# Moscow Mathematical Journal

Volume 16, Issue 1, January–March 2016 pp. 27–44.

Open Intersection Numbers and the Wave Function of the KdV Hierarchy

**Authors**:
A. Buryak

**Author institution:** Department of Mathematics, ETH Zurich, Ramistrasse 101 8092, HG G 27.1, Zurich, Switzerland

**Summary: **

Recently R. Pandharipande, J. Solomon and R. Tessler 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 is a specific solution of a system of PDEs, which they called the open KdV equations. In this paper we show that the open KdV equations are closely related to the equations for the wave function of the KdV hierarchy. This allows us to give an explicit formula for the specific solution in terms of Wittenâ€™s generating series of the intersection numbers on the moduli space of stable curves.

2010 Math. Subj. Class. Primary: 35Q53; Secondary: 14H10.

**Keywords:**Riemann surfaces with boundary, moduli space, KdV equations.

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