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Random supermatrices with an external source

A preprint version of the article is available at arXiv.


In the past we have considered Gaussian random matrix ensembles in the presence of an external matrix source. The reason was that it allowed, through an appropriate tuning of the eigenvalues of the source, to obtain results on non-trivial dual models, such as Kontsevich’s Airy matrix models and generalizations. The techniques relied on explicit computations of the k-point functions for arbitrary N (the size of the matrices) and on an N-k duality. Numerous results on the intersection numbers of the moduli space of curves were obtained by this technique. In order to generalize these results to include surfaces with boundaries, we have extended these techniques to supermatrices. Again we have obtained quite remarkable explicit expressions for the k-point functions, as well as a duality. Although supermatrix models a priori lead to the same matrix models of 2d-gravity, the external source extensions considered in this article lead to new geometric results.


  1. [1]

    E. Brézin and S. Hikami, Random matrix theory with an external source, in Springer briefs in mathematical physics 19, (2016).

  2. [2]

    E. Brézin and S. Hikami, Random Matrix, Singularities and Open/Close Intersection Numbers, J. Phys. A 48 (2015) 475201 [arXiv:1502.01416] [INSPIRE].

  3. [3]

    M. Kontsevich, Intersection theory on the moduli space of curves and the matrix Airy function, Commun. Math. Phys. 147 (1992) 1 [INSPIRE].

  4. [4]

    E. Witten, Algebraic geometry associated with matrix models of two dimensional gravity, in Topological Methods in Modern Mathematics, Publish and Perish, Houston (1993), p. 235.

  5. [5]

    R. Dijkgraaf and E. Witten, Developments in Topological Gravity, arXiv:1804.03275 [INSPIRE].

  6. [6]

    F.A. Berezin, The Method of Second Quantization, Academic press, New-York (1966).

  7. [7]

    E. Witten, Notes On Super Riemann Surfaces And Their Moduli, arXiv:1209.2459 [INSPIRE].

  8. [8]

    R.C. Penner and A.M. Zeitlin, Decorated Super-Teichmüller Space, arXiv:1509.06302 [INSPIRE].

  9. [9]

    J. Alfaro, R. Medina and L.F. Urrutia, The Itzykson-Zuber integral for U(m|n), J. Math. Phys. 36 (1995) 3085 [hep-th/9412012] [INSPIRE].

  10. [10]

    Harish-Chandra, Differential Operators on a Semisimple Lie Algebra Differential Operators on a Semisimple Lie Algebra, Amer. J. Math. 79 (1957) 87.

  11. [11]

    T. Guhr and H. Kohler, Derivation of the supersymmetric Harish-Chandra integral for UOSp(k 1 /2k 2), J. Math. Phys. 45 (2004) 3636 [math-ph/0212060].

  12. [12]

    E. Brézin and S. Hikami, Intersection numbers from the antisymmetric Gaussian matrix model, JHEP 07 (2008) 050 [arXiv:0804.4531] [INSPIRE].

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Correspondence to S. Hikami.

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ArXiv ePrint: 1805.04240

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Brézin, E., Hikami, S. Random supermatrices with an external source. J. High Energ. Phys. 2018, 86 (2018).

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  • Matrix Models
  • Superspaces
  • Topological Field Theories