Abstract
We identify the Kontsevich–Penner matrix integral, for finite size n, with the isomonodromic tau function of a \(3\times 3\) rational connection on the Riemann sphere with n Fuchsian singularities placed in correspondence with the eigenvalues of the external field of the matrix integral. By formulating the isomonodromic system in terms of an appropriate Riemann–Hilbert boundary value problem, we can pass to the limit \(n\rightarrow \infty \) (at a formal level) and identify an isomonodromic system in terms of Miwa variables, which play the role of times of a KP hierarchy. This allows to derive the String and Dilaton equations via a purely Riemann–Hilbert approach. The expression of the formal limit of the partition function as an isomonodromic tau function allows us to derive explicit closed formulæ for the correlators of this matrix model in terms of the solution of the Riemann Hilbert problem with all times set to zero. These correlators have been conjectured to describe the intersection numbers for Riemann surfaces with boundaries, or open intersection numbers.
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Alexandrov, A., Buryak, A., Tessler, R.J.: Refined open intersection numbers and the Kontsevich–Penner matrix model. J. High Energy Phys. 2017(3), 123 (2017)
Alexandrov, A.: Open intersection numbers, Kontsevich–Penner model and cut-and-join operators. J. High Energy Phys. 2015(8), 28 (2015)
Alexandrov, A.: Open intersection numbers, matrix models and MKP hierarchy. J. High Energy Phys. 2015(3), 42 (2015)
Bertola, M., Cafasso, M.: Darboux transformations and random point processes. Int. Math. Res. Not. 2015(15), 6211 (2015)
Bertola, M., Cafasso, M.: The Kontsevich matrix integral: convergence to the Painlevé hierarchy and Stokes’ phenomenon. Commun. Math. Phys. 352(2), 585–619 (2017)
Bertola, M., Dubrovin, B., Yang, D.: Correlation functions of the KdV hierarchy and applications to intersection numbers over \(\overline{\cal{M}}_{g, n}\). Phys. D Nonlinear Phenom. 327, 30–57 (2016)
Bertola, M.: The dependence on the monodromy data of the isomonodromic tau function. Commun. Math. Phys. 294(2), 539–579 (2010)
Brezin, E., Hikami, S.: On an Airy matrix model with a logarithmic potential. J. Phys. A Math. Theor. 45(4), 045203 (2012)
Brezin, E., Hikami, S.: Random matrix, singularities and open/close intersection numbers. J. Phys. A Math. Theor. 48(47), 475201 (2015)
Buryak, A., Tessler, R.J.: Matrix models and a proof of the open analog of Witten’s conjecture. Commun. Math. Phys. 353(3), 1299–1328 (2017)
Buryak, A.: Open intersection numbers and the wave function of the KdV hierarchy. Moscow Math. J. 16(1), 27–44 (2016)
Ince, E.L.: Ordinary Differential Equations. Dover Books on Mathematics. Dover Publications, Mineola (1956)
Itzykson, C., Zuber, J.B.: Combinatorics of the modular group. 2. The Kontsevich integrals. Int. J. Mod. Phys. A 7, 5661–5705 (1992)
Jimbo, M., Miwa, T.: Deformation of linear ordinary differential equations, II. Proc. Jpn. Acad. Ser. A Math. Sci. 56(4), 149–153 (1980)
Jimbo, M., Miwa, T., Ueno, K.: Monodromy preserving deformation of linear ordinary differential equations with rational coefficients. I. General theory and \(\tau \)-function. Phys. D Nonlinear Phenom. 2(2), 306–352 (1981)
Kharchev, S., Marshakov, A., Mironov, A., Morozov, A., Zabrodin, A.: Towards unified theory of 2d gravity. Nucl. Phys. B 380(1), 181–240 (1992)
Kontsevich, M.: Intersection theory on the moduli space of curves and the matrix Airy function. Commun. Math. Phys. 147(1), 1–23 (1992)
Moore, G.: Geometry of the string equations. Commun. Math. Phys. 133(2), 261–304 (1990)
Moore, G.: Matrix models of \(2\)D gravity and isomonodromic deformation. Prog. Theor. Phys. Suppl. 102, 255–285 (1991)
Olver, F.: Asymptotics and Special Functions. AKP Classics. Taylor and Francis, Routledge (1997)
Penner, R.C.: Perturbative series and the moduli space of Riemann surfaces. J. Differ. Geom. 27(1), 35–53 (1988)
Pandharipande, R., Solomon, J.P., Tessler, R.J.: Intersection theory on moduli of disks, open KdV and Virasoro (2014). arXiv:1409.2191
Safnuk, B.: Combinatorial models for moduli spaces of open Riemann surfaces (2016). arXiv:1609.07226v2
Tessler, R.J.: The combinatorial formula for open gravitational descendents (2015). arXiv:1507.04951v3
Wasow, W.: Asymptotic Expansions for Ordinary Differential Equations. Dover Phoenix Editions. Dover, Mineola (2002)
Witten, E.: Two-dimensional gravity and intersection theory on moduli space. Surv. Diff. Geom. 1, 243–310 (1991)
Acknowledgements
The research of M. B. was supported in part by the Natural Sciences and Engineering Research Council of Canada grant RGPIN-2016-06660. G. R. wishes to thank the Department of Mathematics and Statistics at Concordia University for hospitality during which the work was completed.
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Communicated by Jean-Michel Maillet.
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Bertola, M., Ruzza, G. The Kontsevich–Penner Matrix Integral, Isomonodromic Tau Functions and Open Intersection Numbers. Ann. Henri Poincaré 20, 393–443 (2019). https://doi.org/10.1007/s00023-018-0737-8
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DOI: https://doi.org/10.1007/s00023-018-0737-8