Holonomy correspondence for flat connections - Imported Knowlpedia Content

Statement

Let $M$ be a connected smooth manifold with basepoint $x$ , and let $G$ be a Lie group. The holonomy correspondence for flat connections is the bijection

\left\{ \begin{array}{c} \text{flat principal \(G\)-bundles over \(M\)}\\ \text{up to connection-preserving isomorphism} \end{array} \right\} \longleftrightarrow \operatorname{Hom}(\pi_1(M,x),G)/G .

Here a flat principal bundle means a principal $G$ -bundle with a flat connection. The correspondence sends it to its holonomy representation after choosing a point over $x$ ; changing that point conjugates the representation. Conversely, a homomorphism $\rho$ produces the quotient $\widetilde M\times_\rho G$ with the flat connection descended from the product connection. The right-hand quotient is by the conjugation action of $G$ .

Construction from monodromy

Let $\pi_1(M,x)$ act on the universal cover by deck transformations and on $G$ by left multiplication through $\rho$ :

\gamma\cdot(\widetilde x,g) = (\gamma\widetilde x,\rho(\gamma)g).

This action commutes with the principal right $G$ -action, so

P_\rho=(\widetilde M\times G)/\pi_1(M,x)

is a principal $G$ -bundle. The horizontal distribution tangent to $\widetilde M$ descends and is flat. Its holonomy is $\rho$ , up to the inverse or conjugation dictated by path-lifting conventions. This construction and its converse are the classification content behind the standard equivalences for flat principal bundles.

Passage to moduli

The correspondence is a bijection of isomorphism classes. With suitable topologies and hypotheses, it refines to a comparison between the moduli space of flat connections and a topological representation quotient. For compact $G$ over a closed surface, every conjugacy orbit is closed and the quotient behaves well topologically; for noncompact or complex reductive groups, one often replaces the naive orbit space by a closed-orbit or geometric-invariant-theory quotient.

On a fixed principal bundle $P$ , only the representations for which $P_\rho$ is isomorphic to $P$ occur. Thus the correspondence does not say that every representation lies in the flat moduli of one preselected bundle.

Example

For $M=S^1$ , a representation is determined by one element $g\in G$ . Two resulting flat bundles with connection are isomorphic exactly when their elements are conjugate. If $G=U(1)$ , conjugation is trivial and the moduli is $U(1)$ .

Conventions and scope

For complex vector bundles, the same construction is often called the smooth Riemann–Hilbert or monodromy correspondence. It should not be confused with the analytic Riemann–Hilbert correspondence involving regular singular differential equations or perverse sheaves.

References

Shoshichi Kobayashi and Katsumi Nomizu, Foundations of Differential Geometry, Volume I, Wiley Classics, 1996. Publisher record. Relevant: Chapter II, holonomy, flat connections, and bundle reconstruction from monodromy.
William M. Goldman, “The Symplectic Nature of Fundamental Groups of Surfaces,” Advances in Mathematics 54 (1984), 200–225. DOI record. Relevant: §§1–2, surface-group representations and flat-bundle moduli.