DefinitionDefinition: AI-adaptedProof: Not applicablePipeline-generatedprecheck pendingjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

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Smooth and Sobolev configuration and reduced gauge spaces

Definition

Assume the Axiom of Choice. Fix an integer $N \geq 2$ . Write ${Mat}_{N} (C)$ for the complex $N \times N$ matrices, $U^{*} = {\overline{U}}^{T}$ for conjugate transpose, and $I_{N}$ for the identity matrix. Define the special unitary matrix group

$S U (N) : = {U \in {Mat}_{N} (C) : U^{*} U = I_{N}, \det U = 1},$

with matrix multiplication and the following real embedded smooth structure. Identify ${Mat}_{N} (C)$ with $R^{2 N^{2}}$ by the real and imaginary matrix entries, and let ${Herm}_{N} = {H : H^{*} = H}$ have its induced real vector-space structure. Equip $U (N) = {U : U^{*} U = I_{N}}$ with the embedded regular-level-set structure at $I_{N}$ of the smooth map $F : {GL}_{N} (C) \to {Herm}_{N}$ , $F (U) = U^{*} U$ , where ${GL}_{N} (C) = {U : \det U \neq 0}$ . Equip $S^{1} = {z \in C : ∣ z ∣ = 1}$ with its embedded circle structure and $S U (N)$ with the embedded regular-level-set structure at $1$ of $\det : U (N) \to S^{1}$ . These are the embedded structures in the stated real matrix space. Its real Lie algebra is $s u (N) : = {X \in {Mat}_{N} (C) : X^{*} = - X, tr X = 0}$ with bracket $[X, Y] = X Y - Y X$ . The standard unitary representation is the map $ρ : S U (N) \to U (C^{N}, h)$ defined by $ρ (U) v = U v$ , where $h (v, w) = \sum_{j = 1}^{N} \overline{v_{j}} w_{j}$ and $U (C^{N}, h)$ consists of the complex-linear maps preserving $h$ . Conjugation is ${Ad}_{U} z = U z U^{- 1}$ for either $z \in S U (N)$ or $z \in s u (N)$ .

Let $M$ be a compact, connected, oriented, boundaryless smooth manifold of dimension $d \in {3, 4}$ with a smooth Riemannian metric, and fix a smooth right principal $S U (N)$ -bundle $π : P \to M$ , a smooth connection $A_{0}$ , and a point $p_{m} \in P$ over $x_{m} \in M$ . Use the representation $ρ$ and the invariant positive inner product $⟨ X, Y ⟩ = - tr (X Y)$ on $s u (N)$ . Define the adjoint vector bundle $ad P = P \times_{Ad} s u (N)$ and the adjoint group bundle $Ad P = P \times_{Ad} S U (N)$ with associated-bundle relation $[p g, z] = [p, {Ad}_{g} z]$ .

Choose one finite smooth bundle atlas of local sections $(U_{α}, s_{α})$ whose coordinate images are bounded balls, and a subordinate partition of unity with each cutoff compactly supported in its chart. On an overlap write $s_{β} = s_{α} g_{α β}$ for its smooth transition map $g_{α β} : U_{α} \cap U_{β} \to S U (N)$ . Local representatives of an adjoint section $ξ$ and an adjoint group section $u$ are defined by $ξ (x) = [s_{α} (x), ξ_{α} (x)]$ and $u (x) = [s_{α} (x), u_{α} (x)]$ . They obey the transition rules $ξ_{β} = {Ad}_{g_{α β}^{- 1}} ξ_{α}$ and $u_{β} = g_{α β}^{- 1} u_{α} g_{α β}$ .

For each integer $k > d / 2 + 1$ , define $H^{k} (T^{*} M \otimes ad P)$ and $H^{k + 1} (ad P)$ by completion of smooth sections in the resulting finite-atlas norms, using weak derivatives of their local coefficients. Define $H^{k + 1} (Ad P)$ as the corresponding completion of smooth group-bundle sections, represented locally by matrix-valued $H^{k + 1}$ maps satisfying the stated transition rule. After localization by the partition of unity and extension by zero to the coordinate balls, the higher-order Sobolev embedding gives unique continuous representatives (indeed $C^{1}$ ); the group-valued representatives take values in the closed subgroup $S U (N)$ everywhere. This embedding is the point-evaluation convention used below. If $p \in P$ lies over $x$ and $p = s_{α} (x) h$ , define $u (p) : = h^{- 1} u_{α} (x) h$ . The transition rule makes this independent of the chosen local section, and $u (x) = [p, u (p)]$ . In particular, if $p_{m} = s_{α} (x_{m}) h_{α m}$ , then $u (p_{m}) = h_{α m}^{- 1} u_{α} (x_{m}) h_{α m}$ is the value in the $p_{m}$ -trivialization. The map $\tilde{u} : P \to P$ defined by $\tilde{u} (p) : = p u (p)$ is a $C^{1}$ bundle automorphism over ${id}_{M}$ , with inverse $\tilde{u^{- 1}}$ ; it is smooth when $u$ is smooth. Conversely, if $Φ : P \to P$ is a $C^{1}$ bundle automorphism over ${id}_{M}$ commuting with the right $S U (N)$ action, write $Φ (p) = p g_{p}$ for the unique $g_{p} \in S U (N)$ . Equivariance gives $g_{p h} = h^{- 1} g_{p} h$ , so $u (x) : = [p, g_{p}]$ for $p \in P_{x}$ is a well-defined $C^{1}$ adjoint-group section. These assignments are inverse; in the smooth case they identify smooth sections with the smooth gauge group defined by principal-bundle automorphisms.

Define the affine configuration space $A_{k} : = A_{0} + H^{k} (T^{*} M \otimes ad P)$ and the set $G_{k + 1} : = H^{k + 1} (Ad P)$ of Sobolev gauge sections. Its based subset is $G_{k + 1, m} : = {u \in G_{k + 1} : u (p_{m}) = e}$ , equivalently the sections whose associated bundle automorphism fixes $p_{m}$ . Let $Z_{N} = {z I_{N} : z^{N} = 1}$ be the center of $S U (N)$ , viewed as constant gauge sections. Indeed, if $Z = (z_{i j})$ is central and $D = diag (t_{1}, \dots, t_{N}) \in S U (N)$ , then $(t_{j} - t_{i}) z_{i j} = 0$ . For each $i \neq j$ one can choose such a diagonal matrix with $t_{i} \neq t_{j}$ , so $Z$ is diagonal. Commuting with the determinant-one unitary rotation whose block on the $(i, j)$ coordinates is $(\begin{array}{cc} \cos t & \sin t \\ - \sin t & \cos t \end{array})$ for $t \notin π Z$ forces the $i$ th and $j$ th diagonal entries equal. Hence $Z = z I_{N}$ ; determinant one gives $z^{N} = 1$ , and conversely every such scalar is central. Write ${\overset{ˉ}{G}}_{k + 1}$ for the set of $Z_{N}$ -cosets with quotient topology and multiplication $[u] [v] = [u v]$ , well-defined since $Z_{N}$ is central.

Specify its Hilbert Lie structure as follows, using the Hilbert Lie structure on $G_{k + 1}$ established by the calculus lemma. Let $q : G_{k + 1} \to {\overset{ˉ}{G}}_{k + 1}$ be the coset projection. Since $Z_{N}$ is finite and the group is Hausdorff, choose an identity chart neighborhood $V$ with $V^{- 1} V \cap Z_{N} = {e}$ , by continuity of $(v, w) \mapsto v^{- 1} w$ at $(e, e)$ . The map $q$ is open because $q^{- 1} (q (O)) = ⋃_{z \in Z_{N}} O z$ for every open $O$ . Consequently $q ∣_{u V} : u V \to q (u V)$ is a homeomorphism for each $u$ , and transport the translated Hilbert charts through these maps. On an overlap, two lifts differ locally by multiplication by a fixed element of the discrete group $Z_{N}$ , so chart changes are smooth. Distinct cosets have disjoint open neighborhoods: if $v^{- 1} w \notin Z_{N}$ , continuity and finiteness give neighborhoods $O_{v}, O_{w}$ with $O_{v}^{- 1} O_{w} \cap Z_{N} = \emptyset$ , and their images under $q$ are disjoint. Thus the quotient is Hausdorff. Its countable base descends from that of the Sobolev section space by the open map $q$ . Multiplication and inversion are smooth in these charts because they lift locally to the corresponding smooth operations on $G_{k + 1}$ . This defines a Hilbert Lie group with the stated quotient topology, and $q$ is a local diffeomorphism.

Let $A_{\infty} : = A_{0} + Ω^{1} (M, ad P)$ and let $G_{\infty} : = Γ^{\infty} (Ad P)$ be the group of smooth adjoint-group sections under pointwise multiplication, identified above with the smooth gauge group. Give it the $C^{\infty}$ section topology generated by all finite-atlas derivative seminorms. Define ${\overset{ˉ}{G}}_{\infty} : = G_{\infty} / Z_{N}$ . For a connection form $ω$ , write $A_{α} = s_{α}^{*} ω$ for its local form. The right-principal convention gives $A_{β} = g_{α β}^{- 1} A_{α} g_{α β} + g_{α β}^{- 1} d g_{α β}$ . The right gauge action is locally $A_{α}^{u} : = u_{α}^{- 1} A_{α} u_{α} + u_{α}^{- 1} d u_{α}$ . For smooth $u$ , this is the pullback by $\tilde{u}$ : for $X \in T_{x} M$ , $d (\tilde{u} \circ s_{α})_{x} X = d (R_{u_{α} (x)})_{s_{α} (x)} d s_{α} (X) + (u_{α}^{- 1} d u_{α} (X))_{P} (s_{α} (x) u_{α} (x))$ , and the connection equivariance and vertical reproduction identities give the displayed formula. For Sobolev $u$ , the calculus lemma verifies that the local forms patch under the stated transition rules and define a smooth action on the Sobolev spaces above. For an adjoint section $ξ$ , set $d_{A} ξ : = d ξ + [A, ξ]$ using the degree-one Lie-algebra-valued-form bracket. In local coordinates and an orthonormal adjoint frame, define the formal adjoint on an adjoint-valued one-form $a$ by $(d_{A}^{*} a) = - g^{i j} (\nabla_{i}^{L C} a_{j} + [A_{i}, a_{j}])$ , where $\nabla^{L C}$ differentiates the covector index. This is the $L^{2}$ adjoint for the Riemannian volume and stated fiber metric; its local coefficient formula defines a distribution at Sobolev regularity.

For $A \in A_{k}$ , parallel transport of its $C^{1}$ connection form around piecewise smooth loops based at $p_{m}$ defines a subgroup ${Hol}_{p_{m}} (A) \subseteq S U (N)$ . Call $A$ irreducible when this subgroup leaves no nonzero proper complex subspace of $C^{N}$ invariant; changing the chosen frame conjugates the subgroup and leaves this condition unchanged. Write $A_{k}^{*}$ and $A_{\infty}^{*}$ for the irreducible loci. Once the calculus lemma verifies the group action, define orbit equivalence by $A \sim B$ exactly when $B = A^{u}$ for some gauge section $u$ ; all resulting quotients carry the quotient topology, whose open sets are those with open inverse image under the orbit projection.