Slice theorem for the gauge action

Statement

Let $P\to M$ be a principal bundle with compact structure group over a closed Riemannian $n$ -manifold. Use the Sobolev completions of connections in $W^{k,p}$ and gauge transformations in $W^{k+1,p}$ , where $k\geq1$ , $1<p<\infty$ , and $kp>n$ . Fix an Ad-invariant inner product on the Lie algebra and a smooth reference connection $A$ . there is $\varepsilon>0$ such that the Coulomb slice

\mathcal S_{A,\varepsilon} =\{A+a\mid d_A^*a=0,\ \|a\|_{W^{k,p}}<\varepsilon\}

meets every sufficiently nearby gauge orbit. Moreover, a neighborhood of the orbit of $A$ in the connection space is modeled equivariantly by

\mathcal G\times_{\operatorname{Stab}(A)}\mathcal S_{A,\varepsilon}.

Why the Coulomb condition is transverse

The tangent to the gauge orbit at $A$ is $\operatorname{im}d_A$ , while $d_A^*a=0$ selects its $L^2$ -orthogonal complement. Solving for a gauge transformation that places $A+a$ in the slice reduces, after linearization, to the elliptic operator $d_A^*d_A$ . The implicit-function theorem then supplies existence and local uniqueness up to the stabilizer of $A$ .

The offset in Sobolev regularity is essential: a $W^{k+1,p}$ gauge transformation acts on a $W^{k,p}$ connection without losing a derivative. The hypothesis $kp>n$ provides the multiplication and continuity properties required by the nonlinear action.

Quotient structure

Passing to the quotient gives a local model

\mathcal S_{A,\varepsilon}/\operatorname{Stab}(A)

for the moduli problem near $[A]$ . If the stabilizer acts trivially after central symmetries are removed, this behaves like a Banach-manifold chart. Nontrivial effective isotropy can produce an orbifold or a more singular quotient; its presence alone does not force singularity of the coarse quotient. This describes the quotient of all connections. For a gauge-invariant differential equation, one must further intersect the slice with its solution set, and regularity of that equation is an additional issue.

Conventions and scope

References

Daniel S. Freed and Karen K. Uhlenbeck, Instantons and Four-Manifolds, 2nd ed., Springer, 1991. DOI record. Relevant: Chapter 3, especially Theorem 3.4, the slice theorem for the gauge action.
Simon K. Donaldson and Peter B. Kronheimer, The Geometry of Four-Manifolds, Oxford University Press, 1990. DOI record. Relevant: §4.2, Sobolev gauge groups and local slices.