Knowlpedia

Definition

Gauge-fixing condition

A supplementary condition used to choose local representatives of gauge-equivalence classes of fields.

Definition

Let A(P)\mathcal A(P) be the on a fixed and let the G(P)\mathcal G(P) act on it. A gauge-fixing condition on a region U⊆A(P)U\subseteq\mathcal A(P) is an auxiliary equation

χ(A)=0\chi(A)=0

whose solution set is meant to provide representatives for the . It is effective on UU when every relevant orbit has a solution. A local gauge fixing near A0A_0 should additionally cut nearby orbits transversely and uniquely up to the stabilizer of A0A_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 A0A_0, the Coulomb condition has the form

dA0∗(A−A0)=0.d_{A_0}^{*}(A-A_0)=0.

For compact structure group and the regularity hypotheses used in , slice theorems identify a neighborhood of an orbit with applied to such a slice. Stabilizers must still be retained, so the local quotient can have orbifold-like or more singular behavior Freed and Uhlenbeck, chapter 3.

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=0d^{*}A=0 for an abelian connection removes the exact part of AA 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
  1. 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.
  2. 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.