Skip to main content
SpringerLink
Log in

The prequantum line bundle on the moduli space of flat SU(N) connections on a Riemann surface and the homotopy of the large N limit

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

Abstract

We show that the prequantum line bundle on the moduli space of flat SU(2) connections on a closed Riemann surface of positive genus has degree 1. It then follows from work of Lawton and the second author that the classifying map for this line bundle induces a homotopy equivalence between the stable moduli space of flat SU(n) connections, in the limit as n tends to infinity, and \( {\mathbb C}P^\infty \). Applications to the stable moduli space of flat unitary connections are also discussed.

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

Notes

  1. Kirk and Klassen conclude that \(c_1 (L) = -1\). The discrepancy can be explained using the footnote regarding signs in Sect. 3 of the present article.

  2. In [6, Appendix C], the Chern–Weil formula for characteristic classes is stated without signs, because they use a version of Fubini’s theorem with signs [6, p. 304]. Since we have integrated using the usual version of Fubini’s theorem, we need a sign in our formula for \(c_1 (\mathcal {L}')\).

References

  1. Biswas, I., Lawton, S., Ramras, D.: Fundamental groups of character varieties: surfaces and tori. Math. Z. 281(1–2), 415–425 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Cappell, S., Lee, R., Miller, E.: The action of the Torelli group on the homology of representation spaces is nontrivial. Topology 39(4), 851–871 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Drezet, J.-M., Narasimhan, M.: Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques. Invent. Math. 97(1), 53–94 (1989)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  4. Kirk, P., Klassen, E.: Chern–Simons invariants of 3-manifolds decomposed along tori and the circle bundle over the representation space of \(T^2\). Commun. Math. Phys. 153(3), 521–557 (1993)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  5. Lawton, S., Ramras, D.: Covering spaces of character varieties. N. Y. J. Math. 21, 383–416 (2015)

    MathSciNet  MATH  Google Scholar 

  6. Milnor, J., Stasheff, J.: Characteristic Classes. Annals of Mathematical Studies, vol. 76. Princeton University Press, Princeton (1974)

    Google Scholar 

  7. Ramadas, T.R., Singer, I.M., Weitsman, J.: Some comments on Chern–Simons gauge theory. Commun. Math. Phys. 126, 409–420 (1989)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  8. Ramras, D.: The stable moduli space of flat connections over a surface. Trans. Am. Math. Soc. 363(2), 1061–1100 (2011)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The second author thanks Simon Donaldson for suggesting that the results of [8] should be connected to Goldman’s symplectic form. Additionally, we thank Jacques Hurtubise for asking about the higher genus case considered in Sect. 5 and the anonymous referees for helping to improve the exposition.

Author information

Authors and Affiliations

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

    Lisa C. Jeffrey

  2. Department of Mathematical Sciences, IUPUI, Indianapolis, IN, 46202, USA

    Daniel A. Ramras

  3. Department of Mathematics, Northeastern University, Boston, MA, 02115, USA

    Jonathan Weitsman

Authors
  1. Lisa C. Jeffrey
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Daniel A. Ramras
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Jonathan Weitsman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Daniel A. Ramras.

Additional information

Lisa C. Jeffrey was partially supported by a grant from NSERC.

Daniel A. Ramras was partially supported by a grant from the Simons Foundation (#279007).

Jonathan Weitsman was partially supported by NSF Grant DMS-12/11819.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Jeffrey, L.C., Ramras, D.A. & Weitsman, J. The prequantum line bundle on the moduli space of flat SU(N) connections on a Riemann surface and the homotopy of the large N limit. Lett Math Phys 107, 1581–1589 (2017). https://doi.org/10.1007/s11005-017-0956-9

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11005-017-0956-9

Keywords

Mathematics Subject Classification

Search

Navigation