Coupling to Background Gauge Fields and Bundles

A background gauge field is a connection used as a nondynamical source for an ordinary global symmetry. Locally it makes derivatives and Ward identities covariant. Globally it also records a principal bundle, transition functions, and holonomies—data that no single gauge potential can contain. The path integral is evaluated at that prescribed background; the background is not yet summed over.

The discussion assumes an ordinary nonanomalous internal symmetry group $G$, with regulator, state, and boundary data compatible with background transformations. Higher-form backgrounds and detailed differential-cohomology models are outside its scope.

Required background. Current Sources and Generating Functionals supplies the source functional and current-response derivatives. Bundle Connections, Curvature, Gauge Transformations, and Bianchi Identities supplies connection transformations, curvature, and patchwise compatibility.

Helpful background. Vector, Principal, and Associated Bundles supplies the bundle, section, and transition-function language used in the global patching discussion.

A connection is the covariant source

Fix the action of $G$ on the fields. The global group and its representations, not only its Lie algebra, matter. Absorb the coupling into

$$a\equiv gA, \qquad D=\mathrm d-ia.$$

This is the same $a$ used on the preceding page. Restoring $A$ inserts a factor $1/g$ in the inhomogeneous part of its transformation law.

Let a field transform as $\psi\mapsto U\psi$, where $U(x)=e^{i\alpha(x)}$ in the relevant representation. Requiring

$$D^U(U\psi)=U(D\psi)$$

fixes the active transformation of the connection:

$$\begin{aligned} a^U ={}&UaU^{-1} \\ &-i(\mathrm dU)U^{-1}. \end{aligned}$$

For an infinitesimal transformation,

$$\delta_\alpha a =\mathrm d\alpha+i[\alpha,a] \equiv D^{\mathrm{ad}}\alpha,$$

where

$$D^{\mathrm{ad}}\alpha =\mathrm d\alpha-i[a,\alpha].$$

The curvature is

$$f=\mathrm da-ia\wedge a, \qquad D^2=-if,$$

and transforms homogeneously:

$$f^U=UfU^{-1}.$$

Schwartz derives the covariant derivative, the local connection transformation, and the required quadratic scalar coupling at Schwartz 2014, § 8.3, pp. 120–123. That discussion treats the gauge field dynamically in scalar electrodynamics. Only its kinematic connection formulas are imported here; declaring the connection nondynamical is a distinct operation.

Why one gauge potential is not global data

Let $P\to M$ be a principal $G$-bundle and $\{U_i\}$ an open cover. Choose local frames with transition functions

$$u_{ij}:U_i\cap U_j\longrightarrow G, \qquad u_{ij}u_{jk}=u_{ik}$$

on triple overlaps. The local representatives of an associated field and connection obey

$$\begin{aligned} \psi_i &=\rho(u_{ij})\psi_j, \\ a_i &=u_{ij}a_j u_{ij}^{-1} \\ &\quad-i(\mathrm du_{ij})u_{ij}^{-1}. \end{aligned}$$

These laws ensure

$$D_i\psi_i=\rho(u_{ij})D_j\psi_j.$$

The local curvatures patch without an inhomogeneous term,

$$f_i=u_{ij}f_j u_{ij}^{-1},$$

so invariant polynomials in $f$ are globally defined. A different set of local frames changes $a_i$ and $u_{ij}$ together but does not change the bundle with connection.

The globally meaningful argument of the generating functional is therefore

$$W[M;P,a],$$

not merely $W[a_\mu(x)]$ for one globally defined matrix-valued function. A fixed topological sector $P$ can carry physical holonomy data even when every local expression looks pure gauge.

Holonomy and flat backgrounds

Parallel transport around an oriented path $\gamma$ is locally represented by

$$\operatorname{Hol}_\gamma(a) =\mathcal P\exp\!\left( i\int_\gamma a \right),$$

with transition functions inserted when the path crosses between local frames. For a closed loop based at $x$, a background transformation conjugates the result:

$$\operatorname{Hol}_\gamma(a^U) =U(x)\operatorname{Hol}_\gamma(a)U(x)^{-1}.$$

Its conjugacy class, and hence every character $\operatorname{tr}_R\operatorname{Hol}_\gamma(a)$, is independent of the chosen frame.

Flatness, $f=0$, does not imply trivial holonomy. On a spatial circle of circumference $L$, take a $U(1)$ background

$$a_x=\frac{\theta}{L}.$$

Then

$$f=0, \qquad \operatorname{Hol}_{S^1}(a)=e^{i\theta}.$$

The transformation that would set $a_x$ to zero is single-valued only when $e^{i\theta}=1$. A unit-charge scalar mode $e^{2\pi inx/L}$ consequently has covariant momentum

$$k_n^{(a)} =\frac{2\pi n-\theta}{L}.$$

Gaiotto, Kapustin, Seiberg, and Willett describe ordinary symmetries through flat background connections, holonomies, transition functions, and their triple-overlap condition at Gaiotto et al. 2015, § 2, pp. 5–7, esp. p. 7, arXiv PDF. Their discussion also distinguishes fixing a flat background from gauging by summing over such backgrounds. General nonflat connections use the bundle geometry stated above.

The covariant Ward identity

For a bundle automorphism $U$ of fixed $P$, an anomaly-free normalized vacuum functional satisfies

$$\mathcal Z[M;P,a^U]=\mathcal Z[M;P,a].$$

Equivalently, $W=-i\log\mathcal Z$ is invariant modulo its $2\pi$ branch ambiguity. The infinitesimal variation of $W$ is unambiguous.

Charged correlators are covariant rather than numerical invariants, because their insertions transform in associated bundles. Introduce sources $\lambda^r$ for those insertions and identify the Lie algebra with its dual using an invariant pairing $\langle\,\cdot,\cdot\,\rangle_{\mathfrak g}$. The infinitesimal source identity is

$$\begin{aligned} 0=\delta_\alpha W =\int\mathrm d^d x\Bigg[ &\left\langle \overline J^\mu, D_\mu^{\mathrm{ad}}\alpha \right\rangle_{\mathfrak g} \\ &+\frac{\delta W}{\delta\lambda^r} \delta_\alpha\lambda^r \Bigg], \end{aligned}$$

where

$$\overline J_A^\mu(x;a) \equiv\frac{\delta W}{\delta a_\mu^A(x)} =\left\langle [\mathcal J_A^\mu(x;a)]_R \right\rangle_a.$$

With compact support and all charged insertion sources set to zero, covariant integration by parts gives

$$D_\mu^{\mathrm{ad}}\overline J^\mu=0.$$

Differentiating before setting the $\lambda^r$ to zero produces the covariant contact terms for transformed insertions. Explicit breaking sources add their own terms, exactly as on the preceding page.

This infinitesimal equation does not test disconnected or large background transformations. A global anomaly can preserve the local divergence equation while obstructing invariance under a large transformation. A local anomaly instead adds a nonzero functional variation. Regulated Jacobians and Measure Variation begins the anomaly analysis.

The threaded scalar on a line bundle

For the unit-charge complex scalar, a $U(1)$ background is a line bundle $L\to M$ with connection. On overlaps, write

$$u_{ij}=e^{i\chi_{ij}}.$$

The local fields and potentials patch as

$$\phi_i=e^{i\chi_{ij}}\phi_j, \qquad a_i=a_j+\mathrm d\chi_{ij}.$$

Thus $\phi$ is a section of $L$, not necessarily one global complex-valued function, and

$$(D\phi)_i=e^{i\chi_{ij}}(D\phi)_j.$$

The kinetic term is globally defined. Its source derivative is the covariant current

$$\begin{aligned} \mathcal J^\mu(a) =i\Big[ &\phi^\dagger D^\mu\phi \\ &-(D^\mu\phi)^\dagger\phi \Big], \end{aligned}$$

which expands to

$$\mathcal J^\mu(a) =j^\mu+2a^\mu\phi^\dagger\phi.$$

This is precisely the current plus seagull completion derived from the generating functional.

Now add

$$\Delta\mathcal L =h\phi^N+h^*(\phi^\dagger)^N, \qquad N\in\mathbb Z_{\geq2}.$$

The spurion is globally consistent when

$$h_i=e^{-iN\chi_{ij}}h_j.$$

In geometric language, $h$ is a section of $L^{-N}$, so $h\phi^N$ is a scalar. The spurionic $U(1)$ family may retain both $a$ and $h$ as external sources on a general compatible bundle.

A frozen spurion with fixed nonzero norm is nowhere vanishing and trivializes $L^N$. Choose unitary frames in which its representative is the same fixed nonzero number. Compatibility then requires

$$e^{iN\chi_{ij}}=1.$$

The transition functions then lie in $\mathbb Z_N$. This reduction of the bundle does not by itself make an arbitrary $U(1)$ connection flat. A background that preserves the frozen spurion must also satisfy

$$D_a h=(\mathrm d+iNa)h=0.$$

In frames where $h$ is a nonzero constant, this condition removes the local continuous connection and leaves only the discrete $\mathbb Z_N$ holonomy. It is this compatible reduced background—not an arbitrary connection on a bundle with $L^N$ trivial—that is a flat principal $\mathbb Z_N$ background. A finite-group background has transition data but no ordinary Lie-algebra-valued connection one-form. Assuming no other interaction breaks it, this is the global version of the exact residual $\mathbb Z_N$ symmetry identified by the local Ward identity.

What has not been gauged

The notation resembles gauge theory because background covariance is the organizing principle. Nevertheless, this page has introduced none of the defining dynamical operations:

• there is no functional integral over $a$;
• bundle sectors $P$ are not summed;
• no kinetic term for $a$, gauge fixing, or ghosts is required;
• no Gauss constraint or gauge-boson Hilbert space has been added.

A background transformation changes the local presentation of fixed external data. Gauging changes the theory by making appropriate background data dynamical and summing or integrating over them with specified weights. Background Fields versus Dynamical Gauging develops this distinction.

Common pitfalls

Writing one global potential on a nontrivial bundle. A connection is represented by local one-forms plus transition functions. Omitting the latter silently restricts the background sector.

Treating flat as trivial. A flat connection can have nontrivial holonomy around a noncontractible loop.

Using only the Lie algebra. Bundle sectors and allowed holonomies depend on the global group and on which representations occur.

Calling background covariance gauging. A nondynamical connection creates no gauge-field path integral, constraint, or new gauge-boson state.

Applying an infinitesimal Ward identity to a large transformation. Local conservation cannot by itself rule out a global anomaly.

Representing a finite-group background by a smooth Lie-algebra one-form. A discrete group has no nonzero Lie algebra; its flat background information resides in transition functions and holonomies.

Check your understanding

For a unit-charge scalar on a circle, verify that the flat potential $a_x=\theta/L$ can be removed by a single-valued background transformation exactly when $\theta\in2\pi\mathbb Z$. Then derive the shifted covariant momenta.

Check

For $U=e^{i\alpha(x)}$, the Abelian law is $a_x^U=a_x+\partial_x\alpha$. Setting $\alpha(x)=-\theta x/L$ gives $a_x^U=0$. Single-valuedness requires

$$e^{i\alpha(x+L)}=e^{i\alpha(x)},$$

or $e^{-i\theta}=1$. Otherwise the holonomy is nontrivial and the background cannot be removed globally.

For $\phi_n=e^{2\pi inx/L}$,

$$\begin{aligned} D_x\phi_n &=(\partial_x-ia_x)\phi_n \\ &=i\left( \frac{2\pi n-\theta}{L} \right)\phi_n. \end{aligned}$$

Thus $k_n^{(a)}=(2\pi n-\theta)/L$. The spectrum is periodic under $\theta\mapsto\theta+2\pi$, which relabels $n$ by one unit.

What to carry forward

A background connection is local differential data on a globally specified bundle. It covariantizes the source functional and its Ward identities, while holonomy and transition functions probe information invisible to a single local potential. Spurions, Local Counterterms, and Symmetry Response studies the local ambiguities in $W[P,a]$ and the invariant response that survives them. Dynamical gauging remains a later operation.

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