Supersymmetric Calogero-Moser-Sutherland models: Superintegrability structure and eigenfunctions
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Bibliographic record
Abstract
We first review the construction of the supersymmetric extension of the (quantum) Calogero-Moser-Sutherland (CMS) models. We stress the remarkable fact that this extension is completely captured by the insertion of a fermionic exchange operator in the Hamiltonian: sCMS models ({\\it s} for supersymmetric) are nothing but special exchange-type CMS models. Under the appropriate projection, the conserved charges can thus be formulated in terms of the standard Dunkl operators. This is illustrated in the rational case, where the explicit form of the $4N$ ($N$ being the number of bosonic variables) conserved charges is presented, together with their full algebra. The existence of $2N$ commuting bosonic charges settles the question of the integrability of the srCMS model. We then prove its superintegrability by displaying $2N-2$ extra independent charges commuting with the Hamiltonian. In the second part, we consider the supersymmetric version of the trigonometric case (stCMS model) and review the construction of its eigenfunctions, the Jack superpolynomials. This leads to closed-form expressions, as determinants of determinants involving supermonomial symmetric functions. Here we focus on the main ideas and the generic aspects of the construction: those applicable to all models whether supersymmetric or not. Finally, the possible Lie superalgebraic structure underlying the stCMS model and its eigenfunctions is briefly considered.
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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) | 0.001 | 0.001 |
| Meta-epidemiology (broad) | 0.001 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 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 it