupoid or quotient;
• local kinetic terms, theta terms, and other admissible topological weights;
• a regulator and measure for which the integrand descends to gauge equivalence classes;
• the resulting gauge-invariant local and extended observables.

The last descent condition is the gaugeability test. An invariant classical Lagrangian under infinitesimal transformations is insufficient: the renormalized functional must also transform consistently under large gauge transformations and on nontrivial bundles. A nontrivial ’t Hooft anomaly obstructs gauging in the same dimension unless additional degrees of freedom or inflow cancel it.

Different choices in this list can define inequivalent gauged theories. A nonfaithful presentation can be gauged, but it retains extra topological gauge data rather than silently reducing to the faithful quotient. When the matter and operator content is compatible with both global forms, gauging $SU(N)$ and gauging $PSU(N)=SU(N)/\mathbb Z_N$ are not interchangeable: their allowed bundles and line operators differ even though their local Lie algebra is the same. If the operator content does not descend to one global form, that choice is not available without changing the theory.

Continuous gauging

For a continuous group, make the connection dynamical and sum over its global sectors. In the coupling-absorbed notation $D=\mathrm d-ia$, a schematic definition is

$$\begin{aligned} Z_{\mathcal T/G}[M] ={}& \sum_{[P]\in\operatorname{Bun}_G(M)} \frac{1}{\operatorname{Vol}\mathcal G(P)} \\ &\times \int_{\mathcal A(P)}\mathcal Da\, e^{\,iS_{\mathrm{gauge}}[P,a] +iS_{\mathrm{top}}[P,a]} \\ &\times Z_{\mathcal T}[M;P,a]. \end{aligned}$$

Here $\mathcal A(P)$ is the space of connections and $\mathcal G(P)$ the gauge-transformation group. The quotient notation is schematic: perturbation theory represents it by gauge fixing and the corresponding determinant or ghost system, while canonical quantization imposes Gauss’s law. Stabilizers, zero modes, and regularization prevent $1/\operatorname{Vol}\mathcal G(P)$ from being a universal literal measure formula.

Locally, the steps are familiar:

1. introduce a connection and replace derivatives by covariant derivatives;
2. include every source-dependent term required by finite gauge covariance, including scalar seagulls;
3. add allowed gauge-field dynamics or topological terms;
4. identify gauge-related configurations and retain gauge-invariant observables.

Globally, one must also include the allowed isomorphism classes $[P]$. Integrating only globally defined potentials on a trivial bundle is a restricted construction and can omit flux sectors.

Schwartz constructs the Abelian covariant derivative, scalar seagull, dynamical Maxwell term, and gauge redundancy at Schwartz 2014, §§ 8.3–8.6, pp. 120–132. Weinberg develops non-Abelian connections, gauge-invariant local actions, constraints, and a gauge-fixed functional integral for gauge-invariant observables at Weinberg 1996, Vol. II, §§ 15.1–15.4, pp. 2–18. Those local constructions support the continuous steps above; the bundle sum is the separate global completion supplied by the background-bundle framework.

Continuous scalar example

Set the symmetry-breaking coupling to zero and start with the charge-one complex scalar. Gauging its faithful $U(1)$ introduces a dynamical connection $a$ and, in four dimensions, may use

$$\begin{aligned} S[\phi,a] ={}& \int\mathrm d^4x\, \left[ (D_\mu\phi)^\dagger D^\mu\phi -V(\phi^\dagger\phi) \right] \\ &-\frac{1}{4e^2} \int\mathrm d^4x\, f_{\mu\nu}f^{\mu\nu}, \\ D_\mu\phi ={}&(\partial_\mu-ia_\mu)\phi, \qquad f=\mathrm da. \end{aligned}$$

Because the coupling has been absorbed into $a$, the gauge coupling appears in the kinetic normalization. The full definition sums over line bundles and integrates over the connections on them, subject to the selected boundary and flux sectors.

The operator content changes. The section $\phi$ is not by itself a local gauge-invariant observable. The composite $\phi^\dagger\phi$ is local and gauge invariant, while charged excitations require an admissible dressing or an attached line. The gauge field contributes local fluctuations when its action makes them dynamical, and physical states obey Gauss’s law.

A fixed nonzero interaction

$$h\phi^N+h^*(\phi^\dagger)^N$$

is not invariant under the full charge-one $U(1)$. Spurionic covariance of $h$ does not authorize gauging that fixed theory. One may instead set $h=0$, gauge only the exact residual $\mathbb Z_N$ subgroup of the original $U(1)$, or construct a different theory in which the charge-$-N$ object replacing $h$ is dynamical.

Finite gauging as a groupoid sum

For finite $G$, principal bundles are flat local systems: there is no Lie-algebra-valued one-form with local curvature fluctuations. Flat does not mean trivial. On a connected spacetime, a bundle can be represented by a homomorphism

$$\pi_1(M)\longrightarrow G$$

up to conjugation, and nontrivial holonomy produces twisted sectors.

The automorphism weight follows directly from orbit–stabilizer counting. On a finite cell decomposition, let $\mathcal A_{\mathrm{flat}}$ be the finite set of flat edge assignments and $\mathcal G_0$ the vertex gauge group. For any gauge-invariant function $F$,

$$\frac{1}{|\mathcal G_0|} \sum_{A\in\mathcal A_{\mathrm{flat}}}F(A) = \sum_{[A]} \frac{F(A)}{|\operatorname{Stab}(A)|}.$$

Indeed, the orbit of $A$ contains $|\mathcal G_0|/|\operatorname{Stab}(A)|$ representatives. Its stabilizer is the automorphism group of the corresponding flat bundle, which gives the finite analogue of the gauge path integral:

$$\begin{aligned} Z_{\mathcal T/G}[M] ={}& \sum_{[P]\in\operatorname{Bun}^{\mathrm{flat}}_G(M)} \\ &\times \frac{e^{iS_{\mathrm{top}}[P]}} {|\operatorname{Aut}(P)|} Z_{\mathcal T}[M;P]. \end{aligned}$$

The factor $1/|\operatorname{Aut}(P)|$ gives the canonical groupoid-cardinality weight for an isomorphism class. Any additional spacetime-dependent normalization must itself obey locality and gluing and is an allowed invertible local term, not a freely adjustable factor. Likewise, further sector-dependent phases are part of the specified topological action $S_{\mathrm{top}}$, not arbitrary measure choices.

Finite gauging has no photon in this minimal construction. Nevertheless, it changes both states and operators:

• temporal gauge data project onto invariant states;
• spatial holonomy adds twisted or flux sectors;
• gauge-charged local operators cease to be local physical observables by themselves;
• twist and disorder operators can enter the gauged theory.

Gaiotto, Kapustin, Seiberg, and Willett describe finite gauging as a sum over flat backgrounds, the resulting twisted sectors, and discrete-torsion choices at Gaiotto et al. 2015, § 2, pp. 5–10, arXiv PDF.

Circle sectors and the torus sum

The projection-plus-twists mechanism is especially explicit for a two-dimensional theory on a spatial circle. Let $\mathcal H_g$ be the Hilbert space with spatial transition function $g\in G$. Only the centralizer

$$C_g=\{h\in G\mid hg=gh\}$$

preserves that boundary condition. For untwisted finite gauging,

$$\mathcal H_{\mathcal T/G}(S^1) \simeq \bigoplus_{[g]} \mathcal H_g^{\,C_g},$$

where $[g]$ runs over conjugacy classes. A topological term can twist the $C_g$ action, in which case the invariant subspace must include its induced one-dimensional phase line rather than use the bare action shown here.

Equivalently, on a torus,

$$Z_{\mathcal T/G}(T^2) = \frac{1}{|G|} \sum_{\substack{g,h\in G\\gh=hg}} \epsilon(g,h)\,Z_{g,h}.$$

The spatial holonomy $g$ selects the Hilbert space, the temporal holonomy $h$ is inserted in the trace, and $\epsilon(g,h)$ is the permitted topological weight. For trivial weighting, $\epsilon=1$.

The factor $1/|G|$ is the same groupoid normalization written before quotienting commuting pairs by simultaneous conjugation. An orbit of $(g,h)$ has $|G|/|C_G(g,h)|$ representatives, so its total coefficient is $1/|C_G(g,h)|=1/|\operatorname{Aut}(P_{g,h})|$.

In two-dimensional discrete torsion derived from a cocycle $\alpha$, commuting pairs carry

$$\epsilon_\alpha(g,h) =\frac{\alpha(g,h)}{\alpha(h,g)}.$$

Here $\alpha:G\times G\to U(1)$ obeys

$$\alpha(g,h)\alpha(gh,k) =\alpha(h,k)\alpha(g,hk),$$

and rephasing by $\beta:G\to U(1)$ changes it to

$$\alpha'(g,h) =\frac{\beta(g)\beta(h)}{\beta(gh)}\alpha(g,h).$$

The equivalence classes form $H^2(G,U(1))$. The cocycle condition makes the displayed torus weight consistent; arbitrary phases for individual sectors generally do not.

Residual scalar symmetry: a finite worked example

Now take a two-dimensional realization of the same scalar symmetry on a spatial circle. With fixed nonzero

$$\Delta\mathcal L =h\phi^N+h^*(\phi^\dagger)^N, \qquad N\geq2.$$

The exact residual subgroup of the original $U(1)$ is generated by

$$R:\phi\longmapsto e^{2\pi i/N}\phi.$$

The full unitary symmetry can be larger if further transformations preserve the complete action; for example, a compatible reflection can enlarge the cyclic subgroup to a dihedral group. Here only the displayed $\mathbb Z_N\subset U(1)$ is gauged.

Gauging this $\mathbb Z_N$ means summing over flat $\mathbb Z_N$ bundles. Define the circle sector $\mathcal H_k$ by

$$\phi(x+L)=e^{2\pi ik/N}\phi(x), \qquad k=0,\ldots,N-1.$$

The interaction remains single-valued because

$$\phi^N(x+L)=\phi^N(x).$$

Let

$$Z_{k,\ell} =\operatorname{Tr}_{\mathcal H_k} \left( R^\ell e^{-\beta H} \right).$$

Taking Euclidean time to have period $\beta$ gives a rectangular torus. Its partition function and the Hilbert space of the untwisted gauging are

$$\begin{aligned} Z_{\mathcal T/\mathbb Z_N}(T^2) &= \frac{1}{N} \sum_{k,\ell=0}^{N-1}Z_{k,\ell}, \\ \mathcal H_{\mathcal T/\mathbb Z_N} &= \bigoplus_{k=0}^{N-1} \mathcal H_k^{\mathbb Z_N}. \end{aligned}$$

Indeed, on a state of charge $q$ modulo $N$, the temporal sum gives

$$\frac{1}{N} \sum_{\ell=0}^{N-1} e^{2\pi i\ell q/N} = \begin{cases} 1,&q=0\pmod N,\\ 0,&q\ne0\pmod N. \end{cases}$$

Thus the $\ell$ sum performs the projection, whereas the $k$ sum adds the twisted sectors. Keeping only $k=0$ would project the original Hilbert space but would not complete the finite gauging.

For a single unitary cyclic group, let $V$ projectively implement its generator. Projective multiplication implies $V^N=e^{i\vartheta}\mathbf 1$. Replacing $V$ by $e^{-i\vartheta/N}V$ makes $V^N=\mathbf 1$, so the powers of the rephased generator give an ordinary representation and remove the multiplier. Hence $H^2(\mathbb Z_N,U(1))=0$: ordinary two-dimensional discrete torsion is trivial for this group. A nontrivial example requires a different group, such as $\mathbb Z_2\times\mathbb Z_2$, rather than arbitrary phases attached to these cyclic sectors.

Gaugeability and boundary checks

Before accepting either construction, test the complete background functional:

1. Small transformations. The renormalized local Ward identity must have no uncancelled gauge variation.
2. Large transformations. The exponentiated functional must be single-valued under transformations not connected to the identity.
3. Nontrivial bundles. The theory must be consistently defined in every sector being summed.
4. Boundaries. The quotient must specify whether boundary-nonvanishing transformations are redundancies or physical symmetries and what boundary fields or conditions restore consistency.
5. Local choices. Counterterms and topological weights must satisfy locality, quantization, and gluing.

Passing an infinitesimal classical check proves only the first and weakest part of this list. Conversely, a gauge-variant field need not disappear from all useful descriptions: it may create a dressed excitation or appear at the endpoint of an extended operator, even though it is not a standalone local physical observable.

Common pitfalls

Covariantizing derivatives but not quotienting. This produces a background-covariant expression, not yet a gauged theory.

Integrating only the trivial continuous bundle. Globally defined potentials omit allowed flux sectors unless the theory has explicitly been restricted to that sector.

Projecting without twisted sectors. Finite gauging requires both the temporal projection and the spatial bundle sum.

Calling flat finite bundles trivial. A discrete connection has no local curvature fluctuation but can have nontrivial holonomy.

Adding arbitrary sector phases. A valid topological weight must obey locality and gluing; in two dimensions, discrete-torsion phases arise from a cocycle.

Expecting a finite gauge boson. The added finite gauge-field sector is topological and supplies no propagating photon.

Gauging a spurionic symmetry. Transforming a nondynamical breaking coupling organizes a family of theories; it does not make the full group an exact symmetry of a fixed member.

Checking only infinitesimal anomalies. Large transformations and nontrivial bundles can reveal obstructions invisible in the local divergence equation.

Check your understanding

For the $\mathbb Z_N$ scalar construction, explain separately what the sums over $k$ and $\ell$ do. Then verify that the interaction $\phi^N$ is defined in every spatial sector and that a charge-$q$ state survives the temporal sum exactly when $q=0\pmod N$.

Check

The label $k$ specifies the spatial transition function and therefore the twisted Hilbert space $\mathcal H_k$. Summing over $k$ adds all flat bundle sectors. The temporal insertion $R^\ell$ acts on a charge-$q$ state by $e^{2\pi i\ell q/N}$, so

$$\frac1N\sum_{\ell=0}^{N-1}e^{2\pi i\ell q/N} =\delta_{q,\,0\ {\rm mod}\ N}.$$

The $\ell$ sum is therefore the projector onto $\mathbb Z_N$-invariant states in each $\mathcal H_k$. Finally,

$$\left(e^{2\pi ik/N}\phi\right)^N=\phi^N,$$

so the interaction is single-valued in every sector. Projection alone would miss the $k\ne0$ sectors; summing sectors without the temporal average would retain gauge-variant states.

What to carry forward

Continuous gauging integrates connections and adds local gauge dynamics; finite gauging performs a groupoid sum over flat bundles. Both require a gaugeable symmetry, a quotient, global-sector data, admissible local or topological weights, and a newly defined gauge-invariant operator algebra.

Residual, Quotient, and Emergent Dual Symmetries tracks what symmetry survives or appears after gauging. Local perturbative gauge fixing begins with The Faddeev–Popov Construction; Ghosts, Auxiliary Fields, and Gauge-Parameter Dependence and The BRST Differential and Gauge-Fixed Complex develop its quantum structure. Dynamical Gauge Fields and Matter owns model dynamics, while Cosets and Orbifolds owns orbifold consistency. Numerical implementations belong to the lattice volume.

References

• Gaiotto, Davide, Anton Kapustin, Nathan Seiberg, and Brian Willett. “Generalized Global Symmetries.” Journal of High Energy Physics 02 (2015): 172. DOI. Open PDF
• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge: Cambridge University Press, 2014. DOI
• Weinberg, Steven. The Quantum Theory of Fields. Volume II: Modern Applications. Cambridge: Cambridge University Press, 1996. DOI