BANACH CENTER PUBLICATIONS, VOLUME 39 INSTITUTE OF MATHEMATICS POLISH ACADEMY OF SCIENCES
WARSZAWA 1997
THE CONFIGURATION SPACE OF GAUGE THEORY ON OPEN MANIFOLDS OF BOUNDED GEOMETRY
J ¨ U R G E N E I C H H O R N
Fachbereich Mathematik, Universit¨ at Greifswald Jahnstraße 15a, 17487 Greifswald, Germany E-mail: eichhorn@math-inf.uni-greifswald.d400.de
G E R D H E B E R GMD–FIRST
Rudower Chaussee 5, Geb. 13.10, 12489 Berlin, Germany E-mail: heber@first.gmd.de
Abstract. We define suitable Sobolev topologies on the space C
P(B
k, f ) of connections of bounded geometry and finite Yang–Mills action and the gauge group and show that the corresponding configuration space is a stratified space. The underlying open manifold is assumed to have bounded geometry.
1. Introduction. Consider a differential equation Au = 0 (D). If S is the set of all possible solutions and G the automorphism group of (D) then the configuration space S/G plays a decisive role for the solution theory of (D). At the first instance, S/G is a senseless object until there are introduced suitable topologies and performed completions.
In this paper, we present a canonical approach for gauge theory on open manifolds, i.e.
for the equation δ
ωR
ω= 0. For compact manifolds, there exists a nice and complete representation given by Kondracki et al. in [14], [15]. Unfortunately all of the arguments become completely wrong on open manifolds, e.g. the properness of the action of the completed diffeomorphism group on the space of Riemannian metrics, mapping properties of elliptic operators and many other features of elliptic theory. In this paper, we restrict ourselves to the case of bounded geometry and give a complete topological description of the configuration space of gauge theory.
1991 Mathematics Subject Classification: Primary 81T13; Secondary 58A35,58E15.
Research of the second author supported by the Real World Computing Partnership, Japan.
For the second author this was a preliminary study for the application of the PROMOTER programming model in lattice gauge theory.
The paper is in final form and no version of it will be published elsewhere.
[269]
Let (M
n, g) be an open Riemannian manifold satisfying the following two conditions (I) and (B
k),
(I) r
inj(M ) = inf
x∈M
r
inj(x) > 0, (B
k) |(∇
g)
iR
g| ≤ C
i, 0 ≤ i ≤ k, k > n/2 + 1.
Here r
injdenotes the injectivity radius. r
inj> 0 implies the completeness of the metric.
See [6] for a proof.
We motivate (I) and (B
k) as follows. Compact Riemannian manifolds always satisfy (I) and (B
k) for all k. Hence (I) and (B
k) mean that the geometry at infinity is bounded in a certain sense which seems physically reasonable. A second reason for assuming (I) and (B
k) is more technical. (I), (B
k), k ≥ r >
n2+ 1 assure several important and in the sequel needed properties of Sobolev spaces as the embedding theorem 2.2 and the module structure theorem 2.3. Our general idea constructing completed nonlinear spaces is starting with bounded objects (maps, connections) and completing by adding Sobolev perturbations.
Let P (M, G) → M be a G-principal fibre bundle, G compact with Lie algebra g, g
P= P ×
adG, C
P(B
k, fin) the set of G-connections satisfying
(B
k(ω)) |(∇
ω)
iR
ω| ≤ D
i, 0 ≤ i ≤ k, and
YM(ω) := 1 2
Z
M
|R
ω|
2xdvol
x(g) < ∞.
and let G = n
ϕ : P −→P
'ϕ covers id
M, ϕ(p · g) = ϕ(p) · g,
k−1
X
i=0
sup
p∈P
|∇
idϕ|
p< ∞ o , where P is endowed in a canonical manner with a metric of bounded geometry. We introduce in C
P(B
k, fin), G Sobolev topologies, obtain C
Pr(B
k, fin), k ≥ r > n/2 + 1, where C
Pr(B
k, fin) splits into a topological sum of its components,
C
Pr(B
k, fin) = X
i∈I
comp(ω
i) = X
i∈I
ω
i+ Ω
1,r(g
P, ∇
ω).
G
r+1= G
r+1(comp(ω
i)) is adapted to comp(ω
i), is a Hilbert–Lie group and acts smoothly on comp(ω
i).
Our main task is to study
comp(ω
i)/G
r+1(comp(ω
i)).
We show that this is a stratified space with in general uncountable number of strata which are labelled by conjugacy classes of isotropy groups. Here we assume additionally a certain spectral condition.
The paper is organized as follows. In Section 2 we recall some facts on the space
C
Pr(B
k, fin) which are contained in [4]. Section 3 is devoted to the detailed definition of the
completed gauge group G
r+1(ω
0). We show in Section 4 that G
r+1(ω
0) acts smoothly on
comp(ω
0), the orbits are submanifolds and the action admits a slice. Section 5, finally, con- tains our main theorem, the stratification of the configuration space comp(ω
0)/G
r+1(ω
0).
A special feature of the theory on open manifolds is the fact that the isotropy group of a connection is discrete in G
r+1(ω
0). We cannot present here all proofs in detail which are contained in [13]. The general procedure has been essentially modelled in [14].
Up to a great part we follow the schedule of [14], but with modified analytic arguments.
To obtain closed image properties, we must add the spectral condition that the essential spectrum of a certain Laplace operator starts away from zero.
2. The space of connections. We assume (M
n, g) open, complete with (I) and (B
k).
As is well known, there is no obstruction against the existence of a metric g with (I) and (B
k), 0 ≤ k ≤ ∞. Let (E, h) be a Riemannian vector bundle with metric connection ∇
E. Additionally to (I) and (B
k) for (M
n, g) we define the condition
(B
k(E, ∇
E)) |(∇
E)
iR
E| ≤ C
i, 0 ≤ i ≤ k.
The Levi–Civita connection ∇
gand ∇
Einduce metric connections ∇ in all tensor bundles with values in E, T
rq(M ) ⊗ E. Denote by Ω
q(E) the space of E-valued q-forms, Ω
q(E) = C
∞( V
qT
∗M ⊗ E). Now we define several Sobolev spaces and restrict ourselves to the case of E-valued q-forms. The case of E-valued tensor is quite analogous and can be considered as Ω
0(T
qr⊗ E).
Let for 1 ≤ p < ∞, r ≥ 0 Ω
q,pr(E) :=
σ ∈ Ω
q(E)
|σ|
p,r:= Z
M r
X
i=0
|∇
iσ|
pxdvol
x(g)
1/p< ∞
, Ω
q,p,r(E) = completion of Ω
q,pr(E) with respect to | |
p,r,
Ω b
q,p,r(E) = completion of C
0∞(
r
^ T
∗M ⊗ E) with respect to | |
p,rand Ω
q,p,r(E) = {σ
σ measurable distributional q-form such that |σ|
p,r< ∞}.
Moreover define for m ≥ 0
b
m
Ω
q(E) = n
σ ∈ Ω
q(E)
b,m
|σ| :=
m
X
i=0
|∇
iσ|
x< ∞ o ,
b,m
Ω
q(E) = completion of
bmΩ
q(E) with respect to
b,m| | and
b,m
Ω b
q(E) = completion of C
0∞(
q
^ T
∗M ⊗ E) with respect to
b,m| |.
Then
b,m
Ω
q(E) = {σ
σ is a C
m-form with
b,m|σ| < ∞}.
There are natural inclusions
b,m
Ω b
q(E) (
bmΩ
q(E) Ω b
q,p,r(E) ⊆ Ω
q,p,r(E) ⊆ Ω
q,p,r(E).
Proposition 2.1. Assume (M
n, g) open, complete with (I) and (B
k). Then
Ω b
q,p,r(E) = Ω
q,p,r(E) = Ω
q,p,r(E).
Theorem 2.2. Assume (M
n, g) open, complete with (I) and (B
0), r > n/p + m. Then there exists a continuous embedding
Ω
q,p,r(E) ,→
b,mΩ
q(E).
b,m
|σ| ≤ D · |σ|
p,r.
Theorem 2.3. Assume (M
n, g), (E
i, h
i, ∇
Ei) → M , i = 1, 2, with (I) and (B
k), k ≥ r
1, r
2≥ r ≥ 0. If r = 0 assume
r − n
p < r
1− n p
1r − n
p < r
2− n p
2r − n
p ≤ r
1− n p
1+ r
2− n p
21 p ≤ 1
p
1+ 1 p
2
or
r − n
p ≤ r
1− n p
10 < r
2− n p
21
p ≤ 1 p
1
or
0 < r
1− n p
1r − n
p ≤ r
2− n p
21
p ≤ 1 p
2
.
If r > 0 assume
1p≤
p11
+
p12
and
r − n
p < r
1− n p
1r − n
p < r
2− n p
2r − n
p ≤ r
1− n p
1+ r
2− n p
2
or
r − n
p ≤ r
1− n p
1r − n
p ≤ r
2− n p
2r − n
p < r
1− n p
1+ r
2− n p
2
.
Then there exists a continuous embedding
Ω
p1,r1(E
1) ⊗ Ω
p2,r2(E
2) ,→ Ω
p,r(E
1⊗ E
2), where we write Ω
p,r≡ Ω
0,p,r.
The proofs of Proposition 2.1 and Theorem 2.2 are contained in [3], of Theorem 2.3 in [10].
Now we are able to define for C
P(B
k) a uniform structure which allows us to complete this space. Let δ > 0, 1 < p < ∞, k ≥ r > n/p + 1, for η ∈ Ω
1(g
P)
|η|
ω,p,r:= Z
M r
X
i=0
|(∇
ω)
iη|
pxdvol
x(g)
1/pand set
V
δ:= {(ω, ω
1) ∈ C
P(B
k)
2|ω − ω
1|
ω,p,r< δ}.
Theorem 2.4. B = {V
δ}
δ>0is a basis for a metrizable uniform structure U
p,r(C
P(B
k)) on C
P(B
k).
Denote by C
Pp,r(B
k) the completion with respect to this uniform structure and by
comp(ω) the component of ω in C
p,rP(B
k).
Theorem 2.5.
a. comp(ω) = ω + Ω
1,p,r(g
P, ∇
ω).
b. C
P(B
k) has a representation as a topological sum, C
P(B
k) = X
i∈I
comp(ω
i) = X
i∈I
ω
i+ Ω
1,p,r(g
P, ∇
ωi) .
(In the sequel we restrict ourselves to the case p = 2.)
Denote by C
P(B
k, fin) the set of all connections C
P(B
k) such that YM(ω) = 1
2 Z
M
|R
ω|
2dvol
x(g) < ∞.
Then i : C
P(B
k, fin) ,→ C
P(B
k) induces a uniform structure i
−1(U
p,r(C
P(B
k))) and defines a completion C
P2,r(B
k, fin).
Theorem 2.6. C
P2,r(B
k, fin) has a representation as a topological sum, C
2,rP(B
k, fin) = X
j∈J
comp(ω
j) = X
j∈J
ω
j+ Ω
1,2,r(g
P, ∇
ωj) .
The proofs of Theorems 2.4–2.6 are contained in [4] and use essentially Proposition 2.1 and Theorems 2.2 and 2.3.
R e m a r k. The proof that comp(ω) = ω +Ω
1,2,r(g
P, ∇
ω) essentially uses the fact that ω
0∈ comp(ω) implies
Ω
1,2,r(g
P, ∇
ω) = Ω
1,2,r(g
P, ∇
ω0)
as equivalent Sobolev spaces. This proof is nontrivial. Another difficulty arises from the fact that the elements of C
P2,r(B
k, fin) are only C
1but smooth elements are dense. At the first instance, for ω non-smooth (∇
ω)
idoes not make sense for i > 2. But if ω
0is a smooth connection in an ε-neighbourhood of ω then we define
∇
ω:= ∇
ω0+ ∇
ω− ∇
ω0.
It is easy to see that (∇
ω0+ (∇
ω− ∇
ω0))
imakes sense for any i.
In the sequel, we consider one fixed component comp(ω
0) ⊂ C
P2,r(B
k, fin) and write C
r(ω
0) ≡ comp(ω
0).
3. The gauge group. There are three equivalent definitions of the gauge group, G = {ϕ : P −→P
'ϕ covers id
Mand ϕ(p · g) = ϕ(p) · g}, G = { b ϕ : P −→ G b
ϕ(p · g) = g b
−1ϕ(p)g}, b G = C e
∞(P ×
GG),
where G acts on itself by means of the inner automorphism Ad.
G, b G and e G are isomorphic: ϕ ∈ G given defines ϕ by ϕ(p) = p · b ϕ(p) and b ϕ(x) = e [(p, ϕ(p)] ∈ e b G, where p ∈ π
−1(x). Any Sobolev construction for one of the three groups in- duces by extending the isomorphism above such a construction for the other three groups.
We concentrate here to the group G
r+1(ω
0) = b G
r+1(ω
0) adapted to C
r(ω
0) and proceed
as follows. First we construct G
r+1(ω
0) as topological group. Then we endow a neigh- bourhood of the unit element with the structure of a local Lie group, using Campbell–
Hausdorff series. According to a standard theorem, G
r+1(ω
0) then is a Hilbert–Lie group.
To do this, we have to estimate the norm of the commutator by the product of the norm of the factors. This is the key step of our construction. Unfortunately, in general G
r+1(ω
0) does not satisfy the second axiom of countability. Therefore we have to consider later on a smaller group G
•r+1(ω
0). We have to define the elements of the completed gauge group G
r+1(ω
0) and to introduce a natural, suitable Sobolev topology. First we motivate our definition and start with the non-completed group. G acts on C
Pfrom the left as follows:
ϕω := ω ◦ d(ϕ
−1).
In terms of ϕ ∈ b b G this means
ϕω = Ad( ϕ)ω + ( b ϕ b
−1)
∗θ,
where θ denotes the Maurer–Cartan form of G. This implies ϕω − ω = (ϕω − ω) ◦ (proj
ωv+ proj
ωh) = (ϕω − ω) ◦ proj
ωhsince ϕω − ω is horizontal. Hence
ϕω − ω = (ϕω − ω) ◦ proj
ωh= ( ϕ b
−1)
∗θ ◦ proj
ωh. R e m a r k. It is easy to see that ( ϕ b
−1)
∗θ ◦ proj
ωh= −dR
−1ϕ
b ◦ ϕ. b We define
∇
ωϕ := ( b ϕ b
−1)
∗θ ◦ proj
ωh= −dR
−1ϕ
b ◦ ϕ. b Then
ϕω = ω + ∇
ωϕ. b
Similarly as the groups G, b G, e G, we denote at the Lie algebra level by b X : P → g vector fields with
X b
pg= Ad(g
−1)X
pand by e X sections of g
P. b X and e X correspond to each other.
Let ω
0∈ C(B
k, fin) smooth. Then (P, g
ω0) is a manifold satisfying (I) and (B
k), where g
ω0is the Kaluza–Klein metric,
g
ω0(X, Y ) := g
M(π
∗X, π
∗Y ) + hω
0(X), ω
0(Y )i
Killing. This has been proven in [4]. We say ϕ : P → G is bounded up to order r if b
b,r−1
|d ϕ| := b
r−1
X
i=0
sup
p∈P
|∇
id ϕ| b
x< ∞.
Here d ϕ is considered as a section of T b
∗P ⊗ ϕ b
∗T G, endowed with the induced connection.
Then the manifold Ω
r(P, G) of maps is well defined (cf. [5], [11]). f ∈ Ω
r(P, G) if and only if for any ε > 0 there exists f
0∈ Ω
∞(P, G) with
b,r−1|df
0| < ∞ and X ∈ Ω
r(f
0∗T G, f
0∗∇
G) with |X|
r< ε such that
f = exp X = exp
f0X ◦ f
0.
If k ≥ r > n/2 + 1 then f ∈ C
1, according to the Sobolev embedding theorem.
In the sequel, we assume (M
n, g) open complete with (I) and (B
k), k ≥ r > n/2 + 1
and consider C
r(ω
0) ≡ comp(ω
0) ⊂ C
2,r(B
k, fin).
Proposition 3.1. Assume ∇
ω0ϕ ∈ Ω b
1,2,r(g
P, ∇
ω0), ω ∈ C
r(ω
0). Then ϕω ∈ C
r(ω
0).
P r o o f. ω = ω
0+ η, η ∈ Ω
1,2,r(g
P, ∇
ω0)
ϕω = ϕ(ω
0+ η) = ω
0+ η + ∇
ω0+ηϕ. b On the other hand,
ϕω = ϕω
0+ Ad( ϕ)η = ω b
0+ ∇
ω0ϕ + Ad( b ϕ), b hence
∇
ω0+ηϕ = ∇ b
ω0ϕ + (Ad( b ϕ) − I)η. b We are done if we could show
|Ad( ϕ)η| b
ω0,r< ∞.
Since ∇
ω0ϕ ∈ Ω b
1,2,r(g
P, ∇
ω0) and ϕω
0∈ C
r(ω
0) are equivalent, ω
0and ϕω
0generate equivalent Sobolev norms. Hence it suffices to show
|Ad( ϕ)η| b
ϕω0,r< ∞.
But an easy straightforward calculation shows
|Ad( ϕ)η| b
ϕω0,r= |η|
ω0,r< ∞.
This motivates the following definition. Let G
r+1(ω
0) be the set of all ϕ ∈ Ω b
r+1(P, G) such that
(i) ϕ(pg) = g b
−1ϕ(p)g b (ii) ∇
ω0ϕ ∈ Ω b
1,2,r(g
P, ∇
ω0).
Proposition 3.2. G
r+1(ω
0) forms a group under pointwise multiplication.
P r o o f. First we have to show that Ω
r+1(P, G) forms a group under pointwise mul- tiplication, i.e. given any ε > 0 and f, g ∈ Ω
r+1(P, G) there exists h
0∈ C
∞(P, G),
b,r
|dh
0| < ∞, Z ∈ Ω
r+1(h
∗0T G), |Z|
r+1< ε such that f · g = exp Z ≡
h0Z ◦ h
0. This is highly nontrivial and follows from [5], [11]. Next we show that G
r+1(ω
0) is a subgroup.
Condition (i) is trivial. For (ii) we have to show:
a. ϕ, b b ψ ∈ G
r+1(ω
0) implies |∇
ω0( b ψ ϕ)| b
ω0,r< ∞.
b. ϕ ∈ G b
r+1(ω
0) implies |∇
ω0( ϕ b
−1)|
ω0,r< ∞.
The first assertion follows from
(3.1) ∇
ω0( b ψ ϕ) = Ad( b b ψ)∇
ω0ϕ + ∇ b
ω0ψ b and the second from
(3.2) ∇
ω0( ϕ b
−1) = −Ad( ϕ b
−1)∇
ω0ϕ. b Here (3.2) follows from (3.1) by setting b ψ = ϕ b
−1.
The next step is the introduction of a suitable Sobolev topology. We have to establish a filter basis B = B(e) for e ∈ G
r+1(ω
0) satisfying the following conditions:
(i) If U ∈ B then there exists V ∈ B such that V V
−1⊆ U .
(ii) If U ∈ B then there exists V ∈ B such that V
−1⊆ U .
(iii) e ∈ U for each U ∈ B.
(iv) If g ∈ G
r+1(ω
0) and U ∈ B then there exists V ∈ B such that V ⊆ gU g
−1. The proof of the existence of B with (i)–(iv) shall be prepared by a series of proposi- tions.
Observe that Ω
q(g
P) and the space Ω
q(P, g)
G⊂ Ω
q(P, g) of g-valued G-invariant q-forms on P are isomorphic. Let s, t ∈ Ω
0(P, g). Pointwise commutators define [s, t] ∈ Ω
0(P, g). Define for σ ∈ Ω
1(P, g) and X ∈ Ω
0(T P ) ≡ Ω(T P )
[σ ∧ s](X) := [σ(X), s]
[s ∧ σ](X) := [s, σ(X)].
ω ∈ C defines a covariant differential D
ω: Ω
0(g
P) ' Ω(P, g)
G Dω
−→Ω
1(P, g)
Gh' Ω
1(g
P), where ( )
hdenotes horizontal forms.
Lemma 3.3. For s ∈ Ω(P, g)
G' Ω
0(g
P),
(3.3) D
ωs = ds + [ω ∧ s].
Here one writes with a basis {A
i} of g, s = s
iA
i, ds = ds
i⊗ A
i. For a proof see [12].
D
ωsatisfies the following product rule.
Lemma 3.4. For s, t ∈ Ω(P, g)
G,
D
ω[s, t] = [D
ωs ∧ t] + [s ∧ D
ωt].
The proof is an easy calculation with the use of (3.3).
R e m a r k. For higher derivatives, a similar simple formula is not available. It holds only up to permutation in tensor products (cf. [9]). But for norm estimates a weaker product rule is available. Consider the sequence
Ω
0(g
P) −→ Ω
Dω 1(g
P) −→ Ω
∇ω 0((T
∗M )
⊗2⊗ g
P) −→ · · · .
∇ωProposition 3.5. For s, t ∈ Ω
0(g
P), X
1, . . . , X
u∈ Ω(T M ) ≡ Ω
0(T M ),
∇
uXu···X1[s, t] =
u
X
i=0
X
u≥αi≥···≥α1≥1
∇
iXαi···Xα1
s, ∇
u−iXu···X
b
αi···Xb
α1···X1t
. This can be proven by an easy induction, starting with Lemma 3.4.
Theorem 3.6. Let (M
n, g) be open, complete with (I) and (B
k), ω
0with (B
k), k ≥ r > n/2 + 1. Then there exists a Ξ ∈ R, Ξ > 0, such that for all e X, e Y ∈ Ω
0,2,r(g
P, ∇
ω0) ≡ g
rP(ω
0),
(3.4) |[ e X, e Y ]|
ω0,r≤ Ξ · | e X|
ω0,r· | e Y |
ω0,r. P r o o f. For A, B ∈ g,
|[A, B]| ≤ c · |A| · |B|.
Proposition 3.5 then implies for s, t ∈ Ω
0(g
P)
|∇
u[s, t]|
2x≤ C ·
u
X
i=0
|∇
is|
x· |∇
u−it|
x 2.
The module structure theorem 2.3 finally yields the assertion.
R e m a r k. (3.4) is the key inequality for establishing a Campbell–Hausdorff series in our case and for endowing G
r+1(ω
0) with the structure of a local Lie group. For this, we recall some definitions.
Let G a Lie group with Lie algebra g and exponential exp. Assume g to be endowed with a norm | | such that there exists a Ξ ≥ 1 with
(3.5) |[x, y]| ≤ Ξ · |x| · |y|.
Set
CH(x, y) :=
∞
X
k=1
X
µ,ν∈Nk
(−1)
k−1k
1
µ
1+ ν
1+ · · · + µ
kν
k[x
µ1y
ν1· · · x
µky
νk] µ
1!ν
1! · · · µ
k!ν
k! , where
[x
µ1y
ν1· · · x
µky
νk] := [[x, x], x], · · · x]
| {z }
µ1 times
, y], · · · , y]
| {z }
ν1 times
, · · · , x], · · · , x]
| {z }
µk times
, y], · · · , y]
| {z }
νk times
.
Theorem 3.7. Assume (3.3). Then there exists a constant C(Ξ) such that with U :=
{x ∈ g
|x| < C(Ξ)}, CH(·, ·) becomes an analytic mapping U × U → g and
(3.6) exp(x) exp(y) = exp(CH(x, y)).
This is a classical result.
We do not recall the definition of analytic mapping between Banach spaces which is contained in [1].
Theorem 3.8. Let L be a complete normed Lie algebra satisfying (3.5). Assume α ∈ R, 0 < α ≤ 1/(3Ξ) · log 3/2. Set L
α:= {x ∈ L
|x| < α}, Θ := {(x, y) ∈ L
α× L
αCH(x, y) ∈ L
α} and define m = CH|Θ. Then
(i) Θ is open in L
α× L
αand m analytic;
(ii) x ∈ L
αimplies (0, x), (x, 0) ∈ Θ and m(0, x) = m(x, 0) = 0;
(iii) x ∈ L
αimplies −x ∈ L
α, (x, −x), (−x, x) ∈ Θ and m(x, −x) = m(−x, x) = 0;
(iv) x, y, z ∈ L
α, (x, y), (m(x, y), z), (y, z), (x, m(y, z)) ∈ Θ implies m(m(x, y), z) = m(x, m(y, z));
(v) x, y ∈ L
αimplies |CH(x, y)| ≤ −1/Ξ · log(2 − exp Ξ(|x| + |y|)).
For a proof see [2].
Definition. A couple (Γ, e, Θ, m) is called a local Lie group over R if it satisfies the following conditions:
(i) Γ is an analytic manifold modelled by a real Hilbert space;
(ii) e ∈ Γ;
(iii) θ : Γ → Γ is an analytic mapping;
(iv) m is an analytic mapping from an open subset Ω ⊂ Γ × Γ into Γ;
(v) m(e, g), m(g, e) ∈ Γ and m(e, g) = m(g, e) = g for all g ∈ Γ;
(vi) (g, θ(g)), (θ(g), g) ∈ Ω and m(g, θ(g)) = m(θ(g), g) = e for all g ∈ Γ;
(vii) assume g, h, k ∈ Γ arbitrary, (g, h), (h, k), (m(g, h), k), (g, m(h, k)) ∈ Ω, then m(m(g, h), k) = m(g, m(h, k)).
Corollary 3.9. Assume L
α, m as in Theorem 3.8. Then with i : L
α→ L
α, x 7→ −x, (L
α, 0, i, m) is a local Lie group.
Now we are able to endow G
r+1(ω
0) with a topology such that there arises a topological and even a Hilbert–Lie group.
We define a neighbourhood filter basis of e ∈ G
r+1(ω
0) as follows. Assume 0 < ε
0<
min{D · r
inj(M ), 1/(3Ξ) · log 3/2}. (D comes from the Sobolev embedding theorem 2.2.) Set for ε
(3.7) U
ε:= {exp b X
| e X|
ω0,r+1< ε}.
Theorem 3.10. B = {U
ε}
0<ε<ε0is a filter basis for the neighbourhood filter of e such that G
r+1(ω
0) becomes a topological group.
P r o o f. First we have to show exp b X ∈ G
r+1(ω
0), e X ∈ g
Pr+1(ω
0). This follows imme- diately from
∇
ω0exp b X =
∞
X
m=0
(−1)
m+1ad( e X)
m(m + 1)! (∇
ω0X). e
Moreover, the map e X 7→ ∇
ω0exp b X is even analytic. We have to establish properties (i)–(iv) for B (which were formulated after the proof of Proposition 3.2).
(i) Choose ε
0< min{ε
0, 1/(2Ξ) · log 2 − exp(−1/(2Ξ) · ε)}. Using (v) of Theorem 3.8, we see immediately U
ε0U
ε0⊆ U
ε.
(ii) For 0 < ε < ε
0, U
ε−1= U
ε. (iii) e = exp 0.
(iv) Here we have to show: Given any ϕ ∈ G
r+1(ω
0), 0 < ε < ε
0, there exists 0 < ε
0< ε
0such that U
ε0⊆ ϕU b
εϕ b
−1. For this we consider the map G
r+1(ω
0) → L(g
r+1P(ω
0), g
r+1P(ω
0)), ϕ 7→ Ad( b ϕ b
−1). This map is well defined. Write ϕ = exp b
ϕˆ0Z · b ϕ b
0. Then Ad( ϕ b
−10) acts as bounded operator on g
r+1P(ω
0) according to the boundedness con- dition of ϕ b
0. The same holds for Ad((exp
ϕˆ0Z) b
−1) according to the module structure theorem. This is a highly nontrivial fact and we refer to [5]. Denote by kAd( ϕ b
−1)k
ω0,r+1the corresponding operator norm. Choose
ε
0< min n ε
kAd( ϕ b
−1)k
ω0,r+1, ε
0o . Let exp b X ∈ U
ε0and choose e Y := Ad( ϕ b
−1) e X. Then
exp b X = exp Ad( ϕ)Ad( b ϕ b
−1) b X = exp Ad( ϕ) b b Y = ϕ exp b b Y ϕ b
−1∈ ϕU b
εϕ b
−1if we can show exp b Y ∈ U
ε, i.e. | e Y |
ω0,r+1< ε. But
| e Y |
ω0,r+1= |Ad( ϕ b
−1) e X|
ω0,r+1≤ kAd( ϕ b
−1)k
ω0,r+1| e X|
ω0,r+1≤ kAd( ϕ b
−1)k
ω0,r+1· ε
kAd( ϕ b
−1)k
ω0,r+1= ε.
We use the following well known
Theorem 3.11. Let G be a topological group and V ⊂ G an open neighbourhood of the unit element. Assume that V is an analytic manifold and has the structure of a real local Lie group. Then G is a Hilbert–Lie group.
Now we apply this to our situation. Let U
εbe as in (3.7). U
εis homeomorphic to an open neighbourhood from the Hilbert space g
Pr+1(ω
0). Then L
ε:= exp
−1(U
ε) is a real local Lie group: Set for exp b X, exp b Y ∈ U
εm(exp b X, exp b Y ) := exp CH(exp b X, exp b Y ) = exp b X exp b Y θ(exp b X) := exp(− b X) = (exp b X)
−1e := 0.
Hence we obtain
Theorem 3.12. G
r+1(ω
0) is a Hilbert–Lie group.
R e m a r k. G
r+1(ω
0) consists of C
1elements ϕ. Sometimes we write ϕ ∈ G b
r+1(ω
0) which means the corresponding element ϕ : P −→ P .
'4. The action of the gauge group. Recall the action of G
r+1(ω
0) on C
r(ω
0) : ϕω = Ad( ϕ)ω + ( b ϕ b
−1)
∗θ = ω + ∇
ωϕ. b
Theorem 4.1. Assume (M
n, g) open with (I), (B
k), k ≥ r + 1 > n/2 + 2. Then G
r+1(ω
0) acts smoothly on C
r(ω
0).
P r o o f. Let ϕ ∈ G
r+1(ω
0), ω = ω
0+ η ∈ C
r(ω
0), η ∈ Ω
1,2,r(g
P, ∇
ω0) and b X ∈ g
r+1P(ω
0). Then
( ϕ exp b b X)(ω
0+ η) = ω
0+ ∇
ω0( ϕ exp b b X) + Ad( ϕ)Ad(exp b b X)η.
The assertion follows from the smoothness of the map
( b X, η) 7→ ∇
ω0( ϕ exp b b X) + Ad( ϕ)Ad(exp b b X)η and the latter follows from the smoothness of the maps
( b X, η) 7→ ∇
ω0( ϕ exp b b X) ( b X, η) 7→ Ad( ϕ)Ad(exp b b X)η.
If the underlying manifold M
nis compact then the action is proper. This is proba- bly wrong in the open case. Nevertheless, one very important property of the action is preserved.
Theorem 4.2. Assume (M
n, g) open with (I) and (B
k), k ≥ r > n/2 + 2. Then G
r+1(ω
0) acts closed on C
r(ω
0).
P r o o f. We have to show: ϕ
iω → ω
1implies the existence of a ψ ∈ G
r+1(ω
0) such that ω
1= ψω. We endow P with the corresponding Kaluza-Klein metrics. Let U
x0be a normal chart centred at x
0∈ M of radius
0 < ε < min{r
inj(g), r
inj(g
ω), r
inj(g
ω1)}
and for p
0∈ π
−1(x
0)
{h
ω1, . . . , h
ωn, A
]1(p
0), . . . , A
]m(p
0)}
be an orthonormal basis. We choose a subsequence {ϕ
1i}
isuch that (i) ϕ
−11i(p
0) → q
0∈ π
−1(x
0), q
0= p
0g
0, g
0∈ G;
(ii) dϕ
−11i(h
ων) → χ
ν∈ T
q0P .
Then dϕ
−1i(A
]s(p
0)) → A
]s(q
0). Denote by e(ω) the exponential map of g
ω, B
pε0
:= {t ∈ T
p0P
g
ω(t, t) < ε
2}.
We want to define ψ on e(ω)B
pε0
. Let q = e(ω)t, t =
n
X
ν=1
a
νh
ων+
m
X
s=1
b
sA
]s(p
0) and set
ψ
−1(q) ≡ ψ
−1(e(ω)t) := e(ω
1)
nX
ν=1
a
νχ
ν+
m
X
s=1
b
sA
]s(p
0)
Proposition 4.3. If t, t
1∈ B
εp0
, e(ω)t = [e(ω)t
1] · g, then ψ
−1(e(ω)t) = ψ
−1(e(ω)t
1) · g,
i.e. ψ is a gauge transformation on e(ω)B
pε0. Equivariant continuation provides ψ over U
x0.
P r o o f. The proof follows from the following facts:
(i) g
ϕω= ϕ
∗g
ω(ii) ϕ ◦ e(ω) = e(ϕω) ◦ dϕ
(iii) e(ϕ
1iω) → e(ω
1) uniform on compact subsets of T
pP
ε= {t ∈ T
pP
g
ω1(t, t) ≤ ε}.
(iv) π[e(ω)t] = π[ψ
−1e(ω)t].
(i), (ii), (iv) are easy calculations. For (iii) one uses g
ϕiω→ g
ω1in M
rG(ω
0) = certain Sobolev space of metrics (cf. [6]) if ϕ
iω → ω
1in C
r(ω
0) and the boundedness of Christoffel symbols up to a certain order (cf. [7]).
Corollary 4.4. ϕ
−11i→ ψ
−1over U
x0in C
1. Let η = ω − ω
1.
Corollary 4.5. ∇
ωψ = η over U b
x0.
Theorem 4.2 follows by a compact exhaustion K
1⊆ K
2⊆ · · · of M, finite cover of each K
iby normal charts, the construction above and compatibility.
The next step is to show that the orbits are submanifolds of C
r(ω
0). We recall the key Theorem 4.6. Let G be a Lie group, X an analytic manifold , (g, x) 7→ gx an an- alytic left action. Assume that for x ∈ X the orbit map ρ(x) : G → X, g 7→ gx is a subimmersion. Denote by G
xthe isotropy group of x.
(i) G
xis a Lie subgroup and T
e(G
x) = ker(T
eρ(x)).
(ii) The canonical map i
x: G/G
x→ X, [g] 7→ gx is an immersion with image G · x.
(iii) If the orbit G · x is locally closed and G satisfies second countability then G · x
is a submanifold of X and i
xis an isomorphism from G/G
xonto G · x and T
x(G · x) =
im(T
eρ(x)).
R e m a r k. The smooth version of Theorem 4.6 is also valid.
Denote for ω ∈ C
r(ω
0) by S(ω) the symmetry group, S(ω) = {ϕ ∈ G
r+1(ω
0)
ϕω = ω}.
Proposition 4.7. Let ϕ ∈ S(ω). The following conditions are equivalent : (i) ϕ exp b b X ∈ S(ω)
(ii) exp b X ∈ S(ω) (iii) ∇
ωX = 0. e
We omit the easy computational proof.
Corollary 4.8. S(ω) is discrete in G
r+1(ω
0).
P r o o f. (iii) implies ( e X, e X)
x= const. (M
n, g) with (I) and (B
k) imply vol(M
n, g) =
∞, hence e X 6∈ L
2and ϕ is isolated.
Corollary 4.9. G
r+1(ω
0)/S(ω) is an analytic manifold and G
r+1(ω
0) −→ G
π r+1(ω
0)/S(ω) a submersion.
Lemma 4.10. The map
i
ω: G
r+1(ω
0)/S(ω) → C
r(ω
0), [ϕ] 7→ ϕω is smooth.
P r o o f. It is sufficient to show that i
ω◦ π is smooth. But this is nothing else than the restriction of the smooth action map
Φ : G
r+1(ω
0) × C
r(ω
0) −→ C
r(ω
0) to the submanifold G
r+1(ω
0) × {ω}.
Denote by ∆
ω0= (∇
ω)
∗∇
ωthe Laplace operator on 0-forms with values in g
P, by σ
e(∆
ω0) its essential spectrum. This an invariant of C
r(ω
0), i.e. σ
e(∆
ω0) = σ
e(∆
ω00). If (Sp) inf σ
e∆
ω00|(ker(∆
ω00)
⊥) > 0
then im ∆
ω00, im ∆
ω0, im ∇
ω0and im ∇
ωare closed.
Proposition 4.11. Assume k − 1 ≥ r > n/2 + 2, ω ∈ C
r(ω
0) and (Sp). Then i
ωis an injective immersion.
P r o o f. Let o be the class of e ∈ G
r+1(ω
0) in G
r+1(ω
0)/S(ω). We have to show (i) ker T
o= {0};
(ii) im T
oi
ω⊂ T
ωC
r(ω
0) is closed and admits complement.
(i) is very simple. If we consider (ii), an easy calculation shows im T
oi
ω= im T
eΦ
ωand
T
eΦ
ω= −∇
ω. (Sp) implies im ∇
ωclosed. The existence of an L
2-complement is assured
by
Proposition 4.12. Assume k − 1 ≥ r > n/2 + 2 and the spectral condition (Sp) and ω ∈ C
r(ω
0). Then
(4.1) Ω
1,2,r(g
P, ∇
ω0) = im ∇
ωr+1⊕ ker(∇
ω)
∗r, where both spaces on the right-hand side are closed and L
2-orthogonal.
The proof is rather long and nontrivial since ω is nonsmooth and one has to establish Hodge theory for elliptic differential operators with Sobolev coefficients. We refer to [8], [13].
To apply Theorem 4.6 (iii), we need second countability of G
r+1(ω
0) which is in general not satisfied since G
r+1(ω
0) can have uncountably many components. It is easy to see that each element of the centre of G generates one component. Therefore we restrict in the sequel ourselves to the subgroup G
•r+1(ω
0) which consists of the components of G
r+1(ω
0) generated by the centre of G if this centre is countable or which equals to the component of the identity if the centre is uncountable. In the case G = SU (2) the centre consists of two elements, i.e. of two components, each of them satisfies second countability.
The key role in the whole structure of the configuration space is played by the slice theorem which we now start to discuss.
We want to construct an equivariant tubular neighbourhood for each orbit. For this we need a Riemannian metric, a normal bundle and an exponential map. The tangent bundle of C
r(ω
0) is simply C
r(ω
0) × Ω
1,2,r(g
P, ∇
ω0). We define a weak Riemannian metric
ω 7→ ( , )
w,ω= ( , )
was follows: (ω, ξ), (ω, η) ∈ {ω} × Ω
1,2,r(g
P, ∇
ω0),
((ω, ξ), (ω, η))
w:= (ξ, η)
w:=
Z
M
(ξ, η)
xdvol(x).
( , )
wis G
r+1(ω
0)-invariant.
A strong metric ( , )
ston C
r(ω
0) is given by ω 7→
r
X
i=0
(∇
ω)
i·, (∇
ω)
i·
w
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