{"paper":{"title":"Kasteleyn operators from mirror symmetry","arxiv_id":"1810.05985","primary_cat":"math.AG","submitted_at":"2018-10-14T07:08:22+00:00","version_compiled":"v2"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13.0 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v2","sentences":[{"i":0,"p":0.48,"id":"s0","own":true,"role":"contribution","text":"Given a consistent bipartite graph $\\Gamma$ in $T^2$ with a complex-valued edge weighting $\\mathcal{E}$ we show the following two constructions are the same.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.88,"main_claim_p":0.59,"quantitative":false,"quantitative_p":0.24,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.36,"background":0.16,"limitation":0.0,"future_work":0.0,"contribution":0.48,"interpretation":0.0}},{"i":1,"p":0.59,"id":"s1","own":true,"role":"method","text":"The first is to form the Kasteleyn operator of $(\\Gamma, \\mathcal{E})$ and pass to its spectral transform, a coherent sheaf supported on a spectral curve in $(\\mathbb{C}^\\times)^2$.","hedge":{"p":0.96,"label":"asserted"},"own_p":0.74,"main_claim_p":0.15,"quantitative":false,"quantitative_p":0.1,"role_probabilities":{"gap":0.0,"method":0.59,"result":0.05,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.36,"interpretation":0.0}},{"i":2,"p":0.53,"id":"s2","own":true,"role":"method","text":"The second is to form the conjugate Lagrangian $L \\subset T^* T^2$ of $\\Gamma$, equip it with a brane structure prescribed by $\\mathcal{E}$, and pass to its mirror coherent sheaf.","hedge":{"p":0.97,"label":"asserted"},"own_p":0.83,"main_claim_p":0.25,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":0.0,"method":0.53,"result":0.09,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.38,"interpretation":0.0}},{"i":3,"p":0.41,"id":"s3","own":true,"role":"method","text":"This lives on a stacky toric compactification of $(\\mathbb{C}^\\times)^2$ determined by the Legendrian link which lifts the zig-zag paths of $\\Gamma$ (and to which the noncompact Lagrangian $L$ is asymptotic).","hedge":{"p":0.96,"label":"asserted"},"own_p":0.6,"main_claim_p":0.07,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":0.41,"result":0.21,"background":0.06,"limitation":0.0,"future_work":0.0,"contribution":0.26,"interpretation":0.06}},{"i":4,"p":0.71,"id":"s4","own":false,"role":"background","text":"We work in the setting of the coherent-constructible correspondence, a sheaf-theoretic model of toric mirror symmetry.","hedge":{"p":0.87,"label":"asserted"},"own_p":0.29,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.03,"role_probabilities":{"gap":0.0,"method":0.28,"result":0.0,"background":0.71,"limitation":0.0,"future_work":0.0,"contribution":0.01,"interpretation":0.0}},{"i":5,"p":0.89,"id":"s5","own":true,"role":"result","text":"We also show that tensoring with line bundles on the compactification is mirror to certain Legendrian autoisotopies of the 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