Exact solution of Chern-Simons-matter matrix models with characteristic/orthogonal polynomials

  • Miguel TierzEmail author
Open Access
Regular Article - Theoretical Physics


We solve for finite N the matrix model of supersymmetric U(N ) Chern-Simons theory coupled to N f fundamental and N f anti-fundamental chiral multiplets of R-charge 1/2 and of mass m, by identifying it with an average of inverse characteristic polynomials in a Stieltjes-Wigert ensemble. This requires the computation of the Cauchy transform of the Stieltjes-Wigert polynomials, which we carry out, finding a relationship with Mordell integrals, and hence with previous analytical results on the matrix model. The semiclassical limit of the model is expressed, for arbitrary N f , in terms of a single Hermite polynomial. This result also holds for more general matter content, involving matrix models with doublesine functions.


Matrix Models Chern-Simons Theories 


Open Access

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  1. [1]
    V. Pestun, Localization of gauge theory on a four-sphere and supersymmetric Wilson loops, Commun. Math. Phys. 313 (2012) 71 [arXiv:0712.2824] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    V. Pestun, Localization for \( \mathcal{N}=2 \) Supersymmetric Gauge Theories in Four Dimensions, arXiv:1412.7134 [INSPIRE].
  3. [3]
    K. Hosomichi, The localization principle in SUSY gauge theories, PTEP 2015 (2015) 11B101 [arXiv:1502.04543] [INSPIRE].
  4. [4]
    P.J. Forrester, Log-gases and random matrices, Princeton University Press (2010).Google Scholar
  5. [5]
    A. Kapustin, B. Willett and I. Yaakov, Exact Results for Wilson Loops in Superconformal Chern-Simons Theories with Matter, JHEP 03 (2010) 089 [arXiv:0909.4559] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  6. [6]
    A. Kapustin, B. Willett and I. Yaakov, Nonperturbative Tests of Three-Dimensional Dualities, JHEP 10 (2010) 013 [arXiv:1003.5694] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    D.L. Jafferis, The Exact Superconformal R-Symmetry Extremizes Z, JHEP 05 (2012) 159 [arXiv:1012.3210] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  8. [8]
    N. Hama, K. Hosomichi and S. Lee, Notes on SUSY Gauge Theories on Three-Sphere, JHEP 03 (2011) 127 [arXiv:1012.3512] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    A. Barranco and J.G. Russo, Large-N phase transitions in supersymmetric Chern-Simons theory with massive matter, JHEP 03 (2014) 012 [arXiv:1401.3672] [INSPIRE].ADSCrossRefGoogle Scholar
  10. [10]
    J.G. Russo, G.A. Silva and M. Tierz, Supersymmetric U(N) Chern-Simons-Matter Theory and Phase Transitions, Commun. Math. Phys. 338 (2015) 1411 [arXiv:1407.4794] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    L.J. Mordell, The definite integral \( {\displaystyle {\int}_{-\infty}^{\infty}\frac{e^{a{t}^2+bt}}{e^{ct}+d}} \) dt and the analytic theory of numbers, Acta Math. 61 (1933) 322.MathSciNetCrossRefGoogle Scholar
  12. [12]
    G. Giasemidis and M. Tierz, Mordell integrals and Giveon-Kutasov duality, JHEP 01 (2016) 068 [arXiv:1511.00203] [INSPIRE].ADSCrossRefGoogle Scholar
  13. [13]
    Y.V. Fyodorov, Negative moments of characteristic polynomials of random matrices: Ingham-Siegel integral as an alternative to Hubbard-Stratonovich transformation, Nucl. Phys. B 621 (2002) 643 [math-ph/0106006] [INSPIRE].
  14. [14]
    Y.V. Fyodorov and E. Strahov, An Exact formula for general spectral correlation function of random Hermitian matrices, J. Phys. A 36 (2003) 3203 [math-ph/0204051] [INSPIRE].
  15. [15]
    S. Zwegers, Mock Theta Functions, Ph.D. Thesis, Utrecht University (2002) [arXiv:0807.4834] [INSPIRE].
  16. [16]
    M. Tierz, Soft matrix models and Chern-Simons partition functions, Mod. Phys. Lett. A 19 (2004) 1365 [hep-th/0212128] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    P. Deift, Orthogonal Polynomials and Random Matrices: a Riemann-Hilbert Approach, AMS Courant Lecture Notes (2000).Google Scholar
  18. [18]
    S. Hyun and S.-H. Yi, Non-compact Topological Branes on Conifold, JHEP 11 (2006) 075 [hep-th/0609037] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    G. Szegö, Orthogonal Polynomials, fourth edition, Colloquium Publications of the American Mathematical Society, Volume XXIII (1975), section 2.7.Google Scholar
  20. [20]
    Y. Dolivet and M. Tierz, Chern-Simons matrix models and Stieltjes-Wigert polynomials, J. Math. Phys. 48 (2007) 023507 [hep-th/0609167] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    J.S. Christiansen, The moment problem associated with the Stieltjes-Wigert polynomials, J. Math. Anal. Appl. 277 (2003) 218.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    S. Wigert, Sur les polynômes orthogonaux et l’approximation des fonctions continues, Ark. Mat. Astronom. Fysik 17 (1923) 18.zbMATHGoogle Scholar
  23. [23]
    M.K. Atakishiyeva, N.M. Atakishiyev and T.H. Koornwinder, q-Extension of Mehta’s eigenvectors of the finite Fourier transform for q a root of unity, J. Phys. A Math. Gen. 42 (2009) 454004 [arXiv:0811.4100].
  24. [24]
    V. Spiridonov and A. Zhedanov, Zeros and orthogonality of the Askey-Wilson polynomials for q a root of unity, Duke Math. J. 89 (1997) 283 [arXiv:q-alg/9605034].MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    W. Gautschi and J. Wimp, Computing the Hilbert transform of a Jacobi weight function, BIT 27 (1987) 203.MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    W. Gautschi and J. Waldvogel, Computing the Hilbert Transform of the Generalized Laguerre and Hermite Weight Functions, BIT 41 (2001) 490.MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    H. Awata, S. Hirano and M. Shigemori, The Partition Function of ABJ Theory, PTEP 2013 (2013) 053B04 [arXiv:1212.2966] [INSPIRE].
  28. [28]
    Y.T. Li and R. Wong, Global Asymptotics of Stieltjes-Wigert Polynomials, Anal. Appl. 11 (2013) 1350028 [arXiv:1302.5193].MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    D.S. Lubinsky, The Size of (q; q)n for q on the Unit Circle, J. Number Theor. 76 (1999) 217.MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    A. Amariti and M. Siani, Z-extremization and F-theorem in Chern-Simons matter theories, JHEP 10 (2011) 016 [arXiv:1105.0933] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    F. Benini, C. Closset and S. Cremonesi, Comments on 3d Seiberg-like dualities, JHEP 10 (2011) 075 [arXiv:1108.5373] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    N. Hama, K. Hosomichi and S. Lee, SUSY Gauge Theories on Squashed Three-Spheres, JHEP 05 (2011) 014 [arXiv:1102.4716] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    M. Ledoux, Complex Hermite polynomials: from the semi-circular law to the circular law, Comm. Stoch. Anal. 2 (2008) 27.MathSciNetzbMATHGoogle Scholar
  34. [34]
    P.M. Bleher and A. Its, Double scaling limit in the random matrix model: the Riemann-Hilbert approach, Comm. Pure Appl. Math. 56 (2003) 433.MathSciNetCrossRefzbMATHGoogle Scholar
  35. [35]
    J.M. Maldacena, G.W. Moore, N. Seiberg and D. Shih, Exact vs. semiclassical target space of the minimal string, JHEP 10 (2004) 020 [hep-th/0408039] [INSPIRE].
  36. [36]
    Y. Hatsuda, S. Moriyama and K. Okuyama, Exact instanton expansion of the ABJM partition function, PTEP 2015 (2015) 11B104 [arXiv:1507.01678] [INSPIRE].
  37. [37]
    Y. Hatsuda, M. Honda and K. Okuyama, Large-N non-perturbative effects in \( \mathcal{N}=4 \) superconformal Chern-Simons theories, JHEP 09 (2015) 046 [arXiv:1505.07120] [INSPIRE].MathSciNetCrossRefGoogle Scholar

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© The Author(s) 2016

Authors and Affiliations

  1. 1.Departamento de Matemática, Grupo de Física Matemática, Faculdade de CiênciasUniversidade de LisboaLisboaPortugal

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