j. differential geometry 65 (2003) 1-17

HOLOMORPHIC LINE BUNDLES ON THE LOOP SPACE OF THE RIEMANN SPHERE

NING ZHANG

Abstract

The loop space Lℙ_{1} of the Riemann sphere consisting of all C^{k} or Sobolev W^{k,p} maps S^{1} → ℙ_{1} is an infinite dimensional complex manifold. The loop group LP GL(2, ℂ) acts on Lℙ_{1}. We prove that the group of LP GL(2, ℂ)- invariant holomorphic line bundles on Lℙ_{1} is isomorphic to an infinite dimensional Lie group. Further, we prove that the space of holomorphic sections of any such line bundle is finite dimensional, and compute the dimension for a generic bundle.

1. Introduction

Let M be a finite dimensional complex manifold. Its loop space LM with a specified regularity, for example C^{k} (1 ≤ k ≤ ∞) or W^{k,p} (1 ≤ k < ∞, 1 ≤ p < ∞), consists of all maps of the circle S^{1} into M with the given regularity. LM is an infinite dimensional complex manifold. This paper studies holomorphic line bundles on the loop space Lℙ_{1} of the Riemann sphere.

A direct motivation comes from [9], where Millson and Zombro conjecture that there exists a P GL(2, ℂ)-equivariant embedding of Lℙ_{1} into a projectivized Banach/Fréchet space. The conjecture arises in connection with extending Mumford’s geometric invariant theory to an infinite dimensional setting. Another indirect motivation comes from [11], where Witten suggests to study the geometry and analysis of real and complex manifolds through their loop spaces. In finite dimensions it is a problem of fundamental importance to identify the Picard group of holomorphic line bundles on a complex manifold and the space of holomorphic sections of these bundles. Here we address this problem for a class of holomorphic line bundles on the first interesting loop space: Lℙ_{1}, and in particular make some progress toward answering the conjecture by Millson and Zombro.

This research was partially supported by NSF grant DMS-0203072.

Received 07/16/2002.

The following are the main results of this paper:

If a group G acts on a set V, let V^{G} denote the G-fixed subset of V. The loop space of a finite dimensional complex Lie group is a complex Lie group under pointwise group operation (loop group). Let Pic(Lℙ_{1}) be the Picard group of Lℙ_{1}. The group P GL(2, ℂ) acts on ℙ_{1}, so the loop group LP GL(2, ℂ) acts on Lℙ_{1} and on Pic(Lℙ_{1}). Let Lℂ^{∗} be the loop group of ℂ^{∗} = ℂ \ {0}, and Hom(Lℂ^{∗}, ℂ^{∗}) be the group of holomorphic homomorphisms from Lℂ^{∗} to ℂ^{∗}.

Theorem 1.1. Pic(Lℙ_{1})^{LP GL(2,ℂ)} ≅ Hom(Lℂ^{∗}, ℂ^{∗}) as groups.

Note that Hom(Lℂ^{∗}, ℂ^{∗}) is a ℤ-module of infinite rank, while the group of topological isomorphism classes of line bundles on Lℙ_{1} is isomorphic to ℤ (cf. [8]).

Evaluation of loops in ℙ_{1} at t ∈ S^{1} gives rise to a holomorphic map E_{t} : Lℙ_{1} → ℙ_{1}. Let Λ_{t} ∈ Pic(Lℙ_{1}) be the pull back of the hyperplane bundle on ℙ_{1} by E_{t}.

Theorem 1.2. Let Λ ∈ Pic(Lℙ_{1})^{LP GL(2,ℂ)}.

(1) If Λ ≅ Λ_{t_{1}}^{n_{1}} ⊗ ⋯ ⊗ Λ_{t_{r}}^{n_{r}}, where n_{i} ≥ 0 and t_{i} ≠ t_{j} for i ≠ j, then (n_{1} + 1) ⋯ (n_{r} + 1) ≤ dim H^{0}(Lℙ_{1}, Λ) < ∞.

(2) Otherwise H^{0}(Lℙ_{1}, Λ) = 0.

Therefore the sections of no LP GL(2, ℂ)-invariant holomorphic line bundle will give rise to a projective embedding of Lℙ_{1}.

The isomorphism in Theorem 1.1 is gotten by an explicit construction in Section 2. In Section 3 we prove Theorem 1.2(2). In Section 4 we study the space of holomorphic sections or the zero order Dolbeault cohomology group of line bundles defined in Theorem 1.2(1), and in particular prove Theorem 1.2(1). Any such line bundle obviously has holomorphic sections: products of pulled back sections by the evaluation maps. We will show that for a generic bundle of this type these are all sections. Yet there are bundles which have other sections as well; interestingly, in this case dim H^{0}(Lℙ_{1}, Λ_{φ}) depends on the regularity of the loops.

It is natural to ask whether Pic(Lℙ_{1})^{P GL(2,ℂ)} = Pic(Lℙ_{1})^{LP GL(2,ℂ)}. If so, then the conjecture made by Millson and Zombro is answered in the negative.

The author would like to thank Professor László Lempert for many helpful conversations and his continuous encouragement. Thanks are also due to Professor John J. Millson and Professor James McClure for valuable communications on some issues related to this paper.

2. Identification of Pic(Lℙ_{1})^{LP GL(2,ℂ)}

We fix a regularity class F among C^{k} (1 ≤ k ≤ ∞) respectively W^{k,p} (1 ≤ k < ∞, 1 ≤ p < ∞). In this paper we write LM (L_{k}M resp. L_{k,p}M) to denote the F (C^{k} resp. W^{k,p}) loop space of a manifold M. Let M and N be finite dimensional complex manifolds and ϕ : M → N be a holomorphic map. Define Lϕ : LM ∋ x ↦ ϕ ∘ x ∈ LN. Then LM and LN are infinite dimensional complex manifolds locally biholomorphic to open subsets of complex Banach (Fréchet when F = C^{∞}) spaces, and Lϕ is holomorphic. Thus L is a functor from the category of finite dimensional complex manifolds to the category of all complex manifolds. Let t ∈ S^{1}. The evaluation map E_{t} = E_{t}^{LM} : LM ∋ x ↦ x(t) ∈ M is holomorphic. See Section 2 of [6].

We call constant maps S^{1} → M point loops in M. They form a submanifold of LM, which we identify with M.

Next we define a map L : Hom(Lℂ^{∗}, ℂ^{∗}) → Pic(Lℙ_{1})^{LP GL(2,ℂ)}. We will show that L is an isomorphism of groups, which will then prove Theorem 1.1.

In Section 6 of [9] Millson and Zombro construct a holomorphic line bundle on Lℙ_{1}, and a similar idea in fact yields a map from Hom(Lℂ^{∗}, ℂ^{∗}) to Pic(Lℙ_{1}) as follows. Let p : Q → ℙ_{1} be the principal ℂ^{∗}-bundle associated with the hyperplane bundle H → ℙ_{1}. Applying the loop functor we obtain a principal Lℂ^{∗}-bundle Lp : LQ → Lℙ_{1}. Now a homomorphism φ : Lℂ^{∗} → ℂ^{∗} determines a representation of Lℂ^{∗} on ℂ. Recall that, in general, with a principal G-bundle P → B and a representation ρ of G on a vector space V, one can functorially associate a vector bundle E → B with typical fiber V (see Section 12.5 of [4]). If h_{ab} are the G-valued transition functions of P with respect to some trivialization, the corresponding transition functions of E will be ρ(h_{ab}). Accordingly we associate with Lp and φ a line bundle Λ_{φ}. Define the map

L : Hom(Lℂ^{∗}, ℂ^{∗}) → Pic(Lℙ_{1}), φ ↦ Λ_{φ}.

Note that the P GL(2, ℂ) action on ℙ_{1} can be covered by a GL(2, ℂ) action on Q. One way to see this is to pass to the tautological ℂ^{∗}-bundle Q^{−1}, whose total space is ℂ^{2}\{0}, on which the GL(2, ℂ) action is standard.

The GL(2, ℂ) action on Q gives rise to an LGL(2, ℂ) action on LQ. Since LGL(2, ℂ) → LP GL(2, ℂ) is surjective (as follows from the exact homotopy sequence associated with the fibration ℂ^{∗} → GL(2, ℂ) → P GL(2, ℂ) and the lifting of homotopies), this LGL(2, ℂ) action will in fact cover the LP GL(2, ℂ) action on Lℙ_{1}. In particular, γ^{∗}LQ ≅ LQ for γ ∈ LP GL(2, ℂ). Hence γ^{∗}Λ_{φ} ≅ Λ_{φ} and we have proved the following:

Proposition 2.1. The range of L is in Pic(Lℙ_{1})^{LP GL(2,ℂ)}.

Let 𝔘 = {U_{a} = ℙ_{1} \ {a} : a ∈ ℙ_{1}}. Then

L𝔘 = {LU_{a} : a ∈ ℙ_{1}} (2.1)

is an open covering of Lℙ_{1}. Now we introduce a way to construct Čech cohomology classes in H^{1}(L𝔘, O^{∗}). Let O^{G} denote the sheaf of holomorphic maps to the complex Lie group G from a complex manifold. So O^{ℂ^{∗}} = O^{∗}. If c = (c_{ab}) is a Čech 1-cocycle of 𝔘 with values in the sheaf O^{∗} and [c] its cohomology class, then Lc = (Lc_{ab}) is a Čech 1-cocycle of L𝔘 with values in O^{Lℂ^{∗}}. Any φ ∈ Hom(Lℂ^{∗}, ℂ^{∗}) induces a sheaf homomorphism O^{Lℂ^{∗}} → O^{∗}. Since the cohomology class of φ ∘ Lc depends only on [c], we obtain a map

H^{1}(𝔘, O^{∗}) × Hom(Lℂ^{∗}, ℂ^{∗}) → H^{1}(L𝔘, O^{∗}), ([c], φ) ↦ [φ ∘ Lc], (2.2)

a group homomorphism in both variables.

Fix [c] in (2.2) to be the class [c_{H}] of the hyperplane bundle H → ℙ_{1}, where

c_{H} = {g_{ab} ∈ O^{∗}(U_{a} ∩ U_{b}) : a, b ∈ ℙ_{1}, a ≠ b},

g_{ab} = \begin{cases}\frac{z−b}{z−a} & a, b ≠ ∞ \\ z − b & a = ∞ \\ \frac{1}{z−a} & b = ∞ \end{cases} , z ∈ U_{a} ∩ U_{b}. (2.3)

Then we obtain a homomorphism Hom(Lℂ^{∗}, ℂ^{∗}) → H^{1}(L𝔘, O^{∗}), φ ↦ [c_{φ}], where

c_{φ} = {φ ∘ Lg_{ab} ∈ O^{∗}(LU_{a} ∩ LU_{b}) : a, b ∈ ℙ_{1}, a ≠ b}, (2.4)

which has the same range as the map in (2.2).

Proposition 2.2. The line bundle associated to [c_{φ}] is Λ_{φ}. In particular L is a homomorphism.

Proof. From the definitions it follows that Lc_{H} = (Lg_{ab}) is a family of transition functions of Lp, hence c_{φ} in (2.4) is a family of transition functions of Λ_{φ}. q.e.d.

Let t ∈ S^{1} and Λ_{t} ∈ Pic(Lℙ_{1}) be the pull back of the hyperplane bundle on ℙ_{1} by the evaluation map E_{t}^{Lℙ_{1}}. Obviously E_{t}^{Lℂ^{∗}} ∈ Hom(Lℂ^{∗}, ℂ^{∗}), and by Proposition 2.2 L(E_{t}^{Lℂ^{∗}}) = Λ_{t}.

To prove Theorem 1.1 we need the following preparations:

Identify LU_{∞} with Lℂ and let y ∈ Lℂ^{∗}. Define the map Φ_{y} : Lℂ × ℙ_{1} ∋ (x, λ) ↦ x+λy ∈ Lℙ_{1}, where by Φ_{y}(x, ∞) we mean the point loop ∞ ∈ Lℙ_{1}. Since a continuous map h : Ω_{1} → Ω_{2} between open subsets of Fréchet spaces is holomorphic if and only if its restriction to the intersection of Ω_{1} with any affine line is holomorphic, see Sections 2.3, 3.1 of [2], one can easily check that Φ_{y} is holomorphic. Clearly Φ_{y}(x, ·) : ℙ_{1} → Lℙ_{1} is a holomorphic embedding, whose image we will denote by v(x, y). The existence of subvarieties v(x, y) immediately implies that any holomorphic function on Lℙ_{1} is constant.

Let Λ ∈ Pic(Lℙ_{1}). If Λ|_{ℙ_{1}} ≅ H^{n}, then we say that Λ is of order n, or ord(Λ) = n; and we claim that Λ|_{v(x,y)} ≅ H^{n}, x ∈ Lℂ, y ∈ Lℂ^{∗}. This would imply that the only holomorphic section of Λ is the zero section if ord(Λ) < 0. To show the claim, let c_{1} denote the rational first Chern class of a line bundle. According to the first theorem of [10], the inclusion ℙ_{1} → Lℙ_{1} induces an isomorphism H^{2}(Lℙ_{1}, ℚ) ≅ H^{2}(ℙ_{1}, ℚ), so that c_{1}(Λ) is completely determined by c_{1}(Λ|_{ℙ_{1}}) = n. Hence c_{1}(Λ|_{v(x,y)}) , and therefore the degree of Λ|_{v(x,y)}, is also determined by n. That this degree is itself n then follows from computing it in the special case Λ = Λ_{t}^{n}.

Let φ ∈ Hom(Lℂ^{∗}, ℂ^{∗}). The restriction of φ to the subgroup of point loops must be of the type z ↦ z^{n}, z ∈ ℂ^{∗}, where n ∈ ℤ. We call this n the order of φ and denote it by ord(φ). Proposition 2.2 implies that ord(Λ_{φ}) = ord(φ). Let F_{x} be the fiber of Λ at x ∈ Lℙ_{1}.

Proposition 2.3. Let Λ ∈ Pic(Lℙ_{1}), ord(Λ) = 0 and y ∈ Lℂ^{∗}. For any ζ ∈ F_{∞} \{0} there exists a unique non-vanishing section σ = σ_{y,ζ} ∈ H^{0}(LU_{∞}, Λ) such that lim_{λ→∞} σ(x + λy) = ζ, x ∈ LU_{∞}. In particular, Λ|_{LU_{∞}} is holomorphically trivial.

Proof. Since Λ|_{v(x,y)} is trivial, uniqueness is obvious. As to existence, let H be a hyperplane in Lℂ such that y ∉ H and consider the line bundle Λ̃ = Φ_{y}^{∗}Λ|_{H×ℙ_{1}}. Since Φ_{y} ≡ ∞ on H×{∞}, s = Φ_{y}^{∗}ζ is a non-vanishing holomorphic section of Λ̃|_{H×{∞}}. In turn s determines a section σ̃ of Λ̃|_{H×ℙ_{1}} such that σ̃|_{{x}×ℙ_{1}} is the unique non-vanishing holomorphic section of Λ̃|_{{x}×ℙ_{1}} with σ̃(x, ∞) = s(x, ∞), for all x ∈ H. Next we show that σ̃ is holomorphic. Let x_{0} ∈ H. By Proposition 5.1 of [5] there exist a neighborhood x_{0} ∈ U ⊂ H and a section v ∈ C^{∞}(U × ℙ_{1}, Λ̃) such that v is holomorphic on {x} × ℙ_{1} for all x ∈ U, and v|_{{x_{0}}×ℙ_{1}} = σ̃|_{{x_{0}}×ℙ_{1}} ≠ 0. By choosing a sufficiently small U we can assume v ≠ 0. The function σ̃/v is C^{∞} on U × {∞} and is constant on {x} × ℙ_{1}, x ∈ U, hence is C^{∞} on U × ℙ_{1}. So it follows that σ̃ ∈ C^{∞}(H × ℙ_{1}). Since ∂σ̃|_{H×{∞}} = 0 and ∂σ̃|_{{x}×ℙ_{1}} = 0 for all x ∈ H, by Proposition 5.2(ii) of [5] we obtain that indeed ∂σ̃ = 0. Then the desired σ is the pull back of σ̃ by (Φ_{y}|_{H×ℂ})^{−1}. q.e.d.

Since H|_{U_{∞}} is trivial, so is Λ_{t}|_{LU_{∞}}. In general, let Λ ∈ Pic(Lℙ_{1}), ord(Λ) = n. As

(2.5) Λ = Λ_{t}^{n} ⊗ (Λ_{t}^{−n} ⊗ Λ), where ord(Λ_{t}^{−n} ⊗ Λ) = 0,

Proposition 2.3 implies that Λ|_{LU_{∞}} is also trivial. More generally, Λ|_{LU_{a}} is trivial, a ∈ ℙ_{1}, which means

Corollary 2.4. Pic(Lℙ_{1}) ≅ H^{1}(L𝔘, O^{∗}).

If Λ ∈ Pic(Lℙ_{1})^{LP GL(2,ℂ)}, then Proposition 2.3 can be improved: σ there is essentially independent of y, and so is a canonical section of Λ|_{LU_{∞}}.

Proposition 2.5. Let Λ ∈ Pic(Lℙ_{1})^{LP GL(2,ℂ)}, ord(Λ) ≥ 0. Then there exists a non-vanishing section σ_{∞} ∈ H^{0} (LU_{∞}, Λ) such that lim_{λ→∞} σ_{∞}(x + λy) exists for all x ∈ LU_{∞} and y ∈ Lℂ^{∗}. Such a section is unique up to a multiplicative constant.

Proof. Suppose σ_{∞} exists. For any y ∈ Lℂ^{∗}, Λ|_{v(0,y)} ≅ H^{ord(Λ)} has a unique holomorphic section which does not vanish on v(0, y)\{∞} and assumes σ_{∞}(0) at 0; clearly σ_{∞} agrees with this section on v(0, y)\{∞}. In particular, σ_{∞}(y) is uniquely determined by σ_{∞}(0), y ∈ Lℂ^{∗}. Since Lℂ^{∗} is dense in Lℂ = LU_{∞}, σ_{∞} is completely determined by its value at 0, or unique up to a multiplicative constant. Next we show the existence.

First assume ord(Λ) = 0. Let ζ ∈ F_{∞}\{0}, y_{1}, y_{2} ∈ Lℂ^{∗}, and σ_{y_{1},ζ}, σ_{y_{2},ζ} ∈ H^{0}(LU_{∞}, Λ) as in Proposition 2.3. Define

h = h_{y_{1},y_{2}} = \frac{σ_{y_{1},ζ}}{σ_{y_{2},ζ}} ∈ O^{∗}(LU_{∞}),

which is independent of the choice of ζ ∈ F_{∞} \{0}. For fixed x_{1} ∈ Lℂ let μ ∈ LP GL(2, ℂ) be the translation x ↦ x + x_{1}, x ∈ Lℙ_{1}. Clearly μ^{∗}σ_{y_{i},ζ} ∈ H^{0}(LU_{∞}, μ^{∗}Λ), i = 1, 2, is a section of the type as in Proposition 2.3. With a fixed isomorphism μ^{∗}Λ ≅ Λ, μ^{∗}σ_{y_{i},ζ} corresponds to σ_{y_{i},ζ′}, where ζ′ ∈ F_{∞}\{0}. Therefore

h(x + x_{1}) = μ^{∗}h(x) = \frac{μ^{∗}σ_{y_{1},ζ}}{μ^{∗}σ_{y_{2},ζ}} (x) = \frac{σ_{y_{1},ζ′}}{σ_{y_{2},ζ′}} (x) = h(x).

Thus h = h_{y_{1},y_{2}} is a nonzero constant. From the definition of σ_{y_{2},ζ} it follows that σ_{y_{1},ζ} = hσ_{y_{2},ζ} extends to be a section of Λ|_{v(x,y_{2})} for all x ∈ Lℂ, where y_{2} ∈ Lℂ^{∗} is arbitrary. The upshot is σ_{∞} = σ_{y_{1},ζ} with arbitrary y_{1}, ζ will do.

Second assume Λ = Λ_{t}^{n}, n ≥ 0. If s is a section of H^{n}, nonzero on ℙ_{1} \ {∞}, take σ_{∞} = (E_{t}^{Lℙ_{1}})^{∗}s. Finally, these two special cases and (2.5) imply the general case. q.e.d.

Theorem 1.1 follows from:

Theorem 2.6. L : Hom(Lℂ^{∗}, ℂ^{∗}) → Pic(Lℙ_{1})^{LP GL(2,ℂ)} is a group isomorphism.

Proof. i) Injectivity: If L(φ) = Λ_{φ} is trivial, then ord(φ) = 0; and by Proposition 2.2 and (2.4) we can find f_{a} ∈ O^{∗}(LU_{a}), a ∈ ℙ_{1}, such that f_{a} = (φ ∘ Lg_{ab})f_{b} on LU_{a} ∩ LU_{b}. In particular, f_{∞} = φf_{0} on Lℂ^{∗} ⊂ Lℙ_{1}. For any y ∈ Lℂ^{∗} we have

lim_{ℂ∋λ→∞}f_{∞}(λy) = lim_{λ→∞} φ(λy)f_{0}(λy) = φ(y)f_{0}(∞),

since φ(λy) = φ(λ)φ(y) = φ(y). Thus f_{∞}|_{v(0,y)\{∞}} extends to all of v(0, y), and so must be constant. In particular, f_{∞}(y) = f_{∞}(0) and f_{∞} itself is a constant. Similarly f_{0} is also a constant. Then φ is a constant which can only be 1.

ii) Surjectivity: Since Λ_{t}^{n} is in the range of L, by (2.5) we only need to show that any Λ ∈ Pic(Lℙ_{1})^{LP GL(2,ℂ)} with ord(Λ) = 0 is in the range of L.

Let ε(x) = 1/x, x ∈ Lℙ_{1}. The induced bundle ε^{∗}Λ is isomorphic to Λ, so Proposition 2.5 applies to produce a non-vanishing σ̃_{∞} ∈ H^{0}(LU_{∞}, ε^{∗}Λ). Then σ_{0} = σ̃_{∞} ∘ ε ∈ H^{0}(LU_{0}, Λ) is characterized, up to a multiplicative constant, by the fact that lim_{λ→∞} σ_{0} ((x + λy)^{−1}) exists, for all x ∈ LU_{∞}, y ∈ Lℂ^{∗}. To get rid of the ambiguity in the choice of the constant, fix a nonzero s ∈ H^{0}(ℙ_{1}, Λ) and choose σ_{0} and σ_{∞} (as in Proposition 2.5) to agree with s on point loops. Set ϕ = σ_{0}/σ_{∞} : Lℂ^{∗} → ℂ^{∗}. Note that on point loops ϕ = 1. We will show that ϕ is a homomorphism and Λ = Λ_{ϕ}.

For this purpose fix y_{1} ∈ Lℂ^{∗} and define γ(x) = y_{1}x, x ∈ Lℙ_{1}. It is straightforward that the non-vanishing section γ^{∗}σ_{∞} ∈ H^{0}(LU_{∞}, γ^{∗}Λ) satisfies the conditions in Proposition 2.5. Hence under an isomorphism γ^{∗}Λ ≅ Λ, γ^{∗}σ_{∞} corresponds to a constant multiple of σ_{∞}. Similarly, under this isomorphism γ^{∗}σ_{0} corresponds to a constant multiple of σ_{0}. Therefore

ϕ(y_{1}y) = (γ^{∗}ϕ) (y) = \frac{γ^{∗}σ_{0}}{γ^{∗}σ_{∞}} (y) = c\frac{σ_{0}}{σ_{∞}} (y) = cϕ(y).

Letting y = 1 we get c = ϕ(y_{1}), so ϕ is indeed a homomorphism.

Finally with a ∈ ℂ let μ_{a}(x) = x + a, x ∈ Lℙ_{1}. For the bundle Λ′ = μ_{a}^{∗}Λ one can construct corresponding sections σ′_{∞} and σ′_{0}; for the normalization, use s′ = μ_{a}^{∗}s ∈ H^{0}(ℙ_{1}, Λ′). As Λ′ ≅ Λ, σ′_{0}/σ′_{∞} = ϕ. Since (μ_{a}^{−1})^{∗}σ′_{∞} ∈ H^{0}(LU_{∞}, Λ) satisfies the conditions in Proposition 2.5 and agrees with s on point loops, (μ_{a}^{−1})^{∗}σ′_{∞} = σ_{∞}. Let σ_{a} = (μ_{a}^{−1})^{∗}σ′_{0} ∈ H^{0}(LU_{a}, Λ). Since

\frac{σ_{a}}{σ_{∞}} = (μ_{a}^{−1})^{∗} \frac{σ′_{0}}{σ′_{∞}} = ϕ ∘ μ_{a}^{−1},

it is straightforward to check that the transition functions σ_{b}/σ_{a} ∈ O^{∗}(LU_{a}∩ LU_{b}) of Λ agree with those of Λ_{ϕ} given in (2.4); therefore Λ = Λ_{ϕ} is indeed in the range of L. q.e.d.

Let (Lℂ)^{∗} be the space of continuous linear functionals on Lℂ, let φ̃ ∈ (Lℂ)^{∗} be the Lie algebra homomorphism induced by φ ∈ Hom(Lℂ^{∗}, ℂ^{∗}), and x_{0} ∈ Lℂ^{∗} be a fixed loop whose winding number with respect to 0 is 1. The reader can check that the map

Hom(Lℂ^{∗}, ℂ^{∗}) → {ϕ ∈ (Lℂ)^{∗} : ϕ(1) ∈ ℤ} × ℂ^{∗}, φ ↦ (φ̃, φ(x_{0}))

is an isomorphism of groups. Therefore Hom(Lℂ^{∗}, ℂ^{∗}) is a ℤ-module of infinite rank.

The proof of Theorem 2.6 implies the following:

Proposition 2.7. Let φ ∈ Hom(Lℂ^{∗}, ℂ^{∗}), ord(φ) ≥ 0. There is a family of non-vanishing sections {σ_{a} ∈ H^{0}(LU_{a}, Λ_{φ}) : a ∈ ℙ_{1}}, unique up to an overall multiplicative constant, that satisfies σ_{b}/σ_{a} = φ ∘ Lg_{ab} on LU_{a} ∩ LU_{b}. Furthermore, σ_{∞} satisfies the conditions in Proposition 2.5.

Proof. Uniqueness is obvious: if {σ′_{a}} is another family then {σ′_{a}/σ_{a}} defines a holomorphic function on Lℙ_{1}, which, as we have said, must be a constant. When ord(Λ) = 0, the family {σ_{a}} is constructed in the proof of Theorem 2.6. When Λ = Λ_{t}^{n}, take sections τ_{a} of the hyperplane bundle H → ℙ_{1} such that τ_{b}/τ_{a} = g_{ab} (see (2.3)), then σ_{a} = (E_{t}^{Lℙ_{1}})^{∗}τ_{a}^{n}|_{LU_{a}} will do. The case of a general Λ now follows from (2.5). q.e.d.

3. Proof of Theorem 1.2(2)

We start with two results concerning polynomials on Lℂ. For simplicity let E_{t} denote both E_{t}^{Lℂ} ∈ (Lℂ)^{∗} and E_{t}^{Lℂ^{∗}} ∈ Hom(Lℂ^{∗}, ℂ^{∗}) till the end of this paper.

Lemma 3.1. If a homogeneous polynomial h ∈ O(Lℂ) of degree n ≥ 1 does not vanish on Lℂ^{∗}, then h = cE_{t_{1}} ⋯ E_{t_{n}}, where t_{1}, …, t_{n} ∈ S^{1}, and c ≠ 0 is a constant.

Proof. Since L_{∞}ℂ is embedded into Lℂ with a dense image, we only need to show the lemma for the case of C^{∞} loops.

Let W = Lℂ \ Lℂ^{∗}, Z_{h} ⊂ W be the zero locus of h, and Emb(S^{1}, ℂ) ⊂ L_{∞}ℂ be the open subset of embedded loops. Then Z_{h} ∩ Emb(S^{1}, ℂ) ≠ ∅. Otherwise for any x ∈ Emb(S^{1}, ℂ), the polynomial h(x + λ) in λ ∈ ℂ has no zero, hence is constant. In particular, h(x + 1) = h(x) on Emb(S^{1}, ℂ) and therefore on L_{∞}ℂ. So h(1) = h(0) = 0. But 1 ∈ Lℂ^{∗}, contradiction.

Let x_{1} ∈ W ∩Emb(S^{1}, ℂ). Next we show that W is a submanifold of real codimension one near x_{1}. Let t_{1} be the unique element of S^{1} = ℝ/ℤ such that x_{1}(t_{1}) = 0. We can assume that x′_{1}(t_{1}) is not real, otherwise replace x_{1} by ix_{1}. Consider the equation

Im x(s) = 0, x ∈ L_{∞}ℂ, s ∈ S^{1}. (3.1)

This equation can also be considered on the C^{k} loop space L_{k}ℂ, 1 ≤ k < ∞. Note that Im x′_{1}(t_{1}) ≠ 0. Apply the Implicit Function Theorem on Banach spaces (see Theorem 2.5.7 of [1]) to the C^{k} map L_{k}ℂ × S^{1} → ℝ, (x, t) ↦ Im x(t) near (x_{1}, t_{1}). When k = 1 we obtain a neighborhood U_{1} ⊂ L_{1}ℂ of x_{1} consisting of embedded loops, a neighborhood V ⊂ S^{1} of t_{1}, and a C^{1} map ϕ : U → V such that for any x ∈ U, s = ϕ(x) is the unique solution of (3.1) in V. We can shrink U_{1} and V if necessary to ensure that Im x′(t) ≠ 0 for all (x, t) ∈ U_{1} × V. For arbitrary k < ∞ we obtain that ϕ is C^{k} on U_{k} = U_{1} ∩ L_{k}ℂ by the Implicit Function Theorem at any (x, ϕ(x)), x ∈ U_{k}. This implies that ϕ is C^{∞} on U = U_{1} ∩ L_{∞}ℂ. We can choose sufficiently small U so that x(t) ≠ 0 if x ∈ U and t ∉ V; then

x (ϕ(x)) = 0, x ∈ U ∩ W. (3.2)

Let Y be the real hyperplane {y ∈ L_{∞}ℂ : Im y(t_{1}) = 0} of L_{∞}ℂ. Define the C^{∞} map τ : S^{1} × Y → L_{∞}ℂ, (s, x(t)) ↦ x(t − s). The C^{∞} map ρ : U → S^{1} × Y, x(t) ↦ (ϕ(x) − t_{1}, x(t + ϕ(x) − t_{1})) is the local inverse of τ near (t_{1}, x_{1}). Let H_{t} be the kernel of E_{t}. Note that H_{t_{1}} ⊂ Y. Since τ(S^{1} ×H_{t_{1}}) = W and by (3.2) ρ(W ∩U) ⊂ S^{1} ×H_{t_{1}}, W ∩U is a submanifold of real codimension one of U. Its tangent space is

T W_{x} = H_{ϕ(x)} ⊕ {x′ℝ}, x ∈ W ∩ U, (3.3)

if we identify the tangent space of L_{∞}ℂ at x with L_{∞}ℂ.

Now assume x_{1} ∈ Emb(S^{1}, ℂ) ∩ Z_{h}. Let v ∈ H_{t_{1}} and v_{0} ∉ W. Consider the restriction of h to the 2-dimensional affine subspace E = {x_{1} + λ_{1}v + λ_{2}v_{0} : λ_{1}, λ_{2} ∈ ℂ}. Let δ : Δ → Z_{h} ∩ E ∩ U ⊂ W be a holomorphic arc such that δ(0) = x_{1} and 0 ≠ δ′(0) ∈ T W_{x_{1}}. As for all λ ∈ Δ δ′(λ) is in the maximal complex subspace of T_{δ(λ)}W, δ′(λ) ∈ H_{ϕ(δ(λ))} by (3.3). Since ϕ is constant on H_{ϕ(δ(λ))} ∩ U, the derivative of ϕ in the direction of δ′(λ) is zero, hence ϕ ∘ δ ≡ t_{1}. In view of (3.2) δ(Δ) ⊂ H_{t_{1}} ∩ E. It follows that {x_{1} + λ_{1}v} ⊂ Z_{h} and H_{t_{1}} ⊂ Z_{h}.

The local ring O(L_{∞}ℂ)_{x}, x ∈ L_{∞}ℂ, is a unique factorization domain, see Proposition 5.15 of [7]. The germ of E_{t_{1}} at any x ∈ H_{t_{1}} is prime, for the functions e(λ) = E_{t_{1}}(x + λy) vanish to first order at λ = 0 if y(t_{1}) ≠ 0. Applying the Nullstellensatz in Theorem 5.14 of [7] to the prime ideals ((E_{t_{1}})_{x}), x ∈ H_{t_{1}}, we obtain that h = E_{t_{1}} h̃, h̃ ∈ O(L_{∞}ℂ). On any affine line h̃ is a polynomial of degree ≤ n − 1, so it is a polynomial of degree ≤ n − 1 on L_{∞}ℂ, see Section 2.2 of [2]. The zero locus of h̃ is still in W, so that repeating the above process we obtain the conclusion of the lemma.

q.e.d.

Let P^{n}(Lℂ) be the space of holomorphic polynomials of degree ≤ n on Lℂ.

Proposition 3.2. Let h_{1}, h_{2} ∈ O(Lℂ) and φ ∈ O(Lℂ^{∗}) such that φ(λy) = λ^{n}φ(y), λ ∈ ℂ^{∗}, y ∈ Lℂ^{∗}, n ∈ ℕ ∪ {0}. If h_{1}(y) = φ(y)h_{2}(y^{−1}), y ∈ Lℂ^{∗}, then h_{1}, h_{2} ∈ P^{n}(Lℂ). Let h_{j}^{i} be the i-th order homogeneous component of h_{j}, j = 1, 2, i = 0, …, n. Then h_{1}^{i}(y) = φ(y)h_{2}^{n−i}(y^{−1}).

Proof. If x ∈ Lℂ, y ∈ Lℂ^{∗}, then

lim_{λ→∞} \frac{h_{1}(x + λy)}{λ^{n}} = lim_{λ→∞} φ(xλ^{−1} + y)h_{2}((x + λy)^{−1}) = φ(y)h_{2}(0).

Thus h_{1}(x + λy) is a polynomial of degree ≤ n in λ. As Lℂ^{∗} is dense in Lℂ, the same holds for all y ∈ Lℂ. Since a continuous function on a Fréchet space is a polynomial of degree ≤ n if its restriction to any affine line is such a polynomial, see Section 2.2 of [2], we conclude that h_{1} ∈ P^{n}(Lℂ). Similarly h_{2} ∈ P^{n}(Lℂ). Comparing homogeneous components of same order on both sides of the equation h_{1}(y) = φ(y)h_{2}(y^{−1}), we get h_{1}^{i}(y) = φ(y)h_{2}^{n−i}(y^{−1}), i = 0, …, n. q.e.d.

Let σ ∈ H^{0}(Lℙ_{1}, Λ_{φ}), ord(φ) ≥ 0. With sections σ_{a} ∈ H^{0}(LU_{a}, Λ_{φ}), a ∈ ℙ_{1}, in Proposition 2.7, the functions H_{a} = σ/σ_{a} ∈ O(LU_{a}) satisfy

H_{a}(x) = φ ∘ Lg_{ab}(x)H_{b}(x), x ∈ LU_{a} ∩ LU_{b}. (3.4)

Note that Lg_{ab} maps LU_{a} to Lℂ biholomorphically, and let h_{ab} = H_{a} ∘ Lg_{ab}^{−1} ∈ O(Lℂ). Then (3.4) implies that

h_{ab}(y) = φ(y)h_{ba}(y^{−1}), y ∈ Lℂ^{∗}. (3.5)

From Proposition 3.2 it follows that h_{ab} ∈ P^{n}(Lℂ); also

H_{∞} = h_{∞0} ∈ P^{n}(Lℂ). (3.6)

Proof of Theorem 1.2(2).

By Theorem 2.6 each element of Pic(Lℙ_{1})^{LP GL(2,ℂ)} is of the type Λ_{φ}, φ ∈ Hom(Lℂ^{∗}, ℂ^{∗}). Suppose Λ_{φ} has a nonzero holomorphic section σ. We can assume that σ(∞) ≠ 0; this can be arranged by pulling σ back by a suitable element of LP GL(2, ℂ). Then Λ_{φ}|_{ℙ_{1}} ≅ H^{n} also has a nonzero section, so ord(φ) = n ≥ 0. Note that h_{0∞}(0) = H_{0}(∞) ≠ 0. By (3.5), applied with ab = ∞0, and Proposition 3.2

h_{∞0}^{n}(y) = φ(y)h_{0∞}(0) ≠ 0, y ∈ Lℂ^{∗}. (3.7)

Hence h = h_{∞0}^{n} ∈ O(Lℂ) satisfies the condition in Lemma 3.1, so h_{∞0}^{n} = cE_{t_{1}} ⋯ E_{t_{n}}. By (3.7) φ = E_{t_{1}} ⋯ E_{t_{n}}, and Λ_{φ} is of the type as in Theorem 1.2(1). q.e.d.

4. The space of holomorphic sections

In this section we shall study the space H^{0}(Lℙ_{1}, Λ), where Λ is as in Theorem 1.2(1), and we shall prove Theorem 1.2(1). Since dim H^{0}(Lℙ_{1}, Λ) = 1 if Λ is trivial, we fix Λ = Λ_{φ} nontrivial, where

φ = E_{t_{1}}^{n_{1}} ⋯ E_{t_{r}}^{n_{r}}, n_{i} > 0, t_{i} ≠ t_{j} if i ≠ j. (4.1)

With σ_{a} ∈ H^{0}(LU_{a}, Λ_{φ}) as in Proposition 2.7 and σ ∈ H^{0}(Lℙ_{1}, Λ_{φ}), we have seen that σ/σ_{∞} = H_{∞} ∈ P^{n}(Lℂ), cf. (3.6). Define a monomorphism

ℌ_{φ} : H^{0}(Lℙ_{1}, Λ_{φ}) → P^{n}(Lℂ), σ ↦ σ/σ_{∞},

where n = ord(φ) ≥ 0. Let R(ℌ_{φ}) ⊂ P^{n}(Lℂ) be the range of ℌ_{φ}. We shall study H^{0}(Lℙ_{1}, Λ_{φ}) through R(ℌ_{φ}).

The bundle Λ_{φ}, where φ is as in (4.1), has nontrivial holomorphic sections: products of pull back sections by evaluation maps on Lℙ_{1}.

Proposition 4.1. The linearly independent functions E_{t_{1}}^{m_{1}} ⋯ E_{t_{r}}^{m_{r}} are in R(ℌ_{φ}), 0 ≤ m_{i} ≤ n_{i}, 1 ≤ i ≤ r. In particular, (n_{1} + 1) ⋯ (n_{r} + 1) ≤ dim H^{0}(Lℙ_{1}, Λ_{φ}).

Proof. Choose a basis {τ̃_{j} : j = 0, …, n_{1}} of H^{0}(ℙ_{1}, H^{n_{1}}), where τ̃_{0} has a zero of order n_{1} at ∞ ∈ ℙ_{1} and τ̃_{j} = z^{j}τ̃_{0}, z ∈ ℙ_{1}. Let τ_{j} = (E_{t_{1}}^{Lℙ_{1}})^{∗}τ̃_{j} ∈ H^{0}(Lℙ_{1}, Λ_{t_{1}}^{n_{1}}). The section τ_{0}|_{LU_{∞}} satisfies the conditions in Proposition 2.5. Therefore

ℌ_{φ}(τ_{j}) = τ_{j}/τ_{0} = E_{t_{1}}^{∗}z^{j} = E_{t_{1}}^{j}.

Similarly we have pull back sections of Λ_{t_{i}}^{n_{i}}, i = 1, …, r. By taking products of such sections we obtain sections of Λ_{φ}, hence E_{t_{1}}^{m_{1}} ⋯ E_{t_{r}}^{m_{r}} ∈ R(ℌ_{φ}), 0 ≤ m_{i} ≤ n_{i}, 1 ≤ i ≤ r. Another way of obtaining these functions is to pull back monomials on ℂ^{r} by the surjective map (E_{t_{1}}, ..., E_{t_{r}}) : Lℂ → ℂ^{r}, which interpretation proves the claim of linear independence. q.e.d.

The elements of R(ℌ_{φ}) identified in Proposition 4.1 are in the subalgebra of P^{n}(Lℂ) generated by evaluation functions. In other words they are polynomials in finitely many linear functionals. The next proposition shows that this is true in general. For convenience, in Propositions 4.2 and 4.3 we shall restrict our discussion to C^{k} (1 ≤ k ≤ ∞) loop spaces L_{k}ℙ_{1}. Let x^{(ν)}(t) denote the ν-th derivative of x ∈ L_{k}ℂ at t ∈ S^{1} = ℝ/ℤ, ν ≤ k. Note that the function x → x^{(ν)}(t) is in (L_{k}ℂ)^{∗}.

Proposition 4.2. If σ ∈ H^{0}(L_{k}ℙ_{1}, Λ_{φ}) and P = ℌ_{φ}(σ), then P (x) is a polynomial in finitely many derivatives x^{(ν)}(t_{i}), 0 ≤ ν ≤ k, 1 ≤ i ≤ r.

Proof. Let A = {t_{1}, …, t_{r}} ⊂ S^{1}, x_{0} ∈ L_{k}U_{∞}, and denote the k-jet of x ∈ L_{k}ℙ_{1} by j^{k}x. Define

Z = Z(k, A, x_{0}) = {x ∈ L_{k}ℙ_{1} : j^{k}x|_{A} = j^{k}x_{0}|_{A}}.

This is a connected complex submanifold of L_{k}ℙ_{1} and any holomorphic function on it is a constant, see Sections 3, 4 of [6]. Consider the sections σ_{a} of Proposition 2.7. In the proof of that proposition we have shown that, when Λ = Λ_{t}^{n}, σ_{a} can be taken to be the pullback by E_{t}^{Lℙ_{1}} of sections τ_{a} of the hyperplane bundle H → ℙ_{1}, τ_{a} ≠ 0 on U_{a}. When Λ = ⊗Λ_{t_{i}}^{n_{i}}, σ_{a} can be taken as the product of such sections. Then σ_{a}(x) = 0 only if x(t_{i}) = a for some i; in particular σ_{∞} ≠ 0 on Z. It follows that σ/σ_{∞}|_{Z} ∈ O(Z) is constant, i.e., P is constant on any affine subspace

{x ∈ L_{k}ℂ : j^{k}x|_{A} = j^{k}x_{0}|_{A}}, x_{0} ∈ L_{k}ℂ. (4.2)

When k < ∞, the continuous linear functionals x → x^{(ν)}(t_{i}), 0 ≤ ν ≤ k, 1 ≤ i ≤ r, give rise to a surjective linear map J : L_{k}ℂ → ℂ^{(k+1)r}, whose fibers are the affine subspaces in (4.2). What we have shown above implies that P = J^{∗}f, where f is a function on ℂ^{(k+1)r}. In fact f is a holomorphic polynomial of degree ≤ n, for with a linear right inverse I to J we have f = P ∘ I.

Consider the case when k = ∞. Let P^{j} be the j-th order homogeneous component of P ∈ P^{n}(L_{∞}ℂ), j = 1, …, n. These components are also constant on any affine subspace in (4.2). By the definition of a homogeneous polynomial we can find a continuous symmetric j-linear mapping Ψ_{j} : (L_{∞}ℂ)^{j} → ℂ such that P^{j}(x) = Ψ_{j}(x, …, x). Applying the Schwartz Kernel Theorem (see Theorem 5.2.1 of [3]) one can show that there exists a distribution K_{j} in the j dimensional torus T^{j} such that

Ψ_{j}(x_{1}, …, x_{j}) = K_{j}(x_{1}(s_{1}) ⋯ x_{j}(s_{j})), (s_{1}, …, s_{j}) ∈ T^{j}.

So P^{j}(x) = K_{j}(x(s_{1}) ⋯ x(s_{j})). The Polarization Formula (see (2) in Section 2.2 of [2])

K_{j} (x_{1}(s_{1}) ⋯ x_{j}(s_{j})) = \frac{1}{2^{j}j!} ∑_{ε_{1},...,ε_{j}=±1} ε_{1} ⋯ ε_{j}P^{j}(ε_{1}x_{1} + ⋯ + ε_{j}x_{j})

and the fact that P^{j} depends only on j^{∞}x|_{A} imply that K_{j} is supported in A^{j} ⊂ T^{j}. Therefore K_{j} is a (finite) linear combination of partial derivatives at points in A^{j}, see Theorem 2.3.4 of [3]. Hence P^{j} and P are polynomials in finitely many x^{(ν)}(t_{i}), x ∈ L_{∞}ℂ. q.e.d.

For each i let N_{i} be the order of the highest derivative x^{(ν)}(t_{i}) that P (x) depends on, see Proposition 4.2, and m_{i} the degree of P (x) as a polynomial of x^{(N_{i})}(t_{i}). Our next task is to estimate N_{i}, m_{i}.

Proposition 4.3. m_{i} (N_{i} + 1) ≤ n_{i}, where n_{i} is defined in (4.1).

Proof. Fix i. At first assume P = ℌ_{φ}(σ) contains the monomial

c_{1} x^{(N_{i})}(t_{i})^{m_{i}}, (4.3) where c_{1} ≠ 0 is a constant. By (3.6) and (3.5) P = h_{∞0} satisfies h_{0∞}(y) = φ(y)h_{∞0}(y^{−1}) so that Proposition 3.2 implies that

h_{0∞}^{n−m_{i}}(x) = φ(x) [P^{m_{i}}(x^{−1})] , x ∈ L_{k}ℂ^{∗}, (4.4)

where superscripts indicate homogeneous components of the given order. Since P^{m_{i}} is a polynomial in x^{(ν)}(t_{j}), P^{m_{i}}(x^{−1}) is a sum of rational expressions, where the denominators are monomials in x(t_{j}), 1 ≤ j ≤ r, and the numerators are monomials in x^{(ν)}(t_{j}), ν ≥ 1. In this sum the monomial in (4.3) gives rise to the term

c_{1}(−1)^{m_{i}N_{i}}(N_{i}!)^{m_{i}} \frac{x^{(1)}(t_{i})^{m_{i}N_{i}}}{x(t_{i})^{m_{i}(N_{i}+1)}} ,

which is the only term in P^{m_{i}}(x^{−1}) with this high or higher power of x(t_{i}) in the denominator. Hence taking

x(t) = x_{s}(t) = e^{2πi(t−t_{i})} − 1 + s, s ∈ (−1, 0),

in (4.4) and noting that φ(x_{s}) = ∏_{j} x_{s}(t_{j})^{n_{j}}, we obtain

h_{0∞}^{n−m_{i}}(x_{s}) = s^{n_{i}−m_{i}(N_{i}+1)} ∏_{j≠i} x_{s}(t_{j})^{n_{j}} (c_{2} + sg(s)) , (4.5)

where c_{2} ≠ 0 is a constant, g(s) is a rational function in s which is bounded on the interval (−1, 0). In order that the limit of right-hand side of (4.5) exist as s → 0^{−} we must have m_{i}(N_{i} + 1) ≤ n_{i}.

The proof is finished if we can show that for any σ ∈ H^{0}(L_{k}ℙ_{1}, Λ_{φ}) there exists σ′ ∈ H^{0}(L_{k}ℙ_{1}, Λ_{φ}) such that the corresponding N′_{i} = N_{i}, m′_{i} = m_{i}, and ℌ_{φ}(σ′) contains the monomial (4.3).

We write P in the form

P (x) = x^{(N_{i})}(t_{i})^{m_{i}}f_{1}(x) + f_{2}(x), x ∈ L_{k}ℂ,

where f_{1}(x), f_{2}(x) are polynomials in x^{(ν)}(t_{j}), f_{1} ≠ 0 is independent of x^{(N_{i})}(t_{i}) and the degree of f_{2} in x^{(N_{i})}(t_{i}) is strictly less than m_{i}. Let y ∈ L_{k}ℂ^{∗}, and γ ∈ L_{k}P SL(2, ℂ) be the map x ↦ y(x + 1), x ∈ L_{k}ℙ_{1}. Let P′ denote ℌ_{φ}(γ^{∗}σ). From Propositions 2.7 and 2.5 we obtain that γ^{∗}σ_{∞} = c_{3}σ_{∞} (with a fixed isomorphism γ^{∗}Λ_{φ} ≅ Λ_{φ}), where c_{3} ≠ 0 is a constant. Therefore

P′(x) = c_{3} \frac{γ^{∗}σ}{γ^{∗}σ_{∞}} (x) = c_{3}γ^{∗}P (x) = c_{3}P (y(x + 1)) , x ∈ L_{k}ℂ. (4.6)

Computing the right-hand side of (4.6) by the product rule, we find that it contains the monomial

c_{3} y(t_{i})^{m_{i}}f_{1}(y) (x^{(N_{i})}(t_{i}))^{m_{i}} .

Since f_{1} ≠ 0, we can find y such that f_{1}(y) ≠ 0; then we can choose σ′ = γ^{∗}σ. q.e.d.

Proof of Theorem 1.2(1). The lower bound is given in Proposition 4.1. For the case of C^{∞} loops Propositions 4.2 and 4.3 imply that R(ℌ_{φ})) consists of polynomials of degree ≤ ord(Λ_{φ}) in x^{(ν)}(t_{j}), 0 ≤ ν ≤ n_{j} − 1, 1 ≤ j ≤ r, therefore dim H^{0}(L_{∞}ℙ_{1}, Λ_{φ}) < ∞. Since in general L_{∞}ℙ_{1} is continuously embedded into Lℙ_{1} with a dense image, so that the restriction map H^{0}(Lℙ_{1}, Λ_{φ}) → H^{0}(L_{∞}ℙ_{1}, Λ_{φ}) is monomorphic, we conclude that dim H^{0}(Lℙ_{1}, Λ_{φ}) < ∞. q.e.d.

An immediate application of Proposition 4.3 is to identify all holomorphic sections of a “generic” bundle of the type considered in Theorem 1.2(1).

Corollary 4.4. If n_{1} = ⋯ = n_{r} = 1 in (4.1), then dim H^{0}(Lℙ_{1}, Λ_{φ}) = 2^{r}.

Proof. Proposition 4.3 gives that R(ℌ_{φ}) only contains polynomials in evaluation maps E_{t_{1}}, …, E_{t_{r}}, of degree ≤ 1 in each variable, and all such polynomials are indeed in R(ℌ_{φ}) by Proposition 4.1. q.e.d.

There are bundles as in Theorem 1.2(1) which have holomorphic sections other than those identified in Proposition 4.1. We shall show this by an explicit construction.

Proposition 4.5. Let h ∈ O(Lℂ) and φ ∈ Hom(Lℂ^{∗}, ℂ^{∗}). Then h ∈ R(ℌ_{φ}) if and only if there exists a family {f_{a} ∈ O(Lℂ) : a ∈ ℂ} such that

h(x + a) = φ(x)f_{a}(x^{−1}), x ∈ Lℂ^{∗}. (4.7)

Proof. With σ_{a} of Proposition 2.7 (4.7) is equivalent to

h(x)σ_{∞}(x) = f_{a} ((x − a)^{−1}) σ_{a}(x), x ∈ LU_{a} ∩ LU_{∞}, a ∈ ℂ,

which defines a section σ ∈ H^{0}(Lℙ_{1}, Λ_{φ}) such that ℌ_{φ}(σ) = h and vice versa. q.e.d.

From now on, until the last paragraph, we shall work with the space of C^{k} loops. Let h ∈ O(L_{k}ℂ^{∗}) be a rational function of finitely many x^{(ν)}(t_{j}), 0 ≤ ν ≤ k, 1 ≤ j ≤ r. Letting each t_{j} vary in S^{1}, h induces a function χ ∈ C(L_{k}ℂ^{∗} ×(S^{1})^{r}). If h does not depend on the highest derivative x^{(k)}(t_{i}) for some i (this is automatically satisfied if k = ∞), then χ is differentiable in t_{i}, and we define

h^{t_{i}}(x) = \frac{∂}{∂t_{i}} χ(x, t_{1}, …, t_{r}).

Thus h^{t_{i}} ∈ O(L_{k}ℂ^{∗}) is also a rational function in x^{(ν)}(t_{j}), 0 ≤ ν ≤ k, 1 ≤ j ≤ r. If h was a polynomial, so will be h^{t_{i}}. The linear map T^{i} : h ↦ h^{t_{i}} satisfies T^{i}(h_{1}h_{2}) = T^{i}(h_{1})h_{2} + h_{1}T^{i}(h_{2}), T^{i}(h(x^{−1})) = (T^{i}h)(x^{−1}), and T^{i}(h(x + a)) = (T^{i}h)(x + a) if h ∈ O(L_{k}ℂ).

Proposition 4.6. Let φ_{i} = φE_{t_{i}}, and assume that h ∈ R(ℌ_{φ}) does not depend on x^{(k)}(t_{i}) for some i. Then h^{t_{i}} ∈ R(ℌ_{φ_{i}}).

Proof. If f_{a} is defined as in Proposition 4.5, Proposition 3.2 implies f_{a} ∈ P^{n}(Lℂ). Since h and φ depend only on x^{(ν)}(t_{j}), so does f_{a}: it is in fact a polynomial in x^{(ν)}(t_{j}) (ν < k if j = i, ν ≤ k for all other j). Applying T^{i} to (4.7) we obtain

h^{t_{i}}(x + a) = T^{i}(h(x + a))
= n_{i}φ(x)x(t_{i})^{−1}x^{(1)}(t_{i})f_{a}(x^{−1}) + φ(x)f_{a}^{t_{i}}(x^{−1})
= φ_{i}(x)f̃_{a}(x^{−1}),

where f̃_{a}(x) = −n_{i}x^{(1)}(t_{i})f_{a}(x)+x(t_{i})f_{a}^{t_{i}}(x), so that h^{t_{i}} ∈ R(ℌ_{φ_{i}}) by Proposition 4.5. q.e.d.

Finally we shall discuss the order N_{i} of the highest derivative x^{(ν)}(t_{i}) that ℌ_{φ}(σ) can depend on, in the case of the C^{k} loop space L_{k}ℙ_{1}. Recall from Propositions 4.2, 4.3 that N_{i} ≤ min(k, n_{i} − 1). It turns out that this estimate is sharp:

Theorem 4.7. There exists a section of Λ_{φ} for which

N_{i} = min(k, n_{i} − 1).

Proof. With μ_{i} = min(k, n_{i}−1) and 0 ≤ ν ≤ μ_{i} let φ_{ν} = φE_{t_{i}}^{ν−μ_{i}}. Proposition 4.1 gives that E_{t_{i}} ∈ R(ℌ_{φ_{0}}). Repeatedly applying Proposition 4.6 we obtain that the functions h_{ν}(x) = x^{(ν)}(t_{i}) are in R(ℌ_{φ_{ν}}) for ν ≤ μ_{i}. Thus h_{μ_{i}}(x) = x^{(μ_{i})}(t_{i}) is in the range of ℌ_{φ}, and N_{i} = μ_{i} if σ = ℌ_{φ}^{−1}h_{μ_{i}}. q.e.d.

A similar reasoning would apply to the space of W^{k,p} loops, where the largest value for N_{i} turns out to be min(k − 1, n_{i} − 1). A remarkable consequence of this is that while for generic bundles Λ_{φ}, as we have seen in Corollary 4.4, dim H^{0}(Lℙ_{1}, Λ_{φ}) does not vary with the regularity of loops, when at least one n_{i} > 1, this dimension will depend on the regularity class C^{k}, W^{k,p} considered.

References

[1] R. Abraham, J.E. Marsden & T. Ratiu, Manifolds, Tensor Analysis, and Applications, 2nd ed., Springer, New York, 1988, MR 0960687, Zbl 0875.58002.

[2] M. Hervé, Analyticity in Infinite Dimensional Spaces, Walter de Gruyter & Co., Berlin, 1989, MR 0986066, Zbl 0666.58008.

[3] L. Hörmander, The Analysis of Linear Partial Differential Operators. I. Distribution Theory and Fourier Analysis, Springer, Berlin, 1983, MR 0717035, Zbl 0521.35001.

[4] D. Husemoller, Fibre Bundles, 2nd ed., Graduate Texts in Mathematics, 20, Springer, New York, 1975, MR 0370578, Zbl 0307.55015.

[5] L. Lempert, The Dolbeault complex in infinite dimensions, I, J. Amer. Math. Soc. 11 (1998) 485–520, MR 1603858, Zbl 0904.32014.

[6] L. Lempert, Holomorphic functions on (generalized ) loop spaces, Proc. Royal Irish Acad., to appear.

[7] P Mazet, Analytic sets in locally convex spaces, North Holland, Amsterdam, 1984, MR 0756238, Zbl 0588.46032.

[8] J. McClure, Private communication.

[9] J.J. Millson & B. Zombro, A Kähler structure on the moduli space of isometric maps of a circle into Euclidean space, Invent. Math. 123 (1996) 35–59, MR 1376245, Zbl 0859.58007.

[10] M. Vigué-Poirrier & D. Sullivan, The homology theory of the closed geodesic problem, J. Differential Geom. 11 (1976) 633–644, MR 0455028, Zbl 0361.53058.

[11] E. Witten, The index of the Dirac operator in loop space, Elliptic curves and modular forms in algebraic topology, Lecture Notes in Math., 1326, Springer, Berlin, 1988, 161–181, MR 0970288, Zbl 0679.58045.

Department of Mathematics

Purdue University West Lafayette, IN 47906