Selecta Mathematica

, Volume 16, Issue 2, pp 201–240 | Cite as

Vertex operators and the geometry of moduli spaces of framed torsion-free sheaves

Article

Abstract

We define complexes of vector bundles on products of moduli spaces of framed rank r torsion-free sheaves on \({\mathbb{P}^2}\) . The top non-vanishing equivariant Chern classes of the cohomology of these complexes yield actions of the r-colored Heisenberg and Clifford algebras on the equivariant cohomology of the moduli spaces. In this way we obtain a geometric realization of the boson-fermion correspondence and related vertex operators.

Mathematics Subject Classification (2000)

Primary 14C05 17B69 

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References

  1. 1.
    Barth W.: Moduli of vector bundles on the projective plane. Invent. Math. 42, 63–91 (1977)MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Carlsson, E., Okounkov, A.: Exts and vertex operators. arXiv:0801.2565Google Scholar
  3. 3.
    Cox, D.A., Katz, S.: Mirror symmetry and algebraic geometry. In: Mathematical Surveys and Monographs, vol. 68. American Mathematical Society, Providence (1999)Google Scholar
  4. 4.
    Frenkel I., Lepowsky J., Meurman A.: Vertex operator algebras and the Monster. In: Pure and Applied Mathematics, vol. 134. Academic Press, Boston (1988)Google Scholar
  5. 5.
    Frenkel I.B.: Two constructions of affine Lie algebra representations and boson-fermion correspondence in quantum field theory. J. Funct. Anal. 44(3), 259–327 (1981)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Frenkel I.B., Kac V.G.: Basic representations of affine Lie algebras and dual resonance models. Invent. Math. 62(1), 23–66 (1980/81)CrossRefMathSciNetGoogle Scholar
  7. 7.
    Grojnowski I.: Instantons and affine algebras. I. The Hilbert scheme and vertex operators. Math. Res. Lett. 3(2), 275–291 (1996)MATHMathSciNetGoogle Scholar
  8. 8.
    Kac V.G.: Infinite-Dimensional Lie Algebras. Cambridge University Press, Cambridge (1990)MATHGoogle Scholar
  9. 9.
    Kac, V.G., Peterson, D.H.: 112 constructions of the basic representation of the loop group of E 8. In: Symposium on Anomalies, Geometry, Topology (Chicago, IL, 1985), pp. 276–298. World Scientific, Singapore (1985)Google Scholar
  10. 10.
    Lehn M.: Chern classes of tautological sheaves on Hilbert schemes of points on surfaces. Invent. Math. 136(1), 157–207 (1999)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Lepowsky J.: Calculus of twisted vertex operators. Proc. Natl. Acad. Sci. USA 82(24), 8295–8299 (1985)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Lepowsky J., Li H.: Introduction to vertex operator algebras and their representations. In: Progress in Mathematics, vol. 227. Birkhäuser, Boston (2004)Google Scholar
  13. 13.
    Li W.-P., Qin Z., Wang W.: Hilbert schemes and \({\mathcal {W}}\) algebras. Int. Math. Res. Not. 27, 1427–1456 (2002)CrossRefMathSciNetGoogle Scholar
  14. 14.
    Li W.-P., Qin Z., Wang W.: Hilbert schemes, integrable hierarchies, and Gromov-Witten theory. Int. Math. Res. Not. 40, 2085–2104 (2004)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Licata, A.: Moduli spaces of sheaves on surfaces and geometric representation theory. PhD thesis, Yale University (2007)Google Scholar
  16. 16.
    Macdonald I.G.: Symmetric Functions and Hall Polynomials. Oxford Mathematical Monographs. The Clarendon Press/Oxford University Press, New York (1995)Google Scholar
  17. 17.
    Nakajima H.: Instantons on ALE spaces, quiver varieties, and Kac–Moody algebras. Duke Math. J. 76(2), 365–416 (1994)MATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    Nakajima H.: Quiver varieties and Kac–Moody algebras. Duke Math. J. 91(3), 515–560 (1998)MATHCrossRefMathSciNetGoogle Scholar
  19. 19.
    Nakajima, H.: Lectures on Hilbert schemes of points on surfaces. In: University Lecture Series, vol. 18. American Mathematical Society, Providence (1999)Google Scholar
  20. 20.
    Nakajima H., Yoshioka K.: Instanton counting on blowup. I. 4-Dimensional pure gauge theory. Invent. Math. 162(2), 313–355 (2005)MATHCrossRefMathSciNetGoogle Scholar
  21. 21.
    Savage A.: A geometric boson-fermion correspondence. C. R. Math. Acad. Sci. Soc. R. Can. 28(3), 65–84 (2006)MATHMathSciNetGoogle Scholar
  22. 22.
    Segal G.: Unitary representations of some infinite-dimensional groups. Commun. Math. Phys. 80(3), 301–342 (1981)MATHCrossRefGoogle Scholar
  23. 23.
    ten Kroode F., van de Leur J.: Bosonic and fermionic realizations of the affine algebra \({\widehat{\rm gl}_n}\) . Commun. Math. Phys. 137(1), 67–107 (1991)MATHCrossRefGoogle Scholar
  24. 24.
    Vasserot E.: Sur l’anneau de cohomologie du schéma de Hilbert de C 2. C. R. Acad. Sci. Paris Sér. I Math. 332(1), 7–12 (2001)MATHMathSciNetGoogle Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  1. 1.Department of MathematicsStanford UniversityPalo AltoUSA
  2. 2.Department of Mathematics and StatisticsUniversity of OttawaOttawaCanada

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