{"paper":{"title":"$q$-Series Invariants of Three-Manifolds and Knots-Quivers Correspondence","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["hep-th","math.GT","math.MP","math.QA"],"primary_cat":"math-ph","authors_text":"Sachin Chauhan","submitted_at":"2024-12-14T16:31:39Z","abstract_excerpt":"The Gukov-Pei-Putrov-Vafa (GPPV) conjecture is a relationship between two three-manifold invariants: the Witten-Reshetikhin-Turaev (WRT) invariant and the \\(\\widehat{Z}\\) (``Z-hat'') invariant. In fact, WRT invariant is defined at roots of unity, $\\mathbbm{q}\\left(\\exp\\left(\\frac{2\\pi i}{k+2}\\right),~k\\in\\mathbb{Z}_+,~\\text{for}~SU(2)\\right)$, and is generally a complex number, whereas $\\widehat{Z}$-invariant is a $q$-series with integer coefficients such that $|q|<1$. Therefore, $\\widehat{Z}$-invariant can be obtained from WRT-invariant by performing a particular analytic continuation, $\\math"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.10885","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/2412.10885/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"}