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

SIGMA 14 (2018), 127, 46 pages      arXiv:1809.05738      https://doi.org/10.3842/SIGMA.2018.127
Contribution to the Special Issue on Geometry and Physics of Hitchin Systems

Parallels between Moduli of Quiver Representations and Vector Bundles over Curves

Victoria Hoskins
Freie Universität Berlin, Arnimallee 3, Raum 011, 14195 Berlin, Germany

Received September 25, 2018, in final form November 18, 2018; Published online December 04, 2018

Abstract
This is a review article exploring similarities between moduli of quiver representations and moduli of vector bundles over a smooth projective curve. After describing the basic properties of these moduli problems and constructions of their moduli spaces via geometric invariant theory and symplectic reduction, we introduce their hyperkähler analogues: moduli spaces of representations of a doubled quiver satisfying certain relations imposed by a moment map and moduli spaces of Higgs bundles. Finally, we survey a surprising link between the counts of absolutely indecomposable objects over finite fields and the Betti cohomology of these (complex) hyperkähler moduli spaces due to work of Crawley-Boevey and Van den Bergh and Hausel, Letellier and Rodriguez-Villegas in the quiver setting, and work of Schiffmann in the bundle setting.

Key words: algebraic moduli problems; geometric invariant theory; representation theory of quivers; vector bundles and Higgs bundles on curves.

pdf (685 kb)   tex (59 kb)

References

  1. Álvarez Cónsul L., King A., A functorial construction of moduli of sheaves, Invent. Math. 168 (2007), 613-666, math.AG/0602032.
  2. Atiyah M.F., Bott R., The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), 523-615.
  3. Baraglia D., Classification of the automorphism and isometry groups of Higgs bundle moduli spaces, Proc. Lond. Math. Soc. 112 (2016), 827-854, arXiv:1411.2228.
  4. Baraglia D., Schaposnik L.P., Higgs bundles and $(A,B,A)$-branes, Comm. Math. Phys. 331 (2014), 1271-1300, arXiv:1305.4638.
  5. Baraglia D., Schaposnik L.P., Real structures on moduli spaces of Higgs bundles, Adv. Theor. Math. Phys. 20 (2016), 525-551, arXiv:1309.1195.
  6. Bernstein I.N., Gel'fand I.M., Ponomarev V.A., Coxeter functors and Gabriel's theorem, Russian Math. Surveys 28 (1973), no. 2, 17-32.
  7. Białynicki-Birula A., Some theorems on actions of algebraic groups, Ann. of Math. 98 (1973), 480-497.
  8. Biswas I., García-Prada O., Anti-holomorphic involutions of the moduli spaces of Higgs bundles, J. Éc. Polytech. Math. 2 (2015), 35-54, arXiv:1401.7236.
  9. Biswas I., García-Prada O., Hurtubise J., Pseudo-real principal $G$-bundles over a real curve, J. Lond. Math. Soc. 93 (2016), 47-64, arXiv:1502.00563.
  10. Brion M., Representations of quivers, in Geometric Methods in Representation Theory. I, Sémin. Congr., Vol. 24, Soc. Math. France, Paris, 2012, 103-144.
  11. Crawley-Boevey W., Lectures on representations of quivers, 1992, available at https://www.math.uni-bielefeld.de/~wcrawley/quivlecs.pdf.
  12. Crawley-Boevey W., Geometry of the moment map for representations of quivers, Compositio Math. 126 (2001), 257-293.
  13. Crawley-Boevey W., Van den Bergh M., Absolutely indecomposable representations and Kac-Moody Lie algebras (with an appendix by Hiraku Nakajima), Invent. Math. 155 (2004), 537-559, math.RA/0106009.
  14. Davison B., The integrality conjecture and the cohomology of preprojective stacks, arXiv:1602.02110.
  15. Donaldson S.K., A new proof of a theorem of Narasimhan and Seshadri, J. Differential Geom. 18 (1983), 269-277.
  16. Franco E., Jardim M., Marchesi S., Branes in the moduli space of framed sheaves, Bull. Sci. Math. 141 (2017), 353-383, arXiv:1504.05883.
  17. Ginzburg V., Lectures on Nakajima's quiver varieties, in Geometric methods in representation theory. I, Sémin. Congr., Vol. 24, Soc. Math. France, Paris, 2012, 145-219, arXiv:0905.0686.
  18. Harder G., Chevalley groups over function fields and automorphic forms, Ann. of Math. 100 (1974), 249-306.
  19. Harder G., Narasimhan M.S., On the cohomology groups of moduli spaces of vector bundles on curves, Math. Ann. 212 (1975), 215-248.
  20. Hausel T., Kac's conjecture from Nakajima quiver varieties, Invent. Math. 181 (2010), 21-37, arXiv:0811.1569.
  21. Hausel T., Letellier E., Rodriguez-Villegas F., Positivity for Kac polynomials and DT-invariants of quivers, Ann. of Math. 177 (2013), 1147-1168, arXiv:1204.2375.
  22. 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.
  23. Hausel T., Rodriguez-Villegas F., Cohomology of large semiprojective hyperkähler varieties, Astérisque 370 (2015), 113-156, arXiv:1309.4914.
  24. Hitchin N.J., The self-duality equations on a Riemann surface, Proc. London Math. Soc. 55 (1987), 59-126.
  25. Hitchin N.J., Karlhede A., Lindström U., Roček M., Hyperkähler metrics and supersymmetry, Comm. Math. Phys. 108 (1987), 535-589.
  26. Hoskins V., Stratifications associated to reductive group actions on affine spaces, Q. J. Math. 65 (2014), 1011-1047, arXiv:1210.6811.
  27. Hoskins V., Schaffhauser F., Group actions on quiver varieties and applications, arXiv:1612.06593.
  28. Hoskins V., Schaffhauser F., Rational points of quiver moduli spaces, arXiv:1704.08624.
  29. Hua J., Counting representations of quivers over finite fields, J. Algebra 226 (2000), 1011-1033.
  30. Hubery A., Quiver representations respecting a quiver automorphism: a generalisation of a theorem of Kac, J. London Math. Soc. 69 (2004), 79-96, math.RT/0203195.
  31. Huybrechts D., Lehn M., The geometry of moduli spaces of sheaves, Cambridge Mathematical Library, 2nd ed., Cambridge University Press, Cambridge, 2010.
  32. Kac V.G., Infinite root systems, representations of graphs and invariant theory, Invent. Math. 56 (1980), 57-92.
  33. Kac V.G., Some remarks on representations of quivers and infinite root systems, in Representation Theory, II (Proc. Second Internat. Conf., Carleton Univ., Ottawa, Ont., 1979), Lecture Notes in Math., Vol. 832, Springer, Berlin, 1980, 311-327.
  34. Kac V.G., Root systems, representations of quivers and invariant theory, in Invariant Theory (Montecatini, 1982), Lecture Notes in Math., Vol. 996, Springer, Berlin, 1983, 74-108.
  35. Kapustin A., Witten E., Electric-magnetic duality and the geometric Langlands program, Commun. Number Theory Phys. 1 (2007), 1-236, hep-th/0604151.
  36. Kempf G., Ness L., The length of vectors in representation spaces, in Algebraic Geometry (Proc. Summer Meeting, University Copenhagen, Copenhagen, 1978), Lecture Notes in Math., Vol. 732, Springer, Berlin, 1979, 233-243.
  37. King A.D., Moduli of representations of finite-dimensional algebras, Quart. J. Math. Oxford Ser. (2) 45 (1994), 515-530.
  38. Kirwan F.C., Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes, Vol. 31, Princeton University Press, Princeton, NJ, 1984.
  39. Langer A., Semistable sheaves in positive characteristic, Ann. of Math. 159 (2004), 251-276.
  40. Le Bruyn L., Procesi C., Semisimple representations of quivers, Trans. Amer. Math. Soc. 317 (1990), 585-598.
  41. Marsden J., Weinstein A., Reduction of symplectic manifolds with symmetry, Rep. Math. Phys. 5 (1974), 121-130.
  42. Mellit A., Poincaré polynomials of moduli spaces of Higgs bundles and character varieties (no punctures), arXiv:1707.04214.
  43. Milne J.S., Lectures on etale cohomology, 2013, available at http://www.jmilne.org/math/.
  44. Mozgovoy S., Motivic Donaldson-Thomas invariants and McKay correspondence, arXiv:1107.6044.
  45. Mozgovoy S., Schiffmann O., Counting higgs bundles and type a quiver bundles, arXiv:1705.04849.
  46. Mumford D., Fogarty J., Kirwan F., Geometric invariant theory, Ergebnisse der Mathematik und ihrer Grenzgebiete (2), Vol. 34, 3rd ed., Springer-Verlag, Berlin, 1994.
  47. Nakajima H., Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras, Duke Math. J. 76 (1994), 365-416.
  48. Narasimhan M.S., Seshadri C.S., Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. 82 (1965), 540-567.
  49. Ness L., A stratification of the null cone via the moment map (with an appendix by David Mumford), Amer. J. Math. 106 (1984), 1281-1329.
  50. Newstead P.E., Characteristic classes of stable bundles of rank $2$ over an algebraic curve, Trans. Amer. Math. Soc. 169 (1972), 337-345.
  51. Newstead P.E., Introduction to moduli problems and orbit spaces, Tata Institute of Fundamental Research Lectures on Mathematics and Physics, Vol. 51, Tata Institute of Fundamental Research, Bombay, Narosa Publishing House, New Delhi, 1978.
  52. Nitsure N., Construction of Hilbert and Quot schemes, in Fundamental Algebraic Geometry, Math. Surveys Monogr., Vol. 123, Amer. Math. Soc., Providence, RI, 2005, 105-137, math.AG/0504590.
  53. Proudfoot N.J., Hyperkähler analogues of Kähler quotients, Ph.D. Thesis, University of California, Berkeley, 2004, math.AG/0405233.
  54. Reineke M., The Harder-Narasimhan system in quantum groups and cohomology of quiver moduli, Invent. Math. 152 (2003), 349-368, math.QA/0204059.
  55. Reineke M., Every projective variety is a quiver Grassmannian, Algebr. Represent. Theory 16 (2013), 1313-1314, arXiv:1204.5730.
  56. Sala F., Schiffmann O., Cohomological Hall algebra of Higgs sheaves on a curve, arXiv:1801.03482.
  57. Schiffmann O., Lectures on Hall algebras, in Geometric Methods in Representation Theory. II, Sémin. Congr., Vol. 24, Soc. Math. France, Paris, 2012, 1-141, math.RT/0611617.
  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. Schiffmann O., Kac polynomials and Lie algebras associated to quivers and curves, arXiv:1802.09760.
  60. Schiffmann O., Vasserot E., On cohomological Hall algebras of quivers: generators, arXiv:1705.07488.
  61. Serre J.-P., Géométrie algébrique et géométrie analytique, Ann. Inst. Fourier, Grenoble 6 (1956), 1-42.
  62. Seshadri C.S., Space of unitary vector bundles on a compact Riemann surface, Ann. of Math. 85 (1967), 303-336.
  63. Simpson C.T., Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 5-95.
  64. 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.
  65. 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.

Previous article  Next article   Contents of Volume 14 (2018)