Global Form, Matter Representations, and the Faithful Gauge Group

A compact semisimple Lie algebra does not by itself determine a connected gauge group. If $\widetilde G$ is the simply connected compact group with Lie algebra $\mathfrak g$, every connected global form with that algebra is a central quotient $G_\Gamma=\widetilde G/\Gamma$. A representation is allowed for that quotient exactly when $\Gamma$ acts trivially. The common kernel of a specified matter spectrum therefore determines the group that acts faithfully on that matter, but it does not by itself determine the full gauge theory: admissible bundles and extended probes retain additional global information. This page establishes those statements for compact connected semisimple ordinary gauge groups and applies them to $\mathfrak{su}(2)$ Yang–Mills theory on a region with boundary.

Required background. Local Potentials and Global Gauge Configurations supplies the bundle and connection language used below. Representations, Intertwiners, Invariants, and Tensor Decomposition supplies representation kernels, direct sums, and quotient actions.

Helpful background. Compact Lie Groups, Roots, Weights, and Weyl Structure explains how highest weights test whether an infinitesimal representation integrates to a chosen global form.

One Lie algebra admits several connected global forms

Let $\mathfrak g$ be a compact semisimple Lie algebra. There is a simply connected compact Lie group $\widetilde G$, unique up to isomorphism, whose Lie algebra is $\mathfrak g$. Every connected compact Lie group with this Lie algebra has the form

$$G_\Gamma=\widetilde G/\Gamma, \qquad \Gamma\subseteq Z(\widetilde G),$$

where $\Gamma$ is a subgroup of the finite center of $\widetilde G$. The quotient map $q_\Gamma:\widetilde G\to G_\Gamma$ is a local isomorphism, so its differential identifies both Lie algebras with $\mathfrak g$. Infinitesimal gauge transformations, local curvature formulas, and perturbative vertices therefore do not reveal which $\Gamma$ was chosen. The allowed global representations do.

The key group-theoretic step is short. The kernel of a covering homomorphism from $\widetilde G$ is discrete and normal. Conjugating any one kernel element defines a continuous map from the connected group $\widetilde G$ into that discrete kernel, so the map is constant and the kernel element is central. Compact semisimplicity makes the center finite. This classification and its representation consequence are stated explicitly in Aharony, Seiberg, and Tachikawa 2013, arXiv v5, § 1.1, preprint pp. 2–3, Open PDF and reviewed in Bhardwaj et al. 2024, arXiv v2, § 3.3.3, preprint pp. 46–47, Open PDF.

This is already a warning about notation. Writing only “an $\mathfrak g$ gauge theory” fixes the local algebraic data, not the global group. Conversely, writing $G$ commits to more than a choice of basis or generator normalization: it fixes which infinitesimal representations can be exponentiated to honest representations of the gauge group.

A representation descends exactly when the quotient kernel is invisible

Let

$$\rho:\widetilde G\longrightarrow \operatorname{GL}(V)$$

be a representation. To descend to $G_\Gamma$, there must be a representation $\overline\rho:G_\Gamma\to\operatorname{GL}(V)$ satisfying

$$\rho=\overline\rho\circ q_\Gamma.$$

If such a representation exists, then every $\gamma\in\Gamma$ maps to the identity coset, and hence

$$\rho(\gamma)=\overline\rho(e)=\mathbf 1_V.$$

Thus $\Gamma\subseteq\ker\rho$ is necessary. It is also sufficient: if $\Gamma$ lies in the kernel, define

$$\overline\rho([g])=\rho(g).$$

If $g'=g\gamma$ represents the same coset, then $\rho(g')=\rho(g)\rho(\gamma)=\rho(g)$, so the definition is independent of the representative. Therefore

$$\rho\text{ descends to }\widetilde G/\Gamma \quad\Longleftrightarrow\quad \Gamma\subseteq\ker\rho.$$

The same test applies to a dynamical matter multiplet and to an ordinary Wilson probe—the holonomy of a connection along a path, evaluated in a chosen finite-dimensional group representation. A Lie-algebra representation can be perfectly consistent infinitesimally and still fail this global exponentiation test.

The matter kernel gives a faithful action, not a complete theory

Now start with a candidate compact connected group $G$ and a specified list of dynamical matter representations

$$\rho_a:G\longrightarrow\operatorname{GL}(V_a), \qquad a\in I.$$

Their simultaneous action is the direct-sum representation $\rho_{\mathrm{mat}}=\bigoplus_{a\in I}\rho_a$. Its kernel is

$$K_{\mathrm{mat}} =\ker\rho_{\mathrm{mat}} =\bigcap_{a\in I}\ker\rho_a.$$

Because each representation kernel is a closed normal subgroup, $K_{\mathrm{mat}}$ is also closed and normal. The induced action of

$$G_{\mathrm{mat}}^{\mathrm{faithful}} =G/K_{\mathrm{mat}}$$

on $\bigoplus_a V_a$ is faithful: if a coset acts trivially, its representatives lie in every $\ker\rho_a$, so that coset is the identity. This elementary quotient need not preserve the Lie algebra of $G$. If an entire connected factor acts trivially on the matter, then $K_{\mathrm{mat}}$ has positive dimension and quotienting by it removes that factor. Only a discrete kernel can produce another connected global form with the same Lie algebra; for a connected group, such a normal subgroup is central.

To enumerate the same-algebra possibilities, pull the matter representations back to $\widetilde G$ and write them as $\widetilde\rho_a$. The largest central subgroup invisible to all of them is

$$\Gamma_{\mathrm{mat}} =Z(\widetilde G) \cap\bigcap_{a\in I}\ker\widetilde\rho_a.$$

A quotient $\widetilde G/\Gamma$ is compatible with the listed matter exactly when $\Gamma\subseteq\Gamma_{\mathrm{mat}}$. Thus a trivial $\Gamma_{\mathrm{mat}}$ leaves only the simply connected global form, while a nontrivial $\Gamma_{\mathrm{mat}}$ usually leaves several subgroups—and hence several global forms—compatible with the same matter. If the list is empty or every listed representation is trivial, the matter calculation merely says that matter supplies no information about the gauge group.

Dynamical matter can screen center-sensitive probes, and the subgroup left invisible by all included matter representations is described in Bhardwaj et al. 2024, arXiv v2, § 3.3.4, preprint pp. 48–49, Open PDF.

The notation $G_{\mathrm{mat}}^{\mathrm{faithful}}$ must be read literally. It is the faithful group acting on the specified matter multiplets after a candidate $G$ has been supplied. It is not, in general, a reconstruction of the gauge group from local matter. A factor under which every displayed matter field is neutral can still have gauge-boson dynamics. A central element invisible to all local fields can still be detected by an allowed Wilson representation or by bundle topology. The matter list constrains the global form; it need not select a unique full QFT. The distinction between local matter data and the additional line and bundle data is emphasized in Aharony, Seiberg, and Tachikawa 2013, arXiv v5, §§ 1–1.1, preprint pp. 1–3, Open PDF.

SU(2) and SO(3) separate the three questions

The standard example begins with

$$\begin{gathered} \mathfrak g=\mathfrak{su}(2), \\ \widetilde G=\operatorname{SU}(2), \\ Z(\widetilde G)=\{\mathbf 1,-\mathbf 1\}. \end{gathered}$$

There are two connected global forms:

$$\operatorname{SU}(2), \qquad \operatorname{SO}(3) \simeq \operatorname{SU}(2)/\{\mathbf 1,-\mathbf 1\}.$$

Finite-dimensional irreducible $\operatorname{SU}(2)$ representations are labeled by spin $j\in\tfrac12\mathbb Z_{\geq0}$. The nontrivial central element acts as

$$\rho_j(-\mathbf 1)=(-1)^{2j}\mathbf 1_{V_j}.$$

Consequently, $V_j$ descends to an $\operatorname{SO}(3)$ representation exactly when $j$ is an integer. A fundamental doublet with $j=\tfrac12$ forbids the quotient, whereas adjoint matter with $j=1$ permits it.

Suppose the candidate group is $\operatorname{SU}(2)$ and every dynamical matter multiplet has integer spin. Then $K_{\mathrm{mat}}\supseteq\{\mathbf1,-\mathbf1\}$; for a nontrivial adjoint multiplet it is exactly this center, so the faithful group acting on that matter is $\operatorname{SO}(3)$. This conclusion does not say that the full theory must have gauge group $\operatorname{SO}(3)$. Both an $\operatorname{SU}(2)$ theory and an $\operatorname{SO}(3)$ theory can have the same adjoint local fields. Their honest Wilson representations and their allowed bundles distinguish them.

The three questions should therefore be kept separate:

1. Which connected groups integrate the local Lie algebra?
2. Which of those groups are compatible with every included matter representation?
3. Which group, bundle sectors, and extended-operator data define the full theory?

The first two questions constrain the third; they do not replace it.

Bundles and Wilson probes retain the global information

For a principal $G$-bundle $P\to M$, a connection is locally represented by the same $\mathfrak g$-valued gauge potential for every global form with Lie algebra $\mathfrak g$. Globally, however, the choice of $G$ changes which bundles $P$ are admitted.

For $G=\operatorname{SO}(3)$, let $E=P\times_{\operatorname{SO}(3)}\mathbb R^3$ be the associated oriented rank-three vector bundle. The obstruction to lifting $P$ through

$$\operatorname{SU}(2)\longrightarrow\operatorname{SO}(3)$$

is its second Stiefel–Whitney class

$$w_2(P)\equiv w_2(E)\in H^2(M,\mathbb Z_2).$$

The bundle lifts to an $\operatorname{SU}(2)$ bundle if and only if this obstruction vanishes. Thus every $\operatorname{SU}(2)$ bundle projects to an $\operatorname{SO}(3)$ bundle with $w_2=0$, while an $\operatorname{SO}(3)$ path integral may include nonliftable $w_2\neq0$ sectors. This is not a complete bundle classification, and $w_2=0$ does not mean that a bundle is topologically trivial.

Wilson probes see a complementary distinction. An $\operatorname{SU}(2)$ Wilson line may use any spin-$j$ representation. An $\operatorname{SO}(3)$ Wilson line may use only integer-spin representations, because half-integer representations do not descend. Which Wilson probes and bundle sectors are allowed therefore depends on the global group even when the local Lagrangian uses only adjoint fields. The bundle and Wilson consequences are developed in Aharony, Seiberg, and Tachikawa 2013, arXiv v5, §§ 1–1.3, preprint pp. 1–6, Open PDF and reviewed in Bhardwaj et al. 2024, arXiv v2, §§ 3.3.3–3.3.4, preprint pp. 46–49, Open PDF.

The comparison below keeps the local algebra fixed and makes the extra data explicit. It assumes four-dimensional theories on oriented spin manifolds, uses reduced center charges $(e,m)\in\mathbb Z_2^2$, and assumes only integer-spin dynamical matter. A genuine line is one that needs no attached topological surface. The entries $L=\langle(e,m)\rangle$ give the complete maximal mutually local subgroup of reduced genuine charges in the standard three-theory example. The $+$/$-$ label is therefore additional discrete-theta and line-spectrum data, not another connected global form.

Three four-dimensional spin theories with local algebra su(2) and integer-spin dynamical matter

Theory	Connected global group	Honest matter and pure Wilson representations	Admissible bundle sectors	Reduced genuine-line subgroup	Discrete-theta datum
SU(2)	Simply connected SU(2)	Dynamical matter is assumed integer-spin; pure Wilson probes allow every j in one-half times the nonnegative integers, including half-integer representations	SU(2) bundles; the associated SO(3) lift-obstruction class has w2 = 0	L = ⟨(1,0)⟩: the odd electric class is genuine	No SO(3) Pontryagin-square choice
SO(3)+	SO(3) = SU(2)/{+1,−1}	Integer-spin representations only; a half-integer pure Wilson line does not descend	SO(3) bundles, including allowed nonliftable sectors with w2 ≠ 0	L = ⟨(0,1)⟩: the odd pure-magnetic class is genuine	p = 0 in the displayed spin-manifold normalization
SO(3)−	SO(3) = SU(2)/{+1,−1}	Integer-spin representations only; a half-integer pure Wilson line does not descend	The same SO(3) bundle classes as SO(3)+	L = ⟨(1,1)⟩: the odd dyonic class is genuine; its electric label is not a standalone Wilson representation	p = 1, weighting a closed bundle by exp[iπ∫P(w2)/2]

This table does not claim that the global-form page alone derives the last two columns. The representation and bundle columns follow from the group choice; the genuine-line and discrete-theta columns are independent four-dimensional theory data developed on Genuine Line Spectra, Discrete Theta Data, and Theory Specification. On non-spin manifolds the displayed $p\in\mathbb Z_2$ normalization does not apply unchanged. Boundary conditions can also remove bundle sectors or alter which defects may terminate.

This also explains a frequently quoted but limited statement: after matching the local action and parameters, correlation functions of local operators on $\mathbb R^4$ can agree for theories that differ by global form. Such agreement is not equivalence of the full theories. Extended probes or a spacetime supporting nontrivial bundles can distinguish them Aharony, Seiberg, and Tachikawa 2013, arXiv v5, § 1, preprint p. 1, Open PDF.

A bounded Yang–Mills system in three descriptions

Consider Yang–Mills theory with algebra $\mathfrak{su}(2)$ and adjoint local matter on

$$\mathbb R\times\Sigma, \qquad \Sigma=[r_-,r_+]\times S^2.$$

Boundary conditions are part of the system. Assume first that they allow the chosen bundle on each boundary component and that the redundant gauge group $\mathcal G_0(P)$ consists of bundle automorphisms restricting to the identity at the boundary.

Orbit description. The reduced configuration space has the form

$$\mathcal C_G(\Sigma) =\coprod_{[P]\in\mathcal P_G(\Sigma)} \frac{\mathcal A(P)}{\mathcal G_0(P)},$$

where $\mathcal P_G(\Sigma)$ is the set of bundle classes permitted by the boundary conditions. Since $H^2(\Sigma,\mathbb Z_2)\simeq\mathbb Z_2$, an $\operatorname{SO}(3)$ choice can admit a sector with nonzero $w_2(P)$ if its boundary bundle data are allowed. That sector has no $\operatorname{SU}(2)$ lift. Boundary conditions that require a boundary trivialization can remove it, which is why the allowed set $\mathcal P_G(\Sigma)$ was stated explicitly. Within each fixed bundle, Gauge Orbits, Gauss Constraints, and Stabilizers supplies the quotient and stabilizer analysis.

Charge description. Now restrict to the trivial-bundle sector and suppose the boundary conditions allow a constant boundary subgroup isomorphic to $G$. Also suppose its Hamiltonian generators are finite, integrable, and not constant on phase space; only under these conditions is that subgroup a physical boundary symmetry rather than another redundancy. An external Wilson endpoint must transform in an honest representation of this global boundary group. Provided the boundary condition admits termination of that Wilson line, a spin-$\tfrac12$ endpoint is representation-theoretically permitted when $G=\operatorname{SU}(2)$, but not when $G=\operatorname{SO}(3)$. The infinitesimal $\mathfrak{su}(2)$ charge algebra is the same in both cases; whether $-\mathbf1$ in the $\operatorname{SU}(2)$ universal cover acts trivially upon exponentiation supplies the global test. If the boundary conditions eliminate the residual group, there is no such boundary-charge diagnostic.

Gauge-fixed description. In a local trivialization near the trivial connection, the gauge-fixed potential, ghosts, covariant derivatives, and perturbative vertices are all organized by the same algebra $\mathfrak{su}(2)$. Local perturbation theory can therefore look identical for $\operatorname{SU}(2)$ and $\operatorname{SO}(3)$. A gauge condition does not recover the quotient subgroup, decide which bundle sectors are summed over, or turn a half-integer Lie-algebra multiplet into an $\operatorname{SO}(3)$ representation. Global gauge-fixing domains can still depend on the bundle, so this comparison is deliberately restricted to a shared local perturbative patch.

Together, the three descriptions locate the same distinction in different places: the orbit picture sees the set of bundles, the charge picture sees which representations exponentiate at the boundary, and the local gauge-fixed picture does not see either datum without additional global input.

Scope limits

The quotient formula used here is not a classification of every ordinary gauge group.

• Torus factors require lattices. For Abelian factors such as $U(1)$, the character and cocharacter lattices carry global information. The finite central-quotient discussion for compact semisimple groups cannot simply be copied unchanged.
• Disconnected groups require component data. The identity component, the component group, and its action must all be specified.
• Higher groups require additional structure. Mixing ordinary and higher-form gauge data is outside the present group-and-representation framework.
• The global group is still not the whole four-dimensional theory. A complete choice of genuine electric–magnetic lines and discrete theta data can distinguish theories with the same Lie algebra and the same global group. That refinement belongs to the next page.

Common pitfalls

Identifying the Lie algebra with the gauge group. The algebra controls local infinitesimal transformations. Central quotients leave it unchanged while changing exponentiation, representations, and bundles.

Calling $G/K_{\mathrm{mat}}$ the uniquely reconstructed gauge group. This quotient is only the faithful action on the specified matter. Gauge dynamics, bundle sectors, Wilson probes, and other extended data can retain information that the matter action loses.

Reading $w_2(P)=0$ as “the bundle is trivial.” Vanishing $w_2$ removes the obstruction to an $\operatorname{SU}(2)$ lift. It does not rule out other topological data of the lifted bundle.

Promoting local agreement to full equivalence. Matching perturbative fields or local correlators on $\mathbb R^4$ does not match extended operators or path integrals over nontrivial bundle sectors.

Check your understanding

1. A candidate $\operatorname{SU}(2)$ theory contains matter of spins $1$ and $2$. Find $K_{\mathrm{mat}}$ and the faithful matter group. Repeat after a spin-$\tfrac32$ multiplet is added. Which connected global forms are compatible in each case?
2. On $\Sigma=[r_-,r_+]\times S^2$, suppose the boundary conditions allow a nonzero $w_2$ sector and also allow a physical constant boundary group in the trivial sector. Give one orbit-space observation and one boundary-charge observation that distinguish $\operatorname{SU}(2)$ from $\operatorname{SO}(3)$. Explain why matching local gauge-fixed propagators does not contradict either observation.

Solution

For integer $j$, the central element $-\mathbf1$ acts as $(-1)^{2j}=1$. With spins $1$ and $2$, both representations therefore have $\{\mathbf1,-\mathbf1\}$ in their kernels. Their nontrivial irreducible actions have no larger common kernel, so

$$K_{\mathrm{mat}}=\{\mathbf1,-\mathbf1\}, \qquad G_{\mathrm{mat}}^{\mathrm{faithful}}\simeq\operatorname{SO}(3).$$

Both $\Gamma=\{\mathbf1\}$ and $\Gamma=\{\mathbf1,-\mathbf1\}$ are compatible, so the matter permits both $\operatorname{SU}(2)$ and $\operatorname{SO}(3)$ even though its faithful action is $\operatorname{SO}(3)$. For $j=\tfrac32$, $(-1)^{2j}=-1$, so adding that multiplet removes the nontrivial center element from the common kernel. Then $K_{\mathrm{mat}}=\{\mathbf1\}$, the faithful matter group is $\operatorname{SU}(2)$, and only the $\operatorname{SU}(2)$ global form remains compatible with the full matter list.

In the orbit description, an allowed $\operatorname{SO}(3)$ bundle with $w_2(P)\neq0$ contributes a component that has no $\operatorname{SU}(2)$ lift. In the trivial sector with the stated physical boundary symmetry, an $\operatorname{SU}(2)$ boundary can support an external spin-$\tfrac12$ Wilson endpoint, whereas an $\operatorname{SO}(3)$ boundary cannot. Local gauge-fixed propagators are computed from the common $\mathfrak{su}(2)$ action in a shared trivializing patch. They contain neither the choice of bundle sum nor the global exponentiation condition, so their agreement is expected.

Where global form leads

Genuine Line Spectra, Discrete Theta Data, and Theory Specification adds the electric–magnetic line choices that the group alone does not fix. The Global Form of the Standard Model Gauge Group owns the model-specific quotient and charge analysis.

Principal-Bundle Sectors and Large Gauge Transformations owns the theorem-level classification of bundle sectors and disconnected gauge transformations.

Doplicher–Roberts Compact-Gauge Reconstruction develops a theorem-level reconstruction of a compact gauge group from superselection data under substantially stronger hypotheses; the elementary matter-kernel construction on this page is not that theorem.

References

• Aharony, Ofer, Nathan Seiberg, and Yuji Tachikawa. “Reading between the Lines of Four-Dimensional Gauge Theories.” Journal of High Energy Physics 08 (2013): 115. DOI. Open PDF, arXiv:1305.0318v5
• Bhardwaj, Lakshya, Lea E. Bottini, Ludovic Fraser-Taliente, Liam Gladden, Dewi S. W. Gould, Arthur Platschorre, and Hannah Tillim. “Lectures on Generalized Symmetries.” Physics Reports 1051 (2024): 1–87. DOI. Open PDF, arXiv:2307.07547v2