THE INDEX OF PROJECTIVE FAMILIES OF ELLIPTIC OPERATORS

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

Abstract. An index theory for projective families of elliptic pseudodifferential operators is developed. The topological and the analytic index of such a family both take values in twisted K-theory of the parametrizing space. The main result is the equality of these two notions of index when the twisting class is in the torsion subgroup tor(H^{3}(X; ℤ)) and the Chern character of the index class is then computed.

Introduction

In this paper we develop an index theory for projective families of elliptic pseudodifferential operators. Such a family, {D_{b}, b ∈ X}, on the fibers of a fibration

(1)  ϕ : M ⟶ X

with base X, and typical fibre F, is a collection of local elliptic families, for an open covering of the base, acting on finite-dimensional vector bundles of fixed rank where the usual compatibility condition on triple overlaps, to give a global family, may fail by a scalar factor. These factors define an integral 3-cohomology class on the base, the Dixmier-Douady class, Θ ∈ H^{3}(X, ℤ). We show that both the analytic and topological index of D_{b} may be defined as elements of twisted K-theory, with twisting class Θ, and that they are equal. In this setting of finite-dimensional bundles the twisting class is necessarily a torsion class. We also compute the Chern character of the index in terms of characteristic classes. When the twisting class Θ is trivial, these results reduce to the Atiyah-Singer index theorem for families of elliptic operators, [1].

The vector bundles on which D_{b} acts, with this weakened compatibility condition, form a projective vector bundle^{1}. In the torsion case, elements of twisted K-theory may be represented by differences of such projective vector bundles and, after stabilization, the local index bundles of the family give such a difference, so defining the analytic index. The topological index is defined, as in the untwisted case, by push-forward from a twisted K class of compact support on the cotangent bundle. The proof we give of the equality of the analytic and the topological index is a generalization of the axiomatic proof of the families index theorem of [1]. This is done by proving that all normalized, functorial index maps, in twisted K-theory, satisfying excision and multiplicativity coincide and then showing that the analytic index and the topological index satisfy these conditions.

1.0A; Revised: 1-6-2002; Run: June 1, 2002.

V.M. acknowledges support from the Clay Mathematical Institute, R.B.M. from NSF grant DMS-0104116 and I.M.S. from DOE contract # DE-FG02-88ER25066.

1Generally these are called gauge bundles in the Physical literature

Twisted K-theory arises naturally when one considers the Thom isomorphism for real Riemannian vector bundles in the even-rank case and when X is compact. The compactly supported K-theory of E, K_{c}(E), is then isomorphic to the twisted K-theory of X, K(C(X, Cl(E))), where Cl(E) denotes the Clifford algebra bundle of E, and C(X, Cl(E)) denotes the algebra of continuous sections. There is a similar statement with a shift in degree for odd-dimensional vector bundles. The twisting of families of Dirac operators by projective vector bundles provides many examples of elliptic families and, as in the untwisted case, the Chern character has a more explicit formula.

Recently, there has been considerable interest in twisted K-theory by physicists, with elements of twisted K-theory interpreted as charges of D-branes in the presence of a background field; cf. [16], [7].

There is an alternative approach to the twisted index theorem, not carried out in detail here. This is to realize the projective family as an ordinary equivariant family of elliptic pseudodifferential operators on an associated principal PU(n)- bundle. Then the analytic index and the topological index are elements in the U(n)- equivariant K-theory of this bundle. Their equality follows from the equivariant index theorem for families of elliptic operators as in [15]. The proof can then be completed by showing that the various definitions of the analytic index and of the topological index agree for projective families of elliptic operators.

A subsequent paper will deal with the general case of the twisted index theorem when the twisting 3-cocycle is not necessarily torsion. Then there is no known finite-dimensional description of twisted K-theory, and even to formulate the index theorem requires a somewhat different approach.

The paper is organized as follows. A review of twisted K-theory, with an emphasis on the interpretation of elements of twisted K-theory as differences of projective vector bundles is given in §1. The definition of general projective families of pseudodifferential operators is explained in §2, leading to the definition of the analytic index, in the elliptic case, as an element in twisted K-theory. The definition of the topological index is given in §3 and §4 contains the proof of the equality of these two indices. In §5 the Chern character of the analytic index is computed and in §6 the determinant bundle is discussed in this context. Finally, §7 contains a brief description of Dirac operators.

1. Review of twisted K-theory

General references for most of the material summarized here are [11, 12].

1.1. Brauer groups and the Dixmier-Douady invariant. We begin by reviewing some results due to Dixmier and Douady, [4]. Let X be a smooth manifold, let H denote an infinite-dimensional, separable, Hilbert space and let K be the C^{∗}- algebra of compact operators on H. Let U(H) denote the group of unitary operators on H endowed with the strong operator topology and let PU(H) = U(H)/ U(1) be the projective unitary group with the quotient space topology, where U(1) consists of scalar multiples of the identity operator on H of norm equal to 1. Recall that, if G is a topological group, principal G bundles over X are classified up to isomorphism by the first cohomology of X with coefficients in the sheaf of germs of continuous functions from X to G, H^{1}(X, \underline{G}), where the transition maps of any trivialization of a principal G bundle over X define a cocycle in Z^{1}(X, \underline{G}) with fixed cohomology class. The exact sequence of sheaves of groups,

1 ⟶ \underline{U(1)} ⟶ \underline{U(H)} ⟶ \underline{PU(H)} ⟶ 1

gives rise to the long exact sequence of cohomology groups,

(2)  … ⟶ H^{1}(X, \underline{U(H)}) ⟶ H^{1}(X, \underline{PU(H)}) ⟶^{δ_{1}} H^{2}(X, \underline{U(1)}) ⟶ ….

Since U(H) is contractible in the strong operator topology, the sheaf \underline{U(H)} is soft and the higher sheaf cohomology vanishes, H^{j}(X, \underline{U(H)}) = \{0\}, j > 1. Equivalently, every Hilbert bundle over X is trivializable in the strong operator topology. In fact, Kuiper [6] proves the stronger result that U(H) is contractible in the operator norm topology, so the same conclusion holds in this sense too. It follows from (2) that δ_{1} is an isomorphism. That is, principal PU(H) bundles over X are classified up to isomorphism by H^{2}(X, \underline{U(1)}). From the exact sequence of groups,

1 ⟶ ℤ ⟶ ℝ ⟶ U(1) ⟶ 1

we obtain the long exact sequence of cohomology groups,

⋯ ⟶ H^{2}(X, \underline{ℝ}) ⟶ H^{2}(X, \underline{U(1)}) ⟶^{δ_{2}} H^{3}(X, ℤ) ⟶ H^{3}(X, \underline{ℝ}) ⟶ · · · .

Now H^{j}(X, \underline{ℝ}) = \{0\} for j > 0 since \underline{ℝ} is a fine sheaf, therefore δ is also an isomorphism. That is, principal PU(H) bundles over X are also classified up to isomorphism by H^{3}(X, ℤ). The class δ_{2}δ_{1}([P]) ∈ H^{3}(X, ℤ) is called the Dixmier-Douady class of the principal PU(H) bundle P over X, where [P] ∈ H^{1}(X, \underline{PU(H)}).

For g ∈ U(H), let Ad(g) denote the automorphism T ⟶ gT g^{−1} of K. As is well-known, Ad is a continuous homomorphism of U(H), given the strong operator topology, onto Aut(K) with kernel the circle of scalar multiples of the identity where Aut(K) is given the point-norm topology, that is the topology of pointwise convergence of functions on K, cf. [10], Chapter 1. Under this homomorphism we may identify PU(H) with Aut(K). Thus

Proposition 1. [Dixmier-Douady [4]] The isomorphism classes of locally trivial bundles over X with fibre K and structure group Aut(K) are also parametrized by H^{3}(X, ℤ).

Since K ⊗ K ≅ K, the isomorphism classes of locally trivial bundles over X with fibre K and structure group Aut(K) form a group under the tensor product, where the inverse of such a bundle is the conjugate bundle. This group is known as the infinite Brauer group and is denoted by Br^{∞}(X) (cf. [9]). So, essentially as a restatement of Proposition 1

(3)  Br^{∞}(X) ≅ H^{3}(X, ℤ)

where the cohomology class in H^{3}(X, ℤ) associated to a locally trivial bundle E over X with fibre K and structure group Aut(K) is again called the Dixmier-Douady invariant of E and is denoted by δ(E).

In this paper, we will be concerned mainly with torsion classes in H^{3}(X, ℤ). Let tor(H^{3}(X, ℤ)) denote the subgroup of torsion elements in H^{3}(X, ℤ). Suppose now that X is compact. Then there is a well-known description of tor(H^{3}(X, ℤ)) in terms of locally trivial bundles of finite-dimensional Azumaya algebras over X, [5]. Recall that an Azumaya algebra of rank m is an algebra which is isomorphic to the algebra of m × m matrices, M_{m}(C).

Definition. An Azumaya bundle over a manifold X is a vector bundle with fibres which are Azumaya algebras and which has local trivialization reducing these algebras to M_{m}(ℂ).

An example of an Azumaya bundle over X is the algebra, End(E), of all endomorphisms of a vector bundle E over X. Two Azumaya bundles E and F over X are said to be equivalent if there are vector bundles E and F over X such that E ⊗ End(E) is isomorphic to F ⊗ End(F). In particular, an Azumaya bundle of the form End(E) is equivalent to C(X) for any vector bundle E over X. The group of all equivalence classes of Azumaya bundles over X is called the Brauer group of X and is denoted by Br(X). We will denote by δ′(E) the class in tor(H^{3}(X, ℤ)) corresponding to the Azumaya bundle E over X. It is constructed using the same local description as in the preceeding paragraph. Serre’s theorem asserts that

(4)  Br(X) ≅ tor(H^{3}(X, ℤ)).

This is also the case if we consider smooth Azumaya bundles.

Another important class of Azumaya algebras arises from the bundles of Clifford algebras of vector bundles. For a real vector bundle, with fibre metric, the associated bundle of complexified Clifford algebras is an Azumaya bundle for even rank and for odd rank is a direct sum of two Azumaya bundles. In the even-rank case, the Dixmier-Douady invariant of this Azumaya bundle is the third integral Stiefel-Whitney class, the vanishing of which is equivalent to the existence of a spinℂ structure on the bundle.

Thus we see that there are two descriptions of tor(H^{3}(X, ℤ)), one in terms of Azumaya bundles over X, and the other as a special case of locally trivial bundles over X with fibre K and structure group Aut(K). These two descriptions are related as follows. Given an Azumaya bundle E over X, the tensor product E ⊗K is a locally trivial bundle over X with fibre M_{m}(C) ⊗ K ≅ K and structure group Aut(K), such that δ′(E) = δ(E ⊗ K). Notice that the algebras C(X, E) and C(X, E ⊗ K) = C(X, E)⊗K are Morita equivalent. Moreover if E and F are equivalent as Azumaya bundles over X then E ⊗ K and F ⊗ K are isomorphic over X, as locally trivial bundles with fibre K and structure group Aut(K). To see this, recall that by the assumed equivalence there are vector bundles E and F over X such that E ⊗End(E) is isomorphic to F ⊗ End(F). Tensoring both bundles with K(H), we see that E ⊗ K(E ⊗ H) is isomorphic to F ⊗ K(F ⊗ H), where K(E ⊗ H) and K(F ⊗ H) are the bundles of compact operators on the infinite dimensional Hilbert bundles E ⊗H and F ⊗ H, respectively. By the contractibility of the unitary group of an infinite dimensional Hilbert space in the strong operator topology, the infinite dimensional Hilbert bundles E ⊗ H and F ⊗ H are trivial, and therefore both K(E ⊗ H) and K(F ⊗ H) are isomorphic to the trivial bundle X × K. It follows that E and F are equivalent Azumaya bundles over X, if and only if E ⊗K and F ⊗K are isomorphic, as asserted.

Recall that a C^{∗}-algebra A is said to be stably unital if there is a sequence of projections p_{n} ∈ A ⊗ K such that T p_{n} ⟶ T for each T ∈ A ⊗ K. In particular K itself is stably unital since each compact operator can be approximated by finite rank operators. It follows that any unital algebra is stably unital.

Lemma 1. If X is a compact manifold and E is a locally trivial bundle over X with fibre K and structure group Aut(K) then C(X, E), the C^{∗}-algebra of continuous sections of E, is stably unital if and only if its Dixmier-Douady invariant is a torsion element in H^{3}(X, ℤ).

Proof. The assumption that C(X, E) is stably unital where E has fibres isomorphic to K, implies in particular that there is a non-trivial projection p ∈ C(X, E). This must be of finite rank in each fibre. The C^{∗}-algebra pC(X, E)p is a corner in C(X, E) in the sense of Rieffel, so that these algebras are Morita equivalent. They are both continuous trace C^{∗}-algebras with the same spectrum, which is equal to X, and therefore the same Dixmier-Douady invariant by the classification of continuous trace C^{∗}-algebras [10]. Since pC(X, E)p is the C^{∗}-algebra of sections of an Azumaya bundle over X, the Dixmier-Douady invariant is a torsion element in H^{3}(X, ℤ).

The converse is really just Serre’s theorem; given Θ ∈ tor(H^{3}(X, ℤ)) there is a principal PU(m) bundle over X whose Dixmier-Douady invariant is Θ where the order of Θ necessarily divides m. The associated Azumaya bundle A also has the same Dixmier-Douady invariant. Then E = A ⊗ K is a locally trivial bundle over X with fibre K and structure group Aut(K) and with Dixmier-Douady invariant δ(E) = Θ. So there is a non-trivial projection p_{1} ∈ C(X, E) such that p_{1}C(X, E)p_{1} = C(X, A). In fact, one can define a nested sequence of Azumaya bundles, A_{j}, over X defined by A_{j} = A ⊗ M_{j}(ℂ) together with the corresponding projections p_{j} ∈ C(X, E) such that p_{j}C(X, E)p_{j} = C(X, A_{j}) for all j ∈ ℕ. Then {p_{j}}_{j∈ℕ} is an approximate identity of projections in C(X, E), that is, C(X, E) is stably unital. □

Remark. This argument also shows that given a torsion class Θ ∈ tor(H^{3}(X, ℤ)) there is a smooth Azumaya bundle with Dixmier-Doaudy invariant Θ.

1.2. Twisted K-theory. Let X be a manifold and let J be a locally trivial bundle of algebras over X with fibre K and structure group Aut(K). Two such bundles are isomorphic if and only if they have the same Dixmier-Douady invariant δ(J) ∈ H^{3}(X, ℤ). The twisted K-theory of X (with compact supports) has been defined by Rosenberg [12] as

(5)  K_{c}^{j}(X, J) = K_{j}(C_{0}(X, J)) j = 0, 1,

where and K_{•}(C_{0}(X, J)) denotes the topological K-theory of the C^{∗}-algebra of continuous sections of J that vanish outside a compact subset of X. In case X is compact we use the notation K^{j}(X, J). The space K^{j}(X, J) or K_{c}^{j}(X, J) is an abelian group. It is tempting to think of the twisted K-theory of X as determined by the class Θ = δ(J). However, this is not strictly speaking correct; whilst it is the case that any other choice J′ such that δ(J′) = Θ is isomorphic to J and therefore there is an isomorphism K^{j}(X, J) ≅ K^{j}(X; J′) this isomorphism itself is not unique, nor is its homotopy class. However both the abelian group structure and the module structure on K^{0}(X; J) over K^{0}(X, J) arising from tensor product are natural, i.e. are preserved by such isomorphisms. With this caveat one can use the notation K^{j}(X, Θ) to denote the twisted K-theory with Dixmier-Douady invariant Θ ∈ H^{3}(X, ℤ).

In the case of principal interest here, when Θ ∈ tor(H^{3}(X, ℤ)) we will take J = K_{A} = A ⊗ K where A is an Azumaya bundle and use the notation

(6)  K_{c}^{j}(X, A) = K_{j}(C_{0}(X, A ⊗ J)) j = 0, 1

and K^{j}(X, A) in the compact case. As noted above, if A and A′ are two Azumaya bundles with the same Dixmier-Douady invariant, then as bundles of algebras A ⊗ End(V) ≡ A′⊗End(W) for some vector bundles V and W. Whilst there is no natural isomorphism of End(V)⊗J and J these bundles of algebras are isomorphic and Π_{0} of the group of diffeomorphism induces on J is naturally isomorphic to H^{2}(X, ℤ). It follows that the isomorphism between K^{0}(X, A) and K^{0}(X, A′) is determined up to the action of the image of H^{2}(X, ℤ), as isomorphism classes, in K^{0}(X) acting on K^{0}(X, A) through the module structure.

There are alternate descriptions of K^{0}(X, A). A description in terms of the twisted index map is mentioned in [12] and we give a complete proof here. Let Y_{A} be the principal PU(H) = Aut(K) bundle over X associated to K_{A} and let Fred_{A} = (Y_{A} × Fred(H)) / PU(H) be the bundle of twisted Fredholm operators where Fred(H) denotes the space of Fredholm operators on H.

Proposition 2. For any compact manifold X, there is a twisted index map, defined explicitly in (10) below, giving an isomorphism

(7)  index : π_{0}(C(X, Fred_{A})) ⟶^{∼} K^{0}(X, A).

Proof. Consider the short exact sequence of C^{∗}-algebras,

0 ⟶ K ⟶ B ⟶ B/K ⟶ 0

where B/K is the Calkin algebra. It gives rise to the short exact sequence of C^{∗}- algebras of sections,

(8)  0 ⟶ C(X, K_{A}) ⟶ C(X, B_{A}) ⟶ C(X, (B/K)_{A}) ⟶ 0

where the bundles B_{A} and (B/K)_{A} are also associated to Y_{A}. Consider the six term exact sequence in K-theory

(9) \begin{matrix}K^{1}(X, A) & ⟶ & K_{1}(C(X, B_{A})) & ⟶ & K_{1}(C(X, (B/K)_{A})) \\ ↑ & & & & ↓_{index}′ \\ K_{0}(C(X, (B/K)_{A})) & ⟵ & K_{0}(C(X, B_{A})) & ⟵ & K^{0}(X, A) \end{matrix}

arising from (9). By definition,

K_{1}(C(X, (B/K)_{A})) = lim_{→} π_{0} (GL(n, C(X, (B/K)_{A})))
= lim_{→} π_{0}(C(X, GL(n, B/K)_{A}))

where GL(n, A) denotes the group of invertible n × n matrices with entries in the C^{∗}-algebra A. In the case of the Calkin algebra, B(H) ⊗ M_{n}(ℂ) ≅ B(H ⊗ ℂ^{n}) and K(H) ⊗ M_{n}(ℂ) ≅ K(H ⊗ ℂ^{n}) from which it follows that

B(H)/K(H) ⊗ M_{n}(ℂ) ≅ B(H ⊗ ℂ^{n})/K(H ⊗ ℂ^{n}).

Therefore GL(n, B(H)/K(H)) ≅ GL(1, B(H ⊗ ℂ^{n})/K(H ⊗ ℂ^{n})), so

K_{1}(C(X, (B/K)_{A})) ≅ π_{0}(C(X, GL(1, B/K)_{A})).

Let p, q ∈ M_{n}(B) be any two projections. Then p⊕1⊕0 and q ⊕1⊕0 are equivalent in M_{n+2}(B), therefore K_{0}(B) = {0}. By the Stone-von Neumann theorem, one knows that GL(n, B) is connected, and therefore K_{1}(B) = {0}. That is, B is Kcontractible. We want to argue from the version of the Atiyah-Hirzebruch spectral sequence in this context [14] that

K_{•}(C(X, B_{A})) = {0}.

Fix a triangulation of X and let X^{p} denote the p-skeleton of X. Then there is a spectral sequence {E_{r}} converging strongly to K_{•}(C(X, B_{A})), with E_{1} term given by

E_{1}^{p,q} = K_{p+q} (C(X, B_{A})|_{X^{p}\X^{p−1}}) .

Since X^{p} \ X^{p−1} is the disjoint union of interiors of p-simplices, and each p-simplex is homeomorphic to ℝ^{p}, we see that the restriction of the bundle B_{A} to X^{p} \ X^{p−1} is isomorphic to the trivial bundle (X^{p} \ X^{p−1}) × B. Therefore

E_{1}^{p,q} = K_{p+q} (C(X^{p} \ X^{p−1}) ⊗ B) .

By applying the Künneth theorem [13] and the fact that B is K-contractible, we deduce that the E_{1} term of the spectral sequence vanishes. This implies that the spectral sequence {E_{r}} collapses with 0 = E_{1} = K_{•}(C(X, B_{A})) as asserted.

Therefore from (8), we obtain the isomorphism

index′ : π_{0}(C(X, GL(1, B/K)_{A})) ⟶^{∼} K^{0}(X, A).

We now recall the explicit definition of index′. Let u ∈ C(X, GL(1, B/K)_{A}). Then there is lift a ∈ C(X, GL(1, B_{A})) of u and a lift b ∈ C(X, GL(1, B_{A})) of u^{−1}. Then the element

v = (\begin{matrix}2a − aba & ab − 1 \\ 1 − ba & b \end{matrix})

projects to (\begin{matrix}u & 0 \\ 0 & u^{−1} \end{matrix}) in C(X, GL(2, B/K)_{A}). Thus E_{0} = (\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix}) and E_{1} =

vE_{0}v^{−1} are idempotents in M_{2}(C(X, K_{A}^{+})) such that E_{1} − E_{0} ∈ M_{2}(C(X, K_{A})). Therefore [E_{1}] − [E_{0}] ∈ K^{0}(X, A), and the index map is

index′([u]) = [E_{1}] − [E_{0}] ∈ K^{0}(X, A).

A computation show that

E_{1} = (\begin{matrix}2ab − (ab)^{2} & a(2 − ba)(1 − ba) \\ (1 − ba)b & (1 − ba)^{2} \end{matrix}) .

The fibration

K ⟶ Fred ⟶ GL(1, B/K)

shows that Fred is homotopy equivalent to GL(1, B/K), so that we obtain the desired index isomorphism (7). Explicitly, we see that if F ∈ C(X, Fred_{A}), then

(10)  index(F) = [(\begin{matrix}2F Q − (F Q)^{2} & F (2 − QF)(1 − QF) \\ (1 − QF)Q & (1 − QF)^{2} \end{matrix})] − [(\begin{matrix}1 & 0 \\ 0 & 0 \end{matrix})]

where Q ∈ C(X, Fred_{A}) is a parametrix for F, that is, F Q−1, QF −1 ∈ C(X, K_{A}). □

Notice that when Θ = 0 ∈ H^{3}(X, ℤ), then K_{A} ≅ X × K, therefore C_{0}(X, K_{A}) ≅ C_{0}(X) ⊗ K and by stability of K-theory, the twisted K-theory of X coincides with the standard K-theory of vector bundles over X. We wish to extend this description to twisted K-theory, at least when Θ ∈ tor(H^{3}(X, ℤ)). We begin with the following,

Lemma 2. Let X be a closed manifold and A and Azumaya bundle over X then the twisted K-theory K^{0}(X, A) is isomorphic to the Grothendieck group of Murray-von Neumann equivalence classes of projections in C(X, K_{A}).

Proof. This is a consequence of Lemma 1 above, which asserts that is stably unital, and of Proposition 5.5.5. in [2]. □

Thus when Θ is a torsion class, the correspondng twisted K-theory can be described just as if the C^{∗}-algebra C(X, K_{A}) had a unit. In this case the index isomorphism can also be written in familiar form.

Proposition 3. Suppose X is a closed manifold and A is an Azumaya bundle over X, then given a section s ∈ C(X, Fred_{A}) there is a section t ∈ C(X, E_{A}) such that index(s + t) = p_{1} − p_{2}, where p_{1}, p_{2} are projections in C(X, E_{A}) representing the projection onto the kernel of (s+t) and the projection onto the kernel of the adjoint (s + t)^{∗} respectively. Moreover the map

π_{0}(C(X, Fred_{A})) ⟶ K^{0}(X, A)
[[s]] ⟶ [index(s + t)]

is well defined and is independent of the choice of t ∈ C(X, E_{A}).

1.3. Projective vector bundles. Consider the short exact sequence of groups

(11)  ℤ_{n} ⟶ SU(n) ⟶^{π} PU(n), n ∈ ℕ

and the associated (determinant) line bundle L over PU(n). The fiber at p ∈ PU(n) is

(12) L_{p} = {(a, z) ∈ SU(n) × ℂ; π(a) = p}/ ∼,

(a, z) ∼ (a′, z′) if a′ = ta, z′ = tz, t ∈ ℤ_{n}.

This is a primitive line bundle over PU(n) in the sense that there is a natural SU(n) action on the total space of L,

(13)  l_{a} : (g, z) ⟼ (ga, z)

which induces a natural isomorphism

(14)  L_{pq} ≡ L_{p} ⊗ L_{q} ∀ p, q ∈ PU(n),

since if a ∈ SU(n) and π(a) = p then l_{q} : L_{b} ⟶ L_{ab} and l_{a} : L_{Id} ≡ ℂ ⟶ L_{q} combine to give an isomorphism (14) which is independent of choices. From the definition there is an injection

(15)  i_{q} : π^{−1}(q) ↪ L_{q}, q ∈ PU(n)

mapping a to the equivalence class of (a, 1). Thus a ∈ SU(n) fixes a trivialization e_{a} : L_{π(a)} ⟶ ℂ, determined by (a, 1) ⟼ 1.

Let P = P (A) be the PU(n) bundle associated to an Azumaya bundle. Thus the fiber P_{x} at x ∈ X consists of the algebra isomomorphisms of A_{x} to GL(n, ℂ). By a projective vector bundle over X, associated to A, we shall mean a complex vector bundle E over P (A) with a smooth family of linear isomorphisms

(16)  γ_{p} : p^{∗}E ⟶ E ⊗ L_{p}^{−1}, p ∈ PU(n)

with the compatibility condition

(17)  γ_{pp′} = γ_{p} ∘ γ_{p′}

in the sense that on the right γ_{p} : p^{∗}E ⊗ L_{p′}^{−1} ⟶ E ⊗ L_{p}^{−1} ⊗ L_{p′}^{−1} ⟶ E ⊗ L_{pp′}^{−1} using (14). In fact γ lifts to an action of SU(n) :

(18)  γ̃_{a} : p^{∗}E ⟶ E, γ̃_{a} = (Id ⊗e_{a})γ_{π(a)}.

Thus projective vector bundles are just a special case of SU(n)-equivariant vector bundles over P. This has also been studied in the case when P is a bundle gerbe [8] in which case E is known as a bundle gerbe module [3].

A bundle homomorphism between two projective bundles E and F is itself projective if it intertwines the corresponding isomorphisms (16). Since there is a natural isomorphism hom(E_{m}, F_{m}) ≡ hom(E_{m} ⊗ L, F_{m} ⊗ L) for any complex line L, the identifications γ_{p} act by conjugation on hom(E, F) and give it the structure of a PU(n)-invariant bundle. Thus the invariant sections, the projective homomorphisms, are the sections of a bundle hom_{proj}(E, F) over X.

Just as a family of finite rank projections, forming a section of C(X, K), fixes a vector bundle over X, so a section of a model twisted bundle A ⊗ K, with values in the projections, fixes a projective vector bundle over X. Thus, if m ∈ P_{x} then by definition m : A_{x} ⟶ GL(n) is an algebra isomorphism. The projection μ_{x} ∈ A_{x} ⊗ K thus becomes a finite rank projection in GL(n) ⊗ K. Using a fixed identification

(19)  R : ℂ^{n} ⊗ H ⟶ H

and the induced identification Ad(R) of GL(n) ⊗ K and K(H) this projection may be identified with its range E_{m} ⊂ H. The continuity of this operation shows that the E_{m} form a vector bundle bundle E over P. To see that E is a projective vector bundle observe that under the action of p ∈ PU(n) on P_{x}, replacing m by mp, E_{m} is transformed to E_{m′} = R(a ⊗ Id)E_{m} where a ∈ SU(n) is a lift of p, π(a) = p. Using the choice of a to trivialize L_{p}, the resulting linear map

(20)  γ_{p} : E_{m} ⟶ E_{m′} ⊗ L_{p}^{−1}

is independent of choices and satisfies (17).

The direct sum of two projective vector bundles over P is again a projective bundle so there is an associated Grothendieck group of projective K-theory over P, K_{proj}^{0}(P). The discussion above of the equivalence of projections in C(X; A ⊗ K) and projective vector bundles then shows the natural equality of the corresponding Grothendieck groups, just as in the untwisted case.

Lemma 3. If P is the principal PU(n) bundle over X associated to an Azumaya bundle then K_{proj}^{0}(P) is canonically isomorphic to K^{0}(X; A).

Similar conclusions hold for K-theory with compact supports if the base is not compact, we denote the twisted K-groups with compact support K_{c}(X, A).

A projective vector bundle may be specified by local trivializations relative to a trivialization of A and we proceed to discuss the smoothness and equivalence of such trivializations.

Consider a ‘full’ local trivialization of the Azumaya bundle A over a good open cover {U_{a}}_{a∈A} of the base X. Thus, there are algebra isomorphisms

F_{a} : A|_{U_{a}} ⟶ U_{a} × M(n, ℂ)

with lifted transition maps, chosen to be to continuous (or smooth if the G_{ab} are smooth)

(21)  G_{ab} : U_{ab} = U_{a} ∩ U_{b} ⟶ SU(n) such that G_{ab} ≡ F_{a} ∘ F_{b}^{−1} in PU(n).

That is, the transition maps for A over U_{ab} are given by the adjoint action of the G_{ab}. Thus the Dixmier-Douady cocycle associated to the trivialization is

(22)  θ_{abc} = G_{ab}G_{bc}G_{ca} : U_{abc} ⟶ ℤ_{n} ⊂ 𝕊 ⊂ ℂ^{∗}.

Such a choice of full local trivialization necessarily gives a local trivialization of the associated PU(n) bundle, P, and also gives a local trivialization of the determinant bundle L over P.

Projective vector bundle data, associated to such a full trivialization of A, consists of complex vector bundles E_{a}, of some fixed rank k, and transition maps Q_{ab} : E_{b} ⟶ E_{a} over each U_{ab} satisfying the weak cocycle condition

(23)  Q_{ab}Q_{bc} = θ_{abc}Q_{ac}

where θ is given by (22). Two sets of such data E_{a}, Q_{ab} and E′_{a}, Q′_{ab} over the same cover are equivalent if there are bundle isomorphisms T_{a} : E_{a} ⟶ E′_{a} such that Q′_{ab} = T_{a}Q_{ab}T_{b}^{−1} over each U_{ab}.

Associated with the trivialization of A there is a particular set of projective vector bundle data given by the trivial bundles ℂ^{n} over the U_{a} and the transition maps G_{ab}. We will denote this data as E_{τ} where τ denotes the trivialization of A. The Azumaya algebra A may then be identified with hom_{proj}(E_{τ} , E_{τ}) for any of the projective vector bundles E_{τ} .

Lemma 4. Projective vector bundle data with respect to a full trivialization of an Azumaya bundle lifts to define a projective vector bundle over the associated principal PU(n) bundle, all projective bundles arise this way and projective isomorphisms of projective bundles corresponds to equivalence of the projective vector bundle data.

Proof. The given trivialization of A, over each U_{a} defines a section of s_{a} : U_{a} ⟶ P = P (A) over U_{a}. Using this section we may lift the bundle E_{a} to the image of the section and then extend it to a bundle E^{(a)} on the whole of P |_{U_{a}} which is projective, namely by setting

(24)  E(a)_{s_{a}(x)p} = E_{a}(x) ⊗ L_{p} ∀ p ∈ PU(n), x ∈ U_{a}

and taking the ismorphism γ_{p} over U_{a} to be given by the identity on E_{a}(x). Over each intersection we then have an isomorphism Q_{ab}(x) : E_{a}(x) = E^{(a)}(s_{a}(x)) ⟶ E^{(b)}(s_{b}(x)) = E_{b}(x). Now, from the trivialization of P we have s_{b}(x) = s_{a}(x)g_{ab}(x) where G_{ab} : U_{ab} ⟶ SU(n) and g_{ab} is the projection of G_{ab} into PU(n). The choice of G_{ab} therefore also fixes a trivialization of the determinant bundle L_{g_{ab}(x)} ⟶ ℂ. Since E^{(b)}(s_{b}(x)) is identified with E^{(b)}(s_{a}(x)) ⊗ L_{g_{ab}(x)}^{−1} by the primitivity, this allows Q_{ab} to be interpreted as the transition map from E^{(a)} to E^{(b)} over the preimage of U_{ab}. Furthermore the weak cocycle condition (23) now becomes the cocycle condition guaranteeing that the E^{(a)} combine to a globally defined, projective, bundle over P.

This argument can be reversed to construct projective vector bundle data from a projective vector bundle over P and a similar argument shows that projective bundle isomorphisms correspond to isomorphisms of the projective vector bundle data. □

Thus we may simply describe ‘projective vector bundle data’ as a local trivialization of the corresponding projective bundle, where this also involves the choice of a full local trivialization of A. The projective vector bundle data E_{τ} associated to a full trivialization of A thus determines a projective vector bundle, which we may also denote by E_{τ} , over P (A). In particular there are projective vector bundles of arbitrarily large rank over P.

Many of the standard results relating K-theory to vector bundles carry over to the twisted case. We recall two of these which are important for the proof of the index theorem.

Lemma 5. If U ⊂ X is an open set of a compact manifold then for any Azumaya bundle A, over X, there is an extension map

(25)  K_{c}^{0}(U, A_{U}) ⟶ K^{0}(X, A).

Proof. This proceeds in essentially the usual way. An element of K_{c}^{0}(U, A_{U}) is represented in terms of a full local trivialization of A by a pair of sets of projective vector data E_{1}, E_{2} over U and a given bundle isomorphism between them outside a compact set, c : E_{1} ⟶ E_{2} over U \ K. If E_{τ} is the projective vector bundle data associated to the trivialization of A then we may embed E_{2} as a projective subbundle of E_{τ}^{q} for some integer q. To see this, first choose an embedding e of E_{2} as a subbundle of E_{τ}^{l} over P for some large l. Then choose a full trivialization of A and a partition of unity ψ_{a} subordinate on X to the open U_{a}. Over the premimage of each U_{a} e_{a} can be extended uniquely to a projective embedding e_{a} of E in E_{τ}^{l} . The global map formed by the sum over a of the ψ_{a}e_{a} gives a projective embedding into E_{τ}^{q} where q = lN and N is the number of sets in the open cover. Then E_{τ}^{q} = E_{2} ⊕F for some complementary projective bundle F over U. The data E_{1} ⊕ F over U and E_{τ}^{q} over Z \ K with the isomorphism c ⊕ Id_{F} over U \ K determine a projective vector bundle over Z. The element of K^{0}(X; A) represented by the pair consisting of this bundle and E_{τ}^{q} is independent of choices, so defines the extension map. □

Proposition 4. For any real vector bundle π : V ⟶ X and Azumaya bundle A over X a section over the sphere bundle, SV, of V of the isomorphism algebra of the lifts of two projective vector bundles over X, determines an element of K_{c}^{0}(V, π^{∗}A); all elements arise this way and two isomorphisms give the same element if they are homotopic after stabilization with the identity isomorphisms of projective bundles.

Proof. The proof is the same as in the untwisted case. □

If W is a vector bundle over X and E is a projective bundle over a PU(n) bundle over X then E ⊗ π^{∗}W is naturally a projective bundle. This operation extends to make K_{proj}^{0}(P) a module over (untwisted) K^{0}(X) and hence, in view of Lemma 3 gives a module structure

(26)  K^{0}(X) × K^{0}(X; A) ⟶ K^{0}(X; A).

1.4. The Chern character of projective vector bundles. The Chern character

(27)  Ch_{A} : K^{0}(X; A) ⟶ H^{ev}(X)

may be defined in one of several equivalent ways.

To define (27) directly using the Chern-Weil approach we note that the local constancy of the Cěch 2-cocycle θ in (23) allows a connection to be defined directly on such projective vector bundle data. That is, despite the failure of the usual cocycle condition, there exist connections ∇^{a} on each of the bundles E_{a} which are identified by the Q_{ab}. To see this, simply take arbitary connections ˜∇^{a} on each of the E_{a} and a partition of unity ϕ_{a} subordinate to the cover. Now define a new connection on E_{a} by

∇^{a} = ϕ_{a}∇^{a} + ∑_{b≠a} ϕ_{b}Q_{ab}^{∗}∇^{b}.

These are consistent under the transition maps. Thus the curvature of this collective connection is a well-defined section of the endomorphism bundle of the given projective vector bundle data. As such the usual Chern-Weil arguments apply and give the Chern character (27). Lifted to P this connection gives a projective connection on the lift of E to a projective bundle; this also allows the Chern character to be defined directly on K_{proj}^{0}(P).

Either of these approaches to the Chern character show that it distributes over the usual Chern character on K^{0}(X) under the action (26). In particular it follows that (27) is an isomorphism over ℚ.

We may also define the Chern character by reducing to the standard case by taking tensor powers. Thus if E is a projective vector bundle associated to the Azumaya bundle A and n is the order of the associated Dixmier-Douady invariant then

ϕ^{∗}E^{⊗n} ≅ π_{P}^{∗} E^{⊗n}

and therefore E^{⊗n} = π^{∗}(F) for some vector bundle F → X.

Observe that the Chern character of E satisfies,

(28)  Ch(E^{⊗n}) = Ch(E)^{n} = π^{∗}(Ch(F)).

We claim that Ch(E) = π^{∗}(Λ) for some cohomology class Λ ∈ H^{even}(X, ℚ). First observe that (28) implies that the degree zero term, which is a constant term, is of the desired form. Next assume that the degree 2k component of the Chern character satisfies Ch_{k}(E) = π^{∗}(Λ_{k}) for some cohomology class Λ_{k} ∈ H^{2k}(X, ℚ). Then (28) implies that the degree (2k+2) component Ch_{k+1}(E^{⊗n}) = π^{∗}(Ch_{k+1}(F)). But the left hand side is of the form of the Chern character,

(29)  Ch_{k+1}(E^{⊗n}) = a_{0} Ch_{k+1}(E) + ∑_{|I|=k+1,r>1} a_{I} Ch_{I}(E)

where I = (i_{1}, …, i_{r}), |I| = i_{1} + ⋯ + i_{r}, Ch_{I}(E) = Ch_{i_{1}}(E) ∪ ⋯ ∪ Ch_{i_{r}}(E) and a_{0}, a_{I} ∈ ℚ are such that a_{0} ≠ 0. By the induction hypothesis, we deduce that Ch_{k+1}(E) is of the form π^{∗}(Λ_{k+1}) for some cohomology class Λ_{k+1} ∈ H^{2k+2}(X, ℚ). This proves the claim.

Then the Chern character of the projective vector bundle E above is given by

(30)  Ch_{A}(E) = Λ ∈ H^{even}(X, ℚ).

That is, the lift of the Chern character of the projective vector bundle E to P coincides with the ordinary Chern character of E. The following properties of the Chern character of projective vector bundles follow from the corresponding properties of the Chern character of vector bundles.

Lemma 6 (A). Let A be an Azumaya bundle over X and P be the principal PU(n) bundle associated to A. Let E → P be a projective vector bundle over X, associated to A, then the Chern character defined as in (30) above has the following properties.

(1) If E′ → X is another projective vector bundle, then

Ch_{A}(E ⊕ E′) = Ch_{A}(E) + Ch_{A}(E′),

so the Chern character is a homomorphism

Ch_{A} : K^{0}(X, A) → H^{even}(X, ℚ).

(2) The degree 2 component of the Chern character Ch_{A}(E) coicides with the first Chern class of the determinant line bundle det(E) → X.

2. The analytic index

For two ordinary vector bundles, E^{±}, over a compact manifold Z, the space Ψ^{m}(Z; E^{+}, E^{−}) of pseudodifferential operators of order m mapping C^{∞}(Z; E^{+}) to C^{∞}(Z; E^{−}) may be identified naturally with the tensor product

(31)  Ψ^{m}(Z; E^{+}, E^{−}) = Ψ^{m}(Z) ⊗_{C^{∞}(Z^{2})} C^{∞}(Z^{2}; Hom(E^{+}, E^{−}))

where Hom(E^{+}, E^{−}) is the ‘big’ homomorphism bundle over Z^{2} which has fiber hom(E_{z}^{+}, E_{z′}^{−}) at (z′, z) and Ψ^{m}(Z) is the space of pseudodifferential operators acting on functions. The latter is a module over C^{∞}(Z^{2}) through its realization as a space of Schwartz kernels. In particular

Ψ^{m}(Z; E, E) = Ψ^{m}(Z; E) = Ψ^{m}(Z) ⊗_{C^{∞}(Z^{2})} C^{∞}(Z^{2}; Hom(E))

when the two bundles coincide.

For a fibration ϕ : M ⟶ X, with compact boundaryless fibres, the bundle of pseudodifferential operators acting on sections of vector bundles over the total space may be similarly defined. Note that the operators act fibre-wise and so commute with multiplication by functions on the base. If Hom(M_{ϕ}^{2}, E^{+}, E^{−}) denotes the bundle over the fibre product, which is the ‘big’ homorphism bundle on each fibre, then again

Ψ^{m}(M/X; E^{+}, E^{−}) = Ψ^{m}(M/X) ⊗_{C^{∞}(M_{ϕ}^{2})} C^{∞}(M_{ϕ}^{2}; Hom(E^{+}, E^{−}))

is the bundle of operators to which the usual families index theorem applies. Here Ψ^{m}(M/X) is the bundle of pseudodifferential operators acting on functions on the fibres.

Now, let A be an Azumaya bundle over the base of the fibration ϕ. Consider a projective vector bundle, E, over the lift to M of the principal PU(N) bundle associated to A. Given a local trivialization of A there is a bundle trivialization of E with respect to the lift of the trivialization to M. We shall call this a basic bundle trivialization.

Lemma 7. If E^{+} and E^{−} are two projective vector bundles over the lift to M of the PU(N) bundle associated to an Azumaya bundle on X then the big homomorphism bundles Hom(Q_{a}^{+}, Q_{a}^{−}), arising from basic bundle trivializations of the E^{±} define a vector bundle Hom(E^{+}, E^{−}) over M_{ϕ}^{2}.

Proof. As already noted, a local trivialization of A over the base gives a trivialization of the associated PU(N) bundle, and hence of its lift to M. This leads to bundle trivializations Q_{a}^{±}, over the elements ϕ^{−1}(U_{a}) of this open cover, of the projective bundles E^{±}. Since the transition maps act by the adjoint action, the scalar factors cancle and the ‘big’ homomorphism bundles between the Q_{a}^{±} now patch to give a global bundle Hom(E^{+}, E^{−}) over M_{ϕ}^{2}. □

If E_{i}, i = 1, 2, 3, are three such projective bundles then, just as for the usual homomorphism bundles, there is a bilinear product map

(32)  Hom(E_{1}, E_{2}) ⊗ Hom(E_{2}, E_{3})|_{C} ⟶ ψ^{∗} Hom(E_{1}, E_{2})

where C is the central fiber diagonal in M_{ϕ}^{2} × M_{ϕ}^{2} and ψ : C ⟶ M_{ϕ}^{2} is projection off the middle factor. This reduces to the composition law for hom(E_{i}, E_{j}) = Hom(E_{i}, E_{j})|_{Diag} on the diagonal.

We may now simply define the algebra of twisted (fiber-wise) pseudodifferential operators as

(33)  Ψ^{m}(M/X; E^{+}, E^{−}) = Ψ^{m}(M/X) ⊗_{C^{∞}(M_{ϕ}^{2})} C^{∞}(M_{ϕ}^{2}; Hom(E^{+}, E^{−})).

Restricted to open sets in the base over which A is trivialized, this reduces to the standard definition. Thus, the symbol sequence remains exact

(34) 0 ⟶ Ψ^{m−1}(M/X; E^{+}, E^{−}) ↪

Ψ^{m}(M/X; E^{+}, E^{−}) ⟶^{σ_{m}} S^{[m]}(S^{∗}(M/X); ϕ^{∗} hom(E^{+}, E^{−})) ⟶ 0

with the proof essentially unchanged. Here S^{[m]}(S^{∗}(M/X); ϕ^{∗} hom(E^{+}, E^{−})) is the quotient space of symbolic sections of order m, by symbolic sections of order m − 1, of ϕ^{∗} hom(E^{+}, E^{−}) as a bundle over S^{∗}(M/X), the fibre cosphere bundle. Similarly the usual composition properties carry over to this twisted case, since they apply to the local families. For any three projective vector bundles E_{i}, i = 1, 2, 3, over the lift of the same PU(N) bundle from the base

(35) Ψ^{m}(M/X; E_{2}, E_{3}) ∘ Ψ^{m′}(M/X; E_{1}, E_{2}) ⊂ Ψ^{m+m′}(M/X; E_{1}, E_{3}),

σ_{m+m′}(A ∘ B) = σ_{m}(A) ∘ σ_{m′}(B).

.

For any basic bundle trivialization of a projective vector bundle with respect to a local trivialization of A the spaces of sections of the local bundles form infinite-dimensional projective bundle data over the base, associated to the same trivialization of A. More generally, for any fixed real number, m, the spaces of Sobolev sections of order m over the fibres form projective Hilbert bundle data over the base; we will denote the corresponding projective bundle H^{m}(M/X; E). The boundedness of pseudodifferential operators on Sobolev spaces then shows that any A ∈ Ψ^{m}(M/X; E^{+}, E^{−}) defines a bounded operator

(36) A : H^{m_{1}}(M/X, E^{+}) ⟶ H^{m_{2}}(M/X; E^{−}) provided m_{1} ≥ m_{2} + m.

If m_{1} > m_{2} + m this operator is compact.

It is possible to choose quantization maps as in the untwisted case. To do so, choose basic bundle trivializations and quantization maps, that is right inverses for the symbol map, for the local bundles Q_{a}^{±}. Using a partition of unity on the base this gives a global quantization map:

(37) q_{m} : S^{[m]}(S^{∗}(M/X); ϕ^{∗} hom(E^{+}, E^{−})) ⟶ Ψ^{m}(M/X; E^{+}, E^{−}),

σ_{m} ∘ q_{m} = Id, q_{m} ∘ σ_{m} − Id : Ψ^{m}(M/X; E^{+}, E^{−}) ⟶ Ψ^{m−1}(M/X; E^{+}, E^{−}).

By definition a projective family in Ψ^{m}(M/X; E^{+}, E^{−}) is elliptic if σ_{m} is invertible, with inverse in S^{[m]}(S^{∗}(M/X); ϕ^{∗} hom(E^{−}, E^{+})). Directly from the symbolic properties of the algebra, this is equivalent to there being a parameterix B ∈ Ψ^{−m}(M/X; E^{−}, E^{+}) such that A ∘ B − Id ∈ Ψ^{−1}(M/X; E^{−}) and B ∘ A − Id ∈ Ψ^{−1}(M/X; E^{+}). These ‘error terms’ give compact maps, for m_{1} = m_{2} + m, in (36). Thus the elliptic family consists of Fredholm operators. It follows from the discussion in Section 1 that the family defines a twisted K-class using (7). To see this class more concretely, as in the untwisted case, we may perturb the family so that the index bundle gives projective vector bundle data with respect to the given trivialization of A. Locally in the base a bundle map from an auxilliary vector bundle, over the base, may be added to make the family surjective. Choosing this bundle to be part of (some large power) of projective vector bundle data these local maps may be made into global smooth homomorphism into the image bundle

(38)  f : E_{τ}^{N} ⟶ C^{∞}(M/X; E^{−}) such that P + f is surjective.

This necessarily stabilizes the null bundle to projective vector bundle data with respect to the trivialization and we set

(39)  index_{a}(P) = [(P + f) − E_{τ}^{N}] ∈ K^{0}(X, A).

As in the untwisted case this class can be seen to be independent of the precise stabilization used and to be homotopy invariant. In fact adding a further stabilizing bundle is easily seen to leave the index unchanged and stabilizing the family with the additional parameter of a homotopy shows the homotopy invariance.

Proposition 5. For a fibration (1), Azumaya bundle A over X and projective bundles E^{±}, there is a quantization of a given ismorphism b of the lifts of these bundles to S^{∗}(X/M) for which the null spaces, and hence also the null spaces of the adjoint family, define a projective bundle over the base so that the difference class index_{a}(b) ∈ K^{0}(X, A) depends only on the class of b in K^{0}(T (X/M), ρ^{∗}ϕ^{∗}A) and so defines the analytic index homomorphism

(40)  index_{a} : K(T (M/X), ρ^{∗}ϕ^{∗}A) ⟶ K(X; A).

Proof. The stabilization discussed above associates to an elliptic family with principal symbol b an element of K(X, A). This class is independent of the stabilization used to define it and is similarly independent of the quantization chosen, since two such families differ by a family of compact operators. Clearly the element is unchanged if the symbol, or operator, is stabilized by the identity on some other primitve vector bundle defined over the lift of the same PU(N) bundle. Furthermore the homotopy invariance of the index and the existence of a quantization map show that the element depends only on the homotopy class of the symbol. The additivity of the index under composition and the multiplicativity of the symbol map then shows that the resulting map (40) is a homomorphism. □

3. The topological index

In this section we define the topological index map for a fibration of compact manifolds (1)

(41)  index_{t} : K_{c}(T (M/X), ρ^{∗}ϕ^{∗}A) ⟶ K^{0}(X, A)

where ρ : T (X/M) ⟶ M is the projection and A is an Azumaya bundle over X.

We first recall some functorial properties of twisted K-theory. Let f : Y ⟶ Z be a smooth map between compact manifolds. Then the pullback map,

f^{!} : K(Z, A) ⟶ K(Y, f^{∗}A),

for any Azumaya bundle A, is defined as follows. Let V be finite dimensional projective vector bundle data over Z, associated with a trivialization of A. Then f^{∗}V is projective vector bundle data over Y associated to the lifted trivialization and the resulting class in K-theory is independent of choices, so defines f^{!}. Alternatively, if s is a section of the twisted Fredholm bundle of A⊗K over Z, then the pullback f^{∗}s is a section of the corresponding twisted Fredholm bundle over Y. The pull-back map may also be defined directly in terms of the pull-back of projective bundles from the PU(N) bundle associated to A over Z to its pull-back over Y.

Lemma 8. For any Azumaya bundle there is a canonical isomorphism

j_{!} : K(X, A) ≅ K_{c}(X × ℝ^{2N}, p_{1}^{∗}A)

determined by Bott periodicity.

Proof. Recall that K_{c}(X ×ℝ^{2N}, p_{1}^{∗}A) ≅ K(C_{0}(X ×ℝ^{2N}, E_{p_{1}^{∗}A})). Now E_{p_{1}^{∗}A} ≅ p_{1}^{∗}E_{A}, so that C_{0}(X × ℝ^{2N}, E_{p_{1}^{∗}A}) ≅ C(X, E_{A})ˆ⊗C_{0}(ℝ^{2N}). Thus, K_{c}(X × ℝ^{2N}, p_{1}^{∗}A) ≅ K(C(X, E_{A})⊗C_{0}(ℝ^{2N})). Together with Bott periodicity, K(C(X, E_{A})⊗C_{0}(ℝ^{2N})) ≅ K(X, A), this proves the lemma. □

Let f : Y ⟶ Z be a smooth embedding between compact manifolds which is K-oriented, that is T Y ⊕ f^{∗}T Z is a spinℂ vector bundle over Y. We proceed to define the induced Gysin map on twisted K-theory,

(42)  f_{!} : K(Y, f^{∗}A) ⟶ K(Z, A)

for any Azumaya bundle A. As in the untwisted case it is defined by first considering the special case of a spinℂ vector bundle and then applying this to the normal bundle of the embedding.

So consider the embedding of the zero section i : Y ⟶ E, where π : E ⟶ Y is a spinℂ vector bundle over Y. Then the associated Gysin map in K-theory is

\begin{matrix}i_{!} : K(Y, i^{∗}A) & ⟶ K_{c}(E, A) \\ ξ & ⟼ π^{∗}ξ ⊗ (π^{∗}S^{+}, π^{∗}S^{−}, c(v)) \end{matrix}

where ξ is projective vector bundle data over Y associated to a local trivialization of A, (π^{∗}S^{+}, π^{∗}S^{−}, c(v)) is the usual Thom class of the spinℂ vector bundle E, and the right hand side becomes a pair of projective vector bundle data with isomorphism outside a compact set

(43)  Id ⊗c(v)) : π^{∗}(ξ ⊗ S^{+}) ⟶ π^{∗}(ξ ⊗ S^{−}), v ∈ E.

The Thom isomorphism in this context, cf. [5], asserts that i_{!} is an isomorphism.

Now let f : Y ⟶ Z be a smooth embedding between compact manifolds which is K-oriented and let N ⟶ Y denote the normal bundle to the embedding. Then N is diffeomorphic to a tubular neighborhood U of the image of Y; let Φ : U ⟶ N denote this diffeomorphism. By the Thom isomorphism above

i_{!} : K(Y, i^{∗}A) ⟶ K_{c}(N, A) ≅ K_{c}(U, A′),

where A′ = Φ^{∗}A. If Ã is an Azumaya bundle over Z with Ã_{U} = A′ then, as in Lemma 5, the inclusion of the open set U in Z induces a map K_{c}(U, A′) ⟶ K(Z, Ã). The composition of these maps defines the Gysin map, (42), in twisted K-theory.

Let ϕ : M ⟶ X be a fibre bundle of compact manifolds. We know that there is an embedding f : M ⟶ X × ℝ^{N}, cf. [1] §3. Then the fibrewise differential is an embedding Df : T (M/X) ⟶ X × ℝ^{2N}. It induces a Gysin map in twisted K-theory,

Df_{!} : K_{c}(T (M/X), ρ^{∗}π^{∗}A) ⟶ K(X × ℝ^{2N}, p_{1}^{∗}A)

where p_{1} : X×ℝ^{2N} ⟶ X is the projection onto the first factor and ρ : T (M/X) ⟶ X is the projection map. Since π = p_{1} ∘ f it follows that Df^{∗}p_{1}^{∗}A = ρ^{∗}π^{∗}A. Now define the topological index as the map

index_{t} = j_{!}^{−1} ∘ Df_{!} : K_{c}(T (M/X), ρ^{∗}π^{∗}A) ⟶ K(X, A),

where we apply Lemma 8 to see that the inverse j_{!}^{−1} exists.

4. Proof of the index theorem in twisted K-theory

We follow the axiomatic approach of Atiyah-Singer to prove that the analytic index and the topological index coincide.

Definition. An index map is a homomorphism

(44)  index : K_{c}(T (M/X), ρ^{∗}π^{∗}A) ⟶ K(X, A),

satisfying the following:

(1) (Functorial axiom) If M and M′ are two fibre bundles with compact fibres over X and f : M ⟶ M′ is a diffeomorphism which commutes with the projection maps ϕ : M ⟶ X and ϕ′ : M′ ⟶ X then the diagram

(45)  \xymatrix{K_{c}(T (M/X), ρ^{∗}ϕ^{∗}A) \ar[dr]_{index} \ar[rr]^{f^{!}} &  & K_{c}(T (M′/X), ρ^{∗}(ϕ′)^{∗}A) \ar[dl]^{index} \\  & K(X, A)}

is commutative.

(2) (Excision axiom) Let ϕ : M ⟶ X and ϕ′ : M′ ⟶ X be two fibre bundles of compact manifolds, and let α : U ⊂ M and α′ : U′ ⊂ M′ be two open sets with a diffeomorphism g : U ≅ U′ satisfying ϕ′ ∘ g = ϕ used to identify them, then the diagram

(46)  \xymatrix{ & K_{c}(T (M/X), ρ^{∗}ϕ^{∗}A) \ar[dr]^{index} \\ K_{c}(T (U/ϕ(U)), ρ^{∗}α^{∗}ϕ^{∗}A) \ar[ur]^{α_{!}} \ar[dr]_{(α′)_{!}} &  & K(X, A) \\  & K_{c}(T (M′/X), ρ^{∗}(ϕ′)^{∗}A) \ar[ur]_{_{index_{′}}}}

is commutative.

(3) (Multiplicativity axiom) Let V be a real vector space and suppose that i : M ⟶ X ×V is an embedding which intertwines the projection maps ϕ : M ⟶ X and π_{1} : X ×V ⟶ X; the fibrewise differential i_{∗} : T (M/X) ⟶ X × T V also intertwines the projections. The one-point compactification S(V ⊕ℝ) of V is a sphere with the canonical inclusion e : T V ⟶ T S(V ⊕ℝ) inducing the inclusion e′ = Id ×e : X × T V ⟶ X × T S(V ⊕ ℝ). Then the diagram

(47)  \xymatrix{ & K_{c}(X × T V, π_{1}^{∗}A) \ar[d]^{(e′)_{!}} \\ K_{c}(T (M/X), ρ^{∗}ϕ^{∗}A) \ar[dr]_{index} \ar[ur]^{(i_{∗})_{!}} & K_{c}(X × T S(V ⊕ ℝ), π_{1}^{∗}A) \ar[d]^{index} \\  & K(X, A)}

commutes.

(4) (Normalization axiom) If the fibre bundle of compact manifolds ϕ : M ⟶ X has single point fibres, then the index map

(48)  index : K_{c}(T (M/X), ρ^{∗}π^{∗}A) = K(X, A) ⟶ K(X, A)

is the identity map.

The next theorem asserts in particular that such an index map does exist.

Theorem 1. The topological index index_{t} is an index map.

Proof. We proceed to check the axioms above in turn.

If f : M ⟶ M′ is a diffeomorphism as in the statement of the functoriality axiom, let i : M′ ⟶ X × V be an embedding commuting with the projections, where V is a finite dimensional vector space. Then i ∘ f : M ⟶ X × V is also such an embedding. Using these maps, we may identify the topological index maps as index′_{t} = j_{!}^{−1} ∘ (i_{∗})_{!} and index_{t} = j_{!}^{−1} ∘ ((i ∘ f)_{∗})_{!} = j_{!}^{−1} ∘ (i_{∗})_{!} ∘ (f_{∗})_{!} = index′_{t} ∘(f_{∗})_{!}, where j : X ↪ X × V is the zero section embedding. Then the diagram

\xymatrix{K_{c}(T (M/X), ρ^{∗}ϕ^{∗}A) \ar[dr]_{_{index_{t}}} \ar@{-->}[rr]^{((f_{∗})_{!})^{−1}} &  & K_{c}(T (M′/X), ρ^{∗}ϕ′^{∗}A) \\  & K(X, A) \ar[ur]_{_{index_{t}}}}

commutes. Since f is a diffeomorphism, (f_{∗})_{!})^{−1} = f^{!}, which establishes that index_{t} is functorial.

Next consider the excision axiom and let i′ : M′ ⟶ X×V be an embedding, and j : X ↪ X × V be the zero section embedding. Then index′_{t} = j_{!}^{−1} ∘ (i_{∗})_{!}, so that the relevant map in the lower part of the diagram (46) is j_{!}^{−1} ∘ (i_{∗})_{!} ∘ (α′_{∗})_{!} = j_{!}^{−1} ∘ ((i ∘ α′)_{∗})_{!}. But this is merely the topological index K_{c}(T (U/π(U)), ρ^{∗}α^{∗}π^{∗}A) ⟶ K(X, A). That it agrees with the relevant map in the upper part of the diagram, follows from the fact that the topological index is well defined and independent of the choice of embedding. Thus the topological index satisfies the excision property.

The multiplicativity property for the topological index follows from its independent of the choice of embedding, see §4.

For the normalization axiom, note that in case M = X, if i : X ⟶ X × V is an embedding which commutes with the projection maps ϕ : X ⟶ X and π_{1} : X ×V ⟶ X then i is necessarily the trivial embedding Id ×g with g : X ⟶ V constant. Then index_{t} = j_{!}^{−1} ∘ (i_{∗})_{!} = Id since i_{∗} = Id ×0, which shows that the topological index is normalized. □

Theorem 2. The topological index is the unique index map.

Proof. Consider a general index map as in (44); we proceed to show that index = index_{t} .

Suppose that i : M ⟶ X × V is an embedding which intertwines the projection maps ϕ : M ⟶ X and π_{1} : X × V ⟶ X. Then, together with the notation of the Multiplicativity Axiom, set i^{+} = e′ ∘ i_{∗} : T (M/X) ⟶ X × T S(V ⊕ ℝ). Let 0 ∈ T V be the origin and j : {0} ⟶ T V be the inclusion, inducing the inclusion j′ = Id ×j : X × {0} ⟶ X × T V and denote the composite inclusion j^{+} = e′ ∘ j′ : X × {0} ⟶ X × T S(V ⊕ ℝ). Then consider the diagram,

\xymatrix{ & K_{c}(T (M/X), ρ^{∗}ϕ^{∗}A) \ar[dl]_{(i_{∗})_{!}} \ar[d]^{i^{+}} \ar[dr]^{index} \\ K_{c}(X × T V, π_{1}^{∗}A) \ar[r]^{e_{!}} & K_{c}(X × T S(V ⊕ ℝ), π_{1}^{∗}A) \ar[r] \ar[r]^{_{index_{S}}} & K(X, A) \\  & K(X, A) \ar[ur]_{index} \ar[ul]^{j_{!}} \ar[u]_{j^{+}}}

The left side of this diagram commutes by the excision property and the right side by the multiplicative property. By the normalization property, the composite map index ∘j^{+} is the identity mapping, so

index = index^{S} ∘i_{!}^{+} = index^{S} ∘e_{!} ∘ i_{!} = index^{S} ∘j_{!}^{+} ∘ j_{!}^{−1} ∘ i_{!}
= index′ ∘j_{!}^{−1} ∘ i_{!} = j_{!}^{−1} ∘ i_{!} = index_{t} .
□

The following theorem completes the proof of the index theorem in twisted Ktheory.

Theorem 3. The analytic index index_{a} is an index map.

Proof. Again we consider the axioms for an index map in order.

The invariance of the algebra of pseudodifferential operators under diffeomorphism, and the naturality in this sense of the symbol map, show that under the hypotheses of the functoriality axiom, there is an isomorphism of short exact sequences (34):

(49)

Ψ^{−1}(M′/X, E) ⟶ Ψ^{0}(M′/X, E) ⟶ S^{[0]}(T (M′/X), ρ^{∗} End(E))
↓^{f^{∗}}  ↓^{f^{∗}}  ↓^{f^{∗}}
Ψ^{−1}(M/X, E) ⟶ Ψ^{0}(M/X, E) ⟶ S^{[0]}(T (M/X), ρ^{∗} End(E)).

Since the analytic index is by definition the boundary map in the associated 6term exact sequence in K-theory, we see that index_{a}(f^{∗}[p]) = f^{!} index_{a}([p]), for all [p] ∈ K_{c}(T (M/X), ρ^{∗}π^{∗}A). This establishes the functoriality of index_{a} .

For the excision axiom, observe that any element in K_{c}(T (U/π(U)), ρ^{∗}i^{∗}π^{∗}A) may be represented by a pair of projective vector bundle data over U and a symbol q ∈ S^{[0]}(T (U/π(U) with the property that q is equal to the identity homorphism outside a compact set in U. Complementing the second bundle with respect to vector bundle over M, using the discussion following Lemma 3, we may extend both sets of projective vector data to the whole of M, to be equal outside U. This also extends q to an element p ∈ S^{[0]}(T (M/X) by trivial extension. The exactness in (34) shows that there is a projective family of elliptic pseudodifferential operators P of order zero with symbol equal to p, by use of a partition of unity we may take it to be equal to the identity outside U in M. Similarly, q also defines an element p′ ∈ S^{[0]}(T (M′/X) and, from the corresponding exact sequence, there is a projective family of elliptic pseudodifferential operators P′ equal to the identity outside U in M′; we may further arrange that P = P′ in U. We can construct parametrices Q of P and Q′ of P′ such that Q is equal to the identity outside U in M and Q′ is equal to the identity outside U in M′ and Q = Q′ in U. By the explicit formula for the analytic index in terms of the projective family of elliptic pseudodifferential operators and its parametrix, see §3, it follows that the diagram (46) commutes, that is, the analytic index satisfies the excision property.

Under the hypotheses of the multiplicative axiom we need to show for a class [p] ∈ K_{c}(T (M/X), ρ^{∗}ϕ^{∗}A), represented by a symbol p ∈ S^{0}(T (M/X), E), that index_{a}([p]) = index_{a}(h_{!}[p]), where h : T (M/X) ⟶ X×T S(V ⊕ℝ) is the embedding that is obtained as the composition h = e ∘ Di, and h_{!} is the Gysin map. This is done by first embedding M as the zero section of the compactification of its normal bundle to a sphere bundle. In this case argument as in [1], where a family of operators B is constructed on the sphere S^{n} = S(ℝ^{n} × ℝ) to be O(n) invariant, surjective and have symbol equal to the Thom class. Then B can be extended naturally to act on the fibres of the sphere bundle. Having stabilized P, to a projective family with the given symbol p, (and a finite rank term f) we may lift it, as described in [1], to be an operator acting on the lift of E^{±} to the sphere bundle and reducing to P on fibre-constant sections. As in [1] the tensor product of the lifted operator P and B then acts as a Fredholm family

(50)  (\begin{matrix}P & B \\ B^{∗} & P^{∗} \end{matrix}) .

between the bundles E^{+} ⊗ G^{+} ⊕ E^{−} ⊗ G^{−} and E^{−} ⊗ G^{+} ⊕ E^{+} ⊗ G^{−}. Since P and B commute it follows as in the untwisted case that the null space of this surjective operator is isomorphic to the null space of P. Thus has represents the same index class which proves the desired multiplicativity in this case.

The general case now follows by using the excision propety, so the analytic index satisfies the multiplicative property.

The normalization axiom holds by definition; it is important that this is consistent with the proof of the axioms above. □

The equality of the topological and analytic indexes is now an immediate consequence of Theorems 1, 2 and 3:

Theorem 4 (The index theorem in K-theory). Let ϕ : M ⟶ X be a fibre bundle of compact manifolds, let A be an Azumaya bundle over X and P be a projective family of elliptic pseudodifferential operators acting between two sets of projective vector bundle data associated to a local trivialization of A and with symbol having class p ∈ K_{c}(T (M/X); ρ^{∗}ϕ^{∗}A), where ρ : T (M/X) ⟶ M is the projection then

(51)  index_{a}(P) = index_{t}(p) ∈ K(X, A).

5. The Chern character of the index bundle

In this section, we compute the Chern character of the index bundle and obtain the cohomological form of the index theorem for projective families of elliptic pseudodifferential operators. In the process, not surprisingly, the torsion information is lost. We begin with the the basic properties of the Chern character.

The Chern character of projective vector bundles determines a homomorphism

(52)  Ch_{A} : K^{0}(X, A) ⟶ H^{even}(X, ℚ),

and is defined in the appendix in terms of the ordinary Chern character. Hence it can be shown that the Chern character satisfies the following properties.

The Chern character is functorial under smooth maps in the sense that if f : Y ⟶ X is a smooth map between compact manifolds. Then the following diagram commutes,

(53)  \begin{matrix}K^{0}(X, A) & ⟶^{f^{!}} & K^{0}(Y, f^{∗}A) \\ ↓^{Ch}_{A} & & ↓^{Ch}_{f^{∗}A} \\ H^{even}(X, ℚ) & ⟶^{f^{∗}} & H^{even}(Y, ℚ) \end{matrix}

Now K^{0}(X, A) is a K^{0}(X)-module, and the Chern character respects the module structure in the sense that the following diagram commutes

(54)  \begin{matrix}K^{0}(X) × K^{0}(X, A) & ⟶ & K^{0}(X, A) \\ ↓^{Ch × Ch_{A}} & & ↓^{Ch}_{A} \\ H^{even}(X, ℚ) × H^{even}(X, ℚ) & ⟶ & H^{even}(X, ℚ) \end{matrix}

where the top horizontal arrow is the action of K^{0}(X) on K(X, A) given by tensor product and where the bottom horizontal arrow is given by the cup product.

Theorem 5 (The cohomological formula of the index theorem). For a fibration (1) of compact manifolds and a projective family of elliptic pseudodifferential operators P with symbol class p ∈ K_{c}(T (M/X), ρ^{∗}ϕ^{∗}A), where ρ : T (M/X) ⟶ M is the projection, then

(55)  Ch_{A}(index_{a} P) = (−1)^{n}ϕ_{∗}ρ_{∗} {ρ^{∗}Todd(T (M/X) ⊗ ℂ) ∪ Ch_{ρ^{∗}ϕ^{∗}A}(p)}

where the Chern character is denoted Ch_{A} : K(X, A) → H^{•}(M) and Ch_{ρ^{∗}ϕ^{∗}A} : K_{c}(T (M/X), ρ^{∗}ϕ^{∗}A) → H_{c}^{•}(T (M/X)), n is the dimension of the fibres of ϕ ∘ ρ, ϕ_{∗}ρ_{∗} : H_{c}^{•}(T (M/X)) ⟶ H^{•−n}(X) is integration along the fibre.

This theorem will follow fairly routinely from the index theorem in K-theory in §5. The key step to getting the formula (55) is the analog of the Riemann-Roch formula in the context of twisted K-theory, which we now discuss.

Let π : E ⟶ X be a spinℂ vector bundle over X, i : X ⟶ E the zero section embedding, F be a complex projective vector bundle over X that is associated

to the Azumaya bundle A on X. Then we compute,

Ch_{π^{∗}A}(i_{!}F) = Ch_{π^{∗}A}(i_{!}1 ⊗ π^{∗}F)
= Ch(i_{!}1) ∪ Ch_{π^{∗}A}(π^{∗}F),

where we have used the fact that the Chern character respects the K^{0}(X)-module structure. The classical Riemann-Roch formula asserts that

Ch(i_{!}1) = i_{∗}Todd(E)^{−1}.

Therefore we obtain the following Riemann-Roch formula for twisted K-theory,

(56)  Ch_{π^{∗}A}(i_{!}F) = i_{∗} {Todd(E)^{−1} ∪ Ch_{A}(F)} .

The index theorem in K-theory in §5 shows in particular that

Ch_{A}(index_{a} P) = Ch_{A}(index_{t} p).

Now index_{t} p = j_{!}^{−1} ∘ (Di)_{!} where i : M ↪ X × V is an embedding that commutes with the projections ϕ : M ⟶ X and pr_{1} : X × V ⟶ X, and j : X ↪ X × V is the zero section embedding. Therefore

Ch_{A}(index_{t} p) = Ch_{A}(j_{!}^{−1} ∘ (Di)_{!}p)

By the Riemann-Roch formula for twisted K-theory (56),

Ch_{pr_{1}^{∗}A}(j_{!}F) = j_{∗} Ch_{A}(pr_{1}^{∗}F)

since pr_{1} : X × V ⟶ X is a trivial bundle. Since pr_{1}_{∗}j_{∗}1 = (−1)^{n}, it follows that for ξ ∈ K_{c}(X × V, pr_{1}^{∗}A), one has

Ch_{A}(j_{!}^{−1}ξ) = (−1)^{n}pr_{1}_{∗} Ch_{pr_{1}^{∗}A}(ξ)

Therefore

(57)  Ch_{A}(j_{!}^{−1} ∘ (Di)_{!}p) = (−1)^{n}pr_{1}_{∗} Ch_{pr_{1}^{∗}A}((Di)_{!}p)

By the Riemann-Roch formula for twisted K-theory (56),

(58)  Ch_{pr_{1}^{∗}A}((Di)_{!}p) = (Di)_{∗} {ρ^{∗}Todd(N)^{−1} ∪ Ch_{ρ^{∗}ϕ^{∗}A}(p)}

where N is the complexified normal bundle to the embedding Di : T (M/X) ⟶ X × T V, that is N = X × T V/Di(T (M/X)) ⊗ ℂ. Therefore Todd(N)^{−1} = Todd(T (M/X) ⊗ ℂ) and (58) becomes

Ch_{pr_{1}^{∗}A}((Di)_{!}p) = (Di)_{∗} {ρ^{∗}Todd(T (M/X) ⊗ ℂ) ∪ Ch_{ρ^{∗}ϕ^{∗}A}(p)} .

Therefore (57) becomes (59)

Ch_{A}(j_{!}^{−1} ∘ (Di)_{!}p) = (−1)^{n}pr_{1}_{∗}(Di)_{∗} {ρ^{∗}Todd(T (M/X) ⊗ ℂ) ∪ Ch_{ρ^{∗}ϕ^{∗}A}(p)}
= (−1)^{n}ϕ_{∗}ρ_{∗} {ρ^{∗}Todd(T (M/X) ⊗ ℂ) ∪ Ch_{ρ^{∗}ϕ^{∗}A}(p)}

since ϕ_{∗}ρ_{∗} = pr_{1}_{∗}(Di)_{∗}. Therefore

(60)  Ch_{A}(index_{t} p) = (−1)^{n}ϕ_{∗}ρ_{∗} {ρ^{∗}Todd(T (M/X)) ∪ Ch_{ρ^{∗}ϕ^{∗}A}(p)} .

By Theorem 4 and equation (60) above, we deduce the cohomological formula of the index theorem as stated in Theorem 5.

6. Determinant line bundle of the index bundle

In this section, we define the determinant line bundle of the index bundle of a projective family of elliptic pseudodifferential operators, and compute its first Chern class.

We begin with the general construction of the determinant line bundle of a projective vector bundle over X. Let A be an Azumaya bundle over X and P →^{π} X be the principal PU(n) bundle associated to A. Let E → P be a projective vector bundle over X, associated to A, cf. section 1.3. Recall that E satisfies in addition the condition

(61)  ϕ^{∗}E ≅ π_{P}^{∗} E ⊗ π_{PU(n)}^{∗}L

where ϕ : PU(n) × P → P is the action, π_{P} : PU(n) × P → P is the projection onto the second factor, π_{PU(n)} : PU(n)×P → PU(n) is the projection onto the first factor, L → PU(n) is the (determinant) line bundle associated to the principal ℤ_{n} bundle ℤ_{n} → SU(n) → PU(n) as in section 1.3.

Then we observe that

ϕ^{∗}Λ^{n}(E) ≅ π_{P}^{∗} Λ^{n}(E)

and therefore Λ^{n}(E) = π^{∗}(F) for some line bundle F → X. Define det(E) = F to be the determinant line bundle of

the projective vector bundle E. isomorphism of line bundles. In particular, this gives a homomorphism

det : K^{0}(X, A) ⟶ π_{0}(Pic(X))

where Pic(X) denotes the Picard variety of X, and the components of the Picard variety π_{0}(Pic(X)) consist of the isomorphism classes of complex line bundles over X.

Theorem 6 (Chern class of the determinant line bundle of the index bundle). For a fibration (1) of compact manifolds and a projective family of elliptic pseudodifferential operators P with symbol class p ∈ K_{c}(T (M/X), ρ^{∗}ϕ^{∗}A), where ρ : T (M/X) ⟶ M is the projection, then

(62) c_{1}(det(index_{a} P)) = {(−1)^{n}ϕ_{∗}ρ_{∗} {ρ^{∗}Todd(T (M/X) ⊗ ℂ) ∪ Ch_{ρ^{∗}ϕ^{∗}A}(p)}}^{[2]}

where Ch_{ρ^{∗}ϕ^{∗}A} : K_{c}(T (M/X), ρ^{∗}ϕ^{∗}A) ⟶ H_{c}^{•}(T (M/X)) is the Chern character, c_{1} is the first Chern class, N is the dimension of the fibres of ϕ ∘ ρ, ϕ_{∗}ρ_{∗} is integration along the fibres mapping H_{c}^{•}(T (M/X)) to H^{•−n}(X) is and {·}^{[2]} denotes the degree 2 component.

Proof. The proof of the theorem follows from Theorem 5.4 and the once we show that the following diagram commutes,

(63)

K^{0}(X, A) ⟶^{det}  π_{0}(Pic(X))
↓^{Ch}_{A}  ↓^{c_{1}}
H^{even}(X, ℚ) ⟶^{·^{[2]}} H^{2}(X, ℤ) ∩ H^{2}(X, ℚ).

But this is the content of Lemma 6, part 2. □

7. Projective families of Dirac operators

A fiberwise Clifford module on a bundle E over the total space of a fibration is a smooth action of the Clifford bundle of the fibres, Cl_{ϕ}(M) on the bundle. That is to say it is an algebra homorphism

(64)  Cl_{ϕ}(M) ⟶ End(E).

Since the endomorphism bundle of a projective bundle over E, with respect to an Azumaya bundle A, is a vector bundle, this definition can be taken directly over to the projective case. Similarly, the condition that the Clifford module structure be hermitian can be taken over as the condition that (64) be ∗-preserving. The condition that a unitary connection on E be a Clifford connection is then the usual distribution condition, for vertical vector fields,

(65)  ∇_{X}(cl(α)e_{a}) = cl(∇_{X}α)e_{a} + cl(α)(∇_{X}e_{a})

in terms of the Levi-Civita connection on the fibre Clifford bundle and for any sections e_{a} of the bundle trivialization of E with respect to a full trivialization of A.

The Dirac operator associated to such a unitary Clifford connection on a hermitian projective Clifford module is then given by the usual formula over the open sets U_{a} of a given full trivialization of A over the base:

(66)  ð_{a}e_{a} = c̃l(∇e_{a})

where L̃ is the contraction given by the Clifford action from T^{∗}(M/X) ⊗ E_{a} to E_{a}. The invariance properties of the connection and Clifford action show that the ð_{a} form a projective family of differential operators on the projective bundle E.

As in the untwisted case, if ð is a Dirac operator in this sense, acting on a vector bundle F over M and E is a projective vector bundle over M then may twist ð by choosing a unitary connection ∇^{E} on E and extending the Clifford module trivially from F to F ⊗ E (to act as multiples of the identity on E) and taking the tensor product connection on F ⊗ E. The resulting Dirac operator is then a projective family as described above.

In particular, if the bundle T (M/X) is spinℂ and the we consider the family of spinℂ Dirac operators along the fibres twisted by the projective vector bundle E, then we deduce from Theorem 4 that

index_{a}(ð) = ϕ_{!}(E) ∈ K(X, A)

where ϕ_{!} : K(M, ϕ^{∗}A) → K(X, A) is defined as ϕ_{!} = j_{!}^{−1} ∘ f_{!} in the notation of Section 3. By Theorem 5 and standard manipulations of characteristic classes one has

Ch_{A}(index_{a}(ð)) = ϕ_{∗}(Todd(T (M/X)) Ch_{ϕ^{∗}A}(E)) ∈ H^{•}(X).

where ϕ_{∗} : H^{•}(M) → H^{•−ℓ}(X) denotes integration along the fibres, and ℓ denotes the dimension of the fibres.

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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