Stabilizer of a connection - Imported Knowlpedia Content

Definition

Let $P\to M$ be a principal $G$ -bundle, let $\mathcal G(P)$ be its gauge group, and let $A$ be a principal connection. The stabilizer of $A$ is

\operatorname{Stab}(A) := \{u\in\mathcal G(P)\mid u\cdot A=A\},

where $u\cdot A$ denotes the gauge action on connections. It is the stabilizer subgroup of $A$ for this action. Its elements are precisely the gauge symmetries of the connection. A connection with stabilizer larger than the unavoidable central gauge transformations is called reducible in many gauge-theoretic settings.

Description by parallel sections

A gauge transformation may be viewed as a section of the adjoint group bundle $\operatorname{Ad}(P)=P\times_GG$ . It fixes $A$ exactly when that section is parallel for the connection induced by $A$ . Thus a stabilizing transformation is determined by its value at one point on each connected component of $M$ .

If $M$ is connected and $p\in P$ , evaluation at $p$ identifies the stabilizer with the centralizer of the holonomy group:

\operatorname{Stab}(A)\cong \{g\in G\mid gh=hg\text{ for every }h\in\operatorname{Hol}_p(A)\}.

Changing $p$ conjugates both subgroups, so the description is intrinsic up to conjugacy.

Role in gauge quotients

The orbit of $A$ is modeled infinitesimally by gauge directions modulo the Lie algebra of $\operatorname{Stab}(A)$ . When the stabilizer is minimal, a local gauge slice can often produce a manifold-like chart on the quotient. A larger stabilizer produces isotropy, so the orbit space of connections is generally singular or stratified rather than a smooth manifold.

For a connected base, every constant transformation with value in the center $Z(G)$ stabilizes every connection. Gauge theory therefore often quotients by the center, or calls $A$ irreducible when its stabilizer is exactly this central subgroup.

Remarks

The exact terminology depends on the chosen gauge group. Requiring Sobolev regularity, fixing a framing, or fixing the transformation at a basepoint changes the stabilizer. In particular, a based gauge group removes constant central symmetries and can make its action free where the full gauge-group action is not.

References

Daniel S. Freed and Karen K. Uhlenbeck, Instantons and Four-Manifolds, 2nd ed., Springer, 1991. DOI record. Relevant: Chapter 3, gauge-group actions and their stabilizers.