Assumptions, results and research status
An explicit finite mathematical example with declared inputs, conventions and scope.
Assumptions
- The operator, state family or representations described on this page are supplied as model inputs.
- Every exported quantity uses the units and conventions printed beside the calculation.
Results
- The default example, tables and equations remain readable without JavaScript.
- With JavaScript, changing valid inputs recomputes the declared model and enables numerical downloads.
Open questions
- What additional source dynamics and physical matching would turn this example into a prediction of a specified physical system?
The declared model
The initial spectrum contains Q, uᶜ, dᶜ, L and eᶜ, with an optional neutral νᶜ. Every field is written left-handed; the superscript c indicates a conjugate representation. The non-Abelian representations and their multiplicities are fixed inputs. Only the integer Abelian charges q = 6Y change.
The displayed coefficients suppress common nonzero normalization factors so that the exact integer sums test their vanishing. Color cubic, color mixed, weak mixed, gravitational mixed and Abelian cubic terms are listed separately. The fixed list contains four weak doublets after color multiplicity, so it has even Witten parity on the ordinary spin backgrounds assumed here. This ledger does not classify every possible global anomaly of a quotient gauge theory on arbitrary backgrounds. [Conditional unification, anomaly calculation]
Run the calculation
One-generation anomaly ledger
Keep the non-Abelian representations fixed and change the integer Abelian charges. The initial all-left-handed spectrum cancels the displayed local anomalies and passes the global quotient test.
All charges use q = 6Y. Coefficients omit common nonzero normalization factors; only their vanishing is tested.
All displayed local anomalies cancel. All multiplets pass the global quotient test.
- Displayed local anomalies
- All cancel
- Global product-volume quotient
- All rows descend
- Number of weak doublets
- 4
- Witten parity of fixed representations
- Even
| Anomaly | Coefficient | Cancellation |
|---|---|---|
| SU(3)³ | 0 | Zero |
| SU(3)²U(1) | 0 | Zero |
| SU(2)²U(1) | 0 | Zero |
| Gravity²U(1) | 0 | Zero |
| U(1)³ | 0 | Zero |
| Multiplet | q = 6Y | Dimension | d q | d q³ | ℤ₆ residue |
|---|---|---|---|---|---|
| Q: left quark doublet | 1 | 6 | 6 | 6 | 0 |
| uᶜ: conjugate up singlet | -4 | 3 | -12 | -192 | 0 |
| dᶜ: conjugate down singlet | 2 | 3 | 6 | 24 | 0 |
| L: left lepton doublet | -3 | 2 | -6 | -54 | 0 |
| eᶜ: conjugate charged-lepton singlet | 6 | 1 | 6 | 216 | 0 |
| νᶜ: optional neutral singlet | 0 | 1 | 0 | 0 | 0 |
The conjugate singlets uᶜ, dᶜ and eᶜ carry the all-left-handed charges used in the formulas. νᶜ is optional and has no effect at its default zero charge. Giving it a nonzero charge turns it into a charged singlet in this diagnostic; it is then no longer a neutral-neutrino assignment.
Local anomaly cancellation and global quotient descent answer different questions. Reversing only the Abelian charges, while holding the non-Abelian representations fixed, preserves the zero anomaly coefficients but can fail the quotient test.
Equations and conventions used
SU(2)²U(1): 3qQ + qL. Gravity²U(1): Σdiqi. U(1)³: Σdiqi³.
Four weak doublets, including color multiplicity, give an even Witten parity for this fixed list.
The fermion representations and their multiplicities are specified inputs. Cancellation tests consistency for this list; it does not derive the matter spectrum, chirality, number of generations, or uniqueness of hypercharge. The weak global test assumes ordinary spin backgrounds; it is not a complete classification of global anomalies for quotient-group bundles or non-spin backgrounds.
Downloads include the chosen inputs, units and model scope. Values use standard floating-point arithmetic. Read the source and its version notes.
Try these comparisons
- Verify the default sums Σdq = 0 and Σdq³ = 0 from the contribution table.
- Choose the preset that changes only qL from −3 to −2. The weak mixed coefficient becomes one, the gravitational mixed coefficient two, and the Abelian cubic coefficient thirty-eight.
- Reverse only the Abelian charges while keeping all non-Abelian representations fixed. Local polynomial cancellation survives, but some rows fail the chosen global quotient condition.
- Give the optional νᶜ a nonzero charge. It is now a charged singlet in the diagnostic, not the default neutral-neutrino assignment.
The table lists each row’s quotient residue so a failed global test can be traced to the actual multiplet. It also shows the dimension factors used in gravitational and cubic sums.
Interpret the result
Cancellation is a consistency condition on the declared chiral list. The number of generations, the emergence of spacetime chirality, canonical statistics, masses and mixings require separate constructions. Even within a fixed list, claims of unique hypercharge need the exact additional assumptions used to obtain uniqueness.
The global quotient checks whether the kernel of SU(3) × SU(2) × U(1) → S(U(3) × U(2)) acts trivially on each representation. It is distinct from the local triangle-anomaly sums. The matter guide develops both questions and links them to the separately assumed exterior-module construction. Conventional background is available in the anomaly lectures.
Reproduce or extend the calculation
Download JSON to retain the inputs, units, model scope and full computed results. Download CSV for the displayed numerical family or ledger; its opening rows record the input settings and definitions. The tool uses local browser calculations and ordinary floating-point arithmetic.
For an independent check, use the worked example in the exercise collection and the linked source manuscript. A useful extension states which input or assumption changes and which comparison would test the change. See the research collaboration page for focused starting questions.
Continue reading
Sources for this guide
Source manuscripts, version notes, and standard references are listed in the reference library.