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

SIGMA 14 (2018), 129, 18 pages      arXiv:1809.05732
Contribution to the Special Issue on Geometry and Physics of Hitchin Systems

Aspects of the Topology and Combinatorics of Higgs Bundle Moduli Spaces

Steven Rayan
Department of Mathematics & Statistics, McLean Hall, University of Saskatchewan, Saskatoon, SK, Canada S7N 5E6

Received September 23, 2018, in final form December 04, 2018; Published online December 07, 2018

This survey provides an introduction to basic questions and techniques surrounding the topology of the moduli space of stable Higgs bundles on a Riemann surface. Through examples, we demonstrate how the structure of the cohomology ring of the moduli space leads to interesting questions of a combinatorial nature.

Key words: Higgs bundle; Morse-Bott theory; localization; Betti number; moduli space; stability; quiver; partition problem.

pdf (432 kb)   tex (34 kb)


  1. Adams M.R., Harnad J., Previato E., Isospectral Hamiltonian flows in finite and infinite dimensions. I. Generalized Moser systems and moment maps into loop algebras, Comm. Math. Phys. 117 (1988), 451-500.
  2. Álvarez Cónsul L., García-Prada O., Dimensional reduction, ${\rm SL}(2,\mathbb C)$-equivariant bundles and stable holomorphic chains, Internat. J. Math. 12 (2001), 159-201, math.DG/0112159.
  3. Álvarez Cónsul L., García-Prada O., Schmitt A.H.W., On the geometry of moduli spaces of holomorphic chains over compact Riemann surfaces, Int. Math. Res. Pap. 2006 (2006), Art. ID 73597, 82 pages, math.AG/0512498.
  4. Atiyah M.F., Bott R., The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), 523-615.
  5. Baird T.J., Symmetric products of a real curve and the moduli space of Higgs bundles, J. Geom. Phys. 126 (2018), 7-21, arXiv:1611.09636.
  6. Beauville A., Narasimhan M.S., Ramanan S., Spectral curves and the generalised theta divisor, J. Reine Angew. Math. 398 (1989), 169-179.
  7. Białynicki-Birula A., Some theorems on actions of algebraic groups, Ann. of Math. 98 (1973), 480-497.
  8. Biquard O., Boalch P., Wild non-abelian Hodge theory on curves, Compos. Math. 140 (2004), 179-204, math.DG/0111098.
  9. Biswas I., Ramanan S., An infinitesimal study of the moduli of Hitchin pairs, J. London Math. Soc. 49 (1994), 219-231.
  10. Boden H.U., Yokogawa K., Moduli spaces of parabolic Higgs bundles and parabolic $K(D)$ pairs over smooth curves. I, Internat. J. Math. 7 (1996), 573-598, alg-geom/9610014.
  11. Bradlow S.B., Daskalopoulos G.D., Moduli of stable pairs for holomorphic bundles over Riemann surfaces, Internat. J. Math. 2 (1991), 477-513.
  12. Bradlow S.B., García-Prada O., Gothen P.B., Moduli spaces of holomorphic triples over compact Riemann surfaces, Math. Ann. 328 (2004), 299-351, math.AG/0211428.
  13. Bradlow S.B., Wilkin G., Morse theory, Higgs fields, and Yang-Mills-Higgs functionals, J. Fixed Point Theory Appl. 11 (2012), 1-41, arXiv:1308.1460.
  14. Cliff E., Nevins T., Shen S., On the Kirwan map for moduli of Higgs bundles, arXiv:1808.10311.
  15. Daskalopoulos G., Weitsman J., Wentworth R.A., Wilkin G., Morse theory and hyperkähler Kirwan surjectivity for Higgs bundles, J. Differential Geom. 87 (2011), 81-115, math.SG/0701560.
  16. Donagi R., Spectral covers, in Current Topics in Complex Algebraic Geometry (Berkeley, CA, 1992/93), Math. Sci. Res. Inst. Publ., Vol. 28, Cambridge University Press, Cambridge, 1995, 65-86, alg-geom/9505009.
  17. 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.
  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. Donagi R., Pantev T., Langlands duality for Hitchin systems, Invent. Math. 189 (2012), 653-735, math.AG/0604617.
  20. Erdös P., Ginzburg A., Ziv A., Theorem in the additive number theory, Bull. Res. Counc. Israel Sect. F Math. Phys. 10F (1961), 41-43.
  21. Fredrickson L., Perspectives on the asymptotic geometry of the Hitchin moduli space, arXiv:1809.05735.
  22. García-Prada O., Heinloth J., Schmitt A., On the motives of moduli of chains and Higgs bundles, J. Eur. Math. Soc. (JEMS) 16 (2014), 2617-2668, arXiv:1104.5558.
  23. 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.
  24. Gothen P.B., The Betti numbers of the moduli space of stable rank $3$ Higgs bundles on a Riemann surface, Internat. J. Math. 5 (1994), 861-875.
  25. Gothen P.B., The topology of Higgs bundle moduli spaces, Ph.D. Thesis, University of Warwick, 1995.
  26. Gothen P.B., King A.D., Homological algebra of twisted quiver bundles, J. London Math. Soc. 71 (2005), 85-99, math.AG/0202033.
  27. Gothen P.B., Oliveira A.G., Topological mirror symmetry for parabolic Higgs bundles, arXiv:1707.08536.
  28. Groechenig M., Wyss D., Ziegler P., Mirror symmetry for moduli spaces of Higgs bundles via $p$-adic integration, arXiv:1707.06417.
  29. Harder G., Narasimhan M.S., On the cohomology groups of moduli spaces of vector bundles on curves, Math. Ann. 212 (1975), 215-248.
  30. Hausel T., Compactification of moduli of Higgs bundles, J. Reine Angew. Math. 503 (1998), 169-192, math.AG/9804083.
  31. Hausel T., Global topology of the Hitchin system, in Handbook of Moduli, Vol. II, Adv. Lect. Math. (ALM), Vol. 25, Editors G. Farkas, I. Morrison, Int. Press, Somerville, MA, 2013, 29-69, arXiv:1102.1717.
  32. Hausel T., Letellier E., Rodriguez Villegas F., Arithmetic harmonic analysis on character and quiver varieties II, Adv. Math. 234 (2013), 85-128, arXiv:1109.5202.
  33. Hausel T., Rodriguez Villegas F., Mixed Hodge polynomials of character varieties (with an appendix by Nicholas M. Katz), Invent. Math. 174 (2008), 555-624, math.AG/0612668.
  34. Hausel T., Rodriguez Villegas F., Cohomology of large semiprojective hyperkähler varieties, Astérisque 370 (2015), 113-156, arXiv:1309.4914.
  35. Hausel T., Thaddeus M., Mirror symmetry, Langlands duality, and the Hitchin system, Invent. Math. 153 (2003), 197-229, math.AG/0205236.
  36. Hausel T., Thaddeus M., Relations in the cohomology ring of the moduli space of rank 2 Higgs bundles, J. Amer. Math. Soc. 16 (2003), 303-327, math.AG/0003094.
  37. Hausel T., Thaddeus M., Generators for the cohomology ring of the moduli space of rank 2 Higgs bundles, Proc. London Math. Soc. 88 (2004), 632-658, math.AG/0003093.
  38. Hitchin N.J., The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987), 59-126.
  39. Hitchin N.J., Stable bundles and integrable systems, Duke Math. J. 54 (1987), 91-114.
  40. Hitchin N.J., Karlhede A., Lindström U., Roček M., Hyperkähler metrics and supersymmetry, Comm. Math. Phys. 108 (1987), 535-589.
  41. Hitchin N.J., Segal G.B., Ward R.S., Integrable systems: twistors, loop groups, and Riemann surfaces, Oxford Graduate Texts in Mathematics, Vol. 4, The Clarendon Press, Oxford University Press, New York, 1999.
  42. Jovovic V., A131868, On-line encyclopedia of integer sequences, available at, 2007.
  43. Kapustin A., Witten E., Electric-magnetic duality and the geometric Langlands program, Commun. Number Theory Phys. 1 (2007), 1-236, hep-th/0604151.
  44. Macdonald I.G., Symmetric products of an algebraic curve, Topology 1 (1962), 319-343.
  45. Markman E., Spectral curves and integrable systems, Compositio Math. 93 (1994), 255-290.
  46. Markman E., Generators of the cohomology ring of moduli spaces of sheaves on symplectic surfaces, J. Reine Angew. Math. 544 (2002), 61-82, math.AG/0009109.
  47. Mellit A., Poincaré polynomials of moduli spaces of Higgs bundles and character varieties (no punctures), arXiv:1707.04214.
  48. Mozgovoy S., Solutions of the motivic ADHM recursion formula, Int. Math. Res. Not. 2012 (2012), 4218-4244, arXiv:1104.5698.
  49. Mozgovoy S., Schiffmann O., Counting Higgs bundles, arXiv:1411.2101.
  50. Mozgovoy S., Schiffmann O., Counting Higgs bundles and type A quiver bundles, arXiv:1705.04849.
  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. Nitsure N., Moduli space of semistable pairs on a curve, Proc. London Math. Soc. 62 (1991), 275-300.
  53. Noe T., A145855, On-line encyclopedia of integer sequences, available at, 2008.
  54. Rayan S., Co-Higgs bundles on ${\mathbb P}^1$, New York J. Math. 19 (2013), 925-945, arXiv:1010.2526.
  55. Rayan S., The quiver at the bottom of the twisted nilpotent cone on $\mathbb{P}^1$, Eur. J. Math. 3 (2017), 1-21, arXiv:1609.08226.
  56. Rayan S., Sundbo E., Twisted argyle quivers and Higgs bundles, Bull. Sci. Math. 146 (2018), 1-32, arXiv:1803.04531.
  57. Reineke M., Cohomology of quiver moduli, functional equations, and integrality of Donaldson-Thomas type invariants, Compos. Math. 147 (2011), 943-964, arXiv:0903.0261.
  58. Schiffmann O., Indecomposable vector bundles and stable Higgs bundles over smooth projective curves, Ann. of Math. 183 (2016), 297-362, arXiv:1406.3839.
  59. Schmitt A., Moduli for decorated tuples of sheaves and representation spaces for quivers, Proc. Indian Acad. Sci. Math. Sci. 115 (2005), 15-49, math.AG/0401173.
  60. Simpson C.T., Harmonic bundles on noncompact curves, J. Amer. Math. Soc. 3 (1990), 713-770.
  61. Simpson C.T., Nonabelian Hodge theory, in Proceedings of the International Congress of Mathematicians, Vols. I, II (Kyoto, 1990), Math. Soc. Japan, Tokyo, 1991, 747-756.
  62. Simpson C.T., Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. (1992), 5-95.
  63. Sloane N.J.A., A0009900, On-line encyclopedia of integer sequences, available at, 1995.
  64. Thaddeus M., Stable pairs, linear systems and the Verlinde formula, Invent. Math. 117 (1994), 317-353, alg-geom/9210007.
  65. 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.
  66. Wentworth R.A., Wilkin G., Morse theory and stable pairs, in Variational Problems in Differential Geometry, London Math. Soc. Lecture Note Ser., Vol. 394, Cambridge University Press, Cambridge, 2012, 142-181, arXiv:1002.3124.
  67. Wilkin G., Morse theory for the space of Higgs bundles, Comm. Anal. Geom. 16 (2008), 283-332, math.DG/0611113.

Previous article  Next article   Contents of Volume 14 (2018)