Boundary conditions
Group supported orbifold boundary conditions into equivalence classes and compare their apparent symmetries under the stated ordinary-boundary relation.
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 see why listing boundary conditions can overcount theories. The tool walks explicit classes so you can inspect both their size and the different descriptions they contain. It also shows why a group name read at a symmetric point need not identify the whole equivalence class.
First steps
- Select an available orbifold and the gauge-group parameter N.
- Choose a boundary condition and inspect the other members of its class.
- Compare the apparent groups within that class.
- Read the conditions on any vacuum-energy comparison before interpreting it.
Controls and inputs
| Control | Meaning | Try this |
|---|---|---|
| Orbifold selector | Chooses the available geometry and its supported boundary-condition relation. | Compare S¹/Z₂ with T²/Z₃ and read the changed count rule. |
| N and boundary multiplicities | Set the finite condition being classified within the selected family. | Start at a small N and inspect a class with several members. |
| Class-member and energy views | Select equivalent descriptions and inspect the explicitly permitted energy comparisons. | Read What cannot be compared before using an energy ranking. |
Reading the results
| Result | Meaning |
|---|---|
| Equivalence class and size | The boundary descriptions connected by the implemented relation. |
| Class totals | Counts for the chosen group parameter and orbifold, with the relevant published attribution. |
| Apparent group and energy information | Properties of the selected description and calculation; their interpretation requires the displayed comparison conditions. |
The class count is about the specified equivalence relation and orbifold. A boundary description's apparent unbroken group can change within one class. The S¹/Z₂ count must not be transferred to another orbifold.
Worked example
Compare two descriptions of one class
- Choose S¹/Z₂ at a small N.
- Select a class containing more than one condition.
- Compare the apparent group of two members.
- Open One model, every verdict to investigate which calculated quantities remain invariant.
What to expect: Different displayed group labels can occur within the same supported equivalence class.
Formulas and conventions
Number of classes = (N + 1)²
The Haba–Hosotani–Kawamura count for the stated ordinary SU(N) conditions on S¹/Z₂. Other geometries or equivalence hypotheses require their own result.
Scope and limitations
- Conjugate, antilinear boundary conditions belong to the separate Conjugate boundary conditions tool.
- Classifying boundaries does not complete the model's matter content or phenomenology.
- Energy comparisons are meaningful only under the conditions stated by that panel.
If something is unclear
The apparent group changes within a class.
Read it as a property of the selected boundary description and symmetric point, then inspect the invariant diagnostics separately.
Terms used here
Sources and attribution
Current instrument guide. Archived editions retain their own scope.
Back to top