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Loop equations for generalised eigenvalue models

Published 21 Feb 2024 in hep-th | (2402.13835v3)

Abstract: We derive the loop equation for the 1-matrix model with generic difference-type measure for eigenvalues and develop a recursive algebraic framework for solving it to an arbitrary order in the coupling constant in and beyond the planar approximation. The planar limit is solved exactly for a one-parametric family of models and in the general case at strong coupling. The Wilson loop in the N=2* super-Yang-Mills theory and the Hoppe model are used to illustrate our methods.

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References (91)
  1. Y. M. Makeenko and A. A. Migdal, “Exact Equation for the Loop Average in Multicolor QCD”, Phys. Lett. B 88, 135 (1979), [Erratum: Phys.Lett.B 89, 437 (1980)].
  2. A. A. Migdal, “Loop Equations and 1/N Expansion”, Phys. Rept. 102, 199 (1983).
  3. G. ’t Hooft, “A Planar Diagram Theory for Strong Interactions”, Nucl.Phys. B72, 461 (1974).
  4. S. R. Wadia, “On the Dyson-schwinger Equations Approach to the Large N𝑁Nitalic_N Limit: Model Systems and String Representation of Yang-Mills Theory”, Phys. Rev. D 24, 970 (1981).
  5. V. A. Kazakov, “The Appearance of Matter Fields from Quantum Fluctuations of 2D Gravity”, Mod. Phys. Lett. A 4, 2125 (1989).
  6. Y. Makeenko, “Loop equations in matrix models and in 2-D quantum gravity”, Mod. Phys. Lett. A 6, 1901 (1991).
  7. P. Di Francesco, P. H. Ginsparg and J. Zinn-Justin, “2-D Gravity and random matrices”, Phys. Rept. 254, 1 (1995), hep-th/9306153.
  8. E. Brezin, C. Itzykson, G. Parisi and J. B. Zuber, “Planar Diagrams”, Commun. Math. Phys. 59, 35 (1978).
  9. D. Bessis, “A New Method in the Combinatorics of the Topological Expansion”, Commun. Math. Phys. 69, 147 (1979).
  10. D. Bessis, C. Itzykson and J. B. Zuber, “Quantum field theory techniques in graphical enumeration”, Adv. Appl. Math. 1, 109 (1980).
  11. J. Ambjørn, L. Chekhov and Y. Makeenko, “Higher genus correlators and W infinity from the Hermitian one matrix model”, Phys. Lett. B 282, 341 (1992), hep-th/9203009.
  12. J. Ambjørn, L. Chekhov, C. F. Kristjansen and Y. Makeenko, “Matrix model calculations beyond the spherical limit”, Nucl. Phys. B 404, 127 (1993), hep-th/9302014, [Erratum: Nucl.Phys.B 449, 681–681 (1995)].
  13. B. Eynard and N. Orantin, “Invariants of algebraic curves and topological expansion”, Commun. Num. Theor. Phys. 1, 347 (2007), math-ph/0702045.
  14. L. Chekhov and B. Eynard, “Matrix eigenvalue model: Feynman graph technique for all genera”, JHEP 0612, 026 (2006), math-ph/0604014.
  15. L. O. Chekhov, B. Eynard and O. Marchal, “Topological expansion of β𝛽\betaitalic_β-ensemble model and quantum algebraic geometry in the sectorwise approach”, Theor. Math. Phys. 166, 141 (2011), arxiv:1009.6007.
  16. A. Brini, M. Marino and S. Stevan, “The Uses of the refined matrix model recursion”, J. Math. Phys. 52, 052305 (2011), arxiv:1010.1210.
  17. A. Mironov, A. Morozov, A. Popolitov and S. Shakirov, “Resolvents and Seiberg-Witten representation for Gaussian beta-ensemble”, Theor. Math. Phys. 171, 505 (2012), arxiv:1103.5470.
  18. O. Marchal, “One-cut solution of the β𝛽\betaitalic_β ensembles in the Zhukovsky variable”, J. Stat. Mech. 1201, P01011 (2012), arxiv:1105.0453.
  19. N. S. Witte and P. J. Forrester, “Moments of the Gaussian β𝛽\betaitalic_β Ensembles and the large-N𝑁Nitalic_N expansion of the densities”, J. Math. Phys. 55, 083302 (2014), arxiv:1310.8498.
  20. J. Hoppe, “Quantum theory of a massless relativistic surface and a two-dimensional bound state problem”, Soryushiron Kenkyu Electronics 80, no. 3, 145 (1989).
  21. V. A. Kazakov, I. K. Kostov and N. A. Nekrasov, “D particles, matrix integrals and KP hierarchy”, Nucl. Phys. B 557, 413 (1999), hep-th/9810035.
  22. M. Marino, “Chern-Simons theory, matrix integrals, and perturbative three manifold invariants”, Commun. Math. Phys. 253, 25 (2004), hep-th/0207096.
  23. M. Aganagic, A. Klemm, M. Marino and C. Vafa, “Matrix model as a mirror of Chern-Simons theory”, JHEP 0402, 010 (2004), hep-th/0211098.
  24. M. Tierz, “Soft matrix models and Chern-Simons partition functions”, Mod. Phys. Lett. A 19, 1365 (2004), hep-th/0212128.
  25. J. K. Erickson, G. W. Semenoff and K. Zarembo, “Wilson loops in N=4 supersymmetric Yang-Mills theory”, Nucl. Phys. B 582, 155 (2000), hep-th/0003055.
  26. N. Drukker and D. J. Gross, “An Exact prediction of N=4 SUSYM theory for string theory”, J. Math. Phys. 42, 2896 (2001), hep-th/0010274.
  27. V. Pestun, “Localization of gauge theory on a four-sphere and supersymmetric Wilson loops”, Commun. Math. Phys. 313, 71 (2012), arxiv:0712.2824.
  28. A. Brini, B. Eynard and M. Marino, “Torus knots and mirror symmetry”, Annales Henri Poincare 13, 1873 (2012), arxiv:1105.2012.
  29. A. Morozov, A. Popolitov and S. Shakirov, “Harer-Zagier formulas for knot matrix models”, Phys. Lett. B 818, 136370 (2021), arxiv:2102.11187.
  30. J. G. Russo and K. Zarembo, “Large N Limit of N=2 SU(N) Gauge Theories from Localization”, JHEP 1210, 082 (2012), arxiv:1207.3806.
  31. A. Buchel, J. G. Russo and K. Zarembo, “Rigorous Test of Non-conformal Holography: Wilson Loops in N=2* Theory”, JHEP 1303, 062 (2013), arxiv:1301.1597.
  32. J. G. Russo and K. Zarembo, “Evidence for Large-N Phase Transitions in N=2* Theory”, JHEP 1304, 065 (2013), arxiv:1302.6968.
  33. X. Chen-Lin, J. Gordon and K. Zarembo, “𝒩=2∗\mathcal{N}={2}{\ast}caligraphic_N = 2 ∗ super-Yang-Mills theory at strong coupling”, JHEP 1411, 057 (2014), arxiv:1408.6040.
  34. K. Zarembo, “Strong-Coupling Phases of Planar N=2* Super-Yang-Mills Theory”, Theor.Math.Phys. 181, 1522 (2014), arxiv:1410.6114.
  35. J. G. Russo, E. Widén and K. Zarembo, “N𝑁Nitalic_N = 2*{}^{*}start_FLOATSUPERSCRIPT * end_FLOATSUPERSCRIPT phase transitions and holography”, JHEP 1902, 196 (2019), arxiv:1901.02835.
  36. A. V. Belitsky and G. P. Korchemsky, “Circular Wilson loop in N=2* super Yang-Mills theory at two loops and localization”, JHEP 2104, 089 (2021), arxiv:2003.10448.
  37. S.-J. Rey and T. Suyama, “Exact Results and Holography of Wilson Loops in N=2 Superconformal (Quiver) Gauge Theories”, JHEP 1101, 136 (2011), arxiv:1001.0016.
  38. F. Passerini and K. Zarembo, “Wilson Loops in N=2 Super-Yang-Mills from Matrix Model”, JHEP 1109, 102 (2011), arxiv:1106.5763, [Erratum: JHEP 10, 065 (2011)].
  39. M. Beccaria, M. Billò, M. Frau, A. Lerda and A. Pini, “Exact results in a 𝒩𝒩\mathcal{N}caligraphic_N = 2 superconformal gauge theory at strong coupling”, JHEP 2107, 185 (2021), arxiv:2105.15113.
  40. M. Billo, M. Frau, A. Lerda, A. Pini and P. Vallarino, “Three-point functions in a 𝒩𝒩\mathcal{N}caligraphic_N = 2 superconformal gauge theory and their strong-coupling limit”, JHEP 2208, 199 (2022), arxiv:2202.06990.
  41. M. Beccaria, G. P. Korchemsky and A. A. Tseytlin, “Strong coupling expansion in N=2 superconformal theories and the Bessel kernel”, JHEP 2209, 226 (2022), arxiv:2207.11475.
  42. N. Bobev, P.-J. De Smet and X. Zhang, “The planar limit of the 𝒩=2𝒩2\mathcal{N}=2caligraphic_N = 2𝐄𝐄\mathbf{E}bold_E-theory: numerical calculations and the large λ𝜆\lambdaitalic_λ expansion”, arxiv:2207.12843.
  43. A. Pini and P. Vallarino, “Defect correlators in a 𝒩𝒩\mathcal{N}caligraphic_N = 2 SCFT at strong coupling”, JHEP 2306, 050 (2023), arxiv:2303.08210.
  44. M. Beccaria, G. P. Korchemsky and A. A. Tseytlin, “Non-planar corrections in orbifold/orientifold 𝒩𝒩\mathcal{N}caligraphic_N = 2 superconformal theories from localization”, JHEP 2305, 165 (2023), arxiv:2303.16305.
  45. M. Marino, “Lectures on localization and matrix models in supersymmetric Chern-Simons-matter theories”, J. Phys. A 44, 463001 (2011), arxiv:1104.0783.
  46. M. Mariño, “Lectures on non-perturbative effects in large N𝑁Nitalic_N gauge theories, matrix models and strings”, Fortsch. Phys. 62, 455 (2014), arxiv:1206.6272.
  47. D. Anninos and B. Mühlmann, “Notes on matrix models (matrix musings)”, J. Stat. Mech. 2008, 083109 (2020), arxiv:2004.01171.
  48. P. H. Ginsparg, “Matrix models of 2-d gravity”, hep-th/9112013.
  49. F. David, “Planar Diagrams, Two-Dimensional Lattice Gravity and Surface Models”, Nucl. Phys. B 257, 45 (1985).
  50. D. J. Gross, T. Piran and S. Weinberg, “Two-Dimensional Quantum Gravity and Random Surfaces. Proceedings, 8th Winter School for Theoretical Physics, Jerusalem, Israel, December 27, 1990 - January 4, 1991”.
  51. N. Witte and P. Forrester, “Moments of the Gaussian β𝛽\betaitalic_β Ensembles and the large-N expansion of the densities”, Journal of Mathematical Physics 55, 083302 (2014).
  52. J. Ambjorn and G. Akemann, “New universal spectral correlators”, J. Phys. A 29, L555 (1996), cond-mat/9606129.
  53. G. Akemann, “Universal correlators for multiarc complex matrix models”, Nucl. Phys. B 507, 475 (1997), hep-th/9702005.
  54. I. Gradshteyn and I. Ryzhik, “Table of Integrals, Series, and Products”, Elsevier Science (2014).
  55. V. Kazakov and Z. Zheng, “Analytic and numerical bootstrap for one-matrix model and “unsolvable” two-matrix model”, JHEP 2206, 030 (2022), arxiv:2108.04830.
  56. G. Akemann and P. H. Damgaard, “Wilson loops in N𝑁Nitalic_N=4 supersymmetric Yang-Mills theory from random matrix theory”, Phys. Lett. B 513, 179 (2001), hep-th/0101225, [Erratum: Phys.Lett.B 524, 400–400 (2002)].
  57. S. Kawamoto, T. Kuroki and A. Miwa, “Boundary condition for D-brane from Wilson loop, and gravitational interpretation of eigenvalue in matrix model in AdS/CFT correspondence”, Phys. Rev. D 79, 126010 (2009), arxiv:0812.4229.
  58. K. Okuyama, “’t Hooft expansion of 1/2 BPS Wilson loop”, JHEP 0609, 007 (2006), hep-th/0607131.
  59. M. Beccaria and A. A. Tseytlin, “On the structure of non-planar strong coupling corrections to correlators of BPS Wilson loops and chiral primary operators”, JHEP 2101, 149 (2021), arxiv:2011.02885.
  60. N. Drukker, S. Giombi, R. Ricci and D. Trancanelli, “Supersymmetric Wilson loops on S**3”, JHEP 0805, 017 (2008), arxiv:0711.3226.
  61. V. Pestun, “Localization of the four-dimensional N=4 SYM to a two-sphere and 1/8 BPS Wilson loops”, JHEP 1212, 067 (2012), arxiv:0906.0638.
  62. A. S. Alexandrov, A. Mironov and A. Morozov, “Partition functions of matrix models as the first special functions of string theory. 1. Finite size Hermitean one matrix model”, Int. J. Mod. Phys. A 19, 4127 (2004), hep-th/0310113.
  63. J. Ambjorn, J. Jurkiewicz and Y. M. Makeenko, “Multiloop correlators for two-dimensional quantum gravity”, Phys. Lett. B 251, 517 (1990).
  64. S. Giombi, V. Pestun and R. Ricci, “Notes on supersymmetric Wilson loops on a two-sphere”, JHEP 1007, 088 (2010), arxiv:0905.0665.
  65. K. Okuyama, “Connected correlator of 1/2 BPS Wilson loops in 𝒩=4𝒩4\mathcal{N}=4caligraphic_N = 4 SYM”, JHEP 1810, 037 (2018), arxiv:1808.10161.
  66. J. G. Russo and K. Zarembo, “Massive N=2 Gauge Theories at Large N”, JHEP 1311, 130 (2013), arxiv:1309.1004.
  67. N. Beisert, V. Dippel and M. Staudacher, “A Novel long range spin chain and planar N=4 super Yang-Mills”, JHEP 0407, 075 (2004), hep-th/0405001.
  68. N. Beisert, B. Eden and M. Staudacher, “Transcendentality and Crossing”, J. Stat. Mech. 0701, P01021 (2007), hep-th/0610251.
  69. N. Gromov, “Introduction to the Spectrum of N=4𝑁4N=4italic_N = 4 SYM and the Quantum Spectral Curve”, arxiv:1708.03648.
  70. A. Pini, D. Rodriguez-Gomez and J. G. Russo, “Large N𝑁Nitalic_N correlation functions 𝒩=𝒩absent\mathcal{N}=caligraphic_N = 2 superconformal quivers”, JHEP 1708, 066 (2017), arxiv:1701.02315.
  71. M. Beccaria, M. Billò, F. Galvagno, A. Hasan and A. Lerda, “𝒩𝒩\mathcal{N}caligraphic_N = 2 Conformal SYM theories at large 𝒩𝒩\mathcal{N}caligraphic_N”, JHEP 2009, 116 (2020), arxiv:2007.02840.
  72. M. Beccaria, G. V. Dunne and A. A. Tseytlin, “BPS Wilson loop in 𝒩𝒩\mathcal{N}caligraphic_N = 2 superconformal SU(N) “orientifold” gauge theory and weak-strong coupling interpolation”, JHEP 2107, 085 (2021), arxiv:2104.12625.
  73. M. Beccaria, G. V. Dunne and A. A. Tseytlin, “Strong coupling expansion of free energy and BPS Wilson loop in 𝒩𝒩\mathcal{N}caligraphic_N = 2 superconformal models with fundamental hypermultiplets”, JHEP 2108, 102 (2021), arxiv:2105.14729.
  74. M. Billo, M. Frau, F. Galvagno, A. Lerda and A. Pini, “Strong-coupling results for 𝒩𝒩\mathcal{N}caligraphic_N = 2 superconformal quivers and holography”, JHEP 2110, 161 (2021), arxiv:2109.00559.
  75. D. J. Binder, S. M. Chester, S. S. Pufu and Y. Wang, “𝒩𝒩\mathcal{N}caligraphic_N = 4 Super-Yang-Mills correlators at strong coupling from string theory and localization”, JHEP 1912, 119 (2019), arxiv:1902.06263.
  76. S. S. Pufu, V. A. Rodriguez and Y. Wang, “Scattering From (p,q)𝑝𝑞(p,q)( italic_p , italic_q )-Strings in 𝐴𝑑𝑆5×𝑆5subscript𝐴𝑑𝑆5superscript𝑆5\text{AdS}_{5}\times\text{S}^{5}AdS start_POSTSUBSCRIPT 5 end_POSTSUBSCRIPT × S start_POSTSUPERSCRIPT 5 end_POSTSUPERSCRIPT”, arxiv:2305.08297.
  77. M. Billo’, F. Galvagno, M. Frau and A. Lerda, “Integrated correlators with a Wilson line in 𝒩𝒩\mathcal{N}caligraphic_N = 4 SYM”, JHEP 2312, 047 (2023), arxiv:2308.16575.
  78. W. Janke, “Resummation of Divergent Perturbation Series: Introduction to Theory & Guide to Practical Applications”.
  79. C. M. Bender, S. Orszag and S. A. Orszag, “Advanced mathematical methods for scientists and engineers I: Asymptotic methods and perturbation theory”.
  80. M. Beccaria and A. A. Tseytlin, “1/N1𝑁1/N1 / italic_N expansion of circular Wilson loop in 𝒩=2𝒩2\mathcal{N}=2caligraphic_N = 2 superconformal S⁢U⁢(N)×S⁢U⁢(N)𝑆𝑈𝑁𝑆𝑈𝑁SU(N)\times SU(N)italic_S italic_U ( italic_N ) × italic_S italic_U ( italic_N ) quiver”, JHEP 2104, 265 (2021), arxiv:2102.07696.
  81. A. V. Kotikov, L. N. Lipatov and V. N. Velizhanin, “Anomalous dimensions of Wilson operators in N=4 SYM theory”, Phys. Lett. B 557, 114 (2003), hep-ph/0301021.
  82. K. Zarembo, “Quiver CFT at strong coupling”, JHEP 2006, 055 (2020), arxiv:2003.00993.
  83. H. Ouyang, “Wilson loops in circular quiver SCFTs at strong coupling”, JHEP 2102, 178 (2021), arxiv:2011.03531.
  84. N. Drukker and B. Fiol, “All-genus calculation of Wilson loops using D-branes”, JHEP 0502, 010 (2005), hep-th/0501109.
  85. A. E. Lawrence, N. Nekrasov and C. Vafa, “On conformal field theories in four-dimensions”, Nucl. Phys. B 533, 199 (1998), hep-th/9803015.
  86. F. Galvagno and M. Preti, “Chiral correlators in 𝒩𝒩\mathcal{N}caligraphic_N = 2 superconformal quivers”, JHEP 2105, 201 (2021), arxiv:2012.15792.
  87. F. Galvagno and M. Preti, “Wilson loop correlators in 𝒩𝒩\mathcal{N}caligraphic_N = 2 superconformal quivers”, JHEP 2111, 023 (2021), arxiv:2105.00257.
  88. L. Alvarez-Gaume, H. Itoyama, J. L. Manes and A. Zadra, “Superloop equations and two-dimensional supergravity”, Int. J. Mod. Phys. A 7, 5337 (1992), hep-th/9112018.
  89. J. C. Plefka, “Iterative solution of the supereigenvalue model”, Nucl. Phys. B 444, 333 (1995), hep-th/9501120.
  90. J. C. Plefka, “The Supereigenvalue model in the double scaling limit”, Nucl. Phys. B 448, 355 (1995), hep-th/9504089.
  91. Y. Makeenko and K. Zarembo, “Adjoint fermion matrix models”, Nucl. Phys. B 422, 237 (1994), hep-th/9309012.

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