Multiplets & parities
Inspect how supported SU(7) representations split under the boundary symmetries, and how their parity signs determine zero modes and potential terms.
Control names are kept as they appear in the application. Start with a reference, change one input at a time, and retain the assumptions with your result.
What this tool is for
Use this to connect the representation names in the bulk-content editor with the components that survive the boundaries. The cube is a visual index of signs; the component and coefficient tables contain the detailed information.
First steps
- Choose a supported representation and read its parity labels.
- Use the cube and the component table together to identify which fields occupy each corner.
- Read the derived fermion term table and its comparison with the reference coefficients.
- Keep the gauge-sector split and its unresolved physical interpretation separate from the reconstructed fermion tables.
Controls and inputs
| Control | Meaning | Try this |
|---|---|---|
| Representation selector | Chooses the supported multiplet whose decomposition is displayed. | Start with the fundamental representation, then compare a tensor representation. |
| Parity and intrinsic-sign choices | Specify how a component transforms at the boundaries and which sign enters its Fourier term. | Change one available sign and follow the affected components. |
| Parity cube | Groups components by their displayed Z₂ signs. | Rotate it and read the table instead of inferring multiplicity from a drawing alone. |
Reading the results
| Result | Meaning |
|---|---|
| Component decomposition | The subgroup representations and parity assignments obtained from the supported parent representation. |
| Zero-mode information | Identifies components allowed to remain massless by the stated boundary projection before additional mass-generating effects. |
| Derived term table | Checks the Fourier coefficients reconstructed from the component signs against the implemented reference. |
A zero mode is a component allowed by the displayed boundary conditions. Reconstructing fermion coefficients checks that part of the potential; it does not determine the complete physical gauge determinant.
Worked example
Follow one sign through the calculation
- Select the fundamental representation.
- Read a component's boundary signs and zero-mode label.
- Change an available intrinsic sign.
- Compare the component and Fourier-term tables before and after.
What to expect: You can identify which boundary choice changes a component or a potential contribution, without treating the sign cube as a spectrum calculation by itself.
Formulas and conventions
s = η η′ P₅ P′₅
The sign product shown for this potential convention. Read the definitions of the component and intrinsic signs on the panel.
Scope and limitations
- Only the supported representation and boundary conventions are covered.
- Boundary-allowed zero modes can be lifted by interactions that are outside this decomposition.
- The eight fermion tables and the physical gauge-sector question have different evidence status.
If something is unclear
A component exists in the decomposition but has no zero mode.
Check its individual boundary parities; belonging to the parent representation is not sufficient to survive the projection.
Terms used here
Sources and attribution
Current instrument guide. Archived editions retain their own scope.
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