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

SIGMA 16 (2020), 029, 34 pages      arXiv:1908.08348
Contribution to the Special Issue on Integrability, Geometry, Moduli in honor of Motohico Mulase for his 65th birthday

Non-Abelian Hodge Theory and Related Topics

Pengfei Huang ab
a)  School of Mathematics, University of Science and Technology of China, Hefei 230026, China
b)  Laboratoire J.A. Dieudonné, Université Côte d'Azur, CNRS, 06108 Nice, France

Received August 23, 2019, in final form April 08, 2020; Published online April 19, 2020

This paper is a survey aimed on the introduction of non-Abelian Hodge theory that gives the correspondence between flat bundles and Higgs bundles. We will also introduce some topics arising from this theory, especially some recent developments on the study of the relevant moduli spaces together with some interesting open problems.

Key words: non-Abelian Hodge theory; $\lambda$-connection; moduli space; conformal limit; Hitchin section; oper; stratification; twistor space.

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  1. Abbes A., Gros M., Tsuji T., The $p$-adic Simpson correspondence, Annals of Mathematics Studies, Vol. 193, Princeton University Press, Princeton, NJ, 2016.
  2. Alessandrini D., Higgs bundles and geometric structures on manifolds, SIGMA 15 (2019), 039, 32 pages, arXiv:1809.07290.
  3. Amorós J., Burger M., Corlette K., Kotschick D., Toledo D., Fundamental groups of compact Kähler manifolds, Mathematical Surveys and Monographs, Vol. 44, Amer. Math. Soc., Providence, RI, 1996.
  4. Aparicio-Arroyo M., Bradlow S., Collier B., García-Prada O., Gothen P.B., Oliveira A., ${\rm SO}(p,q)$-Higgs bundles and higher Teichmüller components, Invent. Math. 218 (2019), 197-299, arXiv:1802.08093.
  5. Beilinson A., Drinfeld V., Quantization of Hitchin's integrable system and Hecke eigensheaves, unpublished, 1991, available at
  6. Biquard O., Fibrés de Higgs et connexions intégrables: le cas logarithmique (diviseur lisse), Ann. Sci. École Norm. Sup. (4) 30 (1997), 41-96.
  7. Biquard O., Boalch P., Wild non-abelian Hodge theory on curves, Compos. Math. 140 (2004), 179-204, arXiv:math.DG/0111098.
  8. Biswas I., Gómez T.L., Hoffmann N., Logares M., Torelli theorem for the Deligne-Hitchin moduli space, Comm. Math. Phys. 290 (2009), 357-369, arXiv:0901.0021.
  9. Biswas I., Heller S., Röser M., Real holomorphic sections of the Deligne-Hitchin twistor space, Comm. Math. Phys. 366 (2019), 1099-1133, arXiv:1802.06587.
  10. Bradlow S.B., García-Prada O., Mundet i Riera I., Relative Hitchin-Kobayashi correspondences for principal pairs, Q. J. Math. 54 (2003), 171-208, arXiv:math.DG/0206003.
  11. Brantner L., Abelian and nonabelian Hodge theory, unpublished, 2012, available at
  12. Burger M., Iozzi A., Wienhard A., Surface group representations with maximal Toledo invariant, Ann. of Math. 172 (2010), 517-566, arXiv:math.DG/0605656.
  13. Burger M., Iozzi A., Wienhard A., Higher Teichmüller spaces: from ${\rm SL}(2,{\mathbb R})$ to other Lie groups, in Handbook of Teichmüller Theory, Vol. IV, IRMA Lect. Math. Theor. Phys., Vol. 19, Eur. Math. Soc., Zürich, 2014, 539-618, arXiv:1004.2894.
  14. Collier B., Studying deformations of Fuchsian representations with Higgs bundles, SIGMA 15 (2019), 010, 32 pages, arXiv:1809.06786.
  15. Collier B., Wentworth R., Conformal limits and the Białynicki-Birula stratification of the space of $\lambda$-connections, Adv. Math. 350 (2019), 1193-1225, arXiv:1808.01622.
  16. Corlette K., Flat $G$-bundles with canonical metrics, J. Differential Geom. 28 (1988), 361-382.
  17. Deligne P., Various letters to C. Simpson, unpublished.
  18. Donagi R., Pantev T., Geometric Langlands and non-abelian Hodge theory, in Surveys in Differential Geometry, Vol. XIII, Geometry, Analysis, and Algebraic Geometry: Forty Years of the Journal of Differential Geometry, Surv. Differ. Geom., Vol. 13, Int. Press, Somerville, MA, 2009, 85-116.
  19. Donaldson S.K., A new proof of a theorem of Narasimhan and Seshadri, J. Differential Geom. 18 (1983), 269-277.
  20. Donaldson S.K., Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proc. London Math. Soc. 50 (1985), 1-26.
  21. Donaldson S.K., Twisted harmonic maps and the self-duality equations, Proc. London Math. Soc. 55 (1987), 127-131.
  22. Dumitrescu O., Fredrickson L., Kydonakis G., Mazzeo R., Mulase M., Neitzke A., Opers versus nonabelian Hodge, J. Differential Geom., to appear, arXiv:1607.02172.
  23. Faltings G., A $p$-adic Simpson correspondence, Adv. Math. 198 (2005), 847-862.
  24. Franc C., Rayan S., Nonabelian Hodge theory and vector-valued modular forms, arXiv:1812.06180.
  25. Fujiki A., Hyper-Kähler structure on the moduli space of flat bundles, in Prospects in Complex Geometry (Katata and Kyoto, 1989), Lecture Notes in Math., Vol. 1468, Springer, Berlin, 1991, 1-83.
  26. Gaiotto D., Opers and TBA, arXiv:1403.6137.
  27. García-Prada O., Higgs bundles and higher Teichmüller spaces, arXiv:1901.09086.
  28. García-Prada O., Gothen P.B., Mundet i Rierra I., The Hitchin-Kobayashi correspondence, Higgs pairs and surface group representations, arXiv:0909.4487.
  29. García-Raboso A., Rayan S., Introduction to nonabelian Hodge theory: flat connections, Higgs bundles and complex variations of Hodge structure, in Calabi-Yau Varieties: Arithmetic, Geometry and Physics, Fields Inst. Monogr., Vol. 34, Fields Inst. Res. Math. Sci., Toronto, ON, 2015, 131-171, arXiv:1406.1693.
  30. Goldman W.M., Topological components of spaces of representations, Invent. Math. 93 (1988), 557-607.
  31. Gothen P.B., Zúñiga Rojas R.A., Stratifications on the moduli space of Higgs bundles, Port. Math. 74 (2017), 127-148, arXiv:1511.03985.
  32. Greb D., Kebekus S., Peternell T., Taji B., Nonabelian Hodge theory for klt spaces and descent theorems for vector bundles, Compos. Math. 155 (2019), 289-323, arXiv:1711.08159.
  33. Hausel T., Geometry of the moduli space of Higgs bundles, Ph.D. Thesis, University of Cambridge, 1998, arXiv:math.AG/0107040.
  34. Hitchin N.J., The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987), 59-126.
  35. Hitchin N.J., Lie groups and Teichmüller space, Topology 31 (1992), 449-473.
  36. Hitchin N.J., Karlhede A., Lindström U., Roček M., Hyper-Kähler metrics and supersymmetry, Comm. Math. Phys. 108 (1987), 535-589.
  37. Hu Z., Huang P., Flat $\lambda$-connections, Mochizuki correspondence and twistor spaces, arXiv:1905.10765.
  38. Kaledin D., Verbitsky M., Non-Hermitian Yang-Mills connections, Selecta Math. (N.S.) 4 (1998), 279-320, arXiv:alg-geom/9606019.
  39. Labourie F., Anosov flows, surface groups and curves in projective space, Invent. Math. 165 (2006), 51-114, arXiv:math.DG/0401230.
  40. Labourie F., Wentworth R., Variations along the Fuchsian locus, Ann. Sci. Éc. Norm. Supér. (4) 51 (2018), 487-547, arXiv:1506.01686.
  41. Li Q., An introduction to Higgs bundles via harmonic maps, SIGMA 15 (2019), 035, 30 pages, arXiv:1809.05747.
  42. Loray F., Saito M.-H., Simpson C., Foliations on the moduli space of rank two connections on the projective line minus four points, in Geometric and differential Galois theories, Sémin. Congr., Vol. 27, Soc. Math. France, Paris, 2013, 117-170, arXiv:1012.3612.
  43. Migliorini L., Recent results and conjectures on the non abelian Hodge theory of curves, Boll. Unione Mat. Ital. 10 (2017), 467-485.
  44. Mochizuki T., Kobayashi-Hitchin correspondence for tame harmonic bundles and an application, Astérisque 309 (2006), viii+117 pages, arXiv:math.DG/0411300.
  45. Mochizuki T., Kobayashi-Hitchin correspondence for tame harmonic bundles. II, Geom. Topol. 13 (2009), 359-455, arXiv:math.DG/0602266.
  46. Mochizuki T., Wild harmonic bundles and wild pure twistor $D$-modules, Astérisque 340 (2011), x+607 pages, arXiv:0803.1344.
  47. Mochizuki T., Periodic monopoles and difference modules, arXiv:1712.08981.
  48. Mochizuki T., Doubly periodic monopoles and $q$-difference modules, arXiv:1902.03551.
  49. Mochizuki T., Triply periodic monopoles and difference modules on elliptic curves, arXiv:1903.03264.
  50. Mochizuki T., Yoshino M., Some characterizations of Dirac type singularity of monopoles, Comm. Math. Phys. 356 (2017), 613-625, arXiv:1702.06268.
  51. Narasimhan M.S., Seshadri C.S., Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. 82 (1965), 540-567.
  52. Ogus A., Vologodsky V., Nonabelian Hodge theory in characteristic $p$, Publ. Math. Inst. Hautes Études Sci. (2007), 1-138, arXiv:math.AG/0507476.
  53. Pauly C., Peón-Nieto A., Very stable bundles and properness of the Hitchin map, Geom. Dedicata 198 (2019), 143-148, arXiv:1710.10152.
  54. Rayan S., Aspects of the topology and combinatorics of Higgs bundle moduli spaces, SIGMA 14 (2018), 129, 18 pages, arXiv:1809.05732.
  55. Salamon S., Quaternionic Kähler manifolds, Invent. Math. 67 (1982), 143-171.
  56. Simpson C.T., Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J. Amer. Math. Soc. 1 (1988), 867-918.
  57. Simpson C.T., Harmonic bundles on noncompact curves, J. Amer. Math. Soc. 3 (1990), 713-770.
  58. Simpson C.T., Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 5-95.
  59. Simpson C.T., Moduli of representations of the fundamental group of a smooth projective variety. I, Inst. Hautes Études Sci. Publ. Math. 79 (1994), 47-129.
  60. Simpson C.T., Moduli of representations of the fundamental group of a smooth projective variety. II, Inst. Hautes Études Sci. Publ. Math. 80 (1994), 5-79.
  61. Simpson C.T., The Hodge filtration on nonabelian cohomology, in Algebraic Geometry - Santa Cruz 1995, Proc. Sympos. Pure Math., Vol. 62, Amer. Math. Soc., Providence, RI, 1997, 217-281, arXiv:alg-geom/9604005.
  62. Simpson C.T., A weight two phenomenon for the moduli of rank one local systems on open varieties, in From Hodge theory to Integrability and TQFT $tt^*$-Geometry, Proc. Sympos. Pure Math., Vol. 78, Amer. Math. Soc., Providence, RI, 2008, 175-214, arXiv:0710.2800.
  63. Simpson C.T., Iterated destabilizing modifications for vector bundles with connection, in Vector Bundles and Complex Geometry, Contemp. Math., Vol. 522, Amer. Math. Soc., Providence, RI, 2010, 183-206, arXiv:0812.3472.
  64. Uhlenbeck K., Yau S.-T., On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Comm. Pure Appl. Math. 39 (1986), S257-S293.
  65. Wienhard A., An invitation to higher Teichmüller theory, in Proceedings of the International Congress of Mathematicians - Rio de Janeiro 2018. Vol. II. Invited Lectures, World Sci. Publ., Hackensack, NJ, 2018, 1013-1039, arXiv:1803.06870.

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