### Context

#### Differential geometry

differential geometry

synthetic differential geometry

complex geometry

# Contents

## Idea

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

## 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)