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The co-Riemannian Structure of Smooth Loop Spaces

We construct a natural co-Riemannian structure on the manifold of smooth loops in a Riemannian manifold. We show that the smooth loop space of a string manifold is a per-Hilbert-Schmidt locally equivalent co-spin manifold and thus admits a Dirac operator.


Introduction
In [Sta] we introduced the concept of a co-orthogonal structure on an infinite dimensional vector bundle and showed how this led to the construction of an infinite dimensional Dirac operator. In brief, whilst an orthogonal structure on a vector bundle defines a Hilbert completion of each fibre, a co-orthogonal structure defines a dense Hilbert subspace. In this paper we shall show that the space of smooth loops in a Riemannian manifold admits a co-Riemannian structure-that is, a co-orthogonal structure on its tangent bundle-and that this is suitable for the construction of the Dirac operator to work.
We classified co-orthogonal and orthogonal structures in [Sta] according to certain criteria-refining the "weak-strong" classification. In terms of this classification we prove the following theorem.
Theorem A Let M be a finite dimensional Riemannian manifold. Let LM ≔ C ∞ (S 1 , M) be the space of smooth loops in M. This space admits a nuclear, locally equivalent, co-Riemannian structure with structure group Gl res . This definition has the following interpretation. There is a bundle of Hilbert spaces, say E → LM, with structure group Gl res and a vector bundle map E → TLM. There are simultaneous trivialisations of E and of TLM with respect to their structure groups such that E → TLM locally looks like the inclusion of a fixed Hilbert space in the model space of TLM (this model space being LR n ) and that this inclusion is a nuclear map. Under this local trivialisation, the inner product on E is not taken to some fixed inner product on the typical fibre of E. The group Gl res is the restricted general linear group of [PS86].
Theorem A refers to TLM with its standard structure group, namely L Gl n . If we are prepared to work with a slightly larger topological group then we can replace the words "locally equivalent" by "locally trivial" in Theorem A. This means that under the simultaneous trivialisation of E and TLM, the inner product on E is taken to some fixed inner product on the typical fibre of E.
Having shown this, it is straightforward to show that if M is a string manifold then LM has the required structure to define a Dirac operator.

Theorem B Let M be a finite dimensional string manifold. Then LM is an S 1equivariant, per-Hilbert-Schmidt, locally equivalent, co-spin manifold.
Again, if we are prepared to modify the structure group of the tangent bundle of LM we have a locally trivial co-spin manifold.
Combined with the work of [Sta] this yields the following important corollary.
Corollary C Let M be a finite dimensional string manifold. Then LM admits an S 1 -equivariant Dirac operator.
As mentioned above, a co-orthogonal structure on an infinite dimensional vector bundle assigns to each fibre a dense subspace with the structure of a Hilbert space-with, almost certainly, a strictly finer topology than the subspace topology. The classification scheme introduced in [Sta] measures how much it is possible to locally trivialise this structure, whether these Hilbert spaces fit together to form a bundle in their own right (and with what structure group), and whether the map from the Hilbert space to the larger space has any nice properties such as compactness. This classification scheme works equally well for orthogonal structures as co-orthogonal structures.
To illustrate this classification, the standard Riemannian structure on LM is a nuclear, locally trivial, orthogonal structure with structure group O res (the restricted orthogonal group). This means that there is a bundle of Hilbert spaces, say F → LM, with structure group O res and a vector bundle map TLM → F. There are trivialisations of F and of TLM with respect to their structure groups such that TLM → F locally looks like the inclusion of LR n in a fixed Hilbert space (L 2 R n ) and that this inclusion is a nuclear map. In addition, this local trivialisation maps the inner product to the standard one on L 2 R n .
The difference, therefore, is the local triviality of the inner product; this also relates to the fact that we can reduce the structure group of the Hilbert bundle of O res rather than Gl res . The point is that there is only one reasonable loop space on which LO n acts isometrically and that is L 2 R n . Of course, we could always reduce the structure group of E to O res but we couldn't trivialise E with respect to O res at the same time as trivialising TLM with respect to its structure group, L Gl n . The purpose of introducing the larger structure group that we mention above is to enable us to trivialise E with respect to O res and TLM with respect to this larger group simultaneously.
We shall give the construction in three stages. The first stage, in section 2, is the linear case. As the co-orthogonal structure that we wish to construct is intended to have some local triviality properties, we need to decide on a reference structure. That is, we need to choose a fixed Hilbert subspace of LR n . This is not difficult: we take those loops which are analytic on an annulus of radius (r −1 , r) for some fixed r ∈ (1, ∞) and are square integrable on the boundary. We shall denote this space by L 2 r R n . We also need to investigate what group acts on this space. It is easy to show that L Gl n (R) does not preserve any Hilbert subspace of LR n but we can reduce L Gl n (R) to L pol O n , the group of polynomial loops in O n , which does. We also show that the action of L pol O n on L 2 r R n factors through the restricted general linear group. There is an obvious isomorphism of L 2 r R n with L 2 R n and we describe how this fits into the mixture.
The second stage, in section 3, is the universal case. Let G be a connected, compact Lie group. Let ΩG be the based loop group of G. This is a Lie group and its classifying space is again G. Now Ω pol G, the polynomial loop group of G, is homotopy equivalent to ΩG and so G is also the classifying space for Ω pol G. This means that there is a universal Ω pol G-bundle over G and, consequently, an L pol G-bundle and various associated vector bundles. We shall give explicit constructions of these. The difficult part of this section is proving local triviality: as our constructions are explicit we cannot simply appeal to the homotopy equivalence Ω pol G ≃ ΩG to deduce that they are the required universal bundles.
The reason for doing the universal case is simple. Let E → M be a finite dimensional vector bundle with structure group G. A connection on E defines a holonomy map, ΩM → G. This is an explicit classifying map for the bundle ΩE → ΩM. That is, there is a ΩR n -bundle over G which pulls-back to ΩE under the holonomy map. Having constructed a sub-Ω pol R n -bundle we can pull this back as well and obtain a subbundle Ω pol E of ΩE which, on fibres, looks like the inclusion of Ω pol R n in ΩR n .
Thus by starting with a vector bundle with connection, we obtain an explicit classifying map and so the universal construction defines a specific construction on the loop bundle. Compare this with the situation of orthogonal structures on finite dimensional vector bundles. The inclusion O n ⊆ Gl n is a homotopy equivalence but choosing a universal orthogonal structure on the vector bundle over B Gl n does not automatically define an orthogonal structure on every rank n vector bundle-it merely says that these exist.
The final stage, in section 4, is to adapt this to free loop spaces. This is reasonably straightforward. It uses the fact that the free loop space is a fibre bundle over the original manifold with fibre the based loop space. The strategy is to apply the above construction to the fibres of this bundle and twist it suitably over the base.
Finally, let us give a formula for the resulting inner product. Let M be a finite dimensional Riemannian manifold. Let γ : S 1 → M be a smooth loop in M. The tangent space of LM at γ is naturally isomorphic to Γ(γ * TM), the space of sections of γ * TM → S 1 . Let D γ : Γ(γ * TM) → Γ(γ * TM) be the covariant differential operator defined by the Levi-Civita connection on M. This is a skew-adjoint operator with spectrum of the form (is 1 , . . . is n ) + iZ n for some real numbers s j . Although D γ is a differential operator when Γ(γ * TM) is considered as a space of sections, when Γ(γ * TM) is considered as the tangent space of TLM, D γ is more correctly viewed as a linear operator.
Let cos D γ be the densely-defined operator on Γ(γ * TM) defined using the power series for cos. For α, β ∈ Γ(γ * TM) such that cos D γ α and cos D γ β are defined, let The Hilbert subspace of T γ LM = Γ(γ * TM) is the subspace on which this formula makes sense.

Acknowledgements and History
The central idea of this paper-the construction of the polynomial loop bundle-places this paper as the latest in a loosely defined series: [Mor01], [CS04], and [Sta05]. In the first of these, Morava attempted to construct an isomorphism for an almost complex manifold M between the tangent bundle of the loop space, LTM, and a bundle of the form e * 1 TM ⊗ C LC. Here, e 1 : LM → M is the map which evaluates a loop at time 1 and every bundle is considered to be complex. The argument broke down at one crucial step and the paper [Mor01] had to be withdrawn, [Mor03] 1 . The papers [CS04] and [Sta05] grew out of considering the question as to when that crucial step could be made to work. This was found to be highly restrictive and implied, for example, that the tangent bundle of the based loop space of M was trivial.
One consequence which would follow from the existence of an isomorphism LTM e * 1 TM ⊗ C LC would be the existence of a subbundle modelled on the polynomial loop space. In fact, for any class of loops there would be a bundle with the appropriate fibre constructed as e * 1 TM ⊗ C L α C. The construction in this paper was inspired by that of Morava. The basic idea of using parallel transport to select a suitable subbundle of TLM comes directly from [Mor01]. The bulk of this paper is concerned with proving the local triviality of that bundle (this is something that Morava did not have to consider as his bundle was-supposed to be-isomorphic to e * 1 TM and thus automatically locally trivial). The author would like to thank Rafe Mazzeo, Ralph Cohen, and Eldar Straume for helpful conversations, to thank Gerd Laures for asking an interesting question, and to acknowledge the encouragement of Jack Morava.

Notation
In this paper we have two views of the circle. One is as the domain of loops, the other as a Lie group. We regard loops as periodic paths from R and thus wish to identify the domain of loops with R/Z. When thinking of the circle as a Lie group, we think of it as U 1 sitting inside M 1 (C) = C. We shall use the notation S 1 for R/Z and T for U 1 . We shall write t for the parameter in S 1 and z in T, with relationship z = e 2πit .
For a finite dimensional smooth manifold M we define the loop space, LM, and path space, PM, as As we are viewing the circle as a quotient of R we have a natural inclusion LM → PM as the subspace of periodic paths with period 1. Within LM we have a copy of M as the constant loops. Thus we can regard M as a subspace of LM and both as subspaces of PM.

The Linear Case
In this section we shall consider the linear case. We shall find suitable subspaces of LR n and of LC n . To extend this to the bundle case we shall need to examine 1 Although withdrawn, [Mor01] is still available from the website of the journal i Te i 2 is absolutely convergent. The groups introduced above are not topologised as subgroups of Gl(H), rather the topology is strengthened to take into account the Hilbert-Schmidt norm on the commutators. The details can be found in [PS86]. Although we have defined these groups using a specific operator on a specific separable Hilbert space, the groups so defined are-up to isomorphism-independent of these choices. It is therefore customary to drop the "H" or "H R " from the notation when this does not lead to a loss of clarity.
These groups have a strong relationship to loop groups which we shall now introduce. The smooth loop space of a compact Lie group is again a group and we shall refer to it as the smooth loop group. The definition of the polynomial loop group appears in [PS86, §3.5] and we repeat it here.
1. The choice of the embedding of G in U n is immaterial.
2. The space L pol G is the union of the subspaces L pol,N G consisting of those loops with Fourier series of the form: These spaces are naturally compact. The topology on L pol G is the direct limit topology of this union.
3. The free polynomial loop group is the semi-direct product of the based polynomial loop group and the constant loops.
4. The group L pol G does not have an associated Lie algebra, although the Lie algebra L pol g is often linked to it.
5. If G is semi-simple then L pol G is dense in LG.
6. In the case of the circle, Ω pol S 1 = Z and so L pol S 1 = S 1 × Z.
The following is [PS86, proposition 8.6.6]: Proposition 2.4 If G is semi-simple then the inclusion Ω pol G → ΩG is a homotopy equivalence.
Since L pol G Ω pol G × G and LG ΩG × G as spaces (although not generally as groups), this holds for the unbased loops as well.
In passing, let us observe that the topology on L pol G makes questions of continuity very easy to determine. Suppose that a representation of L pol G factors through a representation of the polynomial algebra L pol M n (C) for some n. It is therefore automatically continuous as L pol M n (C) is topologised as the direct limit of its finite dimensional subspaces and so any linear map out of L pol M n (C) into a topological vector space is continuous.
The following result is the starting point of [PS86].

Proposition 2.5
The natural action of L Gl n (C) on H = L 2 (S 1 , C n ) factors through a homomorphism L Gl n (C) → Gl res (H).
We therefore see that if G acts on a finite dimensional real or complex vector space V then the smooth and polynomial loop groups act on H ≔ L 2 (S 1 , V) via a homomorphism into Gl res (H).

Linear Co-Orthogonal Structures
Our main theorem states that if M is a finite dimensional Riemannian manifold then LM admits a nuclear, locally equivalent, co-Riemannian structure with structure group Gl res . This assigns to each tangent space of LM a linear map from a Hilbert space which is injective and has dense image. The term "locally equivalent" means that these look locally like the inclusion of some standard Hilbert space in LR n . In this section we shall define this standard space.
It is also important to consider group actions. The linear interpretation of the phrase "with structure group Gl res " is that there is some subgroup of L Gl n acting on LR n which also acts on the Hilbert subspace and that this action is via Gl res . That we can only say "locally equivalent" and not "locally trivial" means that this action is not by isometries.
We shall define a 1-parameter family of Hilbert subspaces indexed by r ∈ (1, ∞). Contrast this with the opposite structure, that of finding a Hilbert completion of LR n . There is an obvious family of such completions, L k,2 R n , indexed by k ∈ N 0 . In this case the completion corresponding to k = 0 has a mathematical advantage unshared by the others: the action of the loop group LO n is by isometries.
We start by defining our space of interest and a particularly important operator.
Definition 2.6 Let V be a complex vector space of dimension n with an inner product. Let r ∈ (1, ∞). Let L pol V be the space of V-valued polynomial loops, and L 2 r V the space of loops in V which extend holomorphically over an annulus of radii (r −1 , r) and are square integrable on the boundary. Let s ≔ ({v 1 , . . . , v n }, {s 1 , . . . , s n }) where {v 1 , . . . , v n } is a basis for V and s 1 , . . . , s n ∈ R. Let D s : L pol V → L pol V be the operator defined by D s v j z p = i(p + s j )v j z p . Define We state our main result on these.
Theorem 2.7 The group L pol U(V) acts continuously on L 2 r V and this action factors through Gl res .
The map cos r D s extends to an isomorphism L 2 r V → L 2 V which intertwines the actions of Gl res on each space.
Proof. Taking Fourier coefficients allows us to identify all of these spaces with certain spaces of Z-indexed sequences in V. Smooth loops, LV, corresponds to the space of rapidly decreasing sequences; L 2 V to the space of square integrable sequences; polynomial loops, L pol V, corresponds to the space of finite sequences; and L 2 r V corresponds to the space of sequences (a p ) for which the sequence (r |p| a p ) is square integrable.
To show that the polynomial loop group, L pol U(V), acts continuously on all of these spaces it is sufficient to show that the algebra L pol End(V) acts. This is generated as a topological algebra by the subalgebra End(V) and the operator which multiplies by z. The algebra End(V) obviously acts continuously on all of the spaces. The operation corresponding to multiplication by z is the shift operator on the sequence spaces which is also obviously continuous.
For j, p ∈ Z with 1 ≤ j ≤ n, let e p, j be the sequence with v j in the pth slot and zero elsewhere. These form a topological basis for all of the above spaces. On this basis, D s is e p, j → i(p + s j )e p, j .
Hence cos r D s is e p, j → cos i(p + s j ) log(r) e p, j = cosh (p + s j ) log(r) e p, j .
Elementary analysis shows that for all x, t ∈ R cosh(t log(r)) ≥ cosh (x + t) log(r) We therefore see that (a p ) is a sequence such that (r |p| a p ) is square integrable if and only if cos r D s (a p ) is a sequence which is square integrable. Hence cos r D s defines an isomorphism L 2 r V → L 2 V. Thus cos r D s defines a group isomorphism Gl(L 2 r V) Gl(L 2 V). We want to show that this restricts to a group isomorphism Gl res (L 2 r V) Gl res (L 2 V). For the operators J V and J V,r defining Gl res (L 2 V) and Gl res (L 2 r V) respectively we can take the operators characterised by the fact that each takes v j z p to i sign(p)v j z p -this makes sense for both L 2 V and L 2 r V as the set {v j z p } is a topological basis for both spaces (albeit only orthonormal for L 2 V).
As cos r D s takes v j z p to a multiple of itself it is immediate that (cos r D s ) −1 J V (cos r D s ) = J V,r .

Thus for
We did not earlier describe the topology on Gl res ; in brief, it is given by combining the norm topology together with the Hilbert-Schmidt topology on the commutator [A, J]. From this and the above we see that the group isomorphism Gl res (L 2 V) → Gl res (L 2 r V) is continuous.
All of the above has a real counterpart given by taking the underlying real vector spaces.
Although L 2 r V has an obvious inner product, we wish to allow for some variation in this choice. For each choice of s, as described in definition 2.6, we define an inner product on L 2 r V by insisting that cos r D s : A choice of s, as in definition 2.6, isometrically identifies the spaces L 2 r V with L 2 (V) and thus defines a family of representations L pol U(V) → Gl res (L 2 V). We can extend this family to r = 1 by including the standard action on L pol U(V) on L 2 V. It is straightforward to show that this family of representations is continuous in r. As an immediate consequence we have an explicit homotopy between the representation at some fixed r 0 ∈ (1, ∞) and the standard representation corresponding to r = 1.
The main difference between the representations for r ∈ (1, ∞) and r = 1 is that the representation of L pol U(V) is unitary for r = 1 but not for any other r. If we are only concerned with its action on the Hilbert space this is not an issue since we are mainly interested in the induced Gl res -action which has a subgroup that acts unitarily. Moreover, this subgroup is homotopic to Gl res . However, if we are also interested in the action on LV then we must look elsewhere as Gl res does not act on LV.
, with the property that lim k→∞ A k and lim k→−∞ A k exist and are equal. Let EL End(V) be the algebra of rapidly decreasing, Z -indexed sequences in c Z (End(V)); suitably topologised. A sequence (A k ) represents the operator A k z k and the product is defined accordingly. Define the expanded loop group, EL Gl(V), to be the group of units in this algebra with the standard topology making the two maps EL Gl(V) → EL End(V), g → g and g → g −1 , continuous.
This group has a variety of good properties, for example it is a split extension of L Gl(V). The main one-for our purposes-is that for each r ∈ [1, ∞) and s as in definition 2.6 there is a subgroup isomorphic to L pol U(V) which acts unitarily on L 2 r V. Moreover, exactly as for Gl res , the representations L pol U(V) → EL Gl(V) so defined are all homotopic. This group does not play a major rôle in this paper. Its purpose is to show that it is possible to replace the "locally equivalent" co-orthogonal structure with a "locally trivial" one by changing the structure group. Therefore we shall not study it further.

The Universal Case
In this section we consider the universal case. The classifying space of ΩG is (homotopy equivalent to) G. Since ΩG ≃ Ω pol G, this is also the classifying space of Ω pol G. There is therefore a universal Ω pol G-principal bundle over G such that the inclusion Ω pol G → ΩG embeds this Ω pol G-principal bundle in the standard ΩG-principal bundle.
Using the Ω pol G-bundle and ΩG-bundle over G we obtain an L pol G-bundle and an LG-bundle in the obvious way. Given an action of G on a vector space V we can therefore define vector bundles over G with fibres LV, L pol V, L 2 r V, and L 2 V.
In this section we shall explicitly construct the L pol G-bundles over G for G each of U n , SU n , and SO n and the associated vector bundles. We start with some general results on polynomial loops.

Polynomial Loops
In this part we consider the group of polynomial loops in a compact, connected Lie group. This was studied extensively in [PS86] with some further work appearing in [Seg89] in the case of U n .
Although the definition given in section 2.1 of L pol G does not depend on the embedding of G in U n , it is useful to have such an embedding to investigate the structure of L pol G in a little more detail. We consider loops of the form t → exp(tξ) for suitable ξ ∈ g. The main result is the following: Proposition 3.1 Let G be a compact, connected Lie group, g its Lie algebra. For ξ ∈ g, let η ξ : R → G denote the path η ξ (t) = exp(tξ).
As part of the proof of this, we shall prove the following useful result for the unitary group: The proofs of these rely on the simple structure in U n of the centraliser of any particular element. For g ∈ U n , define C(g) and Z(g) to be the centraliser of g and its centre. That is,

Lemma 3.3 For any g ∈ U n , Z(g) is a torus.
Proof. The group C(g) is a closed subgroup of U n , hence its centre is a closed abelian subgroup of U n . In particular, it is compact. Therefore, it is a torus if and only if it is connected.
Recall that two diagonalisable matrices commute if and only if they are simultaneously diagonalisable. This condition does not rely on the eigenvalues of either matrix but only on the eigenspaces.
Let h ∈ Z(g). As h is unitary, it is orthogonally diagonalisable. Let λ 1 , . . . , λ l be the distinct eigenvalues of h with associated eigenspaces E 1 , . . . , E l . For each j, let s j ∈ [−iπ, iπ) be such that e s j = λ j .
Define α : [0, 1] → U n to be the path such that α(t) has eigenvalues e ts j and corresponding eigenspaces E j . Then α(0) = 1 n and α(1) = h so α is a path from 1 n to h. By construction, α(t) for t 0 has the same eigenspaces as h and therefore α(t) commutes with exactly the same elements of U n that h commutes with. Hence as h ∈ Z(g), α(t) ∈ Z(g).
Proof of lemma 3.2. As Z(g) is a torus, it is a connected compact Lie group. Therefore, the exponential map is surjective and so there is some ζ ∈ z(g) ⊆ u n with exp(ζ) = g. As ζ ∈ z(g), exp(tζ) ∈ Z(g) for all t ∈ R.
Using this we can prove proposition 3.1.
Proof of proposition 3.1. Firstly, note that it is sufficient to prove this in the case of the unitary group. For if η −ξ 1 η ξ 2 is a loop in G which is a polynomial loop when G is considered as a subgroup of U n then, by definition, η −ξ 1 η ξ 2 is a polynomial loop in G.
Secondly, note that it is sufficient to consider the case where ξ 2 = 0. This forces exp(ξ 1 ) = 1 n . To deduce the general case from this simpler one, note that by lemma 3.2 that there is some ζ ∈ u n with exp(ζ) = exp(ξ 1 ) (whence also exp(ξ 2 )) such that [ζ, ξ j ] = 0. Then exp(ξ j − ζ) = 1 n so, by assumption, η (ξ j −ζ) is a polynomial loop. The identity: demonstrates that this is a polynomial loop.
Thus we need to show that η ξ is a polynomial loop if exp(ξ) = 1. To show this, we diagonalise ξ. If s is an eigenvalue of ξ then e s is an eigenvalue of exp(ξ) = 1. The eigenvalues of ξ therefore lie in 2πiZ. Hence there is a basis of C n with respect to which η ξ is the path: Since e 2πitk = z k for k ∈ Z, this is a polynomial loop (viewed as a periodic path).
Note that for a general group G, although the loop η ξ 1 η −ξ 2 lies in Ω pol G, there may be no factorisation in G as η ξ 1 −ζ η ζ−ξ 2 since Lemma 3.2 need not hold for a general Lie group.

The Path Spaces
In this part we shall give an explicit construction of the principal L pol G-bundle over G for G each of U n , SU n , and SO n . We shall also construct a similar bundle for the smooth loop group. These bundles will be denoted by P pol G and P per G (the "per" stands for "periodic").
To demonstrate that these are principal bundles with the appropriate fibre we have to show two things: firstly, that the bundles are locally trivial; and secondly, that the fibres have an action of the appropriate loop group which identifies the fibre with that group. The second of these is straightforward, the first is simple for the smooth case but is surprisingly difficult for the polynomial loop group. We shall only consider the cases of U n , SU n , and SO n .
Definition 3.4 Let G be a compact, connected Lie group, g its Lie algebra. We define P per G and P pol G as follows: 2. P pol G ⊆ P per G consists of those paths of the form η ξ γ for some ξ ∈ g and γ ∈ L pol G.
Recall from section 3.1 that for ξ ∈ g the path η ξ : R → G is defined as the path t → exp(tξ).
Observe that a path in P per G is completely determined by its values on the interval [0, 1]. However, not every smooth path [0, 1] → G defines an element of P per G: one needs certain conditions on the derivatives at the endpoints. Defining P per G as we have seems the simplest way to state these conditions. It will sometimes be useful to consider an element of P per G to be a pair (g, α) ∈ G × PG such that α(t + 1) = gα(t). Here PG is all smooth paths R → G. Although g is completely determined by α, this viewpoint makes it more explicit.
We shall now investigate the desired properties of these spaces. Neither is a group (unlike the analogous continuous situation), but the group G acts in two ways: Lemma 3.5 The group G acts on P per G by two actions: These actions restrict to actions on P pol G. For both actions, the action of G on itself by conjugation makes the projection P per G → G G-equivariant (hence also for P pol G → G).
Proof. Let g ∈ G and α ∈ P per G. Both gα and gαg −1 are smooth paths in G so we only need to check the periodicity condition. Let h = α(t + 1)α(t) −1 . Then: This also proves the statement about the induced action on G.
If α ∈ P pol G then α is of the form η ξ γ for some ξ ∈ g and γ ∈ L pol G. Let h be either g −1 or 1 G . Then gη ξ γh = η (Ad g ξ) gγh. As L pol G is closed under left and right multiplication by G, this lies in P pol G as required.
Proposition 3.6 Define an action of LG on P per G by sending (α, γ) ∈ P per G × LG to the path t → α(t)γ(t). This action is well-defined and identifies the fibres of P per G → G with LG. It restricts to an action of L pol G on P pol G and identifies the fibres of P pol G → G with L pol G.
Proof. The path t → α(t)γ(t) is a smooth path R → G (considering γ as a periodic path). We need merely check the periodicity condition. Since γ(t + 1) = γ(t) for all t ∈ R, we have: Hence αγ ∈ P per G. This also shows that αγ lies in the same fibre as α.
Conversely, suppose that α, β ∈ P pol G lie in the same fibre. We need to show that the loop t → α −1 (t)β(t) is a polynomial loop. Let α = η ξ 1 α and β = η ξ 2 β where α and β are polynomial loops. Since α and β lie in the same fibre, exp(ξ 1 ) = exp(ξ 2 ). Thus: By proposition 3.1, the two terms in the centre give a polynomial loop, hence γ is a polynomial loop.
To complete the proof of the proposition, we need to show that no fibre of P pol G → G is empty, whence also no fibre of P per G → G is empty. As G is a compact, connected Lie group, for each g ∈ G there is some ξ ∈ g such that exp(ξ) = g. The path η ξ lies in P pol G (and thus in P per G) and is in the fibre above g. Thus the fibres are non-empty.
Proving that P per G is locally trivial is relatively straightforward. The case of P pol G is harder. Therefore we deal with P per G quickly now before passing to the-for this paper-more relevant case of the polynomial loops in the next section.
Proposition 3.7 The space P per G is locally trivial over G.
Proof. To prove this, we require local sections. Let g ∈ G. Let ξ ∈ g be such that exp(ξ) = g. Let ρ : [0, 1 2 ] → [0, 1 2 ] be a smooth surjection which preserves the endpoints and is constant in a neighbourhood of each endpoint. Let φ : V → U be a chart for G with U a neighbourhood of g such that φ −1 (g) = 0.
For h ∈ U, define a path α h : [0, 1] → G by: By construction, α h is continuous. Since α h is constant in a neighbourhood of 1 2 and is smooth either side, it is smooth. Moreover, as it is constant in neighbourhoods of 0 and 1, the concatenation α h ♯(α h (1)α h ) is smooth. Hence α h extends via the formula: for t ∈ [0, 1) and n ∈ Z, to a smooth path Clearly, α h (1) = h. Also, the assignment h → α h is smooth. Therefore, h → α h is a local section of P per G in a neighbourhood of g.

The Polynomial Path Space
The case of the polynomial path space is harder. Regarding local sections, it would appear from the definition that there are natural local sections, namely g → η ξ where exp(ξ) = g. However, except in the case of the unitary group, there is in general no way to choose ξ smoothly in g for all points g ∈ G (it is always possible to do so for an open dense subset, but this is not good enough).
In fact, we are not able to prove that P pol G → G is locally trivial for all compact, connected G at this time. The methods we employ work on a caseby-case basis. This is sufficient for our needs as we are mainly interested in ordinary vector bundles with inner products and thus in the structure groups U n and SO n . We shall prove that P pol G → G is locally trivial for these groups and also for SU n . There is no a priori reason why the argument for SO n should not extend to Sp n , using quaternionic structures in place of complex structures but we feel that this case is outside the focus of this paper.
The following result will prove useful in examining the structure of P pol G in terms of P pol U n .
Lemma 3.8 Let G be a compact, connected Lie group. Consider G as a subgroup of some U n . Then P pol G = P per G ∩ P pol U n .
Proof. Clearly P pol G ⊆ P per G ∩ P pol U n . For the converse, let α ∈ P per G ∩ P pol U n . Then α = η ξ γ for some ξ ∈ u n and γ ∈ L pol U n . Now exp(ξ) = η ξ (1) = α(1) ∈ G since α ∈ P per G. Choose ζ ∈ g such that exp(ζ) = exp(ξ). Then: By proposition 3.1, η −ζ η ξ is a polynomial loop in U n . Since α and η ζ both take values in G, η −ζ η ξ γ must also take values in G. It thus lies in LG ∩ L pol U n which is, by definition, L pol G. Therefore α is of the form η ζ β with ζ ∈ g and β ∈ L pol G. Hence α ∈ P pol G.

The Unitary Group
In the case of U n , there are local sections of the form g → η ξ where exp(ξ) = g. This will follow from lemma 3.2.
Proposition 3.9 The space P pol U n is locally trivial over U n .
Proof. Let s ∈ iR. Let V s ⊆ U n be the open subset consisting of those operators which do not have −e s as an eigenvalue. Let v s ⊆ u n be the open subset consisting of those operators which have eigenvalue in the interval (s−iπ, s+iπ). The exponential map restricts to a diffeomorphism exp: v s → V s . Let log s : V s → v s be its inverse.
For a direct construction, define the s-logarithm log s : T {−e s } → (s−iπ, s+iπ) as the inverse of the exponential map on this domain (note that this coincides with the above definition putting n = 1). Let g ∈ V s . Let E 1 ⊕ · · · ⊕ E l be the orthogonal decomposition of C n into the eigenspaces of g with eigenvalues λ 1 , . . . , λ l . Then log s g is the operator which acts on E j by multiplication by log s λ j .
It is a simple exercise to show that log s g ∈ Z(g) for any g and s such that log s g is defined, that log s g is locally constant in s, and that V s+2πi = V s and v s+2πi = v s + 2πi1 n .
The local sections of P pol U n → U n are α s : V s → P pol U n given by α s (g)(t) = exp(t log s g).

The Special Unitary Group
The method of the previous section works in U n because every point in U n is exp-regular; that is, is the image of a point in u n such that the exponential map is a diffeomorphism is a neighbourhood of that point. This is not true for a general Lie group. It is straightforward to show that the preimage of −1 ∈ SU 2 under exp: su 2 → SU 2 is a countable number of copies of CP 1 , hence −1 ∈ SU 2 is not exp-regular.
However, we can still prove that P pol SU n → SU n is locally trivial. The strategy is to use the fact that there is a point in u n around which the exponential map is a local diffeomorphism, and then use the fact that SU n → U n → S 1 is split.
Proposition 3.10 The map P pol SU n → SU n is locally trivial.
Proof. Choose a unit vector v ∈ C n . Define the representation σ : T → U n by σ(λ)v = λv and σ(λ) is the identity on v ⊥ .
Let s ∈ iR. Let V s ⊆ U n and v s ⊆ u n be as in the proof of proposition 3.9. Let α s : V s → P pol U n be the local section defined in that proposition.
Define β s : V s ∩ SU n → P per U n by: Recall that det exp(ξ) = e Tr ξ . Thus for g ∈ V S ∩ SU n : det α s (g)(−t) = e Tr(−t log s (g)) = e −t Tr log s (g) .
As g ∈ SU n , e Tr log s (g) = det g = 1 so Tr log s (g) = 2πik for some k ∈ Z. Thus t → e −t Tr log s (g) is the map t → z −k . Hence σ(det α s (g)(−t)) is a polynomial loop in U n . Thus β s (g)(t) ∈ P pol U n . Then as det •σ : T → T is the identity, det β s (g)(t) = 1 for all g, t. Hence β s (g)(t) ∈ SU n for all g, t. Thus by lemma 3.8, β s (g) ∈ P per SU n ∩ P pol U n = P pol SU n .

The Special Orthogonal Group
The situation for SO n is more complicated still. The problem here is with eigenvalue −1. It can be shown that g ∈ SO n is exp-regular if and only if its −1-eigenspace has dimension at most 2. The solution comes from the theory of unitary structures which we now describe.
Definition 3.11 Let E be a real vector space with an inner product. A unitary structure on E is an orthogonal map J : E → E such that J 2 = −1.
Proposition 3.12 Let E be a real even dimensional vector space with an inner product. The properties of unitary structures that we shall need are: 1. E admits a unitary structure. 6. Let g ∈ SO(E) be such that 1 is not an eigenvalue of g. Then log 0 (−g) is of the form ξ − πJ ξ for some ξ ∈ so(E) with exp(ξ) = g.

Let J be a unitary structure on E. Then
In the last property we use the inclusion SO(E) → U(E ⊗ C) to define log 0 : SO(E) ∩ V 0 → u(E). Since log 0 commutes with complex conjugation 2 , the image of SO(E) ∩ V 0 lies in so(E).
Proof. Property 1 is a standard property of complex structures whilst 2 is a simple deduction from the definition of a unitary structure. Therefore we start with property 3. 5. Diagonalise ξ over C. As ξ is a real operator, its eigenvalues and corresponding eigenspaces come in conjugate pairs. As ξ is skew-adjoint, its eigenvalues lie on the imaginary axis in C. Let W ⊆ E ⊗ C be the sum of the eigenspaces of ξ corresponding to eigenvalues of the form is with s > 0. Then W, resp. W ⊥ , is the sum of the eigenspaces of ξ corresponding to eigenvalues of the form is with s < 0, resp. s ≤ 0. The assumption on ξ implies that W = W ⊥ . Define J ξ on E ⊗ C to be the operator with eigenspaces W and W with respective eigenvalues i and −i. By construction, J 2 = −1 and J * J = 1. As the eigenspaces and eigenvalues of J come in conjugate pairs, J is a real operator and thus is a unitary structure.
Since J ξ is defined from the eigenspaces of ξ, it varies smoothly in ξ. Moreover, as the eigenspaces of J ξ decompose as eigenspaces of ξ, J ξ and ξ are simultaneously diagonalisable over C. Hence [ξ, J ξ ] = 0.
It is clear from the construction that if ζ and ξ can be simultaneously diagonalised and the eigenvalues of ζ have the same parity on the imaginary axis as the corresponding ones of ξ then J ζ = J ξ . In particular, J J = J and J ξ+cJ ξ = J ξ for c > 0.
6. Let F be the −1-eigenspace of g. Then E decomposes g-invariantly as F ⊕ F ⊥ . As g does not have 1 as an eigenvalue, log 0 (−g) is well-defined.
Since the decomposition of E is −g-invariant:

Theorem 3.13
The map P pol SO n → SO n is locally trivial.
Proof. We first describe a family of open sets which cover SO n . These will be the domains of the sections of P pol SO n . The family is indexed by the interval [−1, 1] and by elements of SO n . Let r ∈ [−1, 1]. Let W r be the open subset of SO n consisting of those g such that no eigenvalue of g (over C) has real part r. For g ∈ W r there is a g-invariant orthogonal decomposition of R n as E r −1 (g) ⊕ E 1 r (g) where the eigenvalues (over C) of g on E r −1 (g) have real part in the interval [−1, r] and on E 1 r (g) in the interval [r, 1]. Note that g cannot have eigenvalue 1 on E r −1 (g), even if r = 1, so as g ∈ SO n , E r −1 (g) must have even dimension.
Over each W r is a vector bundle with fibre E r −1 (g) at g (this will have different dimension on the different components of W r ). Over most W r 's this bundle is not trivial. Therefore we find smaller open sets over which we can trivialise it.
Let r ∈ [−1, 1] and g ∈ W r . Define W r (g) to be the open subset of SO n consisting of those h ∈ W r for which the orthogonal projection E r −1 (h) → E r −1 (g) is an isomorphism.
Over W r (g), therefore, the aforementioned vector bundle is trivial and of constant even dimension. Hence, we can choose a unitary structure J h on each E r −1 (h) which varies smoothly in h.
Extend J h to a skew-adjoint operator on R n by defining it to be zero on Define β r,g : W r (g) → P per SO n by: This is a smooth path in SO n since both log 0 (ǫ(h)) and J h lie in so n . It varies smoothly in h since both ǫ(h) and J h are smooth in h. Since ǫ(h) = h exp(−πJ h ), β r,g (h)(1) = h so it is a path above h. We need to show that it lies in P pol SO n . Now ǫ(h) respects the decomposition E r −1 (h) ⊕ E 1 r (h) of R n , therefore so does log 0 (ǫ(h)). Accordingly, write log 0 (ǫ(h)) = ξ r −1 + ξ 1 r . Consider the situation on E r −1 (h). Since exp(ξ r −1 ) = ǫ(h) = −h (all restricted to E r −1 (h)), by property 6, ξ r −1 = ζ − πJ ζ for some ζ ∈ so(E r −1 (h)) with exp(ζ) = h. Extend J ζ to R n by defining it to be zero on E 1 r (h). Let ξ = ζ+ξ 1 r . Then exp(ξ) = h, [ξ, J ζ ] = 0, and log 0 (ǫ(h)) = ξ − πJ ζ . Therefore:

The Polynomial Vector Bundles
Now that we have principal bundles, given a representation we can construct vector bundles. Let V be a finite dimensional vector space with an inner product, either real or complex. Let LV be the space of smooth loops in V and L pol V the space of polynomial loops. If V is complex then Let G be a compact, connected Lie group which acts on V by isometries. In the polynomial case, assume that G is one of U n , SU n , or SO n . Then LG acts on LV and L pol G acts on L pol V. Therefore we have vector bundles over G together with a bundle inclusion: We shall now give an alternative view of these vector bundles which will be more enlightening in terms of their structure.
Let PV be the full path space of V. Define τ : PV → PV to be the shift operator: (τβ)(t) = β(t+1). Let D denote the differential operator: (Dβ)(t) = dβ dt (t). There is a strong connection between these operators: D is the infinitesimal generator of the group of translations on PV and exp(D) = τ.
The motivation for considering these operators is that they give simple descriptions of LV and L pol V inside PV. The loop space, LV, is the +1-eigenspace of τ. The space of polynomial loops inside LV is the union of the finite dimensional D-invariant subspaces of LV.
In the complex case, we can write this as the linear span of the eigenvectors of D. This does not carry over to the real case, however, as the only eigenvectors of D are the constant maps.
Theorem 3.14 Let g in G. The fibre of P per V above g is the space of φ ∈ PV such that τφ = gφ.
The fibre of P pol V above g is the union of the finite dimensional D-invariant subspaces of the fibre of P per V above g.
Proof. An element of P per V in the fibre above g is represented by a pair (α, β) with α ∈ P per G above g and β ∈ LV. Any alternative representative is of the form (αγ, γ −1 β) for some γ ∈ LG.
Thus the map φ : R → V defined by φ ≔ αβ depends only on the element of P per V and not on the choice of representative. This satisfies: Hence τφ = gφ.
Now we consider the polynomial path space. We need to show that the fibre of P pol V above g is the union of the finite dimensional subspaces of the fibre of P per V that are D-invariant.
Let ξ ∈ g be such that exp(ξ) = g. This defines two actions on PV. The first is multiplication by η −ξ , α → η −ξ α, which maps P per,g V onto LV. The second is multiplication by ξ, α → ξα, extending the action of g on V to PV. As ξ is a finite dimensional operator, it has a minimum polynomial. This is true also of the second action on PV. Therefore any finite dimensional subspace of PV is contained in a finite dimensional ξ-invariant subspace. Moreover, the action of ξ on PV commutes with that of D so any finite dimensional Dinvariant subspace of PV is contained in a finite dimensional subspace that is both D-invariant and ξ-invariant.
Hence as ξ preserves both P per,g V and LV, when considering the union of finite dimensional D-invariant subspaces in either, it is sufficient to consider those that are in addition ξ-invariant.
We shall now show that W ⊆ LV is ξ and D-invariant if and only if η ξ W is ξ and D-invariant. This will establish the result.
The ξ-invariance is straightforward since ξ commutes with η ξ . Hence If W is ξ and D-invariant, then consider α ∈ η ±ξ W (the ± allows us to consider both directions at once). This is of the form η ±ξ β for some β ∈ W. Then: An immediate corollary of this is that the fibres of P per V and of P pol V are D-invariant. For P per V this follows from the fact that exp(D) = τ so D and τ commute. If we wish to emphasise the fibre, we shall refer to D as D g .
In the complex case, as D g is skew-adjoint, any element of the fibre of P pol V above g is thus the sum of eigenvectors of D g .
When viewing a fibre of P per V or P pol V as a subspace of PV, the corresponding element g ∈ G is not uniquely determined by any one path (contrast with the case of P per G or P pol G). Thus to keep track of the fibre, we shall often use the notation (g, φ).
There is an action of G on P per V and on P pol V given by the following equivalent definitions: We put in both of the top two descriptions to show that the two actions of G on P per G (and thus on P pol G) define the same action on P per V (and P pol V). This action preserves the subbundle P pol V and sends the operator D h to D ghg −1 .

Other Loop Bundles
The groups L pol G and LG act on other linear loop spaces besides the actions on L pol V and LV considered in the previous section. Whenever we have such an action then we can define an associated vector bundle over G. Moreover, whenever we have a map between loop spaces which is L pol G-equivariant or LG-equivariant then we get a similar map between the corresponding bundles.
In particular, we define the periodic L 2 -path space as Using the definitions of section 2 we define Proposition 3.15 The bundle P per V has a nuclear locally equivalent co-orthogonal structure with structure group Gl res .
Proof. The inclusion P 2 per,r V → P per V has fibrewise dense image and so defines a co-orthogonal structure on P per V. As both have structure group L pol G we can simultaneously trivialise both with respect to this group, thus the co-orthogonal structure is locally equivalent. The action on L pol G on L 2 r V factors through Gl res so the co-orthogonal structure has structure group Gl res . Finally, the inclusion L 2 r V → LV is a nuclear map so the co-orthogonal structure is nuclear.
In section 2 we constructed an isomorphism L 2 r V → L 2 V but this isomorphism was not L pol G-equivariant and so we do not automatically get an associated isomorphism of bundles. It is, however, possible to transfer this isomorphism to the bundles.

Lemma 3.16
The operator cos r D defines a bundle isomorphism P pol V → P pol V which extends to an isomorphism P 2 per,r V → P 2 per V.
Proof. This is the global version of theorem 2.7. All we need to do to apply that theorem is to show that we can choose the operator D consistently on the fibres of P pol V. The operator D of the previous section will do.
Corollary 3.17 If we enlarge the structure group of P per V to EL Gl(V) then this co-orthogonal structure becomes locally trivial.
Proof. By viewing the original structure group of P per V as EL Gl(V) we can find a subgroup isomorphic to L pol U(V) which acts unitarily on L 2 r V. Thus the co-orthogonal structure constructed above is locally trivial.
Once again, there is an action of G on all of this structure which maps down to the conjugation action on the base. The action on the vector bundles is as for the vector bundles of smooth and polynomial paths. This action preserves the extra orthogonal structure as the action of G on each of the linear spaces involved preserves this structure.

Free Loop Bundles
The goal of this section is to construct the Hilbert subspace of the vector bundle LE → LM, where E → M is a real or complex vector bundle. The first part of this construction involves defining the polynomial loop bundle, L pol E → LM, and proving that it is a locally trivial vector bundle modelled on L pol F n , for F one of R or C. Once this has been defined, we can thicken it to a bundle L 2 r E modelled on L 2 r F n . This defines the locally equivalent co-orthogonal structure of LE. We then construct a bundle isomorphism L 2 r E → L 2 E similar to the linear one of section 2.2. To prove theorem A we apply this construction with E = TM using the canonical identification of LTM with TLM.
In section 4.4 we discuss the basic properties of the polynomial loop bundle, and thus of L 2 r E. In particular we consider the action of the group of diffeomorphisms of the circle. The natural action on LE does not preserve the polynomial subbundle but it can be modified to an action which does.

Notation
Let M be a smooth finite dimensional manifold without boundary. Let G be one of U n , SU n , or SO n . Let F be the corresponding field. Let Q → M be a principal G-bundle. Let E ≔ Q × G F n be the corresponding vector bundle. As G preserves the inner product on F n , E carries a fibrewise inner product. Let ∇ be a covariant differential operator on E coming from a connection on Q.
We think of a point in a fibre Q p as being an isometry F n → E p . We shall also use the adjoint bundle associated to Q, Q ad ≔ Q × conj G where G acts on itself by conjugation. This is a bundle of groups. A point in a fibre Q ad p is an isometry of E p to itself.
It is a standard result that the loop and path spaces of E, LE and PE, form vector bundles over, respectively, the loop and path spaces of M, LM and PM, with frame bundles the loop and path spaces of Q, LQ and PQ, and adjoint bundles the loop and path spaces of Q ad , LQ ad and PQ ad .
It is not strictly relevant, but should be remarked that whilst LM (and the other loop spaces) are infinite dimensional smooth manifolds, PM (and the other path spaces) are not. They are smooth spaces but are not locally linear. The reason that this does not concern us is that the path spaces are used merely as a "background canvas" on which the interesting structure is painted. See [KM97] for more details on infinite dimensional smooth spaces.

The Holonomy Operator
Recall that we can view M in PM as the subspace of constant paths and LM in PM as the subspace of periodic paths (of period 1). For X each of E, Q, and Q ad , define a bundle P M X → M by restricting PX to this subspace. The fibre of P M X above p ∈ M is thus P(X p ). In terms of the original bundles E, Q, and Q ad we have the following descriptions.
As with M inside LM and PM, G sits inside LG and PG as the constant loops.
In the middle line, the action is as a subgroup, in the third line the action is via conjugation. The covariant differential operator defines a parallel transport operator. To describe this then for t ∈ R let e t : PM → M be the map which evaluates at time t and let X t → PM be the bundle e * t X; thus X t γ = X γ(t) . We use the same notation for the restriction of these bundles to LM. The covariant differential operator defines three compatible families of bundle maps ψ t X : X t → PX. The properties of these maps are: For the second, note that e t ψ s X is a map from X s to X t . This compatibility relation is the statement that if one parallel transports from time s to time t and then on from time t to some when else, it is the same as transporting straight from s to ones final time. For the last, over LM then X t+1 = X t so the domains and codomains of these maps are the same. This property is then an application of the fact that the parallel transport operator is intrinsic to M, therefore the parallel transport from X t to X s is the same as that from X t+1 to X s+1 .
Let P M,t X ≔ e * t P M X, so P M,t α X = P(X α(t) ). The parallel transport operators extend to bundle equivalences: with the property that e s (Ψ t X α) = (e s ψ t X )(α(s)). Note that these equivalences have been chosen such that (Ψ t X α)(s) always lies in X s no matter which t was the starting point.
Let P L X be the restriction of PX to LM. Thus P L X consists of paths in X which project down to loops in M. There is an obvious inclusion of LX in P L X and it is straightforward to recognise this submanifold: LX consists of those paths in P L X which are themselves periodic. Note that for any path β in P L X then β(t + 1) and β(t) both lie in the same fibre of X → M.
Thus in the right hand side of (4.4) (restricted to LM), it is straightforward to recognise the subbundles consisting of the loops. We wish to transfer this recognition principle to the left hand side of (4.4). We do this using the holonomy operator.
Definition 4.1 On LM, define the fibrewise operators h X : X 0 → X 0 by h X = e 1 ψ 0 X . Over PM, e 1 ψ 0 X is a map X 0 → X 1 . Over LM then X 0 = X 1 so h X is as defined. The fibres of Q ad act on each of E, Q, and Q ad : on E the action is by definition, on Q and on Q ad by composition.

Lemma 4.2 The operator h E is a section of Q ad,0 . The operators h E , h Q , and h Q ad satisfy: h Q ad (p)h E = h E p, and h Q (q) = h E q. Thus h E determines both h Q and h Q ad .
Proof. Since e 1 ψ 0 E is a fibrewise isometry E 0 → E 0 , it is a section of Q ad,0 . Then from (4.1), for p ∈ Q ad,0 , q ∈ Q 0 , v ∈ E 0 , and w ∈ F n ⊆ PF n : (h E q)w = (e 1 ψ 0 E q)w = e 1 (ψ 0 E (qw)) = e 1 (ψ 0 Q (q)w) by equation (4.1) = e t ψ 0 X h X . (4.4), the subbundle LX of P L X corresponds to: {α(t) ∈ P M,0 X : h X α(t + 1) = α(t)}.

Corollary 4.4 Under the bundle isomorphism of equation
Proof. An element α ∈ P M,0 X is mapped to a loop in P L X if and only if (Ψ 0 X α)(t + 1) = (Ψ 0 X α)(t) for all t ∈ R. The left hand side of this simplifies to: e t+1 (Ψ 0 X α) = (e t+1 ψ 0 X )(α(t + 1)) = (e t ψ 0 X )(h X α(t + 1)) whilst the right hand side simplifies to: . Since e t ψ 0 X : X 0 → X t is an isomorphism, this implies that Ψ 0 X α is a loop if and only if h X α(t + 1) = α(t) for all t ∈ R.
For X each of E, Q, and Q ad let Y be F n or G as appropriate and define As P per Y → G is a G-equivariant bundle where the G action on the base is by conjugation, P M per X is a fibre bundle over Q ad . Via the evaluation map e 0 : LM → M we therefore get bundles e * 0 P M per X → Q ad,0 . For a section χ : LM → Q ad,0 and X each of E, Q, and Q ad define Corollary 4.5 For X each of Q, Q ad , and E, Ψ 0 X restricts to a bundle isomorphism

Other Loop Bundles
The advantage of the above descriptions of LE, LQ, and LQ ad in terms of the holonomy operator is that they can be generalised using other bundles over G.
All that is needed is to have a G-equivariant bundle over G with respect to the conjugation action on the base. In particular, we can take the bundles that we constructed in section 2.
Definition 4.6 Define the following spaces where in the last two lines we carry over the associated orthogonal structure. For a section χ : LM → Q ad,0 we therefore have associated bundles L χ abc X ≔ χ * e * 0 P M abc X. The following is immediate.

Lemma 4.7
The bundles L χ abc X are locally trivial fibre bundles modelled on L abc Y. When the original linear loop space was a subspace of smooth loops and we take the inverse of the holonomy operator as our section then we can use Ψ 0 X to transfer this bundle to a subbundle of the bundle of smooth loops.
Corollary 4.10 The bundle LE → LM has a nuclear locally equivalent co-orthogonal structure with structure group Gl res . This structure depends naturally on the inner product and compatible connection on E.
The bundle L pol E has a more concrete description in terms of the connection on E. For any path γ : R → M, the connection on E defines a covariant differential operator D γ : Γ R (γ * E) → Γ R (γ * E); that is, D γ : P γ E → P γ E. As the map Ψ 0 E was constructed using parallel transport, it (rather, its inverse) takes D γ to the operator d dt acting on P M,0 E. If γ happens to be a loop, D γ restricts to an operator on L γ E. As Ψ 0 E identifies L γ E with the fibre of P per E → Q ad,0 above h −1 E (γ), it takes D γ to the operator D h −1 E (γ) . Hence L pol,γ E can be constructed from the action of D γ on L γ E in the same fashion as P pol,g F n from P per,g F n , namely as the union of the finite dimensional D γ -invariant subspaces of L γ E. In the complex case, L pol,γ E is the span of the eigenvalues in L γ E of D γ .
This description is more in the spirit of [Mor01]. However, one still has to show that the resulting object is a locally trivial vector bundle over LM and the simplest method for that is by considering principal bundles as above.
We can give an explicit formula for the inner product on the fibres of L 2 r E in terms of this operator D γ . Let ·, · be the inner product on L 2 E. The operator D γ defines an isomorphism cos r D γ : L 2 r E → L 2 E and so the inner product on L 2 r E is given by α, β γ = (cos r D γ ) −1 α, (cos r D γ ) −1 β γ .

Properties of the Polynomial Bundle
The construction of the polynomial loop bundle started from a connection on the original bundle over M. However, it only actually used the map ψ 0 X : X 0 → PX defined by the parallel transport operator. Thus as far as the polynomial loop bundle is concerned, having a connection is overkill. The connection is useful, though, as it implies that the polynomial loop bundle came from structure on the original manifold M and thus one can hope for more structure on the polynomial loop bundle than has yet been described. In this section, we shall investigate this.
Before examining the interesting properties of the polynomial loop bundle, we list some basic ones that are fairly obvious: Proposition 4.11 Let M be a finite dimensional smooth manifold, E 1 , E 2 → M finite dimensional vector bundles over the same field with inner products and connections compatible with the inner products.
1. Let E = E 1 ⊕ E 2 orthogonally and equip E with the direct sum connection. Then 4. Let ψ : E 1 → E 2 be a bundle isomorphism which preserves the inner products and connections. Then ψ defines an isomorphism L pol ψ : L pol E 1 → L pol E 2 .
5. Suppose that E 1 with its inner product is a sub-bundle of E 2 and that the covariant differential operator on E 1 is of the form p∇ where p : E 2 → E 1 is the orthogonal projection and ∇ is the covariant differential operator on E 2 . Then it is not necessarily the case that L pol E 1 = L pol E 2 ∩ LE 1 .
Proof. Only the last of these is not immediate from the construction. Let E 2 be the bundle S 1 × C 2 and E 1 the bundle S 1 × C 1 . Include E 1 in E 2 via the map (t, 1) → (t, 1 √ 2 (1, e 2πit )).
The loop space of E 1 is LS 1 × LC and of E 2 is LS 1 × LC 2 . The polynomial loop space of E 2 is LS 1 × L pol C 2 . The inclusion LE 1 → LE 2 is given by: Therefore LE 1 ∩ L pol E 2 consists of those loops β such that both β and e 2πiγ β are polynomials. We can choose γ such that whenever β is polynomial then e 2πiγ β is not. Hence there is some γ such that above γ the fibres of LE 1 and L pol E 2 intersect trivially.
The advantage of having the polynomial structure defined using a connection on the original bundle is the relationship with the diffeomorphism group of the circle. For σ : S 1 → S 1 smooth (not necessarily a diffeomorphism), γ : S 1 → M, and α ∈ L γ E, the following is a simple application of the chain rule: From this formula, two results can be derived:

Proposition 4.12
1. The action of Diff(S 1 ) on LE does not preserve the sub-bundle L pol E. The subgroup of Diff(S 1 ) which does preserve the sub-bundle L pol E is S 1 ⋊ Z/2 where the non-trivial element in the Z/2-factor is the diffeomorphism t → −t.
2. Let ∇ a and ∇ b be two different connections on E. The two polynomial bundles so defined are different.
Proof. We shall consider the complex case so that we may talk about eigenvectors and eigenvalues of D γ . The real case may be deduced from this.
1. For this, consider the situation over a constant loop. There, LE, resp. L pol E, is E ⊗ LC, resp. E ⊗ L pol C. The action of Diff(S 1 ) on LE is given by its action on LC. Thus if σ ∈ Diff(S 1 ) preserves L pol E then it must preserve L pol C within LC.
The map t → e 2πit lies in L pol C. It is also the identification of S 1 with T. Under σ this transforms to t → e 2πiσ(t) . As σ is a diffeomorphism of S 1 , this map must still be an identification of S 1 with T. The only polynomials which do this are those of the form t → νe ±2πit for ν ∈ T.
The converse is direct from the equation 4.5 since if σ ∈ S 1 ⋊ Z/2 then σ ′ = ±1 so: Hence σ maps eigenvectors of D γ to eigenvectors of D γ•σ and thus preserves L pol E.
2. As ∇ a and ∇ b are different, there will be some loop γ such that D a γ and D b γ differ. The difference will be a section Φ of the bundle u(γ * E) → S 1 , in other words an element of L γ u(E).
If L a pol,γ E = L b pol,γ E then both are preserved under D a γ and D b γ , hence under their difference. Thus Φ must be an element of L pol u(E).
By examining equation 4.5, we see that under the action of a smooth selfmap σ of the circle, Φ transforms to (Φ • σ)σ ′ . It is then a simple matter to find σ such that this is no longer a polynomial. Hence even if we were unlucky enough initially to choose a loop γ with L a pol,γ E = L b pol,γ E then we can find some other loop γ • σ over which the fibres of the polynomial bundles differ.
It is straightforward to show that the result about the action of Diff(S 1 ) on L pol E generalises to the statement that the subgroup of Diff(S 1 ) which preserves L ? E is Diff(S 1 ) ∩ L ? C where the "?" represents some class of regularity of loop.
In the light of this result, it is perhaps surprising that there is an action of Diff(S 1 ) on L pol E which covers the standard action of Diff(S 1 ) on LM. This comes about because the Diff(S 1 )-action preserves the parallel transport operator. Since all else was derived from that, we can make Diff(S 1 ) act on L pol E.
We start with the group Diff + 0 (S 1 ) of orientation and basepoint preserving diffeomorphisms. Since the whole diffeomorphism group is the semi-direct product of this with S 1 ⋊ Z/2, an action of this group together with the above action of S 1 ⋊ Z/2 will give an action of the whole diffeomorphism group.
With these two isomorphisms, the square: does not commute. To make it commute, we need to transfer one action of σ from one side to the other. Clearly, the action of σ on P L E restricts to the standard action on LE which we know does not preserve L pol E.
It is also true that the action of σ on P M,0 E preserves L M,0,h −1 E E and L M,0,h −1 E . Thus is because the holonomy operator h E is equivariant under the action of Diff + 0 (S 1 ). Therefore, the action of σ on P M,0 E when transferred to P L E also restricts to an action on LE and on L pol E.
In formulae, the two actions of Diff + 0 (S 1 ) are as follows: any element of P γ E can be written as j f j ψ 0 E v j where {v 1 , . . . , v n } is a basis for E γ(0) . The usual action is:

Spin Bundles
In finite dimensions, a spin bundle is an oriented bundle with an orthogonal structure together with a lift of the structure group from SO n to a certain Z/2central extension, Spin n .
This can be generalised to infinite dimensions. There is a group SO J which takes the place of SO n . This has a central extension Spin J similar to the central extension Spin n of SO n , although by S 1 rather than Z/2. The group SO J acts naturally on a Hilbert space and is the identity component of another group written O J . The groups SO J and O J are deformation retracts of groups S Gl J and Gl J . These groups all act naturally on a standard separable real Hilbert space. The reason for the notation for SO J and the other groups is that their definition depends on a choice of skew-adjoint operator J acting on a real Hilbert space, say H. The definition is very similar to that of the group Gl res (H) of section 2.1 except that the group Gl res (H) was defined in terms of an operator on the complexification of H. The distinction is important. We know that the completed tangent bundle of a loop space has a SO res -structure but this is not always the same as an SO J -structure. It can be shown that there is a choice of operator J on L 2 (S 1 , R n ) such that SO J = SO res if and only if n is even.
Definition 5.1 Let X be a smooth manifold.
1. Let ζ → X be a bundle of Hilbert spaces over X with structure group SO J . A spin structure on ζ consists of the following data.
(a) A lift of the structure group of ζ from SO J to Spin J .
(b) A connection on the principal Spin J -bundle. If a connection on ζ has already been specified the connection on the Spin J -bundle should be a lift of that on ζ.
2. Let ξ → X be a vector bundle. A (co-)spin structure on ξ consists of the following data.
(a) A (co-)orthogonal structure on ξ with structure group Gl J .
(b) A spin structure on the associated bundle of Hilbert spaces, where the orthogonal structure is used to reduce the structure group of the Hilbert spaces from Gl J to SO J .
3. We further classify (co-)spin structures according to the classification of their (co-)orthogonal structures. 4. A (co-)spin manifold is a manifold together with a choice of (co-)spin structure on its tangent bundle.
Remark 5.2 An astute student of finite dimensional spin theory will have noticed a significant departure in this definition. In finite dimensions, the connection on the spin bundle is assumed to be a lift of a pre-existing connection on the original bundle. Here, we do not assume that such a connection exists (though if it does we require the lift). Whilst we could easily include this assumption for the bundle of Hilbert spaces, it would not necessarily be possible in the more general case. To have the connection on the spin bundle related to a connection on the original bundle we would have to be able to simultaneously trivialise the original bundle and orthogonally trivialise the Hilbert bundle: namely, to have a locally trivial (co-)orthogonal structure. Obviously, it is more desirable to have the connection be related-in some fashion-to structure on the original bundle but we do not insist on this being a requirement.

String Manifolds
There is an obstruction theory to a manifold being spin. In finite dimensions, one needs the first two Stiefel-Whitney classes to vanish. In infinite dimensions one needs a certain class in H 1 (X; Z/2) to vanish (to allow the reduction from O res to SO res ) and a class in H 3 (X; Z) (to allow the lift). When X = LM, these classes are respectively the transgression of the second Stiefel-Whitney class of M and a certain class λ ∈ H 4 (M; Z) which satisfies 2λ = p 1 (M). An interesting question to ask is what structure on M defines a spin structure on LM. There are several ways to answer this, [ST04] being an important one which also gives rise to much more structure. Let us outline a more modest approach.
Let M be an oriented Riemannian manifold. Let Q → M be the principal SO n -bundle of the tangent bundle. The class λ ∈ H 4 (M; Z) is the obstruction to choosing a class in H 3 (Q; Z) which restricts to the canonical generator of H 3 (SO n ; Z) on each fibre of Q → M. Now a class in the third integral cohomology class of a space is represented by a gerbe so let us choose a gerbe on Q which represents this class in H 3 (Q; Z). As explained in [Bry93], there is a natural way to use a gerbe to define a line bundle on the corresponding loop space. Thus we get a line bundle on LQ. If we have chosen our gerbe well, this line bundle-or rather its unit circle-defines the required lift from SO res to Spin res .
The Levi-Civita connection on M loops to a connection on LM, which is also orthogonal and torsion-free so it is reasonable to also call this the Levi-Civita connection. Thus we just need to lift this connection to our principal Spin res -bundle. As is also explained in [Bry93], gerbes can be equipped with connections and this structure loops to a connection on the corresponding line bundle. If we choose this connection data properly on the gerbe, the connection on the corresponding line bundle combines with the Levi-Civita connection on LM to define a connection on the Spin res -bundle. Hence we have a spin structure on LM.
Whatever structure one chooses, the following proposition is immediate.

Proposition 5.3
Let M be a Riemannian manifold of finite even dimension. Suppose that we can give M the structure needed to make LM a spin manifold with its usual orthogonal structure. Then the same structure makes LM a co-spin manifold with the co-orthogonal structure constructed in this paper.
Proof. By assumption, the bundle L 2 TM admits a spin structure. We transfer this to L 2 r TM via the isometric isomorphism L 2 r TM L 2 TM which is an isomorphism of SO J -bundles. Note that as the dimension of M is even, SO res = SO J .
Let us say that a manifold M is a string manifold if it equipped with sufficient structure to define a spin structure on LM, and thus also a co-spin structure.
Using the work of [Sta] we obtain the following result.
Corollary 5.4 Let M be a string manifold. Then LM admits a S 1 -equivariant Dirac operator.