HOMOLOGY AND RESIDUES OF ADIABATIC PSEUDODIFFERENTIAL OPERATORS

SERGIU MOROIANU

Abstract. We compute the Hochschild homology groups of the adiabatic algebra Ψ_{a}(X), a deformation of the algebra of pseudodifferential operators Ψ(X) when X is the total space of a fibration of closed manifolds. We deduce the existence and uniqueness of traces on Ψ_{a}(X) and some of its ideals and quotients, in the spirit of the noncommutative residue of Wodzicki and Guillemin. We introduce certain higher homological versions of the residue trace. When the base of the fibration is S^{1}, these functionals are related to the η function of Atiyah-Patodi-Singer.

1. Introduction

The interplay between analytic properties of manifolds and algebras associated to them is a fruitful topic of current research. Indeed, appropriate algebraic constructions lead to various analytical objects like de Rham cohomology [7], the index of elliptic operators, [17], [11], the residue trace [17] and the η-invariant [18]. One may even speculate that analysis on manifolds could in principle be reduced to “formal” analysis in the spirit of [22].

In the present paper we study algebraic properties of the algebra Ψ_{a}(X) of adiabatic pseudodifferential operators associated to a fibration X → M of closed manifolds, in particular we compute its (topological) Hochschild homology groups. Mazzeo and Melrose [13] have used this algebra in their study of adiabatic families of Riemannian metrics, i.e., time-dependent families of metrics on the total space X such that the base directions blow up at t = 0:

g_{t} = dv^{2} + \frac{dh^{2}}{t^{2}} .

The algebra Ψ_{a}(X) can also be obtained from the methods of [24]. In particular, if X has a spin structure and M = S^{1}, it turns out that the family D_{t} of Dirac operators corresponding to the metric g_{t} is an adiabatic family of differential operators. In [20], [21] we use the adiabatic algebra and some results of this paper to study adiabatic limits of the eta functions of any elliptic adiabatic family of first order differential operators, extending the holonomy formula of Witten [27] and Bismut-Freed [3], whose proof only holds for Dirac operators.

Date: Revised on January 27, 2004.

2000 Mathematics Subject Classification. 58J42,58J20.

Partially supported by ANSTI (Romania).

The definition of the algebra Ψ_{a}(X) is lengthy (see Section 2) so let us try to give in this introduction the simple idea behind it. Let t be a parameter in ℝ_{+} =: [0, ∞), x local coordinates on the base M, y local coordinates on the fiber F. Intuitively, the algebra Ψ_{a}(X) associated to the fibration X → M is made of pseudodifferential operators of the form a(t, x, y, tD_{x}, D_{y}), where a is a classical symbol in the last two variables, smooth in the parameters t, x, y for t ≥ 0. For fixed t > 0 we recover the usual algebra of classical pseudodifferential operators on X. The adiabatic family of Dirac operators discussed above is of this form, and one hopes (as it is indeed the case) that one can reasonably invert it inside the adiabatic algebra. The main point is that D_{0} is no longer elliptic, hence classical elliptic analysis fails. Therefore, the adiabatic algebra gives a better understanding of families of (pseudo)differential operators like D_{t} in the degenerate limit t → 0.

For reasons explained below, we adjoin the inverse of t to the algebra, thus admitting Laurent expansions in t at t = 0. We will denote by Ψ_{a}^{i,j}(X) the space of adiabatic operators of order i and vanishing at least to order −j at t = 0. The second index will be omitted when it is not restricted or it is clear from the context. Thus for instance, Ψ_{a}^{−∞}(X) denotes the ideal of those adiabatic operators which are smoothing for each t > 0.

To give an outline of this work, recall first the description by Melrose and Nistor [17] of the Fredholm index map as the boundary map in Hochschild homology arising from the short exact sequence

0 → Ψ^{−∞}(X) → Ψ(X) → S(T^{∗}X) → 0,

where Ψ(X) is the algebra of classical pseudodifferential operators on X, and S(X) is defined by this sequence. More precisely, HH(Ψ^{−∞}(X)) is concentrated in dimension 0 and isomorphic to ℂ through the trace map. Given an elliptic operator A, they constructed a class in HH_{1}(S(X)) which maps through the boundary map to the index of A.

We consider the analogous short exact sequence in the adiabatic setting

(1)  0 → Ψ_{a}^{−∞}(X) → Ψ_{a}(X) → A_{σ}(X) → 0.

In Section 4 we compute the Hochschild homology of the total symbol algebra A_{σ} := Ψ_{a}(X)/Ψ_{a}^{−∞}(X), by using the spectral sequence argument from [5]. The principal symbols of adiabatic operators live naturally on a modified cotangent bundle ^{a}T^{∗}X → X × ℝ_{+}. There is a natural map ϕ_{a} from T^{∗}X × ℝ_{+} to ^{a}T^{∗}X which is an isomorphism for t > 0 but fails to be an isomorphism at t = 0. To compute the first differential d_{1} in the spectral sequence, we replace the symplectic duality operator ∗ of Brylinski [4] by a (non-canonical) duality ∗_{a} on differential forms on ^{a}T^{∗}X, obtained from ϕ_{a}^{−1}^{∗} ∗ ϕ_{a}^{∗} by means of a connection ∇ in X → M. A completely non-obvious result is the convergence of the spectral sequence. Theorem 1.1. The homology groups of the total symbol algebra A_{σ}(X) are

\begin{aligned}HH_{k}(A_{σ}(X)) & ≅ (H^{2N−k}(S^{∗}X × S^{1}) ⊕ H^{2N+1−k}(S^{∗}X × S^{1})) \\ & ⊗ C^{∞}(ℝ_{+})[t^{−1}]. \end{aligned}

This result is similar to that of Brylinski and Getzler [5]. Note that the theorem would fail if we did not adjoin t^{−1}, since the operator ∗_{a} involves negative powers of t. In particular, the residue trace for positive t extends to A_{σ}(X) as a trace functional singular in t at t = 0.

A remarkable difference from the case of the algebra Ψ(X) is that now the “smoothing” ideal Ψ_{a}^{−∞}(X) has homology in dimensions from 0 to n + 1. In dimension 0, the homology of Ψ_{a}^{−∞}(X) is 1dimensional as a C^{∞}(ℝ_{+})[t^{−1}]-module. In higher dimensions it is concentrated near t = 0, in other words it is inherited from the quotient I_{∂}(X) := Ψ_{a}^{−∞}(X)/Ψ_{a}^{−∞,−∞}(X). This quotient is a L-vector space, where L is the field ℂ[[t, t^{−1}] of formal Laurent series. The homology of I_{∂}(X) is computed in Sections 5 and 6 by using the spectral sequence ∂E with respect to the filtration by the powers of t. Theorem 1.2. The homology of I_{∂}(X) = Ψ_{a}^{−∞}(X)/Ψ_{a}^{−∞,−∞}(X) is

HH_{k}(I_{∂}(X)) ≅ (H^{n−k}(M, OM) ⊕ H^{n−k+1}(M, OM)) ⊗ L

where OM is the orientation bundle of M. Moreover,

HH_{k}(Ψ_{a}^{−∞}(X)) ≅ \begin{cases}HH_{k}(I_{∂}(X)) & if k ≥ 2, \\ H^{n−1}(M, OM) ⊗ L ⊕ C^{∞}(ℝ_{+})[t^{−1}] & if k = 1, \\ C^{∞}(ℝ_{+})[t^{−1}] & if k = 0, \end{cases}

and the map induced in homology by the quotient map Ψ_{a}^{−∞}(X) → I_{∂}(X) is the obvious surjection.

The first isomorphism is realized by the map ∗_{a}χ_{a}, where χ_{a} is a noncommutative analog of the Hochschild-Kostant-Rosenberg map χ defined on Hochschild chains by:

(2)  c_{0} ⊗ … ⊗ c_{k} ↦^{χ_{a}} Tr_{V}(ĉ_{0}∇^{t}ĉ_{1} ∧ … ∧ ∇^{t}ĉ_{k}).

The right-hand side will be explained in Section 6; for the moment, let us only say that Tr_{V} is the fiberwise trace applied to a family of suspended operators in the sense of Melrose [16], ∇^{t} is a ”covariant derivative” which differentiates suspended operators in horizontal directions, and the output is a (2n − k)-form on T^{∗}M whose coefficients are Schwartz along the fibers. We deduce the existence of a unique continuous trace on I_{∂}(X) (up to multiplication by L), which moreover extends the usual operator trace on Ψ_{a}^{−∞}(X) for t > 0.

The ideal Ψ_{a}^{−∞} is H-unital in the sense of [28]. This technical condition is actually fulfilled by all the ideals arising in this paper and we will not mention it further; a detailed proof of this fact is given in [19]. The interested reader should also consult [2] for more on this subject. Thus, there exists a long exact sequence of Hochschild homology groups induced from the short exact sequence (1) and in particular of boundary maps

δ : HH_{k}(A_{σ}(X)) → HH_{k−1}(Ψ_{a}^{−∞}(X)).

Section 7 contains the second type of results of this paper, based on the homological computations from the previous sections. We first describe the boundary maps δ in terms of our identifications of Hochschild homology. Theorem 1.3. The composition

HH_{k}(A_{σ}(X)) →^{δ} HH_{k−1}(Ψ_{a}^{−∞}(X)) → HH_{k−1}(I_{∂}(X))

is given in terms of the isomorphisms from Theorems 1.1 and 1.2 by integration along the fibers of S^{∗}X → M in the first factor, and by the Laurent expansion map C^{∞}(ℝ_{+})[t^{−1}] → L in the second factor.

To prove this, we introduce certain higher analogs of the residue trace, with values in Λ^{∗}(M, OM). Namely, let c = c_{0} ⊗ … ⊗ c_{k} ∈ C_{k}(Ψ_{a}(X)) and define

(3)  R(c) = Res_{z=0} ∫_{^{a}T^{∗}M_{|t=0}/M} ∗_{a}χ_{a}(Q^{z}c)

where Q is a positive adiabatic operator of order 1. We prove that R is the first component (in the filtration by the order of vanishing at t = 0) of some Hochschild cochain. This means that while we restrict the order of vanishing, we allow arbitrary operator orders in c, as in the zeta-function definition of the residue trace. Theorem 1.4. The map

R : C_{k}(Ψ_{a}(X)) → Λ^{n−k}(M, OM)

descends to C_{k}(Ψ_{a}(X)/Ψ_{a}^{−∞}(X)) and has the following properties:

(1) If c ∈ C_{k}(Ψ_{a}(X)/Ψ_{a}^{−∞}(X)) is Hochschild exact, then R(c) is de Rham exact.

(2) If c ∈ C_{k}(Ψ_{a}(X)/Ψ_{a}^{−∞}(X)) is Hochschild closed, then R(c) is de Rham closed.

Therefore R descends to Hochschild homology, with values in the de Rham cohomology of the base M. For comparison, the only such explicit cochain known in Ψ(X) is the residue trace, in dimension 0. If M = {∗} then Ψ_{a}(X) ≅ C^{∞}(ℝ_{+}, Ψ(X)), and our residues are only defined in dimensions 0 and 1. In that case both of them are variants of the residue trace.

Finally, in Section 8 we examine the action of our functionals on the Melrose-Nistor cycle tr(A ⊗ B), where A ∈ Ψ_{a}(X) is elliptic, B is a parametrix of A and tr is the generalized trace functional [12]. We recover essentially the residue of the eta function of a self-adjoint pseudodifferential operator on a closed manifold (this residue is known to vanish by the results of [1] and [8]). Thus the residues R encode highly non-trivial analytic information, and will hopefully find other interesting applications.

Notation. All symbols in this paper are classical (i.e., one-step polyhomogeneous) of possibly complex order. We denote by S_{cl}(V) the classical symbols, by 𝒮(V) the Schwartz (rapidly vanishing) functions, and by S(V) := S_{cl}(V)/𝒮(V) the formal symbols on the total space of a vector bundle V → X. The superscript i in a filtered algebra denotes objects in filtration order i, while [i] denotes homogeneous degree i in the associated graded algebra. For instance,

S^{z}(V) ≅ ∏_{j∈ℕ}S^{[z−j]}(V),

the space of formal sums of homogeneous functions of homogeneity bounded above on the complement of the zero-section.

The field ℂ[[t, t^{−1}] of formal Laurent series is denoted by L.

Acknowledgments. This paper is based on my Ph.D. dissertation [19] supervised by Professor Richard B. Melrose. I wish to thank him for teaching me about the adiabatic algebra and for his inspiring guidance. I am also indebted to Gustavo Granja, Colin Ingalls, Robert Lauter, Andrei Moroianu, Victor Nistor and Ioanid Roşu for their help, and to the referee for careful remarks.

2. Overview of the adiabatic algebra

2.1. The adiabatic algebra. The construction of the adiabatic algebra is based on the general formalism from [15]. It is a particular case of an algebra constructed in [13], and can also be defined as in [24]. For simplicity we describe it in the scalar case, noting that it extends easily to operators between sections of vector bundles.

Let X^{m+n} →^{π} M^{n} be a fibration of compact manifolds. The adiabatic limit of this geometric data means exploding the base while keeping the fibers fixed, so that the points in the base become disconnected, at least in a first approximation. From this point of view, the underlying space of the adiabatic limit as t → 0 is the product X × ℝ_{+}. The central idea in Melrose’s approach is that the limiting value t = 0, where the interesting phenomena occur, is of the same nature as the positive values of t, provided one works in the appropriate ”adiabatic category”.

Denote by T X/M the vertical sub-bundle of T X. A smooth family v : ℝ_{+} → Γ(X, T X) of vector fields is called adiabatic if v(0) ∈ Γ(X, T X/M). Thus, an adiabatic vector field is a family of vector fields on X which in the limit does not “see” the base. Mazzeo and Melrose [13] remarked that the adiabatic vector fields are the sections of a vector bundle over X × ℝ_{+}, called ^{a}T X. If X = F × M → M is a trivial fibration, then ^{a}T X is canonically isomorphic to T X × ℝ_{+}. In general, the vector bundles T X × ℝ_{+} and ^{a}T X are non-canonically isomorphic as bundles over X × ℝ_{+}. When X = M we denote by ^{a}T M the adiabatic tangent bundle of the identity fibration. Definition 2.1. Let ϕ^{a} : ^{a}T X → T X × ℝ_{+} be the tautological map of vector bundles which transforms a section of ^{a}T X into itself, viewed as a time-dependent section of T X.

Denote by ϕ_{a} : T^{∗}X × ℝ_{+} → ^{a}T^{∗}X the dual map.

Following [15], [13] we construct an algebra of pseudodifferential operators in which the adiabatic vector fields are the differential operators of order exactly 1. This algebra is a space of conormal distributions on a radial blow-up of X × X × ℝ_{+}. More precisely, let X ×_{M} X be the fiber diagonal in X × X. Let S^{+} be the positive half-sphere of the normal bundle N(X ×_{M} X × {0}) inside X^{2} × ℝ_{+}. Definition 2.2. Let

X_{a}^{2} = [X^{2} × ℝ_{+}; X ×_{M} X × {0}].

As a set, this is (X^{2} × ℝ_{+} \ X ×_{M} X × {0}) ∪ S^{+}. The front face of the blow-up, denoted ff, is just S^{+}. The set X_{a}^{2} has a natural C^{∞} structure, defined by gluing the two parts along the normal geodesic flow of some Riemannian metric. As manifold with corners, it has one boundary component of codimension 1 if n = 0, three if n = 1 and two otherwise.

There exists a natural blow-down map β : X_{a}^{2} → X^{2} × ℝ_{+}. Let Δ be the diagonal in X^{2}. We define Δ_{a}, the lifted diagonal, to be the closure in X_{a}^{2} of β^{−1}(Δ × (0, ∞)). Note that Δ_{a} is transversal to ff. Let 2X_{a}^{2} be the double of X_{a}^{2} across the front face. By definition, a distribution on X_{a}^{2}, conormal to the lifted diagonal, is smooth to the front face if it is the restriction of a distribution on 2X_{a}^{2}, conormal to 2Δ_{a}.

Let Ω_{a} → X_{a}^{2} be the pull-back via π_{R} ∘ β of the density bundle Ω = Ω^{m+n}(^{a}T^{∗}X), where π_{R} : X^{2} → X is the projection on the second factor.

The set of adiabatic differential operators (defined as compositions of adiabatic vector fields and multiplication operators) corresponds via Schwartz’s kernel theorem to the space of Ω_{a}-valued distributions on X_{a}^{2} which are supported on the lifted diagonal and extendible across the front face. In light of this fact, let Ψ_{a}^{i,0}(X) be the space of those Ω_{a}-valued distributions on X_{a}^{2} which are classical (i.e., 1-step polyhomogeneous) conormal to Δ_{a} of order i, vanish rapidly to the boundary faces of X_{a}^{2} other than ff, and are extendible across ff. This space forms a module over C^{∞}(ℝ_{+}). For j ∈ ℂ, i ∈ ℝ define further Ψ_{a}^{i,j}(X) := t^{−j}Ψ_{a}^{i,0}(X). Definition 2.3. The adiabatic algebra of X → M is defined by

Ψ_{a}^{ℤ,ℤ}(X) := ⋃_{i,j∈ℤ}Ψ_{a}^{i,j}(X).

The fiber of the natural projection X_{a}^{2} → ℝ_{+} for t > 0 is just X^{2}, hence transverse to Δ_{a} and therefore the restriction of adiabatic operators to such a fiber is well-defined. Clearly, such a restriction gives the Schwartz kernel of a classical pseudodifferential operator on X. Thus for all t > 0 we get a map from Ψ_{a}^{ℤ,ℤ}(X) into Ψ^{ℤ}(X).

The algebra structure on Ψ_{a}^{ℤ,ℤ}(X) is given by the following theorem. Theorem 2.4. The space Ψ_{a}^{ℤ,ℤ}(X) has a C^{∞}(ℝ_{+})[t^{−1}]-algebra structure which induces the usual algebra structure on Ψ^{ℤ}(X) for every fixed t > 0.

Proof. The composition rule is defined by constructing a triple blow-up space X_{a}^{3}, such that there exist three maps π_{L}, π_{C}, π_{R} to X_{a}^{2} which are b-submersions [15]. This space is defined as the iterated blow-up

X_{a}^{3} := [X^{3} × ℝ_{+}; Δ_{3} × {0}, D_{1,2} × {0}, D_{1,3} × {0}, D_{2,3} × {0}]

where Δ_{3} is the triple diagonal, and D_{a,b} is the fiber-diagonal in X^{3} in the components a, b ∈ {1, 2, 3}. Using the properties of conormal distributions, set

(4)  A ∘ B := (π_{C})_{∗}((π_{R}^{∗}A)(π_{L}^{∗}B)).

Associativity follows from the action of Ψ_{a}^{ℤ,ℤ}(X) on C^{∞}(X × ℝ_{+}). Alternately, it also follows from the associativity of multiplication on Ψ^{ℤ}(X). □

For E, F vector bundles over X, define Ψ_{a}^{i,j}(X, E, F) by considering distributional kernels as above with values in Ω_{a} ⊗E ⊠F. This is again an algebra if E = F. By Morita equivalence, for homology purposes it is enough to consider scalar operators.

2.2. Ideals, quotients and filtrations. The algebra Ψ_{a}^{ℤ,ℤ}(X) has two increasing filtrations. The first superscript specifies the order of singularity at the lifted diagonal. The ideal Ψ_{a}^{−∞,ℤ}(X) is called the smoothing ideal by abuse of notation; an operator in this ideal has smooth Schwartz kernel and is actually smoothing for fixed t > 0 as an operator on X. while the quotient A_{σ}(X) := Ψ_{a}^{ℤ,ℤ}(X)/Ψ_{a}^{−∞,ℤ}(X) is called the adiabatic symbol algebra.

The second superscript denotes the negative of the order of vanishing at t = 0, so both filtrations are increasing. Let A_{∂}(X) := Ψ_{a}^{ℤ,ℤ}(X)/Ψ_{a}^{ℤ,−∞}(X) be the boundary algebra.

We will use the notation F_{i} for the first filtration, and T_{j} for the second.

Let I_{∂}(X) := Ψ_{a}^{−∞,ℤ}(X)/Ψ_{a}^{−∞,−∞}(X) be the boundary ideal. Let A_{∂,σ}(X) := Ψ_{a}^{ℤ,ℤ}(X)/(Ψ_{a}^{−∞,ℤ}(X) + Ψ_{a}^{ℤ,−∞}(X)) be the boundary symbol algebra. The following diagram summarizes the different ideals and quotients:

(5) \xymatrix{ & 0 \ar[d] & 0 \ar[d] & 0 \ar[d] \\ 0 \ar[r] & Ψ_{a}^{−∞,−∞}(X) \ar[r] \ar[d] & Ψ_{a}^{ℤ,−∞}(X) \ar[r] \ar[d] & I_{σ} \ar[r] \ar[d] & 0 \\ 0 \ar[r] & Ψ_{a}^{−∞,ℤ}(X) \ar[r] \ar[d] & Ψ_{a}(X) \ar[r] \ar[d] & A_{σ}(X) \ar[r] \ar[d] & 0 \\ 0 \ar[r] & I_{∂}(X) \ar[r] \ar[d] & A_{∂}(X) \ar[r] \ar[d] & A_{∂,σ}(X) \ar[r] \ar[d] & 0 \\  & 0 & 0 & 0}

The horizontal and vertical sequences are exact.

2.3. The description of the product. Proposition 2.5. The adiabatic structure map ϕ_{a} induces a canonical splitting

aT X_{|t=0} ≅ T X/M ⊕ π^{∗}^{a}T M_{|t=0}.

Proof. There exists a tautological inclusion map T X/M × ℝ_{+} → ^{a}T X, which views a time-dependent family of vertical vector fields as an adiabatic family. Hence T X/M is a subspace of ^{a}T X_{|t=0}. The nullspace of ϕ_{|t=0}^{a}, π^{∗}T M and π^{∗}^{a}T M_{|t=0} are canonically isomorphic, and form a complement of T X/M in ^{a}T X_{|t=0}. □

Hence, ^{a}T^{∗}X_{|t=0} is canonically isomorphic to (T X/M)^{∗} ⊕ π^{∗}^{a}T^{∗}M_{|t=0}.

From the definition of the blow-up, it follows immediately that the normal bundle to the lifted diagonal in X_{a}^{2} is canonically isomorphic to aT X. In other words, X_{a}^{2} resolves the degeneracy of adiabatic vector fields. Proposition 2.6. The interior of ff is naturally diffeomorphic to the lift of ^{a}T M to X ×_{M} X.

Proof. By definition, ff^{∘} → X ×_{M} X is a fiber bundle with contractible fibers. Define the 0-section as the class of the tangent vector ∂_{t} on X_{a}^{2}. Take an adiabatic vector field v which lives in ^{a}T M at t = 0 (see Proposition 2.5) and lift it via π_{R} ∘ β to X_{a}^{2}. The restriction of the lift ṽ to the front face is tangent to the fibers of ff and depends only on the restriction of v at t = 0. Integrating such vector fields gives the desired identification. □

Thus ff^{∘} has a natural vector bundle structure over X ×_{M} X.

The product structure on Ψ_{a}(X) is given by (4), which is rather inexplicit. We have more explicit descriptions of the product on A_{σ}(X), I_{∂}(X) and Ψ_{a}^{−∞,−∞}(X). First, Ψ_{a}^{−∞,−∞}(X) is isomorphic to the space of rapidly vanishing families of smoothing operators on X endowed with the standard product.

Let A_{σ}^{[i],ℤ} := A_{σ}^{i,ℤ}/A_{σ}^{i−1,ℤ} be the filtration quotients of A_{σ}(X). As in the standard case, the associated graded algebra A_{σ}^{[ℤ],ℤ} is isomorphic to the algebra of homogeneous symbols on the adiabatic cotangent bundle aT^{∗}X.

Let us fix an adiabatic quantization, i.e., a diffeomorphism from an open subset of the radial compactification \overline{^{a}T X} to a subset of X_{a}^{2}, which

(1) extends the diffeomorphism from the zero section to the lifted diagonal;

(2) induces the identity map between the normal bundle to the zero section and the normal bundle to Δ_{a} (they are both canonically isomorphic to ^{a}T X);

(3) extends the natural diffeomorphism from π^{∗}T M_{|t=0} to ff_{ff∩Δ_{a}}^{∘};

(4) induces the identity map on the normal bundle to these two spaces (they are both canonically isomorphic to T X/M).

This choice allows us to find a vector space isomorphism called total symbol map

(6)  A_{σ}(X) → S(^{a}T^{∗}X)[t^{−1}]

in the following way: take an adiabatic operator, cut it off near ff ∪ π^{∗}T M_{|t=0}, pull it back via the quantization and apply Fourier transform in the fibers of ^{a}T X. Only the leading component of the resulting symbol is independent of the choices made.

Let us define the adiabatic Poisson bracket { , }_{t} as the push-forward of the Poisson bracket on T^{∗}X under the canonical map ϕ_{a}. Proposition 2.7. For any choice of adiabatic quantization, the isomorphism (6) induces the canonical isomorphism of graded algebras

A_{σ}^{[ℤ],ℤ}(X) ≅ S^{[ℤ]}(^{a}T^{∗}X)[t^{−1}].

Moreover, the product induced on S(^{a}T^{∗}X)[t^{−1}] by the map (6) is a smooth t-dependent star product, i.e., it is given by a series of bidifferential operators

(7)  ∗ = ∑_{i≥0} P_{i}(t)

such that P_{0}(t)(a, b) = ab, P_{1}(t)(a, b) − P_{1}(t)(b, a) = {a, b}_{t}, and P_{i} is of homogeneity −i in the cotangent fibers.

Proof. The first claim follows directly from the construction of the total symbol map, which starts with the principal symbol. The product structure on Ψ_{a}(X) is local in t since C^{∞}(ℝ_{+})[t^{−1}] is central. Therefore, for any t > 0 the statement follows from the corresponding result for Ψ^{ℤ}(X). Since for every i, the product map Ψ_{a}^{i,i} ⊗ Ψ_{a}^{i,i} → Ψ_{a}^{2i,2i} is continuous, it follows that P_{i}(t) extends down to t = 0 as a smooth differential operator, and that the identities hold down to t = 0. □

Consider now the case where X = M. Adiabatic limits in this setting are called semi-classical limits. Let L = ℂ[[t]][t^{−1}] denote the field of formal Laurent series in the variable t. An adiabatic quantization map gives now vector space isomorphisms

(8) A_{∂}(M) ≅ S_{cl}(^{a}T M_{|t=0}) ˆ⊗L  I_{∂}(M) ≅ 𝒮(^{a}T M_{|t=0}) ˆ⊗L

(the hat means that we allow infinite sums of tensors, provided that they are finite, or equivalently convergent, in each degree). The first isomorphism from (8) is compatible with (6) in the following sense: for an operator A ∈ Ψ_{a}(M), we get a formal symbols with Laurent coefficients over ℝ_{+} from (6), and a Laurent series of classical symbols from (8); these two objects must map to the same formal series in S(^{a}T^{∗}M_{|t=0}) ˆ⊗L. One may ask how this compatibility translates in terms of the product. The answer is given below.

Proposition 2.8. Let M → M be the identity fibration. The product induced on S_{cl}(^{a}T M_{|t=0}) ˆ⊗L by (8) is given by the same formula as the product (7) on formal symbols, specialized to the case X = M.

Proof. In the semi-classical limit algebra, unlike the general adiabatic case, it is not hard to see that the product on A_{∂}(M) is local, hence given by a star-product. The induced product on formal symbols is by definition (7); moreover the bi-differential operators appearing in the product have polynomial coefficients, thus we can recover them from their action on formal symbols. □

In general, the ideal I_{∂}(X) is a module over I_{∂}(M), generated by the algebra of smoothing operators in the fibers of X → M.

A Taylor coefficient at t = 0 of an element in Ψ_{a}(X) becomes, after Fourier transform in the fibers of ^{a}T M_{|t=0}, a classical symbol on aT^{∗}M_{|t=0} taking values in the space of classical pseudodifferential operators on the fibers of X. Note however that adiabatic operators have joint symbolic behavior in the fibers and in the base. Then the product on Laurent series at t = 0 is a combination of a star-product on ^{a}T^{∗}M and of the usual convolution product in the fibers.

Let Ψ_{a}^{ℤ,[0]}(X) be the vertical algebra of Ψ_{a}(X),

Ψ_{a}^{ℤ,[0]}(X) := Ψ_{a}^{ℤ,0}(X)/Ψ_{a}^{ℤ,1}(X).

The algebra Ψ_{a}^{ℤ,[0]}(X) is a fibered version of the suspended algebra [16], [18]. Let Ψ_{a}^{−∞,[0]}(X) := I_{∂}^{ℤ,0}(X)/I_{∂}^{ℤ,1}(X) ↪ Ψ_{a}^{ℤ,[0]}(X) be the vertical smoothing ideal associated to I_{∂}(X). An element in Ψ_{a}^{−∞,[0]}(X) is the restriction to the front face of the Schwartz kernel of a non-singular smoothing adiabatic operator. On ff, the density bundle Ω_{a} decomposes as Ω_{R} ⊗ Ω(^{a}T X), where Ω_{R} is the bundle of densities in the second F factor. We identify such a restriction with its Fourier transform in the fibers of ^{a}T M_{|t=0}. Hence by Fourier transform,

Ψ_{a}^{−∞,[0]}(X) →^{≅} 𝒮(X ×_{M} X ×_{M} ^{a}T^{∗}X, Ω_{R}).

Note that the isomorphisms (8) induce, by passage to the associated graded objects, canonical isomorphisms

Ψ_{a}^{ℤ,[0]}(M) ≅ S_{cl}(^{a}T^{∗}M),
Ψ_{a}^{−∞,[0]}(M) ≅ 𝒮(^{a}T^{∗}M),
Ψ_{a}^{[ℤ],[0]}(M) ≅ S^{[ℤ]}(^{a}T^{∗}M).

3. Hochschild homology and derivations 3.1. Hochschild Homology. Let A be an unital algebra over ℂ. Let C_{i}(A) = A^{⊗i+1} be the space of Hochschild chains. For j = 0, …, i, define b_{j} : C_{i}(A) → C_{i−1}(A) by

a_{0} ⊗ … ⊗ a_{j} ⊗ a_{j+1} ⊗ … ⊗ a_{i} ↦^{b_{j}} a_{0} ⊗ … ⊗ a_{j}a_{j+1} ⊗ … ⊗ a_{i}
a_{0} ⊗ … ⊗ a_{i} ↦^{b_{i}} a_{i}a_{0} ⊗ … ⊗ a_{i−1}.

Define the Hochschild boundary map by b = ∑_{j=0}^{i}(−1)^{j}b_{j}. The homology of the complex (C_{∗}, b) is the Hochschild homology of the algebra A (relative to A).

If A is not unital, let Ā be the augmented algebra A ⊕ ℂ. We define the chain spaces of A by C_{k}(A) = ker(Ā^{⊗k+1} → ℂ^{⊗k+1}). This definition, applied to an unital algebra, gives different chain spaces from the ones above. Nevertheless, there are canonical chain maps which induce isomorphisms on homology.

Benameur-Nistor [2] defined the Hochschild chains for a large class of algebras with topology. We use these chain spaces for the adiabatic algebra. Like the usual algebra of pseudodifferential operators, this is not a topological algebra, in the sense that the product is not jointly continuous. This is however only a minor problem. For every j_{1}, j_{2}, k_{1}, k_{2} ∈ ℂ ∪ {−∞}, the multiplication map

Ψ_{a}^{j_{1},k_{1}}(X) ⊗ Ψ_{a}^{j_{2},k_{2}}(X) → Ψ_{a}^{j_{1}+j_{2},k_{1}+k_{2}}(X)

is continuous with respect to natural Fréchet topologies.

The two filtrations by the order, respectively by the negative of the vanishing order at t = 0, are compatible with multiplication in Ψ_{a}(X). They induce filtrations on the Hochschild chain spaces in the following way: a pure tensor a_{0} ⊗ … ⊗ a_{k} is said to belong to C_{k}^{i,j;l} if a_{0} ∈ Ψ_{a}^{i_{0},j_{0}}(X), …, a_{k} ∈ Ψ_{a}^{i_{k},j_{k}}(X), i_{0} + … + i_{k} = i, j_{0} + … + j_{k} = j and i_{α} < l, j_{α} < l for α = 1, …, k. Then C_{k}^{i,j;l} is defined as the closure of the linear span of pure tensors with respect to the projective tensor product topology in C_{k}^{i,j;l}, and we define

C_{k} := ⋃_{i,j,l∈ℤ}C_{k}^{i,j;l}.

See [2] for more comments on this issue.

The boundary map b is compatible with these two filtrations, hence we get filtrations on the Hochschild complexes of Ψ_{a}(X), Ψ_{a}^{−∞}(X), A_{σ}(X), etc. We will denote the associated spectral sequences by ^{σ}E, respectively ^{∂}E, or simply by E when no confusion can arise.

Note that although A_{∂}(X) is an L–vector space, our definition of Hochschild homology involves only tensors over ℂ.

Let us recall the Hochschild-Kostant-Rosenberg map [10]. Let A be a commutative ℂ-algebra. Define

(9) χ : C_{k}(A) → Ω_{A/ℂ}^{k}  a_{0} ⊗ … ⊗ a_{k} ↦ a_{0}da_{1} ∧ … ∧ da_{k}.

Proposition 3.1. Let Y be a compact manifold, possibly with boundary. The map χ induces isomorphisms

HH_{k}(C^{∞}(Y)) ≅ Λ^{k}(Y)  HH_{k}(Ċ^{∞}(Y)) ≅ Λ̇^{k}(Y),

where the dot denotes rapid vanishing to the boundary of Y.

This result is known as the Hochschild-Kostant-Rosenberg theorem, and it was proved in [7] for the algebra C^{∞}(Y) when Y is a closed manifold. The other cases follow from this one, by considering for instance the double of a manifold with boundary, and the H-unital ideals of rapidly vanishing functions.

3.2. The action of derivations on Hochschild Homology. Let d be a derivation on an algebra A. We define the inner product, respectively the Lie derivative with respect to d on HH(A) by the following chain maps [12]: Definition 3.2.

a_{0} ⊗ a_{1} ⊗ … ⊗ a_{k} ↦^{e_{d}} (−1)^{k+1}d(a_{k})a_{0} ⊗ … ⊗ a_{k−1};
a_{0} ⊗ a_{1} ⊗ … ⊗ a_{k} ↦^{L_{d}} ∑_{i} a_{0} ⊗ … ⊗ d(a_{i}) ⊗ … ⊗ a_{k}.

If d is an inner derivation, then e_{d} and L_{d} both vanish on HH(A) [12]. Proposition 3.3.

(1) [L_{d_{1}}, e_{d_{2}}] = e_{[d_{1},d_{2}]}.

(2) e_{d_{1}}e_{d_{2}} = −e_{d_{2}}e_{d_{1}}.

Proof. The first statement is true at chain level [25]. For the second, we note that d_{1}⊗d_{2} = −d_{2}⊗d_{1} in the graded commutative ring HH^{∗}(A, A). The claim follows from the action of this ring on HH_{∗}(A). □

If d is continuous, then e_{d} and L_{d} extend to the topological chain spaces and to topological Hochschild homology.

3.3. The derivative with respect to t. If we restrict adiabatic operators to t > 0, we remark a canonical derivation preserving the filtrations, namely D := t\frac{d}{dt}. In the blow-up picture, it is given by the Lie derivative in the direction of the lift of the vector field t\frac{d}{dt} to the interior of X_{a}^{2} through the blow-down map. This vector field vanishes on the fiber–diagonal at t = 0, hence its lift extends to ff as a (non-degenerate) vector field, tangent to the fibers of ff^{∘}. In light of Proposition 2.5, we can identify it with the radial vector field in π^{∗}^{a}T M. We can therefore extend the action D to the whole algebra Ψ_{a}(X); by continuity, it will stay a derivation.

Let us choose a connection in X → M. It induces an extension of the splitting ^{a}T^{∗}X_{|t=0} ≅ (T X/M)^{∗} ⊕ π^{∗}^{a}T^{∗}M_{|t=0} to ^{a}T^{∗}X over ℝ_{+}, and defines a splitting T^{∗}X ≅ (T X/M)^{∗} ⊕ π^{∗}T^{∗}M. Recall that ^{a}T^{∗}M and T^{∗}M × ℝ_{+} are canonically isomorphic. Then the map ϕ_{a} (Definition 2.1) takes the form

(10)  ϕ_{a} = [\begin{matrix}1 & 0 \\ 0 & t \end{matrix}]

Let R_{M} be the radial vector field in π^{∗}T^{∗}M. We can lift R_{M} to ^{a}T^{∗}X by using the connection. Proposition 3.4. On A_{σ}^{[ℤ],ℤ} and A_{∂}^{ℤ,[ℤ]}, the derivation D takes the following form:

(11)  D = t\frac{d}{dt} + R_{M}.

Proof. From the above expression of ϕ_{a},

(12)  (ϕ_{a})_{∗} (\frac{d}{dt}) = \frac{d}{dt} + t^{−1}R_{M}.

In any quantization independent of t for t > t_{0} > 0, the action of t\frac{d}{dt} on symbols is just the action of the vector field t\frac{d}{dt} on functions. The blow-down map β induces the isomorphism ϕ_{a} : T^{∗}X × (0, ∞) → aT^{∗}X_{|t>0}. The relation between principal symbols is simply pull-back via ϕ_{a}, hence the induced derivation on A_{σ}^{[ℤ],ℤ} for t > 0 is ϕ_{a}_{∗}(t\frac{d}{dt}) = t\frac{d}{dt} + R_{M}. By continuity, this is valid down to t = 0.

We have implicitly proved that the partial Fourier transform in the horizontal fibers of the operator L_{β^{∗}(t \frac{d}{dt})} equals L_{t \frac{d}{dt} +R_{M}}. This implies

the result for A_{∂}^{ℤ,[ℤ]}. □

We note that (11) is a sum of two well-defined terms on the front face, while in the interior they depend on the connection. Their sum is of course independent of this choice.

Remark that D preserves all ideals like I_{∂}(X), Ψ_{a}^{−∞}(X), etc. As in Definition 3.2, we get an action e_{D} on the Hochschild homology of these algebras. Moreover, D preserves the two algebra filtrations, hence it induces maps of spectral sequences for the Hochschild complexes.

3.4. The conjugation by log Q. Let Q ∈ Ψ_{a}^{1,0}(X) be a positive elliptic adiabatic operator of order 1 acting on the sections of a vector bundle over X. Then the complex powers Q^{z}, z ∈ ℂ, are adiabatic operators of complex order z. Indeed, by the result of Seeley [26], for any fixed t > 0 the complex powers of the restriction Q_{t} are classical pseudodifferential operators. Using the method of Bucicovschi [6] extending a proof by Guillemin [9], one can construct the complex powers of the images of Q in A_{∂}(X), respectively in A_{σ}(X). Moreover, these families are unique. Patching them together gives the family {Q^{z}}_{z∈ℂ}.

We choose Q with σ_{1}(Q) = r where r : ^{a}T^{∗}X → ℝ is the length function of some non-degenerate metric on ^{a}T X.

Let D_{z}(a) := (Q^{−z}aQ^{z} − a)/z, and let D_{Q} be the following derivation on Ψ_{a}(X):

(13)  D_{Q}(a) := lim_{z→0} D_{z}(a)_{|_{z=0}} = \frac{d}{dz} (Q^{−z}aQ^{z})_{|_{z=0}} .

Proposition 3.5. [D_{Q}, D] is an inner derivation.

Proof. Follows easily from the fact that

\frac{d}{dz} (DQ^{z})_{|_{z=0}} ∈ Ψ_{a}(X).  □

Lemma 3.6. D_{Q} decreases the order by 1. If A is an operator of order j, then

σ_{j−1}(D_{Q}A) = \frac{1}{ir}\{r, σ_{j}(A)\}_{t},

where {, }_{t} is the adiabatic Poisson bracket.

Proof. The symbol of Q^{z} is r^{z}. The result follows from Proposition 2.7. □

4. The homology of the algebra of adiabatic symbols

In this section we compute HH_{∗}(A_{σ}(X)) with the spectral sequence σE. Let N = m + n = dim X. Recall that all operators in this paper have classical (i.e., 1-step, polyhomogeneous) symbols.

4.1. The E_{2} term. Recall that the subscript [i] denotes homogeneity i in radial directions. We first compute the E_{1} term. Proposition 4.1. The first term in ^{σ}E(C_{∗}(A_{σ}(X))) is given by

E_{1}^{i,k}(A_{σ}(X)) ≃ Λ_{[i]}^{i+k}(^{a}T^{∗}X \ {0})[t^{−1}].

Proof. From Proposition 2.7, the associated graded algebra GA_{σ}(X) ≅ S(^{a}T^{∗}X)[t^{−1}] is the graded commutative algebra of formal symbols on aT^{∗}X, with t^{−1} adjoined. This implies that E_{1}^{i,k}(A_{σ}(X)) is the component of homogeneity i of HH_{i+k}(S(^{a}T^{∗}X)[t^{−1}])). By Proposition 3.1,

HH_{k}(S(^{a}T^{∗}X)[t^{−1}]) = Λ_{S}^{k}(^{a}T^{∗}X)[t^{−1}].

The isomorphism is given by the map χ. □

Consider the following inclusions of algebras

(14)  Ψ_{a}^{ℤ,−∞}(X) ↪ C^{∞}(ℝ_{+}, Ψ^{ℤ}(X)
(15)  Ψ_{a}^{ℤ,−∞}(X) ↪ Ψ_{a}(X).

The second map is induced by the blow-down map β. Let us examine the induced map on the spectral sequences of the symbol algebras. Note that all the terms of these spectral sequences are HH_{∗}(C^{∞}(ℝ_{+})[t^{−1}])– modules, hence local in t. As in Proposition 4.1, we see that

E_{1}^{i,k}(A_{σ}^{ℤ,−∞}(X)) ≅ Λ̇_{[i]}^{i+k}(^{a}T^{∗}X \ {0})[t^{−1}];
E_{1}^{i,k}(C^{∞}(ℝ_{+}, S(X))) ≅ Λ_{[i]}^{i+k}(ℝ_{+} × T^{∗}X \ {0})[t^{−1}]

where the dot denotes rapid vanishing at t = 0.

For t > 0, the canonical map ϕ_{a} : T^{∗}X×ℝ_{+} → ^{a}T^{∗}X is an isomorphism. Then the map induced by (14), respectively (15) on E_{0} and E_{1} terms is the inclusion, respectively the pull-back by ϕ_{a}. The point of this argument is that we already know the differential d_{1} for the algebra of families of symbols, by the results of [5]. Denote the differential in the spectral sequence of A_{σ}(X) by a superscript a. For t > 0 we get

(16)  d_{1}^{a} = (ϕ_{a}^{∗})^{−1}d_{1}ϕ_{a}^{∗}.

We will extend (16) by continuity down to t = 0. Choose a connection in X → M as in Section 3.3. Let d_{v} = d − dt ⊗ L_{\frac{d}{dt}} be the de Rham differential in the vertical directions in a product decomposition of aT^{∗}X. Let ∗ be the symplectic duality operator on Λ(T^{∗}X) introduced by Brylinski [4]. Let ∗_{ϕ} be the conjugation of ∗ by ϕ_{a}:

∗_{ϕ} = ϕ_{a}^{∗}^{−1} ∗ ϕ_{a}^{∗}.

Proposition 4.2. For t > 0, the differential d_{1}^{a} can be written (up to sign) in terms of the product structure on ^{a}T^{∗}X as

(17)  d_{1}^{a} = ∗_{ϕ}(d_{v} − t^{−1}dt ⊗ L_{R_{M}}) ∗_{ϕ} .

Proof. The differential d_{1} for the algebra S(X) = Ψ^{ℤ}(X)/Ψ^{−∞}(X) was computed in [4], [5]. Up to sign, which is irrelevant for homology, it equals ∗d∗, where d is the de Rham differential on T^{∗}X \ 0. Hence, in the spectral sequence of the families algebra C^{∞}(ℝ_{+}, S(X)), we get d_{1} = ∗(d − dt ⊗ \frac{∂}{∂t})∗. From (16) and (12), it follows that

d_{1}^{a} = ∗_{ϕ}(ϕ_{a}^{∗−1}(d − dt ⊗ L_{\frac{d}{dt}})ϕ_{a}^{∗})∗_{ϕ}

= ∗_{ϕ}(d − dt ⊗ L_{ϕ_{a}_{∗}( \frac{d}{dt})})∗_{ϕ}

= ∗_{ϕ}(d − dt ⊗ (\frac{∂}{∂t} + t^{−1}L_{R_{M}})) ∗_{ϕ} .

(we have used the invariance by pull-back of d). □

Let ^{v}Λ_{[i]}^{k} denote the space ı_{\frac{d}{dt}} Λ_{[i]}^{k}(^{a}T^{∗}X \ \{0\})[t^{−1}] of forms that do not contain dt. Proposition 4.3. The involution ∗_{ϕ} extends to Λ^{∗}(^{a}T^{∗}X)[t^{−1}]. Its restriction to ^{v}Λ_{[i]}^{i+j} acts as follows:

(18) ^{v}Λ_{[i]}^{i+j} →^{∗_{ϕ}} ^{v}Λ_{[N−j]}^{2N−i−j} ⊕ dt ∧^{v}Λ_{[N−j]}^{2N−i−j−1} ⊕ dt ∧^{v}Λ_{[N−j+1]}^{2N−i−j+1}.

Proof. Follows from the following observations: First, the operator ∗ commutes with dt and, as noted in [4], it maps Λ_{[i]}^{i+j}(T^{∗}X) to Λ_{[N−j]}^{2N−i−j}(T^{∗}X). Secondly, ϕ_{a}^{∗} maps ^{v}Λ_{[i]}^{k} onto (1 + t^{−1}dt ⊗ ı_{R_{M}})Λ_{[i]}^{k}, and ϕ^{∗}^{−1} maps Λ_{[i]}^{k} onto (1 − t^{−1}dt ⊗ ı_{R_{M}})^{v}Λ_{[i]}^{k}. □

As a corollary, the identity (17) holds by continuity down to t = 0. Definition 4.4. Fix a product structure on ^{a}T^{∗}X. Define the adiabatic duality operator ∗_{a} on Λ^{∗}(^{a}T^{∗}X) by

(19)  ∗_{a} := (1 + t^{−1}dt ⊗ ı_{R_{M}})∗_{ϕ}.

From the definition, ∗_{a} is an isomorphism with inverse ∗_{ϕ}(1 − t^{−1}dt ⊗ ı_{R_{M}}). Also, it is clear that ∗_{a} consists of the first and third terms of ∗_{ϕ} from the decomposition (18). Proposition 4.5. The first differential d_{a}^{1} is conjugated to the vertical de Rham differential:

∗_{a}d_{1}^{a}∗_{a}^{−1} = d_{v}.

Proof. Direct computation, using (17) and the definition (19). □

Hence d_{1}^{a} : E_{1}^{i,j}(A_{σ}(X)) → E_{1}^{i−1,j}(A_{σ}(X)) is conjugated via ∗_{a} to

(20) ^{v}Λ_{[N−j]}^{2N−i−j} ⊕ dt ⊗^{v}Λ_{[N−j+1]}^{2N−i−j+1} →^{d_{v}} ^{v}Λ_{[N−j]}^{2N−i−j+1} ⊕ dt ⊗^{v}Λ_{[N−j+1]}^{2N−i−j+2}

Let ^{a}S^{∗}X → ℝ_{+} be the sphere bundle of ^{a}T^{∗}X viewed as a bundle over ℝ_{+}. Denote by C^{∞}(ℝ_{+}, H_{v}^{∗}(^{a}S^{∗}X)) the space of sections in the (trivial) fiber cohomology bundle. Theorem 4.6. At the E_{2} level, the spectral sequence with respect to the total operator degree filtration of the Hochschild complex of the formal symbol algebra is isomorphic to a family of de Rham cohomology spaces indexed by ℝ_{+}:

E_{2}^{i,j}(A_{σ}(X)) ≅ \begin{cases}C^{∞}(ℝ_{+}, H_{v}^{N−i}(^{a}S^{∗}X × S^{1}))[t^{−1}] & if j = N, \\ C^{∞}(ℝ_{+}, H_{v}^{N−i}(^{a}S^{∗}X × S^{1}))[t^{−1}]dt & if j = N + 1, \\ 0 & otherwise. \end{cases}

Proof. We have seen that E_{2}^{i,j}(A_{σ}(X)) is isomorphic to the homology of the complex (20). Since d_{v} preserves homogeneity and commutes with dt∧, this complex splits. We claim that

(21)  H_{∗}(Λ_{[k]}^{∗}(^{a}T^{∗}X \ {0})[t^{−1}], d_{v}) = 0 for k ≠ 0.

Indeed, let R be the radial vector field on ^{a}T^{∗}X. Then, for ν ∈ Λ_{[k]}^{∗}(^{a}T^{∗}X \ {0}[t^{−1}]), we have L_{R}ν = kν. Assume ν is d_{v}-closed. Since d/dt is defined using a product decomposition of ^{a}T^{∗}X, we have

(22)  [R, \frac{d}{dt}] = 0.

Thus

kν = L_{R}ν = dı_{R}ν + ı_{R}dν = d_{v}ı_{R}ν + ı_{R}d_{v}ν = d_{v}(ı_{R}ν).

Hence if k ≠ 0 it follows that ν is exact.

On the other hand, it is straightforward to check that

H_{j}(Λ_{[0]}^{∗}(^{a}T^{∗}X \ {0})[t^{−1}], d_{v}) ≅ C^{∞}(ℝ_{+}, H_{v}^{j}(^{a}S^{∗}X × S^{1}))[t^{−1}]. □

By the homotopy invariance of cohomology, the isomorphisms from Theorem 4.6 are independent of the choices made.

4.2. Action of derivations and degeneracy. The derivation D introduced in (11) acts by e_{D} on C_{∗}(A_{σ}(X)). Since it commutes with b and preserves the filtration F_{i}, this action descends to the spectral sequence. Proposition 4.7. The effect of e_{D} on ∗_{a}E_{1}(A_{σ}(X)), and hence on E_{∞}(A_{σ}(X)), is contraction by the vector field t\frac{d}{dt}.

Proof. The following commutations hold trivially on E_{1}(Ψ_{(0,∞)}):

χe_{t \frac{d}{dt}} = ı_{t \frac{d}{dt}} χ  ∗ ı_{t \frac{d}{dt}} = ı_{t \frac{d}{dt}} ∗ .

Since χ is invariant by pull-backs, we can pull it back via the map ϕ_{a}. It follows that

χe_{D} = ı_{(ϕ_{a})_{∗}(t \frac{d}{dt})}χ  ∗_{ϕ}ı_{(ϕ_{a})_{∗}t \frac{d}{dt}} = ı_{(ϕ_{a})_{∗}t \frac{d}{dt}} ∗_{ϕ},

which implies ∗_{ϕ}χ e_{D} = ı_{ϕ_{∗}(t \frac{d}{dt})} ∗_{ϕ} χ, and so

(1 + t^{−1}dt ⊗ ı_{R_{M}}) ∗_{ϕ} χe_{D} = (1 + t^{−1}dt ⊗ ı_{R_{M}})(ı_{t \frac{d}{dt}} + ı_{R_{M}}) ∗_{ϕ} χ

= ı_{t \frac{d}{dt}} (1 + t^{−1}dt ⊗ ı_{R_{M}}) ∗_{ϕ} χ. (23)

This identity extends by continuity down to t = 0. □

Hence the map e_{D} is injective on E_{2}^{∗,m+n+1}(A_{σ}(X)) and vanishes on the rest of E_{2}(A_{σ}(X)). This implies that the spectral sequence ^{σ}E(A_{σ}(X)) degenerates at E_{2}.

Proposition 4.8. The effect of L_{D} on [a] ∈ E_{∞}(A_{σ}(X)) is the Lie derivative L_{t \frac{d}{dt}} [a] applied to the cohomology class [a] representing a at E_{2} = E_{∞} in the presentation of Theorem 4.6. In particular, if a is a class of homogeneity k in t, then L_{D}(a) = ka.

Proof. From the definitions, χ∘L_{D} = L_{(ϕ_{a})_{∗}(t \frac{d}{dt})}χ, i.e L_{D} acts as the Lie derivative L_{(ϕ_{a})_{∗}(t \frac{d}{dt})} on E_{1}, i.e., on forms. This Lie derivative commutes with ∗_{ϕ} and with (1 + t^{−1}dt ⊗ ı_{R_{M}}). Finally, L_{R_{M}} = 0 on cohomology.

□

Let t^{−1}dt∧ be the following operation on C_{∗}(Ψ_{a}(X)):

a_{0} ⊗ … ⊗ a_{n} ⟼^{t^{−1}dt∧} ∑_{i=0}^{n}(−1)^{i}t^{−1}a_{0} ⊗ … ⊗ a_{i} ⊗ t. ⊗ … ⊗ a_{n},

Since t belongs to the center of Ψ_{a}(X), t^{−1}dt∧ is a chain map. Proposition 4.9. Set α := e_{D}t^{−1}dt∧, β := t^{−1}dt ∧ e_{D}. The following identities hold on homology:

(1) (t^{−1}dt∧)^{2} = 0.

(2) e_{D}t^{−1}dt ∧ +t^{−1}dt ∧ e_{D} = 1.

(3) C_{∗}(A_{σ}(X)) and HH(A_{σ}(X)) split as im(e_{D}t^{−1}dt∧)⊕im(t^{−1}dt∧ e_{D}).

Proof. The first two statements follow by direct computation. From part (1), αβ = 0. Then part (2) implies that α = α(α + β) = α^{2} and β = (α + β)β = β^{2}, so α and β are idempotents. Since 1 = α + β = (α + β)^{2} = α + β + βα, we get also βα = 0. This proves part (3). □

Let r^{−1}dr be the generator of H^{1}(S^{1}) in Theorem 4.6. Proposition 4.10. After identification of E_{2} with de Rham cohomology, the derivation D_{Q} acts as an exterior product:

e_{D_{Q}} = r^{−1}dr ∪ .

Proof. First, by Propositions 3.5 and 3.3, e_{D_{Q}} commutes with t^{−1}dt∧ and with e_{D}. This implies that e_{D_{Q}} preserves the spaces of 0 and 1forms in t. Let a ∈ E_{2}^{i,m+n} be a representative for a homology class with no dt. Then, by Theorem 4.6, e_{D_{Q}}(a) is determined by its part in homogeneity i − 1.

From Lemma 3.6, it follows that χ(σ(e_{D_{Q}}(a))) = \frac{dr}{r} ∧ χ(a), which implies the desired formula after applying the conjugation (1 + t^{−1}dt ⊗ ı_{R_{M}})∗_{ϕ}. The case where the class of a is a multiple of dt follows by multiplication with t^{−1}dt. □

4.3. Convergence of the spectral sequence. This section is unavoidably technical because of the crucial role played by the two filtrations F_{i} and T_{j} in the definition of the topological chain spaces for Ψ_{a}(X). Theorem 4.11. The spectral sequence for A_{σ}(X) is convergent, in the sense that there exists an isomorphism

HH_{k}(A_{σ}(X)) ≅ E_{2}^{k−N,N} ⊕ E_{2}^{k−N−1,N+1}.

compatible with the filtration {T_{i}}_{i∈ℤ}, where the E_{2} spaces are given in Theorem 4.6.

A homology class is said to belong to F_{i} if it has a representative in F_{i}. Thus Theorem 4.11 contains what is usually meant by convergence:

F_{i}/F_{i−1}(HH_{k}(A_{σ}(X))) ≅ \begin{cases}E_{2}^{k−N,N} & if i = k − N \\ E_{2}^{k−N−1,N+1} & if i = k − N − 1 \\ 0 & otherwise \end{cases}

but also something more: first, the “residual” homology

⋂_{i∈ℤ}F_{i}(HH_{k}(A_{σ}(X)))

vanishes; secondly, there exists a (natural) splitting of HH_{k}(A_{σ}(X)) as the direct sum of its filtration quotients, as C^{∞}(ℝ_{+})[t^{−1}]-modules.

One can construct abstractly a formal extension of any element in E_{∞} to a closed “chain”. The issue is finding asymptotically summable extensions, i.e., bounding from above the T -order in accordance with the definition of the topological chain spaces (by definition, the T -order of a chain a_{0} ⊗ … ⊗ a_{k} with respect to a increasing filtration T_{i} is at most l if a_{j} ∈ T_{l} for all j). We claim that we can achieve this, and moreover we can find such an extension of the same boundary filtration order as the starting element in E_{∞}.

The proof uses some results and ideas from [18].

We identify elements in Ψ_{a}^{ℤ,[0]} with the Fourier transform of their Schwartz kernel. The algebra of polynomial functions on ^{a}T^{∗}M is central in Ψ_{a}^{ℤ,[0]}. Let A_{σ}^{ℤ,[0]} be the symbol algebra of Ψ_{a}^{ℤ,[0]}. Proposition 4.12. The spectral sequence ^{σ}E(A_{σ}^{ℤ,[0]}) degenerates at E_{2}.

Proof. Let B_{(sus)} ⊂ A_{σ}^{ℤ,[0]} be the ideal consisting of the symbols that vanish rapidly at the vertical sub-bundle of ^{a}T^{∗}X. Let F_{(sus)} denote the quotient A_{σ}^{ℤ,[0]}/B_{(sus)}. These algebras are filtered by the operator order, thus we get spectral sequences for each of the three Hochschild complexes.

Since the polynomial functions on ^{a}T^{∗}M commute with these algebras, we can define the following two types of natural operations on the spectral sequences: contractions by vector fields with polynomial coefficients, and exterior multiplication by forms with polynomial coefficients. These operations define splittings of the homologies and of the spectral sequences, which are preserved by the differentials. We can also consider the action of the Lie derivative in the direction of R_{M}, the radial vector field in ^{a}T^{∗}M. This replaces the dilation argument from [18]. The different spectral sequences split as eigenspaces of this action. By computing the eigenvalues, one can prove as in [18, Propositions 6 and 7 and Lemma 7] that the spectral sequences E(F_{(sus)}) and E(B_{(sus)}) degenerate at E_{2}. The same argument shows that the long exact sequence in homology is determined at E_{2}, hence implying the proposition. □

The convergence of these sequences follows from the fact that the chain spaces are defined as inverse limits, and the following lemma: Lemma 4.13. Let a ∈ C_{k}(C^{∞}(Y) ⊗ L) be a boundary of T -order l. Then there exists c ∈ C_{k+1}(C^{∞}(Y) ⊗ L) of T -order max{0, l} such that a = b(c).

Proof. If Y is 0-dimensional, the statement follows directly. This and the proof of the Eilenberg-Zilber Theorem imply the result for arbitrary Y. □

Note that there is no analog of the index map for the suspended algebra ([18], Lemma 8). Namely, the short exact sequence

0 → Ψ_{a}^{−∞,[0]} → Ψ_{a}^{ℤ,[0]} → A_{σ}^{ℤ,[0]} → 0

induces a long exact sequence in homology (by H-unitality) which actually splits into short exact sequences. See also Proposition 7.15.

Consider now the graded algebra A_{∂}^{ℤ,[ℤ]}(X). It is canonically isomorphic to Ψ_{a}^{ℤ,[0]} ⊗ L. In the statements above, we can replace the algebras by their tensor product with L. A careful analysis of the T -orders shows that the spectral sequences are still convergent.

We have proved Proposition 4.14. Let a ∈ C_{k}^{j,i;l}(Ψ_{a}(X)) be a chain of order j, such that a^{j,[i]} survives at ^{σ}E_{2}^{j,k−j}(Ψ_{a}^{ℤ,[ℤ]}). Then there exists A ∈ C_{k}^{j,i;max{l,0}}(Ψ_{a}(X)) such that A − a ∈ C_{k}^{j,i−1}(Ψ_{a}(X)) and b(A) ∈ C_{k}^{j−1,i−1}(Ψ_{a}(X)). Proposition 4.15. If a ∈ C_{k}^{j,i;l}(Ψ_{a}(X)) survives at E_{2}^{j,k−j}(A_{σ}(X)) and k − j ≠ N, N + 1 then there exists x ∈ C_{k+1}^{j+1,i+1}(Ψ_{a}(X)) such that bx − a ∈ C_{k}^{j−1,i}(Ψ_{a}(X)) and moreover the T -order of x is at most max{T −order(a), 0}.

Proof. Let ν be the form corresponding to a in E_{2}^{j,k−j}(A_{σ}(X)), so that d_{v}(ν) = 0. From (21), it follows that

ν = C^{−1}d_{v}(ı_{R}ν)

where C = k − j − N − 1 or k − j − N, depending on whether ν does or does not contain dt. The assumption on k − j means exactly that C ≠ 0. The passage between E_{1} and E_{2} is done via the isomorphism (1 + t^{−1}dt ⊗ ı_{R_{M}})∗_{ϕ} (Proposition 4.5). Then ı_{R}ν corresponds to

c = C^{−1}∗_{ϕ}(1 − t^{−1}dt ⊗ ı_{R_{M}})ı_{R}ν = C^{−1}R^{#_{ϕ}} ∧ ∗_{ϕ}(1 − t^{−1}dt ⊗ ı_{R_{M}})ν,

where R^{#_{ϕ}} is the 1-form dual to R under ω_{ϕ}. We see that this 1form is in T_{1}, since (ϕ_{a})_{∗}^{−1}R = R, R^{#} = α, the canonical 1-form, and (ϕ_{a})_{∗}α ∈ T_{1}.

Now find a chain x_{0} in C_{k+1}^{j+1,i+1;max{l,0}} representing the class c at E_{1}(A_{σ}(X)). By Proposition 4.14, one can find the desired x. □

Proof of Theorem 4.11. By applying Proposition 4.15 repeatedly, it follows that F_{k−N−1}HH_{k}(A_{σ}(X)) = 0. We must still prove that the edge map F_{i}HH_{k}(A_{σ}(X)) → E_{2}^{i,k−i} is surjective. Start with a ∈ T_{j}F_{i}C_{k} so that σ(a) represents a class in E_{2}^{i,N}(A_{σ}(X)), hence assuming that a contains no dt. This means that we can change the sub-principal symbol of a in T_{j} so that b(a) ∈ F_{i−2}.

From Proposition 4.14, there exists A ∈ T_{j}C_{k}(Ψ_{a}(X)), σ(A) = a, such that b(A) ∈ T_{j−1}. Observe that we can assume that the sub-principal symbols of A and a agree up to a chain μ ∈ T_{j}F_{i−1} with the property b(μ) ∈ F_{i−2}. This follows from the proof of Proposition 4.14. Now b(A) represents a form which is the zero element in E_{2}^{i−2,N+1}(A_{σ}(X)). We can assume that this form contains no dt (if not, project onto the no-dt part). As in Proposition 4.15, it follows that there exists x ∈ T_{j}F_{i−1} such that b(A) − b(x_{1}) ∈ T_{j}F_{i−3} and T − order(x) ≤ max{ord(a), 0}. Repeated applications of Proposition 4.15 will yield an infinite sequence x_{p} ∈ F_{i−p} of chains of uniformly bounded orders and of t-degree j. Due to the decreasing operator orders and bounded T -orders, such a sequence is asymptotically summable in the sense of our definition of chain spaces. It follows that b(A − ∑ x_{p}) = 0. The case where the class a contains dt is similar. □

Corollary 4.16. The group HH(A_{∂,σ}(X)) splits into eigenspaces of the L_{D} action with integer eigenvalues.

Proof. First, note that the spectral sequence associated to A_{∂,σ}(X) also degenerates at E^{2} and is convergent, by the same argument as above. Since α and β commute with b and L_{D}, the splitting form Proposition 4.9 is inherited by Hochschild homology and is preserved by L_{D}. Via the edge inclusion, E_{2}^{k−N−1,N+1}(A_{∂,σ}(X)) is a subspace of HH_{k}(A_{∂,σ}(X)). By Proposition 4.7, β acts the identity and α acts as 0 on this subspace, which therefore coincides with im(β). Then the inclusion im(α) → HH_{k}(A_{∂,σ}(X)), followed by the edge surjection HH_{k}(A_{∂,σ}(X)) → E_{2}^{k−N,N}(A_{∂,σ}(X)), is an isomorphism which commutes with L_{D}. Now use Proposition 4.8. □

5. The semi-classical limit algebra

Consider the case where X = M, i.e., the fiber of the fibration X → M is just one point. In this case, the adiabatic algebra is called the semiclassical limit algebra. Consider the short exact sequence

(24)  0 → I_{∂}(M) ↪^{ı} A_{∂}(M) →^{p} A_{∂,σ}(M) → 0.

This sequence is compatible with the filtration T_{j}, hence it induces a short exact sequence of the associated graded algebras (25)

0 → 𝒮(^{a}T^{∗}M_{|t=0})⊗L ↪^{ı} S_{cl}(^{a}T^{∗}M_{|t=0})⊗L →^{p} S(^{a}T^{∗}M_{|t=0} \{0})⊗L → 0.

In this section, ^{∂}E stands for the spectral sequences with respect to the filtration T_{j} of the Hochschild complexes of I_{∂}(M), A_{∂}(M) and A_{∂,σ}(M). The following Proposition is entirely similar to Proposition 4.1. We have kept the powers of t and dt in order to keep track of the filtration. Proposition 5.1. Let ^{∂}E denote the spectral sequence of the Hochschild complex with respect to the filtration T_{i}. Then the E_{1} terms of the semiclassical algebras are:

∂E_{1}^{i,j}(I_{∂}(M)) ≅ E_{1}^{i,j}(𝒮(^{a}T^{∗}M_{|t=0}) ˆ⊗L)
≅ t^{−i}Λ_{𝒮}^{i+j}(^{a}T^{∗}M_{|t=0}) ⊕ Λ_{𝒮}^{i−j−1}(^{a}T^{∗}M_{|t=0})t^{−i−1}dt;
∂E_{1}^{i,j}(A_{∂}(M)) ≅ E_{1}^{i,j}(S_{cl}(^{a}T^{∗}M_{|t=0}) ˆ⊗L)
≅ t^{−i}Λ_{S_{cl}}^{i+j}(^{a}T^{∗}M_{|t=0}) ⊕ Λ_{S_{cl}}^{i−j−1}(^{a}T^{∗}M_{|t=0})t^{−i−1}dt;
∂E_{1}^{i,j}(A_{∂,σ}(M)) ≅ E_{1}^{i,j}(S(^{a}T^{∗}M_{|t=0}) ˆ⊗L)
≅ t^{−i}Λ_{S}^{i+j}(^{a}T^{∗}M_{|t=0}) ⊕ Λ_{S}^{i−j−1}(^{a}T^{∗}M_{|t=0})t^{−i−1}dt.

Proof. The subscripts 𝒮, S_{cl} and S stand for forms with Schwartz, classical symbol, respectively homogeneous symbol coefficients The first isomorphism for each space follows from the fact that the induced product on A_{∂}^{ℤ,[ℤ]}(M) is the usual commutative product on functions (Proposition 2.8). The second isomorphism for each space is realized by the Hochschild-Kostant-Rosenberg map χ (Definition 9). □

Let d_{1} be the first differential in the above spectral sequences. Proposition 5.2. In the spectral sequences E(I_{∂}(M)), E(A_{∂}(M)) and E(A_{∂,σ}(M)), the first differential takes the form

(26)  d_{1} = (∗_{a})^{−1}d_{v} ∗_{a} .

Proof. We first prove the assertion for the algebra A_{∂,σ}(M). Let a ∈ C_{i+j}^{[s],[i]}(A_{∂,σ}(M)) be a chain of pure homogeneity s in the fibers of ^{a}T^{∗}M, which survives at ^{∂}E_{1}^{i,j}. Then a also survives at ^{σ}E_{1}^{s,i+j−s}(A_{∂,σ}(M)). The identification of both ^{∂}E_{1} and ^{σ}E_{1} with forms on ^{a}T^{∗}M is realized by the map χ. From the definition, d_{1}[a] = χ([b(a)]_{[i−1]}) is represented by the part of b(a) of degree [−i + 1] in t. From the structure of the product on the boundary algebras, this is exactly the part of pure homogeneity s − 1 of b(a), i.e., it is equal to d_{1}^{a}[a]. The claim follows from Proposition 4.5.

Note that d_{1} and (∗_{a})^{−1}d_{v}∗_{a} are operators with polynomial coefficients in the fibers of^{a}T^{∗}M_{|t=0}. We claim that we can recover d_{1} from its action on homogeneous forms, i.e.,^{∂}E_{1}^{i,j}(A_{∂,σ}(M)). Indeed, we can retrieve the coefficients of a differential operator with polynomial coefficients from its action on polynomials. □

We can describe the homogeneity with respect to t of the operator ∗_{a} as follows:

(27) t^{−i}Λ^{k}(^{a}T^{∗}M_{|t=0}) ⊕ t^{−i}dt ∧ Λ^{k−1}(^{a}T^{∗}M_{|t=0})

→^{∗_{a}} t^{−i+k−n}Λ^{2n−k}(^{a}T^{∗}M_{|t=0}) ⊕ t^{−i+k−n−2}dt ∧ Λ^{2n−k+1}(^{a}T^{∗}M_{|t=0}).

The following Proposition is a consequence of (27) and Proposition 5.2. Proposition 5.3. Conjugation by ∗_{a}^{M} on ^{∂}E_{1} induces the following identifications for the E_{2} terms:

(28)  ^{∂}E_{2}^{i,k}(I_{∂}(M)) ≅ t^{k−n}H_{𝒮}^{2n−i−k}(^{a}T^{∗}M_{|t=0})
⊕ t^{k−n−2}dt ⊗ H_{𝒮}^{2n−i−k+1}(^{a}T^{∗}M_{|t=0});
∂E_{2}^{i,k}(A_{∂}(M)) ≅ t^{k−n}H_{S}^{2n−i−k}(^{a}T^{∗}M_{|t=0})
⊕ t^{k−n−2}dt ⊗ H_{S}^{2n−i−k+1}(^{a}T^{∗}M_{|t=0});
(29) ^{∂}E_{2}^{i,k}(A_{∂,σ}(M)) ≅ t^{k−n}H^{2n−i−k}(^{a}S^{∗}M_{|t=0} × S^{1})
⊕ t^{k−n−2}dt ⊗ H^{2n−i−k+1}(^{a}S^{∗}M_{|t=0} × S^{1});

We can now compute the effect of derivations on these E_{2} terms. As in Proposition 4.7, the operator e_{D} acts on E_{2}(I_{∂}(M)), E_{2}(A_{∂,σ}(M)) and E_{2}(A_{∂}(M)) by contraction with the vector field t\frac{d}{dt}. Hence the splitting from Proposition 5.3 is given by the null-space and image of e_{D}.

Proposition 5.4. At the level of E_{2}^{i,k}, the operator L_{D} equals k − n on ker(e_{D}) and k − n − 1 on im(e_{D}).

Proof. Imitate the proof of Proposition 4.8 to obtain that L_{D} acts as the Lie derivative L_{R_{M} +t \frac{d}{dt}}. Hence, e_{D} has eigenvalues k−n and k−n−1 on E_{2}^{i,k}, corresponding to the splitting from Proposition 5.3. This splitting is given by the image and null-space of e_{D}. □

Corollary 5.5. The spectral sequences from Proposition 5.3 degenerate at E_{2}.

Proof. By naturality, the boundary maps in the spectral sequences commute with the maps e_{D} and L_{D}. In particular, they preserve the null-space and the image of e_{D}. On ker(e_{D}) ∩ ^{∂}E_{2}^{i,k}, L_{D} acts as multiplication by k − n. This shows that for s ≥ 2, the map d_{s} : ker(e_{D}) ∩ ^{∂}E_{2}^{i,k} → ker(e_{D}) ∩ ^{∂}E_{2}^{i+s−1,k−s} must vanish. Similarly, d_{s} vanishes on the image of e_{D}. □

Corollary 5.6. Let OM denote the orientation bundle of M. Then

HH_{k}(I_{∂}(M)) ≅ (H^{n−k}(M, OM) ⊕ H^{n−k+1}(M, OM)) ⊗ L.

Proof. The convergence in the sense of Section 4.3 of the spectral sequence is easy, since there is only one filtration to control. In particular there is no residual homology (i.e., in filtration −∞). Since we deal with vector spaces over L, we see that HH_{k}(I_{∂}(M)) is isomorphic to the direct sum of its filtration quotients, which by Corollary 5.5 are just the E_{2} terms computed in Proposition 5.3. In fact, the isomorphism is canonical, corresponding to the splitting in eigenvalues for L_{D}. Finally, we use the Thom isomorphism to integrate the cohomology along the fibers, picking up coefficients in the orientation bundle along the way. □

6. The homology of the boundary ideal

In this section, E(I_{∂}(X)) will represent the spectral sequence with respect to the filtration T_{j} of the Hochschild complex of I_{∂}(X). We define an algebra map ϕ : Ψ_{a}^{−∞}(M)→Ψ_{a}^{−∞}(X), and show that it induces an isomorphism E_{1}(I_{∂}(M)) ≅ E_{1}(I_{∂}(X)). This will imply that ϕ induces an isomorphism in Hochschild homology.

There exists a natural fibration X_{a}^{2} →^{π} M_{a}^{2}. The fibers of this fibration are F × F, where F is the fiber of X → M. Choose a smooth family dv of densities of volume 1 in each fiber of X → M. Definition 6.1. Define ϕ : Ψ_{a}^{−∞}(M) → Ψ_{a}^{−∞}(X) by

A ↦ (π^{∗}A) ⊗ dv_{R},

where dv_{R} is the density dv in the second term of the fiber F × F.

It is immediate that ϕ is a map of algebras, from the structure of the triple spaces. Indeed, X_{a}^{3} fibers naturally over M_{a}^{3}, with fiber F^{3} and the assertion follows from (4).

6.1. A connection on the vertical algebra. Consider the following diagram of fibrations:

(30)  \xymatrix{X \ar[d]^{π} & π^{∗}(^{a}T^{∗}M_{|t=0}) \ar[d]_{π} \ar[l]_{p} \\ M & ^{a}T^{∗}M_{|t=0} \ar[l]_{p}}

Fix a connection in X → M and lift it through p to a connection in π^{∗}(^{a}T^{∗}M_{|t=0}) → ^{a}T^{∗}M_{|t=0}. Over a point x ∈ M, the restricted fibration p^{−1}π^{−1}(x) → p^{−1}(x) is canonically isomorphic to the product X_{x} × aT_{x}^{∗}M_{|t=0}, and the induced connection is the canonical trivial connection on the product.

Let Γ_{S_{cl}}(Λ^{1}(^{a}T^{∗}M_{|t=0})) denote the space of 1-forms on ^{a}T^{∗}M_{|t=0} with symbolic coefficients in the fibers. The connection defines a derivation

𝒮(π^{∗}(^{a}T^{∗}M_{|t=0})) →^{∇} Γ_{S_{cl}}(Λ^{1}(^{a}T^{∗}M_{|t=0})) ⊗ 𝒮(π^{∗}(^{a}T^{∗}M_{|t=0})).

Recall the identification of Ψ_{a}^{−∞,[0]}(X), via Fourier transform, with 𝒮(X ×_{M} X ×_{M} ^{a}T^{∗}M_{|t=0}, Ω_{R}). Let

π_{R}, π_{L} : X ×_{M} X ×_{M} ^{a}T^{∗}M_{|t=0} → X ×_{M} ^{a}T^{∗}M_{|t=0}

be the projections on the first, respectively the second X factor. The action of Ψ_{a}^{ℤ,[0]} on Ψ_{a}^{−∞,[0]} is π_{L}^{∗}(C^{∞}(X))-linear. After Fourier transform, Ψ_{a}^{ℤ,[0]} acts 𝒮(^{a}T^{∗}M_{|t=0})-linearly on 𝒮(X ×_{M} ^{a}T^{∗}M_{|t=0}):

Ψ_{a}^{ℤ,[0]}⊗𝒮(X×_{M}^{a}T^{∗}M_{|t=0}) ∋ (A, f) ↦ π_{R}_{∗}(Aπ_{L}^{∗}(f)) ∈ 𝒮(X×_{M}^{a}T^{∗}M_{|t=0}).

This action is faithful. We extend by duality the derivation ∇ to Ψ_{a}^{ℤ,[0]}:

(∇A)(f) = ∇(Af) − A(∇(f)).

In local coordinates, one sees easily that ∇ preserves Ψ_{a}^{−∞,[0]}.

Let ∇^{t} = ∇ + \frac{∂}{∂t}dt be the extension of ∇ to Ψ_{a}^{ℤ,[ℤ]}(X). Then ∇^{t} commutes with multiplication by 𝒮(^{a}T^{∗}M_{|t=0})⊗L, in the following sense: Let A ∈ Ψ_{a}^{ℤ,[ℤ]}(X), V ∈ Γ_{𝒮}(T (^{a}T^{∗}M_{|t=0})) ⊗ L ⊕ 𝒮(^{a}T^{∗}M_{|t=0}) ⊗ L\frac{∂}{∂t} and g ∈ 𝒮(^{a}T^{∗}M_{|t=0}) ⊗ L. Then

(31)  ∇^{t}_{gV} (A) = g∇^{t}_{V} (A).

By duality, ∇^{t} is a derivation on Ψ_{a}^{ℤ,[0]} ⊗ L.

6.2. The analog of χ. The product in I_{∂}(X) is a deformation of the vertical product. It follows that E_{0}(I_{∂}(X)) ≅ E_{0}(Ψ_{a}^{−∞,[ℤ]}(X)). Moreover, this isomorphism commutes with d_{0}. By using the Künneth formula for continuous Hochschild homology [18], we obtain

E_{1}^{i,j}(I_{∂}(X)) ≅ E_{1}^{i,j}(Ψ_{a}^{−∞,[ℤ]}(X))
= HH_{i+j}(Ψ_{a}^{−∞,[ℤ]}(X))_{[i]}

= t^{−i} ⊗ HH_{i+j}(Ψ_{a}^{−∞,[0]}) ⊕ t^{−i+1}dt ⊗ HH_{i+j−1}(Ψ_{a}^{−∞,[0]}).

There exists a 𝒮(^{a}T^{∗}M_{|t=0})–valued trace functional on the vertical ideal Ψ_{a}^{−∞,[0]}, defined by push-forward under the projection map π of the restriction to the diagonal of each fiber of kernels in Ψ_{a}^{−∞,[0]}. Actually, Tr_{V} extends to Ψ_{a}^{−m−ϵ,ℤ}(X) for ϵ > 0:

A ↦^{Tr_{V}} ∫_{Δ×_{M}^{a}T^{∗}M_{|t=0}/^{a}T^{∗}M_{|t=0}} A_{|_{Δ×_{M}^{a}T^{∗}M_{|t=0}}}.

Let A = A_{0} ⊗A_{1} ⊗…⊗A_{k} ∈ C_{k}(Ψ_{a}^{−∞,[ℤ]}(X)). The following definition was inspired by [18]. Define χ_{a} : C_{k}(Ψ_{a}^{−∞,[0]} ⊗ L) → Λ_{𝒮}(^{a}T^{∗}M_{|t=0}) ⊗ ΛL by

(32)  χ_{a}(A) := Tr_{V}(A_{0}∇^{t}A_{1} ∧ … ∧ ∇^{t}A_{k}).

For V_{1}, …, V_{k} ∈ Γ_{S}(T (^{a}T^{∗}M_{|t=0})) ⊗ L ⊕ S_{cl}(^{a}T^{∗}M_{|t=0}) ⊗ L\frac{∂}{∂t}, we have

χ_{a}(A)(V_{1}, …, V_{k}) = Tr_{V} ( ∑_{σ∈S_{k}} A_{0}∇^{t}_{V_{σ(1)}}(A_{1}) … ∇^{t}_{V_{σ(k)}}(A_{k})) .

Note that χ_{a}(A) can be defined for any A = A_{0} ⊗ A_{1} ⊗ … ⊗ A_{k} ∈ (Ψ_{a}^{ℤ,[ℤ]}(X))^{⊗k+1}, such that the real part of the order of A_{0}∇^{t}A_{1} ∧ … ∧ ∇^{t}A_{k} is strictly less than −m. Proposition 6.2. If A ∈ F_{−m−k−ϵ}C_{k}(Ψ_{a}^{ℤ,[ℤ]}(X)), then χ_{a} ∘ bA = 0.

Proof. From the Leibniz identity for ∇^{t}, it follows that

\begin{aligned}χ_{a}(b(A)) & = (−1)^{k} Tr_{V}[A_{k}A_{0}∇^{t}A_{1} ∧ … ∧ ∇^{t}A_{k−1} \\ & − A_{0}∇^{t}A_{1} ∧ … ∧ ∇^{t}(A_{k−1})A_{k}]. \end{aligned}

Now use the fact that Tr_{V} vanishes on commutators of operators of total order less than −m − ϵ. □

In particular, χ_{a} descends to the Hochschild homology of the vertical ideal I_{∂}(X). Proposition 6.3. For A, B ∈ Ψ_{a}^{−∞,[0]}(M) ⊗ L,

∇^{t}(ϕ(A))ϕ(B) = ϕ(d(A)B).

Proof. Straightforward; however, it is not true that ∇^{t}(ϕ(A)) = ϕ(d(A))! □

Proposition 6.4. The map χ_{a} is related to χ through χ_{a} ∘ ϕ = χ. Hence, χ_{a} is surjective.

Proof. Let A = A_{0} ⊗ A_{1} ⊗ … ⊗ A_{k} ∈ C_{k}(Ψ_{a}^{−∞,[0]}(M) ⊗ L). Then

χ_{a}(ϕ(A)) = Tr_{V}[ϕ(A_{0})∇^{t}ϕ(A_{1}) ∧ … ∧ ∇^{t}ϕ(A_{k})]
= Tr_{V}[∇^{t}ϕ(A_{1}) ∧ … ∧ (∇^{t}ϕ(A_{k}))ϕ(A_{0})]
= Tr_{V}[∇^{t}ϕ(A_{1}) ∧ … ∧ ∇^{t}ϕ(A_{k−1}) ∧ ϕ(d(A_{k})A_{0})]
= … = Tr_{V}[ϕ(d(A_{1}) ∧ … ∧ d(A_{k})A_{0})]
= d(A_{1}) ∧ … ∧ d(A_{k})A_{0}
= χ(A).

We have used Proposition 6.3 and commutativity of the trace. □

6.3. The E_{1} term. Since χ is an isomorphism on Hochschild homology [7], by Proposition 6.4, χ_{a} must be surjective. In order to prove that χ_{a} is an isomorphism at E_{1} we need to prove injectivity. Recall the decomposition b = ∑_{i=0}^{k}(−1)^{i}b_{i} of the Hochschild differential on C_{k}.

Let g be a metric on M and r_{g} the injectivity radius of (M, g). Let ψ be a cut-off function supported inside (−r_{g}, r_{g}) with ψ(0) = 1. For two points x_{1}, x_{2} in M at distance less than r_{g}, let τ_{x_{2}}^{x_{1}} denote the parallel transport along the shortest geodesic from x_{1} to x_{2} in X → M, with respect to the connection ∇, and also in ^{a}T^{∗}M_{|t=0} → M with respect to ∇^{LC}. Fix again a vertical density dv on the fibers of X → M.

Let P_{i} = (ξ_{i}, u_{i}, v_{i}) denote a typical point in ^{a}T^{∗}M_{|t=0} ×_{M} X ×_{M} X over a base point x_{i} ∈ M, i = 0, …, k.h The topological chain space C_{k}(Ψ_{a}^{−∞,[ℤ]}(X)) is just

𝒮((^{a}T^{∗}M_{|t=0} ×_{M} X ×_{M} X)^{k+1}) ⊗ L(t_{0}, …, t_{k}) ⊗ dv_{0} ⊗ … ⊗ dv_{k}.

Let a be a cycle in C_{k}(Ψ_{a}^{−∞,[0]} ⊗ L),

a = A(t_{0}, ξ_{0}, u_{0}, v_{0}, …, t_{k}, ξ_{k}, u_{k}, v_{k})dv_{0} ⊗ … ⊗ dv_{k}.

For l ∈ {0, …, k + 1}, we say that a satisfies the property P_{l} if the following two conditions are satisfied:

(1) b_{j}(a) = 0 for j = 0, …, l − 1.

(2) The function A is independent of v_{0}, …, v_{l−1}. Proposition 6.5. If a satisfies P_{l}, then it is homologous to another cycle a′, which satisfies P_{l+1}.

Proof. Define h_{l} : C_{k}(Ψ_{a}^{−∞,[0]} ⊗ L) → C_{k+1}(Ψ_{a}^{−∞,[0]} ⊗ L) by h_{l}(a)(t_{0}, P_{0}, …, t_{k+1}, P_{k+1})

= ψ(d(x_{l}, x_{l+1}))e^{−∥τ_{x_{l}}^{x_{l+1}}ξ_{l+1}−ξ_{l}∥^{2}}dv_{l}
⊗ a(t_{0}, P_{0}, …, t_{l}, ξ_{l}, u_{l}, τ_{x_{l}}^{x_{l+1}}v_{l+1}, …, t_{k+1}, P_{k+1}).

Note that h_{l}(a) has Schwartz behavior in both ξ_{l} and ξ_{l+1}−ξ_{l}, therefore also in ξ_{l+1}. Let a′ be defined by the following equation:

b(h_{l}(a))

= (−1)^{l}a + (−1)^{l+1}ψ(d(x_{l}, x_{l+1}))e^{−∥τ_{x_{l}}^{x_{l+1}}ξ_{l+1}−ξ_{l}∥^{2}}dv_{l}

⊗ (∫_{X_{x_{l+1}}} A(t_{0}, P_{0}, …, t_{l}, ξ_{l}, u_{l}, τ_{x_{l}}^{x_{l+1}}z, t_{l+1}, ξ_{l+1}, z, v_{l+1}, …)dz

− ∫_{X_{x_{l+1}}} A(t_{0}, P_{0}, …, t_{l}, ξ_{l}, u_{l}, z, t_{l}, ξ_{l}, z, τ_{x_{l}}^{x_{l+1}}v_{l+1}, …,)dz)

− h_{l}(b(a))

= (−1)^{l}(a − a′).

Since, by hypothesis, b(a) = 0, we have h_{l}(b(a)) = 0. In addition, a′ satisfies P_{l} since a does. Thus b_{l}(a′) = 0 and a′ does not depend on v_{l}, hence a′ satisfies P_{l+1}. □

By induction, every cycle a ∈ C_{k}(Ψ_{a}^{−∞,[0]} ⊗ L) is homologous to a cycle which satisfies P_{k}. Definition 6.6. Let a be a cycle in C_{k}(Ψ_{a}^{−∞,[0]} ⊗ L), which satisfies P_{k}. For l ∈ {0, …, k + 1}, we say that a satisfies R_{l} if the the function A is independent of u_{0}, …, u_{l−1}. Proposition 6.7. If a satisfies R_{l}, then it is homologous to another cycle a′ which satisfies R_{l+1}.

Proof. Define q_{l} : C_{k}(Ψ_{a}^{−∞,[ℤ]}(X)) → C_{k+1}(Ψ_{a}^{−∞,[ℤ]}(X)) by q_{l}(a)(t_{0}, P_{0}, …, t_{k+1}, P_{k+1})

:= ψ(d(x_{l}, x_{l+1}))e^{−∥τ_{x_{l}}^{x_{l+1}}ξ_{l+1}−ξ_{l}∥^{2}}dv_{l}
⊗ a(t_{0}, P_{0}, …, t_{l}, ξ_{l}, τ_{x_{l}}^{x_{l+1}}u_{l+1}, …, t_{k+1}, P_{k+1}).

As above, q_{l}(a) is well defined. Since a satisfies P_{k}, it follows that b(q_{l}(a))

= (−1)^{l}a − (−1)^{l}ψ(d(x_{l}, x_{l+1}))e^{−∥τ_{x_{l}}^{x_{l+1}}ξ_{l+1}−ξ_{l}∥^{2}}dv_{l}
⊗ (∫_{X_{x_{l+1}}} A(t_{0}, P_{0}, …, t_{l}, ξ_{l}, τ_{x_{l}}^{x_{l+1}}u_{l+1}, t_{l+1}, ξ_{l+1}, z, …)dz)
= (−1)^{l}(a − a′),

where the last equality is the definition of a′. Note that a′ satisfies P_{k} and R_{l} since a does, and moreover it is independent of u_{l}. □

Therefore, every cycle a ∈ C_{k}(Ψ_{a}^{−∞,[0]} ⊗ L) is homologous to a cycle which satisfies P_{k} and R_{k+1}. Corollary 6.8. If a ∈ C_{k}(Ψ_{a}^{−∞,[0]} ⊗ L) is a cycle, then χ_{a}(a) is independent of ∇.

Proof. We have seen above that a is homologous to some a′ which lies in the image of the map ϕ, i.e a′ = ϕ(c). By Proposition 6.2, we have χ_{a}(a) = χ_{a}(a′). Finally, by Proposition 6.4, it follows that χ_{a}(a′) = χ(c), hence χ_{a}(a) is independent of ∇. □

Proposition 6.9. The map

χ_{a} : E_{1}(I_{∂}(X)) ≅ HH_{∗}(Ψ_{a}^{−∞,[ℤ]}(X)) → Λ_{𝒮}(^{a}T^{∗}M_{|t=0}) ⊗ ΛL

is injective. Together with Proposition 6.4, this implies that χ_{a} is an isomorphism.

Proof. Let a be a cycle in C_{k}(Ψ_{a}^{−∞,[0]} ⊗ L) so that [a] ∈ ker(χ_{a}). We can assume that a satisfies P_{k} and R_{k+1}. This means that

a ∈ im(ϕ : C_{k}(Ψ_{a}^{−∞,[0]}(M) ⊗ L) → C_{k}(Ψ_{a}^{−∞,[0]}(X) ⊗ L).

Say a = ϕ(ā). Then, by Proposition 6.4, we have

0 = χ_{a}(a) = χ_{a}(ϕ(ā)) = χ(ā).

Since χ is an isomorphism, it follows that ā is exact in C_{∗}(Ψ_{a}^{−∞,[0]}(M)⊗ L). As ϕ is an algebra map, it commutes with the Hochschild differential, and so a is also exact. □

Theorem 6.10. The map ϕ induces an isomorphism between the homology groups of the semi-classical and adiabatic smoothing ideals (see Subsection 2.2 and Section 5):

\begin{aligned}HH_{∗}(I_{∂}(M)) & ≅^{ϕ} HH_{∗}(I_{∂}(X)) \\ & ≅ (H^{n−k}(M, OM) ⊕ H^{n−k+1}(M, OM)) ⊗ L. \end{aligned}

Proof. Propositions 6.4 and 6.9 imply that χ_{a} is an isomorphism, hence the map ϕ induces an isomorphism between E_{1}(I_{∂}(M)) and E_{1}(I_{∂}(X)). The result follows using a general property of spectral sequences [14]. We only need to notice that the two spectral sequences are (almost tautologically) convergent. □

The map ϕ and the derivation D commute. From Theorem 6.10 and the similar properties of I_{∂}(M), it follows that e_{D} acts on E_{∞}(I_{∂}(X)) by contraction with the vector field t\frac{d}{dt}. The space E_{∞}(I_{∂}(X)) splits as the direct sum of the null-space and the image of e_{D}. The effect of L_{D} on ker(e_{D}) ∩^{∂}E_{∞}^{i,k}(I_{∂}(X)), respectively on im(e_{D}) ∩^{∂}E_{∞}^{i,k}(I_{∂}(X)), is multiplication by k − n, respectively k − n − 1.

Notice that HH_{0}(I_{∂}(X)) is always a 1-dimensional L-vector space. Thus there exists essentially a unique trace functional on I_{∂}(X). In Subsection 7.3 we will show that this trace extends to a trace on Ψ_{a}^{−∞}(X) which for positive t is simply the operator trace.

7. The residue functional

7.1. Definition of the residue. Since Q^{z} ∈ Ψ_{a}^{z,0}(X), multiplication by Q^{z} descends to Ψ_{a}^{ℤ,[0]} and extends to chains in C_{∗}(Ψ_{a}^{ℤ,[ℤ]}(X)) by Q^{z}(a_{0} ⊗ … ⊗ a_{h}) = (Q^{z}a_{0}) ⊗ … ⊗ a_{h}. We denote by ∗_{a}^{M} the operator ∗_{a} on ^{a}T^{∗}M. Let OM be the orientation bundle of M. Definition 7.1. For z ∈ C, ℜ(z) sufficiently negative, define

F (z) : C_{h}(Ψ_{a}^{ℤ,[ℤ]}(X)) → Λ^{n−h}(M, OM) ⊗ L ⊕ Λ^{n−h+1}(M, OM) ⊗ Ldt
A ↦ ∫_{^{a}T^{∗}M_{|t=0}/M} ∗_{a}^{M}(χ_{a}(Q^{z}A)).

Proposition 7.2. For ℜ(z) < h−N−i, the dt-free part, ı_{\frac{d}{dt}} dt∧F (z), of

F (z) is well-defined and holomorphic on F_{i}C_{h}(Ψ_{a}^{ℤ,[ℤ]}(X)). For ℜ(z) < h−N−i−1 the dt part, ı_{\frac{d}{dt}} F (z), of F (z) is well-defined and holomorphic

on F_{i}C_{h}(Ψ_{a}^{ℤ,[ℤ]}(X)).

Proof. From the definition, for V ⊂ U open sets in M and V relatively compact in U, F (z)(A)_{|_{V}} depends only on A_{|(π^{−1}(U))^{h+1}}. Hence F (z) is local in M.

Lemma 7.3. The operator F (z) is independent of ∇.

Proof. Work over a coordinate patch U ⊂ M. Let (x_{i}, ξ_{j}), (x_{i}, ξ̃_{j}) be local coordinates adapted to the cotangent structure in T^{∗}M and aT^{∗}M_{|t=0} over U. The map ϕ_{a} takes the form (x_{i}, ξ_{j}) ↦ (x_{i}, tξ_{j}). Let Id be the local isomorphism (x_{i}, ξ_{j}) ↦ (x_{i}, ξ_{j}). Let ∗^{M} be Brylinski’s duality operator on Λ(T^{∗}M). We denote by ∗^{U} the conjugate of ∗^{M} by Id. On h-forms,

(33)  ı_{\frac{d}{dt}} dt ∧ (∗_{ϕ}^{U} + t^{−1}dt ⊗ ı_{R_{M}} ∗_{ϕ}^{U}) = t^{h−n} ∗^{U} ı_{\frac{d}{dt}} dt ∧ .

Using this and the fact that the integral along the fiber vanishes on forms that are not multiples of dξ_{1} ∧ … dξ_{n}, we get

\begin{aligned}ı_{\frac{d}{dt}} dt ∧ F (z)(A) & = ∫_{U×ℝ^{n}/U} t^{h−n} ∗^{U} ı_{\frac{d}{dt}} dt ∧ χ_{a}(Q^{z}A) \\ & = t^{h−n} ∫_{U×ℝ^{n}/U} ∗^{U} Tr_{V}({ \atop Q^{z}a_{0}}∇_{ξ}â_{1} ∧ … ∧ ∇_{ξ}â_{h}), \end{aligned} ; (34)

where ∇_{ξ} = ∑ ∇_{∂_{ξ_{i}}}dξ_{i}. Recall that we Fourier-transform the kernels in the horizontal tangent directions. Since the connection is lifted to X ×_{M} X ×_{M} ^{a}T^{∗}M → ^{a}T^{∗}M from X ×_{M} X → M, it follows that ∇_{∂_{ξ_{i}}} is actually independent of ∇. This proves that the dt-free part of F (z) is independent of ∇. The other case follows by replacing formula (33) with (36). □

Using Lemma 7.3, we can assume that we work in local coordinates and that formula (34) holds. From the properties of symbols, the assumption on the order of A, and the assumption ℜ(z) + i − h < −N, it follows that

(35)  { \atop Q^{z}a_{0}}∇_{ξ}â_{1} ∧ … ∧ ∇_{ξ}â_{h} ∈ dξ^{h}F_{z+i−h}Ψ_{a}^{ℤ,[ℤ]}(X)

and hence is of trace class (recall that all operators of order less than −m are of vertical trace class). Taking the trace increases homogeneity by m, hence

Tr_{V}({ \atop Q^{z}a_{0}}∇_{ξ}â_{1} ∧ … ∧ ∇_{ξ}â_{h}) ∈ dξ^{h}S^{z+i−h+m}(^{a}T^{∗}M_{|t=0}) ⊗ L

and so

∗^{M} Tr_{V}({ \atop Q^{z}a_{0}}∇_{ξ}â_{1} ∧ … ∧ ∇_{ξ}â_{h}) ∈ Λ_{[z+i−h+N]}^{2n−h}(^{a}T^{∗}M_{|t=0}) ⊗ L.

Hence for ℜ(z) + i − h + N < 0, this form is of negative homogeneity and therefore integrable along the fibers of ^{a}T^{∗}M_{|t=0}/M. This proves the first assertion. The other one is similar and uses the identity

(36) ı_{\frac{d}{dt}} (∗_{ϕ}^{U} + t^{−1}dt ⊗ ı_{R_{M}} ∗_{ϕ}^{U}) = t^{h−n−1} ∗^{U} ı_{\frac{d}{dt}} + t^{h−n−2}α ∧ ∗^{U}ı_{\frac{d}{dt}} dt∧

instead of (33). Here α is the pull-back via Id^{−1} of the canonical form on T^{∗}M, and has homogeneity 1. □

Remark 7.4. From (34), we see that ı_{\frac{d}{dt}} dt ∧ F (z) shifts the t-degree by h − n on h-chains. Proposition 7.5. Let A_{z} ∈ F_{i}C_{h}(Ψ_{a}^{ℤ,[ℤ]}(X)) be an entire family of hchains in filtration i. Then F (z)(A_{z}) is meromorphic, with at most simple poles at the real integers.

Proof. Denote by H(B) the holomorphic functions in the band B = {−1 < ℜ(z) < 1}. We shall show that F (z)(A_{z}) has at most a simple pole at 0 in the band B. Fix a total symbol map for adiabatic operators, i.e., a map q : Ψ_{a}(X) → S(^{a}T^{∗}X)[t^{−1}], which extends the symbol map. Let A_{z}^{j} be the component of order z+j of q({ \atop Q^{z}a_{0}}(z)⊗â_{1}(z) …⊗â_{h}(z)). By Proposition 7.2, only a finite number of A_{z}^{j}’s are significant for the poles of F (z)(A_{z}) in B. Fix a metric |r| = |(ξ, η)| on ^{a}T^{∗}X. Choose B_{z}^{j} such that q(B_{z}^{j}) = A_{z}^{j}. Let ψ be a cut-off function, ψ(r) ≡ 0 for small r, ψ(r) ≡ 1 for r ≥ 1.

Lemma 7.6. The following identity holds modulo a holomorphic family of forms on Λ_{𝒮}(^{a}T^{∗}M):

∫_{^{a}T^{∗}M_{|t=0}/M} ∗_{a}^{M} Tr_{V}(χ_{a}(B_{z}^{j})) ≡ ∫_{^{a}T^{∗}X_{|t=0}/M} ∗_{a}^{X}ψ(ξ, η)χ(A_{z}^{j}).

Proof. Both terms are local, so we can prove the statement in local coordinates. The left-hand side is independent of the connection. We claim that the part in ∗_{a}^{X}χ(A_{z}^{j}) which is a multiple of the fiber volume form dV of ^{a}T^{∗}X_{|t=0}/M, is also independent of ∇. Indeed,

∗_{a}^{X} = (1 + t^{−1}dt ⊗ ı_{R_{M}})ϕ_{a}^{∗}^{−1} ∗ ϕ_{a}^{∗},

where ϕ_{a} is given by (10). Let μ denote a monomial form in local coordinates. First, if μ does not contain a multiple of dV, then neither does (1 + t^{−1}dt ⊗ ı_{R_{M}})ϕ_{a}^{∗}^{−1}μ. Secondly, if μ contains any dx, dη, dy, then ∗μ does not contain multiples of dV [4]. Finally, if μ contains some dx, dη, dy, then so do all monomials in ϕ_{a}^{∗}μ. Therefore, only those μ that contain only dξ’s survive in ∫_{^{a}T^{∗}X_{|t=0}/M} ψ(ξ, η)∗_{a}^{X} μ. Moreover, the result is seen to be independent of the map A in (10), which can therefore be assumed of the form A(x, y, ξ, η) = ξ. Then, modulo H(ℂ, Λ_{𝒮}(^{a}T^{∗}M)), we have

Tr_{V}(χ_{a}(B_{z}^{j})) = ∫_{^{a}T^{∗}X_{|t=0}/^{a}T^{∗}M_{|t=0}} ψ(ξ, η)χ(A_{z}^{j})ω_{X/M}^{m}.  □

Lemma 7.7. Let μ_{j}(z) ∈ Λ_{[j]}(^{a}T^{∗}X \ 0) be entire in z. Then

∫_{^{a}T^{∗}X_{|t=0}/M} ψ(r)r^{z}μ_{j}(z)

has only one simple pole at z = −j, and

Res_{z=−j} ∫_{^{a}T^{∗}X_{|t=0}/M} ψ(r)r^{z}μ_{j}(z) = − ∫_{^{a}S^{∗}X_{|t=0}/M} ı_{R}μ_{j}(0).

Proof. Let R be the radial vector field on ^{a}T^{∗}X. In polar coordinates,

∫_{^{a}T^{∗}X_{|t=0}/M} ψ(r)r^{z}μ_{j}(z) = ∫_{0}^{∞} ψ(r)r^{z+j−1}dr ∫_{^{a}S^{∗}X_{|t=0}/M} ı_{R}μ_{j}(z).

The first factor equals −\frac{1}{z+j} (mod H(ℂ)). The second one is entire. □

From Lemma 7.6, F (z)(A) has the same poles in B as a finite sum

∑_{j} ∫_{^{a}T^{∗}X_{|t=0}/M} ψ(r)r^{z}μ_{j}(z),

where μ_{j}(z) is entire in z and of homogeneity j in the fibers of ^{a}T^{∗}X_{|t=0}. By Lemma 7.7, the proposition follows. □

Definition 7.8. For A ∈ C_{h}(Ψ_{a}^{ℤ,[ℤ]}(X)), define

R(A) = Res_{z=0}(F (z)(A)).

Definition 7.9. Let A = A_{0} ⊗ … ⊗ A_{k} ∈ C_{k}(Ψ_{a}^{ℤ,[0]} ⊗ L). Define

e_{Q^{z}}(A) = (−1)^{n}zQ^{z} (\frac{Q^{−z}A_{k}Q^{z} − A_{k}}{z} ) A_{0} ⊗ A_{1} ⊗ … ⊗ A_{k−1}.

If A ∈ C_{k}(Ψ_{a}(X)), define e_{Q^{z}}(A) by the same expression, where now all products are in Ψ_{a}(X). Note that e_{Q^{z}} = (−1)^{n}zQ^{z}e_{D_{z}} (see (13)). By direct computation, the following identity holds on C_{∗}(Ψ_{a}^{ℤ,[0]} ⊗ L) and on C_{∗}(Ψ_{a}(X)).

(37)  Q^{z}b = bQ^{z} + zQ^{z}e_{D_{z}}.

Proposition 7.10. The identity R ∘ b = 0 holds on C_{k}(Ψ_{a}^{ℤ,[0]} ⊗ L).

Proof. Using Lemma 37,

R(b(A)) = Res_{z=0} ∫_{^{a}T^{∗}M_{|t=0}/M} ∗_{a}^{M} Tr_{V}(χ_{a}(b(Q^{z}A)))
+ Res_{z=0}(zF (z)(e_{Q^{z}}(A))).

(38)

Note that e_{Q^{z}}(A) is holomorphic, including at z = 0. By Proposition 7.5, the second term vanishes. By Proposition 6.2, the first integrand vanishes for small ℜ(z), so it extends to be identically zero by analytic continuation. □

It follows that R induces a map

(39)  R : HH_{k}(Ψ_{a}^{ℤ,[ℤ]}(X))→Λ(M, OM) ⊗ ΛL.

We claim that on this space, R ∘ e_{D} = ı_{t \frac{d}{dt}} R. Indeed, Ψ_{a}^{ℤ,[ℤ]}(X) is a

module over L ⊗ P olyn(^{a}T^{∗}M), and hence HH_{k}(Ψ_{a}^{ℤ,[ℤ]}(X)) is a module over HH_{k}(L ⊗ P olyn(^{a}T^{∗}M)), i.e., over the ring of differential forms with polynomial coefficients. Then, by exactly the same reasoning as in Chapter 4, we can prove that formula (23) holds with χ replaced by χ_{a}. Note that the claim is false at chain level.

This observation shows that the map (39) takes im(α), im(β) onto forms without dt, respectively multiples of dt. Proposition 7.11. Let A ∈ F_{h−N}HH_{h}(Ψ_{a}^{ℤ,[ℤ]}(X)) ∩ im(α). Then

R(A) = ı_{\frac{d}{dt}} dt ∧ R(A) = − ∫_{^{a}S^{∗}X_{|t=0}/M} ı_{R} ∗_{a}^{X} χ(σ(A)).

Let A ∈ F_{h−N−1}C_{h}(Ψ_{a}^{ℤ,[ℤ]}(X)). Then R(A) is a multiple of dt and

R(A) = − ∫_{^{a}S^{∗}X_{|t=0}/M} ı_{R} ∗_{a}^{X} χ(σ(A)).

Proof. We know that in the first case, R(A) does not contain dt. Apply Proposition 7.2 with i = h − N − 1, respectively i = h − N − 2. It follows that ı_{\frac{d}{dt}} dt ∧ F (z), respectively F (z), is holomorphic for such chains around z = 0, which implies that ı_{\frac{d}{dt}} dt ∧ R(A), respectively R(A) depend only on σ(A). The result follows from Lemmas 7.6 and 7.7. □

Remark 7.12. This Proposition implies that we can improve Remark 7.4 as follows: The map (39) shifts t-degree by h − n when applied to HH_{h}(Ψ_{a}^{ℤ,[ℤ]}(X)) ∩ im(α), and by h − n − 1 when applied to HH_{h}(Ψ_{a}^{ℤ,[ℤ]}(X)) ∩ im(β).

From Proposition 7.2, it follows that R vanishes on Ψ_{a}^{−∞,[ℤ]}(X), so we think about it as being defined on symbols, with a natural extension to vertical operators. Using the projection T_{j}A_{σ}(X) → Ψ_{a}^{ℤ,[0]}, we can extend the definition of R to C_{∗}(A_{σ}(X)). Proposition 7.13. Let A ∈ F_{h−N}C_{h}(A_{σ}(X)) or F_{h−N−1}C_{h}(A_{σ}(X)) represent a class [A] in HH_{h}(A_{σ}(X)). Then R(A) is de Rham closed in M. If A is Hochschild exact, then R(A) is de Rham exact.

Proof. We have seen in Theorem 4.6 that the differential form

[A]_{E_{2}} = ∗_{a}^{X}χ(σ(A))

is closed and of homogeneity 0. Hence L_{R}[A]_{E_{2}} = 0. Using (22), this implies (ı_{R}d_{v} + d_{v}ı_{R})[A]_{E_{2}} = 0, hence ı_{R}[A]_{E_{2}} is closed. Proposition 7.11 shows that

d_{v}R(a) = d_{v} ∫_{^{a}S^{∗}X_{|t=0}/M} ı_{R}[A]_{E_{2}} = ∫_{^{a}S^{∗}X_{|t=0}/M} d_{v}ı_{R}[A]_{E_{2}} = 0.

If [A] = 0, then [A]_{E_{2}} = d_{v}β is exact, and, since d_{v} preserves homogeneity, we can assume that β is also homogeneous of homogeneity 0. Therefore L_{R}β = 0. This implies ı_{R}[A]_{E_{2}} = ı_{R}d_{v}β = −d_{v}ı_{R}β, and so

R(a) = ∫_{^{a}S^{∗}X_{|t=0}/M} ı_{R}[A]_{E_{2}} = −d_{v} ∫_{^{a}S^{∗}X_{|t=0}/M} ı_{R}β.  □

7.2. The boundary map. Let b_{0} denote the differential map in C_{∗}(Ψ_{a}^{ℤ,[ℤ]}(X)). We will derive a formula for the boundary map δ of the short exact sequence (1) in terms of the presentations found in Chapters 4 and 5 for Hochschild homology as de Rham cohomology groups. As with all spectral sequences, we can in principle observe only the top part, say δ_{0} : HH(A_{σ}(X))_{[i]} → HH(I_{∂}(X))_{[i]}, of δ. This might be zero on some element even though δ itself does not vanish on that element. Remark 7.14. The boundary map commutes with e_{D}, L_{D}, t^{−1}dt∧, α and β, since these operations are maps of complexes on C_{∗}(Ψ_{a}(X), b).

This means that δ preserves the dt and dt-free parts, and the thomogeneity in the presentation by cohomology spaces. Proposition 7.15. If dim M ≠ 0 then δ_{0} = 0.

Proof. If M = {pt}, then the adiabatic algebra becomes isomorphic to the algebra Ψ_{ℝ_{+}}(X) of one-parameter families of pseudodifferential operators. In that case, δ = δ_{0} since t is a “flat” parameter. If dim M ≠ 0, we claim that the vertical boundary map vanishes. This is essentially Lemma 8 from [18]. For convenience, we reproduce the proof. Let A ∈ C_{k}(I_{σ}^{ℤ,[ℤ]}(X)) be a cycle. Let Ã ∈ C_{k}(Ψ_{a}^{ℤ,[ℤ]}(X)) be an extension to the full vertical algebra. Then, using Proposition 6.9,

δ_{0}(A) = ∫_{^{a}T^{∗}M_{|t=0}/M} ∗_{a}^{M}χ_{a}(b_{0}Ã).

From Corollary 6.8, this is independent of ∇. The vector fields tangent to the fibers of ^{a}T^{∗}M_{|t=0} → M are outer derivations on the algebra Ψ_{a}^{ℤ,[ℤ]}(X). By naturality, the boundary map commutes with the action L_{V} of such a vector field: [δ_{0}, L_{V}] = 0. But L_{V} (A) has order 1 less than A. After repeated applications, the order of L_{V_{1}} … L_{V_{s}} Ã will be less than −m − k − 1, so, by Proposition 6.2, χ_{a}(b(L_{V_{1}} … L_{V_{s}} Ã)) = 0. Hence, L_{V_{1}} … L_{V_{s}}δ_{0}(A) = 0. Since V_{1}, …, V_{s} were arbitrary and δ_{0}(A) is a Schwartz form on ^{a}T^{∗}M_{|t=0}, the proposition follows. □

Therefore, the first significant part of δ is

δ_{1} : HH(A_{σ}(X))_{[i]} → HH(I_{∂}(X))_{[i−1]},

which decreases the t-filtration by 1. Theorem 7.16. The leading part of the boundary map is essentially integration along the fibers,

δ_{1} = i ∫_{^{a}S^{∗}X_{|t=0}/M} .

Proof. From Remark 7.14, it is enough to prove the claim for a class [a] represented by some a ∈ T_{j}HH_{k}(A_{σ}(X)) ∩ im(α). Such a class can be represented by a chain A ∈ T_{j}F_{k−n−m}C_{k}(A_{σ}(X)). By Hunitality, we can find Ã ∈ T_{j}C_{k}(Ψ_{a}(X)) a pre-image of A, such that bÃ ∈ C_{k−1}(I_{∂}(X)). Since δ_{0}(a) = 0, it follows that there exists γ ∈ T_{j}C_{k}(I_{∂}(X)) such that bÃ−bγ ∈ T_{j−1}. By replacing Ã with Ã−γ, we can assume that δ(a) = [bÃ] is represented by a chain in T_{j−1}. We use now a choice of a vector space isomorphism between A_{σ}(X)/I_{σ}(X) and its graded algebra Ψ_{a}^{ℤ,[ℤ]}(X). Let A_{[j]}, A_{[j−1]} be the components of Ã in C_{k}(Ψ_{a}^{ℤ,[ℤ]}(X))_{[j]}, respectively C_{k}(Ψ_{a}^{ℤ,[ℤ]}(X))_{[j−1]}, where the subscript denotes the negative of the t-degree. Using (37), we see that

δ_{1}(a) = ∫_{^{a}T^{∗}M_{|t=0}/M} ∗_{a}^{M}χ_{a}(bÃ)_{[j−1]}

= (∫_{^{a}T^{∗}M_{|t=0}/M} ∗_{a}^{M}χ_{a}(Q^{z}bÃ)_{[j−1]})_{|z=0}

= (∫_{^{a}T^{∗}M_{|t=0}/M} ∗_{a}^{M}χ_{a}(bQ^{z}Ã)_{[j−1]})_{|z=0} (40)

+ (∫_{^{a}T^{∗}M_{|t=0}/M} ∗_{a}^{M}χ_{a}(zQ^{z}e_{D_{z}} Ã)_{[j−1]})_{|z=0} . (41)

We shall prove that (40) vanishes and (41) equals the desired expression. In (40), we can replace Ã by A_{[j]}+A_{[j−1]} since we are looking only at the part of degree j−1 in t. Let b = b_{0}+b_{1}+… be the expansion of b according to the t-degree. By Proposition 6.2, χ_{a}(b_{0}Q^{z}A_{[j−1]}) vanishes for small ℜ(z), so the analytic continuation of the corresponding part in (40) is 0 at z = 0.

Proposition 7.17. The identity b_{0}(A^{j}) = 0 implies that

(42) ∫_{^{a}T^{∗}M_{|t=0}/M} ∗_{a}^{M}χ_{a}(b_{1}Q^{z}A_{[j]})

− d_{v} ∫_{^{a}T^{∗}M_{|t=0}/M} ∗_{a}^{M}(χ_{a}(Q^{z}A_{[j]})) ∈ zH(B)

thus ∫_{^{a}T^{∗}M_{|t=0}/M} ∗_{a}^{M}χ_{a}(b_{1}Q^{z}A_{[j]}) tends to 0 in Λ^{2n−k+1}(M)/dΛ^{2n−k}(M) as z → 0.

Proof. This would be clear if we knew that b_{0}(Q^{z}A_{[j]}) = 0. Indeed, in this case, for ℜ(z) < 0,

(43)  χ_{a}(b_{1}Q^{z}A_{[j]}) = d_{v}(χ_{a}(Q^{z}A_{[j]}))

(compare with Proposition 5.2). In the semi-classical limit case, this condition holds by commutativity of the vertical algebra Ψ_{a}^{ℤ,[0]}(M). The proof in the general case consists of following an explicit proof of (43) and noting that whenever we have to commute Q^{z}, we get an error term in zH(B). □

Lemma 7.18. Let C ∈ T_{j}F_{k−N}C_{k}(Ψ_{a}(X)). If A_{[j]} = b_{0}C, then

∫_{^{a}T^{∗}M_{|t=0}/M} ∗_{a}^{M}χ_{a}(b_{1}Q^{z}A_{[j]})_{[j−1]}

vanishes at z = 0.

Proof. Notice that

\begin{aligned}χ_{a}(b_{1}Q^{z}A_{[j]})_{[j−1]} & = χ_{a}(b_{1}Q^{z}b_{0}C)_{[j−1]} \\ & = χ_{a}(b_{1}b_{0}Q^{z}C + b_{1}zQ^{z}e_{D_{z}}C). \end{aligned}

The last term belongs to F_{k−N−2+z}C_{h−1} and is a multiple of z, hence like in Proposition 7.2, its integral vanishes at z = 0. From b^{2} = 0, we deduce b_{1}b_{0} = −b_{0}b_{1}. Since b_{1}Q^{z}C ∈ T_{j−1}, it follows from Proposition 6.2 that ∫_{^{a}T^{∗}M_{|t=0}/M} ∗_{a}^{M}χ_{a}(b_{0}b_{1}Q^{z}C)_{[j−1]} vanishes for ℜ(z) sufficiently small, hence it is identically zero. □

Note that the term (41) is by definition R_{j−1}(e_{D_{Q}} Ã). From Remark 7.12, this is a dt-free form of t-degree −j + k − n. From Propositions 4.10 and 7.11,

R(e_{D_{Q}} Ã) = i ∫_{^{a}S^{∗}X_{|t=0}/M} ı_{R}(r^{−1}dr ∧ [a]_{E_{2}}) = i ∫_{^{a}S^{∗}X_{|t=0}/M}[a]_{E_{2}}.

We have assumed that a ∈ T_{j}. We can also ask that [a]_{E_{2}} be of pure homogeneity with respect to L_{D}. This homogeneity must be at least −j + k − n, and if it is bigger, then [a]_{E_{2}} is the class of some a′ ∈ T_{j−1}. So we can also assume that [a]_{E_{2}} has homogeneity −j + k − n. Using again Remark 7.12, we conclude that R_{j−1}(e_{D_{Q}} Ã) = R(e_{D_{Q}} Ã) = R(e_{D_{Q}}a). This ends the proof of Theorem 7.16. □

Recall the decomposition of T_{i}/T_{−∞}HH_{k}(A_{σ}(X)) as eigenspaces of L_{D}. Let a be of pure type with respect to this decomposition. Assume a is dt-free. Since ∗_{ϕ} increases the t degree at most by k − n, it follows that [a] belongs to T_{n+i−h}H^{2N−k}(S^{∗}X × S^{1}) with h ≤ k Hence L_{\frac{d}{dt}} acts on [a] as multiplication with h − i − n. Assume that δ_{1}(a) = 0. This means that δ(a) ∈ T_{i−s}HH_{k−1}(I_{∂}(X)) for some s ≥ 2 minimal with this property. Let δ(a)_{s} be the image of δ(a) in T_{i−s}/T_{i−s−1}HH_{k−1}(I_{∂}(X)) ≅ E_{2}^{i+s,k−1−i−s}(I_{∂}(X)). L_{\frac{d}{dt}} acts as multiplication with s+k −i−1−n on this space. But from h ≤ k and s ≥ 2 we get h−i−n < s+k−i−1−n. Since δ and L_{\frac{d}{dt}} commute, we deduce

Proposition 7.19. Let a ∈ T_{i}HH_{k}(A_{σ}(X)). If δ_{1}(a) = 0, then there exists some ã with a − ã ∈ T_{i−1}HH_{k}(A_{σ}(X)) and δ(ã) = 0.

Theorem 7.16 and Proposition 7.19 completely characterize the boundary map

δ : HH(A_{∂}(X)) → HH(I_{∂}(X)).

7.3. Traces on the adiabatic algebras. From the long exact sequence derived from the short exact sequence

(44)  0 → Ψ_{a}^{−∞,−∞}(X) ↪ Ψ_{a}^{−∞}(X) → I_{∂}(X) → 0,

the ideals I_{∂}(X) and Ψ_{a}^{−∞}(X) have the same homology, except in dimensions 0 and 1. Consider the following map:

Tr : Ψ_{a}^{−∞}(X) → C^{∞}(ℝ_{+})[t^{−1}], A ↦ ∫_{Δ_{a}/ℝ_{+}} A_{|_{Δa}}.

Lemma 7.20. The map Tr generates HH^{0}(Ψ_{a}^{−∞,ℤ}(X)) as a module over C^{∞}(ℝ_{+})[t^{−1}].

Proof. Let first A ∈ Ċ^{∞}(ℝ_{+}, Ψ^{−∞}(X)) ≅ Ψ_{a}^{−∞,−∞}(X) be a rapidly vanishing family of smoothing operators. For any t > 0, the map A ↦ Tr(A)(t) is a trace. Choose any local embedding of T X in X^{2}. Cut A off near the diagonal and pull it back to T X as a compactly supported section in the fiber density bundle. The pull-back of the fiber density bundle to the 0-section of T X is just the density bundle Ω(T X). In local coordinates, and pulling back via the canonical map ϕ_{a}, we get

Tr(A) = \frac{1}{(2π)^{n}} ∫_{T^{∗}X} Âω^{N} = \frac{1}{(2π)^{n}} ∫_{^{a}T^{∗}X} Âω_{a}^{N},

where ω, ω_{a} are the symplectic form, respectively the adiabatic symplectic form. This formula extends to Ψ_{a}^{−∞}(X), and is an extension of the map

\begin{aligned}I_{∂}(X) ∋ A & ↦ \frac{1}{(2π)^{n}} ∫_{^{a}T^{∗}M_{|t=0}} Tr_{V}(Â)(ω_{a}^{M})^{n} \\ & = \frac{1}{(2π)^{n}} ∫_{^{a}T^{∗}M_{|t=0}} ∗_{a}χ_{a}(A) ∈ L, \end{aligned}

which, by Theorem 6.10 and Propositions 6.9 and 5.3, is a generator over L of HH^{0}(I_{∂}(X)). This implies that the boundary map

HH_{1}(I_{∂}(X)) → HH_{0}(Ψ_{a}^{−∞,−∞}(X))

vanishes. □

Recall that e_{D} is an isomorphism from im(β) ⊂ HH_{1}(I_{∂}(X)) to HH_{0}(I_{∂}(X)). A similar statement is valid for Ψ_{a}^{−∞,−∞}(X). From this, it follows that Tr ∘e_{D} is a Hochschild cochain which generates im(β) ⊂ HH_{1}(Ψ_{a}^{−∞}(X)), and that the other boundary map of (44) also vanishes.

Since we have the explicit C^{∞}(ℝ_{+})[t^{−1}]-valued cochain Tr on Ψ_{a}^{−∞}(X), we can imitate the construction of the residue trace to get explicit cocycles on Ψ_{a}(X). Namely, we claim that the maps

Ψ_{a}(X) → C^{∞}(ℝ_{+})[t^{−1}]  A ↦ Res_{z=0} Tr(Q^{z}A)
C_{1}(Ψ_{a}(X)) → C^{∞}(ℝ_{+})[t^{−1}] A ⊗ B ↦ Res_{z=0} Tr(Q^{z}D(B)A)

are cocycles on Ψ_{a}(X). Since e_{D} is a map on homology, it is enough to prove the claim for the first map. We have seen that the residue is well-defined. We want to show that it vanishes on commutators. We have

Res_{z=0} Tr(Q^{z}[A, B]) = Res_{z=0} Tr([Q^{z}A, B] + zQ^{z} \frac{Q^{−z}BQ^{z} − B}{z} A) = 0,

since the first term vanishes for small ℜ(z), and the second term has no residue at zero by Proposition 7.5.

For t > 0, the first map is just Wodzicki’s residue trace on X, hence it equals ∫_{^{a}S^{∗}X}(Â_{−N})ı_{R}ω_{a}^{N}. By continuity, this holds down to t = 0. We note here that for t > 0, ω_{a}^{N} = (ϕ_{a}^{−1})^{∗}(ω^{N}) belongs to T_{−n}Λ^{2}(^{a}T^{∗}X). Indeed, from (10), det(ϕ_{a}) = t^{n}. Therefore, the residue trace of a smooth adiabatic operator is a Laurent function in t of degree −n.

7.4. Properties of the residue functional. We can extend Proposition 7.13 to arbitrary orders. We first treat the semi-classical limit case. Recall that the subscript i means taking the part in T_{i}/T_{i−1}, i.e., the coefficient of t^{−i} in the Laurent expansion at t = 0. Proposition 7.21. If A ∈ T_{i}C_{k}(A_{σ}(M)) is a boundary, then R_{i}(A) is d_{v}-exact.

Proof. The spectral sequence ^{∂}E_{2}, which computes HH(A_{σ}(M)) using the T_{i} filtration, degenerates at E_{2}, so we can assume A = b(P) with P ∈ T_{i−1}C_{k+1}(A_{σ}(M)). The semi-classical limit vertical algebra is commutative so (Q^{z}P)_{i−1} is b-closed (this is not true in the general adiabatic setting). The projection Ψ_{a} → S_{a} is surjective; let P̃ be a pre-image of P. Using (37), we get:

R_{i}(A) = R_{i}(bP)
= Res_{z=0} ∫_{^{a}T^{∗}M_{|t=0}/M} ∗_{a}^{M}(χ_{a}(Q^{z}bP̃))_{i}
= Res_{z=0} ∫_{^{a}T^{∗}M_{|t=0}/M} ∗_{a}^{M}(χ_{a}(b(Q^{z}P̃))_{i}
+ Res_{z=0} ∫_{^{a}T^{∗}M_{|t=0}/M} ∗_{a}^{M}(χ_{a}(zQ^{z}e_{Q^{z}} P̃))_{i}.

By Proposition 7.5 and because of the z factor, the last term vanishes since e_{Q^{z}} P̃ ∈ T_{i} is entire. As for the first term, we claim it is exact.

Lemma 7.22. For ℜ(z) sufficiently small, we have (45)

∫_{^{a}T^{∗}M_{|t=0}/M} ∗_{a}^{M}χ_{a}(b(Q^{z}P))_{i} = d_{v} (∫_{^{a}T^{∗}M_{|t=0}/M} ∗_{a}^{M}(χ_{a}(Q^{z}P))_{i−1}) .

Proof. We notice that Q^{z}P survives at ^{∂}E_{1}^{i,k−i}(Ψ_{a}^{z+Z}(M)) Then (45) is the computation of d_{1}[Q^{z}P] in the spectral sequence of the symbol algebra of order z + ℤ. The identity d_{1} = ∗_{a}^{−1}d_{v}∗_{a} (see (26)) holds in this context. The proof of Proposition 7.17 applies to prove the result. □

By the de Rham theorem, the space of exact forms is closed. A meromorphic family of forms, which takes exact values for z in an open set, must have exact residues. This finishes the proof of Proposition 7.21. □

From Theorem 4.6 and Proposition 7.13 it follows that every Hochschild class in T_{i}HH(A_{σ}(X)) has a representative a such that R_{i}(a) is closed. Together with Proposition 7.21, this proves: Corollary 7.23. The residue R_{i} descends as a map

R : T_{i}HH_{k}(A_{σ}(M))→H^{n−k+j}(M) ⊗ Λ^{j}L.

Proposition 7.21 is a model for the general case. The main difference is that multiplication by Q^{z} destroys the cycle property. Proposition 7.24. Let A ∈ T_{i}C_{k}(Ψ_{a}(X)). If bA ∈ T_{i−1}C_{k−1}(Ψ_{a}(X)), then R_{i−1}(bA) = dR_{i}(A), where d is the de Rham differential on M.

Proof. Q^{z}A_{i} is not a cycle in the vertical algebra, but, since b(A_{i}) = 0, it has the property that b_{0}(Q^{z}A_{i}) is a holomorphic multiple of z. The proof of Proposition 7.17 goes through, with the following modification: replace congruences (mod zH(B)) by equalities of residues at z = 0. □

It follows that if A ∈ T_{i}C_{k}(Ψ_{a}(X)) is Hochschild closed, then R_{i}(A) is de Rham closed. Let now A ∈ T_{i}F_{j}C_{k}(A_{σ}(X)) ∩ im(α). Assume that A is b-exact and that j > k − N. Then, there exists some P ∈ T_{i−1}C_{k+1}(A_{σ}(X)) such that bP − A ∈ T_{i}F_{j−1}C_{k}(A_{σ}(X)). This is a restatement of Proposition 4.15. By Proposition 7.24, R_{i}(A) = R_{i}(A − bP) modulo exact forms. Inductively, R_{i}(A) = R_{i}(A′), where A′ ∈ T_{i}F_{k−N}C_{k}(Ψ_{a}(X)). From Proposition 7.13, R_{i}(A′) is exact. As a consequence, we get the main result of this section:

Theorem 7.25. The residue R_{i} descends as a map on the Hochschild homology of adiabatic symbols with values in the cohomology of M (with twisted coefficients):

T_{i}HH_{k}(A_{σ}(X)) →^{R_{i}} H^{n−k}(M, OM)t^{i+k−n} ⊕ H^{n−k+1}(M, OM)t^{i+k−n−2}dt.

On chains A = A_{0} ⊗…⊗A_{k} ∈ T_{i}C_{k}(Ψ_{a}(X)), R_{i} is given by the explicit formula

R_{i}(A) = Res_{z=0} ∫_{^{a}T^{∗}M_{|t=0}/M} ∗_{a}^{M}(χ_{a}(Q^{z}A^{[i]})).

8. An example

Consider a fibration X → S^{1} and let A ∈ Ψ_{a}(X) be an elliptic adiabatic operator. Choose an inverse b of a modulo Ψ_{a}^{−∞}(X), then the Hochschild 1-chain a ⊗ b defines a cycle in C_{1}(Ψ_{a}(X)/Ψ_{a}^{−∞}(X)). This cycle was introduced by Melrose and Nistor in [17]. The t-degree of this cycle is 0, in other words a ⊗ b ∈ T_{0}C_{1}(A_{σ}(X)).

We can therefore apply the functional R_{0} to this cycle. We examine below the dt-free component f(A) ∈ H^{0}(M) of R_{0}(tr(a ⊗ b)).

Let x, ξ be local coordinates in ^{a}T^{∗}M_{|t=0} (this bundle is canonically trivial in our case). The adiabatic symplectic form has the form ω_{a} = t^{−1}dξ∧dx. Let α, β be the images of a, b in the suspended algebra of the fibers with parameters in the circle M = S^{1} (i.e., in the graded algebra of Ψ_{a}(X) associated to the filtration T_{j}). Following the definition, we get

f(A) = [Res_{z=0} ∫_{ℝ} Tr_{V}(Q^{z}α∂_{ξ}β)dξ ∈ Λ^{0}(S^{1})]_{H^{0}(S^{1})} .

Since we proved in Theorem 7.25 that R_{0}(tr(a⊗b)) is exact as a form on M, it follows that Res_{z=0} ∫_{ℝ_{ξ}} Tr_{V}(Q^{z}α∂_{ξ}β)dξ ∈ Λ^{0}(S^{1}) is constant as a function of x ∈ S^{1}. This constant equals Tr_{R}(α(x)∂_{ξ}β(x)) for all x ∈ S^{1}, where Tr_{R} = Res_{z=0} \overline{Tr} ∘ Q^{z} is the residue trace on the suspended algebra defined in [23]. This motivates the following definition: Definition 8.1. Let Y be a closed manifold. Let α ∈ Ψ_{sus(1)}(Y, E, F) be an elliptic suspended operator in the sense of Melrose [16]. Define

f(α) = Tr_{R} (α\frac{∂β}{∂ξ} ) ,

where β is an inverse of α modulo Ψ_{sus(1)}^{−∞}(Y).

It is straightforward to see that f has the following properties:

(1) f(α) is homotopy invariant inside elliptic suspended operators.

(2) f(α_{1}α_{2}) = f(α_{1}) + f(α_{2}).

(3) When α = D + iξ, where D is a positive first-order differential operator on Y, then f(α) equals the residue at the origin of the meromorphic function η(D, s). From [1], this residue vanishes so f(D + iξ) = 0

(4) If α is differential, or more generally has even symbol in the sense of Gilkey [8], and dim(Y) is even, then f(α) = 0.

The first two properties show that f depends only on the class in K^{0}(T_{sus(1)}^{∗}Y) = K^{1}(T^{∗}Y) of the principal symbol of α. For dim(Y) odd, this K group is generated rationally by the symbols of twisted signature operators, for which f was seen to vanish. Thus f is identically zero when dim(Y) is odd.

On even dimensional manifolds, Gilkey [8] has shown that K^{0}(S^{∗}Y) is generated rationally by self-adjoint polynomial symbols of even degree with values in GL. To such a symbol p of degree k we associate the class p + iξ(p^{2} + ξ^{2k})^{\frac{2k−1}{2k}} ∈ K^{1}(T^{∗}Y); in fact these classes generate K^{1}(T^{∗}Y, ℚ). Let A ∈ Ψ_{sus(1)}^{k}(Y) be the suspended operator defined by

A := P + iξ(P^{2} + ξ^{2k})^{\frac{2k−1}{2k}}

where P is an invertible self-adjoint differential operator on Y with symbol p (such an operator exists since we can add a small real constant to remove the possible eigenvalue 0). We set Q := (A^{∗}A)^{\frac{1}{2}} and B := A^{−1}. The operator A is not differential, nor is its symbol even, so we cannot conclude directly that f(A) = 0. However, for each fixed ξ ∈ ℝ, the operators A, Q and ∂_{ξ}A preserve the eigenspaces of P. We decompose the trace defining f(B) over these eigenspaces; we get

∫_{ℝ} Tr_{V}(Q^{−z}B∂_{ξ}A)dξ = η(P, z)h(z)

where η(P, z) is the APS eta function of P, and h(z) is the integral on ℝ of a smooth function in ξ, depending holomorphically on z, which grows like ξ^{−2− \frac{z(k+1)}{2}} as |ξ| → ∞. This integrand can be made explicit, but we only need to notice that h(z) is regular at z = 0. Together with the regularity of the eta function shown by Gilkey [8], this implies that f(B), and so also f(A), vanish when dim(Y) is even.

Therefore in this example the invariant f vanishes identically. Nevertheless, this vanishing is highly non-trivial. This fact suggests that the homological residues R_{j} introduced above may lead to subtle geometric and analytic properties of elliptic operators.

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Institutul de Matematică al Academiei Române, P. O. Box 1-764, RO-70700, Bucharest, Romania

E-mail address: moroianu@alum.mit.edu