Matrix Models and A Proof of the Open Analog of Witten’s Conjecture

In a recent work, R. Pandharipande, J. P. Solomon and the second author have initiated a study of the intersection theory on the moduli space of Riemann surfaces with boundary. They conjectured that the generating series of the intersection numbers satisfies the open KdV equations. In this paper we prove this conjecture. Our proof goes through a matrix model and is based on a Kontsevich type combinatorial formula for the intersection numbers that was found by the second author.