Publication: Spectral curves in gauge/string dualities: integrability, singular sectors and regularization.
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2013-06-07
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IOP Publishing Ltd
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Abstract
We study the moduli space of the spectral curves y ^2 = W ‘ (z) ^2 + f(z) which characterize the vacua of N = 1 U(n) supersymmetric gauge theories with an adjoint Higgs field and a polynomial tree level potential W(z). The integrable structure of the Whitham equations is used to determine the spectral curves from their moduli. An alternative characterization of the spectral curves in terms of critical points of a family of polynomial solutions W to Euler-Poisson-Darboux equations is provided. The equations for these critical points are a generalization of the planar limit equations for one-cut random matrix models. Moreover, singular spectral curves with higher order branch points turn out to be described by degenerate critical points of W. As a consequence we propose a multiple scaling limit method of regularization and show that, in the simplest cases, it leads to the Painlevè-I equation and its multi-component generalizations.
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LMA and EM are grateful to G Alvarez for many useful conversations on the subject of spectral curves in gauge/string dualities. The financial support of the Universidad Complutense under project GR58/08-910556, the Comision Interministerial de Ciencia y Tecnología under project FIS2011-22566 and PRIN 2008 grant no. 28002K9KXZ are gratefully acknowledged.
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