On the Asymptotics of the Volume of Hitchin Moduli Spaces

Let $X$ be a compact Riemann surface of genus $g\ge 2$, and let $\text{M}$ be the moduli space of rank-two trace-free Higgs bundles with fixed determinant of odd degree. For the normalization of the Hitchin metric used in this paper, we prove that the volume of geodesic ball in the Hitchin moduli space is given by $${\text{Vol}}_{g_{L^2}}{B}_{L^2}(p,R)=\frac{2^{4g-3}{\pi }^{9g-9}}{(3g-3)!}{R}^{6g-6}+o(R^{6g-6})$$ for every $p\in \text{M}$. We also determine the leading asymptotics of Hamiltonian sublevel volumes and exponentially weighted volumes. The proof combines homogeneity of the hyperkähler volume form with metric asymptotics on the regular Hitchin locus. Symplectic reduction and the Prym polarization evaluate the coefficient, which is independently recovered by equivariant localization.