{"paper":{"title":"Nonplanar Integrability","arxiv_id":"1101.5404","primary_cat":"hep-th","submitted_at":"2011-01-27T22:01:39+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":0.92,"id":"s0","own":true,"role":"contribution","text":"In this article we study operators with a dimension $\\Delta\\sim O(N)$ and show that simple analytic expressions for the action of the dilatation operator can be found.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.93,"main_claim_p":0.08,"quantitative":true,"quantitative_p":0.82,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.07,"background":0.01,"limitation":0.0,"future_work":0.0,"contribution":0.92,"interpretation":0.0}},{"i":1,"p":0.63,"id":"s1","own":false,"role":"contribution","text":"The operators we consider are restricted Schur polynomials.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.3,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.03,"role_probabilities":{"gap":0.0,"method":0.08,"result":0.0,"background":0.29,"limitation":0.0,"future_work":0.0,"contribution":0.63,"interpretation":0.0}},{"i":2,"p":0.82,"id":"s2","own":true,"role":"contribution","text":"There are two distinct classes of operators that we consider: operators labeled by Young diagrams with two long columns or two long rows.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.76,"main_claim_p":0.02,"quantitative":true,"quantitative_p":0.67,"role_probabilities":{"gap":0.0,"method":0.05,"result":0.0,"background":0.12,"limitation":0.01,"future_work":0.0,"contribution":0.82,"interpretation":0.0}},{"i":3,"p":0.86,"id":"s3","own":false,"role":"background","text":"The main complication in working with restricted Schur polynomials is in building a projector from a given $S_{n+m}$ irreducible representation to an $S_n\\times S_m$ irreducible representation (both specified by the labels of the restricted Schur polynomial).","hedge":{"p":1.0,"label":"asserted"},"own_p":0.25,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.09,"role_probabilities":{"gap":0.04,"method":0.01,"result":0.0,"background":0.86,"limitation":0.09,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":4,"p":0.86,"id":"s4","own":true,"role":"method","text":"We give an explicit construction of these projectors by reducing it to the simple problem of addition of angular momentum in ordinary non-relativistic quantum mechanics.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.16,"role_probabilities":{"gap":0.0,"method":0.86,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.14,"interpretation":0.0}},{"i":5,"p":0.68,"id":"s5","own":true,"role":"result","text":"The diagonalizationof the dilatation operator reduces to solving three term recursion relations.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.85,"main_claim_p":0.09,"quantitative":true,"quantitative_p":0.79,"role_probabilities":{"gap":0.0,"method":0.26,"result":0.68,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.06,"interpretation":0.0}},{"i":6,"p":0.67,"id":"s6","own":true,"role":"interpretation","text":"The fact that the recursion relations have only three terms is a direct consequence of the weak mixing at one loop of the restricted Schur polynomials.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.77,"main_claim_p":0.03,"quantitative":true,"quantitative_p":0.63,"role_probabilities":{"gap":0.0,"method":0.01,"result":0.32,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.67}},{"i":7,"p":0.64,"id":"s7","own":true,"role":"result","text":"The recursion relations can be solved exactly in terms of symmetric Kravchuk polynomials or in terms of Clebsch-Gordan coefficients.","hedge":{"p":0.96,"label":"asserted"},"own_p":0.85,"main_claim_p":0.07,"quantitative":false,"quantitative_p":0.11,"role_probabilities":{"gap":0.0,"method":0.28,"result":0.64,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.08,"interpretation":0.0}},{"i":8,"p":0.54,"id":"s8","own":true,"role":"result","text":"This proves that the dilatation operator reduces to a decoupled set of harmonic oscillators and therefore it is integrable.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.89,"main_claim_p":0.9,"quantitative":false,"quantitative_p":0.07,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.54,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.46}}]},"compiled":{"main_claim":{"p":1.0,"text":"This proves that the dilatation operator reduces to a decoupled set of harmonic oscillators and therefore it is integrable.","location":"abstract 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