Does the round sphere maximize the free energy of (2+1)-dimensional QFTs?
Bibliographic record
Abstract
A bstract We examine the renormalized free energy of the free Dirac fermion and the free scalar on a (2+1)-dimensional geometry ℝ × Σ, with Σ having spherical topology and prescribed area. Using heat kernel methods, we perturbatively compute this energy when Σ is a small deformation of the round sphere, finding that at any temperature the round sphere is a local maximum. At low temperature the free energy difference is due to the Casimir effect. We then numerically compute this free energy for a class of large axisymmetric deformations, providing evidence that the round sphere globally maximizes it, and we show that the free energy difference relative to the round sphere is unbounded below as the geometry on Σ becomes singular. Both our perturbative and numerical results in fact stem from the stronger finding that the difference between the heat kernels of the round sphere and a deformed sphere always appears to have definite sign. We investigate the relevance of our results to physical systems like monolayer graphene consisting of a membrane supporting relativistic QFT degrees of freedom.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.001 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one teacher head, not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".