Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 15 (2019), 018, 20 pages      arXiv:1809.05735      https://doi.org/10.3842/SIGMA.2019.018
Contribution to the Special Issue on Geometry and Physics of Hitchin Systems

Perspectives on the Asymptotic Geometry of the Hitchin Moduli Space

Laura Fredrickson
Stanford University, Department of Mathematics, 380 Serra Mall, Stanford, CA 94305, USA

Received September 23, 2018, in final form February 25, 2019; Published online March 11, 2019

Abstract
We survey some recent developments in the asymptotic geometry of the Hitchin moduli space, starting with an introduction to the Hitchin moduli space and hyperkähler geometry.

Key words: Hitchin moduli space; Higgs bundles; hyperkähler metric.

pdf (952 kb)   tex (600 kb)

References

  1. Anderson L.B., Esole M., Fredrickson L., Schaposnik L.P., Singular geometry and Higgs bundles in string theory, SIGMA 14 (2018), 037, 27 pages, arXiv:1710.0845.
  2. Baraglia D., Schaposnik L.P., Real structures on moduli spaces of Higgs bundles, Adv. Theor. Math. Phys. 20 (2016), 525-551, arXiv:1309.1195.
  3. Chen G., Chen X., Gravitational instantons with faster than quadratic decay (I), arXiv:1505.01790.
  4. Chen G., Chen X., Gravitational instantons with faster than quadratic decay (III), arXiv:1603.08465.
  5. Conlon R.J., Degeratu A., Rochon F., Quasi-asymptotically conical Calabi-Yau manifolds, arXiv:1611.04410.
  6. Corlette K., Flat $G$-bundles with canonical metrics, J. Differential Geom. 28 (1988), 361-382.
  7. Daskalopoulos G.D., Wentworth R.A., Wilkin G., Cohomology of ${\rm SL}(2,{\mathbb C})$ character varieties of surface groups and the action of the Torelli group, Asian J. Math. 14 (2010), 359-383, arXiv:0808.0131.
  8. Deroin B., Tholozan N., Dominating surface group representations by Fuchsian ones, Int. Math. Res. Not. 2016 (2016), 4145-4166, arXiv:1311.2919.
  9. Donagi R., Markman E., Spectral covers, algebraically completely integrable, Hamiltonian systems, and moduli of bundles, in Integrable Systems and Quantum Groups (Montecatini Terme, 1993), Lecture Notes in Math., Vol. 1620, Springer, Berlin, 1996, 1-119, arXiv:alg-geom/9507017.
  10. Donaldson S.K., A new proof of a theorem of Narasimhan and Seshadri, J. Differential Geom. 18 (1983), 269-277.
  11. Donaldson S.K., Twisted harmonic maps and the self-duality equations, Proc. London Math. Soc. 55 (1987), 127-131.
  12. Dumas D., Neitzke A., Asymptotics of Hitchin's metric on the Hitchin section, Comm. Math. Phys., to appear, arXiv:1802.07200.
  13. Fredrickson L., Exponential decay for the asymptotic geometry of the Hitchin metric, arXiv:1810.01554.
  14. Fredrickson L., Generic ends of the moduli space of ${\rm SL}(n,\mathbb{C})$-Higgs bundles, arXiv:1810.01556.
  15. Fredrickson L., Neitzke A., From $S^1$-fixed points to ${\mathcal W}$-algebra representations, arXiv:1709.06142.
  16. Freed D.S., Special Kähler manifolds, Comm. Math. Phys. 203 (1999), 31-52, arXiv:hep-th/9712042.
  17. Gaiotto D., Moore G.W., Neitzke A., Four-dimensional wall-crossing via three-dimensional field theory, Comm. Math. Phys. 299 (2010), 163-224, arXiv:0807.4723.
  18. Gaiotto D., Moore G.W., Neitzke A., Wall-crossing, Hitchin systems, and the WKB approximation, Adv. Math. 234 (2013), 239-403, arXiv:0907.3987.
  19. Gibbons G.W., Hawking S.W., Gravitational multi-instantons, Phys. Lett. B 78 (1978), 430-432.
  20. Hein H.-J., Gravitational instantons from rational elliptic surfaces, J. Amer. Math. Soc. 25 (2012), 355-393.
  21. Hitchin N.J., Polygons and gravitons, Math. Proc. Cambridge Philos. Soc. 85 (1979), 465-476.
  22. Hitchin N.J., The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987), 59-126.
  23. Joyce D., Asymptotically locally Euclidean metrics with holonomy ${\rm SU}(m)$, Ann. Global Anal. Geom. 19 (2001), 55-73, arXiv:math.AG/9905041.
  24. Joyce D., Quasi-ALE metrics with holonomy ${\rm SU}(m)$ and ${\rm Sp}(m)$, Ann. Global Anal. Geom. 19 (2001), 103-132, arXiv:math.AG/9905043.
  25. Kronheimer P.B., The construction of ALE spaces as hyper-Kähler quotients, J. Differential Geom. 29 (1989), 665-683.
  26. Kronheimer P.B., A Torelli-type theorem for gravitational instantons, J. Differential Geom. 29 (1989), 685-697.
  27. Mazzeo R., Swoboda J., Weiss H., Witt F., Ends of the moduli space of Higgs bundles, Duke Math. J. 165 (2016), 2227-2271, arXiv:1405.5765.
  28. Mazzeo R., Swoboda J., Weiss H., Witt F., Asymptotic geometry of the Hitchin metric, arXiv:1709.03433.
  29. Minerbe V., On the asymptotic geometry of gravitational instantons, Ann. Sci. Éc. Norm. Supér. (4) 43 (2010), 883-924.
  30. Mochizuki T., Asymptotic behaviour of certain families of harmonic bundles on Riemann surfaces, J. Topol. 9 (2016), 1021-1073, arXiv:1508.05997.
  31. Narasimhan M.S., Seshadri C.S., Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. 82 (1965), 540-567.
  32. Neitzke A., Notes on a new construction of hyperkähler metrics, arXiv:1308.2198.
  33. Neitzke A., Moduli of Higgs bundles, 2016, available at http://www.ma.utexas.edu/users/neitzke/teaching/392C-higgs-bundles/higgs-bundles.pdf.
  34. Rayan S., Aspects of the topology and combinatorics of Higgs bundle moduli spaces, SIGMA 14 (2018), 129, 18 pages, arXiv:1809.05732.
  35. Simpson C.T., Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. (1992), 5-95.
  36. Taubes C.H., ${\rm PSL}(2;{\mathbb C})$ connections on 3-manifolds with ${\rm L}^2$ bounds on curvature, Camb. J. Math. 1 (2013), 239-397, arXiv:1205.0514.
  37. Wentworth R.A., Higgs bundles and local systems on Riemann surfaces, in Geometry and Quantization of Moduli Spaces, Adv. Courses Math. CRM Barcelona, Birkhäuser/Springer, Cham, 2016, 165-219, arXiv:1402.4203.
  38. Wolf M., The Teichm\"uller theory of harmonic maps, J. Differential Geom. 29 (1989), 449-479.

Previous article  Next article   Contents of Volume 15 (2019)