geometry, complex numbers, complex line
$dim = 1$: Riemann surface, super Riemann surface
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.
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
equal to (minus) the slope of $E$ (where $\star$ 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
Lecture notes include
A textbook providing much of the background definitions involved is
Related discussion in the context of Hitchin connections is in