Generating functions for intersection numbers on moduli spaces of curves
Andrei Okounkov
TL;DR
The paper establishes a precise, multivariate error-function representation for the n-point functions of psi-class intersection numbers on moduli spaces of curves, revealing a deep link between moduli-space intersection theory and edge scaling in the GUE matrix model. It derives a closed-form formula for the n-point function in terms of an integral function $\mathcal{E}$, its cyclic symmetrization, and a Möbius-connected combination, with the final expression expressed as a symmetric function of $x_i$ and aligned with a GKZ–hypergeometric framework. By connecting Laplace transforms of the Airy kernel to Kontsevich’s ribbon-graph expansion, the paper also provides an alternative path to Witten’s KdV equations via higher Fay identities (ASV), identifying the relevant KdV $\tau$-function and demonstrating how its coefficients reproduce the intersection-number hierarchy. The work highlights cancellations and selection rules that simplify the asymptotics, clarifying which terms contribute to the connected, positive-degree part and establishing a rigorous bridge between random-matrix edge phenomena and the KdV integrable hierarchy with potential applications to moduli-space enumerative geometry. Overall, the results offer a unified viewpoint on n-point functions, map asymptotics, and integrable structures in a single matrix-model framework.
Abstract
Using the connection between intersection theory on the Deligne-Mumford spaces and the edge scaling of the GUE matrix model (see math.CO/9903176, math.AG/0101147), we express the n-point functions for the intersection numbers as n-dimensional error-function-type integrals and also give a derivation of Witten's KdV equations using the higher Fay identities of Adler, Shiota, and van Moerbeke.