Communications in Mathematical Physics

, Volume 367, Issue 1, pp 151–191 | Cite as

Asymptotic Geometry of the Hitchin Metric

  • Rafe MazzeoEmail author
  • Jan Swoboda
  • Hartmut Weiss
  • Frederik Witt


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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Ba.
    Balduzzi D.: Donagi–Markman cubic for Hitchin systems. Math. Res. Lett. 13(5–6), 923–933 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. BC.
    Baues O., Cortés V.: Proper affine hyperspheres which fiber over projective special Kähler manifolds. Asian J. Math. 7(1), 115–132 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  3. BNR.
    Beauville A., Narasimhan M., Ramanan S.: Spectral curves and the generalised theta divisor. J. Reine Angew. Math. 398, 169–179 (1989)MathSciNetzbMATHGoogle Scholar
  4. BL.
    Birkenhake, C., Lange , H.: Complex abelian varieties. 2nd Edn. In: Grundlehren der Mathematischen Wissenschaften, Col. 302. Springer, Berlin (2004)Google Scholar
  5. CM.
    Cortés V, Mohaupt T: Special geometry of Euclidean supersymmetry. III. The local r-map, instantons and black holes. J. High Energy Phys. 7(066), 64 (2009)MathSciNetGoogle Scholar
  6. DH.
    Douady A., Hubbard J.: On the density of Strebel differentials. Invent. Math. 30(2), 175–179 (1975)ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. DN.
    Dumas, D., Neitzke, A.: Asymptotics of Hitchin’s metric on the Hitchin section. Commun. Math. Phys. (2018).
  8. Fr18.
    Fredrickson, L.: Exponential decay for the asymptotic geometry of the Hitchin metric, preprint (2018). arXiv:1810.01554.
  9. Fr.
    Freed D.: Special Kähler manifolds. Commun. Math. Phys. 203(1), 31–52 (1999)ADSCrossRefzbMATHGoogle Scholar
  10. GMN.
    Gaiotto D., Moore G., Neitzke A.: Wall-crossing, Hitchin systems, and the WKB approximation. Adv. Math. 234, 239–403 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  11. GS.
    Guillemin V., Sternberg S.: Symplectic techniques in physics. Cambridge University Press, Cambridge (1990)zbMATHGoogle Scholar
  12. HHP.
    Hertling C., Hoevenaars L., Posthuma H.: Frobenius manifolds, projective special geometry and Hitchin systems. J. Reine Angew. Math. 649, 117–165 (2010)MathSciNetzbMATHGoogle Scholar
  13. Hi87a.
    Hitchin N.: The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc.(3) 55(1), 59–126 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  14. Hi87b.
    Hitchin N.: Stable bundles and integrable systems. Duke Math. J. 54(1), 91–114 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  15. HKLR.
    Hitchin N., Karlhede A., Lindström U., Roček M.: Hyper-Kähler metrics and supersymmetry. Commun. Math. Phys. 108(4), 535–589 (1987)ADSCrossRefzbMATHGoogle Scholar
  16. MSWW14.
    Mazzeo R., Swoboda J., Weiß H., Witt F.: Ends of the moduli space of Higgs bundles. Duke Math. J. 165(12), 2227–2271 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  17. MSWW15.
    Mazzeo, R., Swoboda, J., Weiß, H., Witt, F.: Limiting configurations for solutions of Hitchin’s equation. Semin. Theor. Spectr. Geom. 31, 91–116 (2012–2014)Google Scholar
  18. Mo.
    Mochizuki T.: Asymptotic behaviour of certain families of harmonic bundles on Riemann surfaces. J. Topol. 9(4), 1021–1073 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  19. Ne.
    Neitzke, A.: Notes on a new construction of hyperkahler metrics. Homological mirror symmetry and tropical geometry, 351–375, Lect. Notes Unione Mat. Ital., Vol. 15. Springer, Cham (2014)Google Scholar
  20. Ni.
    Nitsure N.: Moduli space of semistable pairs on a curve. Proc. Lond. Math. Soc. (3) 62(2), 275–300 (1991)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsStanford UniversityStanfordUSA
  2. 2.Mathematisches Institut der Universität MünchenMunichGermany
  3. 3.Mathematisches Seminar der Universität KielKielGermany
  4. 4.Institut für Geometrie und Topologie der Universität StuttgartStuttgartGermany

Personalised recommendations