Matter and the global form of a gauge group

A gauge algebra specifies infinitesimal transformations. To specify the group acting on the fields, one must also choose a global form. For a single compact factor with algebra $\mathfrak{su}(N)$, the possible connected groups are $SU(N)/\mathbb Z_d$ with $d\mid N$. A proposed matter representation must be trivial on the particular subgroup being divided out. The question is therefore concrete: which central elements act trivially on every matter multiplet?

Open the matter and global-form workbench. It accepts the Dynkin labels of several multiplets, computes their common central kernel, and lists every compatible quotient. The calculation concerns finite-dimensional continuous complex representations in characteristic zero. It does not choose a spectrum of magnetic or dyonic lines, a discrete theta parameter, or a quantum-consistent matter theory.

1. Read the central character from a highest weight

Number the simple roots of $A_{N-1}$ consecutively, and write a highest weight as

$$\lambda=\sum_{i=1}^{N-1}a_i\omega_i,\qquad a_i\in\mathbb Z_{\geq0}.$$

The fundamental representation is $(1,0,\ldots,0)$. Fix the central generator

$$z=e^{2\pi i/N}\mathbf1_N.$$

The representation with highest weight $\omega_k$ is the $k$th exterior power of the fundamental, so $z$ acts on it by $e^{2\pi\mathrm{i}k/N}$. Central characters multiply under tensor products. The highest-weight constituent of a product has the sum of the highest weights, which gives

$$\rho_\lambda(z)=\exp\!\left(\frac{2\pi\mathrm{i}}{N}q(\lambda)\right)\mathbf1_{V_\lambda}, \qquad q(\lambda)=\sum_{i=1}^{N-1}i\,a_i\pmod N.$$

This is the $N$-ality of the representation. Equivalently, it is the number of boxes of its Young diagram modulo $N$. Removing a full column of height $N$ changes that number by $N$ and leaves the $SU(N)$ representation unchanged. The weight-lattice description of the center is developed in Conrad’s Centers of Simply Connected Semisimple Groups, pp. 1–2.

2. Test a proposed quotient

For a divisor $d$ of $N$, fix the embedded subgroup

$$\mathbb Z_d=\langle z^{N/d}\rangle\subset Z(SU(N)).$$

Its generator acts on the multiplet by $e^{2\pi i q/d}$. Thus

$$R_\lambda\text{ descends to }SU(N)/\mathbb Z_d \quad\Longleftrightarrow\quad q(\lambda)=0\pmod d.$$

For example, the fundamental has charge one, so it excludes every nontrivial quotient of this single factor. The adjoint has charge zero and descends to every central quotient. A failure of descent does not invalidate the Lie-algebra representation or its dimension: it identifies an incompatibility with the selected global group.

3. Intersect the matter kernels

Let the matter multiplets have charges $q_1,\ldots,q_m$. A central element $z^k$ acts trivially on all of them exactly when $kq_s$ is divisible by $N$ for every $s$. Set

$$h=\gcd(N,q_1,\ldots,q_m).$$

The common kernel is generated by $z^{N/h}$ and has order $h$:

$$K=\bigcap_{s=1}^{m}\ker\!\left(\rho_s|_{Z(SU(N))}\right) =\langle z^{N/h}\rangle\cong\mathbb Z_h.$$

To check the statement directly, every multiple $k=\ell N/h$ satisfies the divisibility conditions. Conversely, Bézout’s identity expresses $h$ as an integer combination of $N$ and the $q_s$; multiplying by $k$ shows that $N\mid kh$. Hence $N/h\mid k$. The compatible quotient orders are precisely the divisors of $h$. The same greatest-common-divisor criterion for the matter-invariant center appears in Cohen, Center symmetry and area laws, property (i) and its derivation.

The empty matter list is useful: the gauge bosons transform in the adjoint and impose no additional restriction on the center. The workbench then uses $h=N$.

4. Worked examples

An antisymmetric tensor of SU(4)

The two-index antisymmetric representation has labels $(0,1,0)$, dimension six, and charge two. Therefore

$$h=\gcd(4,2)=2.$$

Both $SU(4)$ and $SU(4)/\mathbb Z_2$ admit this matter. $PSU(4)=SU(4)/\mathbb Z_4$ does not. In the order-two quotient, the generator $z^2$ acts as $e^{2\pi i}=1$; in the order-four quotient, the generator $z$ acts as $-1$.

Reproduce the SU(4) example. Changing the quotient order to four displays the obstruction while retaining the input.

Two individually nontrivial kernels can have trivial intersection

In $SU(6)$, the antisymmetric representations $\Lambda^2\mathbf6$ and $\Lambda^3\mathbf6$ have charges two and three. Their individual central kernels have orders two and three. Together,

$$\gcd(6,2,3)=1,$$

so they allow no nontrivial central quotient. Looking at one multiplet at a time is insufficient. Replacing the second multiplet by $\Lambda^4\mathbf6$, with charge four, instead leaves a common $\mathbb Z_2$ kernel.

Adjoint matter and N=4 super-Yang–Mills

Every local matter multiplet in the usual $\mathcal N=4$ super-Yang–Mills construction is adjoint-valued. The descent test therefore permits every quotient $SU(N)/\mathbb Z_d$ with $d\mid N$. This does not select a unique quantum gauge theory. The allowed genuine line operators and discrete choices contain further information. Aharony, Seiberg, and Tachikawa treat these choices explicitly in §§1.1 and 2.3.

5. What the result establishes

The largest compatible quotient is the quotient by the common central kernel. Matter alone does not require choosing it: any subgroup of that kernel may be divided out. Nor should the common central kernel automatically be called the entire kernel of the matter action; a list of trivial representations, for example, is trivial on the entire group.

The group-theoretic topology of the chosen quotient is

$$\pi_1\!\left(SU(N)/\mathbb Z_d\right)\cong\mathbb Z_d, \qquad Z\!\left(SU(N)/\mathbb Z_d\right)\cong\mathbb Z_{N/d}.$$

The first statement follows from the simply connected cover and its deck group. These are properties of the group manifold. Their interpretation in a gauge theory requires the relevant bundles, boundary conditions, spacetime, and observable category.

For a product gauge algebra, a central subgroup can be embedded diagonally across factors. Such quotients require the joint central characters, including a declared normalization for abelian charges. The single-factor workbench does not resolve that problem. Likewise, it does not test perturbative or global gauge anomalies, compute a line spectrum, or infer confinement.

Reproduce and cite

The workbench supports $2\leq N\leq12$, at most sixteen matter multiplets, and Dynkin labels from zero to thirty. It uses exact integers, retains every multiplet’s charge and descent result, and exports the input, algorithm version, source locators, and the convention profile su-n-dynkin-center-v1. The accompanying dimension is evaluated by the type-A Weyl product.

Preserve the JSON when using a result in research. A shared URL restores inputs using the currently published implementation. Convention profiles specify the notation; reproducibility explains the version record. Independent human review is tracked separately from source checks and automated verification.