Classify an orbifold
Derive supported orbifold classification data from an integer rotation matrix, then compare the SU, SO and Sp cases under the stated hypotheses.
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 investigate which building blocks and local labels are available for a supported orbifold rotation. The 'alphabet' is the list of representation building blocks; a 'fibre' collects conditions sharing the selected local data. The plan and relief offer two views of the same computed field.
First steps
- Start with a supplied rotation before entering your own integer matrix.
- Check that the matrix passes the finite-order and cyclotomic tests.
- Choose the group family and N, then read the signature, alphabet and count information.
- Inspect a datum-lattice cell and its fibre before comparing the plan and relief views.
Controls and inputs
| Control | Meaning | Try this |
|---|---|---|
| Rotation preset or integer matrix | Defines the torus rotation. Custom matrices are supported up to the stated rank-eight limit and must satisfy the implemented hypotheses. | Classify a supplied example before modifying an entry. |
| SU(N), SO(N), Sp(N) and N | Choose the gauge-group family and parameter whose classification data are displayed. | Keep the rotation fixed while comparing the three real forms. |
| Datum axes, cell and boundary input | Select local multiplicity data and inspect the corresponding conditions. | Compare a cell's class information with the number of conditions in its fibre. |
Reading the results
| Result | Meaning |
|---|---|
| Cone signature and alphabet | The rotation's local singularity orders and the supported building blocks with their weights. |
| Class count, degree and generating information | Describe the computed classification problem and its stated growth properties. |
| Plan, relief and selected fibre | Visualize the same datum field and expose the detailed contents of a selected cell. |
A refused matrix lies outside the implemented theorem's hypotheses; refusal is not a class count of zero. The rotation-matrix dimension and the gauge-group parameter N are different inputs.
Worked example
Change the group while retaining the geometry
- Load a supplied rotation and classify it.
- Read the signature and alphabet.
- Compare SU, SO and Sp at the selected N.
- Inspect the reality-type information to understand why the alphabets differ.
What to expect: The geometry stays fixed while the representation constraints of the group family change.
Scope and limitations
- Infinite-order matrices and matrices outside the stated cyclotomic hypothesis are refused.
- A supported classification result does not calculate a phenomenological vacuum or experimental spectrum.
- Local-data equivalence and the completeness of a proposed move set are separate questions, explored in Name the relations.
If something is unclear
A custom matrix is refused.
Check that it is square with integer entries, then read which finite-order or cyclotomic condition failed.
A plotted cell has no conditions.
Inspect the rank and local multiplicity constraints; a selected coordinate need not be realizable.
Terms used here
Sources and attribution
Current instrument guide. Archived editions retain their own scope.
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