nLab
Narasimhan–Seshadri theorem

Context

Differential cohomology

Complex geometry

Contents

Idea

The Narasimhan–Seshadri theorem (Narasimhan-Seshadri 65) identifies certain moduli spaces of flat connections over a (compact) Riemann surface Σ\Sigma with (compact) complex manifolds of stable holomorphic vector bundles over Σ\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 EE on a Riemann surface (Σ,g)(\Sigma,g) is stable precisely if there is a compatibly unitary connection \nabla on EE with constant central curvature

F =μ(E) \star F_\nabla = - \mu(E)

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

e.g. (Evans, p. 2)

(Atiyah-Bott 83 (8.1))

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)

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)

Revised on July 17, 2014 10:23:21 by Urs Schreiber (82.136.246.44)