Asymptotic Geometry of the Hitchin Metric

Communications in Mathematical Physics volume 367, pages151–191(2019)Cite this article

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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Author information

Affiliations

  1. Department of Mathematics, Stanford University, Stanford, CA, 94305, USA

    Rafe Mazzeo

  2. Mathematisches Institut der Universität München, Theresienstraße 39, 80333, Munich, Germany

    Jan Swoboda

  3. Mathematisches Seminar der Universität Kiel, Ludewig-Meyn-Straße 4, 24098, Kiel, Germany

    Hartmut Weiss

  4. Institut für Geometrie und Topologie der Universität Stuttgart, Pfaffenwaldring 57, 70569, Stuttgart, Germany

    Frederik Witt

Authors
  1. Rafe Mazzeo
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  2. Jan Swoboda
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  3. Hartmut Weiss
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  4. Frederik Witt
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Corresponding author

Correspondence to Rafe Mazzeo.

Additional information

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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