Interplay between Opers, Quantum Curves, WKB Analysis, and Higgs Bundles
Bibliographic record
Abstract
Quantum curves were introduced in the physics literature.We develop a mathematical framework for the case associated with Hitchin spectral curves.In this context, a quantum curve is a Rees D-module on a smooth projective algebraic curve, whose semiclassical limit produces the Hitchin spectral curve of a Higgs bundle.We give a method of quantization of Hitchin spectral curves by concretely constructing one-parameter deformation families of opers.We propose a variant of the topological recursion of Eynard-Orantin and Mirzakhani for the context of singular Hitchin spectral curves.We show that a PDE version of topological recursion provides all-order WKB analysis for the Rees Dmodules, defined as the quantization of Hitchin spectral curves associated with meromorphic SL(2, C)-Higgs bundles.Topological recursion can be considered as a process of quantization of Hitchin spectral curves.We prove that these two quantizations, one via the construction of families of opers, and the other via the PDE recursion of topological type, agree for holomorphic and meromorphic SL(2, C)-Higgs bundles.Classical differential equations such as the Airy differential equation provides a typical example.Through these classical examples, we see that quantum curves relate Higgs bundles, opers, a conjecture of Gaiotto, and quantum invariants, such as Gromov-Witten invariants.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.004 | 0.002 |
| Science and technology studies | 0.002 | 0.008 |
| Scholarly communication | 0.003 | 0.009 |
| Open science | 0.001 | 0.003 |
| Research integrity | 0.001 | 0.002 |
| Insufficient payload (model declined to judge) | 0.005 | 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 source (direct Gemma or distilled Codex), 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".