The Narasimhan–Seshadri theorem identifies certain moduli spaces of flat connections over a (compact) Riemann surface with (compact) complex manifolds of stable vector bundles over .
In Quantum field theory
In gauge field theories such as notably Chern-Simons theory, moduli spaces of flat connections may appear as phase spaces. The Narasimhan–Seshadri theorem then provides a complex structure on these phase spaces and thus a spin^c structure (see there). This is the data required to obatain the geometric quantization by push-forward of the phase space. See also at Hitchin connection.
The original article is
- M. Narasimhan; C. Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Annals of Mathematics. Second Series 82: 540–567, (1965) (JSTOR)
Another proof is in
- Simon Donaldson, A new proof of a theorem of Narasimhan and Seshadri, J. Differential Geom. Volume 18, Number 2 (1983), 269-277. (EUCLID)
Reviews are in
- Jonathan Evans, Narasimhan-Seshadri theorem (pdf I, pdf II)
Related discussision in the context of Hitchin connections is in
- Peter Scheinost, Martin Schottenloher, Metaplectic quantization of the moduli spaces of flat and parabolic bundles, J. reine angew. Mathematik, 466 (1996) (web)
Revised on October 31, 2013 22:24:49
by Urs Schreiber