Skip to main content
SpringerLink
Log in

Equivariant Volumes of Non-Compact Quotients and Instanton Counting

  • Published:
Communications in Mathematical Physics Aims and scope Submit manuscript

Abstract

Motivated by Nekrasov’s instanton counting, we discuss a method for calculating equivariant volumes of non-compact quotients in symplectic and hyper-Kähler geometry by means of the Jeffrey-Kirwan residue formula of non-abelian localization. In order to overcome the non-compactness, we use varying symplectic cuts to reduce the problem to a compact setting, and study what happens in the limit that recovers the original problem. We implement this method for the ADHM construction of the moduli spaces of framed Yang-Mills instantons on \({\mathbb{R}^{4}}\) and rederive the formulas for the equivariant volumes obtained earlier by Nekrasov-Shadchin, expressing these volumes as iterated residues of a single rational function.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Atiyah M.F., Bott R.: The moment map and equivariant cohomology. Topology 23(1), 1–28 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  2. Atiyah M.F., Hitchin N.J., Drinfel’d V.G., Manin Yu.I.: Construction of instantons. Phys. Lett. A 65(3), 185–187 (1978)

    Article  ADS  MathSciNet  Google Scholar 

  3. Braverman, A., Etingof, P.: Instanton counting via affine Lie algebras. II. From Whittaker vectors to the Seiberg-Witten prepotential. In Studies in Lie theory, Volume 243 of Progr. Math., Boston, MA: Birkhäuser Boston, 2006, pp. 61–78

  4. Braverman, A.: Instanton counting via affine Lie algebras. I. Equivariant J-functions of (affine) flag manifolds and Whittaker vectors. In Algebraic structures and moduli spaces, Volume 38 of CRM Proc. Lecture Notes, Providence, RI: Amer. Math. Soc., 2004, pp. 113–132

  5. Bryan, J., Sanders, M.: Instantons on S 4 and \({\overline{\rm CP}^{2}}\) , rank stabilization, and Bott periodicity. Topology 39(2), 331–352 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  6. Berline N., Vergne M.: Classes caractéristiques équivariantes. Formule de localisation en cohomologie équivariante. C. R. Acad. Sci. Paris Sér. I Math. 295(9), 539–541 (1982)

    MATH  MathSciNet  Google Scholar 

  7. Brion M., Vergne M.: Arrangement of hyperplanes. I. Rational functions and Jeffrey-Kirwan residue. Ann. Sci. École Norm. Sup. (4) 32(5), 715–741 (1999)

    MATH  MathSciNet  Google Scholar 

  8. Crawley-Boevey W.: Geometry of the moment map for representations of quivers. Comp. Math. 126(3), 257–293 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  9. Duistermaat J.J., Heckman G.J.: On the variation in the cohomology of the symplectic form of the reduced phase space. Invent. Math. 69(2), 259–268 (1982)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  10. Dolgachev, I.V., Hu, Yi.: Variation of geometric invariant theory quotients. Inst. Hautes Études Sci. Publ. Math. (87):5–56, (1998). with an appendix by N. Ressayre

  11. Dorey N., Hollowood T.J., Khoze V.V., Mattis M.P.: The calculus of many instantons. Phys. Rep. 371(4–5), 231–459 (2002)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  12. Donaldson S.K.: Instantons and geometric invariant theory. Commun. Math. Phys. 93(4), 453–460 (1984)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  13. Freiermuth, H.-G.: On partial compactifications of the space of framed vector bundles on the projective plane. PhD thesis, Columbia University, 2005

  14. Guillemin V., Kalkman J.: The Jeffrey-Kirwan localization theorem and residue operations in equivariant cohomology. J. Reine Angew. Math. 470, 123–142 (1996)

    MATH  MathSciNet  Google Scholar 

  15. Guillemin, V., Lerman, E., Sternberg, S.: Symplectic fibrations and multiplicity diagrams. Cambridge: Cambridge University Press, 1996

  16. Jeffrey L.C., Kirwan F.C.: Intersection pairings in moduli spaces of holomorphic bundles on a Riemann surface. Electron. Res. Announc. Amer. Math. Soc. 1(2), 57–71 (1995)

    Article  MathSciNet  Google Scholar 

  17. Jeffrey L.C., Kirwan F.C.: Localization for nonabelian group actions. Topology 34(2), 291–327 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  18. Jeffrey L.C., Kirwan F.C.: Localization and the quantization conjecture. Topology 36(3), 647–693 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  19. Jeffrey L.C., Kirwan F.C.: Intersection theory on moduli spaces of holomorphic bundles of arbitrary rank on a Riemann surface. Ann. of Math. (2) 148(1), 109–196 (1998)

    Article  MATH  MathSciNet  Google Scholar 

  20. Jeffrey, L., Kogan, M.: Localization theorems by symplectic cuts. In: The breadth of symplectic and Poisson geometry, Volume 232 of Progr. Math., Boston, MA: Birkhäuser Boston, 2005, pp. 303–326

  21. Jeffrey L.C., Kiem Y.-H., Kirwan F., Woolf J.: Cohomology pairings on singular quotients in geometric invariant theory. Transform. Groups 8(3), 217–259 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  22. Kirwan, F.C.: Cohomology of quotients in symplectic and algebraic geometry, Volume 31 of Mathematical Notes. Princeton, NJ: Princeton University Press, 1984

  23. Kirwan F.C.: Partial desingularisations of quotients of nonsingular varieties and their Betti numbers. Ann. of Math. (2) 122(1), 41–85 (1985)

    Article  MathSciNet  Google Scholar 

  24. Lerman E.: Symplectic cuts. Math. Res. Lett. 2(3), 247–258 (1995)

    MATH  MathSciNet  Google Scholar 

  25. Martin, S.K.: Symplectic quotients by a nonabelian group and by its maximal torus. To appear in Annals of Math., http://arXiv.org/list/math.SG/0001001, 2000

  26. Meinrenken E. Equivariant cohomology and the Cartan model. In: Françoise J.-P., Naber G.L., Tsun T.S. editors, Encyclopedia of mathematical physics. Oxford: Academic Press/Elsevier Science, 2006

  27. Moore G., Nekrasov N., Shatashvili S.: Integrating over Higgs branches. Commun. Math. Phys. 209(1), 97–121 (2000)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  28. Nakajima, H.: Lectures on Hilbert schemes of points on surfaces, Volume 18 of University Lecture Series. Providence, RI: Amer. Math. Soc. 1999

  29. Nekrasov N.A.: Seiberg-Witten prepotential from instanton counting. Adv. Theor. Math. Phys. 7(5), 831–864 (2003)

    MATH  MathSciNet  Google Scholar 

  30. Nekrasov, N.A., Okounkov, A.: Seiberg-Witten theory and random partitions. In: The unity of mathematics, Volume 244 of Progr. Math., Boston, MA: Birkhäuser Boston, 2006, pp. 525–596

  31. Nekrasov N., Schwarz A.: Instantons on noncommutative R 4, and (2, 0) superconformal six-dimensional theory. Commun. Math. Phys. 198(3), 689–703 (1998)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  32. Nekrasov N., Shadchin S.: ABCD of instantons. Commun. Math. Phys. 252(1–3), 359–391 (2004)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  33. Nakajima, H., Yoshioka, K.: Lectures on instanton counting. In: Algebraic structures and moduli spaces, Volume 38 of CRM Proc. Lecture Notes, Providence, RI: Amer. Math. Soc. 2004, pp. 31–101

  34. Nakajima H., Yoshioka K.: Instanton counting on blowup, I. 4-Dimensional pure gauge theory. Invent. Math. 162(2), 313–355 (2005)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  35. Nakajima H., Yoshioka K.: Instanton counting on blowup. II. K-theoretic partition function. Transform. Groups 10(3–4), 489–519 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  36. Ortega, J.-P., Ratiu, T.S.: Momentum maps and Hamiltonian reduction, Volume 222 of Progress in Mathematics. Boston, MA: Birkhäuser Boston Inc., 2004

  37. Okonek, C., Schneider, M., Spindler, H.: Vector bundles on complex projective spaces, Volume 3 of Progress in Mathematics. Boston, MA: Birkhäuser Boston, 1980

  38. Paradan P.-E.: The moment map and equivariant cohomology with generalized coefficients. Topology 39(2), 401–444 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  39. Prato E., Wu S.: Duistermaat-Heckman measures in a non-compact setting. Comp. Math. 94(2), 113–128 (1994)

    MATH  ADS  MathSciNet  Google Scholar 

  40. Seiberg N., Witten E.: Electric-magnetic duality, monopole condensation, and confinement in N = 2 supersymmetric Yang-Mills theory. Nucl. Phys. B 426(1), 19–52 (1994)

    Article  MATH  ADS  MathSciNet  Google Scholar 

  41. Thaddeus M.: Geometric invariant theory and flips. J. Amer. Math. Soc. 9(3), 691–723 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  42. Vergne M.: A note on the Jeffrey-Kirwan-Witten localisation formula. Topology 35(1), 243–266 (1996)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Toronto, Toronto, Ontario, M5S 2E4, Canada

    Johan Martens

Authors
  1. Johan Martens
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Johan Martens.

Additional information

Communicated by N.A. Nekrasov

Rights and permissions

Reprints and permissions

About this article

Cite this article

Martens, J. Equivariant Volumes of Non-Compact Quotients and Instanton Counting. Commun. Math. Phys. 281, 827–857 (2008). https://doi.org/10.1007/s00220-008-0501-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00220-008-0501-x

Keywords

Search

Navigation