Product centers, charge lattices, and quotient descent

A central quotient is specified by an embedded subgroup, not only by an abstract name such as $\mathbb Z_2$. In a product group, a central transformation in one factor can be canceled by transformations in the others. Checking each factor separately therefore loses physically relevant global forms.

Open the product gauge-group workbench. This method applies to

$$\widetilde G=\prod_{i=1}^{s}SU(N_i)\times U(1), \qquad s=1\ \text{or}\ 2,$$

with finite-dimensional continuous complex matter representations. The circle has period $2\pi$. This is a compact covering group for the proposed quotient; it is not the universal cover, because the circle already has fundamental group $\mathbb Z$.

1. The input is a character of the full center

Write a multiplet as

$$R_\alpha= \bigotimes_{i=1}^{s}V_{\lambda_{\alpha i}}\otimes\mathbb C_{q_\alpha}, \qquad q_\alpha\in\mathbb Z.$$

Here $\mathbb C_q$ means the one-dimensional circle representation $e^{i\vartheta}\mapsto e^{iq\vartheta}$. If the highest weight of the $i$th factor has Dynkin labels $a_{\alpha i,j}$, its center charge is

$$t_{\alpha i}=\sum_{j=1}^{N_i-1}j\,a_{\alpha i,j}\pmod{N_i}.$$

Choose $z_i=e^{2\pi i/N_i}\mathbf1_{N_i}$. A central element is represented by

$$c=(z_1^{k_1},\ldots,z_s^{k_s},e^{2\pi i\theta}), \qquad k_i\in\mathbb Z/N_i\mathbb Z,\quad\theta\in\mathbb R/\mathbb Z.$$

Its action is the scalar operator

$$\rho_\alpha(c)= \exp\!\left[ 2\pi i\left( \sum_i\frac{t_{\alpha i}k_i}{N_i}+q_\alpha\theta \right) \right]\mathbf1_{R_\alpha}.$$

The $SU(N)$ center formula is the type-$A$ specialization of the weight/root-lattice description in Conrad’s center notes, pp. 1–2. Tensoring the factor representations multiplies the central phases, giving the displayed sum.

For a finite subgroup $\Gamma\subset Z(\widetilde G)$,

$$R_\alpha\text{ descends to }\widetilde G/\Gamma \quad\Longleftrightarrow\quad \sum_i\frac{t_{\alpha i}k_i}{N_i}+q_\alpha\theta\in\mathbb Z \quad\text{for every }c\in\Gamma.$$

It is enough to test a generating set. The implementation additionally evaluates every enumerated element of $\Gamma$, checking that the two tests agree.

2. Why the embedding matters

Consider $SU(2)\times SU(2)\times U(1)$ with matter $(\mathbf2,\mathbf2)_0$. The central action of $(\epsilon_1,\epsilon_2,e^{i\vartheta})$ is $\epsilon_1\epsilon_2$. The diagonal element $(-\mathbf1,-\mathbf1,1)$ acts trivially, while either individual $SU(2)$ center generator acts nontrivially.

Thus the diagonal $\mathbb Z_2$ quotient is compatible with this multiplet even though quotienting either factor separately is obstructed. In the workbench the diagonal generator has center exponents $(1,1)$ and zero circle phase. Its coordinates remain in the result and export.

The abstract invariant factors of $\Gamma$ summarize its group structure. They do not replace these coordinates. Even two subgroups with the same order and invariant factors can define different representation constraints or topology.

3. Computing the entire common central kernel

The matter-invariant center is

$$K=\bigcap_\alpha\ker\!\left(\rho_\alpha|_{Z(\widetilde G)}\right).$$

This is a central kernel. It is not necessarily the full kernel of the matter action: if every multiplet is trivial on a nonabelian factor, that entire factor acts trivially, not merely its center. The workbench keeps the stated covering group fixed and reports the central calculation.

If at least one field has nonzero circle charge, choose a row with $q_*\ne0$. For any fixed finite-center tuple $(k_i)$, its equation permits precisely $|q_*|$ circle phases:

$$\theta= \frac{\ell-\operatorname{sgn}(q_*)\displaystyle\sum_i t_{*i}k_i/N_i} {|q_*|} \pmod1, \qquad \ell=0,\ldots,|q_*|-1.$$

Each candidate is tested against every remaining row. Conversely, any element of $K$ satisfies the chosen row and therefore appears among these candidates. This proves completeness of the finite search.

A common denominator $L|q_*|$, where $L=\operatorname{lcm}(N_1,\ldots,N_s)$, suffices. Every phase test becomes integer divisibility. There is no floating-point evaluation of roots of unity and no numerical tolerance around one. Choosing a field with minimal nonzero $|q_*|$ reduces the search without changing the answer.

If all charges vanish, the circle is unconstrained:

$$K=K_{\mathrm{ss}}\times U(1),$$

where $K_{\mathrm{ss}}$ is obtained by checking the finite-center tuples. An empty matter list has this behavior too. The result explicitly marks a one-dimensional kernel; it does not present a finite sample of circle phases as the whole kernel. The finite component reported in this case is the direct factor $K_{\mathrm{ss}}$.

Repeated copies of an existing representation impose the same central constraint. They can nevertheless matter for other physics, including anomalies and coupling multiplicities.

4. Finite abelian structure and quotient selection

For a finite enumerated subgroup, the engine finds its abstract invariant-factor form

$$K\cong\mathbb Z_{d_1}\times\cdots\times\mathbb Z_{d_r}, \qquad d_1\mid d_2\mid\cdots\mid d_r.$$

One way to recover these factors is to count the elements annihilated by powers of each prime. If the $p$-primary factors are $\mathbb Z_{p^{e_j}}$, then

$$\log_p\left|\{x:p^k x=0\}\right| =\sum_j\min(k,e_j).$$

Successive differences count the factors with $e_j\ge k$. Combining the prime-power factors gives the invariant factors. The implementation checks that their product equals the enumerated order. It also constructs an embedded generating set and verifies that the generators recover the complete kernel.

When $K$ is finite and cyclic, its subgroups are uniquely determined by their orders, so the workbench lists every compatible quotient and a generator for its embedding. For a noncyclic or continuous kernel, it instead asks for the actual finite subgroup through its generators. It does not confuse a divisor list with a classification of all embedded subgroups.

The selected $\Gamma$ is generated independently from the user’s coordinates, whether or not the matter descends. An obstructed choice remains a valid group construction and is reported with the offending fields and their exact phases.

5. The fundamental group: keep the original circle

The universal cover of $\widetilde G/\Gamma$ is

$$\widehat G=\prod_i SU(N_i)\times\mathbb R.$$

Let $\Lambda$ be the inverse image of $\Gamma$ in $Z(\prod_iSU(N_i))\times\mathbb R$. Then $\pi_1(\widetilde G/\Gamma)\cong\Lambda$. The group $\Lambda$ is discrete abelian and contains the integer winding lattice of the original circle.

Its projection to $\mathbb R$ is an infinite cyclic lattice. Its torsion subgroup is exactly

$$T=\Gamma\cap\left(\prod_i Z(SU(N_i))\times\{1\}\right),$$

because a finite-order element in the real covering direction must have coordinate zero. A finitely generated abelian group with this torsion and one free generator is therefore

$$\boxed{\; \pi_1(\widetilde G/\Gamma)\cong\mathbb Z\oplus T. \;}$$

This is an abstract splitting; the engine does not select a canonical free generator. It reports the invariant factors of $T$ and the order of the circle projection of $\Gamma$. The underlying covering-space exact sequence is developed in Conrad’s fundamental-group notes, pp. 1–2.

For example,

$$\begin{aligned} \Gamma_1&=\langle(-\mathbf1,1)\rangle &&\Longrightarrow& \pi_1\!\left((SU(2)\times U(1))/\Gamma_1\right) &\cong\mathbb Z\oplus\mathbb Z_2,\\ \Gamma_2&=\langle(-\mathbf1,-1)\rangle &&\Longrightarrow& \pi_1\!\left((SU(2)\times U(1))/\Gamma_2\right) &\cong\mathbb Z. \end{aligned}$$

Both quotient subgroups are abstractly $\mathbb Z_2$. The first group is $SO(3)\times U(1)$; the second is $U(2)$. Replacing $\pi_1$ by $\Gamma$ would be wrong in both cases, because the starting group is not simply connected.

6. Supported domain and reproducibility

The current exact implementation supports one or two $SU(N_i)$ factors with $2\le N_i\le8$, one compact circle, at most 16 multiplets, Dynkin labels from 0 to 12, and integer charges $|q|\le24$. At most three generators specify $\Gamma$, with circle phases whose input denominators are at most 60. A selected subgroup that would exceed 4,096 elements produces an explicit resource-limit error, without a partial mathematical result.

A nonzero matter charge bounds the common-kernel search by

$$\left(\prod_iN_i\right)|q_*|\le64\cdot24=1536.$$

Exact type-$A$ Weyl products determine each factor dimension; the multiplet dimension is their product. The JSON export preserves the factor sizes, complete highest weights, integer charges, embedded generators, exact rational phases, result, sources, and product-su-u1-center-v1 convention.

This calculation establishes group-representation compatibility and the topology of the selected connected group. Magnetic and dyonic line choices, theta data, anomaly cancellation, spacetime structure, symmetry breaking, and vacuum dynamics require further input. For why global form alone need not finish the specification of a quantum gauge theory, see Aharony–Seiberg–Tachikawa, §1.1 and §2.

The Standard Model guide works through a cyclic kernel and its four quotients. The block-branching reference explains how a subgroup embedding determines correlated nonabelian and circle charges.