Matrix models for β-ensembles from Nekrasov partition functions

  • Piotr SułkowskiEmail author


We relate Nekrasov partition functions, with arbitrary values of ϵ 1, ϵ 2 parameters, to matrix models for β-ensembles. We find matrix models encoding the instanton part of Nekrasov partition functions, whose measure, to the leading order in ϵ 2 expansion, is given by the Vandermonde determinant to the power β = −ϵ 1/ϵ 2. An additional, trigonometric deformation of the measure arises in five-dimensional theories. Matrix model potentials, to the leading order in ϵ 2 expansion, are the same as in the β = 1 case considered in 0810.4944 [hep-th]. We point out that potentials for massive hypermultiplets include multi-log, Penner-like terms. Inclusion of Chern-Simons terms in five-dimensional theories leads to multi-matrix models. The role of these matrix models in the context of the AGT conjecture is discussed.


Matrix Models Extended Supersymmetry Topological Strings 


  1. [1]
    N. Seiberg and E. Witten, Monopole Condensation, And Confinement In N = 2 Supersymmetric Yang-Mills Theory, Nucl. Phys. B 426 (1994) 19 [hep-th/9407087] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  2. [2]
    N. Seiberg and E. Witten, Monopoles, duality and chiral symmetry breaking in N = 2 supersymmetric QCD, Nucl. Phys. B 431 (1994) 484 [hep-th/9408099] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  3. [3]
    N.A. Nekrasov, Seiberg-Witten Prepotential From Instanton Counting, Adv. Theor. Math. Phys. 7 (2004) 831 [hep-th/0206161] [SPIRES].MathSciNetGoogle Scholar
  4. [4]
    N. Nekrasov and A. Okounkov, Seiberg-Witten theory and random partitions, hep-th/0306238 [SPIRES].
  5. [5]
    H. Nakajima and K. Yoshioka, Instanton counting on blowup. I 4-dimensional pure gauge theory, math/0306198 [SPIRES].
  6. [6]
    H. Nakajima and K. Yoshioka, Instanton counting on blowup. II: K-theoretic partition function, math/0505553 [SPIRES].
  7. [7]
    N. Nekrasov and S. Shadchin, ABCD of instantons, Commun. Math. Phys. 252 (2004) 359 [hep-th/0404225] [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  8. [8]
    R. Dijkgraaf and C. Vafa, Matrix models, topological strings and supersymmetric gauge theories, Nucl. Phys. B 644 (2002) 3 [hep-th/0206255] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  9. [9]
    R. Dijkgraaf and C. Vafa, A Perturbative Window into Non-Perturbative Physics, hep-th/0208048 [SPIRES].
  10. [10]
    S. Katz, A. Klemm, C. Vafa, Geometric engineering of quantum field theories, Nucl. Phys. B497 (1997) 173 [hep-th/9609239] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  11. [11]
    M. Aganagic, R. Dijkgraaf, A. Klemm, M. Marino and C. Vafa, Topological strings and integrable hierarchies, Commun. Math. Phys. 261 (2006) 451 hep-th/0312085 [SPIRES].zbMATHCrossRefMathSciNetADSGoogle Scholar
  12. [12]
    A. Iqbal and A.-K. Kashani-Poor, SU(N) geometries and topological string amplitudes, Adv. Theor. Math. Phys. 10 (2006) 1 [hep-th/0306032] [SPIRES].zbMATHMathSciNetGoogle Scholar
  13. [13]
    T.J. Hollowood, A. Iqbal and C. Vafa, Matrix Models, Geometric Engineering and Elliptic Genera, JHEP 03 (2008) 069 [hep-th/0310272] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  14. [14]
    A. Iqbal, C. Kozcaz and C. Vafa, The refined topological vertex, JHEP 10 (2009) 069 [hep-th/0701156] [SPIRES].CrossRefADSGoogle Scholar
  15. [15]
    H. Awata and H. Kanno, Instanton counting, Macdonald functions and the moduli space of D-branes, JHEP 05 (2005) 039 [hep-th/0502061] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  16. [16]
    M. Taki, Refined Topological Vertex and Instanton Counting, JHEP 03 (2008) 048 [arXiv:0710.1776] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  17. [17]
    H. Awata and H. Kanno, Refined BPS state counting from Nekrasov’s formula and Macdonald functions, Int. J. Mod. Phys. A 24 (2009) 2253 [arXiv:0805.0191] [SPIRES].MathSciNetADSGoogle Scholar
  18. [18]
    R. Dijkgraaf, S. Gukov, V.A. Kazakov and C. Vafa, Perturbative analysis of gauged matrix models, Phys. Rev. D 68 (2003) 045007 [hep-th/0210238] [SPIRES].MathSciNetADSGoogle Scholar
  19. [19]
    A. Klemm, M. Mariño and S. Theisen, Gravitational corrections in supersymmetric gauge theory and matrix models, JHEP 03 (2003) 051 [hep-th/0211216] [SPIRES].CrossRefADSGoogle Scholar
  20. [20]
    A. Klemm, M. Marino and M. Rauch, Direct Integration and Non-Perturbative Effects in Matrix Models, arXiv:1002.3846 [SPIRES].
  21. [21]
    A. Klemm and P. Sulkowski, Seiberg-Witten theory and matrix models, Nucl. Phys. B 819 (2009) 400 [arXiv:0810.4944] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  22. [22]
    P. Sulkowski, Matrix models for 2* theories, Phys. Rev. D 80 (2009) 086006 [arXiv:0904.3064] [SPIRES].ADSGoogle Scholar
  23. [23]
    L.F. Alday, D. Gaiotto and Y. Tachikawa, Liouville Correlation Functions from Four-dimensional Gauge Theories, Lett. Math. Phys. 91 (2010) 167 [arXiv:0906.3219] [SPIRES].zbMATHCrossRefADSGoogle Scholar
  24. [24]
    R. Dijkgraaf and C. Vafa, Toda Theories, Matrix Models, Topological Strings and N = 2 Gauge Systems, arXiv:0909.2453 [SPIRES].
  25. [25]
    M. Mehta, Random matrices, Pure and Applied Mathematics Series vol. 142, Elsevier B.V., Amsterdam Netherlands (2004).zbMATHGoogle Scholar
  26. [26]
    B. Eynard and O. Marchal, Topological expansion of the Bethe ansatz and non- commutative algebraic geometry, JHEP 03 (2009) 094 [arXiv:0809.3367] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  27. [27]
    L. Chekhov, B. Eynard and O. Marchal, Topological expansion of the Bethe ansatz and quantum algebraic geometry, arXiv:0911.1664 [SPIRES].
  28. [28]
    B. Eynard, All orders asymptotic expansion of large partitions, J. Stat. Mech. (2008) P07023 [arXiv:0804.0381] [SPIRES].
  29. [29]
    G. Borot, B. Eynard, M. Mulase and B. Safnuk, A matrix model for simple Hurwitz numbers and topological recursion, arXiv:0906.1206 [SPIRES].
  30. [30]
    A. Morozov and S. Shakirov, Generation of Matrix Models by W-operators, JHEP 04 (2009) 064 [arXiv:0902.2627] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  31. [31]
    A. Morozov and S. Shakirov, On Equivalence of two Hurwitz Matrix Models, Mod. Phys. Lett. A 24 (2009) 2659 [arXiv:0906.2573] [SPIRES].MathSciNetADSGoogle Scholar
  32. [32]
    B. Eynard, A Matrix model for plane partitions and TASEP, J. Stat. Mech. (2009) P10011 [arXiv:0905.0535] [SPIRES].
  33. [33]
    B. Eynard and N. Orantin, Invariants of algebraic curves and topological expansion, math-ph/0702045.
  34. [34]
    V. Bouchard, A. Klemm, M. Mariño and S. Pasquetti, Remodeling the B-model, Commun. Math. Phys. 287 (2009) 117 [arXiv:0709.1453] [SPIRES].zbMATHCrossRefADSGoogle Scholar
  35. [35]
    M.-x. Huang and A. Klemm, Holomorphicity and Modularity in Seiberg-Witten Theories with Matter, arXiv:0902.1325 [SPIRES].
  36. [36]
    D. Gaiotto, N = 2 dualities, arXiv:0904.2715 [SPIRES].
  37. [37]
    N. Wyllard, A N−1 conformal Toda field theory correlation functions from conformal \( \mathcal{N} = 2 \) SU(N) quiver gauge theories, JHEP 11 (2009) 002 [arXiv:0907.2189] [SPIRES].CrossRefADSGoogle Scholar
  38. [38]
    A. Mironov and A. Morozov, The Power of Nekrasov Functions, Phys. Lett. B 680 (2009) 188 [arXiv:0908.2190] [SPIRES].MathSciNetADSGoogle Scholar
  39. [39]
    A. Mironov and A. Morozov, On AGT relation in the case of U(3), Nucl. Phys. B 825 (2010) 1 [arXiv:0908.2569] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  40. [40]
    A. Marshakov, A. Mironov and A. Morozov, Zamolodchikov asymptotic formula and instanton expansion in \( \mathcal{N} = 2 \) SUSY N f = 2N c QCD, JHEP 11 (2009) 048 [arXiv:0909.3338] [SPIRES].CrossRefADSGoogle Scholar
  41. [41]
    A. Mironov and A. Morozov, Proving AGT relations in the large-c limit, Phys. Lett. B 682 (2009) 118 [arXiv:0909.3531] [SPIRES].MathSciNetADSGoogle Scholar
  42. [42]
    L.F. Alday, D. Gaiotto, S. Gukov, Y. Tachikawa and H. Verlinde, Loop and surface operators in N = 2 gauge theory and Liouville modular geometry, JHEP 01 (2010) 113 [arXiv:0909.0945] [SPIRES].CrossRefGoogle Scholar
  43. [43]
    N. Drukker, J. Gomis, T. Okuda and J. Teschner, Gauge Theory Loop Operators and Liouville Theory, JHEP 02 (2010) 057 [arXiv:0909.1105] [SPIRES].CrossRefGoogle Scholar
  44. [44]
    H. Awata and Y. Yamada, Five-dimensional AGT Conjecture and the Deformed Virasoro Algebra, JHEP 01 (2010) 125 [arXiv:0910.4431] [SPIRES].CrossRefGoogle Scholar
  45. [45]
    H. Itoyama, K. Maruyoshi and T. Oota, Notes on the Quiver Matrix Model and 2d-4d Conformal Connection, arXiv:0911.4244 [SPIRES].
  46. [46]
    T. Eguchi and K. Maruyoshi, Penner Type Matrix Model and Seiberg-Witten Theory, JHEP 02 (2010) 022 [arXiv:0911.4797] [SPIRES].CrossRefGoogle Scholar
  47. [47]
    R. Schiappa and N. Wyllard, An A r threesome: Matrix models, 2d CFTs and 4d \( \mathcal{N} = 2 \) gauge theories, arXiv:0911.5337 [SPIRES].
  48. [48]
    A. Mironov, A. Morozov and S. Shakirov, Matrix Model Conjecture for Exact BS Periods and Nekrasov Functions, JHEP 02 (2010) 030 [arXiv:0911.5721] [SPIRES].CrossRefGoogle Scholar
  49. [49]
    M. Fujita, Y. Hatsuda and T.-S. Tai, Genus-one correction to asymptotically free Seiberg-Witten prepotential from Dijkgraaf-Vafa matrix model, JHEP 03 (2010) 046 [arXiv:0912.2988] [SPIRES].CrossRefGoogle Scholar
  50. [50]
    A. Marshakov, A. Mironov and A. Morozov, Generalized matrix models as conformal field theories: Discrete case, Phys. Lett. B 265 (1991) 99 [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  51. [51]
    I. Kostov, Conformal field theory techniques in random matrix models, hep-th/9907060 [SPIRES].
  52. [52]
    R. Dijkgraaf, L. Hollands, P. Sulkowski and C. Vafa, Supersymmetric Gauge Theories, Intersecting Branes and Free Fermions, JHEP 02 (2008) 106 [arXiv:0709.4446] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  53. [53]
    R. Dijkgraaf, L. Hollands and P. Sulkowski, Quantum Curves and \( \mathcal{D} - Modules \), JHEP 11 (2009) 047 [arXiv:0810.4157] [SPIRES].CrossRefADSGoogle Scholar
  54. [54]
    K.A. Intriligator, D.R. Morrison and N. Seiberg, Five-dimensional supersymmetric gauge theories and degenerations of Calabi-Yau spaces, Nucl. Phys. B 497 (1997) 56 [hep-th/9702198] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  55. [55]
    Y. Tachikawa, Five-dimensional Chern-Simons terms and Nekrasov’s instanton counting, JHEP 02 (2004) 050 [hep-th/0401184] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  56. [56]
    P. Sulkowski, Deformed boson-fermion correspondence, Q-bosons and topological strings on the conifold, JHEP 10 (2008) 104 [arXiv:0808.2327] [SPIRES].CrossRefMathSciNetADSGoogle Scholar
  57. [57]
    M. Abramowitz and I. Stegun, Handbook of mathematical functions, Applied Mathematics Series vol.55, Dover Publications, New York U.S.A. (1972).zbMATHGoogle Scholar
  58. [58]
    F. Tricomi and A. Erdelyi, The Asymptotic Expansion of a Ratio of Gamma Functions, Pacific J. Math. 1 (1951) 133.zbMATHMathSciNetGoogle Scholar
  59. [59]
    G. Gasper, q-Extensions of Erdelyi’s Fractional Integral Representations for Hypergeometric Functions and Some Summation Formulas for Double q-Kampe de Feriet Series, Contemporary Mathematics vol.254, AMS, Boston U.S.A. (2000).Google Scholar
  60. [60]
    D. Moak, The q-analogue of Stirling’s formula, Rocky Mountain J. Math. 14 (1984) 403.zbMATHCrossRefMathSciNetGoogle Scholar
  61. [61]
    M. Mansour, An asymptotic expansion of the q-gamma function Γq(x), Journal of Nonlinear Math. Phys. 13 (2006) 479.zbMATHCrossRefMathSciNetADSGoogle Scholar

Copyright information

© SISSA, Trieste, Italy 2010

Authors and Affiliations

  1. 1.California Institute of TechnologyPasadenaU.S.A.

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