Irreducible connection

Let $P\to M$ be a principal $G$ -bundle over a connected manifold, with compact structure group $G$ , and let $A$ be a connection. Relative to the full gauge group, $A$ is irreducible if its stabilizer consists exactly of the constant gauge transformations induced by the center:

\operatorname{Stab}(A)=Z(G).

Thus $A$ has no stabilizing gauge transformations beyond this central subgroup. ## Reducibility

If the stabilizer strictly contains $Z(G)$ , the connection is reducible. This convention is designed for nonabelian gauge theory and depends on the chosen gauge group.

Holonomy characterization

Evaluation at a point identifies $\operatorname{Stab}(A)$ with the centralizer of the holonomy group of $A$ . Consequently,

A\text{ is irreducible} \quad\Longleftrightarrow\quad C_G(\operatorname{Hol}(A))=Z(G).

For a unitary connection on a Hermitian vector bundle, a nontrivial parallel orthogonal splitting produces noncentral stabilizing endomorphisms and hence reducibility. Under the standard compactness hypotheses, absence of such a parallel splitting is the corresponding irreducibility criterion.

Role in moduli spaces

Irreducible connections form the locus of minimal isotropy for the gauge action. After dividing out the central subgroup, the action is free there; combined with an analytic gauge slice, this makes gauge quotients locally manifold-like. Reducible connections retain additional isotropy and can produce singular strata; isotropy alone does not force the coarse quotient to be singular.

Examples and conventions

For an $SU(2)$ -connection, the unavoidable stabilizer is $\{\pm I\}$ ; a connection with exactly this stabilizer is irreducible. For nonabelian $G$ , a flat connection with trivial holonomy is reducible: parallel trivialization identifies its stabilizer with $G$ , which strictly contains $Z(G)$ .

The quotient group $\mathcal G(P)/Z(G)$ has trivial stabilizer exactly at irreducible connections. By contrast, a based gauge group has trivial stabilizer at every connection on a connected base: a parallel gauge transformation equal to the identity at one point is the identity everywhere. Based freeness therefore does not detect irreducibility. For abelian $G$ , one has $Z(G)=G$ , and the full-gauge-group convention labels every connection irreducible; the terminology is therefore usually reserved for nonabelian settings.

References

Daniel S. Freed and Karen K. Uhlenbeck, Instantons and Four-Manifolds, 2nd ed., Springer, 1991. Publisher record. Relevant: Chapter 3, stabilizers and irreducible connections.
Simon K. Donaldson and Peter B. Kronheimer, The Geometry of Four-Manifolds, Oxford University Press, 1990. Publisher record. Relevant: §4.2, gauge-group actions and irreducibility.

Ralph L. Cohen, The Topology of Fiber Bundles, Chapter 2, §3, Theorem 2.18 and its proof, pp. 61–62. Author-hosted notes.