The Gribov–Singer Global Gauge-Fixing Obstruction

The Gribov–Singer obstruction is topological: for standard non-Abelian gauge problems, the principal bundle of connections over gauge orbits is nontrivial, so no continuous global section can choose exactly one representative on every orbit. Local Coulomb slices remain valid. What fails is their promotion to one global gauge condition, and the theorem’s hypotheses must not be generalized to every group, base, or boundary problem.

Required background. Local slices and Faddeev–Popov geometry supplies the perturbative slice. Elliptic gauge complexes separates local zero modes from remote copies. Gribov copies and limits of local gauge fixing supplies the physical ambiguity. Helpful background. Gauge fixing, BRST, and Gribov structure explains its effect on functional equations. Lattice gauge fixing and gauge-dependent correlators gives a regulated setting where multiple representatives can be tested.

A global gauge is a section

Let $P\to M$ be fixed and choose a base point $x_0\in M$. The based gauge group

$$\mathcal G_*={u\in\mathcal G(P):u(x_0)=1}$$

acts freely on the affine space $\mathcal A(P)$. Indeed, if $A^u=A$, then $u$ is $A$-parallel; its value at $x_0$ determines it everywhere, so the based condition forces $u=1$. In a Sobolev completion with $k>\dim M/2+1$, the quotient

$$\mathcal G_*\longrightarrow\mathcal A_k \xrightarrow{\pi}\mathcal B_k:=\mathcal A_k/\mathcal G_*$$

is a principal Hilbert-Lie-group bundle. A continuous global gauge choice is exactly a section $s:\mathcal B_k\to\mathcal A_k$ satisfying $\pi\circ s=\operatorname{id}$.

Because $\mathcal A_k$ is affine, it is contractible. The bundle therefore models the universal bundle $E\mathcal G_*\to B\mathcal G_*$. If a section existed, it would trivialize the principal bundle:

$$\mathcal A_k\cong\mathcal B_k\times\mathcal G_*.$$

This is impossible when $\mathcal G_*$ has nontrivial homotopy. Atiyah and Bott identify the classifying-space model for gauge groups in Atiyah and Bott 1983, §2, Proposition 2.4, pp. 541–542. Singer proves the absence of a continuous choice of one connection per orbit for connections over $S^4$ with compact non-Abelian structure group in Singer 1978, §§2–4, pp. 9–12. This is a theorem about the declared bundle and gauge group, not a slogan that every conceivable gauge theory lacks every global coordinate.

The SU(2) connection-space bundle on a three-sphere

The first application is the global assumption tested in Gauge Fixing, BRST, and Gribov Structure. Take the trivial $SU(2)$ bundle on a compact spatial $S^3$. Its based gauge group has the homotopy type of the based mapping space

$$\mathcal G_*\simeq\operatorname{Map}_*(S^3,SU(2)).$$

Since $SU(2)\simeq S^3$,

$$\pi_0(\mathcal G_*) \cong[S^3,SU(2)]_* \cong\pi_3(S^3) \cong\mathbb Z.$$

Thus $\mathcal G_*$ is not contractible, so the universal connection-space bundle cannot have a global section. A Coulomb slice near the trivial connection may be obtained by inverting $-\partial^iD_i$ in a small neighborhood, but it cannot be extended continuously to select one field on every orbit. Somewhere it must miss an orbit, meet an orbit more than once, or cease to be a regular slice.

Gribov’s original analysis exhibits the analytic side of this failure for transverse gauges: the Faddeev–Popov operator develops zero modes at a horizon, and further intersections occur beyond the perturbative region Gribov 1978, §§2–4, pp. 3–13. Singer’s argument is stronger in a different direction: changing from Coulomb gauge to another continuous global prescription cannot trivialize a topologically nontrivial bundle.

What the theorem does and does not say

The obstruction concerns a single continuous choice on the whole orbit space. It does not invalidate local slices, BRST perturbation theory around a background, or gauge-invariant observables defined without a global representative. An atlas of slices with transition gauge transformations is the natural replacement.

Nor does the argument cover every Abelian or boundary problem. For an Abelian connection on a contractible region with gauge transformations fixed at the boundary, a Hodge decomposition can give a global linear gauge representative. Changing boundary conditions changes $\mathcal G_*$ and its topology. Restricting to a smaller open subset of configuration space can also admit a section even though the full space does not.

The independent check for the $S^3$ example is purely homotopical: the degree of a based map $S^3\to S^3$ cannot vary continuously, so the gauge group has disconnected components. No local Faddeev–Popov calculation can remove that integer.

Failure test

Start with the perturbative Coulomb slice at $A=0$ and assert that its unique small representative extends to every orbit. A global extension would be a section of $\mathcal A_k\to\mathcal B_k$ and hence would trivialize the universal bundle, contradicting $\pi_0(\mathcal G_*)\cong\mathbb Z$. This identifies the exact failed premise: the inverse-function theorem supplied a neighborhood, not a global continuation.

The converse boundary is also important. Finding two representatives satisfying one gauge condition proves that condition is not globally unique; it does not by itself establish the full Singer obstruction for a different base, group, or restricted configuration domain. That requires the topology of the corresponding principal bundle.

Exercises

Explain why a global section of a principal $H$-bundle $E\to B$ gives a trivialization, and apply this to $\mathcal A_k\to\mathcal B_k$ for based $SU(2)$ gauge transformations on $S^3$.

Solution

Given a section $s$, the map $B\times H\to E$, $(b,h)\mapsto s(b)h$, is a principal-bundle isomorphism. Here $E=\mathcal A_k$ is contractible, whereas $H=\mathcal G_*$ has components labeled by $\pi_3(SU(2))\cong\mathbb Z$. If $E\cong B\times H$, the inclusion of a fiber and projection onto it make $H$ a retract of a contractible space, forcing all its homotopy groups, including $\pi_0$, to be trivial. Contradiction.

References

• Atiyah, Michael F., and Raoul Bott. “The Yang–Mills Equations over Riemann Surfaces.” Philosophical Transactions of the Royal Society of London A 308 (1983): 523–615. DOI; Open PDF.
• Gribov, Vladimir N. “Quantization of Non-Abelian Gauge Theories.” Nuclear Physics B 139 (1978): 1–19. DOI.
• Singer, Isadore M. “Some Remarks on the Gribov Ambiguity.” Communications in Mathematical Physics 60 (1978): 7–12. DOI; Open PDF.