Random matrix, singularities and open/close intersection numbers

Abstract

The s-point correlation function of a Gaussian Hermitian random matrix theory, with an external source tuned to generate a multi-critical singularity, provides the intersection numbers of the moduli space for the p-th spin curves through a duality identity. For one marked point, the intersection numbers are expressed to all order in the genus by Bessel functions. The matrix models for the Lie algebras of O(N) and Sp(N) provide the intersection numbers of non-orientable surfaces. The Kontsevich-Penner model, and higher p-th Airy matrix model with a logarithmic potential, are investigated for the open intersection numbers, which describe the topological invariants of non-orientable surfaces with boundaries. String equations for open/closed Riemann surface are derived from the structure of the s-point correlation functions. The Gromov-Witten invariants of CP 1 model are evaluated for one marked point as an application of the present method.

Correlation functions of the KdV hierarchy and applications to intersection numbers over M¯g,n

“…It would be interesting to prove directly the equivalence between the formula (1.36), the (explicit) integral/recursive formulae of "n-point functions" given by Okounkov [48], Liu-Xu [36,37], Brézin-Hikami [10,11], and Kontsevich's main identity [34]. We indicate the relation between these three as follows.…”

Random supermatrices with an external source

“…In the GUE case we have used at length a duality between the expectation value of a product of k-characteristic polynomials with N × N random matrices in a source, which is equal to the expectation values of the product of N characteristic polynomials averaged with k × k random matrices [1,2]. We now derive a similar duality for supermatrices.…”

“…We consider here a Gaussian ensemble of supermatrices, a generalized GUE, in the presence of an external matrix source. It presents a number of similarities with the usual case : (1) the k-point function < stre t 1 M · · · stre t k M > are explicitly calculable for random matrices M invariant under the superunitary group U(n|m) or UOSp(n|m), ( 2) there is again a dual representation of < k 1 sdet −1 (x i − M) > valid for arbitrary (n, m) in terms of integrals over matrices of size k × k.…”

“…We have considered at length in the past Hermitian random matrices in the presence of an external matrix source [1,2]. In fact we have limited ourselves to Gaussian models because a specific duality of these models, to be recalled below, made it possible to use the matrix source in order to tune non-trivial models such as Kontsevich's Airy matrix model [3] and generalizations [4].…”

“…In fact we have limited ourselves to Gaussian models because a specific duality of these models, to be recalled below, made it possible to use the matrix source in order to tune non-trivial models such as Kontsevich's Airy matrix model [3] and generalizations [4]. Such models have led to easy calculations of intersection numbers for the moduli space of curves with marked points and boundaries [1,2,5].…”