A Kähler metric on a based loop group and a covariant differentiation

Ichiro Shigekawa^{1} and Setsuo Taniguchi^{2}

1 Department of Mathematics, Graduate School of Science, Kyoto University, Ky-

oto 606-01, Japan 2 Graduate School of Mathematics, Kyushu University, Hakozaki, Fukuoka 812,

Japan

1. Introduction

Loop groups have been attracting many authors recently. In this paper, we are discussing a Kähler metric on a loop group. Let G be a d-dimensional compact Lie group and 𝔤 be its Lie algebra (≡ the space of left invariant vector fields). Then, 𝔤 admits an Ad(G)-invariant inner product (·, ·)_{𝔤} and we fix it through the paper. We denote the G-valued path space on [0, 1] by

P G := {γ: [0, 1]→G; continuous and γ(0) = e}  (1.1)

e being the unit element of G.

On the other hand, our interest is in the based loop group ΩG over G:

ΩG := {γ: [0, 1]→G; continuous and γ(0) = γ(1) = e}.  (1.2)

We develop a differential geometry from an analytic point of view. In particular, we discuss several operators acting on, e.g., tensor fields. Usually, the following Cameron-Martin space H_{0} is regarded as a tangent space:

H_{0} = \{h : [0, 1] → 𝔤 : \begin{matrix}h is absolutely continuous, h(0) = h(1) = 0, and \\ the derivative ḣ satisfies that ∫_{0}^{1} |ḣ(t)|_{𝔤}^{2}dt < ∞ \end{matrix} \} ,

where | · |_{𝔤} = \sqrt{(·, ·)_{𝔤}}. H_{0} is a Hilbert space with the inner product

(h, k)_{H_{0}} = ∫_{0}^{1}(ḣ(t), k̇(t))_{𝔤}dt, h, k ∈ H_{0}.

Using the left translation, this inner product defines a Riemannian metric.

Further, we can introduce an almost complex structure, denoted by J, (see, [11]), but under it, the above metric is not a Kähler metric. The following Kähler form S was introduced by Pressley [11]:

S(h, k) = ∫_{0}^{1}(ḣ(t), k(t))_{𝔤}dt.

The associated Riemannian metric is defined by B(X, Y) = S(X, JY). This metric was discussed by [11, 2] in view of differential geometry. We will discuss it from a probabilistic point of view, in which the pinned Brownian motion measure plays an essential role.

The organization of this paper is as follows. In the section 2, we prepare the fundamental notions of differential geometry. We define vector fields, differential forms, exterior derivatives, etc. We also introduce an almost complex structure. Kähler metric is discussed in the section 3. We will calculate the Levi-Civita covariant derivative and the associated Riemannian curvature. The section 4 is devoted to showing the closability of operators. The Ricci curvature is computed in the section 5.

2. A based loop group and an almost complex structure

In this section, we introduce several notions in differential geometry. The Cameron-Martin space H_{0} is a tangent space of ΩG. Defining a bracket by [h, k](t) = [h(t), k(t)]_{𝔤}, t ∈ [0, 1], H_{0} becomes a Lie algebra.

Thinking of H_{0} as T_{e}(ΩG), the tangent space of ΩG at e, where e(s) ≡ e, one may regard the product space ΩG × H_{0} as the tangent bundle of ΩG. One then defines spaces of tensor fields on ΩG by

FΓ_{b}^{∞}(T_{q}^{p}(ΩG))

= \{u : ΩG→H_{0}^{⊗p} ⊗ (H_{0}^{∗})^{⊗q} : \begin{matrix}u = ∑_{j}^{finite} f_{j}e_{j} for some f_{j} ∈ FC_{b}^{∞}(ΩG) \\ and e_{j} ∈ H_{0}^{⊗p} ⊗ (H_{0}^{∗})^{⊗q} \end{matrix}\} ,

where

FC_{b}^{∞}(ΩG)
= \{u : ΩG → ℝ : \begin{matrix}u(γ) = f(γ(t_{1}), …, γ(t_{n})) for some \\ f ∈ C_{b}^{∞}(G^{n})and 0 ≤ t_{1} < … < t_{n} ≤ 1 \end{matrix} \} .

For X ∈ FΓ_{b}^{∞}(T_{0}^{1}(ΩG)) and u ∈ FC_{b}^{∞}(ΩG), set

Xu(γ) = lim_{ε↓0} \frac{1}{ε}(f(γe^{εX(γ)}) − f(γ)),

where for h ∈ H_{0}, (e^{εh})(t) = e^{εh(t)}, the position of the integral curve along h(t) ∈ 𝔤 at time ε. As is easily seen, one has that

Xu(γ) = ∑_{i=1}^{n}((X(γ))(t_{i})^{(i)}f)(γ(t_{1}), …, γ(t_{n}))  (2.1)
for u(γ) = f(γ(t_{1}), …, γ(t_{n})) ∈ FC_{b}^{∞}(ΩG)

where, for ξ ∈ 𝔤,

ξ^{(i)}f(g_{1}, …, g_{n}) = lim_{ε↓0} \frac{1}{ε}(f(g_{1}, …, g_{i−1}, g_{i}e^{εξ}, g_{i+1}, …, g_{n}) − f(g_{1}, …, g_{n})).

For X, Y ∈ FΓ_{b}^{∞}(T_{0}^{1}(ΩG)), the Lie bracket [X, Y] can be defined as a unique element of FΓ_{b}^{∞}(T_{0}^{1}(ΩG)) so that

[X, Y]u = XY u − Y Xu, X, Y ∈ FΓ_{b}^{∞}(T_{0}^{1}(ΩG)), u ∈ FC^{∞}(ΩG). (2.2)

Defining constant vector field X^{h} ∈ FΓ_{b}^{∞}(T_{0}^{1}(ΩG)), h ∈ H_{0}, by X^{h}(γ) = h, γ ∈ ΩG. Then X^{h} is a left invariant vector field. Due to (2.1), one has that

[X^{h}, X^{k}] = X^{[h,k]}, h, k ∈ H_{0}.

By virtue of (2.2), the Jacobi identity can be seen;

[X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0, X, Y, Z ∈ FΓ_{b}^{∞}(T_{0}^{1}(ΩG)). (2.3)

Let

FΓ_{b}^{∞}(⋀^{p}T^{∗}(ΩG)) = {u ∈ FΓ_{b}^{∞}(T_{p}^{0}(ΩG)) : u(γ) is anti-symmetric}.

The exterior derivative du can be defined, as in the finite dimensional case, for u ∈ FΓ_{b}^{∞}(⋀^{p}T^{∗}(ΩG))

du(X_{1}, …, X_{p+1})

= ∑_{a=1}^{p+1}(−1)^{a−1}X_{a}(u(X_{1}, …, X̂_{a}, …, X_{p+1}))
+ ∑_{1≤a<b≤p+1}(−1)^{a+b}u([X_{a}, X_{b}], X_{1}, …, X̂_{a}, …, X̂_{b}, …, X_{p+1}), (2.4)

for X_{1}, …, X_{p+1} ∈ FΓ_{b}^{∞}(T_{0}^{1}(ΩG)), where X̂_{a} means that X_{a} is omitted. As in the finite dimensional case, one has that

d^{2} = 0.  (2.5)

Indeed, define L_{X} and ι(X) by

L_{X}u(X_{1}, …, X_{p}) = X(u(X_{1}, …, X_{p}))

− ∑_{a=1}^{p} u(X_{1}, …, X_{a−1}, [X, X_{a}], X_{a+1}, …, X_{p}),

ι(X)u(X_{1}, …, X_{p−1}) = u(X, X_{1}, …, X_{p−1}),

for X, X_{1}, …, X_{p} ∈ FΓ_{b}^{∞}(T_{0}^{1}(ΩG)), u ∈ FΓ_{b}^{∞}(⋀^{p}T^{∗}(ΩG)).

An elementary algebraic computation leads one to the identities

\{ \begin{matrix}d ∘ ι(X) + ι(X) ∘ d = L_{X}, \\ d ∘ L_{X} = L_{X} ∘ d,  X ∈ FΓ_{b}^{∞}(T_{0}^{1}(ΩG)). \end{matrix}

These yield that d^{2} ∘ ι(X) = ι(X) ∘ d^{2} from which (2.5) follows by induction on p.

We now introduce an almost complex structure on ΩG following [11]. To do this, put

e_{n}(t) = \frac{1}{2π\sqrt{−1}n}(e^{2π\sqrt{−1}nt} − 1), n ∈ ℤ \ \{0\}

and we take an orthonormal basis {ξ}_{i=1,...,d} in 𝔤. We fix it through the paper. We use the following convention. For α = (n, i), n ∈ {1, 2, …}, i = 1, 2, …, d, we define ᾱ = (−n, i), and

e_{α} = e_{n}(t)ξ_{i}, eᾱ = e_{−n}(t)ξ_{i}.  (2.6)

Every h ∈ H_{0} can be expanded as

h = ∑_{i=1}^{d} ∑_{n≠0}(h, e_{−n,i})_{H_{0}}e_{n,i} in H_{0},

where (h, k_{1} + \sqrt{−1}k_{2})_{H_{0}} = (h, k_{1})_{H_{0}} + \sqrt{−1}(h, k_{2})_{H_{0}}, h, k_{1}, k_{2} ∈ H_{0}. An almost complex structure J : H_{0} → H_{0} on ΩG is defined by

Jh = \sqrt{−1} ∑_{i=1}^{d} ∑_{n>0}(h, e_{−n,i})_{H_{0}}e_{n,i} − \sqrt{−1} ∑_{i=1}^{d} ∑_{n>0}(h, e_{n,i})_{H_{0}}e_{−n,i}.

See [11]. As is easily seen, it holds that

J^{2}h = −h and (Jh, Jk)_{H_{0}} = (h, k)_{H_{0}}, h, k ∈ H_{0}.

Put H_{0}^{ℂ} = H_{0} ⊕ \sqrt{−1}H_{0} and

H_{0}^{(1,0)} = \{η ∈ H_{0}^{ℂ} : Jη = \sqrt{−1}η\}, H_{0}^{(0,1)} = \{η ∈ H_{0}^{ℂ} : Jη = −\sqrt{−1}η\}.

Here we extend J to H_{0}^{ℂ} by complex linearity. Obviously H_{0}^{ℂ} = H_{0}^{(1,0)}⊕H_{0}^{(0,1)}. J^{∗}: H_{0}^{∗} → H_{0}^{∗} satisfies the same property and therefore H_{0}^{∗(1,0)}, H_{0}^{∗(0,1)} can be defined similarly. For h ∈ H_{0}, define

π_{+}h = h^{(1,0)} = \frac{1}{2}(h − \sqrt{−1}Jh),  (2.7)
π_{−}h = h^{(0,1)} = \frac{1}{2}(h + \sqrt{−1}Jh).  (2.8)

Then h^{(1,0)} ∈ H_{0}^{(1,0)}, h^{(0,1)} ∈ H_{0}^{(0,1)}, and h = h^{(1,0)} + h^{(0,1)}. By a straightforward computation one sees that

h^{(1,0)} = ∑_{i=1}^{d} ∑_{n>0}(h, e_{−n,i})_{H_{0}}e_{n,i}, h^{(0,1)} = ∑_{i=1}^{d} ∑_{n>0}(h, e_{n,i})_{H_{0}}e_{−n,i}. (2.9)

Since

[e_{n,i}, e_{m,j}](t) = \frac{1}{2π\sqrt{−1}mn}\{(m + n)e_{m+n}(t) − ne_{n}(t) − me_{m}(t)\}[ξ_{i}, ξ_{j}],

one can conclude from (2.9) that

[h^{(1,0)}, k^{(1,0)}] ∈ H_{0}^{(1,0)} and [h^{(0,1)}, k^{(0,1)}] ∈ H_{0}^{(0,1)} h, k ∈ H_{0}. (2.10)

Let us define the Newlander-Nirenberg tensor N as follows:

N(X, Y) = J[X, Y] − [JX, Y] − [X, JY] − J[JX, JY].

By (2.10), we can easily see N = 0 and in this sense, J is “integrable.” Put

⋀^{p,q} = \underbrace{H_{0}^{∗(1,0)} ∧ ⋯ ∧ H_{0}^{∗(1,0)}}_{p-times} ∧ \underbrace{H_{0}^{∗(0,1)} ∧ ⋯ ∧ H_{0}^{∗(0,1)}}_{q-times}

and define

FΓ_{b}^{∞}(⋀^{p,q}T^{∗}(ΩG)) =\{u : ΩG→⋀^{p,q} : \begin{matrix}u = ∑_{i}^{finite}(f_{i} + \sqrt{−1}g_{i})e_{i} \\ f_{i}, g_{i} ∈ FC_{b}^{∞}(ΩG), e_{i} ∈ ⋀^{p,q} \end{matrix} \} .

Observing that

FΓ_{b}^{∞}(⋀^{p,q}T^{∗}(ΩG)) ⊂ FΓ_{b}^{∞}(T_{p+q}^{0}(ΩG)) ⊕ \sqrt{−1}FΓ_{b}^{∞}(T_{p+q}^{0}(ΩG)),

we can extend the exterior derivative d in (2.1) to FΓ_{b}^{∞}(⋀^{p,q}(T^{∗}(ΩG))), and due to (2.4), obtain that

du(γ) ∈ ⋀^{p+1,q} ⊕ ⋀^{p,q+1}, γ ∈ ΩG, for u ∈ FΓ_{b}^{∞}(⋀^{p,q}T^{∗}(ΩG)).

Thus, operators

∂ : FΓ_{b}^{∞}(⋀^{p,q}T^{∗}(ΩG)) → FΓ_{b}^{∞}(⋀^{p+1,q}T^{∗}(ΩG))
\overline{∂} : FΓ_{b}^{∞}(⋀^{p,q}T^{∗}(ΩG)) → FΓ_{b}^{∞}(⋀^{p,q+1}T^{∗}(ΩG))

can be defined so that for u ∈ FΓ_{b}^{∞}(⋀^{p,q}T^{∗}(ΩG)), ∂u(γ) (resp. \overline{∂}u(γ)) is the projection of du(γ) onto ⋀^{p+1,q} (resp. ⋀^{p,q+1}). Obviously d = ∂ + \overline{∂}. Then, by virtue of (2.2), one obtains that

∂^{2} = 0, \overline{∂}^{2} = 0, and ∂\overline{∂} + \overline{∂}∂ = 0  (2.11)

on FΓ_{b}^{∞}(⋀^{p,q}T^{∗}(ΩG)) for p, q ≥ 0.

3. A Kähler metric

We now introduce a Kähler metric on ΩG following [11]. Define S ∈ FΓ_{b}^{∞}(⋀^{2}T^{∗}(ΩG)) by

S(X^{h}, X^{k}) = ∫_{0}^{1}(ḣ(t), k(t))_{𝔤}dt, = − ∫_{0}^{1}(h(t), k̇(t))_{𝔤}dt, h, k ∈ H_{0}.

Note that

|S(X^{h}, X^{k})| ≤ |h|_{H_{0}}|k|_{H_{0}},

and hence that S is well-defined. By the integration by parts and the Ad(G)- invariance of (·, ·)_{𝔤}, we obtain the following cyclic formula:

S([X^{h}, X^{k}], X^{l}) + S([X^{k}, X^{l}], X^{h}) + S([X^{l}, X^{h}], X^{k}) = 0.

Combine this with (2.1), we can show that dS = 0. Define

B(X, Y) = S(X, JY), X, Y ∈ FΓ_{b}^{∞}(T_{0}^{1}(ΩG)).

We note that S(X, Y) = B(JX, Y). Let T : H_{0}^{ℂ} → H_{0}^{ℂ} be a continuous linear operator so that T e_{n,i} = \frac{1}{2|n|π} e_{n,i}. Observe then that

S(X^{e^{n,i}}, X^{e^{m,j}}) = \frac{1}{2πm\sqrt{−1}}δ_{n,−m}δ_{i,j},

and hence that

B(X^{h}, X^{k}) = (T h, k)_{H_{0}}.  (3.1)

In particular, for X, Y ∈ H_{0},

B(X, X) ≥ 0 and “= 0” if and only if X = 0  (3.2)
B(X, Y) = B(Y, X) ∈ ℝ.  (3.3)

Thus one have obtained the Kähler metric B on ΩG. We denote the completion of H_{0} with respect to B by H_{1}. From now on, we regard H_{0} as a tangent space. Moreover, by noting JT = T J, J also defines an almost complex structure in H_{1}.

We now turn to the Levi-Civita covariant derivative. As in the finite dimensional case, the Levi-Civita covariant derivative is characterized by the following identity:

2B(∇_{X}Y, Z) = XB(Y, Z) + Y B(X, Z) − ZB(X, Y)
+B([X, Y], Z) + B([Z, X], Y) + B(X, [Z, Y]).

In particular, taking left invariant vector fields, we have

2B(∇_{X^{h}}X^{k}, X^{l}) = B([X^{h}, X^{k}], X^{l}) + B([X^{l}, X^{h}], X^{k}) + B(X^{h}, [X^{l}, X^{k}]).

Furthermore, due to the identity (see, e.g., [8, Proposition IX.4.2])

4B((∇_{X}J)Y, Z) = 6dS((X, JY, JY) − 6dS(X, Y, Z) + B(N(Y, Z), JX),

we have ∇J = 0, i.e., the almost complex structure is parallel.

We easily see that

B(∇_{X}Y, Z) = 0 if Y, Z ∈ H^{(1,0)} or Y, Z ∈ H^{(0,1)}.  (3.4)

To see this, we note that the almost complex structure J is parallel. For example, if Y, Z ∈ H^{(1,0)}, then

B(∇_{X}Y, Z) = −\sqrt{−1}B(∇_{X}JY, Z)
= −\sqrt{−1}B(J∇_{X}Y, Z)
= \sqrt{−1}B(∇_{X}Y, JZ)
= −B(∇_{X}Y, Z).

Thus we have (3.4).

Let us calculate the covariant derivative. From the definition,

B(∇_{X^{h}}X^{k}, X^{l})

= B([X^{h}, X^{k}], X^{l}) + B([X^{l}, X^{h}], X^{k}) + B(X^{h}, [X^{l}, X^{k}])
= S(X^{l}, J[X^{h}, X^{k}]) + S([X^{l}, X^{h}], JX^{k}) + S([X^{l}, X^{k}], JX^{h},)
= ∫_{0}^{1}(l̇(t), J[h, k](t))_{𝔤}dt + ∫_{0}^{1}(\frac{d}{dt}[l(t), h(t)], Jk(t))_{𝔤}dt

+ ∫_{0}^{1}(\frac{d}{dt}[l(t), k(t)], Jh(t))_{𝔤}dt

= ∫_{0}^{1}(l̇(t), J[h, k](t))_{𝔤}dt + ∫_{0}^{1}([l̇(t), h(t)], Jk(t))_{𝔤}dt

+ ∫_{0}^{1}([l(t), ḣ(t)], Jk(t))_{𝔤}dt + ∫_{0}^{1}([l̇(t), k(t)], Jh(t))_{𝔤}dt

+ ∫_{0}^{1}([l(t), k̇(t)], Jh(t))_{𝔤}dt

= ∫_{0}^{1}(l̇(t), J[h, k](t))_{𝔤}dt + ∫_{0}^{1}(l̇(t), [h(t), Jk(t)])_{𝔤}dt

+ ∫_{0}^{1}(l(t), [ḣ(t), Jk(t)])_{𝔤}dt + ∫_{0}^{1}(l̇(t), [k(t), Jh(t)])_{𝔤}dt

+ ∫_{0}^{1}(l(t), [k̇(t), Jh(t)])_{𝔤}dt

= ∫_{0}^{1}(l̇(t), J[h, k](t) + [h(t), Jk(t)] + [k(t), Jh(t)])_{𝔤}dt

+ ∫_{0}^{1}(l(t), [ḣ(t), Jk(t)] + [k̇(t), Jh(t)])_{𝔤}dt.

First we consider the case X^{h}, X^{k} ∈ H^{(1,0)} or X^{h}, X^{k} ∈ H^{(0,1)}. We set JX^{k} = ε\sqrt{−1}X^{k}.

2B(∇_{X^{h}}X^{k}, X^{l})

= ∫_{0}^{1}(l̇(t), ε\sqrt{−1}[h, k](t) + ε\sqrt{−1}[h(t), k(t)] + ε\sqrt{−1}[k(t), h(t)])_{𝔤}dt

+ ∫_{0}^{1}(l(t), ε\sqrt{−1}[ḣ(t), k(t)] + ε\sqrt{−1}[k̇(t), h(t)])_{𝔤}dt

= ∫_{0}^{1}(l̇(t), ε\sqrt{−1}[h, k](t))_{𝔤}dt

+ ∫_{0}^{1}(l(t), ε\sqrt{−1}\frac{d}{dt}[h(t), k(t)] − 2ε\sqrt{−1}[h(t), k̇(t)])_{𝔤}dt

= 2 ∫_{0}^{1}(l̇(t), ε\sqrt{−1} ∫_{0}^{t}[h(s), k̇(s)]ds)_{𝔤}dt
= 2S(l, ε\sqrt{−1} ∫_{0}^{·}[h(s), k̇(s)]ds)_{𝔤}
= −2B(l, ε\sqrt{−1}J ∫_{0}^{·}[h(s), k̇(s)]ds)
= 2B(l, ∫_{0}^{·}[h(s), k̇(s)]ds).

Here we used J ∫_{0}^{·}[h(s), k̇(s)]ds = ε\sqrt{−1} ∫_{0}^{·}[h(s), k̇(s)]ds due to the expression of e_{α}. Setting

A(h, k)(t) = ∫_{0}^{t}[h(s), k̇(s)]ds,  (3.5)

we have

∇_{X^{h}}X^{k} = A(h, k).

In the case that X^{h} ∈ H^{(1,0)} and X^{k} ∈ H^{(0,1)}, or X^{h} ∈ H^{(0,1)} and X^{k} ∈ H^{(1,0)}, we set JX^{k} = ε\sqrt{−1}X^{k}. Then

2B(∇_{X^{h}}X^{k}, X^{l})
= ∫_{0}^{1}(l̇(t), J[h, k](t) + ε\sqrt{−1}[h(t), k(t)] − ε\sqrt{−1}[k(t), h(t)])_{𝔤}dt
+ ∫_{0}^{1}(l(t), ε\sqrt{−1}[ḣ(t), k(t)] − ε\sqrt{−1}[k̇(t), h(t)])_{𝔤}dt
= ∫_{0}^{1}(l̇(t), J[h, k](t) + ε\sqrt{−1}[h(t), k(t)] − ε\sqrt{−1}[k(t), h(t)])_{𝔤}dt
− ∫_{0}^{1}(l̇(t), ε\sqrt{−1}[h(t), k(t)])_{𝔤}dt
= S(l, J[h, k](t) + ε\sqrt{−1}[h, k])

= −B(l, J^{2}[h, k](t) + ε\sqrt{−1}J[h, k])
= B(l, [h, k] − ε\sqrt{−1}J[h, k]).

Thus we have (recall the definition of π_{±} in (2.7), (2.8))

∇_{X^{h}}X^{k} = \frac{1}{2}\{[h, k] − ε\sqrt{−1}J[h, k]\} = π_{ε}[h, k].

We sum up into a theorem. Theorem 3.1. The covariant derivatives are given as follows:

∇_{X^{h}}X^{k} = \{ \begin{matrix}A(h, k), & if h, k ∈ H^{(1,0)} or h, k ∈ H^{(0,1)}, \\ π_{+}[h, k], & if h ∈ H^{(0,1)} and k ∈ H^{(1,0)}, \\ π_{−}[h, k], & if h ∈ H^{(1,0)} and k ∈ H^{(0,1)}. \end{matrix}

Now we are ready to compute the Riemannian curvature. The Riemannian curvature is defined by

R(X, Y) := [∇_{X}, ∇_{Y}] − ∇_{[X,Y]}.

From the definition, it is easy to see that R satisfies

B(R(X, Y)Z, W) = −B(R(Y, X)Z, W) = B(R(Z, W)X, Y). Accordingly, the non-trivial term is B(R(X_{α}, Xβ̄)X_{γ}, Xδ̄) where X_{α} = X^{e^{α}}, Xβ̄ = X^{eβ̄}. The Riemannian curvature is given as follows: Theorem 3.2. It holds that B(R(X_{α}, Xβ̄)X_{γ}, Xδ̄)

= −\sqrt{−1} ∫_{0}^{1}(π_{+}[eβ̄, e_{γ}], [ė_{α}, eδ̄])dt − \sqrt{−1} ∫_{0}^{1}(π_{+}[eβ̄, e_{γ}], [e_{α}, ėδ̄])dt
+ \sqrt{−1} ∫_{0}^{1}(A(e_{α}, e_{γ}), [eβ̄, ėδ̄])dt

+ \sqrt{−1} ∫_{0}^{1}(π_{+}[e_{α}, eβ̄], [ė_{γ}, eδ̄])dt − \sqrt{−1} ∫_{0}^{1}(π_{−}[e_{α}, eβ̄], [e_{γ}, ėδ̄])dt.

Proof. By the definition, B(R(X_{α}, Xβ̄)X_{γ}, Xδ̄)

= B(∇_{X_{α}}∇_{Xβ̄}X_{γ} − ∇_{Xβ̄}∇_{X_{α}}X_{γ} − ∇_{[X_{α},Xβ̄]}X_{γ}, Xδ̄)
= B(∇_{X_{α}}π_{+}[Xβ̄, X_{γ}] − ∇_{Xβ̄}A(X_{α}, X_{γ}) − ∇_{π_{+}[X_{α},Xβ̄]}X_{γ}
− ∇_{π_{−}[X_{α},Xβ̄]}X_{γ}, Xδ̄)
= B(A(X_{α}, π_{+}[Xβ̄, X_{γ}]) − π_{+}[Xβ̄, A(X_{α}, X_{γ})] − A(π_{+}[X_{α}, Xβ̄], X_{γ})
− π_{+}[π_{−}[X_{α}, Xβ̄], X_{γ}], Xδ̄)
= −\sqrt{−1}S(A(X_{α}, π_{+}[Xβ̄, X_{γ}]) − [Xβ̄, A(X_{α}, X_{γ})]

− A(π_{+}[X_{α}, Xβ̄], X_{γ}) − [π_{−}[X_{α}, Xβ̄], X_{γ}], Xδ̄)
= \sqrt{−1} ∫_{0}^{1}(A(e_{α}, π_{+}[eβ̄, e_{γ}]), ėδ̄)dt − \sqrt{−1} ∫_{0}^{1}([eβ̄, A(e_{α}, e_{γ})], ėδ̄)dt

−\sqrt{−1} ∫_{0}^{1}(A(π_{+}[e_{α}, eβ̄], e_{γ}), ėδ̄)dt−\sqrt{−1} ∫_{0}^{1}([π_{−}[e_{α}, eβ̄], e_{γ}], ėδ̄)dt

= −\sqrt{−1} ∫_{0}^{1}([e_{α}, \frac{d}{dt}π_{+}[eβ̄, e_{γ}]], eδ̄)dt + \sqrt{−1} ∫_{0}^{1}(A(e_{α}, e_{γ}), [eβ̄, ėδ̄])dt

+ \sqrt{−1} ∫_{0}^{1}([π_{+}[e_{α}, eβ̄], ė_{γ}], eδ̄)dt − \sqrt{−1} ∫_{0}^{1}(π_{−}[e_{α}, eβ̄], [e_{γ}, ėδ̄])dt

= \sqrt{−1} ∫_{0}^{1}(\frac{d}{dt}π_{+}[eβ̄, e_{γ}], [e_{α}, eδ̄])dt + \sqrt{−1} ∫_{0}^{1}(A(e_{α}, e_{γ}), [eβ̄, ėδ̄])dt

+ \sqrt{−1} ∫_{0}^{1}(π_{+}[e_{α}, eβ̄], [ė_{γ}, eδ̄])dt − \sqrt{−1} ∫_{0}^{1}(π_{−}[e_{α}, eβ̄], [e_{γ}, ėδ̄])dt

= −\sqrt{−1} ∫_{0}^{1}(π_{+}[eβ̄, e_{γ}], [ė_{α}, eδ̄])dt − \sqrt{−1} ∫_{0}^{1}(π_{+}[eβ̄, e_{γ}], [e_{α}, ėδ̄])dt
+ \sqrt{−1} ∫_{0}^{1}(A(e_{α}, e_{γ}), [eβ̄, ėδ̄])dt

+ \sqrt{−1} ∫_{0}^{1}(π_{+}[e_{α}, eβ̄], [ė_{γ}, eδ̄])dt − \sqrt{−1} ∫_{0}^{1}(π_{−}[e_{α}, eβ̄], [e_{γ}, ėδ̄])dt

which completes the proof. □

4. Closability

We will show the closability of operators that were introduced in the previous sections. Operators were considered in the framework of L^{2} theory. To define an L^{2} space, we need a measure. So we begin with introducing a measure on the path space P G. To do this, let us consider the following stochastic differential equation. We consider the following stochastic differential equation on G:

\{ \begin{matrix}dγ_{t} & = ∑_{i=1}^{d} ξ_{i}(γ_{t})^{∘}db_{t}^{i}, \\ γ_{0} & = e \end{matrix}  (4.1)

where (b_{t}^{1}, …, b_{t}^{d})_{t∈[0,T]} is a d-dimensional Brownian motion, and ^{∘} stands for the Stratonovich symmetric stochastic integral. Setting b_{t} = ∑_{i} b_{t}^{i}ξ_{i}, (b_{t}) is a Brownian motion on 𝔤. (b_{t}) induces a measure on the space P 𝔤 where P 𝔤 is the 𝔤-valued path space. The measure is called the Wiener measure and is denoted by P^{W}. There exists the unique strong solution to (4.1), i.e., there exists a measurable function I: P 𝔤 → P G such that γ = I(b) is the unique solution to (4.1). We call this map I the Itô map. We denote the image measure of P^{W} under I by μ. Then (P 𝔤, P^{W}) ≅ (P G, μ) as measure spaces. We sometimes regard a function on (P G, μ) as a function on (P 𝔤, P^{W}). We can restrict the measure to ΩG by taking a conditional probability. We set m = E[ · |γ(1) = e]. m is called a pinned measure or a bridge measure.

For function u ∈ FC_{b}^{∞}(ΩG), du is characterized by

⟨du, X⟩ = Xu, for X ∈ FΓ_{b}^{∞}(T_{0}^{1}(ΩG)).

In the section 2, X was an H_{0}-valued function. But we have replaced H_{0} by H_{1}. Since H_{0} ⊆ H_{1}, we have H_{1}^{∗} ⊆ H_{0}^{∗} and du must be in H_{1}^{∗}. Let us give an example of such a function. Set

u(γ) = ∫_{0}^{1} φ(t)f(γ(t))dt

where φ ∈ C^{∞}([0, 1]) and f ∈ C^{∞}(M). Then clearly

⟨du(γ), h⟩ = ∫_{0}^{1} φ(t)⟨df(γ(t)), h(t)⟩dt.

It is now easy to see that du(γ) ∈ H_{1}^{∗}. Since those functions are dense in L^{2}(ΩG), we can define the following pre-Dirichlet form:

E(u, v) := ∫_{ΩG}(du, dv)_{H_{1}^{∗}}m(dγ).  (4.2)

The closability of the pre-Dirichlet form is a fundamental problem. It is equivalent to the closability of the operator d. To show the closability, we are enough to show the existence of the dual operator.

Since |h|_{H_{1}} = |\sqrt{T} h|_{H_{0}}, we easily show that θ ∈ H_{1}^{∗} if and only if θ ∈ Dom(\sqrt{T^{∗−1}}) and

|θ|_{H_{1}^{∗}} = |\sqrt{T^{∗−1}}θ|_{H_{0}^{∗}},

where T^{∗}: H_{0}^{∗} → H_{0}^{∗} is the dual operator of T. We also notice that

(h, k)_{H_{1}} = (T h, k)_{H_{0}} for h, k ∈ H_{1}

and similarly

(θ, η)_{H_{1}^{∗}} = (T^{∗−1}θ, η)_{H_{0}^{∗}} for θ ∈ Ran(T^{∗}), η ∈ H_{1}^{∗}.

Now we have

∫_{ΩG}(du, θ)_{H_{1}^{∗}}dm = ∫_{ΩG}(du, T^{∗−1}θ)_{H_{0}^{∗}}dm = ∫_{ΩG} ud′(T^{∗−1}θ)dm

where d′ is the dual operator with respect to the inner product ( , )_{H_{0}}. This implies that d^{∗} = d′T^{∗−1}.

We shall give a rather explicit expression of d^{∗}. We recall the notion of divergence. For X ∈ FΓ_{b}^{∞}(T_{0}^{1}(ΩG)), div X is characterized by the following identity:

∫_{ΩG} Xudm = − ∫_{ΩG}(div X)udm

If X is left invariant, we can give an explicit formula of div X:

div X^{h} = − ∫_{0}^{1}(ḣ(s), db(s))_{𝔤}.

Notice that this is valid only for h ∈ H_{0}. Because we are given a Riemannian metric, a 1-form can be identified with a vector field, i.e., for any 1-form ω, there exists a unique vector field, which we denote by ω^{♯}, such that

⟨ω, X⟩ = B(ω^{♯}, X).

We take θ ∈ H_{1}^{∗} (a constant 1-form). Then,

∫_{ΩG}(du, θ)_{H_{1}^{∗}}dm = ∫_{ΩG}⟨du, θ^{♯}⟩dm = ∫_{ΩG} θ^{♯}udm = − ∫_{ΩG}(div θ^{♯})udm

which implies d^{∗}θ = − div θ^{♯}. Of course, we have to assume that θ^{♯} ∈ H_{0}. Notice that θ^{♯} ∈ H_{0} if and only if θ ∈ T^{∗}(H_{0}^{∗}) and in this case |θ^{♯}|_{H_{0}} = |T^{∗−1}θ|_{H_{0}^{∗}}.

It is now easy to see that φ = uθ ∈ Dom(d^{∗}) if u ∈ FC_{b}^{∞}(ΩG) and θ ∈ Dom(T^{∗−1}). Here and after, we identify an element of H_{1}^{∗} with a constant 1-form (i.e., a left invariant 1-form) for notational simplicity. Hence Dom(d^{∗}) is dense and thereby E is closable. It is easy to see that the closure of E is a Dirichlet form.

The closability of the exterior differentiation is valid for general p-forms. Of course, we have replaced H_{0} by H_{1} in the argument in the section 2. Hence p-form is a function on ΩG taking values in ⋀^{p} = H_{1}^{∗} ∧ ⋯ ∧ H_{1}^{∗}. Now we notice that T^{∗} can be naturally extended to the tensor product of H_{1}^{∗} as follows:

Γ_{k}(T^{∗})(θ_{1} ⊗ ⋯ ⊗ θ_{k}) = (T^{∗}θ_{1}) ⊗ ⋯ ⊗ (T^{∗}θ_{k}).

We also have

(θ_{1} ⊗ ⋯ ⊗ θ_{k}, η_{1} ⊗ ⋯ ⊗ η_{k})_{H_{1}^{∗⊗k}}
= (Γ_{k}(T^{∗−1})θ_{1} ⊗ ⋯ ⊗ θ_{k}, η_{1} ⊗ ⋯ ⊗ η_{k})_{H_{0}^{∗⊗k}}

for θ_{i} ∈ Ran(T^{∗}), η_{i} ∈ H_{1}^{∗}.

Proposition 4.1. The dual operator of d in ⋀^{k} is given by

d^{∗} = Γ_{k−1}(T^{∗})d′Γ_{k}(T^{∗−1}).

Proof. It is easy to see that

∫_{ΩG} \frac{1}{k!}(ω, dφ)_{H_{1}^{∗⊗k}}dm = ∫_{ΩG} \frac{1}{k!}(Γ_{k}(T^{∗−1})ω, dφ)_{H_{0}^{∗⊗k}}dm
= ∫_{ΩG} \frac{1}{(k − 1)!}(d′Γ_{k}(T^{∗−1})ω, φ)_{H_{0}^{∗⊗k}}dm
= ∫_{ΩG} \frac{1}{(k − 1)!}(Γ_{k−1}(T^{∗})d′Γ_{k}(T^{∗−1})ω, φ)_{H_{1}^{∗⊗k}}dm.

This implies the assertion. □

The closability of ∂, ¯∂ can be shown in a similar manner.

Lastly we consider the covariant differentiation. We show the closability of covariant differentiation for 1-forms. Let ω be a 1-form, i.e., H_{1}^{∗}-valued function on ΩG. Recall that the cotangent bundle T^{∗}(ΩG) is identified with ΩG × H_{1}^{∗}. Hence a section of T^{∗}(ΩG) is nothing but a H_{1}^{∗}-valued function. We define the covariant derivative of 1-form ω by

⟨∇_{X}ω, Y ⟩ = X⟨ω, Y ⟩ − ⟨ω, ∇_{X}Y ⟩.

Setting ∇ω(X, Y) = ⟨∇_{X}ω, Y ⟩, ∇ is a differential operator from Γ (T_{1}^{0}(ΩG)) to Γ (T_{2}^{0}(ΩG)). Let (e_{α}, eᾱ) be a basis in H_{0} defined by (2.6). If θ ∈ H_{1}^{∗(1,0)}, then, by Theorem 3.1, we have ∇θ(·, eβ̄) = 0. Further, we have

∇θ(e_{α}, e_{β}) = −θ(A(e_{α}, e_{β}))
∇θ(eᾱ, e_{β}) = −θ([eᾱ, e_{β}]).

Similar formula holds for θ ∈ H_{1}^{∗(0,1)} (just take a complex conjugate).

We will obtain the dual operator of ∇ acting on 1-forms. To do this, we recall the definition of the divergence operator, which is essentially same as the dual operator of d. Now, taking 1-forms θ, η and ξ ∈ H_{1}^{∗} (constant 1-forms), we have

∫_{ΩG}(θ ⊗ η, ∇ξ)_{H_{1}^{∗}⊗H_{1}^{∗}}dm = ∫_{ΩG}(η, ∇_{θ}_{♯}ξ)_{H_{1}^{∗}}dm
= ∫_{ΩG}{θ^{♯}(η, ξ)_{H_{1}^{∗}} − (∇_{θ}_{♯}η, ξ)_{H_{1}^{∗}}}dm
= ∫_{ΩG}(−(div θ^{♯})η − ∇_{θ}_{♯}η, ξ)_{H_{1}^{∗}}dm.

Thus we have

∇^{∗}(θ ⊗ η) = −(div θ^{♯})η − ∇_{θ}_{♯}η.

Now we can see that the domain of ∇^{∗} is dense.

5. The Ricci curvature

We have obtained the Riemannian curvature. In this section we will get the Ricci curvature. We adopt the following definition of Ricci curvature.

dd^{∗} + d^{∗}d = ∇^{∗}∇ + Ric  (5.1)

To compute the Ricci curvature, we recall the following fact. For h ∈ H_{0}, set

u = ∫_{0}^{1}(ḣ(t), db(t))_{𝔤}.

Then we have (see, e.g., Gross [5, Lemma 3.5])

⟨du, k⟩ = ∫_{0}^{1}(ḣ(t), k̇(t))_{𝔤}dt + ∫_{0}^{1}([ḣ(t), k(t)], db(t))_{𝔤}
= (T^{−1}h, k)_{H_{1}} + ∫_{0}^{1}([ḣ(t), k(t)], db(t))_{𝔤}.  (5.2)

We prepare a proposition for later use.

Proposition 5.1. Set Define Q and Q̃ as follows.

Q(h, k) := \sqrt{−1} ∑_{β} ∫_{0}^{1}([h, ė_{β}], π_{+}[k, T^{−1}eβ̄])_{𝔤}dt

for h ∈ H_{0}^{(0,1)} and k ∈ H_{0}^{(1,0)}, and

Q̃(h, k) = \sqrt{−1} ∑_{β} ∫_{0}^{1}([e_{β}, ḣ], A(T^{−1}eβ̄, k))_{𝔤}dt

for h, k ∈ H_{0}^{(0,1)}. Then it holds that

Q(h, k) = \frac{1}{2π} ∫_{0}^{1} K(Jh, k̇)dt, for h ∈ H_{0}^{(0,1)} and k ∈ H_{0}^{(1,0)}, (5.3)
Q̃(h, k) = 0, for h, k ∈ H_{0}^{(0,1)}  (5.4)

where K is the Killing form: K(ξ, η) = tr_{N(S)}(adξadη).

Proof. We compute Q by using a basis (e_{α}, eᾱ).

Q(e_{−m,j}, e_{l,k})

= \sqrt{−1} ∑_{i=1}^{d} ∑_{n} ∫_{0}^{1}([e_{−m,j}, , ė_{n,i}], π_{+}[e_{l,k}, T^{−1}e_{−n,i}])_{𝔤}dt
= \sqrt{−1} ∑_{i=1}^{d} ∑_{n} ∫_{0}^{1}(e_{−m}ė_{n}[ξ_{j}, ξ_{i}], (2πn)π_{+}\{e_{l}e_{−n}[ξ_{k}, ξ_{i}]\})_{𝔤}dt
= \sqrt{−1} ∑_{i=1}^{d} ∑_{n} ∫_{0}^{1}(\frac{1}{−2πm\sqrt{−1}}(e^{−2π\sqrt{−1}mt} − 1)e^{2π\sqrt{−1}nt}[ξ_{j}, ξ_{i}],

× (2πn)\frac{1}{(2π\sqrt{−1}l)(−2π\sqrt{−1}n)}

× π_{+}\{(e^{2π\sqrt{−1}(l−n)t}−1) − (e^{2π\sqrt{−1}l}−1) − (e^{−2π\sqrt{−1}n}−1)\}[ξ_{k}, ξ_{i}])_{𝔤}dt

= ∑_{n} \frac{1}{4π^{2}ml} ∫_{0}^{1}(e^{−2π\sqrt{−1}mt} − 1)e^{2π\sqrt{−1}nt}

× \{1_{\{l>n\}}(e^{2π\sqrt{−1}(l−n)t} − 1) − (e^{2π\sqrt{−1}l} − 1)\}K(ξ_{j}, ξ_{k})dt

= ∑_{n} \frac{1}{4π^{2}ml} ∫_{0}^{1}\{1_{\{l>n\}}(e^{2π\sqrt{−1}(l−m)t} − e^{2π\sqrt{−1}(n−m)t})

− (e^{2π\sqrt{−1}(n−m+l)} − e^{2π\sqrt{−1}(n−m)t})\}K(ξ_{j}, ξ_{k})dt

= \{\frac{1}{4π^{2}ml} (l − 1)δ_{l,m} − \frac{1}{4π^{2}ml} 1_{\{l>m\}}

− \frac{1}{4π^{2}ml} 1_{\{m>l\}} + \frac{1}{4π^{2}ml} \}K(ξ_{i}, ξ_{j})

= \{\frac{1}{4π^{2}ml} (l − 1)δ_{l,m} + \frac{1}{4π^{2}ml} δ_{l,m}\}K(ξ_{i}, ξ_{j})
= \frac{1}{4π^{2}m}δ_{l,m}K(ξ_{i}, ξ_{j}).

On the other hand,

\frac{1}{2π} ∫_{0}^{1} K(Je_{−m,j}, ė_{l,k})dt

= \frac{−\sqrt{−1}}{2π} ∫_{0}^{1} \frac{1}{−2π\sqrt{−1}m}(e^{−2π\sqrt{−1}mt} − 1)e^{2π\sqrt{−1}lt}K(ξ_{j}, ξ_{k})dt
= \frac{1}{4π^{2}m}δ_{l,m}K(ξ_{i}, ξ_{j}).

Now we have (5.3). Next we show (5.4) Q̃(e_{−m,j}, e_{−l,k})

= \sqrt{−1} ∑_{n,i} ∫_{0}^{1}([e_{n}(t)ξ_{i}, ė_{−m}(t)ξ_{j}], ∫_{0}^{t} 2πn[e_{−n}(s)ξ_{i}, ė_{−l}(s)ξ_{k}]ds)_{𝔤}dt

= \sqrt{−1} ∑_{n,i} ∫_{0}^{1}(\frac{1}{2π\sqrt{−1}n}(e^{2π\sqrt{−1}nt} − 1)e^{−2π\sqrt{−1}mt}[ξ_{i}, ξ_{j}],

\{∫_{0}^{t}(2πn)\frac{1}{−2π\sqrt{−1}n}(e^{−2π\sqrt{−1}ns} − 1)e^{−2π\sqrt{−1}ls}[ξ_{i}, ξ_{k}]ds\})_{𝔤}dt

= −\sqrt{−1} ∑_{n} K(ξ_{j}, ξ_{k}) ∫_{0}^{1} \frac{1}{2πn}(e^{2π\sqrt{−1}nt} − 1)e^{−2π\sqrt{−1}mt}

× \{\frac{−1}{2π\sqrt{−1}(n + l)}(e^{−2π\sqrt{−1}(n+l)t} − 1) + \frac{1}{2π\sqrt{−1}l} (e^{−2π\sqrt{−1}lt} − 1)\}dt

= −\sqrt{−1} ∑_{n} K(ξ_{j}, ξ_{k})

× ∫_{0}^{1}\{\frac{−1}{(2πn)(2π\sqrt{−1}(n + l)}(e^{2π\sqrt{−1}(n−m−n−l)t} − e^{2π\sqrt{−1}(n−m)t})

+ \frac{1}{(2πn)(2π\sqrt{−1}l)}(e^{2π\sqrt{−1}(n−m−l)t} − e^{2π\sqrt{−1}(n−m)t})\}dt

= ∑_{n} K(ξ_{j}, ξ_{k})\{\frac{−δ_{n,m}}{4π^{2}n(n + l)} − \frac{δ_{n,m+l}}{4π^{2}nl} + \frac{δ_{n,m}}{4π^{2}nl} \}
= K(ξ_{j}, ξ_{k})\{\frac{−1}{4π^{2}m(m + l)} − \frac{1}{4π^{2}(m + l)l} + \frac{1}{4π^{2}ml} \}
= 0.

This completes the proof. □

The Ricci curvature is given by the following theorem. Theorem 5.1. Set

Ric(h, k) = (T^{−1}h, k)_{H_{1}} + \frac{1}{2π} ∫_{0}^{1} K(Jh, k̇)dt + (∫_{0}^{1}[ḣ(u), k(u)]du, b(1))_{𝔤}
+ ∫_{0}^{1}(ρ_{+}[ḣ, k], db(t))_{𝔤} − ∫_{0}^{1}(ρ_{−}[h, k̇], db(t))_{𝔤}  (5.5)

for h ∈ H_{0}^{(0,1)} and k ∈ H_{0}^{(1,0)}, and

Ric(h, k) = 0  (5.6)

for h, k ∈ H_{0}^{(0,1)}. If h ∈ H_{0}^{(1,0)}, we define Ric by complex conjugation. Then identity (5.1) holds. Here ρ_{±} in (5.5) is defined by

ρ_{+}l = ∑_{β}{∫_{0}^{1} l(t)ėβ̄(s)ds}ė_{β}  (5.7)
ρ_{−}l = ∑_{β}{∫_{0}^{1} l(t)ė_{β}(s)ds}ėβ̄.  (5.8)

Proof. We first note that dη(X, Y) = ∇η(X, Y) − ∇η(Y, X). So we set ˇ∇η(X, Y) = ∇η(Y, X). Since d^{∗} = ∇^{∗}, we have

dd^{∗}η + d^{∗}dη − ∇^{∗}∇η = dd^{∗}η + ∇^{∗}(∇η − ˇ∇η) − ∇^{∗}∇η
= dd^{∗}η − ∇^{∗} ˇ∇η.

Set η^{♯} = h, u = ∫_{0}^{1}(ḣ(s), db(s))_{𝔤}. Note that h ∈ H_{0}^{(0,1)} since η ∈ H_{1}^{∗(1,0)} (we have assumed h ∈ H_{0}^{ℂ}). As we saw in the previous section, d^{∗}η is given by

d^{∗}η = − div η^{♯} = u

and hence

dd^{∗}η = du.

Let (θ^{α}, θᾱ) be a dual basis of (e_{α}, eᾱ), i.e., ⟨θ^{α}, e_{β}⟩ = δ_{β}^{α}, ⟨θ^{α}, eβ̄⟩ = 0, etc. By noting ∇η(·, eβ̄) = 0, we can expand ∇η as

∇η = ∑_{α,β}{∇η(e_{α}, e_{β})θ^{α} ⊗ θ^{β} + ∇η(eᾱ, e_{β})θᾱ ⊗ θ^{β}}.

Reversing the order, we have

ˇ∇η = ∑_{α,β}{−⟨η, A(e_{α}, e_{β})⟩θ^{β} ⊗ θ^{α} − ⟨η, π_{+}[eᾱ, e_{β}]⟩θ^{β} ⊗ θᾱ}.

Since (θ^{α})^{♯} = T^{−1}eᾱ, we have

∇^{∗}(θ^{β} ⊗ θ^{α}) = −(div T^{−1}eβ̄)θ^{α} − ∇_{T^{−1}eβ̄}θ^{α}.

Therefore

∇^{∗} ˇ∇η = ∑_{α,β}{(div T^{−1}eβ̄)θ^{α} + ∇_{T^{−1}eβ̄}θ^{α}}⟨η, A(e_{α}, e_{β})⟩

+ ∑_{α,β}{(div T^{−1}eβ̄)θᾱ + ∇_{T^{−1}eβ̄}θᾱ}⟨η, π_{+}[eᾱ, e_{β}]⟩.

Hence, if k ∈ H_{0}^{(1,0)},

⟨dd^{∗}η − ∇^{∗} ˇ∇η, k⟩

= ⟨du − ∇^{∗} ˇ∇η, k⟩
= (h, k)_{H_{0}} + ∫_{0}^{1}([ḣ, k], db(t))_{𝔤} − ∑_{α,β}{(div T^{−1}eβ̄)⟨θ^{α}, k⟩⟨η, A(e_{α}, e_{β})⟩

− ∑_{α,β}⟨∇_{T^{−1}eβ̄}θ^{α}, k⟩⟨η, A(e_{α}, e_{β})⟩}

= (T^{−1}(η^{♯}), k)_{H_{1}} + ∫_{0}^{1}([ḣ, k], db(t))_{𝔤} − ∑_{β}(div T^{−1}eβ̄)⟨η, A(k, e_{β})⟩

+ ∑_{α,β}⟨θ^{α}, π_{+}[T^{−1}eβ̄, k]⟩⟨η, A(e_{α}, e_{β})⟩

= ⟨T^{∗−1}η, k⟩ + ∫_{0}^{1}([ḣ, k], db(t))_{𝔤}

− ∑_{β}(div T^{−1}eβ̄)⟨η, A(k, e_{β})⟩ + ∑_{β}⟨η, A(π_{+}[T^{−1}eβ̄, k], e_{β})⟩}.

We first compute the last term.

⟨η, A(π_{+}[T^{−1}eβ̄, k], e_{β})⟩ = (h, A(π_{+}[T^{−1}eβ̄, k], e_{β}))_{H_{1}}
= \sqrt{−1}(Jh, A(π_{+}[T^{−1}eβ̄, k], e_{β}))_{H_{1}}
= \sqrt{−1}S(h, A(π_{+}[T^{−1}eβ̄, k], e_{β}))
= −\sqrt{−1} ∫_{0}^{1}(h, \frac{d}{dt}A(π_{+}[T^{−1}eβ̄, k], e_{β}))_{𝔤}dt
= −\sqrt{−1} ∫_{0}^{1}(h, [π_{+}[T^{−1}eβ̄, k], ė_{β}])_{𝔤}dt
= \sqrt{−1} ∫_{0}^{1}([h, ė_{β}], π_{+}[k, T^{−1}eβ̄])_{𝔤}dt.

By virtue of Proposition 5.1, we can get

∑_{β}⟨η, A(π_{+}[T^{−1}eβ̄, k], e_{β})⟩ = \frac{1}{2π} ∫_{0}^{1} K(Jh, k̇)dt.

Now we turn to the stochastic integral.

(div T^{−1}eβ̄)⟨η, A(k, e_{β})⟩ = (div T^{−1}eβ̄)(h, A(k, e_{β}))_{H_{1}}
= \sqrt{−1}(div T^{−1}eβ̄)(Jh, A(k, e_{β}))_{H_{1}}
= \sqrt{−1}(div T^{−1}eβ̄)S(h, A(k, e_{β}))
= −\sqrt{−1}(div T^{−1}eβ̄) ∫_{0}^{1}(h, [k, ė_{β}])_{𝔤}dt
= −\sqrt{−1}(div T^{−1}eβ̄) ∫_{0}^{1}([h, k], ė_{β})_{𝔤}dt
= −\sqrt{−1}(div T^{−1}eβ̄)S(e_{β}, [h, k])
= −\sqrt{−1}(div T^{−1}eβ̄)(Je_{β}, [h, k])_{H_{1}}
= (div eβ̄)(T^{−1}e_{β}, [h, k])_{H_{1}}
= (div eβ̄)(e_{β}, [h, k])_{H_{0}}
= (div eβ̄) ∫_{0}^{1}([ḣ, k] + [h, k̇], ė_{β})_{𝔤}dt

Further

∫_{0}^{1}([ḣ, k], db(t))_{𝔤} − ∑_{β}(div T^{−1}eβ̄)⟨η, A(k, e_{β})⟩

= ∫_{0}^{1}(ρ_{+}[ḣ, k], db(t))_{𝔤} + ∫_{0}^{1}(ρ_{−}[ḣ, k], db(t))_{𝔤}

+ ∫_{0}^{1}(∫_{0}^{1}[ḣ(u), k(u)]du, db(t))_{𝔤} − ∫_{0}^{1}(ρ_{−}[ḣ, k] + ρ_{−}[h, k̇], db(t))_{𝔤}

= ∫_{0}^{1}(ρ_{+}[ḣ, k], db(t))_{𝔤}+(∫_{0}^{1}[ḣ(u), k(u)]du, b(1))_{𝔤}− ∫_{0}^{1}(ρ_{−}[h, k̇], db(t))_{𝔤}.

Next we consider the case k ∈ H_{1}^{(0,1)}. ⟨dd^{∗}η − ∇^{∗} ˇ∇η, k⟩

= ⟨du − ∇^{∗} ˇ∇η, k⟩
= ⟨T^{∗−1}η, k⟩ + ∫_{0}^{1}([ḣ, k], db(t))_{𝔤}

− ∑_{α,β}{(div T^{−1}eβ̄)⟨θᾱ, k⟩⟨η, π_{+}[eᾱ, e_{β}]⟩+⟨∇_{T^{−1}eβ̄}θ^{α}, k⟩η(π_{+}[eᾱ, e_{β}])}

= ∫_{0}^{1}([ḣ, k], db(t))_{𝔤}

− ∑_{β}(div T^{−1}eβ̄)⟨η, [k, e_{β}]⟩ + ∑_{β}⟨η, [A(T^{−1}eβ̄, k), e_{β}]⟩.

Further,

⟨η, [A(T^{−1}eβ̄, k), e_{β}]⟩ = (h, [A(T^{−1}eβ̄, k), e_{β}])_{H_{1}}
= \sqrt{−1}(Jh, [A(T^{−1}eβ̄, k), e_{β}])_{H_{1}}
= \sqrt{−1}S(h, [A(T^{−1}eβ̄, k), e_{β}])
= \sqrt{−1} ∫_{0}^{1}(ḣ, [A(T^{−1}eβ̄, k), e_{β}])_{𝔤}dt
= \sqrt{−1} ∫_{0}^{1}([e_{β}, ḣ], A(T^{−1}eβ̄, k))_{𝔤}dt.

The stochastic integral vanishes because, for β = (n, i),

(div T^{−1}eβ̄)⟨η, [k, e_{β}]⟩

= (div T^{−1}eβ̄)(h, [k, e_{β}])_{H_{1}}
= \sqrt{−1}(div T^{−1}eβ̄)(Jh, [k, e_{β}])_{H_{1}}
= \sqrt{−1}(div T^{−1}eβ̄)S(h, [k, e_{β}])
= \sqrt{−1}(div T^{−1}eβ̄) ∫_{0}^{1}(ḣ, [k, e_{β}])_{𝔤}dt

= \sqrt{−1}(div T^{−1}eβ̄) ∫_{0}^{1}([ḣ, k], e_{β}])_{𝔤}dt

= \sqrt{−1}(div eβ̄)(2πn) ∫_{0}^{1}([ḣ, k], \frac{1}{2π\sqrt{−1}n}(e^{2π\sqrt{−1}nt} − 1)ξ_{j})_{𝔤}dt

= (div eβ̄) ∫_{0}^{1}([ḣ, k], e^{2π\sqrt{−1}nt}ξ_{j})_{𝔤}dt

= (div eβ̄) ∫_{0}^{1}([ḣ, k], ė_{β})_{𝔤}dt.

Combining this with ∫_{0}^{1}([ḣ, k], ėβ̄)_{𝔤}dt = 0, we have

∑_{β}(div T^{−1}eβ̄)⟨η, [k, e_{β}]⟩ = ∫_{0}^{1}([ḣ, k], db(t))_{𝔤}.

This completes the proof. □

References

1. S. Aida, Sobolev Spaces over Loop Groups, J. Funct. Anal., 127 (1995), 155– 172.

2. D. S. Freed, The geometry of loop groups, J. Diff. Geom., 28 (1988), 223–276.

3. E. Getzler, Dirichlet forms on loop space, Bull. Sci. Math., 113 (1989), 151–174.

4. L. Gross, Logarithmic Sobolev inequalities on loop groups, J. Funct. Anal., 102 (1991), 268–313.

5. L. Gross, Uniqueness of ground states for Schrödinger operators over loop groups, J. Funct. Anal., 112 (1993), 373–441.

6. T. Kazumi and I. Shigekawa, Differential calculus on a submanifold of an abstract Wiener space, I. Covariant derivative, in “Stochastic analysis on infinite dimensional spaces,” ed. by H. Kunita and H.-H. Kuo, pp. 117–140, Longman, Harlow, 1994.

7. T. Kazumi and I. Shigekawa, Differential calculus on a submanifold of an abstract Wiener space, II. Weitzenböck formula, in “Dirichlet forms and stochastic processes,” Proceedings of the International Conference held in Beijing, China, October 25-31, 1993, ed. by Z. M. Ma, M. Röckner and J. A. Yan, pp. 235–251, Walter de Gruyter, Berlin-New York, 1995.

8. S. Kobayashi and K. Nomizu, “Foundations of differential geometry,” I, II, Interscience Publishers, New York-London, 1963, 1969.

9. S. Kusuoka and S. Taniguchi, Pseudoconvex domains in almost complex abstract Wiener spaces, J. Funct. Anal., 117 (1993), 62–117.

10. M.-P. Malliavin and P. Malliavin, Integration on loop groups. I. Quasi Invariant measures, J. Funct. Anal., 93 (1990), 207–237.

11. A. N. Pressley, The energy flow on the loop space of a compact Lie group, J. London Math. Soc. (2) 26 (1982), 557–566.

12. A. Pressley and G. Segal, “Loop groups,” Oxford University Press, New York, 1986.

13. I. Shigekawa, Differential calculus on a based loop group, preprint.