Based gauge group - Imported Knowlpedia Content

Definition

Let $\pi:P\to M$ be a principal bundle, choose $x_0\in M$ , and choose a 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 $u(p_0)=e$ . The chosen point in the total space, not merely its base point $x_0$ , is part of the framing data.

Dependence on the framing

Replacing $p_0$ by $p_0g$ conjugates the evaluation description by $g$ . Thus different choices in the same fiber give naturally isomorphic, though not literally identical, based subgroups. When the bundle is trivialized at $x_0$ , the based condition becomes 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; see Freed and Uhlenbeck, Chapter 3.

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.