The Kontsevich–Penner Matrix Integral, Isomonodromic Tau Functions and Open Intersection Numbers
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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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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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