{"paper":{"title":"Graded decorated marked surfaces: Calabi-Yau-$\\mathbb{X}$ categories of gentle algebras","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.GT","math.SG"],"primary_cat":"math.RT","authors_text":"Akishi Ikeda, Yu Qiu, Yu Zhou","submitted_at":"2020-05-29T18:00:02Z","abstract_excerpt":"Let $\\mathbf{S}$ be a graded marked surface. We construct a string model for Calabi-Yau-$\\mathbb{X}$ category $\\mathcal{D}_\\mathbb{X}(\\mathbf{S}_\\bigtriangleup)$, via the graded DMS (=decorated marked surface) $\\mathbf{S}_\\Delta$. We prove an isomorphism between the braid twist group of $\\mathbf{S}_\\bigtriangleup$ and the spherical twist group of $\\mathcal{D}_\\mathbb{X}(\\mathbf{S}_\\bigtriangleup)$, and $\\mathbf{q}$-intersection formulas. We also give a topological realization of the Lagrangian immersion $\\mathcal{D}_\\infty(\\mathbf{S})\\to\\mathcal{D}_\\mathbb{X}(\\mathbf{S}_\\bigtriangleup)$, where"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2006.00009","kind":"arxiv","version":4},"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/2006.00009/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"}