{"paper":{"title":"K-Decompositions and 3d Gauge Theories","arxiv_id":"1301.0192","primary_cat":"hep-th","submitted_at":"2013-01-02T08:54:42+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-1.13-20260917 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":0.73,"id":"s0","own":true,"role":"contribution","text":"This paper combines several new constructions in mathematics and physics.","hedge":{"p":0.98,"label":"asserted"},"own_p":0.75,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.07,"role_probabilities":{"gap":0.0,"method":0.01,"result":0.0,"background":0.26,"limitation":0.0,"future_work":0.0,"contribution":0.73,"interpretation":0.0}},{"i":1,"p":0.99,"id":"s1","own":true,"role":"contribution","text":"Mathematically, we study framed flat PGL(K,C)-connections on a large class of 3-manifolds M with boundary.","hedge":{"p":0.97,"label":"asserted"},"own_p":0.88,"main_claim_p":0.03,"quantitative":false,"quantitative_p":0.13,"role_probabilities":{"gap":0.0,"method":0.01,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.99,"interpretation":0.0}},{"i":2,"p":0.99,"id":"s2","own":true,"role":"contribution","text":"We define a space L_K(M) of framed flat connections on the boundary of M that extend to M.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.94,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.04,"role_probabilities":{"gap":0.0,"method":0.01,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.99,"interpretation":0.0}},{"i":3,"p":0.63,"id":"s3","own":false,"role":"contribution","text":"Our goal is to understand an open part of L_K(M) as a Lagrangian in the symplectic space of framed flat connections on the boundary, and as a K_2-Lagrangian, meaning that the K_2-avatar of the symplectic form restricts to zero.","hedge":{"p":0.82,"label":"asserted"},"own_p":0.41,"main_claim_p":0.19,"quantitative":false,"quantitative_p":0.1,"role_probabilities":{"gap":0.01,"method":0.01,"result":0.01,"background":0.21,"limitation":0.0,"future_work":0.12,"contribution":0.63,"interpretation":0.01}},{"i":4,"p":0.73,"id":"s4","own":true,"role":"method","text":"We construct an open part of L_K(M) from data assigned to a hypersimplicial K-decomposition of an ideal triangulation of M, generalizing Thurston's gluing equations in 3d hyperbolic geometry, and combining them with the cluster coordinates for framed flat PGL(K)-connections on surfaces.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.1,"quantitative":false,"quantitative_p":0.15,"role_probabilities":{"gap":0.0,"method":0.73,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.27,"interpretation":0.0}},{"i":5,"p":0.93,"id":"s5","own":true,"role":"result","text":"Using a canonical map from the complex of configurations of decorated flags to the Bloch complex, we prove that any generic component of L_K(M) is K_2-isotropic if the boundary satisfies some topological constraints (Theorem 4.2).","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.19,"quantitative":false,"quantitative_p":0.34,"role_probabilities":{"gap":0.0,"method":0.03,"result":0.93,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.04,"interpretation":0.0}},{"i":6,"p":0.79,"id":"s6","own":true,"role":"interpretation","text":"In some cases this implies that L_K(M) is K_2-Lagrangian.","hedge":{"p":0.73,"label":"hedged_possibility"},"own_p":0.85,"main_claim_p":0.1,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.21,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.79}},{"i":7,"p":0.74,"id":"s7","own":true,"role":"method","text":"For general M, we extend a classic result of Neumann-Zagier on symplectic properties of PGL(2) gluing equations to reduce the K_2-Lagrangian property to a combinatorial claim.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.86,"main_claim_p":0.09,"quantitative":false,"quantitative_p":0.12,"role_probabilities":{"gap":0.0,"method":0.74,"result":0.01,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.25,"interpretation":0.0}},{"i":8,"p":0.98,"id":"s8","own":true,"role":"contribution","text":"Physically, we use the symplectic properties of K-decompositions to construct 3d N=2 superconformal field theories T_K[M] corresponding (conjecturally) to the compactification of K M5-branes on M.","hedge":{"p":0.6,"label":"asserted"},"own_p":0.94,"main_claim_p":0.14,"quantitative":false,"quantitative_p":0.18,"role_probabilities":{"gap":0.0,"method":0.02,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.98,"interpretation":0.0}},{"i":9,"p":0.43,"id":"s9","own":false,"role":"background","text":"This extends known constructions for K=2.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.33,"main_claim_p":0.04,"quantitative":true,"quantitative_p":0.74,"role_probabilities":{"gap":0.05,"method":0.0,"result":0.02,"background":0.43,"limitation":0.01,"future_work":0.0,"contribution":0.43,"interpretation":0.06}},{"i":10,"p":0.4,"id":"s10","own":false,"role":"contribution","text":"Just as for K=2, the theories T_K[M] are described as IR fixed points of abelian Chern-Simons-matter theories.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.48,"main_claim_p":0.06,"quantitative":true,"quantitative_p":0.52,"role_probabilities":{"gap":0.0,"method":0.05,"result":0.22,"background":0.23,"limitation":0.0,"future_work":0.0,"contribution":0.4,"interpretation":0.1}},{"i":11,"p":0.75,"id":"s11","own":true,"role":"result","text":"Changes of triangulation (2-3 moves) lead to abelian mirror symmetries that are all generated by the elementary duality between N_f=1 SQED and the XYZ model.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.72,"main_claim_p":0.07,"quantitative":false,"quantitative_p":0.45,"role_probabilities":{"gap":0.0,"method":0.03,"result":0.75,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.19,"interpretation":0.03}},{"i":12,"p":1.0,"id":"s12","own":true,"role":"result","text":"In the large K limit, we find evidence that the degrees of freedom of T_K[M] grow cubically in K.","hedge":{"p":0.5,"label":"hedged_possibility"},"own_p":0.93,"main_claim_p":0.08,"quantitative":true,"quantitative_p":0.94,"role_probabilities":{"gap":0.0,"method":0.0,"result":1.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}}]},"compiled":{"main_claim":{"p":0.37,"text":"Physically, we use the symplectic properties of K-decompositions to construct 3d N=2 superconformal field theories T_K[M] corresponding (conjecturally) to the compactification of K M5-branes on M.","location":"abstract 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