Skip to main content
SpringerLink
Log in

Some comments on Chern-Simons gauge theory

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

Abstract

Following M. F. Atiyah and R. Bott [AB] and E. Witten [W], we consider the space of flat connections on the trivialSU(2) bundle over a surfaceM, modulo the space of gauge transformations. We describe on this quotient space a natural hermitian line-bundle with connection and prove that if the surfaceM is now endowed with a complex structure, this line bundle is isomorphic to the determinant bundle. We show heuristically how path-integral quantisation of the Chern-Simons action yields holomorphic sections of this bundle.

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

  • [A] Atiyah, M.F.: New invariants of three and four manifolds. Proc. Symp. Pure Math. (AMS)48 (1988)

  • [AB] Atiyah, M.F., Bott, R.: The Yang-Mills equation over Riemann surfaces. Phil. Trans. R. Soc. London A308, 523–617 (1982)

    Google Scholar 

  • [DJT] Dunne, G.V., Jackiw, R., Trugenberger, C.A.: Chern-Simons theory in the Schrödinger representation. MIT preprint MIT-CTP 1711 Ann. Phys. (to appear)

  • [FH] Feynman, R.P., Hibbs, A.R.: Quantum mechanics and path integrals. New York: McGraw-Hill 1965

    Google Scholar 

  • [J] Jackiw, R.: Chern-Simons terms and cocycles in physics and mathematics. In Quantum field theory and quantum statistics. Batalin, I., Isham, C., Vikovisky, C. (eds.). Bristol, U.K.: A. Hilger 1987

    Google Scholar 

  • [Q] Quillen, D.: Determinants of Cauchy-Riemann operators over a Riemann surface. Funct. Anal. Appl.19, 31–34 (1986)

    Article  Google Scholar 

  • [M] Mickelsson, J.: String quantisation on group manifolds and the holomorphic geometry of DiffS 1/S 1. Commun. Math. Phys.112, 653–662 (1987)

    Article  Google Scholar 

  • [NS] Narasimhan, M.S., Seshadri, C.S.: Stable and unitary vector on a compact surface. Ann. Math.82, 540–567 (1965)

    Google Scholar 

  • [ND] Narasimhan, M.S., Drezet, J.M.: TiFR preprint, 1988

  • [S] Seshadri, C.S.: Space of unitary vector bundles on a compact Riemann surface. Ann. Math.85, 303–336 (1967)

    Google Scholar 

  • [W] Witten, E.: Quantum field theory and the Jones polynomial. Institute for Advanced Study preprint IASSNS-HEP-88/33 (August 1988)

Download references

Author information

Author notes

  1. T. R. Ramadas

    Present address: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, 400005, Bombay, India

Authors and Affiliations

  1. Department of Mathematics, M.I.T., 02139, Cambridge, MA, USA

    T. R. Ramadas, I. M. Singer & J. Weitsman

Authors
  1. T. R. Ramadas
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. I. M. Singer
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. J. Weitsman
    View author publications

    You can also search for this author in PubMed Google Scholar

Additional information

Communicated by A. Jaffe

I.M.S. and T.R.R. supported by DOE grant DE-FG02-88ER 25066. J.W. supported by NSF Mathematical Sciences post-doctoral research scholarship 8807291

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ramadas, T.R., Singer, I.M. & Weitsman, J. Some comments on Chern-Simons gauge theory. Commun.Math. Phys. 126, 409–420 (1989). https://doi.org/10.1007/BF02125132

Download citation

  • Received:

  • Revised:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02125132

Keywords

Search

Navigation