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Asymptotic Geometry of the Hitchin Metric

Communications in Mathematical Physics volume 367, pages 151–191 (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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References

  1. Balduzzi D.: Donagi–Markman cubic for Hitchin systems. Math. Res. Lett. 13(5–6), 923–933 (2006)

    MathSciNet  Article  MATH  Google Scholar 

  2. Baues O., Cortés V.: Proper affine hyperspheres which fiber over projective special Kähler manifolds. Asian J. Math. 7(1), 115–132 (2003)

    MathSciNet  Article  MATH  Google Scholar 

  3. Beauville A., Narasimhan M., Ramanan S.: Spectral curves and the generalised theta divisor. J. Reine Angew. Math. 398, 169–179 (1989)

    MathSciNet  MATH  Google Scholar 

  4. Birkenhake, C., Lange , H.: Complex abelian varieties. 2nd Edn. In: Grundlehren der Mathematischen Wissenschaften, Col. 302. Springer, Berlin (2004)

  5. 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)

    MathSciNet  Google Scholar 

  6. Douady A., Hubbard J.: On the density of Strebel differentials. Invent. Math. 30(2), 175–179 (1975)

    ADS  MathSciNet  Article  MATH  Google Scholar 

  7. Dumas, D., Neitzke, A.: Asymptotics of Hitchin’s metric on the Hitchin section. Commun. Math. Phys. (2018). https://doiorg.stanford.idm.oclc.org/10.1007/s00220-018-3216-7

  8. Fredrickson, L.: Exponential decay for the asymptotic geometry of the Hitchin metric, preprint (2018). arXiv:1810.01554.

  9. Freed D.: Special Kähler manifolds. Commun. Math. Phys. 203(1), 31–52 (1999)

    ADS  Article  MATH  Google Scholar 

  10. Gaiotto D., Moore G., Neitzke A.: Wall-crossing, Hitchin systems, and the WKB approximation. Adv. Math. 234, 239–403 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  11. Guillemin V., Sternberg S.: Symplectic techniques in physics. Cambridge University Press, Cambridge (1990)

    MATH  Google Scholar 

  12. Hertling C., Hoevenaars L., Posthuma H.: Frobenius manifolds, projective special geometry and Hitchin systems. J. Reine Angew. Math. 649, 117–165 (2010)

    MathSciNet  MATH  Google Scholar 

  13. Hitchin N.: The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc.(3) 55(1), 59–126 (1987)

    MathSciNet  Article  MATH  Google Scholar 

  14. Hitchin N.: Stable bundles and integrable systems. Duke Math. J. 54(1), 91–114 (1987)

    MathSciNet  Article  MATH  Google Scholar 

  15. Hitchin N., Karlhede A., Lindström U., Roček M.: Hyper-Kähler metrics and supersymmetry. Commun. Math. Phys. 108(4), 535–589 (1987)

    ADS  Article  MATH  Google Scholar 

  16. Mazzeo R., Swoboda J., Weiß H., Witt F.: Ends of the moduli space of Higgs bundles. Duke Math. J. 165(12), 2227–2271 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  17. 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)

  18. Mochizuki T.: Asymptotic behaviour of certain families of harmonic bundles on Riemann surfaces. J. Topol. 9(4), 1021–1073 (2016)

    MathSciNet  Article  MATH  Google Scholar 

  19. 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)

  20. Nitsure N.: Moduli space of semistable pairs on a curve. Proc. Lond. Math. Soc. (3) 62(2), 275–300 (1991)

    MathSciNet  Article  MATH  Google Scholar 

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Authors and 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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  • DOI: https://doi.org/10.1007/s00220-019-03358-y