{"paper":{"title":"From Painlev\\'e equations to ${\\cal N}=2$ susy gauge theories: prolegomena","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math-ph","math.MP","nlin.SI"],"primary_cat":"hep-th","authors_text":"Davide Fioravanti, Marco Rossi","submitted_at":"2024-12-30T18:27:38Z","abstract_excerpt":"We study the linear problems in $z,t$ (time) associated to the Painlev\\'e III$_3$ and III$_1$ equations when the Painlev\\'e solution $q(t)$ approaches a pole or a zero. In this limit the problem in $z$ for the Painlev\\'e III$_3$ reduces to the modified Mathieu equation, while that for the Painlev\\'e III$_1$ to the Doubly Confluent Heun Equation. These equations appear as Nekrasov-Shatashvili quantisations/deformations of Seiberg-Witten differentials for $SU(2)$ ${\\cal N}=2$ super Yang-Mills gauge theory with number of flavours $N_f=0$ and $N_f=2$, respectively. These results allow us to conjec"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.21148","kind":"arxiv","version":1},"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/2412.21148/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"}