Coulomb slice for unitary connections ($d_A^{\ast }a=0$)

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