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Singer obstruction to a global continuous gauge representative

Statement

Assume the Axiom of Choice. Let $r \in {3, 4}$ , let $N \geq 2$ , and let $P \to S^{r}$ be any smooth principal right $S U (N)$ -bundle, with the standard round metric and orientation on $S^{r}$ . Here $S^{r} = {x \in R^{r + 1} : ∥ x ∥ = 1}$ , and $S U (N) = {U \in M_{N} (C) : U^{*} U = I, \det U = 1}$ . A principal right bundle means a smooth surjective bundle with free right $S U (N)$ action, locally equivariantly diffeomorphic to $U \times S U (N)$ . Write $s u (N)$ for its Lie algebra, set $ad P = P \times_{Ad} s u (N)$ and $Ad P = P \times_{Ad} S U (N)$ , using $[p g, z] = [p, g z g^{- 1}]$ and ${Ad}_{g} z = g z g^{- 1}$ . A smooth principal connection is a smooth $s u (N)$ -valued one-form $ω$ on $P$ with $ω (ξ_{P}) = ξ$ for each fundamental vertical vector $ξ_{P} (p) = {\frac{d}{d t} ∣}_{t = 0} p \exp (t ξ)$ and $R_{g}^{*} ω = {Ad}_{g^{- 1}} ω$ , where $R_{g} (p) = p g$ . Let $A^{\infty} (P)$ be the affine space of these connections, modeled on $Ω^{1} (S^{r}; ad P)$ , with the $C^{\infty}$ section topology generated by all derivative seminorms in a fixed finite bundle atlas. Let $G^{\infty} (P) = Γ^{\infty} (Ad P)$ be the smooth gauge group with pointwise multiplication and its $C^{\infty}$ section topology. For the Sobolev intermediary choose a smooth reference connection $A_{0}$ (its existence is also exhibited in the proof), fix any integer $k > r / 2 + 1$ , and use $A_{k} (P) = A_{0} + H^{k} (T^{*} S^{r} \otimes ad P)$ and $G_{k + 1} (P) = H^{k + 1} (Ad P)$ , where these Sobolev spaces are completions in the finite-atlas norms from square-integrable derivatives through the indicated order.

Fix $m \in S^{r}$ and a frame $p_{m} \in P_{m}$ . Parallel transport around piecewise smooth loops based at $m$ , expressed in this frame, gives the holonomy subgroup ${Hol}_{p_{m}} (A) \subseteq S U (N)$ . A connection is irreducible when this subgroup leaves no nonzero proper complex subspace of $C^{N}$ invariant; this condition is unchanged by conjugating the chosen frame. Write $A^{\infty, *} (P) \subseteq A^{\infty} (P)$ for this locus. In local sections the right gauge action is $A^{u} = u^{- 1} A u + u^{- 1} d u$ .

Let $Z_{N} = {ζ I_{N} : ζ^{N} = 1}$ be the central subgroup of constant gauge sections and let ${\overline{G}}^{\infty} (P) = G^{\infty} (P) / Z_{N}$ with its quotient topology. Give every orbit space below its quotient topology, and define

q_{*} : A^{\infty, *} (P) ⟶ A^{\infty, *} (P) / {\overline{G}}^{\infty} (P), q : A^{\infty} (P) ⟶ A^{\infty} (P) / G^{\infty} (P)

to be the reduced and full orbit projections. A continuous section of either projection is a continuous right inverse. The natural square

commutes, where $j$ is inclusion and $ι$ sends a reduced orbit to its full orbit. The map $ι$ identifies its domain with the open subset $q (A^{\infty, *} (P))$ of the full orbit space.

Neither $q_{*}$ nor $q$ has a continuous section. Both conclusions hold for every bundle sector, including nontrivial bundles; irreducibility here does not mean that the holonomy equals all of $S U (N)$ .

Statement references

Singer, “Some Remarks on the Gribov Ambiguity” (historical attribution; the proof does not rely on this source)

Configuration-space, gauge-action, and topology conventions

Existence of the reference connection for every bundle sector

Facts & Assumptions

Given: The sphere, bundle, round metric, based frame, smooth section topologies, and orbit projections of the Statement.

[A1]

Assume ZF together with the full Axiom of Choice. The Axiom of Choice

[F1]

Every compact smooth principal $S U (N)$ bundle has a smooth principal connection, constructed locally from equivariant trivializations and a subordinate partition of unity. Existence of a smooth principal connection on a compact SU(N) bundle

[F2]

The configuration definition sets $A_{\infty} = A_{0} + Ω^{1} (S^{r}; ad P)$ with the $C^{\infty}$ section topology from finite-atlas derivative seminorms, defines $G_{\infty} = Γ^{\infty} (Ad P)$ with the same section topology, and fixes the local right action $A^{u} = u^{- 1} A u + u^{- 1} d u$ , the based holonomy irreducibility condition, and the quotient topologies. Smooth and Sobolev configuration and reduced gauge spaces

[F3]

For the stated Sobolev index, the smooth irreducible projection $A_{\infty}^{*} \to A_{\infty}^{*} / {\overline{G}}_{\infty}$ is a locally trivial topological principal ${\overline{G}}_{\infty}$ -bundle with its actual $C^{\infty}$ quotient topology; the Sobolev irreducible locus $A_{k}^{*}$ is open. Sobolev and smooth irreducible gauge-orbit principal bundles

[F4]

For every stated sphere bundle, $A_{\infty}^{*}$ is nonempty and path connected, and $π_{j} (A_{\infty}^{*}, A) = 0$ for every irreducible $A$ and $j \geq 1$ . Weak contractibility of the irreducible connection locus

[F5]

For every $N \geq 2$ and every bundle on $S^{3}$ , the reduced smooth gauge group has infinitely many path components; for every bundle on $S^{4}$ , its fundamental group at the identity is nontrivial. Nontrivial homotopy of reduced sphere gauge groups

[F6]

A section of a nonempty locally trivial principal topological $H$ -bundle makes $H$ a retract of its total space; if the total space is path connected with all positive based homotopy groups trivial, then so is $H$ . A section forces the structure group to retract from a principal bundle

Proof

Given: [A1] and the sphere, bundle, frame, and smooth topology of the Statement.

1.1A1F1F2givenconstruct

Apply [F1] to the specified compact smooth principal bundle $P \to S^{r}$ to obtain a smooth principal connection $A_{0}$ ; fix it as the affine reference used in $A^{\infty} (P)$ and the Sobolev intermediary. This applies to every bundle sector because [F1] uses only local bundle charts and a subordinate partition, not a global trivialization. The based frame and standard metric are already fixed in the Statement.

2.1F2F3F4F6step 1.1

Choose the fixed integer $k > r / 2 + 1$ . The smooth clause of [F3], applied with the configuration data [F2], makes $q_{*}$ a locally trivial principal ${\overline{G}}^{\infty} (P)$ -bundle. By [F4], its total space is nonempty, path connected, and has $π_{j} = 0$ for every $j \geq 1$ , so it is weakly contractible in the sense used in [F6]. For a constant central gauge section $u = ζ I_{N}$ , the local action formula gives $A^{u} = A$ because $u^{- 1} A u = A$ and $d u = 0$ ; hence reduced and full gauge groups have the same orbits on $A^{\infty, *} (P)$ .

3.1F4F5F6step 2.1contradiction

Suppose $q_{*}$ had a continuous section. Apply [F6] to the nonempty principal bundle in step 2.1. Its structure group ${\overline{G}}^{\infty} (P)$ would be path connected and have $π_{j} = 0$ for every $j \geq 1$ . If $r = 3$ , this contradicts the infinitely many components in [F5]. If $r = 4$ , it contradicts the nonzero fundamental group in [F5]. Thus $q_{*}$ has no continuous section.

4.1F2F3step 3.1givenconstruct∎

The locus $A^{\infty, *} (P)$ is gauge invariant: in the chosen frame, the based holonomy subgroup of $A^{u}$ is $u (p_{m})^{- 1} {Hol}_{p_{m}} (A) u (p_{m})$ , so it has the same invariant subspaces up to conjugation. It is open: by [F2], the inclusion $A^{\infty} (P) ↪ A_{k} (P)$ is continuous, as each fixed $H^{k}$ norm is bounded by finitely many smooth derivative seminorms on the finite compact atlas. The smooth irreducible locus is the inverse image of the Sobolev irreducible locus, which is open by [F3]. The full orbit projection $q$ is open because, for any open $U \subseteq A^{\infty} (P)$ , its saturation $q^{- 1} (q (U)) = ⋃_{u \in G^{\infty} (P)} U \cdot u$ is open. For fixed smooth $u$ , the map $A \mapsto u^{- 1} A u + u^{- 1} d u$ is continuous in every finite-atlas derivative seminorm by the Leibniz rule; its inverse is the same formula for $u^{- 1}$ , so each gauge translate is a homeomorphism. Therefore $q (A^{\infty, *} (P))$ is open and $q ∣_{A^{\infty, *} (P)}$ is an open quotient map onto this subset. The reduced and full actions have the same orbits there, so their quotient topologies identify the reduced orbit space homeomorphically with $q (A^{\infty, *} (P))$ ; this proves the claim about $ι$ in the Statement. If $q$ had a continuous section, its restriction to this open subset would take each irreducible orbit to a point in that same orbit, hence into $A^{\infty, *} (P)$ . Via the stated homeomorphism, this would give a continuous section of $q_{*}$ , contradicting step 3.1. Thus $q$ has no continuous section.

Depends on

Dependency tree · 2 levels

Sources

I. M. Singer, Some Remarks on the Gribov Ambiguity