Gauge-fixing condition

Let $\mathcal A(P)$ be the space of principal connections on a fixed principal bundle and let the gauge group $\mathcal G(P)$ act on it. A gauge-fixing condition on a region $U\subseteq\mathcal A(P)$ is an auxiliary equation

\chi(A)=0

whose solution set is meant to provide representatives for the gauge-orbit space. It is effective on $U$ when every relevant orbit has a solution. A local gauge fixing near $A_0$ should additionally cut nearby orbits transversely and uniquely up to the stabilizer of $A_0$ . These requirements are local analytical properties; the equation alone does not assert that a global representative exists on every orbit.

Local slices and analytical role

After completing the configuration and gauge spaces in suitable Sobolev norms, a successful gauge condition often defines a local slice for the gauge action. Near a reference connection $A_0$ , the Coulomb condition has the form

d_{A_0}^{*}(A-A_0)=0.

For compact structure group and the regularity hypotheses used in gauge theory, slice theorems identify a neighborhood of an orbit with gauge transformations applied to such a slice. Stabilizers must still be retained, so the local quotient can have orbifold-like or more singular behavior.

Gauge fixing is therefore an analytical device for studying equations and quotients, not an additional physical field equation. It can make an underdetermined gauge-invariant system elliptic or otherwise suitable for local analysis.

Examples

The Coulomb condition $d^{*}A=0$ for an abelian connection removes the exact part of $A$ locally, while harmonic forms and constant gauge transformations can remain as residual data.

Temporal gauge sets the component of a connection along a chosen time direction to zero. It can often be imposed along an interval by solving an ordinary differential equation, but global topology or periodic time may obstruct a single global choice.

Residual symmetry and global obstructions

Even when every nearby orbit meets a local slice, an orbit can meet the same condition more than once. Such multiple representatives are often called Gribov copies. Conversely, some orbits may fail to meet a proposed global condition. Thus local slice results do not produce a global section of the orbit projection in general.

References

D. S. Freed and K. K. Uhlenbeck, Instantons and Four-Manifolds, 2nd ed., Springer, 1991. DOI record. Relevant: chapter 3, gauge-group actions and local slices.
M. J. D. Hamilton, Mathematical Gauge Theory: With Applications to the Standard Model of Particle Physics, Springer, 2017. DOI record. Relevant: gauge transformations, gauge conditions, and Yang–Mills equations.