# Mirzakhani's hyperbolic method generalized to moduli space of stable maps

I've been learning about Mirzakhani's use of hyperbolic geometry to compute Weil-Petersson volumes of moduli space of curves, and the application to proving Virasoro constraints for a point. Why have these methods not been directly extended to higher dimensional target spaces? I don't know much about the structure of the moduli space of stable maps into a projective variety. Is there expected to exist a reasonable definition of volume for these spaces? I'm further confused by the fact that "topological recursion" introduced by Eyndard-Orantin, and inspired by Mirzakhani's recursion relations, was used to compute Gromov-Witten invariants for toric Calabi-Yau 3-folds. I suppose my question is: to what extent might one expect that intersection theory on the moduli space of stable maps into a projective variety is related to the hyperbolic geometry of the domain Riemann surface?

• I'm not expert enough to know if this is a "good" question, but it's definitely a very interesting question. – Andy Sanders May 7 at 17:57