Open intersection numbers and the wave function of the KdV hierarchy

Abstract

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, that 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 Witten's generating series of the intersection numbers on the moduli space of stable curves.

v3: several typos in the proof of Proposition 4.1 are corrected, 15 pages

References in corpus (5)

• Intersection theory on moduli of disks, open KdV and Virasoro
• Open intersection numbers, Kontsevich-Penner model and cut-and-join operators
• Matrix models and a proof of the open analog of Witten's conjecture
• The partition function of the extended $r$-reduced Kadomtsev-Petviashvili hierarchy
• Random Matrix, Singularities and Open/Close Intersection Numbers

Cited by in corpus (8)

• Double ramification cycles and integrable hierarchies
• Correlation functions of the KdV hierarchy and applications to intersection numbers over ${\overline{\mathcal{M}}}_{g,n}$
• Open intersection numbers, Kontsevich-Penner model and cut-and-join operators
• The partition function of the extended $r$-reduced Kadomtsev-Petviashvili hierarchy
• The hypergeometric functions of the Faber-Zagier and Pixton relations
• Open intersection numbers and free fields
• Random Matrix, Singularities and Open/Close Intersection Numbers
• Open KdV hierarchy of 2d minimal gravity of Lee-Yang series