Publication: Spectral curves in gauge/string dualities: integrability, singular sectors and regularization.
Full text at PDC
Advisors (or tutors)
IOP Publishing Ltd
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.
©IOP Publishing Ltd. 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.
Física-Modelos matemáticos, Física matemática
 Cachazo F, Intriligator K and Vafa C 2001 Nuc. Phys. B 603 3  Dijkgraaf R and Vafa C 2002 Nuc. Phys. B 644 3  Dijkgraaf R and Vafa C 2002 Nuc. Phys. B 644 21  Seiberg N and Witten E 1994 Nuc. Phys. B 426 19  Cachazo F and Vafa C 2002 arXiv:hep-th/020601  Deift P 1999 Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert approach (Providence: American Mathematical Society)  Bleher P 2008 Lectures on random matrix models. The Riemann-Hilbert approach (Amsterdam: North Holland)  Alvarez G, Martínez Alonso L and Medina E 2010 J. Stat. Mech. Theory Exp. 03023  Lazaroiu C I 2003 J. High Energy Phys. 03 044  Felder G and Riser R 2004 Nuc. Phys. B 691 251  Bilal A and Metzger S 2005 J. High Energy Phys. 08 097  Gonchar A and Rakhmanov E A 1987 Math. USSR Sbornik 62  Gonchar A A and Rakhmanov E A 1989 Math. USSR Sbornik 62 305  Bleher P and Its A 1999 Ann. Math. 150 185–266  Bertola M, Eynard B and Harnad J 2006 Commun. Math. Phys. 263 401  Bertola M and Mo M Y 2009 Adv. Math. 220 154  Bertola M 2011 Analysis and Math. Phys. 1 167  Martínez-Finkelshtein A and Rakhmanov E A 2011 Commun. Math. Phys. 302 53  Rakhmanov E A 2012 Orthogonal polynonials and s-curves Recent advances in orthogonal polynomials, special functions and their applications vol 578 of Contemp. Math. (Amer. Math. Soc. Providence, RI.) pp 195–239  Álvarez G, Martínez Alonso L and Medina E 2013 J. High Energy Phys. 03 170  Ferrari F 2003 Phys. Rev. D 67 085013  Krichever I 1994 Comm.Pure Appl. Math. 47 437  Chekhov L and Mironov A 2003 Phys. Lett. B 552 293  Chekhov L, Marshakov A, Mironov A and Vasiliev D 2003 Phy. Lett. B 562 323  Darboux G 1915 Lecons sur la thèorie gènèrale des surfaces II surfaces II, (Gauthier Villars)  Di Francesco P, Ginsparg P and Zinn-Justin J 1995 Phys. Rep. 254 1–133  Konopelchenko B, Martínez Alonso L and Medina E 2011 Phys. Lett. A 375 867–872  Mariñoo M, Pasquetti S and Putrov P 2010 J. High Energy Phys. 10 074  Bertola M and Tovbis A 2011 arXiv:1108.0321  Givental A B 1982 Functional Analysis and its Applications 16 10–14  Whitham G B 1974 Linear and nonlinear waves (J.Wiley and Sons, New-york)  Rozdestvenski B and Yanenko N 1983 Systems of quasilinear their applications to gas dynamics (Math.Monog., v. 55, AMS, Providence, RI.)  Arnold V I 1976 Comm.Pure Appl. Math. 29 557–582  Arnold V I, Varchenko A N and Guseyn-Zade S M 1985 Singularities of differentiable maps (Birkhauser, Boston, Ma.)  Thom R 1989 Structural stability and morphogenesis (Addison-Wesley, P.C. ,CA)  Dubrovin B 2008 On universality of critical behaviour in hamiltonian pdes Geometry, topology and mathematical physics (AMS Transl., 224, Ser. 2, 59-109, AMS, Providence, RI)  Murphy G 1960 Ordinary differential equations and their solutions (Van Vostrand,)