RT Journal Article
T1 Spectral curves in gauge/string dualities: integrability, singular sectors and regularization.
A1 Konopelchenko, Boris
A1 Martínez Alonso, Luis
A1 Medina, Elena
AB 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.
PB IOP Publishing Ltd
SN 1751-8113
YR 2013
FD 2013-06-07
LK https://hdl.handle.net/20.500.14352/34936
UL https://hdl.handle.net/20.500.14352/34936
LA eng
NO [1] Cachazo F, Intriligator K and Vafa C 2001 Nuc. Phys. B 603 3 [2] Dijkgraaf R and Vafa C 2002 Nuc. Phys. B 644 3 [3] Dijkgraaf R and Vafa C 2002 Nuc. Phys. B 644 21 [4] Seiberg N and Witten E 1994 Nuc. Phys. B 426 19 [5] Cachazo F and Vafa C 2002 arXiv:hep-th/020601 [6] Deift P 1999 Orthogonal Polynomials and Random Matrices: A Riemann–Hilbert approach (Providence: American Mathematical Society) [7] Bleher P 2008 Lectures on random matrix models. The Riemann-Hilbert approach (Amsterdam: North Holland) [8] Alvarez G, Martínez Alonso L and Medina E 2010 J. Stat. Mech. Theory Exp. 03023 [9] Lazaroiu C I 2003 J. High Energy Phys. 03 044 [10] Felder G and Riser R 2004 Nuc. Phys. B 691 251[11] Bilal A and Metzger S 2005 J. High Energy Phys. 08 097 [12] Gonchar A and Rakhmanov E A 1987 Math. USSR Sbornik 62 [13] Gonchar A A and Rakhmanov E A 1989 Math. USSR Sbornik 62 305 [14] Bleher P and Its A 1999 Ann. Math. 150 185–266 [15] Bertola M, Eynard B and Harnad J 2006 Commun. Math. Phys. 263 401 [16] Bertola M and Mo M Y 2009 Adv. Math. 220 154 [17] Bertola M 2011 Analysis and Math. Phys. 1 167 [18] Martínez-Finkelshtein A and Rakhmanov E A 2011 Commun. Math. Phys. 302 53 [19] 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 [20] Álvarez G, Martínez Alonso L and Medina E 2013 J. High Energy Phys. 03 170 [21] Ferrari F 2003 Phys. Rev. D 67 085013 [22] Krichever I 1994 Comm.Pure Appl. Math. 47 437 [23] Chekhov L and Mironov A 2003 Phys. Lett. B 552 293 [24] Chekhov L, Marshakov A, Mironov A and Vasiliev D 2003 Phy. Lett. B 562 323 [25] Darboux G 1915 Lecons sur la thèorie gènèrale des surfaces II surfaces II, (Gauthier Villars) [26] Di Francesco P, Ginsparg P and Zinn-Justin J 1995 Phys. Rep. 254 1–133 [27] Konopelchenko B, Martínez Alonso L and Medina E 2011 Phys. Lett. A 375 867–872 [28] Mariñoo M, Pasquetti S and Putrov P 2010 J. High Energy Phys. 10 074 [29] Bertola M and Tovbis A 2011 arXiv:1108.0321 [30] Givental A B 1982 Functional Analysis and its Applications 16 10–14 [31] Whitham G B 1974 Linear and nonlinear waves (J.Wiley and Sons, New-york) [32] Rozdestvenski B and Yanenko N 1983 Systems of quasilinear their applications to gas dynamics (Math.Monog., v. 55, AMS, Providence, RI.) [33] Arnold V I 1976 Comm.Pure Appl. Math. 29 557–582 [34] Arnold V I, Varchenko A N and Guseyn-Zade S M 1985 Singularities of differentiable maps (Birkhauser, Boston, Ma.) [35] Thom R 1989 Structural stability and morphogenesis (Addison-Wesley, P.C. ,CA) [36] 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) [37] Murphy G 1960 Ordinary differential equations and their solutions (Van Vostrand,)
NO ©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.
NO Universidad Complutense de Madrid
NO Comisión Interministerial de Ciencia y Tecnología
DS Docta Complutense
RD 4 may 2024