Coulomb gauge - Knowlpedia

Let $A_0$ be a connection on a principal bundle with compact structure group over a Riemannian manifold, choose an $\operatorname{Ad}$ -invariant positive-definite inner product on its Lie algebra, and write another connection as $A=A_0+a$ . The connection $A$ is in Coulomb gauge relative to $A_0$ when

d_{A_0}^*a=0,

where $d_{A_0}^*$ is the covariant codifferential on adjoint-bundle-valued one-forms. This is a gauge-fixing condition: it makes the displacement $a$ formally $L^2$ -orthogonal to infinitesimal gauge directions $d_{A_0}\phi$ , with $\phi$ compactly supported away from any boundary. For unrestricted gauge directions on a manifold with boundary, an additional boundary condition is needed. It does not by itself assert that every gauge orbit has a representative satisfying the equation, or that such a representative is unique.

Local slice interpretation

The space of connections is affine, so the expression $A-A_0$ is an adjoint-bundle-valued one-form although a connection need not be represented by a single Lie-algebra-valued one-form on the base. Infinitesimally, the tangent to the gauge orbit through $A_0$ is $\operatorname{im}d_{A_0}$ . The Coulomb condition selects its formal $L^2$ -orthogonal complement $\ker d_{A_0}^*$ .

After suitable Sobolev completions and under standard regularity hypotheses, this complement yields a local slice modulo the stabilizer of $A_0$ . It is therefore local analytical structure, not a canonical global section of the gauge-orbit map.

Local trivializations and Uhlenbeck gauge

On a trivial bundle over a coordinate ball, with the product connection as reference, the condition becomes $d^*a=0$ . Uhlenbeck's gauge theorem says, roughly, that a connection with sufficiently small scale-invariant curvature norm can be gauge transformed into such a Coulomb gauge with quantitative Sobolev control.

Boundary versions normally add a condition on the normal component of $a$ . Without that extra condition, integration by parts does not identify $\ker d^*$ as the full orthogonal complement of exact gauge directions.

Residual symmetry

If $A_0$ has a nontrivial stabilizer, its stabilizing gauge transformations preserve the slice condition. Even for an irreducible connection, a Coulomb representative is generally unique only in a sufficiently small neighborhood and after controlling constant or based gauge transformations.

Equivalent codifferential expression

For the invariant inner product fixed above and $a=A-A_0$ , one also has

d_A^*a=d_{A_0}^*a.

Indeed, in an orthonormal frame $(e_i)$ , the difference of the two expressions is

(d_A^*-d_{A_0}^*)a=-\sum_i[a(e_i),a(e_i)]=0.

Thus these two relative Coulomb equations agree. The coordinate condition $d^*a=0$ describes them when the reference connection is the product connection in that trivialization.

References

Daniel S. Freed and Karen K. Uhlenbeck, Instantons and Four-Manifolds, 2nd ed., Springer, 1991. DOI record. Relevant: chapter 3, Sobolev gauge groups and Coulomb slices.
Karen K. Uhlenbeck, “Connections with $L^p$ Bounds on Curvature,” Communications in Mathematical Physics 83 (1982), 31–42. DOI record. Relevant: theorem 1.3, local Coulomb gauges with norm estimates.
Katrin Wehrheim, “Lagrangian Boundary Conditions for Anti-Self-Dual Instantons and the Atiyah–Floer Conjecture,” Journal of Symplectic Geometry 3 (2005), 703–747. Full text. Relevant: §2, pp. 706–709, covariant derivative, formal adjoint, and Theorem 2.2.