Mirzakhani develops a method to integrate geodesic length functions on moduli space by considering circle bundles over moduli space given by level sets of these functions. There are natural circle bundles over representation varieties (for instance, of $\textrm{SU}(n)$) given in a closely analogous way (level sets of trace functions). One could ask, naively, if her methods carry over. Is there some reason such an approach can't work? Any comments are appreciated.
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$\begingroup$ I don't really understand what you are trying to do. Can you give a reference to the work of Mirzakhani you are refering to? Length functions are not well-defined on the moduli space, but rather on certain covers of it. $\endgroup$– NicolastCommented Sep 14, 2021 at 7:33
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