Asymptotic Geometry of the Hitchin Metric

Abstract

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.

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Correspondence to Rafe Mazzeo.

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RM supported by NSF Grant DMS-1105050 and DMS-1608223. JS and HW supported by DFG SPP 2026 ‘Geometry at infinity’. The author(s) acknowledge(s) support from U.S. National Science Foundation Grants DMS 1107452, 1107263, 1107367 "RNMS: Geometric Structures and Representation Varieties" (the GEAR Network).

Communicated by N. Nekrasov

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Mazzeo, R., Swoboda, J., Weiss, H. et al. Asymptotic Geometry of the Hitchin Metric. Commun. Math. Phys. 367, 151–191 (2019). https://doi.org/10.1007/s00220-019-03358-y

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