paper
Tong · Line operators in the Standard Model
David Tong
Use in this reference: Linked source with original summaries of mathematical facts. Consult the source for its own terms; no blanket redistribution license is asserted.
Relevant material: Introduction and Section 3: the Z6 quotient, integer-normalized hypercharge, and the additional choice of line operators.
Dependent assertions
| Record | Assertion | Source locator |
|---|---|---|
| su(3) ⊕ su(2) ⊕ u(1) | Definition | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| su(3) ⊕ su(2) ⊕ u(1) | Real dimension | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| su(3) ⊕ su(2) ⊕ u(1) | Rank | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| su(3) ⊕ su(2) ⊕ u(1) | Lie-algebra center | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| su(3) ⊕ su(2) ⊕ u(1) | Simple | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| su(3) ⊕ su(2) ⊕ u(1) | Global groups and charge normalization | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| su(3) ⊕ su(2) ⊕ u(1) | Color algebra | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| su(3) ⊕ su(2) ⊕ u(1) | Weak algebra | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| su(3) ⊕ su(2) ⊕ u(1) | Abelian algebra | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| su(3) ⊕ su(2) ⊕ u(1) | Compact global form with quotient order 1 | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| su(3) ⊕ su(2) ⊕ u(1) | Compact global form with quotient order 2 | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| su(3) ⊕ su(2) ⊕ u(1) | Compact global form with quotient order 3 | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| su(3) ⊕ su(2) ⊕ u(1) | Compact global form with quotient order 6 | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| SU(3) × SU(2) × U(1) | Definition | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| SU(3) × SU(2) × U(1) | Real dimension | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| SU(3) × SU(2) × U(1) | Rank | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| SU(3) × SU(2) × U(1) | Compact | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| SU(3) × SU(2) × U(1) | Connected | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| SU(3) × SU(2) × U(1) | Fundamental group | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| SU(3) × SU(2) × U(1) | Specified quotient order | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| SU(3) × SU(2) × U(1) | Abstract center | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| SU(3) × SU(2) × U(1) | Matter and the embedded kernel | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| SU(3) × SU(2) × U(1) | Matter and the embedded kernel | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| SU(3) × SU(2) × U(1) | Topology and the universal cover | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| SU(3) × SU(2) × U(1) | Hypercharge and SU(5) | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| SU(3) × SU(2) × U(1) | The common gauge algebra | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| SU(3) × SU(2) × U(1) | Diagonal quotient of order 2 | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| SU(3) × SU(2) × U(1) | Diagonal quotient of order 3 | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| SU(3) × SU(2) × U(1) | Diagonal quotient of order 6 | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(2) | Definition | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(2) | Real dimension | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(2) | Rank | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(2) | Compact | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(2) | Connected | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(2) | Fundamental group | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(2) | Specified quotient order | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(2) | Abstract center | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(2) | Matter and the embedded kernel | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(2) | Matter and the embedded kernel | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(2) | Topology and the universal cover | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(2) | Hypercharge and SU(5) | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(2) | The common gauge algebra | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(2) | Specified compact covering product | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(3) | Definition | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(3) | Real dimension | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(3) | Rank | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(3) | Compact | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(3) | Connected | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(3) | Fundamental group | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(3) | Specified quotient order | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(3) | Abstract center | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(3) | Matter and the embedded kernel | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(3) | Matter and the embedded kernel | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(3) | Topology and the universal cover | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(3) | Hypercharge and SU(5) | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(3) | The common gauge algebra | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(3) | Specified compact covering product | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(6) | Definition | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(6) | Real dimension | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(6) | Rank | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(6) | Compact | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(6) | Connected | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(6) | Fundamental group | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(6) | Specified quotient order | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(6) | Abstract center | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(6) | Matter and the embedded kernel | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(6) | Matter and the embedded kernel | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(6) | Topology and the universal cover | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(6) | Hypercharge and SU(5) | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(6) | The common gauge algebra | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(6) | Specified compact covering product | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(6) | Block subgroup of SU(5) with the specified map | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(1) → G_SM^(2) | Definition | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(1) → G_SM^(2) | Kernel | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(1) → G_SM^(2) | Injective | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(1) → G_SM^(2) | Surjective | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(1) → G_SM^(2) | Matter descent | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(1) → G_SM^(2) | Compact covering product | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(1) → G_SM^(2) | Chosen global form | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(1) → G_SM^(3) | Definition | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(1) → G_SM^(3) | Kernel | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(1) → G_SM^(3) | Injective | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(1) → G_SM^(3) | Surjective | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(1) → G_SM^(3) | Matter descent | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(1) → G_SM^(3) | Compact covering product | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(1) → G_SM^(3) | Chosen global form | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(1) → G_SM^(6) | Definition | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(1) → G_SM^(6) | Kernel | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(1) → G_SM^(6) | Injective | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(1) → G_SM^(6) | Surjective | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(1) → G_SM^(6) | Matter descent | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(1) → G_SM^(6) | Compact covering product | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(1) → G_SM^(6) | Chosen global form | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| SU(3) × SU(2) × U(1)_X → SU(5) | Definition | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| SU(3) × SU(2) × U(1)_X → SU(5) | Kernel | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| SU(3) × SU(2) × U(1)_X → SU(5) | Image | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| SU(3) × SU(2) × U(1)_X → SU(5) | Injective | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| SU(3) × SU(2) × U(1)_X → SU(5) | Surjective onto SU(5) | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| SU(3) × SU(2) × U(1)_X → SU(5) | Physical hypercharge | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| SU(3) × SU(2) × U(1)_X → SU(5) | Keep the charge | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| SU(3) × SU(2) × U(1)_X → SU(5) | From homomorphism to embedding | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| SU(3) × SU(2) × U(1)_X → SU(5) | Underlying compact product, with the X coordinate stated here | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| SU(3) × SU(2) × U(1)_X → SU(5) | Parent group | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| SU(3) × SU(2) × U(1)_X → SU(5) | Image after converting X to Q=−X | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(6) ↪ SU(5) | Definition | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(6) ↪ SU(5) | Kernel | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(6) ↪ SU(5) | Injective | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(6) ↪ SU(5) | Image | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(6) ↪ SU(5) | What the embedding establishes | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(6) ↪ SU(5) | The specified diagonal quotient | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(6) ↪ SU(5) | Parent compact group | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |
| G_SM^(6) ↪ SU(5) | Computable block map in the X convention | §1 and §3, especially pp. 7–9: the embedded Z6, integer hypercharge, and four quotient choices |