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Nonplanar integrability

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Abstract

In this article we study operators with a dimension Δ ∼ O(N) and show that simple analytic expressions for the action of the dilatation operator can be found. The operators we consider are restricted Schur polynomials. There are two distinct classes of operators that we consider: operators labeled by Young diagrams with two long columns or two long rows. 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 × S m irreducible representation (both specified by the labels of the restricted Schur polynomial). 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. The diagonalization of the dilatation operator reduces to solving three term recursion relations. 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. The recursion relations can be solved exactly in terms of symmetric Kravchuk polynomials or in terms of Clebsch-Gordan coefficients. This proves that the dilatation operator reduces to a decoupled set of harmonic oscillators and therefore it is integrable.

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Authors and Affiliations

  1. National Institute for Theoretical Physics, Department of Physics and Centre for Theoretical Physics, University of the Witwatersrand, Wits, 2050, South Africa

    Warren Carlson & Robert de Mello Koch

  2. Stellenbosch Institute for Advanced Studies, Stellenbosch, South Africa

    Robert de Mello Koch

  3. Department of Particle Physics, Facultad de Fisica, Universidad de Santiago de Compostela, 15782, Santiago de Compostela, Spain

    Hai Lin

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  1. Warren Carlson
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  2. Robert de Mello Koch
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  3. Hai Lin
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Correspondence to Robert de Mello Koch.

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ArXiv ePrint: 1101.5404

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Carlson, W., de Mello Koch, R. & Lin, H. Nonplanar integrability. J. High Energ. Phys. 2011, 105 (2011). https://doi.org/10.1007/JHEP03(2011)105

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