$SU(3)\times SU(2)\times U(1)$Standard Model compact covering product

The compact product SU(3) × SU(2) × U(1), with U(1) period 2π and integer hypercharge Q=6Y. It is a covering group of the three nontrivial specified quotients, but it is not the universal cover.

• The Algebra of Grand Unified TheoriesSection 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly.Open source
• Tong · Line operators in the Standard Model§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choicesOpen source
• Brian Conrad, Fundamental Groups of Lie Groupspp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroupOpen source

12

• The Algebra of Grand Unified TheoriesSection 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly.Open source
• Tong · Line operators in the Standard Model§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choicesOpen source
• Brian Conrad, Fundamental Groups of Lie Groupspp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroupOpen source

4

• The Algebra of Grand Unified TheoriesSection 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly.Open source
• Tong · Line operators in the Standard Model§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choicesOpen source
• Brian Conrad, Fundamental Groups of Lie Groupspp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroupOpen source

Yes

• The Algebra of Grand Unified TheoriesSection 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly.Open source
• Tong · Line operators in the Standard Model§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choicesOpen source
• Brian Conrad, Fundamental Groups of Lie Groupspp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroupOpen source

Yes

• The Algebra of Grand Unified TheoriesSection 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly.Open source
• Tong · Line operators in the Standard Model§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choicesOpen source
• Brian Conrad, Fundamental Groups of Lie Groupspp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroupOpen source

Z

• The Algebra of Grand Unified TheoriesSection 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly.Open source
• Tong · Line operators in the Standard Model§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choicesOpen source
• Brian Conrad, Fundamental Groups of Lie Groupspp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroupOpen source

1

• The Algebra of Grand Unified TheoriesSection 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly.Open source
• Tong · Line operators in the Standard Model§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choicesOpen source
• Brian Conrad, Fundamental Groups of Lie Groupspp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroupOpen source

U(1) × Z_6

• The Algebra of Grand Unified TheoriesSection 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly.Open source
• Tong · Line operators in the Standard Model§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choicesOpen source
• Brian Conrad, Fundamental Groups of Lie Groupspp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroupOpen source

A multiplet with color triality t3, SU(2) center charge t2, and integer Q has γ-phase t3/3+t2/2+Q/6 modulo one. Descent to this group requires γ^(6/1) to act trivially. The standard left-handed fermions and Higgs satisfy the condition.

• The Algebra of Grand Unified TheoriesSection 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly.Open source
• Tong · Line operators in the Standard Model§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choicesOpen source
• Brian Conrad, Fundamental Groups of Lie Groupspp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroupOpen source

All four subgroups of the common diagonal Z6 are compatible with the usual matter list. Matter compatibility alone does not determine the physical choice of quotient, a full line-operator spectrum, theta parameters, or anomaly cancellation.

• The Algebra of Grand Unified TheoriesSection 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly.Open source
• Tong · Line operators in the Standard Model§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choicesOpen source
• Brian Conrad, Fundamental Groups of Lie Groupspp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroupOpen source

The universal cover is SU(3) × SU(2) × R. The lift of Γ_1 is infinite cyclic: its finite subgroup at U(1) phase zero is trivial. Therefore the quotient has fundamental group Z, rather than Z_1. The compact product used as the quotient domain already has fundamental group Z.

• The Algebra of Grand Unified TheoriesSection 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly.Open source
• Tong · Line operators in the Standard Model§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choicesOpen source
• Brian Conrad, Fundamental Groups of Lie Groupspp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroupOpen source

Color is listed before the weak factor. The block-branching tool uses X=diag(2,2,2,−3,−3), while this record uses Q=6Y=−X. In the Q convention the map to SU(5) is (A,B,z) ↦ diag(z^−2 A,z^3 B). It has kernel Γ6 and induces an injective map only from the quotient of order six.

• The Algebra of Grand Unified TheoriesSection 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly.Open source
• Tong · Line operators in the Standard Model§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choicesOpen source
• Brian Conrad, Fundamental Groups of Lie Groupspp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroupOpen source

algebra:standard-model-r

• The Algebra of Grand Unified TheoriesSection 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly.Open source
• Tong · Line operators in the Standard Model§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choicesOpen source
• Brian Conrad, Fundamental Groups of Lie Groupspp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroupOpen source

group:standard-model-z2

• The Algebra of Grand Unified TheoriesSection 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly.Open source
• Tong · Line operators in the Standard Model§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choicesOpen source
• Brian Conrad, Fundamental Groups of Lie Groupspp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroupOpen source

group:standard-model-z3

• The Algebra of Grand Unified TheoriesSection 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly.Open source
• Tong · Line operators in the Standard Model§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choicesOpen source
• Brian Conrad, Fundamental Groups of Lie Groupspp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroupOpen source

group:standard-model-z6

• The Algebra of Grand Unified TheoriesSection 3.1: SU(5), exterior powers of its fundamental, and the Standard Model subgroup. Hypercharge conventions must be translated explicitly.Open source
• Tong · Line operators in the Standard Model§1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choicesOpen source
• Brian Conrad, Fundamental Groups of Lie Groupspp. 1–2: the exact sequence for a connected Lie group modulo a discrete central subgroupOpen source