Unitary bundle gauge group ($\mathcal{G}$)

The group of unitary bundle gauge transformations under composition. It acts on connections and their solution spaces. The stabilizer of a connection is the centralizer of its holonomy acting by parallel automorphisms, so a quotient can have singularities when this stabilizer increases.

// Destination: geometry-and-topology.bigb

Near a fixed connection $A$, impose $d_A^{\ast }a=0$ on perturbations. The linearized gauge-fixing equation is $d_A^{\ast }{d}_A\xi$, invertible on the orthogonal complement of its parallel kernel. In Sobolev completions above the multiplication threshold, the implicit function theorem gives a local slice; its remaining identifications are by the compact stabilizer of $A$. This reduces the gauge quotient to a finite-dimensional Kuranishi model when the curvature equation is elliptic.

// Destination: geometry-and-topology.bigb

The subgroup of unitary bundle gauge transformations equal to the identity at a chosen framed base point. Quotienting flat connections by this subgroup retains the full flat holonomy representation; quotienting by the whole group also divides by simultaneous conjugation.

// Destination: geometry-and-topology.bigb