### Context

#### Differential cohomology

differential cohomology

## Ingredients

• cohomology

• differential geometry

• ## Connections on bundles

• connection on a bundle

• curvature

• Chern-Weil theory

• ## Higher abelian differential cohomology

• differential function complex

• differential orientation

• ordinary differential cohomology

• differential K-theory

• differential elliptic cohomology

• differential cobordism cohomology

• ## Higher nonabelian differential cohomology

• Chern-Weil theory in Smooth∞Grpd

• ∞-Chern-Simons theory

• ## Fiber integration

• higher holonomy

• fiber integration in differential cohomology

• ## Application to gauge theory

• gauge theory

• gauge field

• quantum anomaly

• #### Complex geometry

complex geometry

# Contents

## Idea

The Narasimhan–Seshadri theorem (Narasimhan-Seshadri 65) identifies certain moduli spaces of flat connections over a (compact) Riemann surface $\Sigma$ with (compact) complex manifolds of stable holomorphic vector bundles over $\Sigma$.

For the special case of line bundles this may be viewed as a special case of the Hodge-Maxwell theorem, also of Deligne’s characterization of the intermediate Jacobian, see there at Examples – Picard variety. The analogue of the theorem for higher dimensional complex manifolds is the Kobayashi-Hitchin correspondence. The special case of that for Kähler manifolds is the Donaldson-Uhlenbeck-Yau theorem.

## Statement

An indecomposable Hermitian holomorphic vector bundle $E$ on a Riemann surface $(\Sigma,g)$ is stable precisely if there is a compatibly unitary connection $\nabla$ on $E$ with constant central curvature

$\star F_\nabla = - \mu(E)$

equal to (minus) the slope of $E$ (where $\star$ is the Hodge star operator).

e.g. (Evans, p. 2)

## 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.

Related theorems

## References

The original article is

and another proof appeared in

• Simon Donaldson, A new proof of a theorem of Narasimhan and Seshadri, J. Differential Geom. Volume 18, Number 2 (1983), 269-277. (EUCLID)

A good general survey and re-discussion is in

• Michael Atiyah, Raoul Bott, section 8 of The Yang-Mills equations over Riemann surfaces, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences

Vol. 308, No. 1505 (Mar. 17, 1983), pp. 523-615 (jstor, lighning summary)

A recording of a review talk is here

Lecture notes include

A textbook providing much of the background definitions involved is

Related discussion 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)

Last revised on January 5, 2017 at 09:56:04. See the history of this page for a list of all contributions to it.