Gauge Orbit Geometry, Global Obstructions, and Measures
Gauge fields form a quotient with stabilizers, bundle sectors, and possible global obstructions. This chapter separates local slice geometry from a global gauge choice and from the further task of constructing a quantum measure.
Enter this chapter
Section titled “Enter this chapter”Fix the bundle, allowed transformations, and boundary conditions first. Read local slices before the global obstruction theorem. Helpful background. Gauge orbits, Gauss constraints, and stabilizers supplies the physical quotient. Gribov copies and limits of local gauge fixing supplies the local-versus-global distinction. Lattice gauge fixing and gauge-dependent correlators supplies a finite regulator where representative dependence can be tested.
Topics
Section titled “Topics”- Gauge Configuration Groupoids and Moduli — Defines the connection groupoid with its automorphisms.
- Orbit Spaces, Stabilizers, and Stratified Quotients — Explains why stabilizer changes produce different orbit strata.
- Principal-Bundle Sectors and Large Gauge Transformations — Keeps bundle topology and large transformations explicit.
- Local Slices, Gauge Fixing, and Faddeev–Popov Geometry — Constructs local gauge coordinates after treating zero modes.
- Elliptic Gauge Complexes and Gribov Obstructions — Uses elliptic complexes without assuming a global slice.
- The Gribov–Singer Global Gauge-Fixing Obstruction — Identifies the topology obstructing a universal gauge choice.
- Moduli Stacks and Derived Geometry of Gauge Fields — Retains deformation and obstruction data beyond the coarse quotient.
- Determinant Lines, Global Obstructions, and Orientations — Tracks fermionic phases and global trivialization conditions.
- Nonperturbative Gauge Measures: Positivity and Configuration-Space Limits — Distinguishes regulated measures from proved continuum limits.
How the ideas fit together
Section titled “How the ideas fit together”A local slice controls nearby representatives after stabilizers are handled. It does not exclude another intersection elsewhere on the same orbit. Singer’s obstruction makes the failure of a global gauge choice precise in its stated non-Abelian setting Singer 1978, pp. 7–12. Fermion determinants add line-bundle data: curvature and holonomy are distinct tests Freed 1987, §§ 1 and 3 (PDF). Even successful geometric control leaves the measure and regulator-removal problem open for the particular model under study.
Hypotheses and licensed conclusions
Section titled “Hypotheses and licensed conclusions”| Object and domain | Essential hypotheses | Licensed conclusion | Excluded converse or extension | Adversarial check |
|---|---|---|---|---|
| Connection action groupoid | Fixed principal bundle, Sobolev index above the multiplication threshold, and a declared based or boundary gauge group. | Gauge arrows, stabilizers, and families are retained; gauge-invariant functions descend to the coarse orbit space. | The coarse orbit set does not recover automorphisms, boundary symmetries, or gluing data. | On $U(1)$ connections over $S^1$, replace the groupoid by holonomy values and observe that constant $U(1)$ isotropy disappears. |
| Orbit-type stratum | Proper action or gauge-theoretic slice hypotheses, with stabilizer conjugacy type held fixed. | A neighborhood is modeled by a stabilizer quotient of a slice, and orbit dimension includes the stabilizer dimension. | A free-action manifold formula cannot be continued across reducible points where isotropy jumps. | Apply the free quotient formula at $U=\pm\mathbf1$ under $SU(2)$ conjugation; it predicts a nonzero orbit although the orbit is a point. |
| Bundle sector and allowed gauge group | Bundle classification, faithful global form, matter representations, boundary conditions, and allowed components of the gauge group. | Configuration space is a disjoint union over admitted bundle classes, quotiented within each class by the declared transformations. | A large transformation need not change bundle class, and a Lie algebra does not determine the sector set. | Put $c_1=1$ on $S^2$ and demand one global potential, or quotient by a transformation forbidden at the boundary. |
| Coulomb slice and Faddeev–Popov operator | Closed base or elliptic boundary domain, controlled stabilizer, $k>d/2+1$, and invertibility of $d_A^*d_A$ on the chosen gauge algebra. | A unique small transformation carries nearby fields to a local slice, modulo the stabilizer. | Local invertibility does not give one representative on every orbit or exclude distant copies. | Insert $0\neq\xi\in\ker d_A$ at a reducible connection and watch the inverse-function hypothesis fail. |
| Elliptic deformation complex and global gauge bundle | Field equation making a complex, elliptic symbol, Fredholm domains, and separately the topology of the based gauge group. | Stabilizer, tangent, and obstruction groups are finite-dimensional; nontrivial universal-bundle topology can forbid a global section. | Ellipticity and a Fredholm index do not imply unobstructedness, compactness, or global gauge uniqueness. | Promote the perturbative slice on $S^3$ globally despite $\pi_0\operatorname{Map}_*(S^3,SU(2))\cong\mathbb Z$. |
| Derived moduli and determinant or Pfaffian line | Derived mapping-stack hypotheses or a smooth Fredholm family; for a Pfaffian, the required real structure and dimension; gauge-equivariant gluing. | Automorphisms and obstructions occupy separate tangent degrees, while curvature and holonomy locate local and global fermion anomalies. | Derived structure does not remove obstructions, and cancellation of a local anomaly polynomial does not trivialize line holonomy. | Keep only closed points at a reducible representation, or transport one $SU(2)$ doublet around the nontrivial large-gauge loop. |
| Regulated or continuum gauge measure | At finite cutoff, compact integration domain and positive action; for a continuum result, uniform tightness, observable convergence, Euclidean axioms, and reconstruction. | A finite lattice gives a positive gauge-invariant probability law; a continuum QFT follows only when the additional limit theorems are proved. | Finite-cutoff positivity, formal gauge fixing, or a determinant symbol does not imply a regulator-independent interacting measure. | Present one Wilson lattice law as the continuum theory and demand the missing topology, tightness, renormalization, and reconstruction estimates. |
Review the chapter
Section titled “Review the chapter”Does invertibility of the Faddeev–Popov operator near one connection prove that the gauge condition selects exactly one representative on every orbit?
Solution
No. The operator tests local transversality near that connection. Distant intersections, other orbit strata, and global topology remain uncontrolled. A local slice can be valid even when no global continuous section exists.
Where to go next
Section titled “Where to go next”Continue to BRST–BV quantization and anomalies to resolve gauge symmetry cohomologically while keeping these global limitations visible.