Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-10T19:39:23.535Z Has data issue: false hasContentIssue false
  • English
  • Français

Central Extensions of Loop Groups and Obstruction to Pre-Quantization

Published online by Cambridge University Press:  20 November 2018

Derek Krepski
Derek Krepski*
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, ON e-mail: krepskid@math.mcmaster.ca
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

An explicit construction of a pre-quantum line bundle for the moduli space of flat $G$-bundles over a Riemann surface is given, where $G$ is any non-simply connected compact simple Lie group. This work helps to explain a curious coincidence previously observed between Toledano Laredo's work classifying central extensions of loop groups $LG$ and the author's previous work on the obstruction to pre-quantization of the moduli space of flat $G$-bundles.

Keywords

53D22Eloop groupcentral extensionprequantization
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 56 , Issue 1 , 01 March 2013 , pp. 116 - 126
Copyright
Copyright © Canadian Mathematical Society 2013

References

[1] Alekseev, A., Meinrenken, E., and Woodward, C., Formulas of Velinde type for non simply-connected groups. arxiv:math/0005047.Google Scholar
[2] Atiyah, M. F. and Bott, R., The Yang-Mills equations over Riemann surfaces. Philos. Trans. Roy. Soc. London Ser. A 308 (1982), no. 1505, 523–615. http://dx.doi.org/10.1098/rsta.1983.0017 Google Scholar
[3] Bismut, J.-M. and Labourie, F., Symplectic geometry and the Verlinde formulas. In: Surveys in differential geometry: differential geometry inspired by string theory, Surv. Differ. Geom, 5, Int. Press, Boston, MA, 1999, pp. 97–311.Google Scholar
[4] Cannas da Silva, A., Lectures on symplectic geometry. Lecture Notes in Mathematics, 1764, Springer-Verlag, Berlin, 2001.Google Scholar
[5] Krepski, D., Pre-quantization of the moduli space of flat G-bundles over a surface. J. Geom. Phys. 58 (2008), no. 11, 1624–1637. http://dx.doi.org/10.1016/j.geomphys.2008.07.009 Google Scholar
[6] Meinrenken, E. and C.Woodward, Hamiltonian loop group actions and Verlinde factorization. J. Differential Geom. 50 (1998), no. 3, 417–469.Google Scholar
[7] Pressley, A. and Segal, G., Loop groups. Oxford Mathematical Monographs, Oxford Science Publications, The Clarendon Press, Oxford University Press, New York, 1986.Google Scholar
[8] Ramadas, T. R., Singer, I. M., and J.Weitsman, Some comments on Chern-Simons gauge theory. Comm. Math. Phys. 126 (1989), no. 2, 409–420. http://dx.doi.org/10.1007/BF02125132 Google Scholar
[9] Laredo, V. Toledano, Positive energy representations of the loop groups of non-simply connected Lie groups. Comm. Math. Phys. 207 (1999), no. 2, 307–339. http://dx.doi.org/10.1007/s002200050728 Google Scholar