Based gauge group - Knowlpedia

Let $\pi:P\to M$ be a principal bundle, choose $x_0\in M$ , and choose any point $p_0\in P_{x_0}$ . The based gauge group is the subgroup

\mathcal G_{p_0}(P)=\{\Phi\in\mathcal G(P):\Phi(p_0)=p_0\}

of the gauge group $\mathcal G(P)$ . Equivalently, if a gauge transformation is represented by an equivariant map $u:P\to G$ through $\Phi_u(p)=p\,u(p)$ , then $\Phi_u$ is based exactly when $u(p_0)=e$ . Equivariance then gives $u(p_0g)=g^{-1}u(p_0)g=e$ , so $\Phi_u$ fixes every point of the fiber $P_{x_0}$ .

Dependence on the framing

Replacing $p_0$ by $p_0g$ conjugates the evaluation map to $G$ by $g$ , but conjugation fixes the identity. Its kernel, and hence the based subgroup, is therefore literally independent of the choice of $p_0$ in the fiber. One may consequently write $\mathcal G_{x_0}(P)$ . A framing or trivialization at $x_0$ is needed only to identify the evaluation map with a particular copy of $G$ ; under such an identification, the based condition is the familiar requirement that a gauge function $u:M\to G$ satisfy $u(x_0)=e$ .

Action on connections

The based group acts by pullback on the space of principal connections. If $M$ is connected, a gauge transformation preserving a connection is determined by its value at one point through parallel transport. Consequently, its stabilizer inside $\mathcal G_{p_0}(P)$ is trivial. This is why based gauge groups are used to form free gauge quotients.

Conventions and scope

References

Daniel S. Freed and Karen K. Uhlenbeck, Instantons and Four-Manifolds, 2nd ed., Springer, 1991. DOI record. Relevant: Chapter 3, based gauge transformations and free gauge actions.
Mark J. D. Hamilton, Mathematical Gauge Theory, Springer, 2017. DOI record. Relevant: gauge groups, their action on connections, and stabilizers.