Reducible connection - Imported Knowlpedia Content

Definition

Let $P\to M$ be a principal $G$ -bundle over a connected manifold with compact structure group $G$ , and let $A$ be a principal connection. Using the full gauge group, $A$ is reducible when its stabilizer strictly contains the subgroup of constant gauge transformations arising from the center $Z(G)$ :

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

Equivalently, there is a gauge transformation fixed by $A$ whose value is not central. A connection is irreducible in this convention when $\operatorname{Stab}(A)=Z(G)$ . The definition records extra continuous or discrete symmetry of the connection, not merely nontriviality of the center.

Holonomy and preserved reductions

Evaluation at a point identifies the stabilizer with the centralizer of the holonomy group. Thus $A$ is reducible exactly when the holonomy has centralizer larger than $Z(G)$ ; see Freed and Uhlenbeck, Chapter 3. For matrix groups this often means that the holonomy representation preserves a proper decomposition, so $A$ is compatible with a reduction to the corresponding proper subgroup.

For a unitary connection on a Hermitian vector bundle, a parallel orthogonal splitting into nonzero subbundles makes the connection reducible. The converse takes this form when the stabilizing element has suitable eigenspace decomposition.

Role in gauge quotients

Reducible connections are points with nonminimal isotropy for the gauge action. Their orbits therefore produce singular or lower-dimensional strata in the quotient of the space of connections. Gauge-theoretic moduli problems often impose hypotheses excluding reducibles so that a gauge slice has a manifold-like quotient near a solution; this role is developed in Freed and Uhlenbeck, Chapter 3.

For a nonabelian $G$ , the flat product connection with trivial holonomy is reducible because its stabilizer contains all constant $G$ -valued transformations.

Conventions and examples

For an $SU(2)$ -connection on a rank-two Hermitian bundle, a preserved splitting $E=L\oplus L^{-1}$ is the standard reducible situation. By contrast, the central transformations $\{\pm I\}$ stabilize every $SU(2)$ -connection and do not by themselves make it reducible.

References

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