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?
Mirzakhani's computation of volumes of deformation spaces is very heavily based on the work of Greg McShane (McShane's identity). McShane's theory has been extended to other deformation spaces, see, for example
Labourie, François; McShane, Gregory, Cross ratios and identities for higher Teichmüller-Thurston theory, Duke Math. J. 149, No. 2, 279-345 (2009). ZBL1182.30075.
(a search of Google Scholar for "McShane Labourie" reveals a fair bit of other work). So, for the last question, the answer is "not very", but feel free to read the papers...