Global gauge structure · research workbench II

Product gauge groups & matter

Find the common central kernel of matter, then test an explicitly generated quotient of a product gauge group.

Scope: One or two SU(N) factors, 2≤N≤8, and one compact U(1) with period 2π. At most 16 multiplets; Dynkin labels 0–12; integer U(1) charges −24…24; at most 3 quotient generators with phase denominators ≤60; generated subgroup order ≤4096.

Load a worked setup

Exact resultMatter descends

\begin{aligned}G&=(SU(3)\times SU(2)\times U(1))/\Gamma\\K&\cong \mathbb Z_{6}\\\Gamma&\cong \mathbb Z_{6}\\\pi_1(G)&\cong \mathbb Z\end{aligned}

Common central kernel: $\mathbb Z_{6}$
Selected subgroup order: 6
Fundamental group of the quotient: $\mathbb Z$
U(1) projection order of Γ: 6

The U(1) period is 2π and every charge Q is integral. For the Standard Model examples, Q = 6Y. A generator written k₁,k₂ | θ means the central element (ζₙ₁^k₁, ζₙ₂^k₂, exp(2πiθ)).

Matter representations and the selected quotient
Multiplet	SU(3) labels	SU(2) labels	Q	Complex dimension	Descent
Q	$(1,0)$	$(1)$	$1$	$6$	Allowed
u_c	$(0,1)$	$(0)$	$-4$	$3$	Allowed
d_c	$(0,1)$	$(0)$	$2$	$3$	Allowed
L	$(0,0)$	$(1)$	$-3$	$2$	Allowed
e_c	$(0,0)$	$(0)$	$6$	$1$	Allowed
H	$(0,0)$	$(1)$	$3$	$2$	Allowed

A multiplet descends precisely when every generator phase is zero
Multiplet	Phase for γ1 (mod 1)
Q	$0$ trivial
u_c	$0$ trivial
d_c	$0$ trivial
L	$0$ trivial
e_c	$0$ trivial
H	$0$ trivial

Every compatible quotient of this common kernel

All subgroups of the finite cyclic common kernel.

1

\mathbb Z_{2}

\mathbb Z_{3}

\mathbb Z_{6}

Embedded generators and topology

Concrete subgroup generators
Subgroup	Generator: center exponents \| U(1) turns	Element order
Common kernel	`1,1 \| 1/6`	6
Selected Γ	`1,1 \| 1/6`	6

One generating set; this is not an identification of each generator with a particular invariant factor.

Lift the selected subgroup to the universal cover (product SU factors) × R. Its torsion is exactly the subgroup at U(1) phase zero; the real projection is an infinite cyclic lattice.

The JSON export contains the complete finite element lists. If the common kernel is continuous, its additional U(1) factor is stated explicitly.

Kinematic matter descent and topology of the selected connected group. This does not choose line operators, theta parameters, anomaly cancellation, a vacuum, or a complete physical theory.

✓ integer charges✓ exact dimensions✓ kernel generator reconstruction✓ invariant factor orders✓ generator and subgroup descent✓ quotient projection sequence

Save calculation downloads a named JSON file. Loading restores its inputs and recomputes them with the current engine; no account or upload is needed.

Method & interpretation

Solve the central phase conditions exactly

A central element acts on each multiplet by a rational phase. Intersect these conditions to obtain the common kernel, and test every generator of the selected quotient. The fundamental group is computed using the universal cover.

Read the worked guide · Mathematical method

Sources, conventions, and versions

Exact computations run in this browser. Independent human expert review is pending. Calculation JSON preserves inputs, sources, conventions, and the mathematical engine version.

Brian Conrad, Centers of Simply Connected Semisimple Groupspp. 1–2: the type-A center and weight/root lattices
David Tong, Line Operators in the Standard Model§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices
Brian Conrad, Fundamental Groups of Lie Groupspp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroup
Aharony, Seiberg and Tachikawa, Reading between the lines of four-dimensional gauge theories§1.1 and §2: global form and additional magnetic/dyonic line-operator choices

Convention: product-su-u1-center-v1
Algorithm: product-gauge-1.0.0

Convention profiles · Reproduce and cite

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