Asymptotic Geometry of the Hitchin Metric

We study the asymptotics of the natural L2 metric on the Hitchin moduli space with group \({G = \mathrm{SU}(2)}\). Our main result, which addresses a detailed conjectural picture made by Gaiotto et al. (Adv Math 234:239–403, 2013), is that on the regular part of the Hitchin system, this metric is well-approximated by the semiflat metric from Gaiotto et al. (2013). We prove that the asymptotic rate of convergence for gauged tangent vectors to the moduli space has a precise polynomial expansion, and hence that the difference between the two sets of metric coefficients in a certain natural coordinate system also has polynomial decay. New work by Dumas-Neitzke and later Fredrickson shows that the convergence is actually exponential.