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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?

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    $\begingroup$ I'm not expert enough to know if this is a "good" question, but it's definitely a very interesting question. $\endgroup$
    – Andy Sanders
    Commented May 7, 2020 at 17:57

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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...

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    $\begingroup$ Thanks for the answer, but I think I’m confused. How do the spaces in that paper relate to moduli spaces of stable maps into a projective variety (such that appear in Gromov-Witten theory?) $\endgroup$
    – John Rached
    Commented May 7, 2020 at 21:37
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    $\begingroup$ @JohnRached I did not say they did relate. Your question was whether the methods are intrinsically hyperbolic-geometric, and these references show that they are not. $\endgroup$
    – Igor Rivin
    Commented May 7, 2020 at 22:54

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