{"paper":{"title":"Topological expansion of the Bethe ansatz, and quantum algebraic geometry","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.MP"],"primary_cat":"math-ph","authors_text":"B. Eynard, L. Chekhov, O. Marchal","submitted_at":"2009-11-09T13:06:38Z","abstract_excerpt":"In this article, we solve the loop equations of the \\beta-random matrix model, in a way similar to what was found for the case of hermitian matrices \\beta=1. For \\beta=1, the solution was expressed in terms of algebraic geometry properties of an algebraic spectral curve of equation y^2=U(x). For arbitrary \\beta, the spectral curve is no longer algebraic, it is a Schroedinger equation ((\\hbar\\partial)^2-U(x)).\\psi(x)=0 where \\hbar\\propto (\\sqrt\\beta-1/\\sqrt\\beta). In this article, we find a solution of loop equations, which takes the same form as the topological recursion found for \\beta=1. Thi"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"0911.1664","kind":"arxiv","version":2},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/0911.1664/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"}