FRACTIONAL ANALYTIC INDEX

V. MATHAI, R.B. MELROSE, AND I.M. SINGER

1.2I; Revised: 20-2-2004; Run: February 20, 2004

Abstract. For a finite rank projective bundle over a compact manifold, so associated to a torsion, Dixmier-Douady, 3-class, w, on the manifold, we define the ring of differential operators ‘acting on sections of the projective bundle’ in a formal sense. In particular, any oriented even-dimensional manifold carries a projective spin Dirac operator in this sense. More generally the corresponding space of pseudodifferential operators is defined, with supports sufficiently close to the diagonal, i.e. the identity relation. For such elliptic operators we define the numerical index in an essentially analytic way, as the trace of the commutator of the operator and a parametrix and show that this is homotopy invariant. Using the heat kernel method for the twisted, projective spin Dirac operator, we show that this index is given by the usual formula, now in terms of the twisted Chern character of the symbol, which in this case defines an element of K-theory twisted by w; hence the index is a rational number but in general it is not an integer.

The Atiyah-Singer index theorem for an elliptic (pseudodifferential) operator gives an integrality theorem; namely a certain characteristic integral is an integer because it is the index of an elliptic operator. Notably, for a closed spin manifold Z, the Â genus, ∫_{Z} Â(Z) is an integer because it is equal to the index of the Dirac operator on Z. When Z is not a spin manifold, the spin bundle S does not exist, as a vector bundle, and when Z has no spin_{ℂ} structure, there is no global vector bundle resulting from the patching of the local bundles S ⊗ L_{i}, where the L_{i} are line bundles. However, as we show below, S is a always a projective vector bundle associated to the the Clifford algebra of the cotangent bundle T^{∗}Z, ℂl(Z), which is an Azumaya bundle cf. [10]. Such a (finite rank) projective vector bundle, E, over a compact manifold has local trivializations which may fail to satisfy the cocycle condition on triple overlaps by a scalar factor; this defines the Dixmier-Douady invariant in H^{3}(Z, ℤ). If this torsion twisting is non-trivial there is no, locally spanning, space of global sections. The Dixmier-Douady invariant for ℂl(Z) is the third integral Stieffel-Whitney class, W_{3}(Z). In particular, the spin Dirac operator does not exist when Z is not a spin manifold. Correspondingly the Â genus is a rational number, but not necessarily an integer. In this paper we show that, in the oriented even-dimensional case, one can nevertheless define a projective spin Dirac operator, with an analytic index valued in the rational numbers, and prove the analogue of the Atiyah-Singer index theorem for this operator twisted by a general projective bundle. In fact we establish the analogue of the Atiyah-Singer index theorem for a general projective elliptic pseudodifferential operator. In a subsequent paper the families case will be discussed.

For a compact manifold, Z, and vector bundles E and F over Z the Schwartz kernel theorem gives a one-to-one correspondence between continuous linear operators from C^{∞}(Z, E) to C^{−∞}(Z, F) and distributions in C^{−∞}(Z^{2}, Hom(E, F)⊗Ω_{R}). Here Hom(E, F) is the ‘big’ homomorphism bundle over Z^{2} with fibre at (z, z′) equal to hom(E_{z′}, F_{z}) ≡ F_{z} ⊗ E′_{z′}, and Ω_{R} is the density bundle lifted from the right factor. Restricted to pseudodifferential operators of order m, this becomes an isomorphism to the space, I^{m}(Z^{2}, Diag; Hom(E, F)⊗Ω_{R}), of conormal distributions with respect to the diagonal, cf. [7].

This fact motivates our definition of projective pseudodifferential operators when E and F are only projective vector bundles associated to a fixed finite-dimensional Azumaya bundle A. The homomorphism bundle Hom(E, F) is then again a projective bundle on Z^{2} associated to the tensor product A_{L} × A′_{R} of the pull-back of A from the left and the conjugate bundle from the right. In particular if E and F have DD invariant τ ∈ H^{3}(Z; ℤ) then Hom(E, F) has DD invariant π_{L}^{∗}τ − π_{R}^{∗}τ ∈ H^{3}(Z^{2}; ℤ). Since this class is trivial in a tubular neighborhood of the diagonal it is reasonable to expect that Hom(E, F) may be realized as an ordinary vector bundle there. In fact this is the case and there is a canonical choice, Hom^{A}(E, F) of extension. This allows us to identify the space of projective pseudodifferential operators, with kernels supported in a sufficiently neighborhood N_{ϵ} of the diagonal, Ψ_{ϵ}^{•}(Z; E, F) with the space of conormal distributions I_{ϵ}^{•}(N_{ϵ}, Diag; Hom^{A}(E, F)) ⊗ Ω_{R}). Despite not being a space of operators, this has precisely the same local structure as in the untwisted case and has similar composition properties provided supports are restricted to appropriate neighbourhoods of the diagonal. The space of projective smoothing operators Ψ_{ϵ}^{−∞}(Z; E, F) is therefore identified with C_{c}^{∞}(N_{ϵ}; Hom^{A}(E, F) ⊗ π_{R}^{∗}Ω). The principal symbol map is well defined for conormal distributions so this leads directly to the symbol map on Ψ_{ϵ}^{m}(Z; E, F) with values in smooth homogeneous sections of degree m of hom(E, F), the ‘little’ or ‘diagonal’ homomorphism bundle which is a vector bundle. Thus ellipticity is well defined, as the invertibility of this symbol. The ‘full’ symbol map is given by the map to the quotient Ψ_{ϵ}^{•}(Z; E, F)/Ψ_{ϵ}^{−∞}(Z; E, F). This is an algebra and ellipticity, as invertibilty of the principal symbol, implies invertibility of the image of an operator in this quotient. Any lift, B, of the inverse is a parametrix for the given elliptic operator, A. The analytic index of the projective elliptic operator is then defined by

(1)  ind_{a}(A) = Tr([A, B]).

It turns out that ind_{a}(A) is rational; it is constant on path-connected components of projective elliptic operators.

In fact we further extend this discussion by allowing twisting by line bundles defined over the bundle of trivializations of the Azumaya bundle; these are Nth roots of line bundles over the base and have actions of SU(N) arising from fibreholonomy. This allows us to include in the same general framework the case of spin_{ℂ} Dirac operators on non-spin manifolds. Thus we define projective spin_{ℂ} Dirac operators even when there is no spin_{ℂ} structure.

Within the projective pseudodifferential operators, acting between two projective bundles associated to the same Azumaya bundle, there is always a full algebra of differential operators, with kernels supported within the diagonal. On an evendimensional oriented manifold the Clifford bundle is an Azumaya bundle and has associated to it projective spin bundles, S^{±}. The choice of a compatible connection gives a projective spin Dirac operator. Such a projective Dirac operator, ð, can be coupled to any unitary projective vector bundle E over Z associated to an Azumaya algebra A. Thus S ⊗ E is a projective vector bundle associated to the Azumaya algebra ℂl(Z) ⊗ A. This coupled operator, ð_{E}^{+}, is elliptic and its analytic index, in the sense defined above, is

∫_{Z} Â(Z) ∧ Ch_{A}(E)

where Ch_{A} : K^{0}(Z, A) → H^{even}(Z, ℚ) is the twisted Chern character.

When Z is even dimensional, K_{c}^{0}(T^{∗}Z, π^{∗}A) ⊗ ℚ is generated by the classes of symbols of such coupled signature operators. We conclude from this, essentially as in the untwisted case, that for a general projective elliptic pseudodifferential operators T ∈ Ψ_{ϵ}^{•}(Z; E, F) with principal symbol σ_{m}(T),

ind_{a}(T) = ∫_{T^{∗}Z} Td(T^{∗}Z) ∧ Ch_{A}(σ(T)).

This in turn shows the rationality of the analytic index and we conclude by providing several examples where ind_{a} is not an integer, but only a fraction, as justification for the title of the paper.

In the first section below the ‘big’ homomorphism bundle is constructed, near the diagonal, for any two projective bundles associated to the same Azumaya bundle. In Section 2 we discuss a more general construction of such ‘big’ homomorphism bundles which corresponds to twisting by a line bundle, only the Nth power of which is well-defined over the base. The projective spin bundle is discussed in the next section as is its relationship to the spin bundle when a spin structure exists and to spin_{ℂ} bundles when they exist. In Sections 4 and 5 the notion of smoothing, and then general pseudodifferential, operators between projective bundles (and more generally twisted projective bundles) is introduced and for elliptic operators the index is defined. The homotopy invariance of the index is shown directly in Section 6 and in Section 7 projective Dirac operators are defined and the usual local index formula is used to compute the index in that case. Much as in the usual case this formula is extended to general pseudodifferential operators in Section 8. Some examples in which the index is truly fractional are given in the final section.

1. Homomorphism bundles

Let A be a finite-dimensional (star) Azumaya bundle over a compact manifold Z and let E be an associated projective bundle; see [10]. Thus, E may be identified as a projection-valued section of the bundle A ⊗ K over Z; where K is the algebra of compact operators on some fixed Hilbert space, H. By assumption, A has local trivializations as N × N matrix algebras. Let P be the corresponding principal PU(N)-bundle; PU(N), acting by conjugation, being the automorphism group of the star-algebra of N ×N matrices. Then E lifts to a finite-dimensional subbundle, Ẽ ⊂ ℂ^{N} ⊗ H, over P which is equivariant for the standard action of U(N) on ℂ^{N}; interpreted as covering the PU(N) action on P. Since the action of U(N) on hom(Ẽ) is through conjugation this is a bundle over P invariant under the PU(N) action, it therefore descends to hom(E), as a well-defined vector bundle on Z. In fact for any two projective bundles, E, F, associated to the Azumaya algebra the same argument leads to a corresponding bundle hom(E, F). On the other hand the ‘big’ homomorphism bundle Hom(Ẽ, F̃) is only a projective vector bundle over Z^{2}; as noted in the introduction its DD invariant is the difference π_{L}^{∗}w−π_{R}^{∗}w of the lifts of the two DD invariants from the factors. This class is trivial in a neighborhood of the diagonal, hence hom(E, F) can be extended to a bundle near the diagonal. For our purposes it is vitally important that this extension be made in such a way that the composition properties also extend.

For a given metric on Z set

(2)  N_{ϵ} = {(z, z′) ∈ Z^{2}; d_{g}(z, z′) < ϵ}.

The projective unitary group, PU(N), can be written as a quotient of the group, SU(N), of unitary matrices of determinant one:

(3)  ℤ_{N} ⟶ SU(N) ⟶ PU(N).

In the following result, which is the foundation of subsequent developments, we use the discreteness of the fibres of (3).

Proposition 1. Given two projective bundles, E and F, associated to a fixed Azumaya bundle and ϵ > 0 sufficiently small, the exterior homomorphism bundle Hom(Ẽ, F̃), descends from a neighborhood of the diagonal in P × P to a vector bundle, Hom^{A}(E, F), over N_{ϵ} extending hom(E, F). For any three such bundles there is a natural associative composition law

(4) Hom_{(z′′,z′)}^{A}(F, G) × Hom_{(z,z′′)}^{A}(E, F) ∋ (a, a′) ⟼ a ∘ a′ ∈ Hom_{(z,z′)}^{A}(E, G),

(z, z′′), (z′′, z′) ∈ N_{ϵ/2}

which is consistent with the composition over the diagonal.

Remark 1. Applying this result to the projective vector bundle given by Id_{A} ⊗π_{1}, where π_{1} is the projection onto the first basis element of H, gives a bundle, which we denote Â, which extends A from the diagonal to some neighbourhood N_{ϵ} and which has the composition property as in (4)

(5)  Â_{(z′′,z)} × Â_{(z,z′′)} ⟶ Â_{(z,z′)}.

We regard this as the natural extension of A.

Proof. Consider again the construction of hom(E, F), always for two projective bundles associated to the same Azumaya bundle, A. The dual bundle Ẽ′ is associated to the adjoint Azumaya bundle; or as a subbundle of ℂ^{N} ⊗ H over P it is associated with the adjoint action of SU(N) on ℂ^{N}. The external tensor product over the product of P with itself, as a bundle over Z^{2}, F̃ ⊠Ẽ′ is therefore a subbundle of hom(ℂ^{N})⊗hom(H) over P ×P equivariant for the action of SU(N)×SU(N) over PU(N) × PU(N). Restricted to the diagonal, P × P has the natural diagonal subbundle P. The restriction of F̃ ⊠ Ẽ′ to this submanifold has a PU(N) action, and so descends to the bundle hom(E, F) over Diag ≡ Z, since for A ∈ PU(N) we can take the same lift Ã to SU(N) in each factor and these different diagonal lifts lead to the same operator through conjugation.

In a neighborhood, N_{ϵ}, of the diagonal there is a corresponding ‘near diagonal’ submanifold of P×P; for instance we can extend P over the diagonal to a subbundle P̃ ⊂ P × P by parallel transport normal to the diagonal for some connection on A. Now, any two points of P̃ in the same fibre over a point in N_{ϵ} are related by the action of (A′, A) ∈ PU(N) × PU(N) where A′A^{−1} is in a fixed small neighborhood of the diagonal, only depending on ϵ. It follows from the discreteness of the quotient SU(N) ⟶ PU(N) that, for ϵ > 0 sufficiently small, on lifting A to Ã ∈ SU(N) there is a unique neighboring lift, Ã′, of A′. The conjugation action of these lifts on F̃ ⊠ Ẽ′ is therefore independent of choices, so defining Hom^{A}(E, F) over N_{ϵ}. This bundle certainly restricts to hom(E, F) over the diagonal.

In fact this construction is independent of the precise choice of P̃. Namely if Hom(Ẽ, F̃) ≡ F̃ ⊠ Ẽ′ is restricted to a sufficiently small open neighbourhood, N, of P as the diagonal in P × P, then the part of PU(N) × PU(N) acting on the fibres of N lifts to act linearly on Hom(Ẽ, F̃), so defining Hom^{A}(E, F) as a bundle over the projection of N into Z^{2}. It follows that this action is consistent with the composition of Hom(Ẽ, F̃) and Hom(F̃, G̃) for any three projective bundles associated to A. This leads to the composition property (4). □

As a bundle over P, the projective bundle Ẽ can be given an SU(N) invariant connection. A choice of such connections on Ẽ and F̃ induces, as in the standard case, a connection on Hom(Ẽ, F̃) over P×P and hence a connection on Hom^{A}(E, F) over N_{ϵ}.

Remark 2. Since H is a fixed Hilbert space we can also identify K as a trivial bundle over Z^{2}. The construction about then identifies Hom^{A}(E, F) as a subbundle of Â ⊗ K, as a bundle over N_{ϵ}, with the composition (4) induced from (5).

Remark 3. A particularly important case of an Azumaya bundle is the Clifford bundle on any oriented even-dimensional manifold (in the odd-dimensional case the complexified Clifford bundle is not quite an Azumaya bundle but rather the direct sum of two). On a manifold of dimension 2n this is locally isomorphic to the algebra of 2^{n} × 2^{n} matrices. Letting P_{ℂl} be the associated prinicpal PU(2^{n})- bundle of trivializations, we call the trivial bundle, ℂ^{2^{n}}, over P_{ℂl} the projective spin bundle; the relationship to the usual spin bundle is explained in Section 3. Proposition 1 and Remark 1, give an extension of the Clifford bundle to a bundle, ℂ̂l, in neighborhood of the diagonal as the ‘big’ homomorphism bundle of this projective spin bundle. The discussion below shows that this allows us to define a projective spin Dirac operator even when no spin (or even spin_{ℂ}) structure exits. However, it is an element of an algebra of ‘differential operators’ which does not have any natural action.

2. Twisting by a line bundle over P

In Proposition 1 we have described a canonical extension of a given Azumaya algebra A to a bundle Â near the diagonal. In general this canonical extension, the existence of which is based on the discreteness of the cover of PU(N) by SU(N), is not unique as an extension with the composition property (5). Rather, it is based on the selection, natural as it is, of the trivial bundle Ẽ = P × ℂ^{N} with its natural SU(N) action, as generating A through hom(Ẽ). In this section we consider the possibility of other choices of bundle in place of Ẽ and hence other extensions of A.

To motivate this discussion, consider the case of an Azumaya bundle which is trivial, in the sense that it is isomorphic to hom(W) for an Hermitian vector bundle W. The frame bundle W of W is a principal U(N) bundle to which hom(W) lifts to be the trivial bundle of N × N matrices on which U(N) acts through conjugation. Thus the center acts trivially, so hom(W) can also be identified with the trivial bundle of N × N matrices over P = W/ U(1). The circle bundle, L, over P with total space W has an induced SU(N) action and W can be identified with the bundle L_{W} ⊗ℂ^{N} over P, where L_{W} is the line bundle corresponding to L. Abstracting this situation we arrive at the corresponding notion for a general Azumaya bundle.

Definition 1. A representing bundle for a star Azumaya bundle A is a vector bundle Ṽ over P equipped with an action of SU(N) which is equivariant for the PU(N) action on P with the center acting as scalars and with an isomorphism, as bundles of algebras, of A and hom(Ṽ) as a bundle over the base.

When appropriate we consider the unique U(N) action on Ṽ for which the center acts also acts as scalars and such that the only elements of the center acting trivially are elements of ℤ_{N} ⊂ SU(N). Note that this is consistent with the ‘trivial’ case discussed above.

We consider two such representing bundles, Ṽ_{1}, Ṽ_{2}, to be equivalent if there is a bundle isomorphism between them which intertwines the SU(N) actions and projects to intertwine the isomorphisms with A. To understand the non-equivalent representing bundles we study the line bundles on P. The fibres of P are diffeomorphic to PU(N) so all line bundles over P have flat connections over the fibres.

Proposition 2. (cf. Kostant [9] and Brylinski [3]). The total space of any line bundle on P admits a ‘fibre holonomy’ action by SU(N) which is equivariant for the PU(N) action on P, is linear between the fibres and in which the centre acts as the fibre holonomy; this canonical SU(N) action is unique up to conjugation by a bundle isomorphism.

As for representing bundles we consider the unique U(N) action on the line bundle for which the center acts also acts as scalars and such that the only elements of the center acting trivially are elements of ℤ_{N} ⊂ SU(N).

Proof. Let L be a given line bundle on P; choose some connection, ∇, on it with curvature ω ∈ C^{∞}(P; Λ^{2}). Each fibre of P is diffeomorphic to PU(N). Since H^{2}(PU(N), ℝ) = {0} the restriction of ω to each fibre is exact. Thus we can find a smooth 1-form, α, on P such that ω − dα vanishes on each fibre. It follows that the connection ∇ − α has vanishing curvature on each fibre of P.

Now, the SU(N) action on L is given by parallel transport with respect to such a connection. For each smooth curve c : [0, 1] ⟶ PU(N) with c(0) = Id and each point l ∈ L_{p} consider the curve c(t)p in the fibre through p ∈ P and let s(l) ∈ L_{c(1)p} be obtained by parallel transport along L over c(t)p. This certainly gives a smooth map s(c) on the total space of L which is linear on the fibres. Furthermore, since the curvature on each fibre vanishes, s(c) depends only on the homotopy class of c in PU(N) as a curve from Id to g = c(1) ∈ PU(N) and composition of curves leads to the composite map. Thus in fact s is an action of the universal covering group, SU(N), of PU(N) on the total space of L as desired. The centre, ℤ_{N}, of SU(N) gives the fibre holonomy essentially by definition.

Any two connections on L which are fibre-flat differ by a 1-form β which is closed on each fibre. Again, since H^{1}(PU(N), ℝ) = {0}, we may choose f ∈ C^{∞}(P) such that β − df vanishes on each fibre. Parallel transport along curves in PU(N) as discussed above, for the two connections, is then intertwined by the bundle isomorphism exp(f). Thus the SU(N) action defined by parallel transport on the fibres is well-defined up to bundle isomorphism. □

Remark 4. An alternate proof of Proposition 2 uses Cheeger-Simons characters, and will be described here. As in the proof of Proposition 2, (1) given a line bundle L over P, we can always find a connection with curvature

F with the property that F restricted to the fibers is trivial.

(2) SU(N)-actions on L that cover the PU(N) action on P are obtained from characters of ℤ_{N} via the holonomy of flat connections on line bundles along PU(N).

(3) Consider the exact sequence

(6)  0 → H^{1}(P, ℝ)/H^{1}(P, ℤ) → Ȟ^{2}(P) → A^{2}(P) → 0

where Ȟ^{2}(P) denotes the Cheeger-Simons characters of 1-cycles on P and

A^{2}(P) = {(c_{1}, F) : where F is a closed 2-form on P representing c_{1}, which is in
the image of H^{2}(P, ℤ) in H^{2}(P, ℝ)},

cf. page 25, middle formula of equation (3.3) in [6]. Take any pair (c_{1}(L), F) as in (1) above. Then there exists a Cheeger-Simons

character χ : Z_{1}(P) → ℝ/ℤ, whose value on a closed curve is the holonomy of some connection with curvature equal to F. Now the exact sequence (6) when restricted to any fiber of P reduces to,

0 → Ȟ^{2}(PU(N)) → H^{2}(PU(N), ℤ) → 0.

Therefore the Cheeger-Simons characters of 1-cycles on PU(N), are in one-to-one correspondence with line bundles on PU(N). So we deduce that the map from line bundles on P to SU(N)-actions on L that cover the PU(N) action on P, is simply given by the map

H^{2}(P, ℤ) ∋ c_{1} → r^{∗}c_{1} ∈ H^{2}(PU(N), ℤ)

where r is the restriction map to any fiber.

One consequence of Proposition 2 is that any line bundle on P is necessarily the Nth root of a line bundle on the base.

Lemma 1. If L̃ is a line bundle on P for a given Azumaya bundle then L̃^{⊗N} descends to a line bundle L over the base Z.

Proof. It follows from the equivariance that the center ℤ_{N} of SU(N) acts on the fibre L̃_{q}, q ∈ P, at each point as multiplication by Nth roots of unity. Thus, in the induced action on L̃^{⊗N} the center acts trivially, so L̃^{⊗N} has an induced PU(N)-action over P and so descends to a bundle on the base. □

Lemma 2. Any line bundle over P has a connection with curvature which is the lift of the form \frac{1}{N} π^{∗}ω_{L} from the base where ω_{L} is the curvature on the base of a connection on L = L̃^{⊗N}.

Proof. Following the discussion in the proof of Proposition 2 a given line bundle L carries a connection with curvature, ω, which is PU(N)-invariant and vanishes on the fibres. Taking a smooth lift of any vector field, v, to a vector field v^{∗} on P the form i_{v^{∗}}ω is well-defined independent of the lift, and closed on each fibre. Since H^{1}(PU(N), ℝ) = {0}, it follows that the connection may be further modified, by a smooth 1-form which vanishes on each fibre, so that the curvature is a basic and PU(N)-invariant form, i.e. is the lift of a form from the base. Computing in any local trivialization of P gives the curvature in terms of the induced connection on the Nth power. □

Remark 5. Given a line bundle L over a compact manifold M, the problem of finding an N-th root of L which is also a line bundle on M can be approached as follows. One can take the N-th root of the transition functions of L with respect to a good cover U_{a}, g_{ab}^{1/N} : U_{ab} → U(1). On triple overlaps, this gives a cocycle

(7)  t_{abc} = g_{ab}^{1/N}g_{bc}^{1/N}g_{ca}^{1/N} : U_{abc} → ℤ_{N}

where ℤ_{N} = ker(s), s being the second homomorphism in the short exact sequence

(8)  ℤ_{N} → U(1) → U(1)

which is given by s(z) = z^{N}. Then the obstruction to the existence of an N-th root for L is given by the connecting homomorphism in the corresponding long exact sequence in cohomology

(9) ⋯ → H^{1}(M, \underline{U(1)}) →^{β} H^{2}(Z, ℤ_{N}) → H^{2}(M, \underline{U(1)}) → H^{2}(M, \underline{U(1)}) → ⋯

i.e. the obstruction class is β(L) ∈ H^{2}(M, ℤ_{N}). In fact, β(L) = c_{1}(L) (mod N). If L has an Nth root L_{0} then all of the other Nth roots of L are of the form L_{0} ⊗ R, where R is a line bundle on Z such that R^{⊗N} = 1. Hence the set of Nth roots of L is a ℤ_{N}-affine space with associated vector space H^{1}(M, ℤ_{N}).

Applying this to the case M = P it follows that all of the line bundles which have Nth powers a given bundle L over the base are of the form L̃ ⊗ R, where R^{⊗N} = 1, so form a ℤ_{N}-affine space with associated vector space H^{1}(P, ℤ_{N}).

Recall that a principal PU(N) bundle π : P → M has an invariant, t(P) ∈ H^{2}(M, ℤ_{N}), which measures the obstruction to lifting P to a principal SU(N) bundle. This obstruction is obtained via the connecting homomorphism of the exact sequence in cohomology associated to the short exact sequence of sheaves of groups on M,

1 → ℤ_{N} → \underline{SU(N)} → \underline{PU(N)} → 1,

namely t(P) = δ(P), where δ : H^{1}(M, \underline{PU(N)}) → H^{2}(M, ℤ_{N}) and is the mod N analogue of the 2nd Stieffel-Whitney class. Then a theorem of Serre (cf. [4]) asserts that given any class t ∈ H^{2}(M, ℤ_{N}), there is a principal PU(mN) bundle π′ : Q → M (for some m ∈ ℕ) such that t(Q) = t ∈ H^{2}(M, ℤ_{N}). Such a bundle Q is by no means unique.

In particular, given any line bundle L on M, by the theorem of Serre [4], discussed above we know that there is a principal PU(mN) bundle π′ : Q → M (for some m ∈ ℕ) such that t(Q) = β(L) ∈ H^{2}(M, ℤ_{N}). Since t(Q) and β(L) are characteristic classes,

0 = π^{′∗}(t(Q)) = t(π^{′∗}(Q)) = β(π^{′∗}(L)) ∈ H^{2}(Q, ℤ_{N}),

that is, there is a line bundle L̃ on Q which isomorphic to an Nth root of the lifted line bundle π^{′∗}(L) on Q, i.e. L̃^{⊗N} ≅ π^{′∗}(L).

The proof of Proposition 1 applies just as well to the line bundle L̃ ⊠ L̃^{−1} over P × P.

Lemma 3. If L̃ is a line bundle over P, the bundle L̃ ⊠ L̃^{−1} descends from a neighbourhood of the diagonal submanifold of P × P to a well-defined line bundle L̂ over a neighborhood of the diagonal in Z^{2}.

In general a line bundle over P with its SU(N) action, and the corresponding U(N) action, represents a ‘partial trivialization’ of the Azumaya bundle. If the ℤ_{N} action on the fibres of L̃, arising from the centre of SU(N), is injective then in fact the circle bundle associated to L̃ is a lift of the principal PU(N)-bundle, P, to a principal U(N)-bundle. If, at the other extreme, this ℤ_{N} action is trivial then L̃ is simply the lift of a line bundle from the base.

Proposition 3. Any representing bundle for an Azumaya bundle is equivalent to L̃ ⊗ Ẽ with the induced SU(N) action, where Ẽ is the trivial bundle with standard SU(N) action and L̃ is a line bundle on P with its fibre-holonomy SU(N) action.

Proof. Let Ṽ be a representing bundle for the Azumaya bundle A. By assumption Ṽ is a bundle over P. Let F be the frame bundle for Ṽ . This is the principal U(N)- bundle with fibre at a point p ∈ P the space of trivializations of Ṽ_{p}. Now, as part of the data of a representing bundle, we are given an identification of A with hom(Ṽ) as a bundle over the base. Since a point of P is an identification of the fibres of A with N × N matrices, this data picks out a U(1) subbundle L ⊂ F, consisting of the isomorphisms between Ṽ_{p} and ℂ^{N} which realize this identification at that point. Since the equivariant U(N) action on Ṽ has center acting as scalars, L has an induced equivariant U(N) action coming from the equivariant U(N) action on Ṽ and the standard U(N) action on the trivial bundle. If we let L̃ be the line bundle over P associated to L then it has a U(N) action and the restriction of this to SU(N) must be the SU(N) action. The frame bundle of L̃^{−1} ⊗ Ṽ has a natural U(N)-invariant section over P, so Ṽ is equivalent, as a representing bundle, to L̃ ⊗ Ẽ where Ẽ is the standard, trivial, representing bundle. □

In view of this result we generalize projective bundles slightly by allowing twisting by line bundles over P.

Definition 2. For any Azumaya bundle A and line bundle, L̃, over P an associated (L̃−)twisted projective bundle is a subbundle of (L̃ ⊗ ℂ^{N}) ⊗ H which is invariant under the tensor product SU(N) action, arising from the SU(N) action on L̃ and the standard SU(N) action on ℂ^{N} interpreted as covering the PU(N) action on P.

Proposition 4. Any choice of representing bundle, Ṽ ≡ L̃ ⊗ Ẽ, for an Azumaya bundle A over Z gives rise to a vector bundle ÂL̃ which is defined in a neighborhood of the diagonal of Z^{2}, extends A = hom(V) from the diagonal, has the composition property (5) and lifts canonically to Hom(Ṽ) over a neighborhood of the diagonal on P × P; with its composition maps (5) it is isomorphic to Â⊗ L̂ where L̃ is given by Proposition 3.

Proof. The proof of Proposition 1 may be used directly, since no use is made of the fact that the SU(N) action there is the standard one. □

Remark 6. The same argument also gives an extension Hom^{A,L̃}(E, F) of hom(E, F) for any two L̃-twisted projective bundles Ẽ and F̃ for the same line bundle over P.

Remark 7. Applying Proposition 4 to the ‘trivial’ case of an Azumaya bundle A = hom(W) for a vector bundle W we recover recover Â_{W} = Hom^{A,L̃} = Hom(W) in a neighborhood of the diagonal.

3. Trivialization and spin structures

Corresponding to (3), the principal PU(N)-bundle P of local trivializations of the Azumaya bundle A may have a lift to a principal SU(N)-bundle, or to a principal U(N)-bundle,

(10)  \xymatrix{SU(N) \ar[r] \ar[d] & P_{SU(N)} \ar[d] & or & U(N) \ar[r] \ar[d] & P_{U(N)} \ar[d] \\ PU(N) \ar[r] & P &  & PU(N) \ar[r] & P.}

In either case the Dixmier-Douady invariant of A vanishes; conversely the vanishing of the invariant implies the existence of such a lifting to a U(pN)-bundle for A ⊗ hom(G) for some bundle G (and so with rank a multiple of N.)

Since it is of primary concern below, consider the special case of the Clifford bundle. Namely, a choice of metric on Z defines the bundle of Clifford algebras, with fibre at z ∈ Z the (real or complexified) Clifford algebra

(11)  \begin{matrix}Cl_{z}(Z) = (⨁_{k=0}^{∞}(T_{z}^{∗}X)^{k}) /⟨α ⊗ β + β ⊗ α − 2(α, β)_{g}, α, β ∈ T_{z}^{∗}Z⟩, \\ ℂl_{z}(Z) = ℂ ⊗ Cl_{z}(Z). \end{matrix}

If dim Z = 2n, this complexified algebra is isomorphic to the matrix algebra on ℂ^{2^{n}}. A local smooth choice of orthonormal basis over an open set Ω ⊂ Z identifies T^{∗}Ω with Ω × ℝ^{2n} and so identifies Cl(Ω) with Ω × Cl(ℝ^{2n}) as Azumaya bundles. Choosing a fixed identification of ℂl(2n) with the algebra of complex 2^{n} × 2^{n} matrices therefore gives a trivialization of ℂl(Z), as an Azumaya bundle, over Ω. As noted in [10], its Dixmier-Douady invariant is W_{3}(Z).

In particular the Clifford bundle is an associated bundle to the metric coframe bundle, the principal SO(2n)-bundle F, where the action of SO(2n) on the Euclidean Clifford algebra Cl(2n) is through the spin group. Thus, the spin group may be identified within the Clifford algebra as

(12)  Spin(2n) = {v_{1}v_{2} ⋯ v_{2k} ∈ Cl(2n); v_{i} ∈ ℝ^{2n}, |v_{i}| = 1}.

The non-trivial double covering of SO(2n) comes through the mapping of v to the reflection R(v) ∈ O(2n) in the plane orthogonal to v

(13)  p : Spin(2n) ∋ a = v_{1} ⋯ v_{2k} ⟼ R(v_{1}) ⋯ R(v_{2k}) = R ∈ SO(2n).

Thus P may be identified with the bundle associated to F by the action of SO(2n) on ℂl(2n) (or in the real case Cl(2n)) where R in (13) acts by conjugation by a

(14)  Cl(2n) ∋ b ⟼ aba^{−1} ∈ Cl(2n).

We therefore have a map of principal bundles

(15)  F ⟶ P.

Recall that the projective spin bundle on P is just the bundle associated to the natural action of Cl(n) on itself; it can therefore be identified with the trivial bundle over F with an equivariant SU(N) action, where N = 2^{n}.

Now, a spin structure on Z, corresponds to an extension, F_{S}, of the coframe bundle to a Spin bundle,

(16)  \xymatrix{Spin(2n) \ar[r] \ar[d]^{p} & F_{S} \ar[d] \\ SO(2N) \ar[r] & F.}

Since Spin(2n) ⊂ SU(N), where SU(N) ⊂ ℂl(2n), this in turn gives rise to a lift of P to a principal SU(N) bundle:

(17)  [figure: SU(N); P_{SU(N)}; Spin(2n); p; F_{S}; SO(2N); F; P.]

Thus the projective bundle naturally associated to the Clifford bundle can reasonably be called the projective spin bundle since a spin structure on the manifold gives a lift of Ẽ ⊗ M, where M is the ℤ_{2} bundle given by the spin structure, to the usual spin bundle.

As in the standard case, the Levi-Civita connection induces a natural, SU(N) equivariant, connection on the projective spin bundle over P. We use this below to define the projective spin Dirac operator; a choice of spin structure, when there is one, identifies it with the spin Dirac operator.

Note that similar remarks apply to a spin_{ℂ} structure on the manifold Z. The model group

(18) Spin_{ℂ}(2n) = {cv_{1}v_{2} ⋯ v_{2k} ∈ Cl(2n); v_{i} ∈ ℝ^{2n}, |v_{i}| = 1, c ∈ ℂ, |c| = 1}

= (Spin × U(1))/±,

is a central extension of SO(2n),

(19)  U(1) ⟶ Spin_{ℂ}(2n) ⟶ SO(2n)

where the quotient map is consistent with the covering of SO(2n) by Spin(2n).

Thus a spin_{ℂ} structure is an extension of the coframe bundle to a principal Spin_{ℂ}(2n)-bundle;

(20)  \xymatrix{U(1) \ar[r] \ar[d] & L \\ Spin_{ℂ}(2n) \ar[r] \ar[d] & F_{L} \ar[d] \\ SO(2n) \ar[r] & F.}

where F_{L}, the Spin_{ℂ}(2n) bundle, may be viewed as a circle bundle over F. Since Spin_{ℂ}(2n) ↪ U(N) (but is not a subgroup of SU(N)) this gives a diagram similar to (17) but lifting to a principal U(N) bundle

(21)  [figure: U(N); P_{U(N)}; Spin_{ℂ}(2n); p; F_{L}; SO(2n); F; P.]

In this case the spin_{ℂ} bundle over Z is the lift of S ⊗ L from P to P_{U(N)}.

Note that the existence of a spin structure on Z is equivalent to the condition w_{2} = 0. The Clifford bundle is then the homomorphism bundle of the spinor bundle, so the existence of a spin structure implies the vanishing of the Dixmier-Douady invariant of the Clifford bundle (which is W_{3}, the Bockstein of w_{2}); the vanishing of W_{3} is precisely equivalent to the existence of a spin_{ℂ} structure (without any necessity for stabilization).

In the general case, even when W_{3} ≠ 0 and there is no spin_{ℂ} structure, we shall show below that we can still introduce the notion of a ‘projective spin_{ℂ} Dirac operator’ starting from the following notion.

Definition 3. On any even-dimensional, oriented manifold a projective spin_{ℂ} structure is a choice of representing bundle, in the sense of Definition 1, for the complexified Clifford bundle.

Thus, by Proposition 3 such a representing bundle is always equivalent to, and hence can be replaced by, S̃ ⊗ L̃ where L̃ is a line bundle over the bundle of trivializations P_{ℂl} of the Clifford bundle and S̃ is the projective spin bundle. As remarked above, this is consistent with the standard case in which there is a spin_{ℂ} structure and Ṽ then descends to a bundle on the base.

Remark 8. Any line bundle over P_{ℂl} is necessarily a square root of a line bundle from the base. This follows by restricting the line bundle to the frame bundle, as a subbundle of P_{ℂl}, and the U(N), N = 2^{n}, action to Spin(2n) showing that the centre acts through the subgroup ℤ_{2} ⊂ ℤ_{N}.

4. Smoothing operators

For two vector bundles E and F the space of smoothing operators Ψ^{−∞}(Z; E, F) between sections of E and sections of F may be identified with the corresponding space of kernels

(22)  Ψ^{−∞}(Z; E, F) = C^{∞}(Z^{2}; Hom(E, F) ⊗ π_{R}^{∗}Ω)

where the section of the density bundle allows invariant integration. Thus, such kernels define linear maps C^{∞}(Z; E) ⟶ C^{∞}(Z; F) through

(23)  Au(z) = ∫_{Z} A(z, z′)u(z′).

Operator composition induces a product

(24)  \begin{matrix}Ψ^{−∞}(Z; F, G) ∘ Ψ^{−∞}(Z; E, F) ⊂ Ψ^{−∞}(Z; E, G), \\ A ∘ B(z, z′′) = ∫_{Z} A(z, z′)B(z′, z′′) \end{matrix}

using the composition law (4). The right density factor in A is used in (24) to carry out the integral invariantly.

Given the extensions in Proposition 1 and Proposition 4 of the homomorphism bundles it is possible to define the linear space of smoothing operators with kernels supported in N_{ϵ} for any pair E, F of projective bundles (or twisted projective bundles) associated to a fixed Azumaya bundle (and twisting) as

(25)  Ψ_{ϵ}^{−∞}(Z; E, F) = C_{c}^{∞}(N_{ϵ}; Hom^{A,L̃}(E, F) ⊗ π_{R}^{∗}Ω)

where in case E and F are projective bundles, without twisting, L̃ is trivial so is dropped from the notation. Note that the projective and possibly twisted nature of E and F is implicit in the notation. Although there is no action analogous to (23) the composition law (4) allows (24) to be extended directly to define

(26)  Ψ_{ϵ/2}^{−∞}(Z; F, G) ∘ Ψ_{ϵ/2}^{−∞}(Z; E, F) ⊂ Ψ_{ϵ}^{−∞}(Z; E, G)

in the case of three projective bundles associated to the fixed A. For sufficiently small supports this product is associative

(A ∘ B) ∘ C = A ∘ (B ∘ C)

if A ∈ Ψ_{ϵ/4}^{−∞}(Z; G, H), B ∈ Ψ_{ϵ/4}^{−∞}(Z; F, G), C ∈ Ψ_{ϵ/4}^{−∞}(Z; E, F).

The trace functional extends naturally to these spaces

(27)  Tr : Ψ_{ϵ}^{−∞}(Z; E) = ∫_{Z} tr A(z, z)

and vanishes on appropriate commutators

(28)  Tr(AB − BA) = 0 if A ∈ Ψ_{ϵ/2}^{−∞}(Z; F, E), B ∈ Ψ_{ϵ/2}^{−∞}(Z; E, F)

as follows from Fubini’s theorem.

5. Pseudodifferential operators

Just as the existence of the bundle Hom^{A,L̃}(E, F) over the neighbourhood N_{ϵ} of the diagonal allows smoothing operators to be defined, it also allows arbitrary pseudodifferential operators, with kernels supported in N_{ϵ} to be defined as the space of kernels

(29)  Ψ_{ϵ}^{m}(Z; E, F) = I_{c}^{m}(N_{ϵ}; Diag) ⊗_{C_{c}^{∞}(N_{ϵ})} C_{c}^{∞}(N_{ϵ}; Hom^{A,L̃}(E, F)).

Here, E, F are either L̃-twisted projective bundles associated to some Azumaya bundle A. These are just the conormal sections of Hom^{A,L̃}(E, F) ⊗ π_{R}^{∗}Ω with support in N_{ϵ}. Notice that for any small δ < ϵ,

(30)  Ψ_{δ}^{m}(Z; E, F) + Ψ_{ϵ}^{−∞}(Z; E, F) = Ψ_{ϵ}^{m}(Z; E, F).

The singularities of these kernels are unrestricted by the support condition so there are the usual short exact sequences

(31)

Ψ_{ϵ}^{m−1}(Z; E, F) ⟶ Ψ_{ϵ}^{m}(Z; E, F) ⟶^{σ_{m}} C^{∞}(S^{∗}Z; hom(E, F) ⊗ N_{m})

Ψ_{ϵ}^{−∞}(Z; E, F) ⟶ Ψ_{ϵ}^{m}(Z; E, F) ⟶^{σ} ρ^{−m}C^{∞}(S^{∗}Z; hom(E, F))[[ρ]].

In the first case N_{m} is the bundle over S^{∗}Z of the smooth functions on T^{∗}Z\0 which are homogeneous of degree m; this sequence is completely natural and independent of choices. In the second sequence ρ ∈ C^{∞}(\overline{T^{∗}Z}) is a defining function for the boundary and the image space represents Taylor series at the boundary, with an overall factor of ρ^{−m}; this sequence depends on choices of a metric and connection to give a quantization map.

The product for pseudodifferential operators extends by continuity (using the larger spaces of symbols with bounds, rather than the classical symbols implicitly used above) from the product for smoothing operators and leads to an extension of (26)

(32)  \begin{matrix}Ψ_{ϵ/2}^{m}(Z; F, G) ∘ Ψ_{ϵ/2}^{m′}(Z; E, F) ⊂ Ψ_{ϵ}^{m+m′}(Z; E, G), \\ σ_{m+m′}(A ∘ B) = σ_{m′}(B)σ_{m}(A). \end{matrix}

The induced product on the image space given by the second short exact sequence in (31) is a star product as usual.

The approximability of general pseudodifferential operators by smoothing operators, in the weaker topology of symbols with bounds, also shows, as in the standard case, that (28) extends to

(33)  Tr([A, B]) = 0 if A ∈ Ψ_{ϵ/2}^{m}(Z; F, E), B ∈ Ψ_{ϵ/2}^{−∞}(Z; E, F)

for any m.

Now, the standard symbolic constructions of the theory of pseudodifferential operators carry over directly since these are all concerned with the diagonal singularity and the symbol map.

Theorem 1. For any two projective bundles associated to the same Azumaya bundle (or twisted projective bundles associated to the same Azumaya bundle and the same line bundle over P), if A ∈ Ψ_{ϵ/2}^{m}(Z; E, F) is elliptic, in the sense that σ_{m}(A) is invertible (pointwise), then there exists B ∈ Ψ_{ϵ/2}^{−m}(Z; E, F) such that

(34) B ∘ A = Id −E_{R}, A ∘ B = Id −E_{L}, E_{R} ∈ Ψ_{ϵ}^{−∞}(Z; E), E_{L} ∈ Ψ_{ϵ}^{−∞}(Z; F)

and any two such choices B′, B satisfy B′ − B ∈ Ψ_{ϵ/2}^{−∞}(Z; E, F).

Proof. Now absolutely standard. □

If B′ and B are two such parametrices it follows that B_{t} = (1 − t)B′ + tB, t ∈ [0, 1], is a smooth curve of parametrices. Furthermore

(35)  \frac{d}{dt}[A, B_{t}] = [A, (B − B′)]

so, by (33), it follows that for any two parametrices

(36)  Tr([A, B′]) = Tr([A, B])

since B′ − B is smoothing.

Definition 4. For an elliptic pseudodifferential operator A ∈ Ψ_{ϵ/2}^{m}(Z; E, F) acting between projective bundles associated to a fixed Azumaya bundle, or more generally between twisted projective bundles corresponding to the same twistting line bundle over P, we define

(37)  ind_{a}(A) = Tr([A, B])

for any parametrix as in Theorem 1.

6. Homotopy invariance

Proposition 5. The index (37) is constant on a 1-parameter family of elliptic operators.

Remark 9. Given the rationality proved in the next section this follows easily. Here we use the homotopy invariance to prove the rationality!

Proof. For a smooth family A_{t} ∈ C^{∞}([0, 1]; Ψ_{ϵ/2}^{m}(Z; E, F)) of elliptic operators as discussed above, it follows as in the standard case that there is a smooth family of parametrices, B_{t} ∈ C^{∞}([0, 1]; Ψ_{ϵ/2}^{−m}(Z; E, F)). Thus the index, defined by (37) is itself smooth, since [B_{t}, A_{t}] is a smooth family of smoothing operators. To prove directly that this function is constant we use the residue trace of Wodzicki, see [15], with an improvement to the definition due to Guillemin [5] and also the trace defect formula from [11].

For a classical operator A of integral order, m, in the usual calculus the residue trace is defined by ‘ζ-regularization’ (following ideas of Seeley) using the entire family of complex powers of a fixed positive (so self-adjoint) elliptic operator of order 1 :

(38)  Tr_{R}(A) = lim_{z→0} z Tr(AD^{z})

where Tr(AD^{z}) is known to be meromorphic with at most simple poles at z = −k−dim Z+{0, 1, 2, … }. One of Guillemin’s innovations was to show that the same functional results by replacing D^{z} by any entire family D(z) of pseudodifferential operators of complex order z which is elliptic and has D(0) = Id .

One way to construct such a family, which is useful below, is to choose a generalized Laplacian on the bundle in question, which is to say a second order selfadjoint differential operator, L, with symbol |ξ|^{2} Id, the metric length function, and to construct its heat kernel, exp(−tL). This is a well-defined (locally integrable) section of the homomorphism bundle on [0, ∞)_{t} × Z^{2} which is singular only at {t = 0} × Diag(Z) and vanishes with all derivatives at t = 0 away from the diagonal. If L = D^{2} is strictly positive the heat kernel decays exponentially as t → ∞ and the complex powers of L are given by the Mellin transform

(39)  L^{z} = D^{2z} = \frac{1}{Γ(z)} ∫_{0}^{∞} t^{−z−1} exp(−tD)dt,

where the integral converges for Re z << 0 and extends meromorphically to the whole of the complex plane. The fact that D^{0} = Id arises from the residue of the integral at z = 0, so directly from the fact that exp(−tL) = Id at t = 0. It follows that if χ ∈ C_{c}^{∞}([0, ∞) × Z^{2}) and χ ≡ 1, in the sense of Taylor series, at {0} × Diag then

(40)  D(2z) = \frac{1}{Γ(z)} ∫_{0}^{∞} t^{−z−1}H(t)dt, H(t) = χ exp(−tL)

is an entire family as required for Guillemin’s argument, that is D is elliptic and D(0) = Id . Since the construction of the singularity of the heat kernel for such a differential operator is completely symbolic (see for instance Chapter 5 of [12]), quite analogous to the construction of a parametrix for an elliptic operator, it can be carried out in precisely the same manner in the projective case, so giving a family of the desired type via (40).

Alternatively, for any projective vector bundle, such a family can be constructed using an explicit linear quantization map, with kernels supported arbitrarily close to the diagonal

(41)  D(z) ∈ Ψ_{ϵ/4}^{z}(Z; E).

Thus we may define the residue trace and prove its basic properties as in the standard case; in particular it vanishes on all operators of sufficiently low order. It is also a trace functional

(42)  Tr_{R}([A, B]) = 0, A ∈ Ψ_{ϵ/4}^{m}(Z; E, F), B ∈ Ψ_{ϵ/4}^{m′}(Z; F, E).

The additional result from [11], see also [13], that we use here is that the regularized trace, defined to be

(43)  \overline{Tr}_{D}(A) = lim_{z→0} (Tr(AD(z)) − \frac{1}{z} Tr_{R}(A))

which does depend on the regularizing family, satisfies

(44)  \overline{Tr}_{D}([A, B]) = Tr_{R}(Bδ_{D}A)

where δ_{D} is a derivation acting on the full symbol algebra in (31) and which also satisfies

(45)  Tr_{R}(δ_{D}a) = 0 ∀ a.

From these formulæ the homotopy invariance of the index in the projective case follows. Namely

(46)

\frac{d}{dt} ind_{a}(A_{t}) =\frac{d}{dt}\overline{Tr}_{D}([A_{t}, B_{t}])
=\overline{Tr}_{D}([Ȧ_{t}, B_{t}]) + \overline{Tr}_{D}([A_{t}, Ḃ_{t}])
= Tr_{R}(a_{t}^{−1}δ_{D}ȧ_{t}) + Tr_{R}((\frac{d}{dt}a_{t}^{−1})δ_{D}a_{t},)
= − Tr_{R}(ȧ_{t}δ_{D}a_{t}^{−1}) − Tr_{R}(a_{t}^{−1}ȧ_{t}a_{t}^{−1}δ_{D}a_{t}) = 0.

Here, a_{t} is the image of A_{t} in the full symbol algebra in which the image of B_{t} is a_{t}^{−1} and (45) has been used. □

Remark 10. A similar argument also proves the multiplicativity of the index. Thus if A_{i} for i = 1, 2 are two elliptic projective operators with the image bundle of the first being the same as the domain bundle of the second, they can be composed if their supports are sufficiently small. Let B_{i} be corresponding parametrices, again with very small supports. Then B_{1}B_{2} is a parametrix for A_{2}A_{1} and the index of the product is given by (44) in terms of the ‘full symbols’ a_{i} of the A_{i}

(47) ind_{a}(A_{2}A_{1}) = \overline{Tr}_{D}([A_{2}A_{1}, B_{1}B_{2}]) = Tr_{R}(a_{1}^{−1}a_{2}^{−1}δ_{D}(a_{2}a_{1}))

= Tr_{R}(a_{1}^{−1}δ_{D}a_{1}) + Tr_{R}(a_{2}^{−1}δ_{D}a_{2}) = ind_{a}(A_{1}) + ind_{a}(A_{2}).

7. Projective Dirac operators

The space of differential operators ‘acting between’ two projective bundles associated to the same Azumaya algebra is well defined, since these are precisely the pseudodifferential operators with kernels with supports contained in the diagonal; we denote by Diff^{k}(Z; E, F) the space of these operators of order at most k.

Of particular interest is that, in this projective sense, there is a ‘spin Dirac operator’ on every oriented even-dimensional compact manifold. As discussed in Section 3 above, the projective bundle associated to the spin representation is the projective spin bundle of Z, which we denote by S; if Z is oriented it splits globally as the direct sum of two projective bundles S^{±}. There are natural connections on ℂl(Z) and S^{±} arising from the Levi-Civita connection on T^{∗}Z. As discussed in Proposition 1, the homomorphism bundle of S, which can be identified with ℂl(Z), has an extension to ℂ̂l(Z) in a neighbourhood of the diagonal, and this extended bundle also has an induced connection. The projective spin Dirac operator may then be identified with the distribution

(48)  ð = cl ·∇_{L}(κ_{Id}), κ_{Id} = δ(z − z′) Id_{S} .

Here κ_{Id} is the kernel of the identity operator in Diff^{∗}(Z; S) and ∇_{L} is the connection restricted to the left variables with cl the contraction given by the Clifford action of T^{∗}Z on the left. As in the usual case, ð is elliptic and odd with respect to the ℤ_{2} grading of S and locally the choice of a spin structure identifies this projective spin Dirac operator with the usual spin Dirac operator.

More generally we can consider projective twists of this projective spin Dirac operator. If E is any unitary projective vector bundle over Z, associated to an Azumaya bundle A and equipped with a Hermitian connection then S ⊗ E is a projective bundle associated to ℂl(Z) ⊗ A and it is a Clifford module in the sense that

ℂl(Z) ⊂ hom(S ⊗ E).

The direct extension of (48), using the tensor product connection, gives an element ð_{E} ∈ Diff^{1}(Z; S ⊗ E) which is again ℤ_{2} graded. In the special case that S ⊗ E is a bundle a related construction is given by M. Murray and M. Singer [14].

The relation between the index of twisted, projective spin Dirac operators (or more generally, projective elliptic operators) and the distribution index of transversally elliptic operators, will be discussed in a subsequent paper.

Theorem 2. The positive part, ð_{E}^{+} ∈ Diff^{1}(Z; E ⊗ S^{+}, E ⊗ S^{−}) of the projective spin Dirac operator twisted by a unitary projective vector bundle E, has index

(49)  ind_{a}(ð_{E}^{+}) = ∫_{Z} Â(Z) ∧ Ch_{A}(E)

where Ch_{A} : K^{0}(Z; A) ⟶ H^{even}(Z; ℚ) is the Chern character in twisted K-theory.

Proof. The proof via the local index formula, see [2] and also [12], carries over to the present case. As discussed above, the heat kernel H, formally representing exp(−tð_{E}^{2}), is well-defined as a smooth kernel on Z^{2} × (0, ∞), with values in Hom^{ℂl⊗A}(S ⊗ E) ⊗ Ω_{R}, near the submanifold Diag_{Z} ×{0} modulo an element of Ċ^{∞}(Z^{2} × [0, ∞); Hom^{ℂl⊗A}(S ⊗ E) ⊗ Ω_{R}); that is vanishing to all orders at t = 0. Then we claim that the analogue of the McKean-Singer formula holds and

(50)  ind_{a}(ð_{E}^{+}) = lim_{t↓0} STr(H_{t})

where STr is the supertrace, the difference of the traces on E^{+} and E^{−}. The local index formula, as a result of rescaling, asserts the existence of this limit and its evaluation (49).

In the standard case the McKean-Singer formula (50), for the actual heat kernel, follows by comparison with the limit as t → ∞, which explicitly gives the index. Indeed then the function STr(exp(−tð^{2})) is constant in t. In the present case the approximate kernel gives a parametrix for ð through analytic continuation to z = 0,

(51)  B = B(0)ð, B(z) = ∫_{0}^{∞} t^{z}H_{t}dt

where this integral is regularized by holomorphic extension. Inserting this as the parametrix in the definition, (37), of the index gives

(52)

ind_{a}(ð^{+}) = Tr([ð^{+}, B(0)])
= Tr([ð^{+}, ∫_{0}^{∞} t^{z}H_{t}^{+}ð_{−}dt|_{z=0}])
= − STr(∫_{0}^{∞} (t^{z} \frac{d}{dt}H_{t}) dt|_{z=0}
= lim_{z→0} z ∫_{0}^{∞} STr(t^{z−1}H_{t})dt.

Here in the passage from the second to the third line, smoothing error term (\frac{d}{dt} + ð^{2})H_{t} contributes nothing to the trace. The residue in the last line is just the value at z = 0 of the supertrace of the approximate heat kernel so we arrive at (50) and hence (49). □

Let P be the principal PU(N) bundle associated to A and P′ be the principal PU(N′) bundle associated to ℂl(Z), cf. Section 3. Twisting by P and P′-twisting line bundles L̃ and L̃′ respectively, does not affect the local discussion, only the final formula. Thus if S̃L̃′ = S̃ ⊗ L̃′ is a projective spin_{ℂ} bundle in the sense of Definition 3 and ẼL̃ = Ẽ⊗L̃ is a projective vector bundle we may define the twisted projective spin_{ℂ} Dirac operator on it by choice of an SU(N′)-invariant connection on the twisting bundle L̃′ and SU(N)-invariant connection on the twisting bundle L̃, the Levi-Civita connection on S̃ and an SU(N)-invariant connection on Ẽ. As usual we think of these bundles, S_{L′}, E_{L} as twisted projective bundles over the manifold, although they are in fact bundles over P′ and P respectively.

Theorem 3. The positive part, ð_{L′,E_{L}}^{+} ∈ Diff^{1}(Z; E_{L} ⊗ S_{L′}^{+}, E_{L} ⊗ S_{L′}^{−}) of the projective spin_{ℂ} Dirac operator corresponding to a general projective spin_{ℂ} strucuture and twisted by a unitary projective vector bundle E has index

(53)  ind_{a}(ð_{L′,E_{L}}^{+}) = ∫_{Z} Â(Z) ∧ exp(\frac{1}{2}ω_{L′}) ∧ Ch_{A}(E) ∧ exp(\frac{1}{N} ω_{L})

where Ch_{A} : K^{0}(Z; A) ⟶ H^{even}(Z; ℚ) is the Chern character in twisted K-theory, ω_{L} is the curvature of L, the Nth power of the line bundle and ω_{L′} is the curvature of L′, the square of the line bundle over P′.

8. Index formula

Theorem 4. Given an Azumaya bundle, A, over an even-dimensional compact manifold Z the analytic index defines a map

(54)  ind_{a} : K_{c}^{0}(T^{∗}Z; π^{∗}A) ⟶ ℚ

where ind_{a}(A) = ind_{a}(σ(A)) for elliptic elements of Ψ_{ϵ}(Z; E, F) for projective vector bundles associated to A and

(55)  ind_{a}(b) = ∫_{T^{∗}Z} Td(T^{∗}Z) ∧ Ch_{A}(b) ∀ b ∈ K_{c}(T^{∗}Z; π^{∗}(A)).

Proof. It has been shown above that ind_{a}(A), for elliptic elements of Ψ_{ϵ}(Z; E, F) is additive, homotopy invariant and multiplicative on composition. Thus it does descend to a map as in (54), just as in the standard case, but with possibly real values. As such a real-valued additive map on the twisted K-space K_{c}(T^{∗}Z; π^{∗}(A)), ind_{a} must factor through the Chern character, since it is an isomorphism over ℝ (or ℚ). Thus

(56)  ind_{a}(b) = ĩnd_{a}(Ch_{A})(b)), ĩnd_{a} : H_{c}^{even}(T^{∗}Z; ℚ) ⟶ ℝ

being a well-defined map. However we may construct such elliptic projective psedodifferential operators by twisting the signature operator by a projective vector bundle associated to A. For these (49) gives the index. From the Thom isomorphism in cohomology, we know that these elements generate H_{c}^{even}(T^{∗}Z; ℚ) so suffice to compute the map ĩnd_{a} in (56). Thus it suffices to show that the Riemann-Roch formula (49) is consistent with (55), but this follows from the standard case of the index formula and linearity. □

In the non-oriented case we can pass to the oriented cover and deduce the same formula. Similarly if we consider pseudodifferential operators acting between L̃ twisted projective vector bundles corresponding to a line bundle L̃ over the bundle of trivializations of an Azumaya bundle A, and with Nth power L over the base, we arrive at the analogous twisted formula generalizing (55) and (53) (57)

ind_{a}(Q) = ∫_{T^{∗}Z} Td(T^{∗}Z) ∧ Ch_{A}(σ(Q)) ∧ exp(\frac{1}{N} c_{1}(L)), ∀ Q ∈ Ψ_{ϵ}(Z; E, F) elliptic.

Remark 11. The analytic index map (54) takes rational values, as follows from (55), all the characteristic classes being rational. In fact we can say a little more about the denominators. Namely for a given torsion class w ∈ H^{3}(Z; ℤ), of order p there is an integer q such that

(58)  pq ind_{a} : K_{c}^{0}(T^{∗}Z; A) → ℤ, 𝔻(A) = w.

Namely, the integer pq is the minimal rank of a flat virtual projective bundle with DD class w. This virtual bundle can be realized as a difference of projective bundles and when twisted with the dual of these bundles any projective elliptic pseudodifferential, associated to A, becomes untwisted and hence has integral index. This index is, from (55), just pq times the original index and (58) follows.

In the odd-dimensional case we may use suspension to reduce to the evendimensional case and again arrive at (57). Namely take the exterior tensor product with an untwisted operator of index one on the circle. To do this it is necessary to generalize the discussion in Section 1 to 6 to such ‘product type’ operators, including the homotopy invariance, enough to show that this exterior tensor product can be deformed, through elliptic operators in the product sense, to a true (projective) elliptic pseudodifferential operator. This is essentially a smooth analogue of arguments already present in [1] and we forgo the details, since geometrically the even dimensional case is the more interesting one.

9. Fractions and the index formula

On an oriented even-dimensional manifold, the vanishing of W_{3} is equivalent to the existence of a spin_{ℂ} structure (cf. [8]); in particular this follows if the manifold is almost complex. In the almost complex case there is no spin structure unless the canonical bundle has a square root. Nevertheless, there is always a projective spin Dirac operator and Theorem 2 applied in this case case gives the usual formula

ind_{a}(ð^{+}) = ∫_{Z} Â(Z).

We recall some well known examples of oriented but non-spin manifolds where ∫_{Z} Â(Z) is a fraction, justifying the title of the paper. The simplest is Z = ℂP^{2}, in which case ∫_{Z} Â(Z) = −\frac{1}{8}.

Also in the almost complex case with Hermitian metric, we have the spin_{ℂ} Dirac operator

(59)  \overline{∂} + \overline{∂}^{∗} : Λ^{0,even}Z ⟶ Λ^{0,odd}Z.

Its index is ∫_{Z} Â(Z)e^{\frac{1}{2} c_{1}} where c_{1} = c_{1}(Z) is the Chern class of the canonical line bundle. The integral is the formula for the top term in the Todd polynomial written in terms of Â and c_{1}.

An amusing corollary of Theorem 3 is that we can now interpret the integral as the index of the projective Dirac operator coupled to a line bundle which is a square root of the canonical bundle. Previously this interpretation was only possible when Z was itself spin, when this square root bundle exists as an ordinary line bundle on Z.

Another important class of examples is the following. Let V^{2n}(2d + 1) be hypersurfaces in ℂP^{2n+1}. That is, in the homogeneous coordinates [Z_{0}, …, Z_{2n+1}] for ℂP^{2n+1},

\begin{aligned}V^{2n}(2d + 1) & = & \{[Z_{0}, …, Z_{2n+1}] ∈ ℂP^{2n+1} : P (Z_{0}, …, Z_{2n+1}) = 0, \\ & & ∇P (Z_{0}, …, Z_{2n+1}) ≠ 0, (Z_{0}, …, Z_{2n+1}) ≠ 0\} \end{aligned}

where P (Z_{0}, …, Z_{2n+1}) is a homogeneous polynomial of degree 2d + 1. Then it is known that V^{2n}(2d + 1) is not a spin manifold, and that

∫_{V^{2n}(2d+1)} Â(V^{2n}(2d + 1)) = \frac{2^{−2n}(2d + 1)}{(2n + 1)!} ∏_{k=1}^{n}((2d + 1)^{2} − (2k)^{2}).

It is straightforward to see that for d ≥ n, the right hand side is equal to a non-zero fraction that is not an integer.

Note that ℂP^{2} has positive scalar curvature and the Bochner-Lichnerowicz formula holds for the projective operator ð^{2}, yet ind_{a}(ð) = ∫_{Z} Â(Z) = −\frac{1}{8} ≠ 0! The usual argument, by contradiction, to the vanishing of the index, and hence Â genus, is not applicable since in the twisted case there is no notion of global section of the projective spinor bundle and therefore no way to construct harmonic spinors.

As we have observed above, Z has no spin_{ℂ} structure if W_{3} ≠ 0. Nevertheless the projective Dirac operator exists and can have a nonzero (rational) index. We thank M.J. Hopkins for examples of Z with both W_{3} and ∫_{Z} Â(Z) nonzero. Here is one of his examples. Let 𝕊^{2} ⟶ ℂℙ^{4} be an embedding of degree 2. Surgery on the 𝕊^{2} produces a manifold Z with W_{3} ≠ 0 and

∫_{Z} Â(Z) = ∫_{ℂℙ^{4}} Â(ℂℙ^{4}) = \frac{3}{128}.

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Department of Mathematics, University of Adelaide, Adelaide 5005, Australia

E-mail address: vmathai@maths.adelaide.edu.au

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass 02139

E-mail address: rbm@math.mit.edu

Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Mass 02139

E-mail address: ims@math.mit.edu