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Mirror symmetry and Langlands duality in the non-Abelian Hodge theory of a curve

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Geometric Methods in Algebra and Number Theory

Part of the book series: Progress in Mathematics ((PM,volume 235))

Summary

The paper surveys the mirror symmetry conjectures of Hausel-Thaddeus and Hausel-Rodriguez-Villegas concerning the equality of certain Hodge numbers of SL(n, ℂ) vs. PGL(n, ℂ) flat connections and character varieties for curves, respectively. Several new results and conjectures and their relations to works of Hitchin, Gothen, Garsia-Haiman and Earl-Kirwan are explained. These use the representation theory of finite groups of Lie-type via the arithmetic of character varieties and lead to an unexpected conjecture for a Hard Lefschetz theorem for their cohomology.

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References

  1. D. Alvis —The duality operation in the character ring of a finite Chevalley group”, Bull. Amer. Math. Soc. (N.S.)1 (1979), no. 6, 907–911.

    Article  MATH  MathSciNet  Google Scholar 

  2. M.F. Atiyah and R. Bott —The Yang-Mills equations over Riemann surfaces”, Philos. Trans. Roy. Soc. London Ser. A308 (1982), 523–615.

    MathSciNet  Google Scholar 

  3. V.V. Batyrev and D. Dais —Strong McKay correspondence, string-theoretic Hodge numbers and mirror symmetry”, Topology35 (1996), 901–929.

    Article  MathSciNet  MATH  Google Scholar 

  4. K. Corlette —Flat G-bundles with canonical metrics”, J. Differential Geom.28 (1988), no. 3, 361–382.

    MATH  MathSciNet  Google Scholar 

  5. C. Curtis —Truncation and duality in the character ring of a finite group of Lie type”, J. Algebra62 (1980), no. 2, 320–332.

    Article  MATH  MathSciNet  Google Scholar 

  6. P. Deligne —Équations différentielles á points singuliers réguliers”, Lecture Notes in Math., vol. 163, Springer, Berlin-New York, 1970.

    MATH  Google Scholar 

  7. —,Théorie de Hodge II, Inst. Hautes Études Sci. Publ. Math. 40 (1971), 5–57.

    Google Scholar 

  8. R. Earl and F. Kirwan —The Pontryagin rings of moduli spaces of arbitrary rank holomorphic bundles over a Riemann surface”, J. London Math. Soc. (2) 60 (1999), no. 3, 835–846.

    Article  MathSciNet  MATH  Google Scholar 

  9. C. Faber —A conjectural description of the tautological ring of the moduli space of curves”, inModuli of curves and abelian varieties”, 109–129, Aspects Math., E33, Vieweg, Braunschweig, 1999.

    Google Scholar 

  10. D. Freed and F. Quinn —Chern-Simons theory with finite gauge group”, Comm. Math. Phys.156 (1993), no. 3, 435–472.

    Article  MathSciNet  MATH  Google Scholar 

  11. G. Frobenius and I. Schur —Über die reellen Darstellungen der endlichen Gruppen”, Sitzungsberichte der königlich preussichen Akademie der Wissenschaften (1906), 186–208

    Google Scholar 

  12. A.M. Garsia and M. Haiman —A remarkable q,t-Catalan sequence and q-Lagrange inversion”, J. Algebraic Combin.5 (1996) no. 3, 191–244.

    Article  MathSciNet  MATH  Google Scholar 

  13. W.M. Goldman and E.Z. Xia —Rank One Higgs Bundles and Representations of Fundamental Groups of Riemann Surfaces”, 2004, math.DG/0402429.

    Google Scholar 

  14. J.A Green —The characters of the finite general linear groups”, Trans. Amer. Math. Soc.80 (1955), 402–447.

    Article  MATH  MathSciNet  Google Scholar 

  15. P. Griffiths, J. Harris — Principles of algebraic geometry. New York, Wiley 1978.

    MATH  Google Scholar 

  16. P.B. Gothen —The Betti numbers of the moduli space of rank 3 Higgs bundles”, Internat. J. Math.5 (1994), 861–875.

    Article  MATH  MathSciNet  Google Scholar 

  17. M. Haiman —t,q-Catalan numbers and the Hilbert scheme”, Discrete Math. 193 (1998) no. 1–3, 201–224.

    Article  MATH  MathSciNet  Google Scholar 

  18. G. Harder and M.S. Narasimhan —On the cohomology groups of moduli spaces of vector bundles over curves”, Math. Ann.212 (1975), 215–248.

    Article  MathSciNet  MATH  Google Scholar 

  19. T. Hausel —Compactification of moduli of Higgs bundles”, J. Reine Angew. Math.503 (1998), 169–192.

    MATH  MathSciNet  Google Scholar 

  20. —, Geometry of the moduli space of Higgs bundles”, Ph.D. thesis, University of Cambridge, 1998, math. AG/0107040.

    Google Scholar 

  21. —,Quaternionic Geometry of Matroids”, 2003, math.AG/0308146.

    Google Scholar 

  22. T. Hausel and N. Proudfoot —Abelianization for hyperkähler quotients”, (to appear in Topology), math.SG/0310141.

    Google Scholar 

  23. T. Hausel and F. Rodriguez-Villegas —Mirror symmetry, Langlands duality and representations of finite groups of Lie type”, (in preparation).

    Google Scholar 

  24. T. Hausel and B. Sturmfels —Toric hyperkähler varieties”, Doc. Math.7 (2002), 495–534.

    MathSciNet  MATH  Google Scholar 

  25. T. Hausel and M. Thaddeus —Generators for the cohomology ring of the moduli space of rank 2 Higgs bundles”, Proc. London Math. Soc.88 (2004), 632–658.

    Article  MathSciNet  MATH  Google Scholar 

  26. —,Relations in the cohomology ring of the moduli space of rank 2 Higgs bundles”, J. Amer. Math. Soc., 16 (2003), 303–329.

    Article  MathSciNet  MATH  Google Scholar 

  27. —,Examples of mirror partners arising from integrable systems”, C. R. Acad. Sci. Paris, 333(4) (2001), 313–318.

    MathSciNet  MATH  Google Scholar 

  28. —,Mirror symmetry, Langlands duality and Hitchin systems”, Invent. Math., 153, no. 1 (2003), 197–229.

    Article  MathSciNet  MATH  Google Scholar 

  29. N. Hitchin —The self-duality equations on a Riemann surface”, Proc. London Math. Soc. (3) 55 (1987), 59–126.

    MATH  MathSciNet  Google Scholar 

  30. —,Stable bundles and integrable systems”, Duke Math. J. 54 (1987), 91–114.

    Article  MATH  MathSciNet  Google Scholar 

  31. —,Lectures on special Lagrangian submanifolds”, Winter School on Mirror Symmetry, Vector Bundles and Lagrangian Submanifolds (Cambridge, MA, 1999), 151–182, Amer. Math. Soc., 2001.

    Google Scholar 

  32. H. Jordan —Group characters of various types of linear groups”, Amer. J. Math, 29 (1907), 387.

    MATH  MathSciNet  Google Scholar 

  33. L. Katzarkov and T. Pantev —Nonabelian (p,p) classes”, (in Motives, polylogarithms and Hodge theory, Part II (Irvine, CA, 1998)), Int. Press Lect. Ser., 3, II, 625–715.

    Google Scholar 

  34. M. Kontsevich —Motivic Integration”, Lecture at Orsay, 1995.

    Google Scholar 

  35. G. Lehrer —The characters of the finite special linear groups”, J. Algebra26 (1973), 564–583.

    Article  MATH  MathSciNet  Google Scholar 

  36. G. Lusztig —Fermionic form and Betti numbers”, math.QA/0005010.

    Google Scholar 

  37. A.D. Mednykh —Determination of the number of nonequivalent coverings over a compact Riemann surface”, Soviet Mathematics Doklady19 (1978), 318–320.

    MATH  MathSciNet  Google Scholar 

  38. M. Mehta —Hodge structure on the cohomology of the moduli space of Higgs bundles”, math.AG/0112111.

    Google Scholar 

  39. E. Markman —Generators of the cohomology ring of moduli spaces of sheaves on symplectic surfaces”, J. Reine Angew. Math.544 (2002), 61–82.

    MATH  MathSciNet  Google Scholar 

  40. H. Nakajima —Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras”, Duke Math. J.76 (1994) no. 2, 365–416.

    Article  MATH  MathSciNet  Google Scholar 

  41. M.S. Narasimhan and S. Ramanan —Generalised Prym varieties as fixed points”, J. Indian Math. Soc.39 (1975), 1–19.

    MathSciNet  MATH  Google Scholar 

  42. I. Schur —Untersuchungen über die Darstellung der endlichen Gruppen durch Gebrochene Lineare Substitutionen”, J. Reine Angew. Math.132 (1907), 85.

    MATH  Google Scholar 

  43. J-P. Serre —Exemples de variétés projectives conjuguées non homéomorphes”, C. R. Acad. Sci. Paris258 (1964), 4194–4196.

    MATH  MathSciNet  Google Scholar 

  44. C.T. Simpson —Nonabelian Hodge theory”, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990), 747–756, Math. Soc. Japan, Tokyo, 1991.

    Google Scholar 

  45. —,Higgs bundles and local systems”, Inst. Hautes Études Sci. Publ. Math. 75 (1992), 5–95.

    MATH  MathSciNet  Google Scholar 

  46. —,Moduli of representations of the fundamental group of a smooth projective variety”, I, Inst. Hautes Études Sci. Publ. Math. 79 (1994), 47–129.

    MATH  MathSciNet  Google Scholar 

  47. —,Moduli of representations of the fundamental group of a smooth projective variety”, II, Inst. Hautes Études Sci. Publ. Math. 80 (1995), 5–79.

    MATH  Google Scholar 

  48. —,The Hodge filtration on nonabelian cohomology”, Algebraic geometry — Santa Cruz 1995, Proc. Symp. Pure Math. 62, ed. J. Kollár, R. Lazarsfeld, and D. Morrison, American Math. Soc., 1997.

    Google Scholar 

  49. A. Strominger, S.-T. Yau, and E. Zaslow —Mirror symmetry is T-duality”, Nuclear Phys. B479 (1996), 243–259.

    Article  MathSciNet  MATH  Google Scholar 

  50. C. Vafa —String vacua and orbifoldized LG models”, Modern Phys. Lett. A4 (1989), 1169–1185.

    Article  MathSciNet  Google Scholar 

  51. E. Verlinde —Fusion rules and modular transformations in 2D conformal field theory”, Nucl. Phys. B300 (1988), 360.

    Article  MathSciNet  Google Scholar 

  52. D. Zagier —Elementary aspects of the Verlinde formula and of the Harder-Narasimhan-Atiyah-Bott formula”, (in Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry (Ramat Gan, 1993), 445–462, Israel Math. Conf. Proc., 9, Bar-Ilan Univ., Ramat Gan, 1996.

    Google Scholar 

  53. E. Zaslow —Topological orbifold models and quantum cohomology rings”, Comm. Math. Phys.156 (1993), 301–331.

    Article  MATH  MathSciNet  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Texas, Austin, TX, 78712, USA

    Tamás Hausel

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  1. Tamás Hausel
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Editors and Affiliations

  1. Department of Mathematics Courant Institute of Mathematical Sciences, New York University, New York, NY, 10012, USA

    Fedor Bogomolov

  2. Department of Mathematics, Princeton University, Princeton, NJ, 08544, USA

    Yuri Tschinkel

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Hausel, T. (2005). Mirror symmetry and Langlands duality in the non-Abelian Hodge theory of a curve. In: Bogomolov, F., Tschinkel, Y. (eds) Geometric Methods in Algebra and Number Theory. Progress in Mathematics, vol 235. Birkhäuser Boston. https://doi.org/10.1007/0-8176-4417-2_9

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