j. differential geometry

59 (2001) 269-300

LOCAL FORMULA FOR THE INDEX OF A FOURIER INTEGRAL OPERATOR

ERIC LEICHTNAM, RYSZARD NEST & BORIS TSYGAN

Abstract

Let X and Y be two closed connected Riemannian manifolds of the same dimension and ϕ : S^{∗}X ↦ S^{∗}Y a contact diffeomorphism. We show that the index of an elliptic Fourier operator Φ associated with ϕ is given by ∫_{B^{∗}(X)} e^{θ_{0}} Â(T^{∗}X)−∫_{B^{∗}(Y )} e^{θ_{0}} Â(T^{∗}Y) where θ_{0} is a certain characteristic class depending on the principal symbol of Φ and, B^{∗}(X) and B^{∗}(Y) are the unit ball bundles of the manifolds X and Y. The proof uses the algebraic index theorem of Nest-Tsygan for symplectic Lie Algebroids and an idea of Paul Bressler to express the index of Φ as a trace of 1 in an appropriate deformed algebra.

In the special case when X = Y we obtain a different proof of a theorem of Epstein-Melrose conjectured by Atiyah and Weinstein.

1. Introduction

Let X and Y be two smooth closed connected Riemannian manifolds of the same dimension such that there exists a contact diffeomorphism ϕ : S^{∗}X ↦ S^{∗}Y between the two unit cotangent bundles which induces a homogeneous symplectomorphism, still denoted by ϕ, from T^{∗}X \ X onto T^{∗}Y \ Y.

We first recall the definition of the index of ϕ, which is defined only when dim X ≥ 3, following [15]. We will denote by Ω_{\frac{1}{2}} the half-density bundle over X or Y. Let C_{ϕ} be the graph of ϕ^{−1} in (T^{∗}Y \Y)×(T^{∗}X\X) and L_{C_{ϕ}} be the associated Maslov bundle. Let A : L^{2} (X, Ω_{\frac{1}{2}} ) →

L^{2} (Y, Ω_{\frac{1}{2}} ) be an elliptic Fourier integral operator of order zero whose canonical relation is C_{ϕ} and whose principal symbol is an invertible

Received April 18, 2000

section of the bundle Ω_{\frac{1}{2}} ⊗ L_{C_{ϕ}} → C_{ϕ} (see [15], [7]) for details). Sup-

pose that B : L^{2} (Y, Ω_{\frac{1}{2}} ) → L^{2} (X, Ω_{\frac{1}{2}} ) is an elliptic Fourier integral operator of order zero whose canonical relation is C_{ϕ^{−1}}. Then B ∘ A : L^{2} (X, Ω_{\frac{1}{2}} ) → L^{2} (X, Ω_{\frac{1}{2}} ) is an elliptic scalar pseudo-differential operator of order zero. Since dim X ≥ 3 there exists a smooth non vanishing function x ∈ X → a(x) ∈ ℂ^{∗} such that the principal symbol of B ∘ A is homotopic to (x, ξ) ∈ T^{∗}X → a(x) ∈ ℂ^{∗}. In particular the index of B ∘ A is zero. Thus IndB = −IndA for any Fourier integral operators A and B as above, and, as the corollary of this fact, IndA does not depend on the choice of A. Since it only depends on the transformation ϕ, it is called the index of ϕ and denoted by Ind ϕ. A. Weinstein has proved (see [15]) that the integer Ind ϕ naturally appears if one wants to compare the spectrum ( λ_{k}(X) )_{k∈ℕ} of the Laplace Beltrami operator Δ_{X} of X (associated with the metric g_{X}) with the one of Δ_{Y} (associated with the metric g_{Y}); for instance if T^{∗}X \ X is simply connected then the sequence (λ_{k}(X) − λ_{k−Indϕ}(Y))_{k∈ℕ} is bounded.

The goal of this paper is to provide a geometric formula for the index of an elliptic Fourier integral operator Φ of order zero whose canonical relation is C_{ϕ} (we do not assume dim X ≥ 3).

Let us first fix some notation. Given a smooth manifold X, we will use T^{∗}X to denote the cotangent bundle of X and \overline{B}^{∗}X to denote the projective compactification of T^{∗}X. We will use M to denote the smooth manifold obtained by glueing at infinity \overline{B}^{∗}X and \overline{B}^{∗}Y with the help of the map

ϕ′ : (x, ξ) → ϕ(x, −ξ).

Let S^{0}(T^{∗}X) and S^{0}(T^{∗}Y) denote the algebras of asymptotic symbols of pseudodifferential operators of order at most zero on X and Y. Given an element a ∈ S^{0}(T^{∗}X), we denote by a_{ℏ} the symbol a scaled by ℏ in the cotangent direction and by Op(a) the pseudodifferential operator associated to a. Given a pseudodifferential operator A, we denote by σ(A) its full symbol (for the precise definition see the next section).

The general strategy is as follows. We interpret conjugation by Φ as an isomorphism of the algebras of pseudodifferential operators (modulo the smoothing ones) on X and Y. Translated into terms of formal deformations of the cotangent bundles, this allows us to construct a formal deformation 𝔸^{ℏ}(M) of C^{∞}(M) which represents on T^{∗}X and T^{∗}Y the calculus of differential operators, while on the common cosphere at infinity it represents the calculus of pseudodifferential operators. While the symplectic structures on T^{∗}X and T^{∗}Y do not glue together (so there is in general no almost complex structure on B^{∗}X ∪_{ϕ′} B^{∗}Y), there is a (noncanonical) symplectic Lie algebroid structure (E, [·, ·], ω) over M and 𝔸^{ℏ}(M) is a deformation associated to it in the sense of [10]. The usual traces on the algebras of smoothing operators on X and Y give rise to a trace τ_{can} on 𝔸^{ℏ}(M) such that ind Φ = τ_{can}(1). An application of the general algebraic index theorem from [10] gives the local formula for the index.

The content of the paper is given below.

• In the first section we recall the relation between the calculus of smoothing operators on X and a formal deformation of T^{∗}X which is basically given by the full symbol of a pseudodifferential operator.

• The manifold \overline{B}^{∗}X carries a structure of symplectic Lie algebroid (E_{X}, [,], ω) described in Section 2. The symbolic calculus of pseudodifferential operators gives rise to a formal deformation of the sphere at infinity of \overline{B}^{∗}X which, together with the formal deformation of T^{∗}X given above, gives rise to a formal deformation 𝔸^{ℏ}(X) of \overline{B}^{∗}X associated to (E_{X}, [,], ω).

• Let us fix an almost unitary elliptic Fourier integral operator Φ whose canonical relation is given by the graph of ϕ^{−1}. In Section 4 we show how to glue together the deformations 𝔸^{ℏ}(X) and 𝔸^{ℏ}(Y) into a formal deformation 𝔸^{ℏ}(M) of M associated to a symplectic Lie algebroid structure (E, [,], ω) on M. The construction is based on the following strengthening of the Egorov theorem (see Theorem 2).

1. The map which to any a ∈ S^{0}(T^{∗}X) associates the asymptotic expansion at ℏ = 0 of (σ(ΦOp(a_{ℏ})Φ^{∗}))_{ℏ^{−1}} induces an algebra isomorphism

Φ̃ : S^{0}(T^{∗}X) → S^{0}(T^{∗}Y).

2. For each k ∈ ℕ^{∗}, there exists an E_{X}−differential operator D_{k} on \overline{B}^{∗}X such that, for any a ∈ S^{0}(T^{∗}X), the following identity holds:

Φ̃(a) = (a + ∑_{k≥1} ℏ^{k}D_{k}(a)) ∘ ϕ^{−1}.

Egorov’s theorem corresponds to the leading term in the above expansion.

The real symplectic vector bundle E is isomorphic to T M (as a vector bundle over M) and hence T M is the realification of a complex vector bundle on M which will be denoted by E_{ℂ}.

• In Section 5 we identify the space traces on 𝔸^{ℏ}(M) and relate it to the traces on the algebras of smoothing and of pseudodifferential operators.

• In Section 6 we identify the index of the Fourier integral operator with the value of the trace τ_{can} on the unit 1 in the formal deformation algebra 𝔸^{ℏ}(M).

The local index formula for Ind Φ follows from the algebraic index theorem of [10], the class θ_{0} being the coefficient of ℏ^{0} in the characteristic class θ (cf. [5], [10]) of the deformation. Notice that the first algebraic type index theorem was proved in [5] where θ appears as the curvature of a suitable connection.

The main result can be formulated as follows:

3. Let Φ be a Fourier integral operator and 𝔸^{ℏ}(M) the formal deformation of M associated to it as in Definition 3. Then

ind Φ = ∫_{M} e^{θ_{0}} Â(M),

where θ_{0} denotes the characteristic class of the deformation of the Lie Algebroid (E, [ , ], ω) given by 𝔸^{ℏ}(M).

4. Let ∇_{X} be a connection on the tangent bundle T (\overline{B}^{∗}X) and Â(T^{∗}X) an associated representative form of the Â-class of ∇_{X}. The symplectomorphism ϕ induces a connection ϕ_{∗}(∇_{X}) on the tangent space of \overline{B}^{∗}Y \ B^{∗}(Y). Let ∇_{Y} denote its extension to a connection of T (\overline{B}^{∗}Y) and Â(T^{∗}Y) an associated representative differential form of the Â-class of ∇_{Y}. Then

ind Φ = ∫_{B^{∗}(X)} e^{θ_{0}} Â(T^{∗}X) − ∫_{B^{∗}(Y)} e^{θ_{0}} Â(T^{∗}Y). (1)

Remark 1. It is easy to see that our local formula implies the following fact:

If ϕ extends as a symplectomorphism : T^{∗}X → T^{∗}Y up to the zero section, then Ind Φ = 0.

• The computation of the characteristic class of the deformation is given in the last section, where we simultaneously construct a deformation of M and the Fourier integral operator whose index is given by the trace of the 1 in the deformed algebra. As the starting point we give a somewhat nonstandard definition of the characteristic class of a formal deformation which is more amenable to computations in the case of deformations associated to (twisted) differential or pseudodifferential operators.

As a corollary we get the following result:

5. There exists an almost unitary Fourier integral operator Φ_{0} whose canonical relation is C_{ϕ} and such that:

ind Φ_{0} = ∫_{M} Â(M)e^{\frac{1}{2} c_{1}(E_{ℂ})}.

6. If the dimension of M is at least three, then

ind(ϕ) = ∫_{M} Â(M)e^{\frac{1}{2} c_{1}(E_{ℂ})}

(cf. [4], [14]).

In the case when X = Y Epstein and Melrose ([4]) had shown that ind Φ_{0} is equal to the index of the Dirac operator associated to the Spin^{c}-structure of the mapping torus of (X, ϕ) then, using the Atiyah-Singer index formula they proved the following Theorem which provides a geometric formula for ind Φ_{0}. We show how to derive this Theorem from our previous result.

Theorem 1 ([4], Section 7). Let ϕ be a contact automorphism of S^{∗}X, denote by Z_{ϕ} the mapping torus of (S^{∗}X, ϕ) which, as explained in Section 3 of [4], is endowed with a “positive” almost complex structure. Then:

ind Φ_{0} = ∫_{Z_{ϕ}} exp (\frac{c_{1}(T^{1,0}Z_{ϕ})}{2}) Â(Z_{ϕ}). (2)

Proof. Recall that the mapping torus of ϕ is given by

Z_{ϕ} = {S^{∗}X × [0, 1]}/{(ξ, 0) ∼ (ϕ(ξ), 1)}.

We will write it as the manifold obtained from glueing together two copies of \{ξ ∈ \overline{B}^{∗}X | 1 ≤ ||ξ|| ≤ ∞\}, where at the sphere ||ξ|| = 1 we use the identity map, while at ||ξ|| = ∞ we use ϕ. We can choose two connections ∇_{1} and ∇_{2} on T^{∗}X which coincide at ||ξ|| ≤ \frac{1}{2} and are related by ϕ at ||ξ|| ≥ 2 They glue together to a connection ∇ on the mapping torus Z_{ϕ}. One can use the Mayer-Vietoris exact sequence for the cohomology of Z_{ϕ} associated to the open covering by two open sets — a neighbourhood of ||ξ|| = 1 and a neighbourhood of ||ξ|| = ∞ — to evaluate the right hand side of (2). But, provided that the above ∇ is used to compute the characteristic classes appearing, the result becomes the same as the left hand side of (1) expressed with connections ∇_{1} and ∇_{2}. q.e.d.

Remark 2. The methods of this paper extend in a fairly straightforward manner to the case of a Fourier integral operator Φ between L^{2} sections of vector bundles E and F of the same dimension on X and Y. In the case when both X and Y possess a metalinear structure, the corresponding index formula is given by the expression

ind Φ = ∫_{B^{∗}(X)} ch(L)Â(T^{∗}X) − ∫_{B^{∗}(Y)} ch(L)Â(T^{∗}Y).

Here L is the vector bundle over M obtained by glueing together pullbacks (by the canonical projections π_{X}^{∗} and π_{Y}^{∗}) to the cotangent bundles of X (resp. Y) of the bundles Λ^{\frac{n}{2}} (X) ⊗ E and Λ^{\frac{n}{2}} (Y) ⊗ F with the help of the symbol of Φ.

Note that existence of an isomorphism of π_{X}^{∗}(Λ^{\frac{n}{2}} (X) ⊗ E) and π_{Y}^{∗} (Λ^{\frac{n}{2}} (Y) ⊗ F) over ϕ′ is equivalent to existence of an elliptic Fourier integral operator from L^{2}(X, E) to L^{2}(Y, F).

Acknowledgements. The authors are very grateful to the referees, whose detailed observations and comments have allowed us to improve the exposition of this paper.

2. Symbolic calculus for ΨDO’s and formal deformations

2.1 Deformation of T^{∗}X

We will recall the pertinent facts from [12].

Let χ be a smooth, nonnegative function on X × X satisfying the following conditions:

(1) χ(x, y) = χ(y, x).

(2) χ ≡ 1 on an open set containing the diagonal in X × X.

(3) For each x ∈ X, the set D_{x} = {y ∈ X/ (x, y) ∈ supp χ} is geodesically convex.

We denote by Exp_{x}^{−1} the unique smooth inverse to the exponential map:

Exp_{x} : T_{x} X → X

defined on D_{x} and such that Exp_{x}^{−1}(x) = 0.

Given x ∈ X, y ∈ D_{x}, let z denote the midpoint of the unique geodesic joining x and y within D_{x}, and let v ∈ T_{z}X be given by

v/2 = Exp_{x}^{−1}(y) = −Exp_{z}^{−1}(x). (3)

Now, denote by S^{m}(T^{∗}X) the space of classical symbols of order m on X, i.e., smooth functions θ on T^{∗}X satisfying estimates of the form:

sup_{(x,ξ)} |∂_{x}^{α} ∂_{ξ}^{β} θ (x, ξ)| ≤ C_{α,β} (1 + |ξ|^{2})^{\frac{m−|β|}{2}}

S^{m}(T^{∗}X) is given the topology of (Frechet) topological vector space by the “best” C_{α,β}.

We will denote by S^{∞}(T^{∗}X) = ∪_{m∈ℝ}S^{m}(T^{∗}X) the set of all classical symbols on T^{∗}X.

With the above notation (3), the map:

Op : S^{m}(T^{∗}X) → End (C^{∞}(X))

given by

Op(θ) (u) (x) = ∫_{T_{z}^{∗}X} dξ ∫ dy χ(x, y) e^{i ξ.v} θ(z, ξ) u(y)

defines a pseudo-differential operator. Conversely, if P is a pseudodifferential operator on X we define its complete symbol to be:

σ(P)(z, ξ) = P_{y}( χ(x, y) e^{i ξ.v})|_{x=y=z} (4)

where z is the midpoint of the geodesic joining x and y and v satisfies (3). We observe that P − Op(σ(P)) is a smoothing operator whose Schwartz kernel vanishes to infinite order on the diagonal. Now, for a given θ ∈ C^{∞}(T^{∗}X), we set: θ_{ℏ}(x, ξ) = θ(x, ℏξ).

Following [12], we endow the algebra

𝔸^{ℏ}(T^{∗}X) = C^{∞}(T^{∗}X) ⊗_{ℂ} ℂ[[ℏ]]

with a star product ⋆_{X} by defining, for any symbols θ^{1}, θ^{2}, ∈ S^{m}(T^{∗}X), θ^{1} ⋆_{X} θ^{2} to be the asymptotic expansion at ℏ = 0 of:

(5) σ( Op(θ_{ℏ}^{1}) ∘ Op(θ_{ℏ}^{2}) )_{ℏ^{−1}}(x, ξ) = σ( Op(θ_{ℏ}^{1}) ∘ Op(θ_{ℏ}^{2}) )(x, ℏ^{−1}ξ).

One sees immediately that ⋆_{X} extends to 𝔸^{ℏ}(T^{∗}X).

Recall that there exists, unique up to normalization, a canonical trace on (𝔸^{ℏ}(T^{∗}X), ⋆_{X}), Tr_{can}^{X}, given by:

∀a ∈ S^{−∞}(T^{∗}X),

Tr_{can}^{X} (a) = Tr( Op(a_{ℏ}) ) = \frac{1}{n! ℏ^{n}} ∫_{T^{∗}X} a (ω^{X})^{n} ∈ ℂ[ℏ^{−1}, ℏ]].

(See Proposition 2.5 (3) of [12], the uniqueness is addressed in [5, p. 172] and in [11]).

2.2 Lie algebroid structure and deformation quantization of the projective completion \overline{B}^{∗}X

For any x ∈ X, we set B_{x}^{∗}X = \frac{ℝ_{+}⊕T_{x}^{∗}X}{ℝ_{+}^{∗}} \ \{(0, 0)\}, and embed T_{x}^{∗}X in

\overline{B}^{∗}_{x}X by sending ξ to the class of 1 ⊕ ξ. We view B_{x}^{∗}X as a compactification of T_{x}^{∗}X. Then we consider the fiber bundle \overline{B}^{∗}X over X defined by \overline{B}^{∗}X = ∪_{x∈X}B_{x}^{∗}X. Therefore \overline{B}^{∗}X is a compactification of T^{∗}X and a smooth compact manifold with boundary: ∂\overline{B}^{∗}X = \overline{B}^{∗}X \ T^{∗}X. Similarly one defines the bundle \overline{B}^{∗}Y over Y. We observe that the map from S^{∗}X into \overline{B}^{∗}X given by ξ → 0⊕ξ defines an isomorphism between S^{∗}X and \overline{B}^{∗}X \ T^{∗}X. For any ξ = (x, ξ_{x}) ∈ T_{x}^{∗}X we will define −ξ to be (x, −ξ_{x}) ∈ T_{x}^{∗}X. Clearly, ϕ induces a natural smooth isomorphism of manifolds with boundary:

ϕ′ : \overline{B}^{∗}X \ X ↦ \overline{B}^{∗}Y \ Y

defined by

ϕ′(λ ⊕ ξ) = (λ ⊕ ϕ(−ξ)) if ξ ∈ T^{∗}X \ X and λ ≥ 0,
∀λ ∈ ℝ^{+∗}, ϕ′(λ ⊕ 0) = (λ ⊕ 0).

By glueing \overline{B}^{∗}X and \overline{B}^{∗}Y along the boundary \overline{B}^{∗}X \ T^{∗}X with the help of ϕ′, we define the following smooth compact manifold M:

M = \overline{B}^{∗}X ∪_{ϕ′} \overline{B}^{∗}Y. (6)

Let Π_{X} : \overline{B}^{∗}X → X be the projection map. We denote by Ξ^{X} the set of smooth vectors fields of \overline{B}^{∗}X which are tangent to all the submanifolds Π_{X}^{−1}(x)∩(\overline{B}^{∗}X \T^{∗}X), x ∈ X. Let (x, ξ) = (x_{1}, …, x_{n}; ξ_{1}, …, ξ_{n}) be a local chart of T^{∗}X and (ρ, θ) be the polar coordinates: ρ = ||ξ||, θ = \frac{ξ}{||ξ||}, where || · || denotes the Euclidean norm of T^{∗}X. Then a local chart of \overline{B}^{∗}X near \overline{B}^{∗}X \ T^{∗}X is given by

(x_{1}, …, x_{n}; t = \frac{1}{ρ}, θ = (θ_{1}, …, θ_{n−1})) t ≥ 0, θ ∈ S^{n−1}. (7)

In this local chart, Ξ^{X} is generated by the vector fields t\frac{∂}{∂x_{j}}, t\frac{∂}{∂t}, \frac{∂}{∂θ_{l}}, where 1 ≤ j ≤ n, 1 ≤ l ≤ n − 1. We will make several applications of the following lemma. It is easily proved using polar coordinates.

Lemma 1. The vector fields \frac{∂}{∂ξ_{j}} (1 ≤ j ≤ n) belong to the C^{∞}(\overline{B}^{∗}X)-module tΞ^{X} generated by t^{2} \frac{∂}{∂t}, t\frac{∂}{∂θ_{l}} (1 ≤ l ≤ n − 1).

Moreover we observe that the set of classical symbols of order zero on T^{∗}X is nothing else but C^{∞}(\overline{B}^{∗}X).

Before we continue, let us recall the definition of a symplectic Lie algebroid (see for instance [8], [10]).

Definition 1.

1) A symplectic Lie algebroid on M is a quadruple (E, ρ, [ , ], ω) on M, where E is a smooth vector bundle on M, [ , ] is a Lie algebra structure on the sheaf of sections of E, ρ is a smooth map of vector bundles,

ρ : E → T M,

such that the induced map

Γ(ρ) : C^{∞}(M, E) → C^{∞}(M, T M)

is a Lie algebra homomorphism and, for any sections σ and τ of E and any smooth function f on M, the following identity holds:

[ σ, fτ ] = ρ(σ)(f).τ + f[σ, τ].

Lastly, ω is a closed E-two form on M such that the associated linear map:

C^{∞}(M, E) × C^{∞}(M, E) ∋ (U, V) ↦ ω(U, V) ∈ C^{∞}(M)

defines a symplectic structure on E.

2) The ring of E-differential operators is by definition the ring generated by smooth functions on M and smooth sections of E.

3) We denote by ^{E}Ω^{•} = C^{∞}(M, Λ^{•}E^{∗}) the set of smooth sections on M of the bundle of alternating multilinear forms on E.

The following proposition will allow us to construct the relevant Lie algebroid E over M.

Proposition 1. For any p ∈ \overline{B}^{∗}X, we set

E_{p}^{X} = \frac{Ξ^{X}}{I_{p}Ξ^{X}}

where I_{p} is the set of smooth real-valued functions on \overline{B}^{∗}X which vanish at p. Then:

1) (E_{p}^{X})_{p∈\overline{B}^{∗}X} form a smooth vector bundle, denoted E^{X}, over \overline{B}^{∗}X such that the set of smooth sections over \overline{B}^{∗}X of E^{X} coincides with Ξ^{X}. If U, V ∈ Ξ^{X} then the Lie bracket [U, V] also belongs to Ξ^{X}.

2) The fundamental two-form ω^{X}(= ∑_{j=1}^{n} dξ_{j} ∧dx_{j}) of T^{∗}X induces a smooth form, still denoted ω^{X}, in C^{∞}(\overline{B}^{∗}X;^{E}^{X}Ω^{2} ). Moreover, (E^{X}, [,], ω^{X}) defines a symplectic Lie algebroid over \overline{B}^{∗}X.

Proof. We shall use the local coordinates (7).

1) Near a point p^{0} = (x^{0}; 0 = t, θ^{0}), each vector field U of Ξ^{X} is of the form:

∑_{j=1}^{n} A_{j}(x; t, θ)t\frac{∂}{∂x_{j}} + A_{n+1}(x; t, θ)t\frac{∂}{∂t} + ∑_{j=n+2}^{2n} A_{j}(x; t, θ)t\frac{∂}{∂θ_{j−n−1}} .

Then by considering the Taylor formulas:

A_{j}(x; t, θ) = A_{j}(x^{0}; 0, θ^{0}) + ∑_{k=1}^{n}(x_{k} − x_{k}^{0})B_{j,k}(x; t, θ)

+ tC_{j,0}(x; t, θ) + ∑_{l=1}^{n−1}(θ_{l} − θ_{l}^{0})C_{j,l}(x; t, θ)

one sees immediately that E_{p}^{X} defines a smooth vector bundle over \overline{B}^{∗}X whose set of smooth sections coincides with Ξ^{X}. Next, working with the vector fields

t\frac{∂}{∂x_{j}} , t\frac{∂}{∂t}, \frac{∂}{∂θ_{l}}

one checks immediately that for any U, V ∈ Ξ^{X}, [U, V] is still in Ξ^{X}.

2) The fundamental form ω^{X} is not defined on the boundary of T \overline{B}^{∗}X but, using the polar coordinates (7), one checks easily for any U, V ∈ Ξ^{X}, ω^{X}(U, V) is well defined up to the boundary of \overline{B}^{∗}X (= {t = 0}). One then immediately gets the proposition. q.e.d.

Proposition 2. The star product ⋆_{X} on T^{∗}X extends to a star product, still denoted ⋆_{X}, on \overline{B}^{∗}X such that for any f, g ∈ C^{∞}(\overline{B}^{∗}X) we have:

f ⋆_{X} g = fg + ∑_{n≥1} ℏ^{n}A^{(n)}(f , g)

where the A^{(n)} are E^{X}-bidifferential operators.

Proof. Let (x, ξ) = (x_{1}, …, x_{n}; ξ_{1}, …, ξ_{n}) be a local chart of T^{∗}X. Then for any f, g ∈ C^{∞}(\overline{B}^{∗}X) and (x, ξ) in the domain of this local chart we have:

f ⋆_{X} g(x, ξ) = ∑_{α,β∈ℕ^{n}, |β|≤|α|} \frac{ℏ^{|α|}}{α!} c_{α,β}(x)D_{ξ}^{α}f(x, ξ)\frac{∂^{β}}{∂^{β}x}g(x, ξ)

then, using the local coordinates (7) and Lemma 1, one gets easily all the results of the proposition. q.e.d.

Proposition 1 allows us to formulate the following definition:

Definition 2.

1) A smooth real-vector bundle E^{Y} over \overline{B}^{∗}Y is defined by setting E_{|T^{∗}Y}^{Y} = T (T^{∗}Y) and E_{|\overline{B}^{∗}Y \Y}^{Y} = ϕ_{∗} (E_{|\overline{B}^{∗}X\X}^{X}).

2) By glueing E^{X} and E^{Y} along \overline{B}^{∗}X \ T^{∗}X with the help of ϕ′, one defines a smooth vector bundle E over M which is isomorphic to T M. A smooth exact differential form ω ∈ C^{∞}(M;^{E}Ω^{2} ) is defined by setting ω_{|\overline{B}^{∗}X} = ω^{X}, ω_{|\overline{B}^{∗}Y \Y} = (ϕ^{−1})^{∗} (ω_{|\overline{B}^{∗}X\X}^{X}) and

ω_{|T^{∗}Y} = ω^{Y} (where ω^{Y} is the canonical two form of T^{∗}Y).

3) The natural injection

C^{∞}(M, E) → C^{∞}(M, T M)

is induced by a bundle map ρ : E → T M as in Proposition 1 and (E, ρ, [,], ω) defines a symplectic Lie algebroid which will be denoted (E, [,], ω) in the sequel.

3. Regularized index formula for a Fourier integral operator

Let C_{ϕ} be the graph of ϕ^{−1} in (T^{∗}Y \Y)×(T^{∗}X \X) and L_{C_{ϕ}} be the

associated Maslov bundle over C_{ϕ}. We fix Φ : L^{2}(X, Ω_{\frac{1}{2}} ) → L^{2} (Y, Ω_{\frac{1}{2}} ) an elliptic Fourier integral operator of order zero whose canonical relation is C_{ϕ} and whose principal symbol a is a unitary section of the bundle Ω_{\frac{1}{2}} ⊗ L_{C_{ϕ}} → C_{ϕ}: this means that a is homogeneous of degree zero (i.e., constant on each ray) and that a\overline{a} ≡ 1: see [15]. We can, and will, assume in the sequel that ΦΦ^{∗} − Id and Φ^{∗}Φ − Id are smoothing. As observed in [15] Φ is Fredholm, with index defined by ind Φ = dim ker Φ − dim cokerΦ. In order to give a formula “via regularization” for ind Φ we introduce the following algebra A which will have a “regularized” trace:

A = \{(A, B) ∈ Ψ^{0}(X, Ω_{\frac{1}{2}} ) × Ψ^{0}(Y, Ω_{\frac{1}{2}} )| A − Φ^{∗}BΦ is smoothing\}.

Proposition 3.

1) The map τ : A → ℂ given by

(8) ∀(A, B) ∈ A, τ(A, B) = Tr(A − Φ^{∗}BΦ) − Tr(B(Id −ΦΦ^{∗}))

is a trace.

2) ind Φ = τ(Id, Id).

Proof. 1). If (A, B), (A′, B′) ∈ A are smoothing operators then one immediately checks that

τ( (A, B) · (A′, B′) ) = Tr(AA′) − Tr(BB′) = τ( (A′, B′) · (A, B) ). (9)

Now if the operators (A, B), (A′, B′) are not smoothing one considers operators of the form (A ∘ χ (\frac{1}{N} \sqrt{Δ_{X}}) , B ∘ χ (\frac{1}{N} \sqrt{Δ_{Y}} )) where χ ∈ C^{∞}(ℝ, [0, 1]) is equal to 1 on [−1, 1] and to zero on [−2, 2]. By applying Equation (9) and letting N go to ∞ one immediately gets part 1). Part 2) means that

ind Φ = Tr(Id −Φ^{∗}Φ) − Tr(Id −ΦΦ^{∗}),

which is well-known. q.e.d.

Remark 3. τ(A, B) is a ”regularization” of TrA − TrB.

4. Algebraization of a Fourier integral operator

We are going to use the following (deformed quantized algebra), where the manifold Z is equal to X or Y:

𝔹^{ℏ}(\overline{B}^{∗}Z) = \frac{C^{∞}(\overline{B}^{∗}Z)}{C_{0}^{∞}(\overline{B}^{∗}Z)} ⊗_{ℂ} ℂ[[ℏ]]

where C_{0}^{∞}(\overline{B}^{∗}Z) denotes the set of smooth functions which vanish to infinite order to \overline{B}^{∗}Z \T^{∗}Z. We observe that ⋆_{Z} induces a star-product, still denoted ⋆_{Z}, on 𝔹^{ℏ}(\overline{B}^{∗}Z).

Theorem 2.

1) The map which to any a ∈ S^{0}(T^{∗}X) associates the asymptotic expansion at ℏ = 0 of (σ(ΦOp(a_{ℏ})Φ^{∗}))_{ℏ^{−1}} induces an algebra isomorphism Φ̃ from (𝔹^{ℏ}(\overline{B}^{∗}X), ⋆_{X}) onto (𝔹^{ℏ}(\overline{B}^{∗}Y), ⋆_{Y}).

2) For each k ∈ ℕ^{∗}, there exists an E-differential operator D_{k} on \overline{B}^{∗}X such that for any a ∈ C^{∞}(\overline{B}^{∗}X) which is identically zero in a neighborhood of the zero section, we have the following identity:

Φ̃(a) = (a + ∑_{k≥1} ℏ^{k}D_{k}(a)) ∘ ϕ^{−1}

in the vector space 𝔹^{ℏ}(\overline{B}^{∗}Y).

Before proving this theorem we state the next proposition which is an easy consequence of Proposition 2 and Theorem 2:

Proposition 4. The star product ⋆_{Y} on T^{∗}Y extends to a star product, still denoted ⋆_{Y}, on \overline{B}^{∗}Y such that for any f, g ∈ C^{∞}(\overline{B}^{∗}Y) we have:

f ⋆_{Y} g = fg + ∑_{n≥1} ℏ^{n}B^{(n)}(f , g)

where the B^{(n)} are ^{E}^{Y} -bidifferential operators.

Proof of Theorem 2. Let us first assume part 2). Then, using the results of Section 2.1 and the fact that ΦΦ^{∗} − Id and Φ^{∗}Φ − Id are smoothing, one proves easily that Φ̃ is an isomorphism whose inverse is given by:

b ∈ S^{0}(T^{∗}Y) → (σ(Φ^{∗}Op(b_{ℏ})Φ))_{ℏ^{−1}}.

Now let us prove part 2). Following [7] page 26, we recall that the Schwartz kernel of Φ is the finite sum of a smooth function and of oscillatory integrals (supported in small coordinates charts) of the following type:

K(y, x) = ∫_{ℝ^{n}} e^{i(φ(y,η)−x·η)} b(y, η)dη (10)

where b(y, η) ∈ S^{0}(T^{∗}Y) vanishes for ||η|| ≤ 1, φ(y, η) is an homogeneous phase function parametrizing locally the graph C_{ϕ} of ϕ^{−1} which satisfies det\frac{∂^{2}φ}{∂y∂η} ≠ 0 so that locally we have:

{(y, φ′_{y}(y, η); φ′_{η}(y, η), η)} = C_{ϕ}

and ϕ^{−1}(y, φ′_{y}(y, η)) = (φ′_{η}(y, η), η). Notice moreover that (y, η) → (y, φ′_{y}(y, η)) and (y, η) → (φ′_{η}(y, η), η) are local diffeomorphisms.

With these notations, the Schwartz kernel of Φ^{∗} is the finite sum of a smooth function and of oscillatory integrals (supported in small coordinates charts) of the following type:

K^{∗}(x, y) = ∫_{ℝ^{n}} e^{−i(φ(y,η)−x·η)} b_{1}(y, η)dη. (11)

Let a ∈ C^{∞}(\overline{B}^{∗}X) which is identically zero in a neighborhood of the zero section. Recall that our goal is to study the operator Φ∘Op(a_{ℏ})∘Φ^{∗} and its complete symbol. We shall first analyze Φ ∘ Op(a_{ℏ}), to this aim it is enough to study the operator K ∘ Op(a_{ℏ}) where K denotes the operator whose Schwartz kernel is given by (10).

The Schwartz kernel of K ∘ Op(a_{ℏ}) is given by

T (y, z) = ∫_{ℝ^{n}} ∫_{ℝ^{n}} ∫_{ℝ^{n}} e^{i(φ(y,η)−x·η)} b(y, η) a(x, ℏξ)e^{i(x−z)·ξ}dξdxdη.

In this integral we replace a(x, ℏξ) by its Taylor expansion:

∑_{α∈ℕ^{n}} \frac{1}{α!}∂_{x}^{α}a(z, ℏξ)(x − z)^{α}.

local formula for the index of fio 283

Using the following two identities

(x − z)^{α}e^{i(x−z)·ξ} = D_{ξ}^{α}(e^{i(x−z)·ξ})
∫_{ℝ^{n}} e^{ix·(ξ−η)}dx = (2π)^{n}δ_{ξ=η}

and integrating by parts we see that T (y, z) is the sum of a smooth function and of H(y, z) =

∫ ∫ ∫ e^{i(φ(y,η)−x·η)} b(y, η) ∑_{α∈ℕ^{n}} \frac{1}{α!}(−ℏ)^{|α|}∂_{x}^{α}D_{ξ}^{α}a(z, ℏξ)e^{i(x−z)·ξ}dξdxdη

= (2π)^{n} ∫_{ℝ^{n}} e^{i(φ(y,η)−z·η)} b(y, η) ∑_{α∈ℕ^{n}} \frac{1}{α!}(−ℏ)^{|α|}∂_{x}^{α}D_{ξ}^{α}a(z, ℏη)dη.

Now for α ∈ ℕ^{n} we set

c_{α}(z; ℏη) = ∂_{x}^{α}D_{ξ}^{α}a(z, ℏη)

and we consider

H_{α}(y, z) = ∫_{ℝ^{n}} e^{i(φ(y,η)−z·η)} b(y, η) c_{α}(z, ℏη)dη.

If we replace c_{α}(z, ℏη) by its Taylor expansion

∑_{β∈ℕ^{n}} \frac{1}{β!}∂_{z}^{β}c_{α}(φ′_{η}; ℏη)(z − φ′_{η})^{β}

then, using integrations by parts as above and the formula

∂_{η}^{β} (e^{i(φ(y,η)−z·η)}) = (−1)^{|β|}(z − φ′_{η})^{β}e^{i(φ(y,η)−z·η)},

it follows easily that H_{α}(y, z) is the sum of a smooth function and of

∫_{ℝ^{n}} e^{i(φ(y,η)−z·η)} ∑_{β∈ℕ^{n}} \frac{1}{β!}D_{η}^{β}( b(y, η) ∂_{z}^{β}c_{α}(φ′_{η}; ℏη) )dη.

We observe that if we apply the Leibniz rule for the term D_{η}^{β}(....) in the previous integral then the following differential operators will appear D_{η}^{β−γ}b(y, η) D_{η}^{γ−γ′}(φ′_{η})D_{η}^{γ′}∂_{z}^{β}. (12)

It is clear from Lemma 1 that, expressed in the coordinates (φ′_{η}(y, η), η), these differential operators (12) are E-differential operators. Therefore we have just proved that T (y, z) is the sum of a smooth function and of:

∫_{ℝ^{n}} e^{i(φ(y,η)−z·η)} ∑_{k∈ℕ}ℏ^{k}P_{k}(a)(φ′_{η}(y, η), ℏη)dη

where the P_{k} are E-differential operators.

Now we recall that the Schwartz kernel of Φ^{∗} is the finite sum of a smooth function and of terms of the type (11). So in order to study the complete symbol of Φ ∘ Op(a_{ℏ}) ∘ Φ^{∗} it is enough to study the operator K ∘ Op(a_{ℏ}) ∘ K^{∗} whose Schwartz kernel is the finite sum of a smooth function and of integrals of the type

∫ ∫ ∫ e^{i(φ(y,η)−x·η)}e^{−i(φ(y′,η′)−x·η′)}P_{k}(a)(φ′_{η}(y, η), ℏη)b_{1}(y′, η′)dxdη′dη

= (2π)^{n} ∫_{ℝ^{n}} e^{i(φ(y,η)−φ(y′,η)}P_{k}(a)(φ′_{η}(y, η), ℏη)b_{1}(y′, η)dη. (13)

Moreover we can write φ(y, η) − φ(y′, η) = (y − y′).η̂(y, y′, η) where η̂(y, y, η) = φ′_{y}(y, η) and we can assume (at the expense of shrinking the local coordinates charts) that η → η̂(y, y′, η) is a local diffeomorphism whose inverse is denoted η̂ → η(y, y′, η̂). With these notations, we set:

A_{k}(y, y′, ℏ, η̂) = P_{k}(a)(φ′_{η}(y, η) , ℏη)b_{1}(y′, η).

Then a change of variable formula allows us to see that the oscillatory integral (13) is equal to

(2π)^{n} ∫_{ℝ^{n}} e^{i(y−y′)·η̂} A_{k}(y, y′, ℏ, η̂) |\frac{Dη}{Dη̂}| dη̂.

We observe that, expressed in the coordinates (φ′_{η}(y, η), η), the vector fields ∂_{η}(φ′_{η})∂_{y} are E-differential operators. Therefore one proves easily the assertion of part 2) of the Theorem by replacing A_{k}(y, y′, ℏ, η̂) by its Taylor expansion

∑_{β∈ℕ^{n}} \frac{1}{β!}∂_{y′}^{β}A_{k}(y, y, ℏ, η̂)_{|y=y′} (y′ − y)^{β}

and using, as before, integration by parts. q.e.d.

5. The formal deformation and traces on \overline{B}^{∗}X ∪_{ϕ} \overline{B}^{∗}Y and regularized traces on ΨDO’s

Recall first that C^{∞}(M) is exactly the set of functions (f, g) ∈ C^{∞}(\overline{B}^{∗}X) × C^{∞}(\overline{B}^{∗}Y) such that f − g ∘ ϕ′ vanish of infinite order to the boundary of \overline{B}^{∗}X.

We are going to use the ⋆-products denoted ⋆_{X}, ⋆_{Y} on \overline{B}^{∗}X and \overline{B}^{∗}Y defined in Propositions 2 and 4. We set 𝔸^{ℏ}(\overline{B}^{∗}X) = C^{∞}(\overline{B}^{∗}X)⊗_{ℂ}ℂ[[ℏ]] and 𝔸^{ℏ}(\overline{B}^{∗}Y) = C^{∞}(\overline{B}^{∗}Y) ⊗_{ℂ} ℂ[[ℏ]].

Let 𝔸^{ℏ}(M) be the vector space given by

\{(a, b) ∈ 𝔸^{ℏ}(\overline{B}^{∗}X) × 𝔸^{ℏ}(\overline{B}^{∗}Y) | Φ̃(\overline{a}) = \overline{b}\}

where \overline{a} (resp. \overline{b}) denotes an element of 𝔹^{ℏ}(\overline{B}^{∗}X) (resp. 𝔹^{ℏ}(\overline{B}^{∗}Y)) induced by a (resp. b). Theorem 2 shows that 𝔸^{ℏ}(M) is an algebra with respect to the diagonal product: (⋆_{X}, ⋆_{Y}). In particular, pairs of the form (σ(Φ^{∗}Φ), σ(ΦΦ^{∗})) belong to 𝔸^{ℏ}(M).

In the statement of the next proposition we will use the notations of Theorem 2.

Proposition 5.

1) Let χ ∈ C^{∞}(T^{∗}X, [0, 1]) be such that χ(x, ξ) = 0 for ||ξ|| ≤ 1/2 and χ(x, ξ) = 1 for ||ξ|| ≥ 1. For any f ∈ C^{∞}(\overline{B}^{∗}X) we have

Φ̃(χf) − (χf) ∘ ϕ^{−1} ∈ ℏ𝔹^{ℏ}(\overline{B}^{∗}Y).

2) For each b ∈ C^{∞}(\overline{B}^{∗}Y) one defines b^{−} ∈ C^{∞}(\overline{B}^{∗}Y) by setting b^{−}(η) = b(−η) for any η ∈ \overline{B}^{∗}Y. Then the formula

∀(a, b) ∈ 𝔸^{ℏ}(M), U(a, b)

= (a + ∑_{k≥1} ℏ^{k} D_{k}(a), b^{−}) ∈ C^{∞}(M) ⊗_{ℂ} ℂ[[ℏ]]

defines a ℂ[[ℏ]]-linear isomorphism U from 𝔸^{ℏ}(M) to C^{∞}(M) ⊗_{ℂ} ℂ[[ℏ]].

3) The product U(∗_{X}, ∗_{Y}) defines an E-deformation of M (or a star product) associated to the symplectic Lie algebroid (E, [,], ω) (see [10] Section 3.3).

Proof. Part 1) is a consequence of the theorem of Egoroff (see [7]). Part 2) is an easy consequence of Theorem 2. Part 3) is an easy consequence of 2) and of Theorem 2 (2). q.e.d.

Definition 3.

• 𝔸^{ℏ}(M) denotes the formal deformation of M associated to the symplectic Lie algebroid (E, [,], ω) constructed in Proposition 5.

• The linear functional

τ_{can} : 𝔸^{ℏ}(M) → ℂ[ℏ^{−1}, ℏ]]

is given by

∀(a, b) ∈ 𝔸^{ℏ}(M), (14)

τ_{can}(a, b) = \{ \begin{matrix}asymptotic expansion at ℏ = 0 of \\ ℏ ↦ τ(Op(a_{ℏ}), Op(b_{ℏ})) \end{matrix} \}

where τ is the trace defined in Proposition 3. It follows immediately from the definition that τ_{can} is a trace.

Computation of τ.

Since the space of traces on 𝔸^{ℏ}(M) may be very big we introduce the following algebra:

𝔻^{ℏ}(M) = 𝔸^{ℏ}(M) [( χ ||ξ||, (χ ||ξ|| + ∑_{k≥1} ℏ^{k}D_{k}(χ ||ξ||)) ∘ ϕ^{−1})] .

Another way of describing 𝔻^{ℏ}(M) is given by glueing from

ϕ̃: 𝔸^{ℏ}(\overline{B}^{∗}X)[χ||ξ||] → 𝔸^{ℏ}(\overline{B}^{∗}Y)[χ||ξ||]

where χ is as in Proposition 5. It is easily seen that τ_{can} defines, by the same formula as (14), a trace on 𝔻^{ℏ}(M).

Next proposition describes the space of traces on 𝔻^{ℏ}(M).

Proposition 6 The space of traces with values in ℂ[ℏ^{−1}, ℏ]] on the algebra 𝔻^{ℏ}(M) is two dimensional over ℂ[ℏ^{−1}, ℏ]]. A basis is given by (τ_{can}, τ_{1}) where for any (a, b) ∈ 𝔻^{ℏ}(M) τ_{1}(a, b) = Res_{W} (a). Here Res_{W} denotes Wodzicki’s noncommutative residue.

Proof. For Z = X or Y we set

C_{0}^{∞}(\overline{B}^{∗}Z) ⊗_{ℂ} ℂ[[ℏ]] = 𝔸_{0}^{ℏ}(T^{∗}Z)

where C_{0}^{∞}(\overline{B}^{∗}Z) denotes the set of smooth functions which vanish of infinite order at \overline{B}^{∗}Z \ T^{∗}Z. Then we have the following exact sequence of ℂ[[ℏ]]-algebras:

0 → 𝔸_{0}^{ℏ}(T^{∗}X) ⊕ 𝔸_{0}^{ℏ}(T^{∗}Y) → 𝔻^{ℏ}(M) → T (M) → 0.

Here T (M) denotes the induced formal E-deformation of the algebra of transversal Laurent series with coefficients given by smooth functions on the sphere at infinity. A more direct construction of this deformation may be described as follows. Let P^{i} denote the space of pseudodifferential operators on, say X, of order ≤ i modulo the smoothing operators. The space SP of doubly infinite sequences

{P_{i}}_{i∈ℤ}, P_{i} ∈ P^{i}, and there exists i_{0} such that P_{i} ∈ P^{i_{0}} for i large

is a flat module over ℂ[[ℏ]], where the multiplication by ℏ acts as the right translation. We endow SP with the product

{P_{i}}_{i∈ℤ}{Q_{i}}_{i∈ℤ} = { ∑_{i+j=n} P_{i}Q_{j}}_{n∈ℤ} .

Note that, given the particular sequences {P_{i}}_{i} and {Q_{i}}_{i}, there exists N such that all the pseudodifferential operators involved have order bounded by N, hence the doubly infinite sums ∑_{i+j=n} P_{i}Q_{j} make sense. In fact, given n, the top order of operators appearing as products P_{i}Q_{n−i} are at most 2N, and given any integer k ≤ 2N, there are at most finitely many terms in the sum of the order k.

The space SP is immediately seen to be isomorphic to T (M). Any trace τ on T (M) is given by a sequence of ℂ-linear, ℂ[[ℏ]]-valued functionals τ_{n} on P^{n} such that

τ({P_{i}}) = ∑ τ_{i}(P_{i}).

The ℏ-linearity of τ implies that τ_{i+1} = ℏτ_{i} and the trace condition on τ implies that each τ_{i} is a trace on the algebra of pseudodifferential operators modulo the smoothing operators. Recall that, on this latter algebra, the Wodzicki residue res is the unique trace up to multiplicative constant. Thus τ is, up to multiplicative constant, uniquely determined by τ_{−n} = res, and hence the space of traces on T (M) is one-dimensional.

We recall that 𝔸_{0}^{ℏ}(T^{∗}X) is H-unital (in the sense of Wodzicki, see [17]), so we have the following long exact sequence in cyclic cohomology:

0 → HC^{0}(T (M)) → HC^{0}(𝔻^{ℏ}(M)) → HC^{0}( 𝔸_{0}^{ℏ}(T^{∗}X) ⊕ 𝔸_{0}^{ℏ}(T^{∗}Y) ) →

HC^{1}(T (M)) → · · · .

From Section 2.1 we recall that the space of ℂ[[ℏ]]-linear traces (with values in ℂ[ℏ^{−1}, ℏ]]) on 𝔸_{0}^{ℏ}(T^{∗}X) is one dimensional and generated by Tr_{can}^{X}. By above, HC^{0}(T (M)) is one-dimensional. The connecting map

δ : HC^{0}( 𝔸_{0}^{ℏ}(T^{∗}X) ⊕ 𝔸_{0}^{ℏ}(T^{∗}Y) ) → HC^{1}(T (M))

is given by taking a trace on 𝔸_{0}^{ℏ}(T^{∗}X) ⊕ 𝔸_{0}^{ℏ}(T^{∗}Y), extending it to a linear functional on 𝔻^{ℏ}(M) and taking its Hochschild boundary. In particular, it is not zero because V. Nistor has shown in [13] that the nonvanishing of this Hochschild boundary is equivalent to the existence of a pseudodifferential operator with nonzero index!. This implies that HC^{0}(𝔻^{ℏ}(M)) is either one or two dimensional. Since, with the notations of the Proposition, τ_{can}, τ_{1} are two linearly independent elements of the vector space of traces on 𝔻^{ℏ}(M), the rest of the statement of Proposition 6 follows. q.e.d.

6. The algebraic index theorem for the Lie algebroid E

The following algebraic index theorem is proved in [10] (see Theorem 6.1) and is an extension to the symplectic Lie algebroid (E, [,], ω) of the Riemann Roch theorem (on symplectic manifolds) for periodic cochains of [1], [9]. We recall that the set of E-differential forms ^{E}Ω^{∗} is endowed with a de Rham differential ^{E}d (see Definition 2.2 in [10]) so that there is an associated de Rham cohomology ^{E}H^{∗}(M).

Theorem 3. The following diagram is commutative:

\xymatrix{CC_{∗}^{per}(𝔸^{ℏ}(M)) \ar[r]^{σ} \ar@{-->}[dr]_{μ^{ℏ}} & CC_{∗}^{per}(C^{∞}(M)) \ar[d]^{μ∪Â(M)∪e^{θ}} \\  & (^{E}Ω^{2n−∗}(M)[ℏ^{−1}, ℏ]], d)}

where σ is the specialization map at ℏ = 0, μ is the transposed of the map constructed by A. Connes (see Lemma 45 of [3]) in order to identify the continuous Hochschild cohomology groups of C^{∞}(M) with the space of de Rham currents, μ^{ℏ} is the trace density map defined in Section 6 of [10] and θ = \frac{1}{\sqrt{−1}ℏ}ω + ∑_{k≥0} ℏ^{k}θ_{k} ∈ ^{E}H^{2}(M, ℂ[[ℏ]]) is the characteristic class of the deformation of the symplectic Lie algebroid (E, [,], ω) ([10]).

Remark 4.

1) The first index theorem of algebraic index type has been proved by Fedosov (see [5] page 189) for a symplectic manifold N (and Lie algebroid T N), it computes the index as an element of ℂ[ℏ^{−1}, ℏ]]. Theorem 3 is a far reaching generalization of Fedosov’s theorem since it involves higher cyclic cocycles. In fact, the next proposition and theorem implicitly show how Theorem 3 implies Fedosov’s index theorem.

2) In the case of a regular affine algebra A, the analogue of the above map μ had been constructed by Hochschild-Kostant-Rosenberg (see [6]) in order to identify the Hochschild homology groups of A with spaces of differential forms.

The natural injection 𝔸^{ℏ}(M) → 𝔻^{ℏ}(M) induces a natural map

CC_{∗}^{per}(𝔸^{ℏ}(M)) → CC_{∗}^{per}(𝔻^{ℏ}(M)).

Since the traces τ_{can} and τ_{1} of Proposition 6, and the trace density map μ^{ℏ} extend to CC_{∗}^{per}(𝔻^{ℏ}(M)), they can be identified using the following result.

Proposition 7.

1) The ℂ-vector space ^{E}H^{2n}(M, ℂ) is two-dimensional. The vector space of ℂ-linear forms on ^{E}Ω^{2n}(M) which vanish on the range of the E-exterior derivative ^{E}d admits a unique linear basis (^{reg}∫ , ∫_{1}), characterized by the following properties: For any (α, β) ∈ ^{E}Ω^{2n}(M) such that α (resp. β) is zero in a neighborhood of the boundary of \overline{B}^{∗}X (resp. \overline{B}^{∗}Y)

reg∫ (α, β) = ∫_{T^{∗}X} α − ∫_{T^{∗}Y} β, ∫_{1}(α, β) = 0.

Moreover ∫_{1} ∘μ^{ℏ} = Res_{W}.

2) There exists a constant C such that

τ_{can} = ^{reg}∫ ∘ μ^{ℏ} + C ∫_{1} ∘μ^{ℏ}.

Moreover ∫_{1} ∘μ^{ℏ}(1, 1) = 0 and for any (a, b) ∈ 𝔻^{ℏ}(M) such that a is zero in a neighborhood of the boundary of \overline{B}^{∗}X, ∫_{1} ∘μ^{ℏ}(a, b) = 0.

Proof.

1) A standard Mayer-Vietoris sequence argument shows that EH^{2n}(M, ℂ) is indeed two dimensional. The fact that (^{reg}∫ , ∫_{1}) defines a basis is left to the reader.

2) This is an easy consequence from part 1) and of the properties (see [9], [10] ) of the trace density map μ^{ℏ}. To begin with, ^{reg}∫ ∘ μ^{ℏ} and τ_{can} coincide on 𝔸_{0}^{ℏ}(T^{∗}X), while res_{W} vanishes there. In particular reg∫ ∘ μ^{ℏ} is not proportional to res_{W} and, since the space of traces on 𝔻^{ℏ}(M) is two dimensional,

τ_{can} = ^{reg}∫ ∘ μ^{ℏ} + C ∫_{1} ∘μ^{ℏ}

holds. Since res_{W} (f) depends only on the jet of f at the cosphere at infinity, and here it is given by the Wodzicki residue of the corresponding pseudodifferential operator, the rest of the statement follows immediately. q.e.d.

7. Local formula for the index of a Fourier integral operator

Theorem 4.

• Let Φ be a Fourier integral operator and 𝔸^{ℏ}(M) the formal deformation of M associated to it as in Definition 3. Then

ind Φ = ∫_{M} e^{θ_{0}} Â(M),

where θ_{0} denotes the characteristic class of the deformation of the Lie Algebroid (E, [ , ], ω) given by 𝔸^{ℏ}(M).

• Let ∇_{X} be a connection ∇_{X} on the tangent bundle T (\overline{B}^{∗}X) and Â(T^{∗}X) an associated representative form of the Â-class of ∇_{X}. The symplectomorphism ϕ induces a connection ϕ_{∗}(∇_{X}) on the tangent space of \overline{B}^{∗}Y \ B^{∗}(Y). Let ∇_{Y} denote its extension to a connection of T (\overline{B}^{∗}Y) and Â(T^{∗}Y) an associated representative differential form of the Â-class of ∇_{Y}. Then

ind Φ = ∫_{B^{∗}(X)} e^{θ_{0}} Â(T^{∗}X) − ∫_{B^{∗}(Y)} e^{θ_{0}} Â(T^{∗}Y).

Proof. Proposition 3 shows that

ind Φ = τ(Id, Id) = τ_{can}(1, 1).

Proposition 7 (2) shows that

τ_{can}(1, 1) = ^{reg}∫ ∘ μ^{ℏ}(1, 1).

The result follows then from Theorem 3 by letting ℏ → 0^{+}, provided that the involved characteristic classes of vector bundles on M are in fact standard de Rham cohomology classes. We will prove this below and see that the regularized integral coincides with the orientation class of M. q.e.d.

The previous formula shows that if ϕ extends as a symplectomorphism T^{∗}X → T^{∗}Y up to the zero section then ind Φ = 0.

For a deformation associated with a Fourier integral operator (as in Proposition 5) the characteristic class θ of Theorem 3 is in fact of the form

θ = \frac{1}{\sqrt{−1}ℏ}ω + θ_{0}

where θ_{0} ∈ H^{2}(M, ℂ) is a closed differential form (not only an E-differ-ential form). In order to do this and to identify the relevant characteristic class we will give below a slightly nonstandard description of a formal deformation.

7.1 General construction of the characteristic class of a formal deformation

Let us start with some notation.

Let 𝔸^{ℏ} denote the Weyl algebra of the symplectic vector space ℝ^{2n} with the standard symplectic structure, i.e., the algebra generated by the vectors x̂_{l}, ξ̂_{l} (1 ≤ l ≤ n) satisfying the relations [ξ̂_{k}, x̂_{l}] = \sqrt{−1}ℏδ_{k,l}. The algebra 𝔸^{ℏ} is completed in the topology associated to the ideal generated by {x̂_{l}, ξ̂_{l}, ℏ; 1 ≤ l ≤ n} and has the grading induced by

deg x̂_{l} = deg ξ̂_{l} = 1, deg ℏ = 2.

The corresponding Lie algebra \frac{1}{ℏ}𝔸^{ℏ} will be denoted by 𝔤̃. We set

𝔤 = Der (𝔸^{ℏ}) = 𝔤̃/center

and

G = Aut(𝔸^{ℏ}) = exp(𝔤_{≥0}).

We set

G̃ = \{g ∈ \frac{1}{ℏ}𝔸^{ℏ} | g ∈ 𝔰p(2n, ℝ) mod 𝔤_{≥1}\}

and will endow it with the group structure coming from the exponential map. Note that G̃ is an extension of G associated to the (Lie algebra) central extension 𝔤̃ of 𝔤.

We endow the bundle ℝ^{2n} × 𝔸^{ℏ} with the obvious fiber-wise action of G̃ and with the g̃-valued (Fedosov) connection

˜∇^{0} = ∑_{l=1}^{n} (dξ_{l} (∂_{ξ_{l}} − \frac{1}{\sqrt{−1}ℏ}x̂_{l}) + dx_{l} (∂_{x_{l}} + \frac{1}{\sqrt{−1}ℏ}ξ̂_{l}))

where (dx_{1}, …, dx_{n}; dξ_{1}, …, dξ_{n}) denote the local (usual) dual basis of T^{∗}ℝ^{n}. Let us recall (see Section 2.2) that a local chart of \overline{B}^{∗}ℝ^{n} near \overline{B}^{∗}ℝ^{n} \ T^{∗}ℝ^{n} is given by:

(x_{1}, …, x_{n}; t = \frac{1}{||ξ||}, θ = (θ_{1}, … θ_{n−1})) t ≥ 0, θ ∈ S^{n−1}. (15)

By using the local coordinates (15) one checks easily that ˜∇^{0} extends as an E^{ℝ^{n}}-connection, still denoted ˜∇^{0}, of \overline{B}^{∗}ℝ^{n} × 𝔸^{ℏ}.

The description given below of a formal deformation of a symplectic Lie algebroid structure on M is just the representation of the Fedosov construction in terms of the bundle of jets on M with the fiber-wise product structure induced by the *-product (which is isomorphic to Weyl bundle) .

Local description of the characteristic class θ of a formal deformation.

The deformation is described by a local (Darboux) cover {U_{i}}_{i∈I} of (M, ω), a collection of functions {g_{i,j} : U_{i} ∩ U_{j} → G̃} and a collection of 𝔤̃-valued E-connections ˜∇_{i} on U_{i} ×𝔸^{ℏ} which, when expressed in terms of local Darboux coordinates (x_{1}, …, x_{n}, ξ_{1}, …, ξ_{n}) (resp (15)) if U_{i} does not meet (resp. meets) the boundary at infinity, are equal to ˜∇^{0} modulo 𝔤̃_{≥1} and so that the three following conditions hold:

1) The cocycle condition holds:

g_{i,j}g_{j,i} = 1 and g_{i,j}g_{j,k} = g_{i,k} on U_{i} ∩ U_{j} ∩ U_{k}.

In particular {g_{i,j} : U_{i} ∩ U_{j} → G̃} define a smooth bundle W of algebras over M with fiber isomorphic to 𝔸^{ℏ} and the structure group G̃.

2) The local connections ˜∇_{i} define a 𝔤̃-valued connection ˜∇ on the

bundle W, i.e.,

g_{i,j} ˜∇_{j} = ˜∇_{i}g_{i,j}.

3) The induced 𝔤-valued connection ∇ on the bundle W is flat, i.e., θ = ˜∇^{2} is a globally defined differential form on M with values in the center of 𝔤̃, necessarily of the form

\frac{1}{\sqrt{−1}ℏ}ω + θ_{0} where θ_{0} ∈ Ω^{2}(M, ℂ[[ℏ]]).

The algebra of ∇-flat sections of W is a formal deformation of (M, ω) whose characteristic class is θ.

7.2 Local canonical liftings

We endow ℝ^{2n} with its canonical symplectic structure ω = ∑_{l=1}^{n} dξ_{l} ∧ dx_{l}. Given any smooth, ℂ[[ℏ]]-valued function H on ℝ^{2n}, we define the following sections of the bundle ℝ^{2n} × 𝔸^{ℏ} (over ℝ^{2n}):

(1) H_{0} = H(x, ξ)|_{ℏ=0}, H_{1} = ∑_{l=1}^{n}(x̂_{l}∂_{x_{l}}H_{0} + ξ̂_{l}∂_{ξ_{l}}H_{0}).

(2) H̃ = ∑ \frac{x̂^{α}ξ̂^{β}}{α!β!} ∂_{x}^{α}∂_{ξ}^{β}H.

Using those, we can associate to H the 𝔤̃-lift of the Lie derivative L_{{H, }} given by

D_{H} = L_{\{H, \}} + \frac{1}{ℏ} (H̃ − H_{0} − H_{1} + \frac{1}{2}ℏ ∑_{l=1}^{n} ∂_{x_{l},ξ_{l}}^{2}H_{0}) .

We can think of it as an element of the Lie algebra of the semidirect product of C^{∞}(ℝ^{2n}, G̃) by the pseudogroup of local diffeomorphisms of ℝ^{2n}. The D_{H}’s form a Lie algebra, in fact

[D_{H}, D_{K}] = D_{\frac{1}{ℏ}(H∗K−K∗H)}, (16) and they satisfy

[D_{H}, ˜∇^{0}] = −\frac{1}{2}d (∑_{l=1}^{n} ∂_{x_{l},ξ_{l}}^{2}H_{0}) . (17)

We will also have an occasion to use

D_{H}^{0} = L_{\{H, \}} + \frac{1}{ℏ}( H̃ − H_{0} − H_{1} ), (18)

which satisfies

[ ˜∇^{0}, D_{H}^{0}] = 0. (19)

The identities (16), (17) and (19) follow from the definition of the quantities involved (and a completely straighforward computation).

7.3 The cotangent bundle case

The deformation of T^{∗}X associated to the sheaf of differential operators on X can be now described as follows.

Locally on a coordinate domain U ⊂ X we use coordinates on U to give an explicit symplectomorphism

T^{∗}U → ℝ^{2n}

and use Weyl deformation of ℝ^{2n} to construct the deformation of T^{∗}U. This amounts to the choice of a (𝔤̃-valued) connection given in our local coordinates (x_{1}, …, x_{n}) on U and the induced local coordinates (x_{i}, ξ_{i})_{i=1,...n} on T^{∗}U ≃ U × ℝ^{n} by

˜∇^{0} = d + \frac{1}{ℏ} ∑_{l=1}^{n}(dx_{l}ξ̂_{l} − dξ_{l}x̂_{l}).

The infinitesimal change of coordinates on U is given by a vector field of the form ∑_{l=1}^{n} X_{l}∂_{x_{l}} and the associated infinitesimal symplectomorphism of T^{∗}U is given by the Hamiltonian vector field {∑_{l=1}^{n} X_{l}ξ_{l}, ·}.

It is immediate to see that the map

∑_{l} X_{l}∂_{x_{l}} ↦ D_{∑_{l} X_{l}ξ_{l}}

is the Lie algebra homomorphism.

The associated local diffeomorphisms (coordinate changes) exp ∑_{l} X_{l}∂_{x_{l}} lift to a local isomorphisms of the bundle T^{∗}U × 𝔸^{ℏ} given by exp D_{∑_{l} X_{l}ξ_{l}}.

Given a local coordinate cover {U_{i}}_{i∈I} of X it is now immediate to construct the associated G̃-valued cocycle {g_{ij}} glueing the bundles together. Note that, since D’s do not commute with the connection ˜∇^{0}, the corresponding collection of connections

˜∇_{i} = ˜∇^{0} in i’th coordinate system on T^{∗}U_{i}

do not glue together. But it is not difficult to check that (see for instance [5] page 260)

g_{ij} ˜∇_{i}g_{ji} = ˜∇_{j} + \frac{1}{2}d log det Dg_{ij},

where Dg_{ij} is the induced action of g_{ij} on the tangent bundle. The cocycle

\frac{1}{2}d log det Dg_{ij} (20)

is in Č^{1}(T^{∗}X, Ω^{1}(T^{∗}X)) and represents the pull back to T^{∗}X of one half of the first Chern class of the complexified tangent bundle of X over X. Since this class vanishes (with coefficients in ℝ), we can find a collection α_{i} of closed one-forms on T^{∗}X satisfying

\frac{1}{2}d log det Dg_{ij} = α_{i} − α_{j}.

Setting

∇_{i} = ˜∇_{i} + α_{i},

we get a globally defined 𝔤̃-connection ∇ whose curvature, and hence also the characteristic class of the associated deformation, is represented in the complex Č^{∗}(T^{∗}X, Ω^{1}(T^{∗}X)) by the cocycle \frac{1}{2}d log det Dg_{ij}.

It is immediate that the deformation constructed in this way coincides with the one associated to the calculus of differential operators on X, while its jet at ξ = ∞ gives the deformation associated to the calculus of pseudodifferential operators on X.

If we recall that the real symplectic vector bundle E|_{T^{∗}X} (see Definition 2) is the realification of a restriction of a complex vector bundle E_{ℂ} to T^{∗}X, the characteristic class of this deformation can be described as the jet of \frac{1}{2}c_{1}(E_{ℂ}|_{T^{∗}X}) at ξ = ∞ and represented by the explicit cochain given by the formula (20).

7.4 The Lie algebroid case

Recall now that the Lie algebroid (on M) (E, [ , ], ω) is given by glueing (at infinity) the two cotangent bundles (T^{∗}X, ω^{X}) and (T^{∗}Y, ω^{Y}) by the symplectomorphism ϕ′. To construct the deformation in this case, we will use the following data, whose existence follows immediately from the compactness of the co-sphere bundles of X and Y:

(1) A local coordinate cover {U_{i}}_{i∈I} of X and an open relatively compact neighborhood U_{X} of the zero section in T^{∗}X.

(2) A local coordinate cover {V_{i}}_{i∈I} of Y and an open relatively compact neighborhood U_{Y} of the zero section in T^{∗}Y.

(3) For each i ∈ I we can choose local coordinates on U_{i} and V_{i}, so that both T^{∗}U_{i} and T^{∗}V_{i} become identified with open subsets of (the same) copy of ℝ^{2n} = T^{∗}ℝ^{n} with its standard symplectic structure and ϕ_{i} = ϕ|_{T^{∗}U_{i}\U_{X}} is the induced symplectomorphism between two open subsets of ℝ^{2n}.

(4) For each i ∈ I there exists a smooth real-valued function H_{i} on ℝ^{2n} × [0, 1], which is one-homogeneous in the cotangent variables and such that ϕ_{i} can be obtained by integrating the (time dependent) hamiltonian flow L_{H_{i}}.

The existence of such a function H_{i} follows from the fact that any symplectomorphism between two open subsets of a connected symplectic manifold is locally given as ϕ_{1} for a 1-parameter Hamiltonian flow ϕ_{t}, t ∈ [0, 1].

Using the above data, we can construct cocycles

T^{∗}U_{i} ∩ T^{∗}U_{j} ∋ p → g_{ij}(p) ∈ C^{∞}(T^{∗}U_{i} ∩ T^{∗}U_{j}, G̃)

and

T^{∗}V_{i} ∩ T^{∗}V_{j} ∋ p → h_{ij}(p) ∈ C^{∞}(T^{∗}V_{i} ∩ T^{∗}V_{j}, G̃)

intertwining the flat connections ˜∇_{i}^{0} up to the term \frac{1}{2}d log det Dg_{ij} (\frac{1}{2}d log det Dh_{ij} respectively) as in the cotangent bundle case.

Let Ψ_{i} denote the lifting of ϕ_{i} given by:

exp D_{H_{i}}^{0} : Γ(T^{∗}U_{i} \ U_{X}, 𝔸^{ℏ}) → Γ(T^{∗}V_{i} \ U_{Y} , 𝔸^{ℏ}).

We can and will view Ψ_{i} as local isomorphisms of jets at infinity of the G̃-bundles on compactified cotangent bundles of X and Y constructed from the cocycles g_{i,j} and h_{i,j}.

While both g_{i,j}’s and h_{i,j}’s do satisfy the cocycle conditions on T^{∗}(X) (T^{∗}(Y) respectively), however

λ_{ij} = Ψ_{j}^{−1}h_{ji}Ψ_{i}g_{ij} ≠ 1

and hence we do not yet have the data necessary to construct the bundle W over M.

The following facts are easy corollaries of the construction:

(1) λ_{ij} = 1 mod G̃_{≥1}.

(2) λ_{ij} ˜∇_{j}^{0}λ_{j,i} = ˜∇_{i}^{0} − \frac{1}{2}d log det Dg_{ij} − \frac{1}{2}d log det Dh_{ij}.

(3) λ_{ij} form a two-cocycle with values in G̃.

To begin with, recall that both \frac{1}{2}d log det Dg_{ij} and \frac{1}{2}d log det Dh_{ij} as cohomology classes on T^{∗}X \ X and T^{∗}Y \ Y represent (under our symplectomorphism) the same cohomology class, to wit half of the first Chern class of the tangent bundle with the complex structure induced by the symplectic form. Since these vanish, we can find a zero-Čech cochain τ_{i} with the coefficient in the sheaf of smooth nowhere vanishing functions and such that τ_{i}λ_{ij}τ_{j}^{−1} intertwines the (local) flat connections ˜∇_{i}^{0} and ˜∇_{j}^{0}.

In particular, τ_{i}λ_{ij}τ_{j}^{−1} are given by exponentials of jets of ˜∇_{i}^{0}-flat sections of the bundle T^{∗}U_{i} × 𝔸^{ℏ} and, using a partition of unity, they can be written in the form τ_{i}λ_{ij}τ_{j}^{−1} = λ_{i}λ_{j}^{−1}, where λ_{i} is a jet of a flat section of the Weyl bundle supported on T^{∗}U_{i} \ U_{X}. We now define an operator Ψ, acting on the set of sections of the Weyl bundle W, by setting for each i ∈ I:

Ψ_{|T^{∗}U_{i}\U_{X}} = Ψ_{i}τ_{i}M_{λ_{i}}.

Here M_{λ_{i}} stands for the operator of multiplication with the flat section λ_{i}. It is easy to see that Ψ descends to an isomorphism of jets at the sphere at infinity of the deformations of cotangent bundles constructed above so that

∀a ∈ C^{∞}(\overline{B}^{∗}X), Ψ(a) = ( a + ∑_{k≥1} ℏ^{k}D̂_{k}(a) ) ∘ ϕ^{−1}

holds in 𝔹^{ℏ}(\overline{B}^{∗}Y), where the D̂_{k} are E-differential operators. Hence, as in Proposition 5, Ψ induces a deformation of the Lie algebroid (E, [ , ], ω).

7.5 The characteristic class of 𝔸^{ℏ}(M)

The characteristic class of the deformation constructed above can now be easily obtained as follows. The collection g_{ij}, h_{ij}, and the jet at infinity of Ψ_{i}M_{λ_{i}} give a cocycle with values in G̃, and it commutes with local flat connections up to the Cech cocycle given by the collection of differential forms

(21) \frac{1}{2}d log det Dg_{ij}, d log τ_{i}, \frac{1}{2}d log det Dh_{ij}.

As in the case of cotangent bundle, we can correct local connections by a scalar term and the characteristic class θ_{0} of the deformation is given by (21) as a cochain in Č^{1}(M, Ω^{1}(T^{∗}M)). Moreover, in the case that both X and Y admit metalinear structures, the collection {τ_{i}}_{i∈I} can be thought of as glueing of the pulled back)of the half-top form bundles of X and Y along the graph of the symplectomorphism into a line bundle L over M and, in this case,

θ_{0} = c_{1}(L).

7.6 The Fourier integral operator

To get the Fourier integral operator we will work locally. We will dispense with the half-density bundles (trivial in any case) for the sake of simplicity of notation. We will begin by constructing, for each i, an operator on L^{2}(ℝ^{n}) as follows. Choosing local coordinates on U_{i} and V_{i}, we can assume that H_{i} (introduced at the end of Section 7.4) is a smooth function on T^{∗}ℝ^{2n} which is 1-homogeneous in the cotangent direction. The differential equation

\frac{d}{dt}T_{i}(t) = Op(\sqrt{−1}H_{i}) ∘ T_{i}(t), T_{i}(0) = 1

has a solution given by a smooth family of bounded operators. Using the fact that {τ_{i}}_{i∈I} is a Čech zero - cochain with coefficients in the sheaf of everywhere nonzero functions and proceeding as in Section 7.3.2 of [5] one checks that T_{i}(1) satisfies

Ad (T_{i}(1)Op(λ_{i}))Op(f_{ℏ}) ∼ Op(Ψ(f)_{ℏ})

mod ℏ^{∞} as ℏ → 0 whenever suppf ⊂ T^{∗}U_{i} \ X (recall that λ_{i} is introduced in Section 7.4). In other words, the deformation constructed above is associated (in the sense of Proposition 5) to the almost unitary Fourier integral operator Φ = ∑_{i∈I} T_{i}(1)Op(λ_{i}) whose canonical relation is C_{ϕ}. Moreover, the index of this operator Φ_{0} is given by

∫_{M} Â(M)e^{θ_{0}}.

Remark 5. The result above depends on the choice of the τ_{i}’s which in turn determine the homotopy class of the symbol of the Fourier integral operator. Note however that in the case of scalar coefficients and the dimension of the underlying manifold greater then two this class is unique, since any invertible function on the cosphere bundle is homotopic to the constant one.

Moreover, since the characteristic classes involved are given by differential forms associated to connections on a vector bundle over M and Ω ⊂ ^{E}Ω, the E-classes involved in the index formulas are in fact identical with corresponding standard characteristic classes.

Let us recall that the real vector bundle E ≃ T M is given by realification of a complex vector bundle E_{ℂ} on M (the almost complex structure coming from the symplectic vector bundle structure on E). Moreover, it is easy to see, that there exists a choice of the τ_{i}’s such that the associated characteristic class of the deformation coincides with \frac{1}{2}c_{1}(E_{ℂ}). This gives the following result (compare with [4] and [15]):

Theorem 5. There exists an almost unitary Fourier integral operator Φ_{0} (as in Section 3) whose canonical relation is C_{ϕ} and such that:

ind Φ_{0} = ∫_{M} Â(M)e^{\frac{1}{2} c_{1}(E_{ℂ})}.

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