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Superconformal block quivers, duality trees and Diophantine equations

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Abstract

We generalize previous results on \( \mathcal{N} \) = 1, (3 + 1)-dimensional superconformal block quiver gauge theories. It is known that the necessary conditions for a theory to be superconformal, i.e. that the beta and gamma functions vanish in addition to anomaly cancellation, translate to a Diophantine equation in terms of the quiver data. We re-derive results for low block numbers revealing an new intriguing algebraic structure underlying a class of possible superconformal fixed points of such theories. After explicitly computing the five block case Diophantine equation, we use this structure to reorganize the result in a form that can be applied to arbitrary block numbers. We argue that these theories can be thought of as vectors in the root system of the corresponding quiver and superconformality conditions are shown to associate them to certain subsets of imaginary roots. These methods also allow for an interpretation of Seiberg duality as the action of the affine Weyl group on the root lattice.

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Authors and Affiliations

  1. Theoretical Physics Group, The Blackett Laboratory, Imperial College, Prince Consort Road, London, SW7 2AZ, U.K.

    Amihay Hanany

  2. Department of Mathematics, City University, London, Northampton Square, London, EC1V 0HB, U.K.

    Yang-Hui He

  3. Merton College, University of Oxford, Merton Street, Oxford, OX1 4JD, U.K.

    Yang-Hui He

  4. Department of Physics, NanKai University, CN-300 071, Tianjin, Nankai, P.R. China

    Yang-Hui He & Chuang Sun

  5. Rudolf Peierls Centre for Theoretical Physics, 1 Keble Road, Oxford, OX1 3NP, U.K.

    Chuang Sun

  6. Department of Physics, King’s College London, Strand, London, WC2R 2LS, U.K.

    Spyros Sypsas

Authors
  1. Amihay Hanany
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  2. Yang-Hui He
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  3. Chuang Sun
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  4. Spyros Sypsas
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Correspondence to Spyros Sypsas.

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ArXiv ePrint: 1211.6111

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Hanany, A., He, YH., Sun, C. et al. Superconformal block quivers, duality trees and Diophantine equations. J. High Energ. Phys. 2013, 17 (2013). https://doi.org/10.1007/JHEP11(2013)017

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