Abstract

We derive from Witten's KdV equation a simple formula of the $n$-point functions for intersection numbers on moduli spaces of curves, generalizing Dijkgraaf's two-point function and Zagier's three-point function. This formula uncovers many new identities about integrals of $\psi$ classes and provides an elementary and more efficient algorithm to compute intersection numbers other than the celebrated Witten-Kontsevich theorem.