Conjugate boundary conditions
Compare conjugate boundary conditions on S¹/Z₂ under two explicit hypotheses about the allowed transformations at the endpoints.
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 understand why conjugate boundary conditions require a different classification from ordinary linear twists. They transform by congruence, and the allowed relation between endpoint transformations changes which information can be removed.
First steps
- Choose N and a symmetric or antisymmetric twist at each endpoint.
- For odd N, read why the antisymmetric choice is unavailable.
- Compare the independent-endpoint and linked-transformation conclusions side by side.
- Read the Wilson-line and unbroken-group diagnostics with the hypothesis attached.
Controls and inputs
| Control | Meaning | Try this |
|---|---|---|
| N | The gauge-group parameter; its parity affects which twist types exist. | Compare a small even N with a small odd N. |
| Twist at y = 0 and y = πR | Choose a symmetric or, when allowed, antisymmetric conjugate twist at each endpoint. | Inspect all allowed combinations at even N. |
| Hypothesis comparison and Wilson-line card | Show the consequences of independent versus linked endpoint transformations and the additional dimension-count diagnostic. | Read the hypothesis before quoting a class count. |
Reading the results
| Result | Meaning |
|---|---|
| Counts under explicit hypotheses | A discrete classification under one assumption and retained continuous data under the other are different answers to different problems. |
| Wilson-line diagnostic | Examines the stated degrees of freedom; it does not silently transfer the ordinary-boundary conclusion. |
| Unbroken-group information | Must be interpreted with the selected twist and transformation hypothesis, not from a short label alone. |
Independent endpoint transformations give four classes at even N and one at odd N in the displayed classification. Linked transformations retain continuous information. The laboratory keeps these hypotheses explicit instead of treating one as an unconditional physical answer.
Worked example
See why the hypothesis must accompany the count
- Choose an even N and inspect the allowed endpoint types.
- Read both hypothesis columns.
- Switch to an odd N and observe the unavailable antisymmetric choice.
- Save a result with the hypothesis retained in its description.
What to expect: You can state precisely which classification problem produced a number.
Scope and limitations
- The calculation does not choose the correct transformation hypothesis for every physical construction.
- Ordinary-boundary class counts do not automatically extend to antilinear twists.
- The displayed structural diagnostic is not a full vacuum or phenomenological analysis.
If something is unclear
The antisymmetric option is refused at odd N.
This is the stated existence condition, not a missing interface feature.
Two columns give different kinds of answers.
Compare the permitted endpoint transformations; the columns intentionally describe different equivalence hypotheses.
Terms used here
Sources and attribution
Current instrument guide. Archived editions retain their own scope.
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