The Clifford Algebra Bundle on Loop Space

We construct a Clifford algebra bundle formed from the tangent bundle of the smooth loop space of a Riemannian manifold, which is a bundle of super von Neumann algebras on the loop space. We show that this bundle is in general non-trivial, more precisely that its triviality is obstructed by the transgressions of the second Stiefel-Whitney class and the first (fractional) Pontrjagin class of the manifold.


Introduction
The bundle of Clifford algebras Cl(X) constructed from the tangent bundle over a Riemannian manifold X is fundamental to spin geometry.In particular, X has a spin c structure if and only if it is oriented and the Dixmier-Douady class of Cl(X) vanishes.In particular, the spin c condition is related to (partial) triviality of Cl(X).The purpose of this paper is to obtain similar results for the analogous bundle on the loop space LX of an oriented Riemannian manifold, which is fundamental to string geometry (i.e., to spin geometry on the loop space).
The Riemannian metric on X (which we assume to be oriented throughout) induces a natural metric on the smooth loop space.Forming the (infinite-dimensional) algebraic Clifford algebra on each tangent space is unproblematic, but in order to make the setting amenable to analysis, we must complete these algebras in some way.
It is a fact that the infinite-dimensional Clifford algebra has a unique C * norm, and completing the fibers with respect to this norm yields a bundle of C * -algebras.However, it turns out that this bundle is always trivial, hence does not encode any information on whether the loop space satisfies a spin condition.
Instead, we consider a fiberwise completion in a suitable weak topology, which leads to a continuous bundle A LX of von Neumann algebras.The canonical grading of the Clifford algebra carries over to the von Neumann completion, and in fact, the fibers are super factors of type I, meaning that the fibers are type I von Neumann algebras with trivial graded center.
Super factors of type I come in two stable isomorphism classes: Those of even kind, which are stably isomorphic to C, and those of odd kind, which are stably isomorphic to Cl 1 .It turns out that if the dimension of X is even, then the fibers of A LX are of even kind, while otherwise, they are of odd kind.
The classifying space of the automorphism group Aut(A) of a non-trivially graded, properly infinite super factor of type I turns out to be a product of Eilenberg-MacLane spaces, BAut(A) ≃ K(Z, 3)×K(Z 2 , 1).Hence bundles A → X with typical fiber A are classified by two characteristic classes DD(A) ∈ H 3 (X, Z), or(A) ∈ H 1 (X, Z 2 ), which we call the Dixmier-Douady class and the orientation class.The first is an analog of the class first defined in [10].The second class comes from the fact that we work with bundles of super algebras, and that all automorphisms considered are required to respect the grading.
Our main result is the calculations of these characteristic classes for the loop space Clifford algebra bundle A LX , which in particular shows that it is non-trivial in many cases.Explicitly, we find: Main Theorem 1.1.Let X be an oriented Riemannian manifold of dimension d ≥ 5. Then where τ denotes transgression, and where p 1 (X) and w 2 (X) are the first Pontrjagin class, respectively the second Stiefel-Whitney class.Moreover, if X is spin, then DD(A LX ) equals the transgression of the fractional Pontrjagin class 1 2 p 1 (X).In fact, we have a more refined version of the above theorem (see Theorem 4.16): There is a canonical characteristic class S(X) ∈ H 3 (LX, Z) on the loop space such that 2 • S(X) = τ (p 1 (X)), which we call loop spin class (see Definition 4.12), and DD(A LX ) is expressed in terms of this class.The interesting point is that while the fractional Pontrjagin class 1  2 p 1 (X) only exists when X is spin, the corresponding class S(X) on the loop space always exists.
The typical fiber of the bundle A LX is a suitable completion A d of the algebraic Clifford algebra on H d = L 2 S 1 , R d .The loop group LSO(d) acts naturally on the von Neumann completion A d by Bogoliubov automorphisms, and it turns out that A LX can be written as an associated bundle to the looped frame bundle LSO(X) of X.Our proof of Theorem 1.1 is then based on the fact that the map ΩSO(d) → Aut(A d ) induces an isomorphism on π k for k ≤ 2; in other words, Aut(A d ) is the Postnikov truncation of ΩSO(d).
The relation of our Clifford von Neumann algebra bundle to other objects from loop space spin geometry, such as the transgression of the Chern-Simons gerbe [34] and the loop space spinor bundle [16,17,18] is best understood using the language of 2-vector bundles [15].This point of view is discussed in [19, Section 1].
Recall that a spin manifold X is called a string manifold if the fractional Pontrjagin class 1 2 p 1 (X) vanishes.Our theorem therefore implies in particular that the loop space Clifford algebra bundle A LX of a string manifold X is trivializable.Hence A LX admits a bundle of irreducible modules, the loop space spinor bundle, so that LX is spin.The converse of the above statement is false, as the transgression τ 1 2 p 1 (X) may vanish without 1  2 p 1 (X) being zero.It is a general fact that such converses require the extra condition of fusion [31,32,33].In the present context, it turns out that the bundle A LX is the transgression of a certain bundle of free fermion conformal nets [12], which (on a spin manifold) is classified by 1  2 p 1 (X).This will be discussed in future work.

Bundles of super von Neumann algebras
In this section, we define bundles of von Neumann algebras with fiber a super factor of type I, a notion that will be explained in the next subsection.Throughout the paper, all Hilbert spaces are assumed to be separable and all von Neumann algebras are σ-finite.

Super factors of type I and their classification
A super von Neumann algebra is a von Neumann algebra A together with a normal (i.e., ultraweakly continuous) involutive * -automorphism γ.Such an automorphism gives a direct sum decomposition The graded center of A is defined as where the graded commutator is defined by [a, b] = ab − (−1) |a||b| ba on homogeneous elements and extends to all of A by bilinearity.
Definition 2.1 (super factor).We say that a von Neumann algebra A is a super factor of type I if it is type I (as an ungraded von Neumann algebra) and its graded center is equal to C. We say that A is properly infinite if its even part A 0 is properly infinite in the usual sense.
Lemma 2.2.Let A be a super factor.Then the ungraded center Z un (A) is a graded subalgebra, which is isomorphic to either C (trivially graded) or to C ⊕ C (with the grading operator given by swapping the two summands).
Proof .Let a = a 0 + a 1 ∈ Z un (A).Then comparing the odd and even components of ab and ba, for b homogeneous, shows that both a 0 , a 1 ∈ Z un (A).Hence Z un (A) is a graded subalgebra.The subalgebra Z un (A) is an abelian von Neumann algebra, hence isomorphic to L ∞ (X) for some measure space X.As it is a graded subalgebra, we can write Z un (A) = Z 0 ⊕ Z 1 for its graded components.That A is a super factor implies that Z 0 = C • 1, as any even element in the ungraded center is also an element of the graded center, which is trivial by assumption.Hence Then there exists a projection p ̸ = 0, 1 in Z un (A).Write p = λ1 + p 1 with λ ∈ C, p 1 ∈ Z 1 (A).Then λ ̸ = 0, as projections cannot be purely odd.Now, for any other non-zero projection q = µ1 + q 1 with pq = 0, we have Considering the odd part, we obtain λq 1 + µp 1 = 0. Hence (as both λ, µ ̸ = 0) q 1 is a multiple of p 1 , so that q lies in the span of 1 and p.As von Neumann algebras are generated by their projections, this implies that 3 (even/odd kind).We say that a super factor A of type I is of even kind if its ungraded center Z un (A) is trivial, and of odd kind otherwise.
Example 2. , where Cl 1 is the complex Clifford algebra of degree one.
Theorem 2.7.Let A be a super factor of type I.If A is of even kind, then it is isomorphic to B(H), with the grading operator given by conjugation with some unitary involution Γ of H.
If A is of odd kind, then it is isomorphic to B(H) ⊕ B(H), with grading operator given by exchanging the two summands.
Proof .We distinguish by the two cases of Lemma 2.2.(i) If Z un (A) = C, then A is an ordinary type I factor, hence isomorphic to B(H).As any automorphism of B(H) is inner, the grading automorphism γ is given by conjugation with a unitary Γ, which must satisfy Γ 2 = z • 1 for some z ∈ U(1) as γ is an involution.If w is some square root of z, then Γ = wΓ also implements γ and satisfies Γ2 = 1.
(ii) If Z un (A) = C ⊕ C, then (as A is type I), we have A = B(H) ⊕ B(K) for Hilbert spaces H and K.The grading operator γ of A is then given by conjugation with a unitary Γ = u z x w on H ⊕ K. Since the restriction of γ to Z un (A) swaps the two factors, we have This implies that u and w are zero, hence x and z are unitary.After modifying Γ by an element of U(1) as above, we may assume that Γ2 = 1.Then x and z are inverses to each other, and identifying K with H using x gives an isomorphism of A to a super factor of type I of the form claimed. ■ Remark 2.8.A reformulation of the previous theorem is that each super factor is isomorphic to either B(H) for some super Hilbert space H or to B(H) ⊗ Cl 1 for some ungraded Hilbert space H.
Remark 2.9.By the isomorphism Cl 1 ⊗ Cl 1 ∼ = B C 2 , the previous remark implies that if A and B are two super factors of type I, then their spatial super tensor product A ⊗ B is again a super factor of type I.Here the product in A ⊗ B is on homogeneous elements defined by Remark 2.10.The isomorphism classification of super factors of type I is complicated by the fact that some Hilbert spaces involved may be finite-dimensional.In particular, in the case that A = B(H) for some super Hilbert space H, there is a variety of possibilities, as the even and the odd part of H may have different dimensions.
Calling type I super factors A, B stably isomorphic when A ⊗ B(H) ∼ = B ⊗ B(H) for some super Hilbert space H, the set of equivalence classes forms a group (isomorphic to Z 2 and generated by Cl 1 ), which is an infinite-dimensional version of the graded Brauer group1 considered in [35].

The automorphism group of super factors of type I
We will now calculate the automorphism group (i.e., the group of grading-preserving * -automorphisms) of a non-trivially graded, properly infinite super factor A of type I. 2 To this end, let H be a Hilbert space and define involutions Γ ev and Γ odd on H ⊕ H by

.1)
Set moreover with grading automorphisms given by conjugation with Γ ev , respectively Γ odd .Then A ev is of even kind, while A odd is of odd kind.One can also show that any non-trivially graded, properly infinite super factor of type I is isomorphic to either A ev or A odd , assuming that the Hilbert space H is infinite-dimensional.
Proposition 2.11.The automorphism groups of A ev and A odd are given as follows.
Here the letter P denotes the corresponding projective group, i.e., the quotient by U(1) (where U(1) acts diagonally on U(H) × U(H)).
Proof .(i) It is well known that the group of (not necessarily grading-preserving) normal *automorphisms of A ev is PU(H ⊕ H), the projective unitary group of H ⊕ H, which acts on A ev by conjugation.That the conjugation with a unitary U ∈ U(H ⊕ H) is grading preserving is equivalent to the requirement Hence Γ ev U * Γ ev U is in the (ungraded) center of A ev , which consists only of multiples of the identity.Consequently, there exists λ ∈ U(1) such that Γ ev U Γ ev = λU .Writing which implies that λ ∈ {±1} and, moreover, this implies that either x = z = 0 (when λ = 1) or u = v = 0 (when λ = −1).Hence the non-zero entries must be unitary, and we obtain where There is an obvious short exact sequence which is split by sending the generator of Z 2 to the operator Γ odd defined in (2.1).This realizes Aut(A ev ) as a semidirect product of P(U(H) × U(H)) with Z 2 .
(ii) It is straightforward to see that the group of not necessarily grading-preserving automorphisms of A odd is precisely Aut(A ev ).The additional requirement that such an automorphism preserves the grading operator means that U Γ odd aΓ odd U * = Γ odd U aU * Γ odd for all a ∈ A odd , where U is a representing unitary.This is the case if and only if and only if Γ odd U Γ odd = λU for some λ in the (ungraded) center of A odd , which in this case is generated by the identity operator and Γ ev .As U is either diagonal or off-diagonal, this means that where The automorphism group Aut(A) of a von Neumann algebra A will always be endowed with Haagerup's u-topology (see [14,Section 3]).Under the identification of Proposition 2.11, this coincides with the (quotient of the) strong topology on G ev /U(1), respectively G odd /U(1) [14, Corollary 3.8].
Remark 2.12.The subgroups G ev and G odd of U(H ⊕ H) defined in (2.2) and (2.3) both have two connected components.It is clear that the continuous group homomorphism i from G ev/odd to Aut(A) (sending a unitary U to the automorphism given by conjugation with U ) induces an isomorphism on π 0 .Remark 2.13.Observe that G ev equals the set of homogeneous unitaries inside A ev .It follows that all automorphisms of A ev are inner.As A odd ∩ G odd = (G odd ) 0 , the identity component of G odd , only the automorphisms in the identity component Aut(A odd ) 0 are inner.
Observe that the automorphism group Aut(A odd ) is a subgroup of the automorphism group Aut(A ev ).We will need the following lemma.
Lemma 2.14.If H is infinite-dimensional, then the inclusion Aut(A odd ) → Aut(A ev ) is a weak homotopy equivalence.
with exact rows.It is clear that the inclusion G odd → G ev is a weak homotopy equivalence, as it is on π 0 by inspection, and the connected components of G ev and G odd are contractible.We obtain that the first two vertical maps induce isomorphisms on π k for all k, hence so must the third.■ Corollary 2.15.The classifying space of the automorphism group of a non-trivially graded, properly infinite super factor A of type I has the homotopy type and the map induced by the inclusion Aut(A) 0 → Aut(A) of the identity component is trivial in the second component.
Proof .If A is of odd kind, then the result follows from the isomorphism Aut(A) ∼ = PU(H) × Z 2 (Proposition 2.11 (ii)) and the fact that PU(H) is a K(Z, 2), hence its classifying space is a K(Z, 3).If A is of even kind, its automorphism group is homotopy equivalent (as a group) to the automorphism of an odd factor, i.e., to PU(H) × Z 2 , by Lemma 2.14.This induces a weak homotopy equivalence between the classifying spaces.■ In particular, we obtain that the homotopy groups of Aut(A), for A a non-trivially graded, properly infinite super factor of type I, are given by otherwise. (2.4)

Bundles of graded type I factors
Let S be a topological space and let A s , s ∈ S be a collection of super von Neumann algebras.For a subset U ⊂ S, we write A| U for the disjoint union of all A s , s ∈ U .By a local trivialization, we mean a map φ : A| U → U × A, A a super von Neumann algebra, that restricts to grading preserving * -homomorphisms A s to {s} × A for each s ∈ U .Two local trivializations are called compatible if the corresponding transition function is continuous as a map U ∩ V → Aut(A) (endowed with Haagerup's u-topology).
Definition 2.16.A collection A as above together with a maximal compatible collection of transition functions is called a (continuous) von Neumann algebra bundle with typical fiber A.
If P is a principal Aut(A)-bundle over S, then the associated bundle Cohomology classes over BAut(A) provide characteristic classes for bundles with typical fiber A via pullback.If A is a non-trivially graded, properly infinite super factor of type I, Corollary 2.15 implies that Denote by or the generator of H 1 (BAut(A), Z 2 ).There is also a preferred generator DD of H 3 (BAut(A), Z), defined as the transgression of the first Chern class of the canonical U(1)bundle over Aut(A) (which over the identity component is U A 0 → Aut(A) 0 ); see Appendix A for more details.

Definition 2.18 (characteristic classes).
Let A be a non-trivially graded, properly infinite super factor of type I and let A be a von Neumann algebra bundle with typical fiber A and classifying map f : S → BAut(A).The characteristic classes will be called the Dixmier-Douady class, respectively the orientation class of A.
The terminology for DD(A) follows that for the analogous class for bundles with typical fiber the algebra of compact operators, first defined by Dixmier and Douady [10].
By Corollary 2.15, for a non-trivially graded, properly infinite super factor of type I, BAut(A) is a product of Eilenberg-MacLane spaces.As these are classifying spaces for cohomology, we obtain the following result.
Proposition 2.19.Suppose that S has the homotopy type of a CW complex and let A be a von Neumann algebra bundle with typical fiber A over S, where A is a super factor of type I. Then A is trivializable if and only if the characteristic classes DD(A) and or(A) are zero.
Remark 2.20.For CW-complexes S, the characteristic classes of Definition 2.18 can be conveniently described using Čech cohomology, as follows.Over a suitable open cover {O α } α∈I , we can choose super Hilbert spaces H α and grading-preserving * -isomorphisms ϕ α : A| Oα → B(H α ).Over two-fold intersections, we can choose families β is given by conjugation with U αβ .A Z 2 -valued Čech 1-cocycle is obtained by defining ε αβ = {±1}, depending on whether U αβ is grading preserving or grading reversing.Over O α ∩ O β ∩ O γ , we have 1), so we obtain a U(1)-valued Čech cochain {λ αβγ } αβγ∈I .One checks that this cochain is closed with respect to the Čech coboundary, and that a cochain obtained from another choice of unitaries { Ũαβ } αβ∈I differs from this cochain by a coboundary.Hence we obtain a well-defined element in Ȟ2 (S, U(1)).Under the Bockstein homomorphism for the sequence Z → R → U(1), this element corresponds to the Dixmier-Douady class.
The Dixmier-Douady class can also be defined for bundles A with typical fiber a finitedimensional non-trivially graded super factor A of type I.One way to do this is through the Čech picture from Remark 2.20.A second, equivalent, way is via stabilization: We replace A by the bundle A ⊗ B(H) for some infinite-dimensional super Hilbert space H such that both H 0 and H 1 are infinite-dimensional and take the characteristic classes in the sense of Definition 2.18 of this bundle.
Remark 2.21.Let A = A ev or A odd be one of the super factors from Section 2.1, constructed in terms of a finite-dimensional Hilbert H.In this case, it turns out that where n = dim(H).We get a group homomorphism Aut(A) → Aut(A ⊗B(H ′ )) (where H ′ is an infinite-dimensional Hilbert space as above) by sending φ → φ ⊗ id, and one can show that pullback along this homomorphism induces an isomorphism on H 1 and is reduction mod n on H 3 .If now A is a bundle over S with typical fiber A, the classifying map for A ⊗B(H ′ ) factors through BAut(A), which shows that the Dixmier-Douady class of such a bundle is n-torsion.
Remark 2.22.A bundle A with typical fiber a properly infinite super factor of type I is trivial if and only if A = B(H) for some bundle H of super Hilbert spaces over S. For the class DD(A) to be zero it suffices that H exists as a bundle of Hilbert spaces which is not necessarily globally graded (i.e., H does split locally into two subbundles but not necessarily globally).The class or(A) is zero if H splits into the direct sum of two subbundles, but may only be a projective bundle.In the ungraded setting, projective bundles of Hilbert spaces were discussed in [3].
Let A and B be two von Neumann algebra bundles over a space S with typical fibers type I super factors A and B. Then the fiberwise spatial super tensor product A ⊗ B has a canonical structure of a von Neumann algebra bundle with typical fiber A ⊗ B, which is again a type I super factor (see Remark 2.9).The proof of the following result is analogous to that of Lemma 9 (respectively, Lemma 4) in [11].where β : is the Bockstein homomorphism.

Clifford von Neumann algebras
In this section, we explain the construction of the von Neumann algebra completion of the algebraic Clifford algebra, given the choice of an equivalence class of (sub-)Lagrangians, and we recall the action by restricted orthogonal transformations on this algebra.General references for the theory of Clifford (and, closely related, CAR) algebras and Fock representations are, e.g., [2,24,26,27,28,29,36].The completion of the Clifford algebra to a hyperfinite factor of type II 1 is considered in [24, Section 1.3], but the subsequent discussion of a completion to a factor of type I ∞ seems to be new.

Clifford algebras and Fock spaces
Let H be a real Hilbert space and denote its complexification by H C .Let Cl alg (H) be the algebraic Clifford algebra, generated by elements of H C , subject to the relation In order to make the situation accessible to analysis, we have to complete Cl alg (H) to C * -algebra, using the * -operation given by In fact, any * -representation of Cl alg (H) induces the same C * -norm on Cl alg (H) [28, Proposition 1], and it follows that the Clifford algebra has a unique norm-completion Cl(H) to a C * -algebra, which turns out to be isomorphic to the infinite tensor product It is moreover a Real C * -algebra, as the complex conjugation of H extends to an anti-linear * -automorphism of Cl(H).The situation is quite different when we ask for completions of Cl alg (H) to a von Neumann algebra.Such a completion can be obtained by the choice of a Lagrangian, which is a complex subspace L ⊂ H C such that L ⊥ = L.The Clifford algebra Cl alg (H) then has a natural representation π L on the Fock space F L = ΛL (the Hilbert space exterior power of L) where elements v ∈ L ⊂ H C act by exterior multiplication and elements v ∈ L ⊂ H C act by contraction.We can therefore take the von Neumann completion in the space B(F L ) of bounded operators on F L .The Fock space is a super Hilbert space with its even/odd grading and the Fock representation is a graded representation, hence Cl L (H) is naturally a super von Neumann algebra.However, the real structure on Cl alg (H) does not extend to a real structure on Cl L (H) if H is infinite-dimensional.It follows from irreducibility of the Fock representation F L [24, Theorem 2.4.2] that any bounded operator on F L commuting with the Clifford action is scalar, hence is a super factor of type I, of even kind.The choice of Lagrangian can be partially eliminated as follows: Two Lagrangians L 1 , L 2 ⊂ H are equivalent if the difference P L 1 − P L 2 is a Hilbert-Schmidt operator, where P L i denotes the orthogonal projection onto L i .By the Segal-Shale equivalence criterion, two Fock representation π L 1 and π L 2 are unitarily equivalent if and only if L 1 and L 2 are equivalent [24, Theorem 3.4.1].Moreover, the unitary implementing the equivalence is grading-preserving if and only if dim L 1 ∩ L 2 is even (see [25,Theorem 1.22] and [24, Theorem 3.5.1]).We denote the equivalence class of a Lagrangian L by [L].

M. Ludewig
For any L ′ ∈ [L], Cl L ′ (H) is a completion of Cl alg (H) with respect to the pullback via π L ′ of the weak operator topology on B(F L ′ ).However, as all representations π L ′ , L ′ ∈ [L], are equivalent, all these topologies coincide.As any two completions of a topological vector space are canonically isomorphic, we obtain a universal Clifford algebra associated to an equivalence class of Lagrangians.Remark 3.1.An explicit description of this von Neumann algebra is as the set of equivalence classes (a Another approach is to take the abstract completion of Cl alg (H) with respect to the ultraweak topology induced by any π L , defined in terms of equivalence classes of Cauchy nets.
is a Hilbert-Schmidt operator.Associated to an equivalence class of sub-Lagrangians, we still have a canonical completion of Cl alg (H), constructed as follows.First we need the following lemma.
⊥ have the same parity.
Proof .Consider the operators i commute with complex conjugation, hence are the complex linear extension of operators on H denoted by J i .Observe that these operators are skew-adjoint, hence by [4], they have a well-defined index ind However, by the assumption that L 1 and L 2 are equivalent, the difference J 1 − J 2 is a Hilbert-Schmidt operator, in particular compact.This implies that ind(J 1 ) = ind(J 2 ), so the lemma follows.■ For a sub-Lagrangian L, consider the complex subspace K = L ⊕ L ⊥ of H C .The construction of the desired completion of Cl alg (H) depends on the dimension of K.
(i) If K is even-dimensional, we can find a Lagrangian F ⊂ K, and L+F ∈ [L] is a Lagrangian in H C .This yields the completion Cl L+F (H) of Cl alg (H).If L ′ is a sub-Lagrangian equivalent to L, then by Lemma 3.2, K ′ = L ′ ⊕ L ′ ⊥ is still even-dimensional, and for any Hence we obtain the completion Cl L+F (H) of Cl alg (H ⊕R) ∼ = Cl alg (H)⊗Cl 1 .In particular, we get a completion of Cl alg (H), as a closed subalgebra of Cl L+F (H).

2, and
for any Lagrangian is canonically isomorphic to Cl L+F (H), and the isomorphism induces an isomorphism between the corresponding completions of Cl alg (H).
We denote by Cl [L] (H) the canonical von Neumann completion of Cl alg (H), determined by the equivalence class [L] of sub-Lagrangians, as constructed above.
Observe that in the case that K is even-dimensional, (3.1) implies that Cl

The restricted orthogonal group
The algebraic Clifford algebra Cl alg (H) has an action of the orthogonal group O(H) by Bogoliubov automorphisms, by its universal property.This action extends to an action on the C * -Clifford algebra Cl(H).In contrast, by the Segal-Shale equivalence criterion, the Clifford algebra Cl [L] (H) does no longer have an action of the entire orthogonal group O(H).That O res (H, [L]) acts on Cl [L] (H) is well known in the case that L is equivalent to a Lagrangian (see, e.g., [2,Section 6]).To get the same statement in the odd case, embed with the upper left corner embedding.Then L ⊕ 0 is equivalent to a Lagrangian and the latter group now acts on the Clifford von Neumann algebra Cl We always consider O res (H, [L]) with the coarsest topology finer than the norm topology induced from O(H) that makes the group homomorphism The proof uses the bundle of implementers, defined as follows.Depending on whether the equivalence class [L] contains a Lagrangian or not, we may choose a Lagrangian L either in H C or in H C ⊕ C and let F L be the corresponding Fock space.For g ∈ O res (H, [L]), define By irreducibility of the Fock representation, Imp g is a U(1)-torsor.It follows from the proof of Proposition 2.11 that U is either even or odd.Let Imp be the union of all Imp g , a subgroup of U(F L ).Then Imp can be equipped with the structure of a Banach Lie group such that the map Imp → O res (H) is a central extension of Banach Lie groups (where the fiber over g is Imp g ), see [18,Section 3.5].
Proof .First suppose that Cl [L] (H) is of even kind.In this case, we may assume that L is a Lagrangian, so that Cl [L] (H) ∼ = B(F L ).Now, the group O res (H) is well known to have the homotopy type of the based loop space of the infinite orthogonal group [26,Proposition 12.4.2].In particular, the first few homotopy groups are Comparing with (2.4), we observe that it has the same homotopy groups as Aut(Cl [L] (H)) for k ≤ 5.For k / ∈ {0, 2}, the statement of the theorem is therefore automatic, and it remains to consider the cases k = 0 and k = 2.
For k = 0, we use that when g does not lie in the identity component of O res (H), then dim gL ∩ L is odd [24, Theorem 3.5.1].Hence the (projectively unique) unitary U : F gL → F L with π gL (a) = U π L (a)U * (which exists as gL and L are equivalent) is parity reversing.Let Λ g : F L → F gL be the unitary map given by taking the exterior power of g.As Λ g is parity preserving, the unitary U Λ g on F L is still parity reversing.By Remark 2.12, this implies that the * -automorphism of Aut(Cl [L] (H)) ∼ = B(F L ) given by conjugation with U Λ g lies in the nonidentity component of Aut(Cl [L] (H)).But U Λ g implements θ(g), hence [θ(g)] is the non-trivial element in π 0 (Aut(Cl [L] (H))).
We now consider k = 2.Here we use the fact that Imp is a generator for the group of line bundles over the identity component O res (H) 0 , i.e., the first Chern class of Imp is a generator for On the other hand, Imp is (by definition) the pullback of the canonical line bundle over Aut(Cl [L] (H)) given over the identity component by U A 0 , the first Chern class of which is a generator for But this implies that θ is an isomorphism on H 2 , hence also on π 2 (applying the Hurewicz isomorphism to the identity component).
This finishes the proof in the even case, so we now discuss the odd case.Then there exists a sub-Lagrangian L in the fixed equivalence class such that The Clifford algebra Cl [L] (H ′ ), is then of even kind, where H ′ ⊂ H is the real subspace of K ⊥ .We now have the commutative diagram where the right vertical map is θ → θ ⊗ id Cl 1 (using the isomorphism Cl

The loop space Clifford algebra bundle
In this section, we define the loop space Clifford algebra bundle and calculate its characteristic classes.We also discuss transgression and define the loop spin class.

Definition of the bundle
Let X be an oriented Riemannian manifold of dimension d and let LX = C ∞ S 1 , X be its smooth loop space.LX is an infinite-dimensional manifold, modeled on the nuclear Fréchet space C ∞ S 1 , R d .Its tangent space T γ LX at a loop γ ∈ LX can be identified with the space C ∞ S 1 , γ * T X of vector fields along γ.It has a natural inner product coming from the standard parametrization of S 1 and the Riemannian metric of X, turning it into a pre-Hilbert space.As we will form completed Clifford algebras (which are insensitive to whether the underlying pre-Hilbert space is complete or not [24]), it is natural to consider the completion of the tangent space.These Hilbert spaces fit together to a bundle H of Hilbert spaces over LX.
To describe the bundle structure of H, let SO(X) be the oriented frame bundle of X and LSO(X) its loop space.LSO(X) is a principal LSO(d)-bundle as X is orientable. 3Now, LSO(d) acts on the Hilbert space by pointwise multiplication, and we have the canonical identification Here we interpret elements q ∈ LSO(X) as orthogonal transformations H d → H γ .As the group LSO(d) acts smoothly on H d , this bundle is a smooth bundle of Hilbert spaces. 4or each γ ∈ LX, we can form the algebraic Clifford algebra Cl alg (H γ ) and its canonical C *completion Cl(H γ ).These algebras fit together to a continuous bundle of C * -algebras, which, as above, can be identified with the associated bundle LSO(X) As an infinite tensor product algebra, Cl H d is an example of a so-called strongly self-absorbing C * -algebra, which has a contractible automorphism group [8,9].Hence any bundle with typical fiber Cl H d must in fact be contractible.However, this argument does not take into account the grading or the real structure, while the above argument also shows that Cl(H) is trivial as a bundle of graded, real C * -algebras.
In order to obtain a non-trivial bundle, we now construct a suitable completion of Cl alg (H) to a bundle of von Neumann algebras.To this end, we observe that the model space H d C admits a canonical sub-Lagrangian The space L d +L d ⊥ is just the space of constant functions on the circle, which has dimension d.
By the discussion in Section 3.1, we therefore obtain a canonical von Neumann completion which is of even kind when d is even and of odd kind when d is odd.
To obtain a von Neumann completion A γ of Cl alg (H γ ) for γ ∈ LX, we observe that any lift q ∈ LSO(X) gives a Lagrangian qL d ⊂ H γ .It is now crucial that the multiplication action of LSO(d)  .Hence for any two lifts q, q ′ ∈ LSO(X) of γ, the Lagrangians qL d and q ′ L d are equivalent.We therefore obtain a well-defined von Neumann completion , independent of the choice of q.These algebras can be canonically identified with the fibers of the bundle The

Transgression and the loop spin class
For a manifold Y and a coefficient group R, transgression is the composition where the left map is pullback with the evaluation map ev : LY × S 1 → Y and the right map is fiber integration over the S 1 factor.Transgression is natural, in the sense that for a smooth map f : Y → Y ′ , the diagram The classifying spaces BG and the universal bundle EG admit an infinite dimensional manifold model, so that their smooth loop space LBG is welldefined.As LEG is again contractible and LG acts freely on it, the quotient LBG is a model for the classifying space BLG.We therefore have transgression homomorphisms Remark 4.7.Of course, transgression is also defined for general topological spaces, using the continuous loop space instead of the smooth version.However, as we work with the smooth loop space throughout, we presented the construction in this case.We recall that the smooth loop space of a manifold is homotopy equivalent to the continuous loop space, and the same is true for based loop spaces [6, Proposition 7.1].
The base loop group ΩG is the kernel of the evaluation-at-zero map ev 0 : LG → G, so we have the short exact sequence

ΩG
LG G, which is split via the inclusion ι : G → LG as constant loops.This induces a fibration of the corresponding classifying spaces, and an exact sequence Proof .As (4.8) is split exact, with G = SO(d) including into LG as constant loops, the corresponding fibration of classifying spaces admits a section Bι.This induces a left split of (4.9).That the sequence (4.9) is exact in the middle follows from the Serre spectral sequence for the classifying space fibration of (4.8).The right map in (4.9) is surjective by the following argument.By Corollary 4.6, the map B θ : BΩSO(d) → BAut(A d ) induces an isomorphism on π k for k ≤ 3. It is moreover trivially surjective on π 4 as π 4 (BAut(A d )) = 0. Hence pullback induces an isomorphism On the other hand, the group homomorphism θ : ΩSO(d) → Aut(A d ) extends to all of LSO(d), hence the above isomorphism factors through H 3 (BLSO(d), Z).This shows that the right map in (4.9) must be surjective.■ We are now interested in the following commutative diagram For the proof, we need the following lemma.

Proof . Observe that ΩSpin(d) is canonically identified with the identity component of ΩSO(d).
We therefore obtain the commutative diagram The horizontal maps of this diagram induce isomorphisms on π k for k ≤ 2, by Corollary 4.6.It follows that in the corresponding diagram on classifying spaces, the horizontal maps induce isomorphisms on π k for k ≤ 3.They are moreover trivially surjective on π 4 (since this group is trivial for the right-hand side).By Whitehead's theorem [30,Theorem 10.28], these maps also induce isomorphisms on H k , for k ≤ 3 (and a surjection on H 4 ).From the universal coefficient theorem and the five lemma, we obtain that they also induce isomorphisms on H k , for k ≤ 3 (and a surjection on H 4 ).Moreover, the map The bottom horizontal map is an isomorphism by Lemma 4.10.The left horizontal map is surjective by Lemma 4.8.Now the counterclockwise composition is surjective, hence so must be the clockwise composition.As the two rightmost groups are isomorphic to Z, we obtain in particular that the top horizontal map must be surjective, as claimed.■ We conclude that the square (4.10) takes the following form:

Proof of the main theorem
In this section, we express the characteristic classes of A LX in terms of transgression and hence prove the main theorem from the introduction.Let X be an oriented Riemannian manifold of dimension d ≥ 5 and let f : X → BSO(d) be the classifying map for its oriented frame bundle SO(X).Then the looped map Lf : LX → LBSO(d) = BLSO(d) classifies the principal LSO(d)-bundle LSO(X).
Definition 4.13 (loop spin class).The loop spin class of X is Let A LX be the Clifford von Neumann algebra bundle on LX constructed in Section 4.1.Its typical fiber is the Clifford von Neumann algebra A d from (4.3).By Proposition 2.17, the bundle is therefore classified by a map The characteristic classes of A LX are then by definition the pullback along h of the universal classes or and DD on BAut(A d ).On the other hand, by (4.5),A LX is an associated bundle to LSO(X), via the action (4.6) of LSO(d) on A d .This means that the classifying map h of the bundle A LX admits the factorization Proof of Main Theorem 1.1.We first consider the orientation class.To this end, we consider the following commutative diagram: Here all groups independent of X are Z 2 , and all the maps independent of X are isomorphisms: Indeed, it is well known that The above discussion implies that B θ * or = τ (w 2 ).Hence using commutativity of the left rectangle in (4.12), we obtain where we recall that the pullback f * w 2 is by definition the second Stiefel-Whitney class w 2 (X) of X.This proves the claim.
For the third integer cohomology, we consider the commutative diagram Proof .Consider the following commutative diagram: The left vertical map is an isomorphism by Corollary 4.6.The bottom horizontal map is an isomorphism by Lemma 4.10.Hence the diagonal map in (4.14) is an isomorphism.The fact that all groups involved are isomorphic to Z then implies that also the right vertical map in (4.14) is an isomorphism.■ Both the top and bottom right groups in (4.13) are canonically isomorphic to Z, with generators the universal Dixmier-Douady class DD, respectively the fractional universal Pontrjagin class 1  2 p 1 .By Lemma 4.14 and the fact that the bottom right map is an isomorphism (see the discussion below (4.10)), we observe that the transgression of 1  2 p 1 equals the pullback of DD along the top right vertical map in (4.13), up to a possible sign.This sign turns out to be +1, but establishing this involves rather intricate calculations which we defer to the appendix.By a diagram chase, we then get that 2 • B θ * DD = τ (p 1 ), hence Proof .With a view on Lemma 4.8, it follows from Main Theorem 1.1 that We have ι * X S(X) = 0 and ι * X ev * 0 W 3 (X) = W 3 (X), where ι X : X → LX is the inclusion as constant loops and ev 0 : LX → X is evaluation at zero.For the proof of Theorem 4.16, it is therefore left to show that ι * X DD(A LX ) = W 3 (X).By naturality of the Dixmier-Douady class, we have ι * X DD(A LX ) = DD(ι * X A LX ).Now, the bundle ι * X A LX over X can be written as an associated bundle, It is well known that DD(Cl(X)) = W 3 (X) (see [11,Lemma 7] and [20]).We show that A ′ is trivializable, which finishes the proof.To this end, observe that as A ′ is associated to SO(X), its classifying map X → BAut(A ′ ) factors through BSO(d).We now show that the map BSO(d) → BAut(A ′ ) is contractible.To this end, observe that the action of ι(SO(d)) ⊂ LSO(d) preserves L d .This means that for each q ∈ SO(d), multiplication by ι(q) commutes with the complex structure and so is the induced map on classifying spaces.This proves the claim.■

A twisted Clifford algebra bundle
There is a variant for the construction of the loop space Clifford algebra bundle which takes the model space H d S = L 2 S 1 , R d ⊗ S as input, where S is the Möbius bundle over S 1 .This has been considered, e.g., in [16,Section 6.2].The main difference here is that this space has a canonical Lagrangian L d S , which under the identification of H d S with 2π-anti-periodic functions on R can be written as Proof .This is shown analogously to the proof of Theorem

A Sign discussion
In this appendix, we fix the sign indeterminacy present in the proof of Theorem 4.16 above.Precisely, we prove the following.For definiteness, we emphasize that we use the (standard) convention for Pontrjagin classes that for any complex line bundle L with underlying real bundle L R , we have p We recall the definition of the universal Dixmier-Douady class DD.To begin with, recall that for a topological group G with classifying space fibration G → EG → BG, there is a homomorphism τ : natural in G, which is called transgression and should not be confused with the notion of transgression discussed in Section 4.2. 5 In the case of G = Aut(A d ) and k = 3, the transgression homomorphism τ : is an isomorphism, and the universal Dixmier-Douady class DD is by definition the class that is sent to the first Chern class As the classes we are interested in are not torsion, we may work over real coefficients.We consider the diagram which commutes by naturality of the transgression maps (A.1).By the results of Section 4 and the universal coefficient theorem, all groups in this diagram are isomorphic to R and all maps are isomorphisms.In a first step, we observe that the pullback of the bundle G d under the map θ is precisely the implementer bundle Imp → O res H d , so that the statement of Proposition A.1 is equivalent to the equality 2 • j * c 1 (Imp) = τ (τ (p 1 )).
Since the three groups on the right are all Fréchet Lie groups, we may work with de Rham cohomology instead of singular cohomology.Here the transgression homomorphisms τ may be described as follows.Let α ∈ H k dR (BG) and let π * α ∈ H k dR (EG) be its pullback.Then since EG is contractible, π * α = dβ for some β ∈ Ω k (EG).The transgression of α is then defined by where ι : SO(d) → ESO(d) is the inclusion of a fiber.The de Rham cohomology groups H k dR (SO(d)) are best understood using the isomorphism with the Lie algebra cohomology group H k (so(d), R), which identifies a Lie algebra k-cocycle α with the left-invariant 3-form α on SO(d) that coincides with α at the identity.We now have the following general statement.
The proof uses the theory of Chern and Simons [7], which we recall now.Let ω be a connection 1-form on a principal G-bundle E over B with curvature Ω.Let P ∈ Sym 2 (g * ) G be an invariant polynomial on g.Then the corresponding Chern-Simons form is TP (ω) ∈ H 3 (E, R) defined by see [7, formula (3.5)].Denoting by ι : G → E the inclusion of a fiber (for some fixed base point e ∈ E), one uses that the curvature form Ω is horizontal, so that ι * Ω = 0, while ι * ω = ω G , the Maurer-Cartan form of G. Hence which differs from the formula (3.11) in [7] by a factor of 2. Going through the conventions used in [7] for the wedge product and commutator of g-valued differential forms (see [7, p. 50]), one obtains that ι * TP (ω) is the left-invariant form corresponding to the Lie algebra cocycle Proof .We take G = SO(d) and E = ESO(d), B = BSO(d), the universal bundles.Choosing models for these that are infinite-dimensional Fréchet manifolds (for example, the infinite Grassmannian and Stiefel manifold), we may choose a connection 1-form ω ∈ H 1 (ESO(d), so(d)).
By the usual Chern-Weil formulas for Pontrjagin classes, we see that this equals the pullback π * p 1 of the de Rham representative of the universal first Pontrjagin class along the bundle projection π : ESO(d) → BSO(d).
Using the description (A.2) of the transgression homomorphism, we see that τ (p 1 ) = ι * TP (ω).As discussed above, the pullback ι * TP (ω) is the left invariant differential form that corresponds to the Lie algebra cocycle α given by (A.3).For our particular choice of P , we get This result can be found as [26,Proposition 4.4.4],but with incorrect prefactors and an incomplete proof, which is why we repeat the proof below.
Proof .Tangent vectors X, Y at γ ∈ LSO(d) can be identified with those elements of LMat d×d such that X ∈ Lso(d), where X(t) = γ(t) −1 X(t).We now calculate Let β ∈ Ω 1 (LSO(d)) be given by For the exterior derivative of β, we find where Γ = i(P L − P L ) is the complex structure determined by the Lagrangian L.
When trying to apply these results to the specific Hilbert space H d defined in (4.1), we face the difficulty that the subspace L d ⊂ H d defined in (4.2) is only a sub-Lagrangian.We deal with this issue as follows: Given an element g ∈ LSO(d), the element j(g) acts on H d ⊕R through multiplication by g on the first summand and the identity on the second, and consequently, for X ∈ Lso(d), the operator j * X acts by multiplication with X on the first summand and by zero on the second.
To have a uniform notation, we write H for either the Hilbert space H d or for H d ⊕ R in the case that d is odd and let L ⊂ H be the Lagrangian described above.
A similar computation, for the Lagrangian (4.15) instead of L, can be found in [18]; unfortunately, the result is off by a factor of ±i.Proposition 6.7.1 in [26] is an analogous result for the basic central extension of restricted unitary group U res (H) of a polarized complex Hilbert space H.
Proof .We will establish the cocycle identity 2 • j * Ω = −ω, (A.6) which gives the result by (A.4).By continuity and bilinearity, it suffices to verify that both Lie algebra cocycles evaluate identically on the specific Lie algebra elements X, Y ∈ Lso(d) of the form We observe that j * X sends V n to V n+k and j * Xj * Y sends V n to V n−k−ℓ .On the other hand, P L and P L preserve the subspaces V n for n ̸ = 0 and send V 0 to K (where K = V 0 if d is even, while K = V 0 ⊕C if d is odd).We conclude that both xy Comparing with (A.7), this establishes (A.6).■

Definition 3 . 3 (
restricted orthogonal group).The restricted orthogonal group of H with respect to an equivalence class [L] of sub-Lagrangians, denoted by O res (H, [L]), consists of those orthogonal transformations g of H such that the commutator [g, P L ] with the orthogonal projection P L onto L is a Hilbert-Schmidt operator.If the equivalence class [L] is clear from the context, we write just O res (H).

Theorem 3 . 4 .
continuous, which sends an orthogonal transformation to its Bogoliubov automorphism.In fact, O res (H, [L]) is a Banach Lie group with this topology [26, Sections 6.2 and 2.4].The map θ from (3.2) induces an isomorphism on π k for k ≤ 5.

. 5 )Remark 4 . 3 .Lemma 4 . 5 .
This is a continuous bundle of von Neumann algebras as the homomorphism θ : LSO(d) O res H d Aut(A d ) (4.6) obtained by composing (4.4) with the Bogoliubov action (3.2) is continuous.An alternative construction of the von Neumann completion of Cl alg (H γ ) is the following.Let D γ = i ∇ dt be the operator acting on the bundle γ * T X ⊗ C using the pullback of the Levi-Civita connection on T X.Let L γ be the Hilbert space direct sum of eigenspaces to negative eigenvalues of D γ .This is a sub-Lagrangian in H C γ , which can be shown to be equivalent to qL d for any q ∈ LSO(d).For details, see[19, Section 1.3].Remark 4.4.A closely connected bundle of Clifford algebras on LX has recently been considered in the somewhat different context of rigged von Neumann algebra bundles by Kristel and Waldorf [16].For a Lie group G, we denote by ΩG ⊂ LG the based loop space of G, i.e., the set of smooth loops γ : S 1 → G with γ(0) = e, the neutral element of G.For G = SO(d), the restriction of (4.4) to based loops gives a continuous group homomorphism ΩSO(d) −→ O res H d .If d ≥ 5, the above homomorphism induces an isomorphism on π k for k ≤ 2. Proof .For d even, this statement is well known: By [26, Proposition 12.5.2],for any m ∈ N, the map ΩSO(2m) → O res (H 2m ) is (2m−3)-connected.This shows the claim in even dimensions d = 2m ≥ 6.For d ≥ 5 odd, we consider the commutative diagram ΩSO(d) O res H d ΩSO(d + 1) O res H d+1 , (4.7) the bottom map of which is an isomorphism on π k for k ≤ d − 3 by the discussion for the even case.The right vertical map in (4.7) is the restriction of the map

(4. 10 )Proposition 4 . 9 .
where d ≥ 5.The bottom left group is Z, generated by the universal Pontrjagin class p 1 .The bottom right group is also Z, generated by the fractional Pontrjagin class 1 2 p 1 , and the bottom horizontal map has been shown to be multiplication by two on generators by McLaughlin [21, proof of Lemma 2.2], more specifically it sends p 1 to the class 2 • 1 2 p 1 .The right transgression map is an isomorphism as BSpin(d) is 2-connected [21, p. 149].If d ≥ 5, the top horizontal map in (4.10) is surjective.
generated by the universal Stiefel-Whitney class w 2 .That the right transgression map is an isomorphism has been shown by McLaughlin [21, proof of Proposition 2.1].This implies H 1 (BLSO(d), Z 2 ) ∼ = Z 2 .That also H 1 (BAut(A d ), Z 2 ) = Z 2 follows from (2.4) and the Hurewicz isomorphism.The rightmost vertical map is an isomorphism by Corollary 4.6.It follows that the map H 1 (BLSO(d), Z 2 ) → H 1 (BΩSO(d), Z 2 ) is surjective.That it is also injective is clear as all groups involved are Z 2 .

13 )
Here we use that LSpin(d) acts on A d along the homomorphism Lp : LSpin(d) → LSO(d), which induces a map BLSpin(d) → BAut(A d ).The bottom right square is just (4.10).Lemma 4.14.The top right vertical map in (4.13) is an isomorphism.

Remark 4 . 15 . 1 . 4 . 16 .
This finishes the proof of Main Theorem 1.1.■ Replacing the Lagrangian L d with L d ⊥ results in a different von Neumann algebra bundle ÃLX , satisfying DD ÃLX = −DD(A LX ).This follows from the fact that the Lie algebra cocycle for the group extension Imp → O res H d is replaced by its negative under this change, by the calculation in [2, Theorem 6.10].Hence any choice of sign for the Dixmier-Douady class (in comparison to τ (p 1 )) can be achieved by a modification of the Clifford algebra construction.The following is a more refined version of Main Theorem 1.Theorem Let d ≥ 5. Then we have or(A LX ) = τ (w 2 ), DD(A LX ) = S(X) + ev * 0 W 3 (X).
SO(d) acts on A d through Bogoliubov automorphisms, induced by the multiplication with constant loops on H d .Denote by K ⊂ H d the subspace of constant functions.Then both K C and L d are invariant under the action of SO(d) and L d is a Lagrangian in H ′ = K ⊥ C .Let A ′ d = B(F L d ) be the von Neumann algebra completion of Cl alg (H ′ ) with respect to this Lagrangian.Then identifying Cl(K) ∼ = Cl d , we have A d = A ′ ⊗ Cl d and ι * X A LX splits as a tensorproduct, ι * X A LX ∼ = A ′ ⊗Cl(X), where A ′ = SO(X) × SO(d) A ′ and Cl(X) is the usual complex Clifford algebra bundle on X.By Theorem 2.23, we have DD(ι * X A LX ) = DD(A ′ ) + DD(Cl(X)) + β(or(A ′ ) ⌣ or(Cl(X))).

. 15 )
Hence the corresponding Clifford von Neumann algebraA S d = Cl [L d S ] H d S isof even kind in any dimension d.In a similar fashion to before, we obtain a bundle A S LX of super type I factors.Theorem 4.17.Let d ≥ 5. Then the characteristic classes of A S LX are or A S LX = τ (w 2 (X)), DD A S LX = S(X).

Proposition A. 1 .
For d ≥ 5, the transgression τ (p 1 ) ∈ H 3 (LBSO(d), Z) = H 3 (BLSO(d), Z) of the universal first Pontrjagin class equals two times the pullback of the universal Dixmier-Douady class DD ∈ H 3 (BAut(A d ), Z) along the map on classifying spaces induced by the composition θ : LSO(d) O res H d Aut(A d ).

1 (
t)] dt for suitable extensions of X and Y to vector fields on LSO(d).Integrating by parts, we see thatdβ = −2 • 2π • ω − 8π 2 • τ (σ), hence 2π • τ (σ) and −ω are cohomologous.■ Let G → G be a central U(1)-extension of a Fréchet Lie groups, inducing a Lie algebra homomorphism g → g.Choosing a linear section of the Lie algebra homomorphism g → g gives an identification g = g ⊕ R. With respect to this choice, the Lie bracket of g is given by[(X, λ), (Y, µ)] = ([X, Y ], Ω(X, Y ))for some continuous Lie algebra cocycle 2-cocycle Ω, which represents a class in H 2 c (g, R).The first Chern class of the principal U(1)-bundle G → G is then given by c the associated left-invariant 2-form, see for example [26, Proposition 4.5.6].In the case that G = O res (H, [L]) and G = Imp L for some real Hilbert space with a Lagrangian L ⊂ H C , the relevant Lie algebra cocycle has been computed in several places; see, e.g., [2, Theorem 6.10], [22, Theorem 10.2] or [23, Theorem 6].The result is Ω(X, Y ) = 1 8 tr(J[J, X][J, Y ]), X, Y ∈ o res H d , (A.5) (i) If d is even we let K = L d ⊕ Ld ⊥ be the even-dimensional subspace of constant functions and choose a Lagrangian L 0 ⊂ K.Then L = L d + L 0 ⊂ H d C is a Lagrangian.(ii) If d is odd, we let K = L d ⊕ L d ⊥ ⊕ C and again choose a Lagrangian L 0 ⊂ K. Then L = L d + L 0 is a Lagrangian in H d C ⊕ C.By definition, the implementer bundle over O res H d is the restriction of the implementer bundle over O res H d ⊕ R , hence the group cocycle associated to this extension is the restriction of the cocycle (A.5) to o res H d ⊂ o res H d ⊕R .
t) = be −iℓt , a, b ∈ so(d),k, ℓ ∈ Z.On these elements, the right-hand side of (A.6) is given byω(X, Y ) = 1 2π 2π 0 tr(X(t)Y ′ (t))dt = − iℓ 2π 2π 0 tr(ab)e −i(k+ℓ)t dt = −iℓ • tr(ab) k + ℓ = 0, 0 otherwise.(A.7)To calculate the left-hand side of (A.6), we writej * X = x ′ x x x ′ , j * Y = y ′y y y ′ with respect to the decomposition H C = L ⊕ L. In other words, x ′ = P L XP L and x = P L XP L and similarly for y ′ and y, while x denotes the conjugation of x by the real structure of H C .Then since j * X and j * Y are restricted, the off-diagonal entries x and y are Hilbert-Schmidt operators and a straightforward calculation gives Ω(j * X, j * Y ) = i 2 tr(xy − yx).(A.8)
where LSO(d) acts on Cl H d through O H d , by Bogoliubov automorphisms.Here it is important that the homomorphism LSO(d) → O H d → Aut Cl H d is continuous when O H d carries the norm topology [1, Proposition 4.35]; this implies that Cl(H) has the structure of a continuous bundle of C * -algebras.However, this bundle seems rather uninteresting for loop space spin geometry, by the following.Proof .The action of LSO(d) extends to an action of the orthogonal group O H d of H d , which is contractible by Kuiper's theorem.So Cl(H) is an associated bundle for the principal O H d -bundle LSO(X) × LSO(d) O H d , which must be trivial by contractibility of O H d and its classifying space BO H d .Hence Cl(H) is trivial as well.
on H d is in fact by elements of the restricted orthogonal group O res H d = O res H d , L d and that we get a continuous group homomorphism left vertical map of the diagram is induced by the canonical embedding SO(d) → SO(d + 1), which induces an isomorphism on π k for k ≤ 3.By [6, Proposition 7.1], the inclusion of ΩSO(d) into the continuous based loop space of SO(d) is a homotopy equivalence.Hence ΩSO(d) → ΩSO(d + 1) induces an isomorphism on π k for k ≤ 2.By the above considerations and the commutativity of (4.7), the mapπ k (ΩSO(d)) → π k O res H dis injective, and the right vertical map in (4.7) is surjective on π k , for k ≤ 2. As seen in (3.3), the first few homotopy groups of O res H d are Z 2 , 0 and Z, which implies that all maps in the diagram must be isomorphisms on π k .■Combining the above result with Theorem 3.4, we obtain the following result.
d) induces an isomorphism on π k for k ≤ 2.
9) on the corresponding cohomology groups.The first group here is Z 2 , generated by the third universal integral Stiefel-Whitney class W 3 (i.e., the Bockstein image of the second universal Stiefel-Whitney class w 2 ).As BΩSO(d) ≃ SO(d), the right group equals H 3 (SO(d), Z) ∼ = Z.Lemma 4.8.If d ≥ 5, the sequence (4.9) is split exact, hence we have a canonical isomorphism an isomorphism by Corollary 2.15.The lemma follows.■ Proof of Proposition 4.9.Consider the commutative diagram