8.4 Vector-Valued Lagrangian Dynamics
Let Q be a smooth manifold, letC(Q) be the space of paths τ : I Q for some fixed interval I = [a, b]⊆ R, and let L : T Q V be a smooth function which we will call the Lagrangian. The aim of this section is to investigate those paths τ : I ⊆ R Q which constitute the critical points of the V -valued Lagrangian action,
S(τ ) =
b a
L( ˙τ ) dt, τ ∈ C(Q)
As a matter of terminology, a critical point of the action S is any path γ ∈ C(Q) for which dsdS(γs)
s=0 = 0 with respect to every variation γs that fixes the endpoints of γ.
Our approach is to consider instead the spaceC′(Q)⊆ C(T Q) of paths γ : I T Q which arise as the velocity vector field ˙τ for some curve τ ∈ C(Q). That is, C′(Q) is the image of the space of paths C(Q) by the map
C(Q) − C(T Q) τ − ˙τ
Note that this map is injective and thus forms an equivalence between C(Q) and
C′(Q). Under this identification, we consider the Lagrangian action on C′(Q),
8.4. VECTOR-VALUED LAGRANGIAN DYNAMICS
If X is tangent to a variation with fixed endpoints, then the boundary term vanishes and d ˙L(π∗X) = 0 for all variations X tangent to C′(Q) along a critical point γ of the
Then −H is a Hamiltonian function for the vector field ˙γ along γ.
Proof. We will adapt the approach of [61] to our V -valued context. Define the function S : I V by
Since ˙γ is an infinitesimal variation of γ, Theorem 8.7 implies that
dS(t) = θL
˙γ(a)
− θL
˙γ(t)
where θL=FL∗θ. Thus,
L˙γθL =−d dtθL
γ(t))
= d
dtdS(t) = dL γ(t)
and so
−ι˙γωL= ι˙γdθL =L˙γθL− dι˙γωL= d(L− ι˙γθL)
We conclude that L− ι˙γθL= L− FL(γ) γ is a Hamiltonian function for ˙γ along γ.
Part III
The Moduli Space of Flat
Connections
In the final part of this dissertation, we turn our attention once more to the space of connections. While Chapter 9 employs the framework of Part II, and Section 10.1 requires Chapter 3, the remaining material can be read immediately after Chapter 2 and a knowledge of only the definition of the symplectic volume.
In Chapter 9 we apply the theory of Part II to characterize the space of connections A(P ) on a G-principal bundle P over a manifold M of dimension at least 3. In particular, we show that A(P ) possesses a natural vector-valued symplectic structure, that the moduli space of flat connections M(P ) is the symplectic reduction of A(P ) with respect to the action of the gauge group, and that the reduced form takes values in H2(M ). Utilizing the language of characteristic forms, we obtains similar results for a variant of this vector-valued symplectic structure. In Chapter 10 we compute the volume of the moduli spaceMG(M ) of flat G-connections on M , first in the case that G is abelian, and second in the case that G is semisimple and π1M is free abelian.
In Chapter 11 we show that if Σ ⊆ M is a distinguished embedded surface in M, then there is a symplectic immersion of the moduli space MG(M ) intoMG(Σ), thus yielding information on the possible structure of MG(M ). We refer to the second half of Section 1.1 for an outline of the main results.
Chapter 9
The Reduction of the Space of Connections
Let M be a smooth manifold of dimension greater than 2, let G be a Lie group with Ad-invariant metric 〈 , 〉 on its Lie algebra g, and let P be a fixed G-principal bundle on M .
Before proceeding, let us briefly review the relevant notation from Chapter 9. We denote byA(P ) the Ω1(M, adP )-affine space of connections on P . For each A∈ A(P ), we frequently utilize the identification
Ω1(M, adP )− T∼ AA(P )
given by
αA= d
dtA + tα
t=0, A∈ A(P ), α ∈ Ω1(M, adP )
where + denotes the action of Ω1(M, adP ) on A(P ). In other words, we identify α∈ Ω1(M ) ∼= T0Ω1(M ) with the induced vector field α∈ X(A(P )).
The exterior covariant derivative dA: Ωk(M, g) Ωk+1(M, g) is given by
dAσ(X1, . . . , Xk+1) = dσ(hAX1, . . . , hAXk+1SS)
where hA : T P A is the fiberwise projection induced by the splitting A⊕ V (P ), where V (P ) is the vertical tangent bundle of P . Since dA preserves the subspace of tensorial forms in Ω∗(P, g) of type AdP , we may also consider dA : Ωk(M, adP ) Ωk+1(M, adP ). Many of the results of this chapter rely on the property that
d(α∧ β) = dAα∧ β + (−1)deg αα∧ dAβ
See [2]. Finally, we write Bk(M ) for the space of k-coboundaries on M . That is,
Bk(M ) = d Ωk−1(M )≤ Ωk(M )
9.1. THE MODEL SPACE
Ω1(M ),∧
9.1 The Model Space
Ω
1(M ), ∧
In this section we will consider two distinct vector-valued symplectic structures on Ω1(M ).
Let Σ be a compact oriented surface. We have seen that the vector space Ω1(Σ) carries a natural symplectic structure: namely,
ω(α, β) =
Σ
α∧ β, α, β ∈ Ω1(Σ)
The aim of this chapter is to determine a suitable generalization of this symplectic form to the case where dim M ≥ 3.
The most natural vector-valued symplectic structure on Ω1(M ) is the wedge prod-uct ∧. The following proposition establishes that ∧ is indeed a Ω2(M )-valued sym-plectic form on Ω1(M ).
Proposition 9.1. Let M be a manifold with boundary of dimension at least 2. The wedge product
U ⊆ M, and let αi ∈ C∞(U ) be given by
α =
i
αidxi
For each k ≤ n,
0 = α∧ dxk =
i
αidxi∧ dxk
Since n ≥ 2, for each i ≤ n there is a k ≤ n with k ∕= i, and thus dxi ∧ dxk ∕= 0 so that αi = 0. Since our choice of U was arbitrary, we conclude that α = 0.
It turns out that this symplectic structure is too fine for our purposes. The action of C∞(M ) on Ω1(M ) given by
f · α = df + α
is not in general Hamiltonian with respect to symplectic structure ω obtained by lifting
∧ to the fibers of T Ω1(M ). The issue is settled by reducing the space of coefficients from Ω2(M ) to Ω2(M )/B2(M ). To show this, we first establish a technical lemma.
Lemma 9.1. (i) Let U be a vector space with dim U ≥ 3 and let w ∈ Λ2U . If u∧ w = 0 for all u ∈ U then w = 0.
(ii) Let M be a manifold with dim M ≥ 3. If θ ∈ Ω2(M ) satisfies d(f θ) = 0 for all f ∈ C∞(M ), then θ = 0.
ij
9.1. THE MODEL SPACE
Ω1(M ),∧
(i, j ≤ n) so that
w =
i,j≤N
wijei∧ ej
For each k ≤ n, we have
0 = ek∧ ω =
i,j≤n
wijek∧ ei∧ ej
Since n≥ 3, for every pair of distinct i, j ≤ n we can find a k ≤ n with k ∕= i, j.
Consequently, ek∧ ei∧ ej ∕= 0 and thus wij = 0.
(ii) Since d(1· θ) = 0, we have
df ∧ θ = d(fθ) = 0
for all f ∈ C∞(M ). Fix p∈ M and observe that α ∧ θp = 0∈ Λ3(Tp∗M ) for all α = dfp ∈ Tp∗M . Now part (i) yields θp = 0.
Proposition 9.2. Let M be a compact manifold of dimension at least 3. The assign-ment
ω : Ω1(M )⊗ Ω1(M )− Ω2(M )/B2(M )
defined by
ω(α, β) = α∧ β + B2(M ), α, β ∈ Ω1(M )
is an Ω2(M )/B2(M )-valued symplectic structure on Ω1(M ) if and only if dim M ≥ 3 or M is a closed orientable surface.
Proof. The cases where dim M = 0, 1 are clear. Suppose for the moment that dim M ≥ 3. Let α ∈ Ω1(M ) and assume that α∧ γ ∈ B2(M ) for all γ ∈ Ω1(M ). Let β ∈ Ω1(M ) and observe that
d(α∧ fβ) = d
f (α∧ β)
∈ B2(M )
for all f ∈ C∞(M ). Thus α∧ β = 0 by Lemma 9.1. Since our choice of β was arbitrary, the nondegeneracy of the wedge product yields α = 0.
Finally, suppose that M = Σ is a closed compact orientable surface and equip Σ with an orientation and a Riemannian structure g. Let ∗ : Ω1(Σ) Ω1(Σ) denote the Hodge star operator. Let α∈ Ω1(Σ) and observe that if
α∧ β ∈ B2(Σ)
for all β ∈ B2, then, in particular,
α2dvol = α∧ ∗α ∈ B2(Σ)
and thus