Duke Mathematical Journal

Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras

Hiraku Nakajima

Full-text: Access denied (no subscription detected) We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Article information

Source
Duke Math. J. Volume 76, Number 2 (1994), 365-416.

Dates
First available in Project Euclid: 20 February 2004

Permanent link to this document
http://projecteuclid.org/euclid.dmj/1077286968

Mathematical Reviews number (MathSciNet)
MR1302318

Zentralblatt MATH identifier
0826.17026

Digital Object Identifier
doi:10.1215/S0012-7094-94-07613-8

Subjects
Primary: 53C25: Special Riemannian manifolds (Einstein, Sasakian, etc.)
Secondary: 17B67: Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras 58D27: Moduli problems for differential geometric structures 58E15: Application to extremal problems in several variables; Yang-Mills functionals [See also 81T13], etc.

Citation

Nakajima, Hiraku. Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras. Duke Math. J. 76 (1994), no. 2, 365--416. doi:10.1215/S0012-7094-94-07613-8. http://projecteuclid.org/euclid.dmj/1077286968.


Export citation

References

  • [At] M. F. Atiyah, Convexity and commuting Hamiltonians, Bull. London Math. Soc. 14 (1982), no. 1, 1–15.
  • [AB] M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A. 362 (1982), 523–615.
  • [ADHM] M. F. Atiyah, N. J. Hitchin, V. G. Drinfeld, and Yu. I. Manin, Construction of instantons, Phys. Lett. A 65 (1978), no. 3, 185–187.
  • [Ba] S. Bando, Einstein-Hermitian metrics on noncompact Kähler manifolds, Einstein metrics and Yang-Mills connections (Sanda, 1990) eds. T. Mabuchi and S. Mukai, Lecture Notes in Pure and Appl. Math., vol. 145, Dekker, New York, 1993, pp. 27–33.
  • [BLM] A. Beilinson, G. Lusztig, and R. MacPherson, A geometric setting for the quantum deformation of $\rm GL\sb n$, Duke Math. J. 61 (1990), no. 2, 655–677.
  • [BM] W. Borho and R. MacPherson, Partial resolutions of nilpotent varieties, Analysis and topology on singular spaces, II, III (Luminy, 1981), Astérisque, vol. 101, Soc. Math. France, Paris, 1983, pp. 23–74.
  • [Bo] R. Bott, Nondegenerate critical manifolds, Ann. of Math. (2) 60 (1954), 248–261.
  • [Do] S. K. Donaldson, Instantons and geometric invariant theory, Comm. Math. Phys. 93 (1984), no. 4, 453–460.
  • [DK] S. K. Donaldson and P. B. Kronheimer, The geometry of four-manifolds, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, New York, 1990.
  • [Fr] T. Frankel, Fixed points and torsion on Kähler manifolds, Ann. of Math. (2) 70 (1959), 1–8.
  • [Ga] P. Gabriel, Unzerlegbare Darstellungen. I, Manuscripta Math. 6 (1972), 71–103.
  • [Gi] V. Ginzburg, Lagrangian construction of the enveloping algebra $U(\rm sl\sb n)$, C. R. Acad. Sci. Paris Sér. I Math. 312 (1991), no. 12, 907–912.
  • [GN] T. Gocho and H. Nakajima, Einstein-Hermitian connections on hyper-Kähler quotients, J. Math. Soc. Japan 44 (1992), no. 1, 43–51.
  • [Go] R. Goto, On toric hyper-Kähler manifolds given by the hyper-Kähler quotient method, Infinite analysis, Part A, B (Kyoto, 1991), Adv. Ser. Math. Phys., vol. 16, World Sci. Publishing, River Edge, NJ, 1992, pp. 317–338.
  • [Hi] N. J. Hitchin, The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3) 55 (1987), no. 1, 59–126.
  • [HKLR] N. J. Hitchin, A. Karlhede, U. Lindström, and M. Roček, Hyper-Kähler metrics and supersymmetry, Comm. Math. Phys. 108 (1987), no. 4, 535–589.
  • [Kac] V. G. Kac, Infinite-dimensional Lie algebras, Cambridge University Press, Cambridge, 1990.
  • [KS] M. Kashiwara and P. Schapira, Sheaves on manifolds, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 292, Springer-Verlag, Berlin, 1990.
  • [Kin] A. D. King, Moduli of representations of finite dimensional algebras, to appear in Quart. J. Math. Oxford Ser. (2).
  • [Ki] F. C. Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Mathematical Notes, vol. 31, Princeton University Press, Princeton, NJ, 1984.
  • [KP] H. Kraft and C. Procesi, Closures of conjugacy classes of matrices are normal, Invent. Math. 53 (1979), no. 3, 227–247.
  • [Kr1] P. B. Kronheimer, The construction of ALE spaces as hyper-Kähler quotients, J. Differential Geom. 29 (1989), no. 3, 665–683.
  • [Kr2] P. B. Kronheimer, Instantons and the geometry of the nilpotent variety, J. Differential Geom. 32 (1990), no. 2, 473–490.
  • [KN] P. B. Kronheimer and H. Nakajima, Yang-Mills instantons on ALE gravitational instantons, Math. Ann. 288 (1990), no. 2, 263–307.
  • [La] G. Laumon, Correspondance de Langlands géométrique pour les corps de fonctions, Duke Math. J. 54 (1987), no. 2, 309–359.
  • [L1] G. Lusztig, Quantum deformations of certain simple modules over enveloping algebras, Adv. in Math. 70 (1988), no. 2, 237–249.
  • [L2] G. Lusztig, Canonical bases arising from quantized enveloping algebras, J. Amer. Math. Soc. 3 (1990), no. 2, 447–498.
  • [L3] G. Lusztig, Quivers, perverse sheaves, and quantized enveloping algebras, J. Amer. Math. Soc. 4 (1991), no. 2, 365–421.
  • [L4] G. Lusztig, Affine quivers and canonical bases, Inst. Hautes Études Sci. Publ. Math. (1992), no. 76, 111–163.
  • [Mc] R. D. MacPherson, Chern classes for singular algebraic varieties, Ann. of Math. (2) 100 (1974), 423–432.
  • [Na1] H. Nakajima, Moduli spaces of anti-self-dual connections on ALE gravitational instantons, Invent. Math. 102 (1990), no. 2, 267–303.
  • [Na2] H. Nakajima, Homology of moduli spaces of instantons on ALE spaces (I), to appear in J. Differential Geom.
  • [Na3] H. Nakajima, Resolutions of moduli spaces of ideal instantons on $\mathbbR^4$, proceedings of the 1993 Taniguchi Symposium on Low-dimensional Topology and Topological Field Theory, to appear.
  • [Na4] H. Nakajima, Gauge theory on resolutions of simple singularities and simple Lie algebras, Internat. Math. Res. Notices (1994), no. 2, 61–74.
  • [Nee] A. Neeman, The topology of quotient varieties, Ann. of Math. (2) 122 (1985), no. 3, 419–459.
  • [Ne] L. Ness, A stratification of the null cone via the moment map, Amer. J. Math. 106 (1984), no. 6, 1281–1329.
  • [Ri] C. M. Ringel, Hall algebras and quantum groups, Invent. Math. 101 (1990), no. 3, 583–591.
  • [Ro] M. Rosso, Finite-dimensional representations of the quantum analog of the enveloping algebra of a complex simple Lie algebra, Comm. Math. Phys. 117 (1988), no. 4, 581–593.
  • [Sc] G. W. Schwarz, The topology of algebraic quotients, Topological methods in algebraic transformation groups (New Brunswick, NJ, 1988), Progr. Math., vol. 80, Birkhäuser Boston, Boston, MA, 1989, pp. 135–151.
  • [Si] C. T. Simpson, Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J. Amer. Math. Soc. 1 (1988), no. 4, 867–918.
  • [SL] R. Sjamaar and E. Lerman, Stratified symplectic spaces and reduction, Ann. of Math. (2) 134 (1991), no. 2, 375–422.
  • [S1] P. Slodowy, Four lectures on simple groups and singularities, Communications of the Mathematical Institute, Rijksuniversiteit Utrecht, vol. 11, Rijksuniversiteit Utrecht Mathematical Institute, Utrecht, 1980.
  • [S2] P. Slodowy, Simple singularities and simple algebraic groups, Lecture Notes in Mathematics, vol. 815, Springer, Berlin, 1980.