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

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 $n\to\infty$ (at a formal level) and identify an isomonodromic system in terms of the Miwa variables, which play the role of times of the 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 formul\ae\ 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.