Superconformal Block Quivers, Duality Trees and Diophantine Equations

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Hanany, Amihay ; He, Yang-Hui ; Sun, Chuang ; Sypsas, Spyros (2012)

We generalize previous results on 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.
  • References (50)
    50 references, page 1 of 5

    [1] W. Crawley-Boevey. Lectures on representations of quivers. available at www.maths.leeds.ac.uk/ pmtwc/quivlecs.pdf.

    [2] H. Derksen and J. Weyman. Quiver representations. Notices Amer. Math. Soc, 52:200{206, 2005.

    [3] Alistair Savage. Finite dimensional algebras and quivers. Encyclopedia of Mathematical Physics, 2:313{ 320, 2005.

    [4] Michel Brion. Representations of quivers. Notes de l' ecole d' ete \Geometric Methods in Representation Theory", 2008.

    [5] Ibrahim Assem, Andrzej Skowronski, and Daniel Simson. Elements of Representation Theory of Associative Algebras, Vol. 1. Cambridge University Press, 2006.

    [6] Michael R. Douglas and Gregory W. Moore. D-branes, quivers, and ale instantons. 1996.

    [7] Sergio Benvenuti and Amihay Hanany. New results on superconformal quivers. JHEP, 0604:032, 2006.

    [8] Yang-Hui He. Some remarks on the nitude of quiver theories. In.J.Math.Math.Sci., 1999.

    [9] Amihay Hanany and Yang-Hui He. NonAbelian nite gauge theories. JHEP, 9902:013, 1999.

    [10] David Berenstein and Samuel Pinansky. The Minimal Quiver Standard Model. Phys.Rev., D75:095009, 2007.

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