Standard Model global group with diagonal Z6 quotient
A compact connected twelve-dimensional global group with algebra su(3) ⊕ su(2) ⊕ u(1), quotient order 6, and fundamental group Z. Usual Standard Model matter descends in the stated Q=6Y convention.
The group (SU(3) × SU(2) × U(1))/Γ_6, where Γ_6=⟨γ^1⟩ and γ=(exp(2πi/3) I₃, −I₂, exp(2πi/6)). The U(1) action on a matter multiplet is z^Q with integer Q=6Y.
Definition sourcesReference properties
Matter and the embedded kernel
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^(6/6) to act trivially. The standard left-handed fermions and Higgs satisfy the condition. [sources]
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. [sources]
Topology and the universal cover
The universal cover is SU(3) × SU(2) × R. The lift of Γ_6 is infinite cyclic: its finite subgroup at U(1) phase zero is trivial. Therefore the quotient has fundamental group Z, rather than Z_6. The compact product used as the quotient domain already has fundamental group Z. [sources]
Hypercharge and SU(5)
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. [sources]
Names and aliases
SM Z6 · Standard Model quotient 6 · S(U(3) × U(2))
Related objects and maps
- The common gauge algebra
- Specified compact covering product
- Block subgroup of SU(5) with the specified map
Claims and sources
Each assertion has a stable identifier within this record and a source locator or derivation. Cite the dataset edition when referring to a corrected or historical claim.
Definition
The group (SU(3) × SU(2) × U(1))/Γ_6, where Γ_6=⟨γ^1⟩ and γ=(exp(2πi/3) I₃, −I₂, exp(2πi/6)). The U(1) action on a matter multiplet is z^Q with integer Q=6Y.
- 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#definitionReal dimension
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
group:standard-model-z6#dimensionRank
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
group:standard-model-z6#rankCompact
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
group:standard-model-z6#compactConnected
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
group:standard-model-z6#connectedFundamental 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
group:standard-model-z6#pi1Specified quotient order
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
group:standard-model-z6#quotient-orderAbstract center
U(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
Premises: source:conrad-pi1-lie · Γd projects injectively to both the diagonal finite center and the U(1) factor.
group:standard-model-z6#centerMatter and the embedded kernel
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^(6/6) 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
group:standard-model-z6#representations-1Matter and the embedded kernel
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
group:standard-model-z6#representations-2Topology and the universal cover
The universal cover is SU(3) × SU(2) × R. The lift of Γ_6 is infinite cyclic: its finite subgroup at U(1) phase zero is trivial. Therefore the quotient has fundamental group Z, rather than Z_6. 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
group:standard-model-z6#topology-1Hypercharge and SU(5)
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
group:standard-model-z6#conventions-1The common gauge algebra
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-z6#relation-1Specified compact covering product
group:standard-model-z1
- 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#relation-2Block subgroup of SU(5) with the specified map
group:su-5
- 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#relation-3