nLab
Narasimhan–Seshadri theorem

Context

Differential geometry

differential geometry

synthetic differential geometry

Axiomatics

Models

Concepts

Theorems

Applications

Complex geometry

Contents

Idea

The Narasimhan–Seshadri theorem identifies certain moduli spaces of flat connections over a (compact) Riemann surface Σ\Sigma with (compact) complex manifolds of stable vector bundles over Σ\Sigma.

Applications

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.

References

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 (82.169.114.243)