Communications in Number Theory and Physics

Volume 10 (2016)

Number 4

Topological recursion for open intersection numbers

Pages: 833 – 857



Brad Safnuk (Department of Mathematics, Central Michigan University, Mount Pleasant, Mich., U.S.A.)


We present a topological recursion formula for calculating the intersection numbers defined on the moduli space of open Riemann surfaces. The spectral curve is $x = \frac{1}{2} y^2$, the same as the spectral curve used to calculate intersection numbers for closed Riemann surfaces, but the formula itself is a variation of the usual Eynard–Orantin recursion. It looks like the recursion formula used for spectral curves of degree $3$, and also includes features present in $\beta$-deformed models. The recursion formula suggests a conjectural refinement to the generating function that allows for distinguishing intersection numbers on moduli spaces with different numbers of boundary components.

Published 3 April 2017