Gauge groups, centers, and representations

A theorem about $SU(N)$ is not automatically a theorem about every compact simple gauge group. Even two groups with the same Lie algebra can admit different probes. Keep the group and the representation in the statement, not in a footnote.

Prerequisites: physical states and matrices used in elementary gauge theory.

After this page: distinguish a group from its Lie algebra, test whether a representation descends to a quotient, and identify when a center-based criterion applies.

Local algebra, global group

The Lie algebra describes infinitesimal transformations and the commutator in the local curvature. The group also specifies how finite transformations fit together globally. A representation $R:G\to U(V)$ tells us how group elements act on a vector space $V$; its dimension is not the spacetime dimension.

For $SU(N)$, the fundamental representation acts on $\mathbb C^N$. Its Lie algebra has dimension $N^2-1$; its rank, the dimension of a maximal commuting torus, is $N-1$. The adjoint representation acts on the algebra by $X\mapsto gXg^{-1}$. Thus “fundamental,” “adjoint,” “rank” and “number of generators” name different things.

The official problem, section 4, quantifies over every compact simple gauge group. In the physical convention used here, a compact connected group with simple Lie algebra can have a finite center. A quotient by an allowed central subgroup retains the local algebra but changes global data.

A center calculation

The center consists of elements commuting with every group element. In the defining representation of $SU(N)$ its elements are

$$z_k=e^{2\pi i k/N}I,\qquad k=0,\ldots,N-1.$$

They are unitary and have determinant $(e^{2\pi i k/N})^N=1$. Conjugation by $z_k$ leaves every matrix $X$ unchanged, so the adjoint representation is center-neutral. The fundamental vector instead acquires a phase.

For $SU(N)$, the phase with which the generator of the center acts on an irreducible representation can be written $e^{2\pi i q/N}$. The integer $q$ modulo $N$ is its $N$-ality. Tensor products add these charges modulo $N$. This is a statement about how a representation transforms, not a guarantee about its long-distance potential.

Worked example: why the doublet does not descend

The groups $SU(2)$ and $SO(3)\simeq SU(2)/\{I,-I\}$ share the Lie algebra $\mathfrak{su}(2)$. In a quotient, $g$ and $-g$ represent the same group element. A representation descends only if it gives them the same matrix.

The $SU(2)$ doublet has $R(-I)=-I$, so it fails that test. The adjoint has $R(-I)X=(-I)X(-I)^{-1}=X$, so it passes. Equivalently, the spin-$j$ representation has $R(-I)=(-1)^{2j}I$: integer-spin representations descend, half-integer-spin representations do not.

Global group	Lie algebra	Center	Example of an allowed probe
$SU(2)$	$\mathfrak{su}(2)$	$\mathbb Z_2$	Fundamental doublet Wilson loop
$SO(3)$	$\mathfrak{su}(2)$	Trivial	Three-dimensional vector/adjoint Wilson loop
$SU(N)$	$\mathfrak{su}(N)$	$\mathbb Z_N$	Fundamental Wilson loop
$PSU(N)=SU(N)/\mathbb Z_N$	$\mathfrak{su}(N)$	Trivial	Center-neutral representations

Tong’s Gauge Theory, section 2.6.2, printed pages 92–95 explains this distinction through electric and magnetic line operators. We use only the elementary representation test here; discrete theta data and the full classification of line operators require further specification.

Why this matters for a theorem

Chatterjee’s v5, Theorem 2.2 requires an irreducible representation acting nontrivially on the center. A center-neutral adjoint loop cannot simply be substituted. A centerless group has no representation meeting that particular condition.

That restriction does not remove centerless compact simple groups from the Yang-Mills target. It limits the applicability of this criterion. Nor does an adjoint field’s center neutrality make it a gauge-invariant local observable: it still transforms under noncentral gauge transformations.

Screening also depends on available dynamical matter. An external probe is not the same as adding a propagating quark field to the action. Use the confinement and screening guide to keep those physical questions separate.

Exercise: a neutral tensor product

In $SU(3)$ the fundamental has center charge 1 and its conjugate has charge $-1$ modulo 3. What is the charge of their tensor product? Does center neutrality alone imply that every vector in it is gauge invariant?

Solution: the total charge is 0. The tensor product contains both a singlet and an adjoint component; only the singlet is invariant under the full group. The center test is necessary for descent to $PSU(3)$ but is weaker than invariance under all gauge transformations.

Return to physical states or use the notation crosswalk before comparing group-normalized actions.