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 × 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 → ∞ (at a formal level) and identify an isomonodromic system in terms of the 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 formulae 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.Lemma 2.22. The jump matrix J(λ) (2.44) admits formal meromorphic extension in D + . More precisely

Laguerre Ensemble: Correlators, Hurwitz Numbers and Hodge Integrals

“…This is easily seen from the linear recursions of Lemma 2.11. For the coefficients E , the initial datum of the recursion is 11) and the recursion reads 12) and the claim follows by induction, as the initial datum is odd and the recursion is even in N . Similarly, for the coefficients D , F , the initial datum of the recursion is odd in N…”

“…The following proof is reported for the sake of completeness; it has appeared in the literature several times, e.g., see [7,8,11,30]. The only slight difference here is that we consider two different set of times and correspondingly the residues are taken at two different points.…”

Brezin–Gross–Witten tau function and isomonodromic deformations

“…Following the strategy already applied in [BC17,BR17], we consider a dressing of the bare ODE (1.25). This is conveniently expressed in terms of the Riemann-Hilbert problem (RHP) 1.8 below.…”

“…1.12; the approach is exactly parallel to that in [BC17, App. A], which we refer to for further details (see also [BR17]).…”

“…1.1 follows from the same algebraic manipulations first introduced in [BDY16] which have subsequently appeared many times, e.g. in [BDY15,DY17,BR17] and it is explained in detail for the reader's convenience.…”