Background Fields versus Dynamical Gauging

A background field probes a theory; gauging constructs a new theory. In the first operation one chooses a bundle with connection and evaluates the generating functional there. In the second one declares appropriate gauge data dynamical, quotients by gauge redundancy, and sums or integrates over global sectors with a specified measure and action. That change alters the physical operator algebra, the state space, and often the symmetry that remains.

The comparison is structural. Detailed gauge-fixing machinery, interacting gauge dynamics, and lattice or Hamiltonian implementations are deferred to their application volumes.

Required background. Coupling to Background Gauge Fields and Bundles supplies the prescribed bundle-with-connection data and background covariance. Symmetry, Gauge Redundancy, and Duality supplies the distinction between a physical global action and a redundancy of description.

Fixed backgrounds and dynamical gauge variables

The quickest diagnostic is:

Is the connection an argument of the functional, or an integration variable?

If it is an argument, it is a background. If gauge-equivalence classes of connections or bundles are summed with dynamical weights, the symmetry has been gauged. A kinetic term alone does not answer the question if the field remains fixed; the integration and quotient are decisive.

Fixed background: one prescribed sector

For a theory $\mathcal T$ with global symmetry $G$, write

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

for its partition functional on spacetime $M$ with a prescribed principal $G$-bundle $P$ and connection $a$. The matter fields are integrated over, but $(P,a)$ is held fixed. Varying $a$ generates current response. Changing its holonomy or bundle sector compares the theory in different external environments.

Background gauge covariance identifies different local presentations of the same external data. Charged operators can still be retained as covariant insertions: their correlators transform with their representation and live in associated bundles. No Gauss constraint projects them out because the background transformation has not become a redundancy acting on dynamical gauge data.

Gauging: quotient, integration, and sector sum

A schematic continuous gauging is

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

Here $\mathcal A(P)$ is the space of connections and $\mathcal G_P$ the gauge-transformation group. The formula is structural rather than a universal measure prescription: stabilizers, zero modes, regularization, and gauge fixing require theory-specific treatment.

The higher-form gauging process map shows the corresponding $(p+1)$-background specialization, including the anomaly stop, global-sector sum, operator attachment, and conditional dual symmetry.

This operation needs data not contained in the background functional alone:

• the subgroup and global form to be gauged;
• the allowed bundles, singular sectors, and boundary conditions;
• the measure and quotient by gauge transformations;
• a gauge-field action or topological weighting;
• finite counterterms, theta data, and line-operator choices;
• a regulator and anomaly-cancellation mechanism;
• a rule for gauge transformations at physical boundaries.

Different choices can define inequivalent theories even when they start from the same local current algebra.

Finite gauging: projection plus twisted sectors

For a finite group there is no Lie-algebra-valued connection one-form and usually no ordinary kinetic term. A useful schematic expression is

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

Here $[P]$ ranges over isomorphism classes in $\operatorname{Bun}^{\mathrm{flat}}_G(M)$. The factor $1/|\operatorname{Aut}(P)|$ is the canonical groupoid-cardinality weight. The normalization $\mathcal N(M)$ is fixed as part of the convention; any spacetime dependence must itself satisfy locality and gluing and amounts to an allowed invertible local factor. Further sector-dependent phases belong to the specified topological action $S_{\mathrm{top}}$ and are not arbitrary measure choices.

On a spatial manifold $\Sigma$, the corresponding state-space picture is schematically

$$\begin{aligned} \mathcal H_{\mathrm{gauged}}(\Sigma) &\simeq\bigoplus_{[P]}\mathcal V_P, \\ \mathcal V_P &\equiv\Bigl( \mathcal H_{\mathcal T}(\Sigma;P) \\ &\qquad \otimes\ell_{S_{\mathrm{top}}}(P) \Bigr)^{\operatorname{Aut}(P)}. \end{aligned}$$

The one-dimensional topological line $\ell_{S_{\mathrm{top}}}(P)$ carries the phase or projective automorphism action induced by the topological term. For trivial $S_{\mathrm{top}}$ it is the trivial line and $\mathcal V_P$ reduces to the ordinary invariant subspace. With discrete torsion, temporal holonomy implements a twisted projection. Thus gauging does two things:

1. temporal gauge data impose the projection onto invariant states;
2. spatial bundle data add twisted or flux sectors.

Keeping only the invariant part of the untwisted sector is generally incomplete. Gaiotto, Kapustin, Seiberg, and Willett distinguish a fixed flat background from summing over flat connections and discuss the accompanying twisted sectors and discrete-torsion choices at Gaiotto et al. 2015, § 2, pp. 5–10, esp. p. 7, arXiv PDF.

Continuous gauging: a new dynamical field

For a continuous group the connection has local fluctuations. In four-dimensional Abelian language, a standard dynamical term is

$$S_{\mathrm{gauge}}[a] =-\frac{1}{4e^2} \int\mathrm d^4x\, f_{\mu\nu}f^{\mu\nu}.$$

The path integral now includes $\mathcal Da$ and, perturbatively, a representation of the quotient through gauge fixing and the appropriate determinant or ghost system. Canonical quantization imposes Gauss’s law on physical states. These are not features of a fixed source.

Schwartz constructs scalar electrodynamics from the covariant matter coupling through the dynamical Maxwell theory, and separates local gauge redundancy from the surviving global content, at Schwartz 2014, §§ 8.3–8.6, pp. 120–132. Detailed gauge fixing, BRST structure, and interacting gauge dynamics belong to the gauge-theory volume.

The operator algebra changes

Let $\mathcal O_R(x)$ transform in a nontrivial representation $R$ of the group being gauged.

With a background connection, $\mathcal O_R$ remains a valid covariant local insertion. After gauging, it is not by itself a local gauge-invariant observable. It can instead appear in a neutral composite, at an endpoint paired with suitable dressing, or as part of a nonlocal operator. Wilson lines, flux operators, and twist or disorder sectors become part of the gauged theory’s observable structure.

The same change appears in the state space. Gauge-variant vectors are removed by the constraint or projection, while new flux and bundle sectors can enter. Gauging is therefore not a relabelling of the original Hilbert space.

For finite symmetries, these operator and twisted-sector changes are described in the bounded setting of Gaiotto et al. 2015, § 2, pp. 5–10, arXiv PDF. Their precise form depends on dimension, boundary conditions, global form, and topological weights.

Threaded scalar: three different operations

For the complex scalar with exact $U(1)$, a fixed-background functional is

$$Z_{\mathrm{bg}}[a] =\int\mathcal D\phi\,\mathcal D\phi^\dagger\, e^{iS[\phi;a]}.$$

The connection $a$ is prescribed. The insertion $\phi(x)$ is charged but meaningful as a section-valued, background-covariant operator. The source derivative gives the current response.

To gauge this $U(1)$, set aside any incompatible fixed charged coupling, add gauge dynamics and global-sector data, and integrate:

$$\begin{aligned} Z_{\mathrm{gauge}} &=\sum_{[P]}\mathcal I_P, \\ \mathcal I_P &\equiv \frac{1}{\operatorname{Vol}\mathcal G_P} \\ &\quad\times\int\mathcal Da \\ &\quad\times e^{iS_{\mathrm{gauge}}[P,a] +iS_{\mathrm{top}}[P,a]} \\ &\quad\times Z_{\mathrm{bg}}[P,a]. \end{aligned}$$

Now $\phi$ alone is not in the local physical operator algebra, whereas $\phi^\dagger\phi$ is gauge invariant. The gauge field carries local degrees of freedom when its action makes them dynamical, and Gauss’s law constrains the physical states.

Next include

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

Three possibilities must not be conflated:

• Fixed $h$. The physical global symmetry is only $\mathbb Z_N$, assuming no other breaking term. Gauging that residual symmetry means summing over flat $\mathbb Z_N$ bundles and projecting onto $\mathbb Z_N$-invariant states.
• Spurionic $h$. Letting $h$ transform organizes a covariant $U(1)$ family, but a prescribed transforming source is not a new dynamical field and does not make the fixed member $U(1)$ symmetric.
• Dynamical charge-$-N$ field. Replacing the spurion by a field with its own dynamics can define a $U(1)$ gauge theory with a Higgs regime that leaves $\mathbb Z_N$ gauge structure. That is a different theory, not a reinterpretation of the fixed coupling.

A full $U(1)$ gauge redundancy cannot coexist with a fixed nonzero charge-$-N$ numerical coupling: the term is not gauge invariant. One must gauge only the exact subgroup, remove the coupling, or supply additional dynamical structure.

Consistency conditions and boundaries

If the background functional transforms anomalously,

$$\mathcal Z[P,a^U] \neq\mathcal Z[P,a],$$

then the integrand does not descend to gauge-equivalence classes. Gauging is obstructed unless the anomaly is cancelled by additional fields, counterterms when possible, or an inflow construction. The background remains useful precisely because its transformation diagnoses the obstruction.

An infinitesimal local Ward identity is not enough: large transformations and nontrivial bundles can reveal global anomalies. Physical boundaries add another qualification. Transformations required to vanish at the boundary are redundancies, while nonvanishing boundary transformations may act as genuine boundary symmetries with surface charges. Those choices must be made before defining the quotient.

Common pitfalls

Adding a kinetic term but never integrating. A derivative term in a functional of a fixed source does not create a gauge-field Hilbert space.

Integrating only over the trivial bundle. On nontrivial spacetime this can omit flux and twisted sectors and define a restricted theory rather than the intended gauging.

Projecting without adding twisted sectors. For finite gauging, both operations are required by the spacetime sum over flat bundles.

Keeping charged local operators unchanged. After gauging, a charged field must be combined into a gauge-invariant composite or dressing.

Gauging an anomalous symmetry. If the background functional fails to descend to gauge orbits, the proposed gauge path integral is not well defined without further cancellation data.

Treating gauge fixing as physical input. Gauge fixing represents the quotient; different valid gauges should not define different physical theories.

Forgetting discrete-theta choices. Distinct topological weights can produce inequivalent gauged theories even when the summed background sectors are the same.

Check your understanding

Assume trivial topological weighting and let the scalar have exact global symmetry $\mathbb Z_N$ on a spatial circle. Label a fixed flat background by $m\in\mathbb Z_N$ with holonomy $e^{+2\pi im/N}$. Explain why choosing one $m$ is not gauging, and why a gauged Hilbert space requires both a sum over $m$ and a projection in each sector.

Check

The holonomy specifies a twisted boundary condition,

$$\phi(x+L) =e^{-2\pi im/N}\phi(x).$$

Choosing $m$ selects one external background and one twisted Hilbert space $\mathcal H_m$. Gauging makes the spatial bundle dynamical, so all allowed $m$ contribute. Temporal holonomy implements the average over gauge transformations, projecting each sector onto its invariant subspace. Thus

$$\mathcal H_{\mathrm{gauged}} \simeq \bigoplus_{m\in\mathbb Z_N} \mathcal H_m^{\mathbb Z_N},$$

with the ordinary automorphism projection in this trivial-weight case. The charged field $\phi$ no longer defines a local physical operator by itself, while twist and flux sectors absent from the untwisted projection are retained.

What to carry forward

A fixed background supplies probes and Ward identities. Gauging adds a quotient, a measure, dynamical or topological weights, global-sector sums, and a new physical operator/state space. Gauging Continuous and Finite Symmetries develops the constructions in detail; Residual, Quotient, and Emergent Dual Symmetries tracks what symmetry survives or appears afterward. Gauge orbits, constraints, and observables are developed in the next gauge-structure chapter.

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