Witten’s Problems, Set One — N^{o}. 1 (solution written by D. Freed)

1. We denote the Riemannian metric on M by (− , −) and so write the lagrangian density as

ℒ = \frac{1}{2}(Ẋ, Ẋ)dt .

Here Ẋ = \frac{dX}{dt} is the tangent to the curve.

(a) We compute the variation by considering a path of X’s, denoted X_{u}, u ∈ (−ε, ε). The computation takes place on (−ε, ε) × ℝ: There is a map X: (−ε, ε) × ℝ → M and we use the pullback metric, the pullback tangent bundle, and the pullback of the Levi-Civita connection, which we denote ∇. We use the notation δ = \frac{∂}{∂u}.

Since [∂_{t}, δ] = 0 and ∇ is torsion free, we have^{∗}

(1.1)  ∇_{t}δX = ∇_{δ}Ẋ .

So we make the computation

(1.2)  \begin{aligned}δℒ & = \frac{1}{2}δ(Ẋ, Ẋ)dt \\ & = (∇_{δ}Ẋ, Ẋ) dt \\ & = (∇_{t}δX, Ẋ) dt \\ & = d (δX, Ẋ) − (δX, ∇_{t}Ẋ) dt \end{aligned}

Now the first term in the final line is a total derivative, and the second is linear over the functions (as a function of δX). So the variational derivative is

\frac{δℒ}{δX(t)} = −∇_{t}Ẋ dt

\frac{∂^{2}x^{i}}{∂t∂u} + Γ_{jk}^{i} \frac{∂X}{∂t} \frac{∂X}{∂u}

∗ If x^{i} are local coordinates, and Γ_{jk}^{i} the corresponding Christoffel symbols on M, then the pullback by X makes the x^{i} functions of t, u and equation (1.1) follows, since is symmetric in t, u. (In other words, Γ_{jk}^{i} is symmetric in j, k). A derivation without local coordinates is given at the end of the exercise in an appendix.

The space of critical points is then

W = {X: ∇_{t}Ẋ = 0} .

The image of any such X is a geodesic; it is parametrized at constant speed.

(b) We assume that M is complete. Then the standard theory of ordinary differential equations shows that there is a unique solution X to ∇_{t}Ẋ given an initial position and velocity, i.e., a point of T M.

(c) To compute the symplectic structure we use the exact term in formula (1.2). Set

(1.3)  γ = (δX, Ẋ) _{t_{0}} ,

where t_{0} is some fixed time. Then the symplectic structure is computed as the differential of γ in the space of paths:

(1.4)  ω = δγ = − (δX, ∇_{δ}Ẋ) _{t=t_{0}} = (∇_{δ}Ẋ, δX) _{t=t_{0}}

We elaborate: If ξ_{1}, ξ_{2} are sections of X^{∗}(T M) — in other words, are tangent at X to the space of paths — then

ω(ξ_{1}, ξ_{2}) = (∇_{ξ_{1}} Ẋ(t_{0}), ξ_{2}(t_{0})) − (∇_{ξ_{2}} Ẋ(t_{0}), ξ_{1}(t_{0})) .

Now the standard symplectic structure on T^{∗}M is the differential of the 1-form

(1.5)  θ_{α}(η) = α(π_{∗}η),  α ∈ T^{∗}M, η ∈ T_{α}(T^{∗}M),

where π: T^{∗}M → M is projection. We identify X ∈ W with the tangent vector ⟨X(t_{0}), Ẋ(t_{0})⟩ ∈ T M. Then

(1.6)  γ(η) = (π_{∗}η, Ẋ(t_{0})) ,  η ∈ T_{X(t_{0})}(T M).

Under the map T M → T^{∗}M given by the metric, it is easy to see that (1.5) pulls back to (1.6) at the point ⟨X(t_{0}), Ẋ(t_{0})⟩ ∈ T M.

(d) The vector field v on W may be described by the equation

v_{X} = Ẋ ∈ C^{∞}(ℝ, X^{∗}(T M)) .

To compute ι_{v}ω we substitute δX = Ẋ into (1.4):

ι_{v}ω = (∇_{t}Ẋ, δX) _{t_{0}} − (∇_{δ}Ẋ, Ẋ) _{t_{0}}
= − (∇_{δ}Ẋ, Ẋ) _{t_{0}} ,

since X satisfies the eqution of motion ∇_{t}Ẋ = 0. It is easy to see that ι_{v}ω = dH where H is the function

H(X) = −\frac{1}{2} |Ẋ(t_{0})|^{2} .

(Of course, this is independent of t_{0}.)

Remark on Equation (1.1) (by P. Deligne).

An affine space A gives rise to a vector space V, the translations of A, and it is convenient (barycentric calculus) to view A and V as living in the same vector space B of dimension one more:

0 ⟶ V ⟶ B ⟶^{m} ℝ ⟶ 0, A = m^{−1}(1) .

If A is an affine space bundle over a manifold M (structural group the affine group GL(n) ⋉ ℝ^{n}), a connection ∇ on A allows us to define ∇_{X}a, for a a section of A, as a section of the associated vector bundle (vector bundle of translations of A), structural group GL(n), obtained by applying GL(n) ⋉ ℝ^{n} → GL(n).

Let T^{aff} be the tangent bundle T to M, viewed as a bundle of affine space. The vector bundle T is the vector bundle associated to T^{aff}. A connection ∇ on T defines as follows a connection ∇^{aff} on T^{aff}: ∇^{aff} induces ∇ on T and for z the zero section of T^{aff}, one has ∇_{X}^{aff}z = −X [connexion sans glissement].

The torsion of ∇ is the translation part of the curvature of ∇^{aff}. View X as a map of ℝ^{2} to the zero section of th tangent bundle of M. If ∇ is torsion free, vanishing of the translation part of the curvature gives

∇_{t}∇_{u}^{aff}X − ∇_{u}∇_{t}^{aff}X = 0 , i.e.
−∇_{t}∂_{u}X + ∇_{u}∂_{t}X = 0