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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.

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  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).

  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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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).

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