The Narasimhan–Seshadri theorem (Narasimhan-Seshadri 65) identifies certain moduli spaces of flat connections over a (compact) Riemann surface with (compact) complex manifolds of stable holomorphic vector bundles over .
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.
An indecomposable Hermitian holomorphic vector bundle on a Riemann surface is stable precisely if there is a compatibly unitary connection on with constant central curvature
equal to (minus) the slope of (where is the Hodge star operator).
e.g. (Evans, p. 2)
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
The original article is
and another proof appeared in
A good general survey and re-discussion is in
Vol. 308, No. 1505 (Mar. 17, 1983), pp. 523-615 (jstor, lighning summary)
A recording of a review talk is here
Lecture notes include
Narasimhan-Seshadri theorem (lecture 13 pdf, lecture 14 pdf, lecture 15 pdf, lecture 16 pdf)
A textbook providing much of the background definitions involved is
Related discussion in the context of Hitchin connections is in
Last revised on July 26, 2024 at 15:06:09. See the history of this page for a list of all contributions to it.