THE LOCAL INDEX FORMULA IN

NONCOMMUTATIVE GEOMETRY

Alain CONNES and Henri MOSCOVICI

Abstract

In noncommutative geometry a geometric space is described from aspectral point of view, as a triple (A,H, D) consisting of a ∗-algebra Arepresented in a Hilbert space H together with an unbounded selfadjointoperator D, with compact resolvent, which interacts with the algebra in abounded fashion. This paper contributes to the advancement of this pointof view in two significant ways: (1) by showing that any pseudogroup oftransformations of a manifold gives rise to such a spectral triple of finitesummability degree, and (2) by proving a general, in some sense universal,local index formula for an arbitrary spectral triple of finite summabilitydegree, in terms of the Dixmier trace and its residue-type extension.

Many of the tools of the differential calculus acquire their full power only whenformulated at the level of variational calculus, where the original space X oneis dealing with is replaced by a functional space F(X) of functions or fields onX . The space X itself is involved only indirectly in F(X), for instance to writedown the right hand side F (ϕ) of a nonlinear evolution equation,

dϕ

dt= F (ϕ), ϕ ∈ F(X),

with the right hand side usually involving the pointwise product of functions ϕon X and partial differentiation.

The essence of noncommutative geometry is the existence of many situations inwhich F(X) makes perfectly good sense, while X itself is no longer an ordinaryspace, described set-theoretically by means of points p ∈ X and coordinates.When X is given by a set, the basic structure on the space F(X) of (real orcomplex valued) functions on a set X is the pointwise product of functions.Given two functions f1, f2 one forms a new function f1f2 by:

(1) (f1f2)(p) = f1(p) f2(p), ∀p ∈ X .

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In noncommutative geometry one still has a product on F(X) but the commu-tativity property of (1)

(2) f1f2 = f2f1, ∀fj ∈ F(X) .

is dropped. It is precisely this commutativity property which signals that Xis an ordinary set. When dropped, one no longer deals with just a set X ,but essentially with a set endowed with relations between different points. Forinstance, if one considers a set Y consisting of two points {1, 2} and the relationwhich identifies 1 and 2, then F(Y, rel) is the space M2(C) of 2 × 2 complexmatrices with the product

(3) (f1f2)(i, j) =∑

f1(i, k) f2(k, j)

i.e. the usual product of matrices.

In this simple example the ordinary space {1, 2}, given by the two points with-out any relation, is described by the subalgebra of diagonal matrices. It is the

“off-diagonal” matrices, such as e12 =[

0 10 0

]or e21 =

[0 01 0

], which describe

the relation. This type of construction of an algebra F(X) is rather general.It extends to a pseudogroup of transformations of a manifold and also to theholonomy pseudogroup of a foliation (see [2]). The resulting noncommutativealgebra encodes the structure of the “space with relations”. We shall later dis-cuss in detail the case of a smooth manifold together with its full diffeomorphismgroup.

As another simple example we can consider the case of a single point dividedby a discrete group Γ. Then the corresponding algebra F is the group ringattached to Γ, whose elements f are functions (with finite support) on Γ,

(4) g → fg ∈ C,

with the product given by linearization of the group law g1, g2 → g1g2 in Γ:

(5) (f1f2)g =∑

g1g2=g

f1,g1 f2,g2 .

So far, in describing the functional space F(X) associated to an ordinary spaceX we have ignored the degree of regularity of the elements f ∈ F(X) as functionsof p ∈ X . To various degrees of regularity correspond various branches of thegeneral theory of noncommutative associative algebras. The latter are assumedto be algebras over C, which moreover are involutive, i.e. endowed with anantilinear involution

(6) f → f∗ , (f1f2)∗ = f∗2 f∗

1 .

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The two kinds of regularity assumptions for which the corresponding algebraictheory is satisfactory are:

measurability, which corresponds to the theory of von Neumann algebras;continuity, which corresponds to the theory of C∗-algebras.

In both theories the Hilbert space plays a key role. Indeed, both types ofalgebras are faithfully representable as algebras of operators in Hilbert spacewith suitable closure hypothesis. One can trace the role of Hilbert space to thesimple fact that positive complex numbers are those of the form

(7) λ = z∗z .

In any of the above algebras, functional analysis provides the existence, viaHahn-Banach arguments, of sufficiently many linear functionals L which arepositive

(8) L(f∗f) ≥ 0

From such an L, one easily constructs a Hilbert space together with a represen-tation, by left multiplication, of the original algebra.

Next, many of the tools of differential topology, such as the de Rham theory ofdifferential forms and currents, the Chern character etc. . ., are well captured (see[2]) by cyclic cohomology applied to pre C∗-algebras, i.e. to dense subalgebrasof C∗-algebras which are stable under the holomorphic functional calculus:

(9) f → h(f) =1

2iπ

∫h(z)f − z

dz

where h is holomorphic in a neighbourhood of Spec(f). The prototype of suchan algebra is the algebra C∞(X) of smooth functions on a manifold X . Thecyclic cohomology construction then recovers the ordinary differential forms, thede Rham complex of currents and so on. More significantly, this constructionalso applies to the highly noncommutative example of group rings, in whichcase the group cocycles give rise to cyclic cocycles with direct application tothe Novikov conjecture on the homotopy invariance of the higher signatures ofnon-simply connected manifolds with given fundamental group. (For a morethorough discussion, see [2]).

If one wants to go beyond differential topology and reach the geometric structureitself, including the metric and the real analytic aspects, it turns out that themost fruitful point of view is that of spectral geometry. More precisely, whileour measure theoretic understanding of the space X was encoded by a (vonNeumann) algebra of operators A acting in the Hilbert space H, the geometricunderstanding of the space X will be encoded, not by a suitable subalgebra ofA, but by an operator in Hilbert space:

(10) D = D∗, selfadjoint unbounded operator in H .

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In the compact case, i.e. X compact, the operatorD will have discrete spectrum,with (real) eigenvalues λn, |λn| → ∞, when n→ ∞.

Formulating the precise conditions to which the triples (A,H, D) should besubjected is tantamount to devising the axioms of noncommutative geometry.If we let F and |D| be the elements of the polar decomposition of D,

(11) D = F |D| , |D|2 = D2 , F = Sign D

then the operators F and |D| play a similar role to the measurements of anglesand, respectively, of length in Hilbert’s axioms of geometry. In particular theoperator F = Sign D captures the conformal aspect while D describes the fullgeometric situation.

Considering F alone, the quantized calculus was developed (cf. [2]) based onthe following dictionary.

Classical Quantum

Complex variable Bounded operator in Hilbert space H

Real variable Selfadjoint operator

Infinitesimal Compact operator

Infinitesimal of order Compact operator whose characteristicα > 0 values µn satisfy µn = O(n−α)

Differential dT = [F, T ] = FT − TF

Integral of infinitesimal of order 1 Dixmier trace Trω(T )

We refer to Appendix A for a thorough treatment of the Dixmier trace. For ahost of applications of the quantized calculus, including Julia sets, the quantumHall effect and the analysis of group rings, the reader is referred to [2]. A furtherapplication, namely the construction of a 4-dimensional conformal invariantanalogue of the 2-dimensional Polyakov action, is discussed in [4].

Our goal in the present paper is to use the quantized calculus to develop ge-ometry from a spectral point of view. In more precise terms, our initial datumwill be a triple (A,H, D) where A is an involutive algebra represented in theHilbert space H and D is a selfadjoint operator in H with compact resolvent,which almost commutes with any a ∈ A, to the extent that

[D, a] is bounded for any a ∈ A .

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The basic example of such a triple is provided by the Dirac operator on a closedRiemannian (Spin) manifold. In that case, H is the Hilbert space of L2 spinorson the manifold M , A is the algebra of (smooth) functions acting in H bymultiplication operators and D is the (selfadjoint) Dirac operator. One caneasily check that no information has been lost in trading the geometric spaceM for the spectral triple (A,H, D). Indeed (see [2]), one recovers

(i) the space M , as the spectrum Spec(A), of the norm closure of the algebraA of operators in H;

(ii) the geodesic distance d on M , from the formula:

d(p, q) = Sup {|f(p) − f(q)| ; ‖[D, f ]‖ ≤ 1} , ∀p, q ∈M .

The right hand side of the above formula continues to make sense in generaland the simplest non-Riemannian example where it applies is the 0-dimensionalsituation in which the geometric space is finite. In that case both the algebra Aand the Hilbert space H are finite dimensional, so that D is a selfadjoint matrix.For instance, for a two-point space, one lets A = C⊕C act in the 2-dimensionalHilbert space H by

f ∈ A →[f(a) 0

0 f(b)

],

and one takes D =[

0 µµ 0

]. The above formula gives d(a, b) = 1/µ.

As a slightly more involved 0-dimensional example, one can consider the alge-braic structure provided by the elementary Fermions, i.e. the three families ofquarks. Thus, one lets H be the finite dimensional Hilbert space with orthonor-mal basis labelled by the left-handed and right-handed elementary quarks suchas ur

L, ubR, . . .. The algebra A is C⊕H, where the complex number λ in (λ, q) ∈ A

acts on the right-handed part by λ on “up” particles and λ on “down” parti-cles. The isodoublet structure of the left-handed (up, down) pairs allows thequaternion q to act on them by the matrix[

α β−β α

]q = α+ βj ; α, β ∈ C .

Then the Yukawa coupling matrix of the standard model provides the selfadjointmatrix D.

In [5] the theory of matter fields was developed in the above framework, underthe finite-dimensionality hypothesis that the characteristic values of D−1 areO(n−1/d), for some finite d.

This allows to define the action functional of Quantum Electrodynamics at thesame level of generality (cf. [2]). The striking fact there is that if one replaces theusual picture of space-time by its product by the above 0-dimensional example,

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the QED action functional gives the Glashow-Weinberg-Salam standard modelLagrangian with its Higgs fields and symmetry breaking mechanism. In thedevelopment of this theory, the tools of the quantized calculus, in particular theDixmier trace as the substitute for the Lebesgue integral, played an essentialrole.

The matter field Lagrangian involves the metric gµν but does not involve anyderivative of gµν . This indicates that the difficulty involved in developing theanalogue of gravity in the above context is of a different scale. In order toovercome it, one needs both a good list of examples of spectrally defined spacesand a difficult mathematical problem to solve. By a spectrally defined spacewe mean a triple (A,H, D) as above; the involutive algebra A is not necessarilycommutative. We shall also refer to them as spectral triples.

Before running through the list of examples, let us state the mathematical prob-lem:

compute by a local formula the cyclic cohomology Chern character of (A,H, D).

More specifically, the representation of A in H together with the operator Dallows to set up an index problem:

IndD : Kj(A) → Z

where j = 0 in the Z/2-graded (or even case) and j = 1 otherwise. The indexmap turns out to be polynomial and given, in the above generality, by thepairing of Kj(A) with the following cyclic cocycle

τ(a0, a1, . . . , an) = Trace(a0[F, a1] . . . [F, an]

), ∀aj ∈ A

where n has the same parity as j and n > d− 1. In the even case, one replacesthe trace by the supertrace, i.e. one uses the Z/2-grading γ of H to write

τ(a0, a1, . . . , an) = Trace(γa0[F, a1] . . . [F, an]

), ∀aj ∈ A .)

The class of τ in the cyclic cohomology HCn(A) is the Chern character of(A,H, D). We refer to [2] for more details as well as for the appropriate nor-malizations.

The general problem is to compute the class of τ by a local formula. A partialanswer to this problem was already obtained in [2], by means of a general localformula for the Hochschild class of τ as the Hochschild n-cocycle:

(12) ϕ(a0, . . . , an) = Trω

(a0[D, a1] . . . [D, an] |D|−n

), ∀aj ∈ A,

where n is as above and, in the even case, with γ inserted in front of a0 .

In the above formula Trω is the Dixmier trace, which when evaluated on a givenoperator T only depends upon the asymptotic behavior of its eigenvalues. More

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precisely, for T ≥ 0, with µn(T ) the nth eigenvalue of T in decreasing order,one has (cf. Appendix A):

Trω(T ) = limω

1logN

N∑0

µn(T );

this is insensitive to the perturbation of µn by any sequence εn = o(

1n

), i.e.

such that nεn → 0, n→ ∞.

For a classical pseudodifferential operator P with distributional kernel k(x, y)the Dixmier trace is given by the Wodzicki residue

Trω(T ) =∫a(x)

where k(x, y) has an asymptotic expansion near the diagonal of the form

k(x, y) = a(x) log(d(x, y)) + b(x, y),

with b bounded.

In particular, when one evaluates Trω on a product T1 . . . Tn of such operatorsthe result is expressed as an integral in a single variable x of a local quantity.This is in sharp contrast with what happens for the ordinary trace, which whenevaluated on T1 . . . Tn involves a multiple integral, of the form∫

k1(x1, x2) k2(x2, x3) . . . kn(xn, x1),

where the xj ’s vary arbitrarily in the manifold.

While the expression (12) of the Hochschild cocycle ϕ is local in full generality, itonly accounts for the Hochschild class of the Chern character of (A,H, D), whichis not sufficient to recover the index map. In the manifold case for instance, itonly gives the index of D with coefficients in the Bott K-theory class supportedby an arbitrarily small disk.

In Section II of this paper we shall obtain a general local formula for all thecomponents of the cyclic cocycle τ . This will be achieved by adapting theWodzicki residue, the unique extension of the Dixmier trace to pseudodifferentialoperators of arbitrary order, to all our examples. For spectrally defined spaces(A,H, D), we shall see that the usual notion of dimension is replaced by adimension spectrum, which is a subset of C. Under the assumption of simplediscrete dimension spectrum, the Wodzicki residue makes sense and defines atrace on the algebra of the pseudodifferential operators of (A,H, D). The latteralgebra is obtained by analysing the one-parameter group σt = |D|it · |D|−it ina manner very similar to Tomita’s analysis of the modular automorphism group

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of von Neumann algebras. When the dimension spectrum is discrete but notsimple, the analogue of the Wodzicki residue is no longer a trace; it satisfies,however, cohomological identities which relate it to higher residues.

Under the sole hypothesis of discreteness of the dimension spectrum, we shallobtain a universal local formula for the Chern character of a spectral triple(A,H, D), expressing the components of the Chern character in terms of fi-nite linear combinations, with rational coefficients, of higher residues applied toproducts of iterated commutators of D2 with [D, aj ], aj ∈ A. A noteworthyfeature of the proof is the use of renormalization group techniques to removethe transcendental coefficients which arise when the dimension spectrum hasmultiplicity. In the manifold case, this formula reduces, of course, to the clas-sical local index formula. In general however it is necessarily more intricate, inseveral respects, because of its large domain of applicability, which encompassesfor instance the diffeomorphisms-equivariant situation described in Section I.

We conclude the introduction with a list of spectral triples corresponding togeometric or group-theoretic spaces.

1. Riemannian manifolds (with some variations allowing for Finsler metricsand also for the replacement of |D| by |D|α, α ∈ ]0, 1]).

2. Manifolds with singularities. For this, the work of J. Cheeger on conicalsingularities is very relevant. In fact, the spectral triples are stable undertheoperation of “coning”, which is easy to formulate algebraically.

3. Discrete spaces and their product with manifolds (as in the discussion in[2] of the standard model). The spectral triples are of course stable underproducts.

4. Cantor sets. Their importance lies in the fact that they provide examplesof dimension spectra which contain complex numbers (cf. Section II).

5. Nilpotent discrete groups. The algebra A is the group ring of the dis-crete group Γ, and the nilpotency of Γ is required to ensure the finite-summability condition D−1 ∈ L(p,∞). We refer to [2] for the constructionof the triple for subgroups of Lie groups.

6. Transverse structure for foliations. This example, or rather the intimatelyrelated example of the Diff-equivariant structure of a manifold will betreated in detail in Section I of this paper.

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1 Diffeomorphism invariant geometry

1.1 Diffeomorphism invariant geometric structures

Let W be a smooth manifold and Diff(W ) its group of diffeomorphisms. Givenan arbitrary subgroup Γ of Diff(W ) one can form the crossed product of thealgebra of functions on W by the action of Γ (cf. [2]) and describe in this waythe measure theory and topology of the noncommutative space encoding theidentifications of points in W by the action of Γ.

The basic idea which we developed in [3], in order to obtain invariants of K-theory of the C∗-algebra C0(W )>Γ, is to relate the general case of an arbitraryΓ acting on W to a “type II” situation, in which the action of Γ preserves acertain G-structure which we shall describe and use at great length below.

In general, and for instance if we take Γ = Diff(W ), the action of Γ on Wpreserves no structure at all. Thus, if we take W = S1 there is no Γ-invariantmeasure in the Lebesgue measure class, and even at the measure theory level thecrossed product L∞(S1)>Γ is of type III. The basic structure theory of typeIII factors, as crossed products of type II by an action of R∗

+, is easy to interpretin this example (cf. e.g. [2]) as the replacement of the manifold W = S1 by thetotal space P of the R∗

+-principal bundle of (positive) 1-densities on S1.

On this total space P the group Diff(W ) is still acting and now there is on Pa tautological invariant measure for the action of Diff(W ), so that the crossedproduct of P by Γ is of type II. Furthermore the group R∗

+ acts on this crossedproduct and gives back (up to Morita equivalence) the original crossed productof W by Γ.

Even though Γ acting on P preserves a natural density, it does not preserve aRiemannian geometry, since it would then have to be contained in the Lie groupof isometries of that geometry. Let us describe the natural geometric structureon P preserved by the action of Diff(S1). By construction, P is an R∗

+ principalbundle over S1 and we let π : P → S1 be the canonical projection. Givenx ∈ S1, a point p ∈ π−1{x} is the same thing as a unit of length in the tangentspace Tx(S1). Moreover, the canonical action of R∗

+ on P also specifies a unitof length in Tp

(π−1{x}), given by the vertical vector field d

dt (et p).

We thus have a natural integrable subbundle V of the tangent bundle of Ptogether with Euclidean metrics on both V and N = T/V , where at any p ∈ Pwe use the identification of Np with Tx(S1), x = π(p), to define the metric at p.

Since this construction is completely canonical it is invariant under the actionof Diff(S1).

If we make the (non canonical) choice of section dθ of P we can label the pointsof P by (θ, λ), θ ∈ S1, λ ∈ R∗

+ or equivalently (θ, s), s ∈ R, λ = es. In these

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coordinates, the vertical metric is ds2 and the transverse one is (es dθ)2. Adiffeomorphism ϕ of S1 acts in the obvious way, namely:

ϕ(θ, s) = (ϕ(θ), s + logϕ′(θ)) .

If we perform the measure theory construction of the principal R∗+ bundle in

higher dimension, we obtain the correct description of the corresponding typeII algebra but since we use only the (absolute value of the) determinant of theJacobian matrix of ϕ we only control the volume distortion by ϕ but not thegeometric distortion.

To describe the latter, we take for P the bundle over W whose fiber Px ateach x ∈ W is the space of all Euclidean metrics on the vector space Tx(W ).Thus, a point p of P is given by a point x ∈ W together with a non-degeneratequadratic form, gµν dx

µ dxν in local coordinates, on Tx(W ). Equivalently, wecan describe P as the quotient of the frame bundle of W , whose fiber at x ∈Wis the space of linear isomorphisms Rn → Tx W , by the action of the subgroupO(n) ⊂ GL(n,R). On the symmetric space GLn/O(n) we use the naturalinvariant Riemannian metric, which on the tangent space at the unit matrix,identified with the space of symmetric matrices, is given by the Hilbert-Schmidtnorm. Once transported to the fiber Px, x ∈ W , this metric gives an Euclideanstructure on the vertical bundle V ⊂ TP . Given a vertical path p(t), p(0) = pits square length at t = 0 is simply the trace of (p−1 p)2.

Also, exactly as above, we can identify the transverse space Np = Tp P/Vp

with the tangent space Tx(W ), x = π(p), so that the quadratic form p providesus with a natural Euclidean structure on Np. In order to have a convenientterminology, we introduce the following definition:

Definition I.1. By a triangular structure on a manifold M we mean an inte-grable subbundle V of the tangent bundle TM together with Euclidean metricson both V and N = T/V .

We can summarize the above discussion as follows:

Proposition I.1. Given a manifold W , the space P of all metrics on W ,defined above, has a canonical triangular structure, invariant under the actionof Diff(W ).

This construction was used in [3] to prove analytic properties of cyclic cocyclessuch as the transverse fundamental class or Gelfand-Fuchs cohomology classes.We refer to [3] for purely geometric corollaries of this technique. To obtainthem, it was crucial to relate the K-theory of the C∗-algebras obtained (fromcrossed products by subgroups Γ of Diff W ) using W and using P . This followedfrom the “dual Dirac” construction of a bivariant Kasparov class (cf. [3]). In

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[9] M. Hilsum and G. Skandalis went further and constructed the transversefundamental class in K-homology for the space P , using hypoelliptic operators.They did this at the level of homotopy classes of Fredholm modules and thecentral theme of the first part of this paper will be to refine their constructionin order to describe the geometry of the crossed products (of P by Γ) by aspectral triple (A,H, D) satisfying all the conditions of our general spectralgeometry.

Thus, we shall now free ourselves from the particular features of the exampleP above and discuss, in the context of triangular manifolds, the construction ofthe corresponding spectral triple.

One of the new features will be the quartic aspect of the discussion, as opposedto the quadratic feature of Riemannian geometry.

Finally one should keep in mind that the crossed product of P by a givensubgroup Γ of Diff(W ) is only the “type II” counterpart of the crossed productof W by Γ. To obtain the latter from the former, one needs to take a crossedproduct by “GLn/O(n)”, operation in which the non amenability of the Liegroup GLn, n > 1 comes into play. In this paper we shall content ourselveswith the type II discussion.

1.2 The spectral triple (A,H, D) of a triangular structureon a manifold

We let M be a smooth (not necessarily compact) manifold together with anintegrable subbundle V of its tangent bundle. We let N = TM/V be thetransverse bundle and assume that both N and V are oriented Euclidean evendimensional vector bundles.

Our first aim is to construct a hypoelliptic differential operator Q correspond-ing to the signature of M , which modulo lower order only depends upon theEuclidean structures of both V and N but not upon a choice of Riemannianmetric on M . This will be done by combining a longitudinal signature operatorof order 2 with the usual signature operator in the transverse direction. Then,using Q, we shall define a first-order operator D by the equation

(1) Q = D|D| .

Let us first see how one can replace the usual signature operator by an equivalentoperator of order 2.

α) Second order signature operatorLet V be a smooth (and, for simplicity, compact) oriented even-dimensionalRiemannian manifold. On the bundle ∧T ∗

Cof exterior differential forms on V

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one has a natural Z/2-grading γ, γ2 = 1, γ = γ∗ , given by the ∗-operation,such that

(2) d∗ = −γ d γ ;

thus, the signature operator d+ d∗ anticommutes with γ. Let

∆ = (d+ d∗)2 = dd∗ + d∗d.

It commutes with both d and d∗ and we can thus consider, for λ ∈ [0, 1], theoperators

(3) Uλ = ∆1/2 + λ(d − d∗) , U∗λ = ∆1/2 + λ(d∗ − d) .

One hasUλU

∗λ = ∆ − λ2(d− d∗)2 = (1 + λ2)∆

and similarlyU∗

λUλ = (1 + λ2)∆.

Lemma I.1. 1) Uλ commutes with the Z/2-grading γ.2) Uλ(d+ d∗)U∗

λ = 2∆1/2(dd∗ − d∗d) for λ = 1.

Proof. 1) Both ∆ and d− d∗ = d+ γdγ commute with γ.One has

(∆1/2 + (d− d∗)) (d+ d∗) (∆1/2 − (d− d∗)) =

∆(d+d∗)+(d−d∗) (d+d∗) ∆1/2+∆1/2(d+d∗) (d∗−d)−(d−d∗) (d+d∗) (d−d∗) .On the other hand,

(d− d∗) (d+ d∗) = −d∗d+ dd∗, (d+ d∗) (d∗ − d) = −d∗d+ dd∗

and

−(d−d∗) (d+d∗) (d−d∗) = (dd∗−d∗d) (d∗−d) = −dd∗d−d∗dd∗ = −∆(d+d∗).

It follows, ignoring finite rank operators and using the operators Uλ ∆−1/2 whichare bounded, that one gets a homotopy between the signature operator and theoperator ∆−1/2(dd∗ − d∗d) with the same Z/2 grading. This latter operator isan elliptic pseudodifferential operator of order 1 defined by the equation:

(4) D|D| = dd∗ − d∗d .

The second order operator dd∗ − d∗d, with the Z/2-grading γ, thus representsthe signature class on M .

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Let us now combine it with d+ d∗ in the above context.

β) The mixed signature operator

We let M , V ⊂ TM and N = TM/V be as above. We consider over M thehermitian vector bundle E with fiber

(5) E = ∧∗ V ∗C ⊗ ∧∗ N∗

C .

Its metric comes from the metrics of V and N , together with the orientationsthese yield Z/2-grading operators γV and γN . It also yields a natural volumeelement, i.e. a section of

∧v V ∗ ⊗ ∧n N∗ = ∧d T ∗M

where v = dimV , n = dimN , d = dimM = v + n.

Thus the Hilbert space L2(M,E) of sections of E has a natural inner product,independent of any additional choice.

Using the canonical flat connection of the restriction of the bundle N to theleaves of the foliation by V , we can define the longitudinal differential dL, as anoperator of degree (1, 0) with respect to the obvious bigrading, satisfying

(6) d2L = 0 .

The operator d∗L is, by definition, the adjoint of dL. It is of the form

(7) d∗L = −γV dL γV + Order 0

with the additional term of order 0 uniquely prescribed without any extra choice.

This means that the following operator is a well defined longitudinal ellipticoperator:

(8) QL = dLd∗L − d∗LdL .

By the discussion of section α) this operator describes at the K-theory levelthe longitudinal signature class. To obtain the full signature of M we needto combine it with a transverse signature operator, which is of order 1 as adifferential operator.

Our next step will thus be to define the operator dH +d∗H , where dH is of degree(0, 1) in the bigrading of E and corresponds to transverse differentiation.

This step will require an additional choice of a (non integrable) subbundle H ofTM transverse to V , dimH = n. It is crucial that such a choice does not affectthe principal symbol of the operator as a hypoelliptic operator (see below).

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The choice of H provides a natural isomorphism

(9) jH : ∧V ∗x ⊗ ∧N∗

x → ∧T ∗x , ∀x ∈M,

and for ω ∈ C∞(M,∧r V ∗ ⊗ ∧s N∗) we define dH(ω) as the component ofbidegree (r, s+ 1) of

(10) j−1H d(jH ω) .

To understand the ambiguity in the choice of H we consider locally a functionf which is leafwise constant, i.e. dLf = 0. Then dHf is independent of H andgiven as a section of N∗. We can then define the transverse symbol of dH usingits commutation with such f :

(11) dH(fω) − fdH(ω) = df ∧ ω ∀f, dL f = 0;

in the right hand side we use the natural algebra structure for ∧V ∗ ⊗ ∧N∗.

Thus, (11) means that the transverse symbol is independent of the choice of H .We let:

(12) QH = dH + d∗H

where the ∗ is taken relative to the inner product in L2(M,E). We now combineQL andQH , using the parity (−1)∂N in the transverse direction which commuteswith QL and anticommutes with QH , to define

(13) Q = QL (−1)∂N +QH .

We should remark that one can use (−1)∂N instead of 1⊗γN , without changingthe homotopy class of the operators, since these two gradings are homotopicamong operators which anticommute with QH .

The selfadjoint operator D is now uniquely defined by the equation:

(14) D|D| = Q .

Note that Q is formally selfadjoint by construction. We shall not discuss herethe problem of selfadjointness of Q in the noncompact case. This issue will needto be adressed eventually, in connection with our main example, i.e. the totalspace P in I.1.

The following theorem shows that the operator D constructed above gives riseto a spectral triple (A,H, D) for the crossed product A = C∞

c (M)>Γ, whereΓ is any group of diffeomorphisms preserving the triangular structure.

Theorem I.1. 1) [D, f ] is bounded for any f ∈ C∞c (M), and both f and [D, f ]

belong to ∩n≥1 Dom δn, where δ = [|D|, ·].

14

2) If M is compact D has compact resolvent, in all cases

f(D − λ)−1 is compact f ∈ C∞c (M) , λ /∈ R .

3) Changing the choice of H only affects D and |D| by bounded operators (locallyin the noncompact case).4) Let ϕ ∈ Diff(M) preserve the foliation V and be isometric on both V and N .Let Uϕ be the corresponding unitary in H. Then [D, fUϕ] is bounded, and bothfUϕ and [D, fUϕ] belong to ∩n≥1 Dom δn for all f ∈ C∞

c (M).

We shall also give the precise summability in 2) by showing that f(D− λ)−1 ∈L(p,∞), p = v + 2n, and compute the Dixmier trace of the product f |D|−p (cf.[2]).

1.3 Preliminaries on the ΨDO′ calculus

As a technical tool in the proof of Theorem I.1, we shall describe the pseudod-ifferential calculus which is adapted to the situation. It is just a special case ofthe pseudodifferential calculus on Heisenberg manifolds (cf. [1]), which is how-ever sufficiently different from the ordinary ΨDO calculus to deserve a carefultreatment. The reader familiar with [1] can skip this section.

Recall that M is foliated by the integrable subbundle V . We shall only usecharts, i.e. local coordinates xi, which are foliation charts, that is

(15) V is generated by∂

∂xj= ∂j , j = 1, . . . , v .

Thus the plaques, i.e. the leaves of the restriction of the foliation, are Rv × pt.In such coordinates we shall use the ordinary formula to pass from a symbolσ(x, ξ) to the corresponding operator:

(16) Pσ = (2π)−m

∫ei〈(x−y),ξ〉 σ(x, ξ) dmξ m = v + n .

One has ξ ∈ (Rv × Rn)∗ = Rv × Rn and one defines a (coordinate dependent)notion of homogeneity of symbols using

(17) λ · ξ = (λ ξv, λ2 ξn) for ξ = (ξv, ξn) , λ ∈ R∗+ .

The natural length for ξ which is homogeneous of degree 1 is

(18) ‖ξ‖′ =(‖ξv‖4 + ‖ξn‖2

)1/4.

Let us start with a symbol σ, smooth on Rm × Rm\{0} and homogeneous ofdegree q for the dilations (17), i.e.

(19) σ(x, λ · ξ) = λq σ(x, ξ) .

15

In order to control the operator Pσ defined in (16), one needs to control thepartial derivatives

∂αx ∂β

ξ σ(x, ξ) .

When we apply ∂αx , the homogeneity property (19) is preserved. The action of

∂iξv

, i = 1, . . . , v, lowers q by 1 while the action of ∂iξn

, i = 1, . . . , n, lowers it by2. Thus, if we let

(20) 〈β〉 =v∑1

βi + 2n∑1

βv+j ,

we see that ∂αx ∂β

ξ σ is homogeneous of degree q − 〈β〉. It follows that, for x ina compact subset of Rm and α, β fixed,

(21)∣∣∣∂α

x ∂βξ σ(x, ξ)

∣∣∣ ≤ Cα,β (1 + ‖ξ‖′)q−〈β〉.

To employ the usual classes of ΨDO one needs to relate the right hand side tothe expression

(1 + ‖ξ‖)q− 12 |β| .

With a = ‖ξv‖, b = ‖ξn‖, one has

‖ξ‖2 = a2 + b2, ‖ξ‖′4 = a4 + b2,

so that‖ξ‖′ ≤ (

1 + ‖ξ‖2)1/2 ≤ 1 + ‖ξ‖ , 1 + (‖ξ‖′)4 ≥ ‖ξ‖2,

(22) ‖ξ‖1/2 ≤ 1 + ‖ξ‖′ ≤ 2 + ‖ξ‖ .It thus follows that, if q ≥ 0, a homogeneous symbol σ of degree q is of classSq

0,1/2, i.e.

(23)∣∣∣∂α

x ∂βξ σ(x, ξ)

∣∣∣ ≤ Cα,β (1 + ‖ξ‖)q− 12 |β| ,

while for q < 0 it is of class Sq/20,1/2.

This implies in particular that for q ≤ 0 the operator Pσ is bounded in L2 (cf.[1]).

We can now introduce the relevant class of symbols for the proof of TheoremI.1. We consider symbols σ such that:

there exists a sequence (σq) of homogeneous symbols,

σq of degree q, with σ ∼∑q≤q0

σq(24)

16

where ∼ means that for any N the difference

σN = σ−∑

N<q≤q0

σq

satisfies the inequalities (21) for q = N .

Definition I.2. We shall say that an operator P is a ΨDO′ if it is a Pσ, withσ as in (24).

Let us now describe the composition Pσ ◦ Pσ′ of two ΨDO′. Denoting as usual

(25) Dαx = (−i)|α| ∂α

x ,

the action of Pσ can be formally expanded as

Pσ =∑ 1

α!∂α

ξ σ(x, ξ) Dαx .

Let us consider the expansion:

(26)∑ 1

α!∂α

ξ σq1 (x, ξ) Dαx σ′

q2(x, ξ) .

The degree of homogeneity of each of the terms is

q1 + q2 − 〈α〉,

where 〈α〉 is defined in (20).

This shows that (26) makes sense as an asymptotic expansion and correspondsto the product Pσ ◦ Pσ′ .

Note that every differential operator P is a ΨDO′; the only difference is in thenotion of degree, since while ∂/∂xj = ∂j has degree 1 for j = 1, . . . , v it hasdegree 2 for j = v + 1, . . . , v + n. It is not true however that an ordinary ΨDOis a ΨDO′, even in the order 0 case.

We need to define the notions of principal symbol and ellipticity for ΨDO′. Tothis end we shall first consider what happens under a change of foliation chart.

Let ϕ be such a change of charts. It defines a local Rn diffeomorphism ϕn, insuch a way that, with the notations x = (xv, xn), y = (yv, yn), one has

(27) ϕ(x) = (ϕv(xv, xn), ϕn(xn)) .

Similarly, ψ = ϕ−1 is of the same form, ψ = (ψv, ψn). Given a covector ξ ∈ T ∗x

the corresponding covector at y = ϕ(x) is

(28) η = ψt∗(ξ)

17

i.e. 〈η, Y 〉 = 〈ξ, ψ∗(Y )〉, ∀ Y ∈ Ty.

With σ = σ(y, η) a homogeneous symbol, let us consider the composition

(29) σ(x, ξ) = σ(ϕ(x), ψt

∗,ϕ(x)(ξ)).

To compare it with σ(x, λ · ξ) we just need to understand the linear map ψt∗ at

ϕ(x) = y. This map, L, preserves the natural subspace N∗ ⊂ T ∗ of covectorsorthogonal to the leaves. On the subspace N∗ = {(0, ξn)} one has λ · ξ = λ2ξ.Thus

L =[Lvv 0Lnv Lnn

]L(λ · ξ) =

(λ Lvv(ξv), λ2 Lnn(ξn) + λ Lnv(ξv)

).

With obvious notation, we have

σ(L(ξ)) = σ (Lvv(ξv), Lnn(ξn) + Lnv(ξv))

=∑ 1

α!∂α

ηvσ (Lvv(ξv), Lnn(ξn)) (Ln,v(ξv))α

,

which gives the desired expansion of σ ◦ L as a sum of homogeneous symbols.This shows that formula (29) defines a transformation of symbols.

To obtain the symbol of the operator Pσ in the new coordinates one writes itskernel as

(30) k(x, x′) = (2π)−m

∫ei〈ϕ(x)−ϕ(x′),η〉 σ(ϕ(x), η) dmη .

One first changes variables from η to ξ using (28) and the fact that k is a1-density, so that no Jacobian enters in the change of variables, due to theinvariance of the symplectic Liouville measure. Thus,

(31) k(x, x′) = (2π)−m

∫ei(〈x−x′,ξ〉+α(x,x′,ξ)) σ(x, ξ) dmξ,

where α(x, x′, ξ) is the non linearity at x of the map ϕ:

(32) 〈ϕ(x) − ϕ(x′), η〉 = 〈x− x′, ξ〉 + α(x, x′, ξ) .

The variable ξ appears linearly in α and the coefficient of ξn only invokesϕn(xn) − ϕn(x′n) and not xv or x′v. By construction, α and its first deriva-tives in x′ vanish at x′ = x. This implies that the Taylor expansion at x′ = xof eiα(x,x′,ξ) is of the form

(33) eiα(x,x′,ξ) =∑

Pβ(x, ξ)(x − x′)β ,

18

where Pβ(x, ξ) is a polynomial in ξ whose degree, in the sense that ξα has degree〈α〉, is smaller than 1

2 〈β〉. Thus, using ∂βξ ei〈x−x′,ξ〉 = i|β| (x − x′)β ei〈x−x′,ξ〉

and integrating by parts, one gets the full symbol for Pσ:

(34) σ(x, ξ) =∑

i−|β| ∂βξ (Pβ(x, ξ) σ(x, ξ)) .

In particular, at a given x the value of σ(x, ξ) only involves σ restricted to T ∗x

and not its restriction to any T ∗y , y �= x. To this restriction σ one applies a

differential operator with polynomial coefficients.

Let us now turn to the principal symbol. Let σ be a symbol of order q; then itsprincipal part is

(35) limλ→∞

λ−q σ(x, λ · ξ) .

We see that it gives a well defined function on the bundle

V ∗ ⊕N∗,

the direct sum of the subbundle N∗ ⊂ T ∗ and of V ∗ = T ∗/N∗. The abovelimit exists and under a change of coordinates it behaves, in view of the aboveformulae, as a function on V ∗ ⊕N∗.As an example, let us compute the principal symbol of

[|D|, Pσ]

where σ is of order 0 and where the principal symbol of |D| is

(36) σ1(x, ξ) = ‖ξ‖′ (cf. 18) .

We can use formula (26) in local coordinates. It gives

(37)∑

〈α〉=1

(∂α

ξ σ1(x, ξ) Dαx σ(x, ξ) − ∂α

ξ σ(x, ξ) Dαx σ1(x, ξ)

).

Note that since 〈α〉 = 1, this formula only involves the longitudinal differentia-tion Dα

xv.

1.4 Proof of Theorem I.1.

The operator D is defined by equation (14), where Q is a selfadjoint differentialoperator by construction. We shall first show that D is a ΨDO′ of order 1.

As noted before, any differential operator is ΨDO′ in the above sense , butthe notions of degree and of principal symbol differ from the usual ones. In

19

particular, Q is elliptic of degree 2 and its principal symbol is, for ξv ∈ V ∗,ξn ∈ N∗, the endomorphism of ∧V ∗ ⊗ ∧N∗

σ2(ξv, ξn) = (eξv iξv − iξv eξv) ⊗ (−1)∂ + 1 ⊗ (i eξn + (i eξn)∗) .

The two sides anticommute, so that when we square σ2 we get

σ22(ξv, ξn) =

(‖ξv‖4 + ‖ξn‖2) · 1,

which shows that Q2 is a ΨDO′ of order 4, elliptic and with principal symbol amultiple of the identity.

As in the ordinary pseudodifferential calculus, this is enough (cf. [8] p.52) toconstruct an asymptotic ΨDO′, R(µ) which is an asymptotic resolvent for Q2.

We shall use for D and |D| the following formulas:

(38) |D| =√

22π

∫ ∞

0

Q2

Q2 + µµ−3/4 dµ

(39) D =√

22π

∫ ∞

0

Q

Q2 + µµ−1/4 dµ .

Let us replace, in these two formulae, (Q2 + µ)−1 by the asymptotic resolventR(µ). Then R(µ)(Q2 +µ)−1 is smoothing of any order, with the correspondingnorms controlled by (1+µ)−k. Thus, when we replace Q2/(Q2 +µ) by Q2R(µ),we use

Q2

Q2 + µ

(1 − (Q2 + µ) R(µ)

)=

Q2

Q2 + µ−Q2 R(µ),

which is therefore also smoothing of any order if µ ≥ 0. Using the boundednessof

µ1/2 Q2

Q2 + µfor µ ≥ 0,

a similar statement holds for

µ1/2

(Q

Q2 + µ−Q R(µ)

).

Since the asymptotic symbol of (Q2 + µ)−1 is, at the principal level,

σ−4 =(‖ξv‖4 + ‖ξn‖2 + µ

)−1

one obtains the principal symbols of the operators

(40) |D|r =√

22π

∫ ∞

0

Q2 R(µ) µ−3/4 dµ,

20

(41) Dr =√

22π

∫ ∞

0

Q R(µ) µ−1/4 dµ

as the integrals

(42)√

22π

∫ ∞

0

‖ξ‖′4

‖ξ‖′4 + µµ−3/4 dµ = ‖ξ‖′

(43)√

22π

∫ ∞

0

σ2(Q)(‖ξ‖′4 + µ

)−1

µ−1/4 dµ =σ2(Q)(ξ)

‖ξ‖′ .

This is enough to show that both |D| and D are ΨDO′ of order 1 and to givein local coordinates the asymptotic expansion of their symbol.

Since the principal symbol of |D| is ‖ξ‖′ · 1, a multiple of the identity matrix,it commutes with the symbol of any ΨDO′ of order 0. This shows that, with δdenoting the derivation

δ(T ) = [|D|, T ],

one has:

Lemma I.2. Any ΨDO′ of order 0 belongs to ∩n≥1

Dom δn.

This applies to the multiplication operator f as well as [D, f ] and proves The-orem I.1 1).

For assertion 2) of the theorem, we shall prove the more precise result

(44) |D|−1 ∈ L(v+2n,∞) ,

which in turn will follow from:

Lemma I.3. Let P be a ΨDO′ of order −(v + 2n). Then

µk(P ) = O(k−1)

where µk(P ) is the kth characteristic value of P .

Proof. It is enough to check this locally, so that one can assume without lossof generality that

M = Tv × Tn and P =(∆2

v ⊗ 1 + 1 ⊗ ∆n + 1)− v+2n

4 .

One just has to bound the number of eigenvalues λ > ε of P by C/ε for someC < ∞. Equivalently, one has to bound the number of eigenvalues λ < ε−

4v+2n

of ∆2v ⊗ 1 + 1 ⊗ ∆n + 1 by C/ε. But this number is less than Nv(E) Nn(E),

21

where E = ε−4

v+2n , Nv(E) is the number of eigenvalues of ∆2v less than E, while

Nn(E) stands for the number of eigenvalues of ∆n less than E.

One has Nv(E) ≤ Cv Ev/4, Nn(E) ≤ Cn En/2 and we get the required bound.

For Theorem I.1 3), we just note that the choice ofH does not affect the principalsymbols of either D or |D|, while any ΨDO′ of order 0 is bounded.

It remains to prove assertion 4). The operators Uϕ D U−1ϕ , Uϕ |D| U−1

ϕ areΨDO′ of order 1, with the same principal symbols as D and |D| respectively.This shows that the following are ΨDO′ of order 0:

[D,Uϕ] U−1ϕ , [|D|, Uϕ] U−1

ϕ .

In particular, they are bounded and belong to ∩n≥1 Dom δn. Thus, Uϕ ∈ Dom δ,δ(Uϕ) U−1

ϕ ∈ Dom δ, hence Uϕ ∈ Dom δ2. By induction, using δ(Uϕ) · U−1ϕ ∈

Dom δn, one gets Uϕ ∈ ∩n≥1 Dom δn and thus [D,Uϕ] =([D,Uϕ] U−1

ϕ

) ·Uϕ ∈∩n≥1 Dom δn.

1.5 The Dixmier trace of ΨDO′ of order −(v + 2n)

We shall now describe the kernels k(x, y) for operators of order −(v + 2n) andcompute their Dixmier trace. As we shall see, the relevant question is that ofextending a homogeneous symbol σ(ξ), ξ ∈ Rv+n\{0},

(45) σ(λ · ξ) = λ−(v+2n) σ(ξ) ∀λ ∈ R∗+

to a homogeneous distribution.

The degree of homogeneity q = −(v + 2n) considered in (45) is the limit casefor integrability both near 0 and near ∞. Indeed, the Jacobian of ξ → λ · ξ isλv+2n and on each orbit of the flow F ,

Fs(ξ) = es · ξ,

the measure σ(ξ) dv+n ξ is proportional to dλλ . It thus has a logarithmic diver-

gency both at 0 and at ∞. They turn out to be intimately related and we shallinvestigate the divergency at 0, following closely ([1]).

We consider the linear form on the space{f ∈ S(Rv+n) ; f(0) = 0

}= S0

given by

L(f) =∫f(ξ) σ(ξ) dv+n ξ .

22

It makes sense because σ is bounded at ∞ (polynomial growth would be enough)and f(0) = 0 takes care of the non integrability at 0.

With fλ(ξ) = f(λ−1 · ξ) the homogeneity of σ means:

(46) L(fλ) = L(f) ∀f ∈ S0 .

By the Hahn-Banach theorem the linear form L extends from the hyperplaneS0 to all of S as a continuous linear form and we get a one dimensional affinesubspace of S′:

(47) E = {τ ∈ S′ ; τ |S0 = L} ,

the corresponding linear space is the space of multiples of δ0 the Dirac mass at0.

The dilations θλ,〈θλ(τ), f〉 = 〈τ, fλ〉

act on E; since they act trivially on the associated linear space, their action onE is given, for some constant c, by

(48) θλ τ = τ + c logλ δ0 , ∀τ ∈ E , λ ∈ R∗+ .

To determine c, let ψ ∈ C∞c ([0,∞[) be identically 1 near 0, and let τ be given

by

(49) τ(f) := L (f − f(0) ψ (‖ ‖′)) =∫

(f(ξ) − f(0) ψ (‖ξ‖′)) σ(ξ) dξ .

One has,

τ(fλ) − τ(f) =∫ (

f(λ−1 · ξ) − f(0) ψ (‖ξ‖′)) σ(ξ) dξ

−∫

(f(ξ) − f(0) ψ (‖ξ‖′)) σ(ξ) dξ

= f(0)∫

(ψ (‖ξ‖′) − ψ (λ‖ξ‖′)) σ(ξ) dξ

= f(0) cσ∫ ∞

0

(ψ(µ) − ψ(λµ))dµ

µ,

where cσ is obtained as the pairing between any transversal cycle ‖ξ‖′ = constantto the foliation of Rv+n\{0} by the orbits of the flow F and the closed de Rhamcurrent obtained as the contraction ie(σ dξ) of the differential form σ dξ by thevector field e = λ−1 d/dλ generating the flow.

Letting µ = eu, λ = es one gets∫ ∞

0

(ψ(µ) − ψ(λµ))dµ

µ=

∫ ∞

−∞

(ψ(eu) − ψ(eu+s)

)du = s = logλ .

23

Thus we have shown that c = cσ is exactly the obstruction to extending σ as ahomogeneous distribution on Rv+n (cf. [1]).

We can write, in a formal way,

(50) cσ =∫‖ξ‖′=1

ie σ(ξ) dξ .

Let us now relate this obstruction to the behavior of the inverse Fourier trans-form of σ

(51) σ(y) = (2π)−(v+n)

∫ei〈y,ξ〉 σ(ξ) dξ .

We first need to relate the oscillatory integral definition (51) to the Fouriertransform for tempered distributions.

For y �= 0, the oscillatory integral is defined as the value of the convergentintegral

(52) (2π)−v+n

∫ei〈y,ξ〉 (P k σ(ξ)) dξ k ≥ 1

where the symbol σ(ξ) has been smoothed for ξ small, and Py is a differentialoperator of degree 1 in ∂α

∂ξα such that

P ty(ei〈y,ξ〉) = ei〈y,ξ〉 .

The smoothing of σ near ξ = 0 introduces an ambiguity of the addition ofarbitrary elements of (C∞

c ) ⊂ S. But this does not affect the behavior aty = 0, which we are after.

Having chosen an extension τ of σ as a distribution, we need to check that theinverse Fourier transform τ , a tempered distribution, is represented by a locallyintegrable tempered function h whose behavior at y = 0 is the same as for theoscillatory integral.

Since the Fourier transform of a distribution with compact support is repre-sented by a nice smooth function, we just need to know that if σ1 is a symbol,smooth on all of Rv+n, then its Fourier transform as a tempered distribution isgiven by the oscillatory integral expression (52).

To check this, let P(y, ∂

∂ξ

)= 1

i

∑yj ∂

∂ξj ; then

P ei〈y,ξ〉 = ‖y‖2 ei〈y,ξ〉 .

24

With f ∈ S(Rn+v)0, we can write:∫f(ξ) σ1(ξ) dξ = (2π)−(n+v)

∫ ∫f(y) ei〈y·ξ〉 dy σ1(ξ) dξ

= (2π)−(n+v)

∫ ∫P (ei〈y·ξ〉) ‖y‖−2 f(y) σ1(ξ) dy dξ

= (2π)−(n+v)

∫ ∫ei〈y,ξ〉 ‖y‖−2 f(y) (Pσ1)(ξ) dy dξ

= (2π)−(n+v)

∫ ∫ei〈y,ξ〉 ‖y‖−2 Pσ1(ξ) dξ f(y) dy .

Thus, we know that the distribution τ is represented outside 0 by a smooth func-tion with tempered growth. This function is then unique and the homogeneityproperty (48) implies, using 〈λ · ξ, y〉 = 〈ξ, λ · y〉,

(53) τ(λ−1 · y) = (θλ τ)∨(y) = τ (y) + c′ logλ .

Thus,τ (y) = τ (y/‖y‖′) − c′ log (‖y‖′) , c′ = (2π)−(n+v) c .

We are now ready to deal with the Dixmier trace of ΨDO′ of order −(v + 2n).

We let (M,V ) be as above, with M compact.

Proposition I.2. Let T be a ΨDO′ of order −(v + 2n). Then1) T is measurable, T ∈ L(1,∞), with its Dixmier trace Trω(T ) independent ofω and given by

Trω = (2π)−(n+v) 1v + 2n

∫‖ξ‖′=1

σ(x, ξ) ie (dx dξ) ,

where e is the generator of the flow

Fs(ξv, ξn) = (es ξv, e2s ξn)

and the choice of transversal ‖ξ‖′ = 1 is irrelevant. Also dx dξ corresponds tothe symplectic volume form.2) The kernel k(x, y) for T has the following behavior near the diagonal

k(x, y) = c(x) log (‖x− y‖′) + 0(1) ,

where the 1-density c(x) is given by the formula

c(x) = (2π)−(n+v)

∫‖ξ‖′=1

σ(x, ξ) ie dξ .

25

Remark I.1. Before beginning the proof, let us note that 1) reduces to theusual formula (cf. [2]) for the Dixmier trace of ordinary ΨDO when either n = 0or v = 0; in the latter case the exponent 2s accounts for the 2n. Note also thatin 2) the choice of the local distance function ‖x− y‖′ has no effect on the valueof c(x). The statement 2) is a special case of [1].

Proof. 1) By Lemma I.2, one has T ∈ L(1,∞) so that Trω(T ) is well defined.Since operators of lower order are (by the argument of Lemma I.2) of trace class,it follows that Trω(T ) only depends upon the principal symbol σ(T ). The map

(54) σ → Trω(Tσ)

is then a positive linear form on the space F of homogeneous symbols of order−(v + 2n). It is therefore given by a positive measure on the (non canonical)unit sphere

(55) S∗ {(x, ξ) ∈ V ∗ ⊕N∗ ; ‖ξ‖′ = 1} .

The unitary invariance of Trω together with the use of translations in a folia-tion chart, show that this measure is absolutely continuous with respect to thesmooth measure dx on M . The diffeomorphism invariance of the ΨDO′-calculusshows that the conditional measures on the fibers of

(56) p : S∗ →M

must be, in the appropriate sense, invariant under all maps

(ξv, ξn) → (Lv ξv, Ln ξn)

with both Lv and Ln invertible.

Such maps do not act transitively on V ∗x ⊕N∗

x . The only two invariant subspacesare V ∗

x ⊕ 0, 0 ⊕ N∗x . But they do not carry any measure with the correct

homogeneity.

This implies that there exists a constant a(ω) such that

(57) Trω(T ) = a(ω)∫‖ξ‖′=1

σ(x, ξ) ie dx dξ .

To show that this constant a(ω) does not depend on ω and to determine it, onejust needs to compute Trω(T ) for one specific example for each value of v andn. With the notations of Lemma I.2, we take:

(58) T =(∆2

v ⊗ 1 + 1 ⊗ ∆n + 1)−( v+2n

4 ).

In order to compute Trω(T ), we just use the following general fact (cf. AppendixA).

26

(59) Let ∆ be a positive (unbounded) operator such that ∆−1 ∈ L(p,∞) for somep ≥ 1, and

tp Trace (e−t∆) →t→0

L .

Then Trω(∆−p) is independent of ω and given by

Trω(∆−p) =L

Γ(p+ 1).

We take ∆ = ∆2v ⊗ 1 + 1 ⊗ ∆n + 1, p = v+2n

4 . To compute L it is enough todetermine

limt→0

tpv Trace (e−t∆2v) = Lv , pv = v/4 ,

limt→0

tpn Trace (e−t∆n) = Ln , pn = n/2

which then gives L = Lv Ln.

Using (59) one has Lv = Γ(pv +1) Trω(∆−v/2v ) and Ln = Γ(pn +1) Trω(∆−n/2

n ).Choosing the standard metric

∑dθ2j on both Tv and Tn gives, with |Sk| the

volume of the k-dimensional sphere,

(60) Lv = Γ(v

4+ 1

) 1v|Sv−1| , Ln = Γ

(n2

+ 1) 1n

|Sn−1|

and

Trω(∆−p) = Γ(v + 2n

4+ 1

)−1

Lv Ln .

The principal symbol of ∆−p is

(61) σ(x, ξ) = (‖ξ‖′)−(v+2n).

If we let|S|′ =

∫‖ξ‖′=1

ie (dξ),

we have the equalities

(62)∫‖ξ‖′=1

σ(x, ξ) ie (dx dξ) = (2π)n+v |S|′

and

(63)∫f (‖ξ‖′) dξ = |S|′

∫ ∞

0

f(ρ) ρv+2n dρ

ρ,

whenever both sides make sense.

27

Using f(ρ) = e−λρ4yields

(64) |S|′ =12

Γ(v + 2n

4

)−1

Γ(v

4

)|Sv−1| Γ

(n2

)|Sn−1| .

Together with (60) and (62), this gives the normalization:

(65) Trω(∆−p) = (2π)−(n+v) (v + 2n)−1

∫‖ξ‖′=1

σ(x, ξ) ie (dx dξ) .

2) The kernel k(x, y) of T is given by

k(x, y) = (2π)−(v+n)

∫ei〈x−y,ξ〉 σ(x, ξ) dξ ,

so that, for fixed x, it is, as a function of y−x, the Fourier transform of σ(x, ξ).Thus, using (53) we get the required answer. For a more detailed proof, thereader is referred to [1] .

Proposition I.2 extends immediately to non scalar operators, with σ(x, ξ) re-placed by its trace tr(σ(x, ξ)) taken in the fiber over x. We can therefore applyit to the operator |D|−(v+2n) of Theorem I.1, whose symbol is the identity matrix(for ‖ξ‖′ = 1) on a space of dimension 2(v+n). We get:

(66) Trω

(f |D|−(v+2n)

)= π−(n+v) (v + 2n)−1 |S|′

∫f(x) dx

where |S|′ is given by (64) and dx is the volume form on M corresponding tothe given Euclidean structures on both V and N .

1.6 The analogue of the Wodzicki residue for ΨDO′ oper-ators

Let us now go back to the obstruction cσ (cf. (50)), and exploit its definitioninvolving the behavior of T near 0 rather than (the ultraviolet behavior) near∞.

Lemma I.4. ([1]) Let σ ∈ C∞(Rn+v\{0}) be homogeneous of order q, i.e.σ(λ ·ξ) = λq σ(ξ) , ∀ξ �= 0, ∀λ ∈ R∗

+, with (q ∈ C).a) If q /∈ {−(v+2n)− k ; k ∈ N} then σ extends to a homogeneous distributionon Rn+v.b) If q = −(v+2n)− k, then the obstruction to homogeneous extension is givenby the cξασ, |α| = k.

28

Proof. Let ψ be as in (48) and k the integral part of

−Re(q) − (v + 2n) = a .

The size of σ(ξ) for ξ small is comparable to (‖ξ‖′)Req = ‖ξ‖′−(v+2n+a). Thusσ(ξ) ξα is locally integrable if |α| > k and the following is an extension of σ:

(67) τ(f) =∫ f(ξ) −

∑|α|≤k

ξα

α!f (α)(0) ψ(ξ)

σ(ξ) dξ .

One gets then,

τ(fλ) − λq+(v+2n) τ(f) =∫ f(λ−1 · ξ) −

∑|α|≤k

λ−|α|ξα

α!f (α)(0) ψ(ξ)

σ(ξ) dξ

− λq+(v+2n)

∫ f(ξ) −∑|α|≤k

ξα

α!f (α)(0) ψ(ξ)

σ(ξ) dξ

= λq+(2n+v)∑|α|≤k

fα(0)α!

(∫‖ξ‖′=1

ξα σ(ξ) ie dξ

)ρα

ρα =∫ ∞

0

(ψ(µ) − ψ(λµ)) µq+(v+2n)+|α| dµ/µ .

To prove a), it is enough to choose ψ so that the ρα all vanish. With ψ(µ) =h(logµ), we thus look for h′ ∈ C∞

c (R),∫h′(s) ds = −1, such that

(68)∫ ∞

−∞h′(s) ebs ds = 0 ∀b = q + (v + 2n) + |α| , 0 ≤ |α| ≤ k .

One just lets h′ = Π(d/ds + b)f , f ∈ C∞c (R), with

∫f(s) ds = (Πb)−1. This

is possible since b �= 0 by hypothesis.

Assertion b) can be proved in a similar fashion, since for q = −(v+ 2n)− k onecan get ρα = 0 for any α, |α| < k.

The ambiguity in the extension of σ is, a priori, of the form∑

|α|≤k

aα ∂α δ0. By

a), if q /∈ {−(v + 2n) − N}, the homogeneous extension exists and is unique,except when q ∈ N.

Corollary I.1. ([1]) Let σ ∈ C∞(Rn+v) have an asymptotic expansion σ ∼∑k≤q

σk in homogeneous symbols (for λ·). Then the Fourier transform σ (of σ smooth

29

at 0) has a behavior at y = 0 of the form:

σ(y) =0∑

−(q+(v+2n))

aj(y) + c log ‖y‖′ + 0(1) ,

where aj is homogeneous of degree j.

Proof. We can neglect the difference σ− ∑−(v+2n)≤k≤q

σk since it is integrable

at ∞ and yields a 0(1) contribution for small y. By Lemma I.4, each σk,k > −(v + 2n) extends to a homogeneous distribution, so that the Fouriertransform σk of σk (smoothed at 0) has the indicated divergency at 0. The casek = v + 2n follows from (53).

We see that

(69) the coefficient of log ‖y‖′ only depends on σ−(v+2n) and is equal to

(2π)−(n+v)

∫‖ξ‖′=1

σ−(v+2n) (ξ) ie dξ .

One can check directly, using (29) and (34), that the density

(70) (2π)−(n+v)

∫‖ξ‖′=1

σ−(v+2n) (x, ξ) ie dξ

is invariantly defined under the action of diffeomorphisms on the ΨDO′ calculus.The obtained density is the coefficient of log ‖x− y‖′, in the expansion near thediagonal, of the kernel k(x, y) for P = Pσ.

Proposition I.3. The Dixmier trace Trω has a canonical extension to a traceon the algebra of ΨDO′ operators of arbitrary order. It is given globally by theequality

Trω(T ) =1

v + 2n

∫M

c(x) ,

where c(x) is the 1-density occuring in the expansion of the kernel k of T nearthe diagonal, as a coefficient of log ‖x − y‖′. In local coordinates it is given by(v + 2n)−1 times the expression (70).

The fact that the asserted extension is a trace is proved in greater generality inSection II (cf. Prop. II.1 ).

We proceed to show that there is a natural extension, similar to [10], of theordinary trace of operators to ΨDO′ of complex order.

30

To this end, we go back to the space of symbols, where we need to define whatis meant by a holomorphic map z → σz with values in the space of symbols.We want the order f(z) to be holomorphic in z, the bounds in the asymptoticexpansion σ(z) ∼ ∑

σf(z)−p to be uniform and the pointwise values of σ(z)(ξ),σ(z) ∈ C∞(Rv+n), to be holomorphic in the variable z. Then each σf(z)−p(ξ)is also holomorphic in z. The functional

(71) L(σ) = σ(0) = (2π)−(v+n)

∫σ(ξ) dξ

is well defined on symbols of order < −(v + 2n) (for the real part of the order)and it is holomorphic inasmuch as L(σz) is a holomorphic function of z for anyholomorphic map z → σz .

Lemma I.5. The functional L has a unique holomorphic extension L to thespace of symbols of non integral order (z /∈ Z). The value of L on σ ∼ ∑

σz−p

is given by

L(σ) = (2π)−(v+n)

∫ (σ −

N∑0

τz−p

)(ξ) dξ , N ≥ Re(z) + (v + 2n)

where τz−p is the unique homogeneous extension of σz−p.

This unique extension τ of σ is given by Lemma I.4 a).

Proof. First, the value of N used in Lemma I.5 is irrelevant, since the Fouriertransform of a homogeneous distribution such as τz−p vanishes at 0 if p >Rez + (v + 2n).

Also the uniqueness is clear since any σ of order z can be connected to integrableorder by a holomorphic path.

It remains to show that if z → σ(z) is holomorphic, with order f(z) /∈ Z thenL(σ(z)) is holomorphic.

For large ξ the pointwise value of σ(z)−N∑0σf(z)−p is holomorphic in z and has

uniformly integrable behavior at ∞; thus it is enough to control the behaviorin z of ∫ (

σ −N∑0

τf(z)−p

)ϕ(ξ) dξ ,

where ϕ has compact support. In fact we can consider separately the term∫τf(z)−p ϕ(ξ) dξ, which is holomorphic in z by the very construction of τ (cf.

(67) and (68)).

31

Remark I.2. To a symbol of order z and to any given p ∈ N one can assignthe number ∫

‖ξ‖′=1

σz−p(ξ) ie dξ

and the functional thus obtained is holomorphic, but it does not vanish on σwith integrable order and cannot be added to L to yield another extension.

Let us now return to the (compact) manifold (M,V ) as above, and considerthe product Ψ0 × C of the space of ΨDO′ of order 0 by C, endowed with theproduct structure of complex manifold. We shall adapt the method of [10] toour context to obtain:

Proposition I.4. The function (P, z) → Trace(P |D|−z) is holomorphic onΨ0 × {z ; Rez > (v + 2n)} and extends uniquely to a holomorphic function TRon Ψ0 × (C\Z).

Proof. In a local chart, the trace of a ΨDO′ P = Pσ of order < −(v + 2n) isgiven by

(72) Trace(Pσ) = (2π)−(n+v)

∫σ(x, ξ) dx dξ ,

where the total symbol σ is smooth.

Thus, in a local chart, the following formula provides the required extension ofTrace to ΨDO′ of arbitrary order z /∈ Z

(73) TR(Pσ) =∫L(σ(x, ·))dx .

The ambiguity of smoothing operators does not alter the existence of this ex-tension globally on all ΨDO′ of order z /∈ Z and yields the required extension ofTrace, provided one knows that |D|−z is a ΨDO′ of order −z and a holomorphicfunction of z. This follows from the proof of Theorem I.1 (see (38)), using theasymptotic resolvent of Q2 (cf.[12]).

Let us now relate the value of Trω(P ), P ∈ ΨDO′ to the residue at z = 0 of thefunction:

(74) ζ(z) = TR(P |D|−z

).

We can work first at the level of symbols and consider a fixed symbol σ ofintegral order q. We let σz , σz(ξ) = σ(ξ) (‖ξ‖′)−z and investigate the behaviorof L(σz) near z = 0. Let N ≥ q + (v + 2n), then

L(σz) = (2π)−(v+n)

∫ (σ −

N∑0

σq−k

)(ξ) (‖ξ‖′)−z

dξ

32

where σq−k(ξ) ‖ξ‖1−z is replaced near 0 by its unique extension as a homoge-neous distribution.

The singularity at z = 0 comes from ξ in the neighborhood of 0. Whenq − k > −(v + 2n), σq−k(ξ) dξ is integrable at 0 and the unique extensionof σq−k(ξ) ‖ξ‖′−z dξ is holomorphic in z at z = 0. Thus none of these termscontribute to the singularity of L(σz) at z = 0.

We can choose N = q+(v+2n) since, by Lemma I.5, any larger value gives thesame answer. We thus need to understand the behavior at z = 0 of

(75)∫‖ξ‖′≤a

σ−(v+2n)(ξ) ‖ξ‖′−z dξ

where σ−(v+2n)(ξ) (‖ξ‖′)−zdξ is extended uniquely as a homogeneous distribu-

tion at ξ = 0. But for Rez < 0 one has integrability near 0 so that this uniqueextension is the obvious one and one can write (75) as

(76)

(∫‖ξ‖′=1

σ−(v+2n)(ξ) ie dξ

) (∫ a

0

µ−z dµ

µ

).

As∫ a

0 µ−z dµ

µ = −[

µ−z

z

]a

0= −a−z/z, one gets that the singularity of L(σz) at

z = 0 is a simple pole with residue:

(77) (2π)−(v+2n)

∫‖ξ‖′=1

σ−(v+2n)(ξ) ie dξ .

Next, when we investigate (74) the situation is more complicated since in a local(foliation) chart we have an intricate expression for the total symbol of |D|−z.

Let us first remark that the above discussion of the behavior of L(σz) continuesto hold when σz is of the form

(78) σz = σ(ξ, z) (‖ξ‖′)−z,

where z → σ(·, z) is a holomorphic map to symbols of fixed order q. Moreover,the residue at z = 0 is given by (77), with σ(ξ, 0). Using this, we should be ableto replace |D| by the operator |D1| which in the given (foliation) chart involvesthe flat metric

(79) ‖ξ‖′4 = ‖ξv‖4 + ‖ξn‖2 ,

independently of x = (xv, xn).

Since the total symbol of D41, assumed to be given by (79), does not involve

x, the corresponding (differential) operator is translation invariant. So are the

33

complex powers |D1|z and their total symbol is given by (a smoothed versionof)

(80) σz(x, ξ) = (‖ξ‖′)z.

The computation of the total symbol of P |D1|−z for any ΨD0′, is then obvious,by (26), and the above discussion shows that the function ζ1(z) = TR (P |D1|−z)has a simple pole at z = 0, which is given by (77). This continues to hold forany holomorphic map z → Pz to symbols of fixed order q.

We can now write

(81) P |D|−z = P U(z)|D1|−z , U(z) = |D|−z |D1|z

and it just remains to show that U(z) is a holomorphic map to operators oforder 0, with U(0) = 1.

Thus, the principal symbol of U(z) will be σ(ξ)−z/4 σ1(ξ)z/4. By construction,U(z) is the ΨDO′ product of two holomorphic maps and hence is holomorphic.We thus proved the following result:

Theorem I.2. Let P be a ΨDO′ of integral order q. Then the function z →Trace (P |D|−z) is holomorphic for Rez > q + (v + 2n) and admits a (unique)analytic continuation to C\Z with at most simple poles at integers k ≤ q+ (v+2n). Its residue at z = 0 is given by

Resz=0 Trace(P |D|−z

)= (v + 2n) Trω(P ),

where Trω is defined in Proposition I.3.

34

2 The universal local index formula

2.1 Dimension spectrum

In this section, we shall describe a general local index formula in terms of theDixmier trace, extended to operators of arbitrary order, for our spectral triples:

(1) (A,H, D) .

Contrary to the standard practice, we shall focus on the odd case, the pointbeing that in the even case there is a natural obstruction to express the (cycliccocycle) character (cf. [2]) of the triple (1) in terms of a residue or Dixmiertrace. Indeed, the latter vanishes on any finite rank operator and thus will givethe result 0 whenever H is finite dimensional. Since it is easy to construct finitedimensional (i.e. dimH < ∞) even triples with Ind(D) �= 0 one cannot expectto cover this case as well. However, one can convert any even triple into an oddone by crossing it with S1, i.e. with the triple

(2)(C∞(S1) , L2(S1) , D =

1i

∂

∂θ

).

Thus, there is no real loss of generality in treating the odd case only.

The next point is that the usual notion of dimension (cf. [2]) for spectral triples,provided by the degree of summability

(3) D−1 ∈ L(p,∞) ,

gives only an upper bound on dimension and cannot detect the dimensions of thevarious pieces of a space constructed as a union of pieces of different dimensions(Ak,Hk, Dk) , k = 1, ..., N ,

(4) A = ⊕Ak , H = ⊕Hk , D = ⊕Dk .

In [2] we gave a formula for the p-dimensional Hochschild cohomology class ofthe character, namely:

(5) τ(a0, . . . , ap) = Trω

(a0[D, a1] . . . [D, ap] |D|−p

).

Clearly, this Hochschild cocycle cannot account for lower dimensional pieces ina union such as (4).

As it turns out, the correct notion of dimension is given not by a single realnumber p but by a subset

(6) Sd ⊂ C

which shall be called the dimension spectrum of the given triple. We shallassume that Sd is a discrete subset of C, condition which can be incorporatedin the definition of Sd, as follows:

35

Definition II.1. A spectral triple (1) has discrete dimension spectrum Sd, ifSd ⊂ C is discrete and for any element of the algebra B generated by the δn(a),a ∈ A, the function

ζb(z) = Trace(b|D|−z

)extends holomorphically to C\Sd.

Here δ denotes the derivation δ(T ) = [|D|, T ] and we assume that A ⊂ ∩n>0

Dom δn (see also Appendix B). The operator b|D|−z of Definition II.1 is thenof trace class for Re z > p, with p as in (3). On the technical side, we shallassume that the analytic continuation of ζb is such that Γ(z) ζb(z) is of rapiddecay on vertical lines z = s+ it, for any s with Re s > 0.

It is not difficult to check that Sd has the correct behavior with respect to theoperations of sum and product for spectral triples:

(7) Sd (Sum of two spaces) = ∪Sd(Spaces)

(8) Sd (Product of two spaces) = Sd(Space1) + Sd(Space2) ;

more precisely, (8) holds with the exception of Sd ∩ −N.

According to Theorem I.2 of Section I, the dimension spectrum of the hypoel-liptic triple considered there is contained in

(9) {q ∈ N ; q ≤ v + 2n} .

It is easy to give many examples of spectral triples with discrete dimensionspectrum, but we shall now concentrate on the general theory of such spaces.

Our first task will be to extend the Wodzicki residue to this general framework,or equivalently, to extend the Dixmier trace to operators P |D|−z of arbitraryorder, where P is an element of B. In fact it is more convenient (cf. AppendixB) to introduce the algebra Ψ∗(A) of operators which have an expansion:

(10) P � bq|D|q + bq−1|D|q−1 + · · · , bq ∈ B ,

where the equality with∑

−N<n≤q

bn |D|n holds modulo OP−N .

To see that it is an algebra one uses Theorem B.1 of Appendix B, which givesan identity of the form:

(11) |D|α b �∞∑0

cα,k δk(b) |D|α−k ,

36

where cα,k is the coefficient of εk in the expansion of

(12) (1 + ε)α =∞∑0

α(α− 1) · · · (α− k + 1)k!

εk ,

with ε(b) = δ(b) |D|−1.

We shall say that the dimension spectrum Sd is simple, when the singularitiesof the functions ζb(z) of Definition II.1 at z ∈ Sd are at most simple poles.Similarly, we say that Sd has finite multiplicity k when ζb has at most a poleof order k. We shall assume for simplicity that Sd has finite multiplicity in thissection.

Proposition II.1 Let p <∞ be the degree of summability of D.a) For P ∈ Ψ∗(A) the function h(z) = Trace

(P |D|−2z

)is holomorphic for

Rez > 12 (Order P+p) and extends to a holomorphic function on the complement

of a discrete subset of C.b) Let τk(P ) be the residue at 0 of zk h(z), k ≥ 0; then

τk(P1P2 − P2P1) =∑n>0

(−1)n−1

n!τk+n(P1 L

n(P2)),

where L is the derivation L = 2 log(1 + ε).

Proof. a) The statement follows immediately from Definition II. 1, for anyfinite sum of operators bn |D|n. Furthermore, if P is of order less than −N thenh(z) is holomorphic if Rez > 1

2 (p−N), and for any given z this is achieved forN large enough.b) First, the derivation L = 2 log(1 + ε) makes sense as a power series in ε andcan be viewed at the formal level as implemented by log |D|2.One has, for any P , an expansion near 0

(13) Trace(P |D|−2z

)=

∑k≥0

τk(P ) z−(k+1) + 0(1) .

We can then write

(14) Trace(P2P1|D|−2z

)= Trace

(P1(1 + ε)−2z (P2) |D|−2z

)and, since

(15) (1 + ε)−2z = exp(−zL) ,

we get

(16) Trace(P2P1|D|−2z

)=

∑ (−z)n

n!Trace

(P1 L

n(P2) |D|−2z).

37

By (13) we can expand:

Trace(P1 L

n(P2) |D|−2z)

=∑

τq (P1 Ln(P2)) z−(q+1) + 0(1)

and, when multiplied by zn, we see that we get the exponent z−(k+1) for n−q =−k. Thus, the coefficient of z−(k+1) in the expansion (16) is∑

n=q−k

(−1)n

n!τq (P1 L

n(P2)) =∑ (−1)n

n!τn+k (P1 L

n(P2)) .

Therefore, we obtain:

(17) τk(P2P1) − τk(P1P2) =∑n>0

(−1)n

n!τn+k (P1 L

n(P2)) .

It follows, of course, that if q is the multiplicity of Sd, i.e. the highest order ofpoles, then τq is a trace.

By Appendix A, in the case of simple spectrum the trace τ = τ0 is an extensionof the Dixmier trace, the latter being defined only when the operator P ∈ Ψ∗(A)belongs to OP−p.

2.2 Local formula for the Chern character

Before giving the general local formula for the Chern character of a triple(A,H, D) with discrete dimension spectrum, we need to recall a few basic defi-nitions from [2].

First the cyclic cohomology HCn(A) is defined as the cohomology of the com-plex of cyclic cochains, i.e. those satisfying

(18) ψ(a1, . . . , an, a0) = (−1)n ψ(a0, . . . , an) , ∀aj ∈ A ,

under the coboundary operation b given by:

(bψ)(a0, . . . , an+1) =(19)n∑0

(−1)j ψ(a0, . . . , aj aj+1, . . . , an+1)+(−1)n+1 ψ(an+1 a0, . . . , an) , ∀aj ∈ A .

Equivalently, HCn(A) can be described in terms of the second filtration of the(b, B) bicomplex of arbitrary (non cyclic) cochains on A, where B : Cm → Cm−1

is given by

(B0 ϕ)(a0, . . . , am−1) = ϕ(1, a0, . . . , am−1) − (−1)m ϕ(a0, . . . , am−1, 1) ,

B = AB0 , (Aψ)(a0, . . . , am−1) =∑

(−1)(m−1)j ψ(aj , . . . , aj−1) .(20)

38

To an n-dimensional cyclic cocycle ψ one associates the (b, B) cocycle ϕ ∈Zp(F q C), n = p− 2q given by

(21) (−1)[n/2] (n!)−1 ψ = ϕp,q

where ϕp,q is the only non zero component of ϕ.Given a spectral triple (A,H, D), with D−1 ∈ L(p,∞), its Chern character incyclic cohomology is obtained from the following cyclic cocycle τn, n ≥ p, nodd,

(22) τn(a0, . . . , an) = λn Tr′(a0[F, a1] . . . [F, an]

), ∀aj ∈ A ,

where F = SignD, λn =√

2i (−1)n(n−1)

2 Γ(

n2 + 1

)and

(23) Tr′(T ) =12

Trace(F (FT + TF )) .

In [2] we obtained the following general formula for the Hochschild cohomologyclass of τn in terms of the Dixmier trace :

(24) ϕn(a0, . . . , an) = λn Trω

(a0[D, a1] . . . [D, an] |D|−n

), ∀aj ∈ A .

Our local formula for the cyclic cohomology Chern character, i.e. for a cycliccocycle cohomologous to (22), will be expressed in terms of the (b, B) bicomplex.Bearing this in mind we see that if we want to regard the cochain ϕn of (24) as acochain of the (b, B) bicomplex, we should use, instead of λn, the normalizationconstant

(25) µn = (−1)[n/2] (n!)−1 λn =√

2iΓ(

n2 + 1

)n!

(for n odd) .

Let us now state the result. We let (A,H, D) be a spectral triple with discretedimension spectrum and D−1 ∈ L(p,∞). We shall use the following notations:

(26) da = [D, a] , ∀a operator in H ,

(27) ∇(a) = [D2, a] ; a(k) = ∇k(a) , ∀a operator in H .

Theorem II.1. a) The following formula defines a cocycle in the (b, B) bicom-plex of A:

ϕn(a0, . . . , an) =√

2i∑

q≥0,kj≥0

cn,k,qτq

(a0(da1)(k1) . . . (dan)(kn)|D|−(n+2Σkj)

),

39

where

cn,k,q = (−1)k1+···+kn (k1! . . . kn!)−1 Γ(q)(k1 + · · · + kn +

n

2

)×

1q!

((k1 + 1)(k1 + k2 + 2) . . . (k1 + · · · + kn + n)

)−1,

with Γ(q) the qth derivative of the Γ function.b) The cohomology class of the cocycle (ϕn), n odd, in HCodd(A) coincides withthe cyclic cohomology Chern character ch∗(A,H, D).

Before starting the proof let us note that the term τq(Tn,k) with coefficient cn,k,q

in the above sum vanishes when

(28) n+∑

kj > p ,

since the operator Tn,k is in L(1,∞)0 when (28) holds. This implies that for fixed

n the sum involved contains only finitely many terms. (We assume that Sd hasfinite multiplicity so that only finitely many q’s are involved.) It also impliesthat

(29) ϕn = 0 , if n > p .

Assertion b) is the cyclic cohomology analogue of Theorem IV.2.8 of [2]. Itimplies in the same way that if the cyclic homology is Hausdorff then ϕ∗ iscohomologous to ch∗(A,H, D).Note also that all the operators Tn,k involved in the above formula are homoge-neous of degree 0 in D, i.e. they are unaffected by the scaling

(30) D → λD , λ ∈ R∗+ .

Finally, let us remark that assertion a), i.e. the equality

(31) b ϕn +B ϕn+2 = 0 , ∀n ,is actually a consequence of our proof of b). However, it is an instructive exerciseto check it directly. We shall do so by making use of the following properties(with τ = τ0) :

(32) D da+ da D = ∇(a) , ∀a operator in H

(33) τ((da)(k) |D|−q

)= 0 , ∀a ∈ A , ∀k ≥ 0 , ∀q

(34) τ(∇(T ) |D|−q

)= 0 , ∀T , ∀q

40

(35) D(k) b =∞∑0

(−1)�

�!b(�) D(k+�) , ∀b ∈ B ,

where by definition D(k) = Γ(k) |D|−2k.

The meaning of (32) is that, if we view the graded commutator with D as agraded derivation in the appropriate way, then d2 = ∇. The meaning of (33) and(34) is that integration by parts is possible, since both d and ∇ are derivations.Finally (35) follows from Theorem B.1 of Appendix B and will be used onlyunder the trace τ so that only finitely many non zero terms of the sum of theright hand side will appear.

Proof. a) We shall perform the computations under the simplifying assumptionof simple spectrum. Only minor modifications will be required to treat thegeneral case.

Let us first show that B ϕ1 = 0. One has (up to the overall factor√

2i whichwe shall ignore) :

(36) (B0 ϕ1)(a) =∑

c1,k τ((da)(k) |D|−1−2k

),

hence, using (33), one gets

B0 ϕ1 = 0 and B ϕ1 = 0.

We shall now compare b ϕ1 and −B ϕ3. The Leibnitz rule gives, in general,

(37)1q!

(b1 . . . bn)(q) =∑qj≥0Σqj=q

b(q1)1

q1!· · · b

(qn)n

qn!.

One has

(38) ϕ1(a0, a1) =∑

k

(−1)k

(k + 1)!τ(a0(da1)(k) D(k+ 1

2 )), ∀aj ∈ A .

To compute b ϕ1 we need to apply τ , for each k, to(a0a1(da2)(k) − a0 d(a1a2)(k)

)D(k+ 1

2 ) + a0(da1)(k) D(k+ 12 ) a

2 .

Using d(a1a2) = a1 da2 + (da1)a2, (37) and (35) we thus get

−∑

k1+k2=kk1 �=0

k! a0 (a1)(k1)

k1!(da2)(k2)

k2!D(k+ 1

2 )

−∑

k1+k2=k

k! a0 (da1)(k1)

k1!(a2)(k2)

k2!D(k+ 1

2 )

+∑�≥0

(−1)� a0 (da1)(k) (a2)(�)

�!D(k+�+ 1

2 ) .

41

Thus, if we introduce the cochains(39)

τ(1; k1, k2) (a0, a1, a2) = (−1)k1+k2 τ(a0(a1)(k1) (da2)(k2) D(k1+k2+ 1

2 )),

τ(2, k1, k2) (a0, a1, a2) = (−1)k1+k2 τ(a0(da1)(k1) (a2)(k2) D(k1+k2+ 1

2 )),

we can express b ϕ1 as follows:

b ϕ1 = −∑k1 �=0kj≥0

(k1 + k2 + 1)−1

k1! k2!τ(1, k1, k2)(40)

+∑kj≥0

((k1 + 1)−1 − (k1 + k2 + 1)−1

)k1! k2!

τ(2, k1, k2) .

We shall now express B ϕ3 in a similar manner, as a linear combination of thecochains (39). We have(41)B0 ϕ3(a0, a1, a2) =

∑c3,k τ

((da0)(k1) (da1)(k2) (da2)(k3) |D|−(3+2|k|)

),

where |k| = k1 + k2 + k3.

The cochain B ϕ3 is the sum of the three cochains obtained from B0 ϕ3 bycyclic permutations,

(42) B ϕ3 = B0 ϕ3 + (B0 ϕ3)′ + (B0 ϕ3)′′ ,

where(B0 ϕ3)′(a0, a1, a2) = (B0 ϕ3)(a1, a2, a0)

and(B0 ϕ3)′′(a0, a1, a2) = (B0 ϕ3)(a2, a0, a1).

Using integration by parts (i.e. (33) and (34)), we can express B0 ϕ3 as∑kj≥0,�≥0

(−1)k1+1 αk1�!

1(k1 − �)!

1k2!

1k3!

τ(1, k2 + 1 + �, k3 + k1 − �)(43)

+∑

(−1)k1 αk1�!

1(k1 − �)!

1k2!

1k3!

τ(2, k2 + �, k3 + 1 + k1 − �) ,

whereαk = (k1 + 1)−1 (k1 + k2 + 2)−1 (k1 + k2 + k3 + 3)−1.

Let us now compute (B0 ϕ3)′ :

(B0 ϕ3)′(a0, a1, a2) =(44)∑(−1)|k| αk

1k1!

1k2!

1k3!

τ((da1)(k1) (da2)(k2) (da0)(k3) D(3/2+|k|)

).

42

In order to obtain terms of the form (39) we must move (da0)(k3) in front, using(35), and then integrate by parts. We use the trace property of τ and apply(35), for each term, with

(45) b = (da1)(k1) (da2)(k2) .

Thus, we get

(B0 ϕ3)′(a0, a1, a2) =(46)∑(−1)|k|+� αk

1k1!

1k2!

1k3!

1�!τ

((da0)(k3)

((da1)(k1) (da2)(k2)

)(�)

D( 32+|k|+�)

).

Integration by parts gives:

τ

((da0)(k3)

((da1)(k1) (da2)(k2)

)(�)

D(3/2+|k|+�)

)= (−1)k3 τ

(da0

((da1)(k1) (da2)(k2)

)(�+k3)

D(3/2+|k|+�)

)= (−1)k3+1 τ

(a0

((a1)(k1+1) (da2)(k2)

)(�+k3)

D(3/2+|k|+�)

)+ (−1)k3 τ

(a0

((da1)(k1) (a2)(k2+1)

)(�+k3)

D(3/2+|k|+�)

).

We then use (37) to get ((a1)(k1+1) (da2)(k2)

)(�+k3)

=∑ (�+ k3)!

m!(�+ k3 −m)!(a1)(k1+1+m) (da2)(k2+�+k3−m)

((da1)(k1) (a2)(k2+1)

)(�+k3)

=∑ (�+ k3)!

m!(�+ k3 −m)!(da1)(k1+m) (a2)(k2+1+�+k3−m) .

This gives the following formula for (B0 ϕ3)′:

(B0 ϕ3)′ =∑

kj≥0,�,m≥0

(−1)k3+1 αk1k1!

1k2!

1k3!

1�!

1m!

(47)

(�+ k3)!(�+ k3 −m)!

τ(1, k1 + 1 +m, k2 + �+ k3 −m)+

∑kj≥0,�,m≥0

(−1)k3 αk1k1!

1k2!

1k3!

1�!

1m!

(�+ k3)!(�+ k3 −m)!

τ(2, k1 +m, k2 + 1 + �+ k3 −m) .

43

The computation of (B0 ϕ3)′′ is completely similar and gives:

(B0 ϕ3)′′ =∑

kj≥0,�,m≥0

(−1)k2+1 αk1k1!

1k3!

1�!

1m!

1(k2 −m)!

(48)

τ(1, k3 + 1 +m, k1 + �+ k2 −m)+

∑(−1)k2 αk

1k1!

1k3!

1�!

1m!

1(k2 −m)!

τ(2, k3 +m, k1 + 1 + �+ k2 −m) .

In order to compare b ϕ1 with B0 ϕ3 +(B0 ϕ3)′+(B0 ϕ3)′′ = B ϕ3 we just needto compare the coefficients of τ(j, k1, k2) in the formulae (40) and (43)+ (47) +(48). The most convenient way to proceed is to introduce generating functionsfj(x, y), j = 1, 2 , in which we replace τ(j, k1, k2) by xk1 yk2 .

Let us first compute fj(x, y) for the expression (40); we get

(49)f b1(x, y) = −

∑k1 �=0

(k1 + k2 + 1)−1 xk1

k1!yk2

k2!,

f b2(x, y) =

∑((k1 + 1)−1 − (k1 + k2 + 1)−1

) xk1

k1!yk2

k2!.

Thus, f b1(x, y) = − ∫ 1

0(eu(x+y) − euy) du, hence

(50) f b1(x, y) =

1 − ex+y

x+ y+ey − 1y

.

Similarly,

(51) f b2(x, y) =

1 − ex+y

x+ y+ ey

(ex − 1x

).

Let us next compute f1 and f2 for the expression (43); we get

(52)f1(x, y) =

∑(−1)k1+1 αk

1�!

1(k1 − �)!

1k2!

1k3!

xk2+1+� yk3+k1−� ,

f2(x, y) =∑

(−1)k1 αk1�!

1(k1 − �)!

1k2!

1k3!

xk2+� yk3+k1+1−� .

In order to compute them, we introduce the function

(53) f(x, y, z) =∑

αkxk1

k1!yk2

k2!zk3

k3!.

With this notation,

(54) f1(x, y) = −f(−(x+ y), x, y)x ,f2(x, y) = f(−(x+ y), x, y)y .

44

With the selfexplanatory notation f ′1, f

′2 for (47), we have

(55)

f ′1(x, y) =

∑(−1)k3+1 αk

1k1!

1k2!

1k3!

1�!

1m!

(�+ k3)!(�+ k3 −m)!

xk1+1+m yk2+�+k3−m

f ′2(x, y) =

∑(−1)k3 αk

1k1!

1k2!

1k3!

1�!

1m!

(�+ k3)!(�+ k3 −m)!

xk1+m yk2+1+�+k3−m .

It follows that

(56) f ′1(x, y) = −e(x+y) f(x, y,−(x+ y))x ,f ′2(x, y) = ex+y f(x, y,−(x+ y))y .

Similarly, for f ′′1 , f

′′2 , we have

(57)

f ′′1 (x, y) =

∑(−1)k2+1 αk

1k1!

1k3!

1�!

1m!

1(k2 −m)!

xk3+1+m yk1+�+k2−m

f ′′2 (x, y) =

∑(−1)k2 αk

1k1!

1k3!

1�!

1m!

1(k2 −m)!

xk3+m yk1+1+�+k2−m

which gives

(58) f ′′1 (x, y) = −ey f(y,−(x+ y), x)x ,f ′′2 (x, y) = ey f(y,−(x+ y), x)y .

We can thus express the generating functions for B ϕ3 as follows:

(59)

fB1 (x, y) = −(

f(−(x+ y), x, y) + ex+y f(x, y,−(x+ y))+ ey f(y,−(x+ y), x)

)x

fB2 (x, y) =

(f(−(x+ y), x, y) + ex+y f(x, y,−(x+ y))

+ ey f(y,−(x+ y), x))y .

Let us compute f(x, y, z); one has

f(x, y, z) =∫

0≤u1≤u2≤u3≤1

e(u1x+u2y+u3z) du1 du2 du3 ,

since ∫0≤u1≤u2≤u3≤1

uk11 uk2

2 uk33 du1 du2 du3 = αk .

45

We then obtain:

(60) f(x, y, z) =ex+y+z − 1

x(x + y)(x+ y + z)− ez − 1x(x + y)z

− (ey+z − 1)xy(y + z)

+(ez − 1)xyz

.

Since only the restriction of f to the hyperplane x + y + z = 0 is involved, wecan use the equality ea−1

a = 1 for a = 0, to handle the first term. We get

f(−(x+ y), x, y) =(61)1

−(x+ y)(−y) +(ex+y − 1)

(x+ y)x(x + y)+ (ey − 1)

(1

−(x+ y)xy− 1

(x+ y)y2

),

(62) ex+y f(x, y,−(x+ y)) =ex+y

x(x+ y)+ey − ex+y

x2y+ex+y − 1(x+ y)2y

,

(63) ey f(y,−(x+ y), x) = − ey

xy+

ey − 1y2(x+ y)

+(ex − 1)ey

x2(x+ y).

Let us first compute the coefficient of ex+y in the sum (61) + (62) + (63). Wehave

1x(x+ y)2

+1

x(x + y)− 1x2y

+1

(x+ y)2y+

1x2(x+ y)

=1

x(x+ y)+

(x2y(x+ y)2

)−1 (xy − (x+ y)2 + x2 + y(x+ y)

)=

1x(x+ y)

.

The coefficient of ey is given by

− 1(x+ y)y2

− 1xy(x+ y)

+1x2y

− 1xy

+1

y2(x+ y)− 1x2(x+ y)

= − 1xy

+(x2y2(x + y)

)−1 (−x2 − xy + y(x+ y) + x2 − y2)

= − 1xy

.

The rational term which remains is then

1(x+ y)y

− 1x(x+ y)2

+1

xy(x+ y)+

1y2(x + y)

− 1y(x+ y)2

− 1y2(x + y)

=1

(x+ y)y+

(xy2(x+ y)2

)−1 (−y2 + y(x+ y) + x(x + y) − xy − x(x + y))

=1

(x+ y)y.

46

Thus, the sum (61) + (62) + (63) is equal to

(64)ex+y

x(x + y)− ey

xy+

1y(x+ y)

.

When we multiply (64) by −x we get

(65)1 − ex+y

x+ y−

(1 − ey

y

).

When we multiply (64) by y we get:

(66) −ey

x+

1x+ y

+ex+y y

x(x+ y).

Comparing these expressions with (50) and (51) we then obtain

(67) f b1 + fB

1 = 0 , f b2 + fB

2 = 0 .

This shows that in the expression for b ϕ1 + B ϕ3 the coefficient of any of theτ(j; k1, k2) vanishes for j = 1, 2 and any k1, k2.

We have thus shown that B ϕ1 = 0 and that b ϕ1 + B ϕ3 = 0. The proof forthe general identity

b ϕn +B ϕn+2 = 0

is based on a similar computation.The above discussion only covers the case of simple poles. The general casefollows in exactly the same way, by introducing the expression

D(z) = Γ(z)|D|−2z,

for complex values of z, and performing the above manipulations with τ andD(k) replaced respectively by the usual trace Trace and by D(k+ε), where ε is asmall complex number. Taking the residue at ε = 0 gives the desired identities.

b) With a0, . . . , an ∈ A fixed, we let

ζ(z0, . . . , zn) =(68)Trace

(a0 |D|−2z0 da1 |D|−2z1 da2 . . . |D|−2zn−1 dan |D|−2zn

).

This expression makes sense if∑

Re zi >p2 and we shall first express it in terms

of the following functions of a single complex variable:

(69) hk(z) = Trace(a0(da1)(k1) (da2)(k2) . . . (dan)(kn) D−2Σkj−2z

),

where k = (k1, . . . , kn) is a multi-index.

47

As in (35), one has

(70) |D|−2z da =∑ (−1)k

k!z(k) (da)(k) |D|−2z−2k ,

where z(k) = z(z + 1) . . . (z + k − 1).

We can then write the expansion

(71) ζ(z0, . . . , zn) =∑

Pk(z0, . . . , zn−1) hk

(n∑0

zj

),

where the polynomial Pk is given by

Pk(z0, . . . , zn−1) =(−1)k1+···+kn

k1! . . . kn!z(k1)0 (z0 + k1 + z1)(k2)(72)

(z0 + k1 + z1 + k2 + z2)(k3) . . . (z0 + k1 + z1 + k2 + z2 + k3 + · · · + zn−1)(kn) .

In the expansion (71), if we sum the terms for which |k| > p, they contribute by afunction of (z0, . . . , zn) which is holomorphic and bounded for

∑Re zi > 0. This

follows from the Holder inequality and the control (Theorem B.1 of AppendixB) of the remainder in the Taylor expansion (70).

Let λ ∈ R+, λ > p2(n+1) ; then using the equality

(73) e−uD2=

12π

∫ ∞

−∞Γ(λ+ is) |D|−2(λ+is) u−(λ+is) ds , ∀u > 0 ,

we obtain:

Trace(a0 e−u0D2

da1 e−u1D2da2 . . . e−un−1D2

dan e−unD2)

(74)

= (2πi)−(n+1)

∫Cn+1

λ

Γ(z0) . . .Γ(zn) u−z ζ(z0, . . . , zn) dz0 . . . dzn ,

where uj > 0, u−z = u−z00 . . . u−zn

n , and Cλ = {λ+ is ; s ∈ R}.We let θ(u0, u1, . . . , un) be the function defined by (74) and we want to computethe coefficient of ε−n/2 in the expansion of

(75) θ(ε v0, ε v1, . . . , ε vn) ,∑

vi = 1 , ε→ 0 .

From (71) and the hypothesis on the dimension spectrum we see, using theboundedness of Γ(z0) . . .Γ(zn) Γ(z0 + · · · + zn)−1 on Cn+1

λ , that except forfinitely many values of λ the function

Γ(z0) . . .Γ(zn) ζ(z0, . . . , zn)

48

is integrable on Cn+1λ (for λ > 0 say). It follows then that the right hand side

of (74), when evaluated for such a λ, at (ε vi) is a O(ε−(n+1)λ). It is not equalto (75) because of the contribution of the residues of the differential form

(76) ω = Γ(z0) . . .Γ(zn) u−z ζ(z0, . . . , zn) dz0 . . . dzn .

Using the expansion (71), we let

(77) c(k, q) = coeff of ε−q in hk

(n2

+ ε)

at ε = 0

and concentrate on the contribution of the residue of the differential form

(78) ω = Γ(z0) . . .Γ(zn) Pk(z0, . . . , zn−1) u−z(z0 + · · · + zn − n

2

)−q n

Π0dzi

(which then needs to be multiplied by c(k, q)).

If we denote by X the differential operator

(79) X =1

n+ 1

n∑0

∂

∂zj,

the contribution of the residue is given by

(80)∫

Σzi=n2

Xq−1

(q − 1)!(Γ(z0) . . .Γ(zn) Pk(z0, . . . , zn−1) u−z

) n

Π1dzi ,

where Re zi = λ = n2(n+1) .

Thus, with uj = ε vj ,n∑0vj = 1, the coefficient of ε−n/2 is given by (80)

with v instead of u, since the derivatives of ε−Σzi contribute by terms involvingε−n/2(log ε)k.

Introducing an additional variable t ∈ R we can rewrite (using the Fourierexpansion of the δ function) the result as(81)

(2π)−1

∫tq−1

(q − 1)!Γ(z0) . . .Γ(zn) Pk(z0, . . . , zn−1) v−z e−t(Σzi−n

2 ) n

Π0dzi dt

where Re zi = λ = n2(n+1) and one integrates in the n+ 2 remaining variables.

Before taking care of the polynomial Pk, we can already compute, for fixed t,

(82)∫

Rezi=λ

Γ(z0) . . .Γ(zn) v−z e−tΣzin

Π0dzi = (2πi)(n+1)

n

Π0e−vjet

et n2 ,

which holds for any value of vj > 0.

49

Next, we have

(83)(

∂

∂v0

)k

v−z00 = (−1)k z

(k)0 v

−(z0+k)0 ,

or better

(84)(∂

∂u

)k

(uv0)−z0 = (−1)k z(k)0 u−(z0+k) v−z0

0 .

This means that the effect of Pk(z0, . . . , zn−1) in the above integral is obtainedas follows, for the term

(−1)Σkj z(k1)0 (z0 + k1 + z1)(k2) . . . (z0 + k1 + z1 + · · · + zn−1)(kn).

One starts with the integral (82) written as f(v0, . . . , vn), then one applies

(85)(∂

∂u

)k1

f(uv0, v1, . . . , vn) = f1(u, v0, v1, . . . , vn)

and one continues with(∂

∂u

)k2

f1(u, v0, uv1, v2, . . . , vn) = f2(u, v0, . . . , vn) ,(86) (∂

∂u

)k3

f2(u, v0, v1, uv2, . . . , vn) = f3(u, v0, . . . , vn) ,

· · ·(∂

∂u

)kn

fn−1(u, v0, . . . , uvn−1, vn) = fn(u, v0, . . . , vn) ,

which is finally evaluated at u = 1.

Using (82), we are just applying this rule to

f(v0, . . . , vn) = e−(Σvj)et

.

We get

f1(u, v0, . . . , vn) = (−v0 et)k1 f(uv0, v1, . . . , vn) ,

f2(u, v0, . . . , vn) = (−v0 et)k1(−(v0 + v1)et

)k2f(uv0, uv1, v2, . . . , vn) ,

f3(u, v0, . . . , vn) = (−v0 et)k1(−(v0 + v1)et

)k2 (−(v0 + v1 + v2)et)k3

f(uv0, uv1, uv2, v3, . . .)

fn(1, v0, . . . , vn) = (−1)Σkj etΣkj vk10 (v0 + v1)k2 . . . (v0 + · · · + vn−1)kn

f(v0, . . . , vn) .

50

We can thus write (81) as

(2π)−1 (2πi)n+1 (−1)Σkj

k1! . . . kn!

∫tq−1

(q − 1)!et(Σkj+ n

2 )(87)

e−(Σvj)et

dt vk10 (v0 + v1)k2 . . . (v0 + · · · + vn−1)kn .

We then have to integrate the result on the simplex∑vj = 1. The first task is

thus to compute

(88)∫ ∞

−∞

tq−1

(q − 1)!etα e−et

dt , α =∑

kj +n

2.

It is obtained from the Taylor expansion at α of∫ ∞

−∞etα e−et

dt = Γ(α)

and is given by

(89)Γ(q−1) (α)(q − 1)!

.

The remaining part of the integral is∫n∑0

vj=1

vk10 (v0 + v1)k2 . . . (v0 + · · · + vn−1)kn Π dvi

= (k1 + 1)−1 (k1 + k2 + 2)−1 . . . (k1 + · · · + kn + n)−1 .

We can now complete the proof of 3 b).

We use the Chern character of (A,H, D) in entire cyclic cohomology (cf. [2])given in the most efficient manner by the JLO formula, which defines the com-ponents of an entire cocycle in the (b, B) bicomplex:

ψn(a0, . . . , an) =√

2i∫

n∑0

vi=1,vi≥0

(90)

Trace(a0 e−v0D2

[D, a1] e−v1D2. . . e−vn−1D2

[D, an] e−vnD2), ∀aj ∈ A

where n is odd.

We introduce a parameter ε by replacing D2 by εD2, which yields a cocycle ψεn

which is cohomologous to ψn. One has moreover

(91) ψεn(a0, . . . , an) =

√2i

∫n∑0

vi=1

θ(ε v0, . . . , ε vn) π dvi

εn/2 ,

51

where θ was defined by (74). The coefficient εn/2 comes from the n terms [D, aj ]in the formula (90). Since n is odd, n/2 �= 0. The above computation of thebehavior of εn/2 θ(ε v0, . . . , ε vn) from the residues of the differential form ω(76) gives an expansion as a finite sum of terms:

(92) θ(ε v0, . . . , ε vn) =∑

αm,� ε−pm(log ε)� +O(ε−n/2) ,

where the pm correspond to the poles of hk whose real part is larger than n2 .

Moreover we have computed above the coefficient of ε−n/2 in this expansion,after integration of the result in vi we get:

(93) ψεn(a0, . . . , an) =

∑βm,� ε

n/2−pm(log ε)� +O(1) ,

β0,0 =√

2i∑

k1,...,kn,q

(−1)Σkj1

k1! . . . kn!(k1 + 1)−1 (k1 + k2 + 2)−1 . . .

(k1 + · · · + kn + n)−1 Γ(q)(k1 + · · · + kn + n

2

)q!

c(k, q) .(94)

When we pair the (b, B) cocycle ψε with a cyclic cycle c = (cn) in entire cyclichomology (cf. [2]), the pairing gives a scalar independent of ε and written as asum of terms of the form (93). The total contribution of the terms ψε

n, n > pconverges to 0 by the argument of [6].

Thus, we can assert that

(95)∑n≤p

〈ψεn, cn〉 →

ε→0〈ch∗(H, D), c〉 .

Using (93) this is possible only if the asymptotic expansion of the left hand sidein terms of ε−pm(log ε)�, Re pm ≤ 0, only contains a constant term, and by (94)we know the value of this constant term, it is given by the pairing 〈ϕ, c〉 of thecyclic cocycle of Theorem II.1 a) with c.

To prove that the class of (ϕn) actually coincides with ch∗(A,H, D), we needto recall that there is a canonical transgressed cochain (ψk) such that

(96)d

dtψε

n = bψεn−1 +Bψε

n+1 ,

and to note that (ψk) also has an asymptotic expansion of the form (93). Itthen follows from [CM2, §4] that (ψn) is cohomologous with the finite part of(ψε

n), which in turn, from the above discussion, can be seen to give precisely thecocycle (ϕn).

52

2.3 Renormalization

There is one unpleasant feature of the formula II.1. a) for the cyclic cocycle ϕ,namely the occurence of the transcendental numbers which enter in the Taylorexpansion of the Γ function at the points Γ

(12 + q

), q ∈ N. Also the sum

(97)∑ Γ

(|k| + n2

)(q)

q!Ress=0

(sq ζ(s))

is an infinite sum when ζ is not meromorphic at s = 0. We can of course rewriteit as

(98) Ress=0

Γ(|k| + n

2+ s

)ζ(s) .

We shall however proceed to show how to obtain a modified cyclic cocycle ϕ′,giving the same result Thm. II.1. b) as ϕ, but involving a finite linear combi-nation with rational coefficients of the terms

(99)√

2i Γ(

12

)τq

(a0(da1)(k1) . . . (dan)(kn) |D|−2Σkj−n

).

To achieve this, we shall exploit the freedom of replacing the operator D byµ−1D, µ ∈ R∗

+ without affecting Thm. II.1. b). The effect of this transformationon the functionals τq is as follows:

(100) τµq =

∑ (logµ)m

m!τq+m .

This implies that for any integer m ≥ 1 the following formula defines the com-ponents of a cyclic cocycle which pairs trivially with cyclic homology:(101)ϕ(m)

n (a0, . . . , an) =∑

cn,k,q τq+m

(a0(da1)(k1) . . . (dan)(kn) |D|−(n+2|k|)

).

What we shall do now is add a suitable linear combination of countertermsϕ(m) in order to cancel all the transcendental coefficients occuring in the Taylorexpansion of Γ(1/2)−1Γ(s) at half integers. Even though we could right awaywrite down the list of the coefficients βm needed in front of ϕ(m), we shall ratherexplain carefully how they are obtained.

To begin with, there is no problem at all if Sd is simple, i.e. if one has at mosta simple pole. In that case one simply writes

(102) Γ(

12

+ q

)=

12

32. . .

(12

+ q − 1)

Γ(

12

)and since all τq’s with q ≥ 1 vanish one gets the desired answer.

53

Let us see what happens when Sd has multiplicity two, i.e. when we have atmost a double pole. In that case by Proposition II.1 we know that τ1 is atrace, while τq = 0 for q ≥ 2. This means that the formula for ϕn involves thecombination

(103) Γ(|k| + n

2

)τ0(A) + Γ′

(|k| + n

2

)τ1(A) ,

where A is some operator. Now since the Hochschild coboundary b τ0 is givenrationally in terms of τ1 (cf. Proposition II.1) we do not expect to need thetranscendental coefficient

(104)Γ′ ( 1

2 +m)

Γ(

12 +m

)in order to compensate for the lack of trace property of τ0. If we replace the termΓ′ (|k| + n

2

)τ1(A) by Γ

(|k| + n2

)τ1(A), then we get exactly the components of

ϕ(1)n which we can subtract from ϕ without affecting Thm. II.1. b). Thus, we

shall look for a coefficient λ such that

(105) Γ′(

12

+m

)= λ Γ

(12

+m

)+ cm Γ

(12

+m

), m ∈ N ,

where the cm are rational numbers.

To obtain (105) one just uses the equality

(106)Γ′ ( 1

2 +m+ ε)

Γ(

12 +m+ ε

) =m−1∑a=0

112 + a+ ε

+Γ′ ( 1

2 + ε)

Γ(

12 + ε

) ,

which we write with ε for later use.

Thus, the constant λ is

Γ′ ( 12

)Γ(

12

) = −(γE + 2 log 2),

where γE is Euler’s constant.

If we replace ϕ by ϕ−λϕ(1) then using (105) we find that in the formula giving ϕthe terms Γ′ (|k| + n

2

)τ1(A) should be replaced simply by c|k|+ n−1

2Γ(|k| + n

2

)τ1(A), where :

(107) c� =�−1∑a=0

112 + a

.

Let us consider the next case, when Sd has multiplicity 3, i.e. when we havetriple poles. This time we shall get the combination:

(108) Γ(|k| + n

2

)τ0(A) + Γ′

(|k| + n

2

)τ1(A) +

Γ′′

2!

(|k| + n

2

)τ2(A) .

54

We want to use a further subtraction, say of λ2 ϕ(2), to get coefficients for τ1(A)

and τ2(A) of the form Γ(|k| + n

2

)× Q. From (106) we get the formula

(109) Γ′(

12

+m+ ε

)= Rm(ε) Γ

(12

+m+ ε

)+ f(ε) Γ

(12

+m+ ε

),

where Rm is the rational fraction

(110) Rm(ε) =m−1∑a=0

112 + a+ ε

and where the function f is given by

(111) f(ε) =Γ′ ( 1

2 + ε)

Γ(

12 + ε

) .

If we differentiate (109) we get:

Γ′′(

12

+m+ ε

)= R′

m(ε) Γ(

12

+m+ ε

)+Rm(ε) Γ′

(12

+m+ ε

)+f ′(ε) Γ

(12

+m+ ε

)+ f(ε) Γ′

(12

+m+ ε

).(112)

We have to transform the term Rm(ε) Γ′ ( 12 +m+ ε

)because it involves at the

same time a function of m, Rm(ε) and a derivative of Γ. To do this we replaceΓ′ ( 1

2 +m+ ε)

by its value (109) which yields:

(113) Rm(ε)2 Γ(

12

+m+ ε

)+Rm(ε) f(ε) Γ

(12

+m+ ε

)and we use again (109) to replace the second term of the formula by

(114) f(ε) Γ′(

12

+m+ ε

)− f(ε)2 Γ

(12

+m+ ε

).

Coming back to all the terms of (112) we thus proved:

Γ′′(

12

+m+ ε

)=

(R′

m(ε) +Rm(ε)2)

Γ(

12

+m+ ε

)(115)

+(f ′(ε) − f(ε)2

)Γ(

12

+m+ ε

)+ 2 f(ε) Γ′

(12

+m+ ε

).

This shows that, if we replace ϕ by ϕ− λ1 ϕ(1) − λ2 ϕ

(2), where λ1 = λ = f(0)as above and

(116) λ2 =12!f2(0) , f2(ε) = f ′(ε) − f(ε)2 ,

55

then the combination (108) gets replaced by

Γ(|k| + n

2

)τ0(A) + c|k|+ n−1

2Γ(|k| + n

2

)τ1(A)(117)

+c′|k|+ n−12

Γ(|k| + n

2

)τ2(A) ,

where the rational number c′m is 12! R

(1)m (0), with

(118) R(1)m (ε) = R′

m(ε) +Rm(ε)2 .

We can now proceed by induction to the general case. One proves by inductionon � the following formula on the �th derivative Γ(�) of the Γ function:

Γ(�)

(12

+m+ ε

)= R(�−1)

m (ε) Γ(

12

+m+ ε

)(119)

+�∑

j=1

Cj� fj(ε) Γ(�−j)

(12

+m+ ε

)

where R(�)m and fj are defined inductively by

(120) R(�+1)m (ε) = R(�)

m (ε)′ +Rm(ε) R(�)m (ε) ,

(121) fj+1(ε) = f ′j(ε) − f(ε) fj(ε) .

The proof uses the same patern as above and is straightforward. It shows thatif we replace ϕ by

(122) ϕ′ = ϕ− λ1 ϕ(1) − λ2 ϕ

(2) − · · · − λ� ϕ(�) · · ·

where λ� = 1�! f�(0), then the combination of terms:

(123)∑ 1

q!Γ(|k| + n

2

)(q)

τq(A)

in the expression of the cocycle ϕ, gets replaced by

(124) Γ(|k| + n

2

)∑ 1q!R(q−1)

m (0) τq(A) m = |k| + n− 12

.

Now the functions R(�)m (ε) are easy to compute, since

Rm(ε) =m−1∑a=0

Ta(ε) with T ′a(ε) = −Ta(ε)2 , Ta(ε) =

112 + a+ ε

.

56

One gets that R(�)m (ε) is the (�+1)th symmetric function of the m terms 1

12 +a+ε

:

(125)m−1∏a=0

(1 +

z12 + a+ ε

)=

∑R�

m(ε) z�+1 .

We can then easily compute the product Γ(

12 +m

)R

(q−1)m (0) which appears in

(124) and get

(126) Γ(

12

+m

)R(q−1)

m (0) = Γ(

12

)σm−q(m) ,

where σj(m) is the jth symmetric function of the first m odd 1/2 integers:

(127)m−1∏�=0

(z +

(2�+ 1)2

)=

∑zj σ(m−j) (m) .

What is remarkable now is that these coefficients vanish if q is larger than mso that not only we transformed them to elements of Γ

(12

)Q but we have also

eliminated all but a finite number.We note that the function fj(ε) is not difficult to compute, and by induction weget the formula

(128) fj(ε) = −Γ(

12

+ ε

) (1

Γ(

12 + ε

))(j)

as can be seen by interpreting the transformation

(129) T (h) = h′ − fh

as T (h) = Γ(

hΓ

)′, and using (121).

It follows then that

(130) λ� = −Γ(

12

)1�!

(1

Γ(

12 + ε

))(�)ε=0

and that the above operation of subtraction has a very simple interpretation,namely the following. In the proof of Theorem II.1 a), we were applying thelinear form Ress=0 to meromorphic functions of the form

(131) Γ(|k| + n

2+ s

)Trace

(A |D|−2s

)= ζ(s) ,

where A is an operator. But any other linear form such as

(132) ζ → Ress=0 g(s) ζ(s) ,

57

with g holomorphic at 0, would have worked equally well. What the subtractionof

∑λ� ϕ

(�) is doing is exactly to take

g(s) =Γ(1/2)

Γ(1/2 + s).

If we combine this with

(133)Γ(

12 +m+ s

)Γ(

12 + s

) =m−1∏

0

(12

+ a+ s

),

we can summarize the above discussion by the following variant of TheoremII.1.

Theorem II.2. The statements of Theorem II.1 are true for the cocycle ϕ′n

given, for n-odd, n ≤ p, by the formula

ϕ′n(a0, . . . , an) =

√2πi

∑k,q

(−1)|k|

k1! . . . kn!αk

1q!σm−q(m) τq

(a0(da1)(k1) . . . (dan)(kn) |D|−(2|k|+n)

),

with m = |k| + n−12 , α−1

k = (k1 + 1)(k1 + k2 + 2) . . . (k1 + · · · + kn + n) and σdefined in (127).

Not only the coefficients are all rational multiples of the overall factor√

2πi,but also the total number of terms in the formula is now finite and bounded interms of p alone and not the order of the poles. Indeed,

q ≤ |k| + n− 12

and|k| + n ≤ p,

so that the formula does not involve more than p terms in the Laurent expansion.Let us see what this formula looks like for small values of p.

p = 1. Then only ϕ′1 is non zero; we have k = 0 and q = 0, also

(134) ϕ′1(a

0, a1) =√

2πi τ0(a0 da1 |D|−1

).

This shows that, even if we had poles of arbitrary order for the function ζ(s) =Trace

(a0 da1 |D|−1−2s

)at s = 0, they do not contribute to ϕ′

1 except for theresidue of ζ at s = 0.

If we had used the formula of Theorem II.1 we would be taking the residue ofΓ(

12 + s

)ζ(s) at s = 0 which involves infinitely many of the functionals τk.

Note also that here τ0 is a trace.

58

p = 2. Again, only ϕ′1 is non zero, but now we can have k1 = 1 and also q = 1

if k1 = 1. Thus, we get three terms:

ϕ′1(a0, a1) =

√2πi

(τ0

(a0 da1 |D|−1

)− 12τ0

(a0(da1)(1) |D|−3

)(135)

−12τ1

(a0(da1)(1) |D|−3

)).

This time τ0 is no longer a trace, as one can see using Proposition II.1, and theformula involves τ1, i.e. the coefficient of s−2 in some ζ-function. However, nohigher order coefficient is involved, unlike the formula for ϕ in Thm. II.1. a).

p = 3. Let us look at ϕ′3 in this case. Here, we must have k = 0 but since

q ≤ |k| + n−12 , we can have q = 1. Thus, we get two terms for ϕ′

3:

ϕ′3(a0, a1, a2, a3) =(136) √

2πi(τ0

(a0 da1 da2 da3 |D|−3

)+ τ1

(a0 da1 da2 da3|D|−3

)).

This shows that, even for ϕ′3, the coefficient of s−2 in the expansion of the

ζ-function is playing a role, i.e. that τ1 enters into play.

2.4 Local index formula

To get more insight into the content of Theorems II.1 and II.2, we shall nowwrite down a corollary whose statement does not involve cyclic cohomology ornoncommutative geometry but computes a Fredholm index (called spectral flow)as a sum of residues of ζ-functions attached to the problem.

To formulate the problem we just need a pair (D,U) of operators in Hilbertspace, where D is selfadjoint with discrete spectrum, while U is unitary. Themain assumption that we need is that [D,U ] is bounded, which implies immedi-ately that the compression PUP of U of the positive part of D,

(P = 1+F

2 , F =Sign D

)is a Fredholm operator. The index

(137) Index PUP = dim Ker PUP − dim Ker PU∗P

can be interpreted as spectral flow, i.e. as the number of eigenvalues which crossthe origin in the natural homotopy between D and UDU∗ = D + U [D,U∗]. Inany case, it is an integer, and we shall compute it as a sum of residues.

We make the following hypotheses:

(138)

a) If S is the spectrum of D (with multiplicity), then∑λ∈S

|λ|−s <∞ for some finite s .

(We call p the lower bound of such s).

59

b) The operators U and [D,U ] are in the domain of δk, δ = [|D|, ·] for1 ≤ k ≤ N , N >> 0.

c) The following functions, holomorphic for Res >> 0, are meromorphic,with finitely many poles for Res > −ε,

ζ(k,n)(s) = Trace(U−1 [D,U ](k1) [D,U−1](k2) . . . [D,U ](kn) |D|−2|k|−n−s

),

where we use the notation X(k) = ∇k(X), ∇(X) = [D2, X ].

In c) only finitely many functions are involved because of the inequality |k|+n ≤p. At the technical level, we need to assume that Γ(s) ζ(s) restricted to verticallines is of rapid decay.

Corollary II.1. Let D and U be as above. Then

Index PUP =∑n≤p

(−1)n−1

2

(n− 1

2

)!∑k,q

(−1)|k|

k1! . . . kn!αk

1q!σm−q(m) Ress=0 s

q ζ(k,n)(s) ,

with m = |k| + n−12 .

Proof. We just apply Theorem II.2 to the special case when A = C∞(S1),acting on H by the unique representation which sends the function f(eiθ) tof(U). We use the formula for the pairing between K1-theory and odd cycliccohomology, together with the index formula (cf. [2]),

(139) 〈ϕ′, U〉 =1√2πi

∑n odd

(−1)n−1

2

(n− 1

2

)! ϕ′

n(U−1, U, . . . , U−1, U) .

The proof of Theorem II.1 b) shows that the hypothesis (138) is sufficient toconclude.

At this point we should stress the considerable freedom that one has in applyingCorollary II.1. The data is a discrete subset (perhaps with multiplicity) of R,

(140) S = Spectrum D ,

together with a unitary matrix, u(λ, λ′)λ,λ′∈S which signifies a “unitary corre-spondence” on the list S. The main hypothesis is that when D is shifted by thiscorrespondence (i.e. UDU∗ is considered), it stays at bounded distance fromD. Then one writes down a finite number of ζ-functions, the ζ(k,n) above, whichcan be expressed as Dirichlet series of the form

(141)∑

a±n |λ±n |−s

60

when one computes the trace in the basis of eigenvectors

(142) D e±n = λ±n e±n

for the operator D.

The statement is that a certain rational combination of residues of these func-tions gives the index of PUP or spectral flow. In particular, one has:

Corollary II.2. If Index PUP �= 0, at least one of the functions ζ(k,n)(s) hasa non trivial pole at s = 0.

2.5 Concluding remarks

We shall now briefly discuss the analogues of Theorems II.1 and II.2 in the evencase, i.e. when we have a Z/2-grading γ of the Hilbert space H such that:

(143) γa = aγ ∀a ∈ A , Dγ = −γD .

Let us come back to Theorem II.1 and consider for n even, n > 0, the cochaingiven by the similar formula:

ϕn(a0, . . . , an) =(144) ∑q≥0,kj≥0

cn,k,q τq

(γa0(da1)(k1) . . . (dan)(kn) |D|−(n+2Σkj)

),

∀aj ∈ A, where the cn,k,q are given by:

cn,k,q = (−1)k1+···+kn(145)

(k1! . . . kn!)−1 Γ(q)(k1 + · · · + kn +

n

2

) 1q!

((k1 + 1)(k1 + k2 + 2) . . .)−1 .

One can compute Bϕ2 exactly as we did in the proof of Thm. II.1. a). Oneobtains:

(146) Bϕ2(a0, a1) = −∑k,qk≥1

(−1)k Γ(k)(q)

k!q!τq

(γa0(da1)(k) |D|−1−2k

).

One checks as in the proof of a) that bϕn + Bϕn+2 = 0 for all n > 0 (n even),but to get the correct expression for ϕ0 we need to go back to the proof of b),in the case n = 0.

In this case we get:

(147) ψε0(a) = Trace(γa e−εD2

) ∀a ∈ A

61

and this time we look for the constant terms in the asymptotic expansion at 0of this function of θ(ε). It is of course related to the analogue of (68) for n = 0,

(148) ζ(z) = Trace(γa |D|−2z) ;

however, this time it is no longer the residue of (148) which matters, but ratherits value at 0, which governs the residue of the product Γ(z)ζ(z)

(149) ϕ0(a) = Ress=0

(Γ(s) Trace(γa |D|−2s)) .

The right hand side is equal to

∑q≥0

Γ(1)(q)

q!τq−1(γa) ,

where we need to define τ−1 following Proposition II.1, by

(150) τ−1(b) = Ress=0

s−1 Trace(b |D|−2s) .

We can now simplify the cocycle ϕ as we did in Theorem II.2. The expression(144) for ϕn involves

(151) Ress=0

Γ(|k| + n

2+ s

)Trace(A|D|−2s) ,

for some operator A. As in the proof of Theorem II.2 we replace this by

(152) Ress=0

Γ(1 + s)−1 Γ(|k| + n

2+ s

)Trace(A|D|−2s)

and we use the equality

(153) Γ(m+ s) Γ(1 + s)−1 =m−1∑

0

σj(m) sj ,

where the σj are the elementary symmetric functions of the numbers 1, 2, . . . ,m−1.

We thus obtain the following analogue of Theorem II.2 :

Theorem II.3. a) The following formula defines a cocycle in the (b, B) bicom-plex of A:

ϕn(a0, . . . , an) =∑k,q

(−1)|k|

k1! . . . kn!αk σq

(|k| + n

2

)τq

(γa0(da1)(k1) . . . (dan)(kn) |D|−(2|k|+n)

)

62

for n even, �= 0, whileϕ0(a0) = τ−1(γa0) .

b) The cohomology class of the cocycle (ϕn), n even, in HCeven(A) coincideswith the cyclic cohomology Chern character ch∗(A,H, D).

Remark II.1. To see where the classical index theorem for manifolds fits inthis picture, let us consider the spectral triple (A,H, D) consisting of the DiracoperatorD acting on the Hilbert space H of L2-spinors over a closed RiemannianSpin manifold M , of dimension p, and with A = C∞(M). Then:

a) the dimension spectrum of the triple (A,H, D) is simple and contained in theset

{n ∈ N ; n ≤ p};

b) with τ = τ0 and otherwise using the above notation, one has

τ(a0(da1)(k1) . . . (dan)(kn) |D|−(n+2|k|)

)= 0 , for |k| �= 0 ;

c) for k1 = k2 = . . . = kn = 0, one has

τ(a0[D, a1] . . . [D, an] |D|−n

)= νn

∫M

A(R) ∧ a0 da1 ∧ . . . ∧ dan ,

where νn is a numerical factor and A(R) stands for the A-form of the Rieman-nian curvature of M .

Assertion a) follows from the standard pseudodifferential calculus, while b) andc) can be easily checked, e.g. by means of the symbolic calculus for asymptoticoperators, as in [6], §3.

A much more interesting example for the general local index formalism devel-oped in this section is provided by the spectral triples of Section 1, associatedto triangular structures on manifolds. In that case, the foliation-preserving dif-feomorphisms contribute (via Gelfand-Fuchs classes) to the “higher residues”corresponding to |k| �= 0. This application will be discussed in a forthcomingpaper.

63

Appendix A. Inequalities for eigenvalues and the

Dixmier trace

Let H be a (separable) Hilbert space. We let K ⊂ L(H) be the ideal of compactoperators and L1 ⊂ K the ideal of trace class operators:

(1) T ∈ L1 ⇔∞∑0

µn(T ) <∞

where µn(T ) = Inf{‖T/E⊥‖ ; dimE = n

}is the nth characteristic value of T .

Note that:

(2) µn(T ) = dist(T,Rn) , Rn = {operators of rank ≤ n}

(3) µn(T ) = n+ 1’th eigenvalue of |T |.

(This last equality is the minimax principle.)

One defines Trace(T ) for T ∈ L1 as:

(4) Trace(T ) =∑

〈Tξn, ξn〉 (ξn) orthonormal basis .

This converges because T is an �1 sum of rank one operators |η〉〈η| and it isindependent of the basis as can be seen for such an operator.

Lemma A.1.∞∑0

µn(T ) = Sup |Trace(TX)| ; ‖X‖ ≤ 1 .

Proof. First note that |Trace(T )| ≤∞∑0µn(T ) using the polar decomposition

T = U |T |, |T | =∑µn|ηn〉〈ηn| and

∣∣Trace |Uηn〉〈ηn|∣∣≤ 1.

Then |Trace(TX)| ≤∞∑0µn(TX) ≤

∞∑0µn(T ), if ‖X‖ ≤ 1. For the converse use

U∗ = X .

One lets ‖T ‖1 = Trace(|T |). It is a norm, the trace norm.

Definition A.1. For each integer N ≥ 1, let, for T ∈ K

σN (T ) = Sup {‖TE‖1 ; E subspace , dimE = N} .

64

By construction, σN is a norm,

(5) σN (T1 + T2) ≤ σN (T1) + σN (T2) ∀Tj ∈ K .

Proposition A.1.

σN (T ) =N−1∑

0

µn(T ) .

Proof. We can assume T ≥ 0. Then taking E = EN (T ) the spectral projectionon the spectral subspace corresponding to the first N eigenvalues one gets the

inequality ≥. Conversely if dimE = N then ‖TE‖1 =∞∑0µn(TE) =

N−1∑0

µn(TE) ≤N−1∑

0µn(T ) (since dist(TE,RN) = 0).

We shall now extend the definition of σN to a function σλ of the positive “scale”parameter λ as follows:

Lemma A.2. Let T ∈ K. The following function of λ ∈ R∗+ agrees with σN (T )

at integer values:

σλ(T ) = Inf {‖x‖1 + λ‖y‖∞ ; x+ y = T } .

Proof. One can assume T ≥ 0. Let N ∈ N∗, we compare σN (T ) of DefinitionA.1 with the r.h.s. of 4.

Assume T = x+ y, with ‖x‖1 +N‖y‖∞ ≤ 1. Then

σN (T ) ≤ σN (x) +N‖T − x‖∞ ≤ ‖x‖1 +N‖y‖∞ ≤ 1 .

Conversely write T = (T − µN (T )1) EN + (µN (T ) EN + T (1 − EN )) = x + y.One has ‖x‖1 = σN (T )−NµN(T ) and ‖y‖∞ = µN (T ). Thus ‖x‖1 +N‖y‖∞ =σN (T ).

By construction, the function σλ(T ) is increasing as a function of λ. Also theunit ball for the norm σλ is the convex hull of the unit ball of L1 and λ−1 timesthe unit ball of K.

The slope of σλ, such as σN+1 − σN = µN , is decreasing with λ, thus σλ is aconcave function of λ.

65

Between 0 and 1 one has σλ(T ) = λ‖T ‖ as follows from:

‖x‖∞ ≤ ‖x‖1 .

One can check as in the proof of Lemma A.2 that σλ is affine between N andN + 1 for any N (see Fig.). In particular (5) holds for all real values λ > 0.

For a positive operator T we view σλ(T ) as the trace of T cutoff at the (inverse)scale λ.

We shall now investigate the additivity of σλ for T ≥ 0.

Lemma A.3. For T1, T2 ≥ 0 and λ1, λ2 ∈ R∗+

σλ1+λ2(T1 + T2) ≥ σλ1(T1) + σλ2(T2) .

Proof. Let us assume that N1, N2 are integers. First for T ≥ 0 one has:

(6) σN (T ) = Sup {Trace(TE) ; dimE = N} .

(The r.h.s. is smaller than ‖TE‖1 and hence than σN (T ), the other inequalityis clear.)

Then Trace(T1E1)+Trace(T2E2) ≤ Trace(T1E)+Trace(T2E) whereE = E1∨E2

has dimension ≤ N1 +N2. One can then deduce the result for arbitrary λ1, λ2

by piecewise linearity.

We shall now concentrate on the log λ divergence of σλ(T ) and average thecoefficient of logλ over various scales by considering the following function:

(7) τλ(T ) =1

logλ

∫ λ

a

σu(T )log(u)

du

u(we fix a > e) .

By construction τλ is subadditive (using 5). Let us now evaluate a lower boundfor τλ(T1 + T2), Tj ≥ 0.

Lemma A.4. Let T1, T2 be such that σλ(Tj) ≤ Cj logλ ∀λ ≥ a. If Tj ≥ 0,then:

|τλ(T1 + T2) − τλ(T1) − τλ(T2)| ≤ (C1 + C2) ((log logλ+ 2) log 2)/ logλ .

Proof. One has σ2u(T1 + T2) ≥ σu(T1) + σu(T2). Thus:

τλ(T1) + τλ(T2) ≤ 1logλ

∫ λ

a

σ2u(T1 + T2)log u

du

u=

1logλ

∫ 2λ

2a

σu(T1 + T2)log(u/2)

du

u.

66

On [a,+∞[ one has σu(T1 + T2) ≤ (C1 + C2) log u, thus:∣∣∣∣∣ 1logλ

∫ λ

a

σu(T1 + T2)log u

du

u− 1

logλ

∫ 2λ

2a

σu(T1 + T2)log u

du

u

∣∣∣∣∣ ≤ (C1+C2)2 log 2logλ

.

Next,1

logλ

∫ 2λ

2a

∣∣∣∣σu(T1 + T2)log(u/2)

− σu(T1 + T2)log u

∣∣∣∣ du

u

≤ (C1 + C2)1

logλ

∫ 2λ

2a

log u− log u/2log u/2

du

u∫ 2λ

2a

log 2log(u/2)

du

u= log 2

∫ λ

a

du

u log u= log 2(log logλ− log log a) .

Let A be the space of functions h(λ); λ ∈ [a,∞[, which are bounded and aretaken modulo those of order 0(log logλ/ logλ). The latter form an ideal for theobvious pointwise product and thus A is an algebra.

By Lemma A.4 we have a well defined additive map:

(8) τ : L(1,∞) → A

defined by τ(T ) = class of (τλ(T ))λ≥a for any T ≥ 0. Here we used:

Definition A.2. L(1,∞) is the normed ideal with norm:

Supu≥a

σu(T )log u

= ‖T ‖1,∞ .

Note that the image by τ of the ideal L1 of trace class operators is {0}.The above definition of τ(T ) for T ≥ 0 has been extended to any T usinglinearity. For instance if we write T = T ∗ in two ways as T1 − T2 = T ′

1 − T ′2 (all

Ti, T′j ≥ 0) we have:

(9) τ(T1) − τ(T2) = τ(T ′1) − τ(T ′

2)

since the equality T1 + T ′2 = T2 + T ′

1 can be combined with Lemma A.4.

Proposition A.2. τ is a linear positive map from L(1,∞) to A such that forany bounded operator S in H:

τ(ST ) = τ(TS) ∀T ∈ L(1,∞) .

67

Proof. One has for any unitary U and T ≥ 0 in L(1,∞) the equality:

τ(UTU∗) = τ(T ) .

This extends by linearity to arbitrary T ’s and then using TU instead of T onegets τ(UT ) = τ(TU) and the conclusion by linearity.

Corollary A.1. For T ∈ L(1,∞) the class τ(T ) only depends upon the locallyconvex topology of H, not the inner product.

Proof. τ(STS−1) = τ(T ) for any invertible S.

Let us now consider states ω on the C∗ algebra:

(10) A = Cb ([a,∞[) /C0 ([a,∞[) .

Lemma A.5. Let f ∈ Cb ([a,∞[) then f(λ) has a limit for λ → ∞ iff ω(f) isindependent of ω.

Proof. If f → L then f − L ∈ C0 on which any ω vanishes. Conversely if fhas two distinct limit points, one gets two states ω1, ω2 whose values on f aredifferent.

Remark A.1. For any separable C∗-subalgebra of Cb ([a,∞[) /C0 the construc-tion of states can be effectively performed without using the axiom of uncount-able choice. So whenever we apply Trω (cf. below) to any separable subspaceof L(1,∞) we may assume that it is effectively constructed.

Definition A.3. For any ω we let:

Trω(T ) = ω(τ(T )) ∀T ∈ L(1,∞) .

One has, using Proposition A.2:

Proposition A.3. a) Trω is a positive linear form on L(1,∞) and Trω(ST ) =Trω(TS) ∀S ∈ L(H).b) Trω(T ) is independent of ω iff τ(T ) converges for λ → ∞ (and the limit isthen equal to Trω(T )).

By construction Trω is continuous for the norm ‖ ‖1,∞ and vanishes on theclosure L(1,∞)

0 of finite rank operators in this norm.

One has:

(11) T ∈ L(1,∞)0 ⇔ σλ(T ) = o(log λ) .

68

In particular one has:

(12) µn(T ) = o

(1n

)⇒ T ∈ L(1,∞)

0 .

There is an easy case where Trω is independent of ω and can be computed:

Proposition A.4. Let T ≥ 0, µn(T ) = 0(1/n) and ζ(s) = Trace(T s). Thenthe following are equivalent:

a) (s− 1) ζ(s) → L as s→ 1+ , b) (logN)−1N−1∑

0

µn(T ) → L as N → ∞ .

When this holds one has Trω(T ) = L independently of ω.

In particular take T = ∆−p, p > 0, where one knows the asymptotic behaviorof Trace(e−t∆) ∼ L t−p for t→ 0.

Assume ∆ ≥ c > 0 and write:

(13) ∆−s =1

Γ(s)

∫ ∞

0

e−t∆ tsdt

t.

With s = p(1 + ε) and applying the trace on both sides we get:

ζ(1 + ε) =1

Γ(p+ εp)

∫ ∞

0

Trace(e−t∆) tp+εp dt

t

and for ε→ 0 it is equivalent to:

1Γ(p)

L

∫ 1

0

tεp dt

t=

1ε

1Γ(p+ 1)

L .

Thus we get in that case:

(14) Trω(∆−p) =1

Γ(p+ 1)limt→0

tp Trace(e−t∆) .

If we let ∆ be the Laplacian on the n torus Tn where the length of T is 2π, onecan use ∑

Z

e−tk2 ∼ π1/2

√t

t→ 0

and get Trω(∆−n/2) = πn/2

Γ(n2 +1) .

More generally for ordinary pseudodifferential operators (ΨDO) on a manifoldthe Dixmier trace is given by the Wodzicki residue,

69

Proposition A.5. Let V be an n-dimensional manifold and P ∈ OP−n(V ) aΨDO of order −n then

1) P ∈ L(1,∞) 2) Trω(P ) =1n

(2π)−n

∫S∗a−n(x, ξ) dnx dn−1ξ .

Recall that in local coordinates x a ΨDO is written as

(Pη) (x) = (2π)−n

∫ei(x−y)·ξ α(x, ξ) η(y) dy dξ

where α ∼ aq + aq−1 + · · · and a�(x, ξ) is homogeneous of degree � in ξ.

The principal symbol aq(x, ξ) is invariantly defined as a function on the cotan-gent bundle T ∗V by:(

P(eiτϕ η

))(x) ∼ τq aq (x, dϕ) η(x) ∀η .

Next, let us consider on the complement of the 0-section V in T ∗V the measuredx dξ associated to the symplectic structure. For any homogeneous function ofdegree −n, a−n(x, ξ) dx dξ is now invariant under ξ → λξ.

If we let E = r ∂∂r be the vector field generating this one parameter group we

have ∂E µ = 0, d iE µ = 0, where µ is viewed as a form of top degree. ThusiE µ is a form of degree 2n− 1 and its integral on any two homologous cyclesof dimension 2n− 1 gives the same result.

In particular we can choose a Riemannian metric on V and take as cycle its unitsphere bundle S∗. Then we get∫

S∗a−n(x, ξ) dnx dn−1ξ

where dn−1ξ is the volume form on the n− 1 sphere with its induced metric. Ifwe compute it for the constant 1 we get |Sn−1| ∫

V dn x, where

|Sn−1| =2πn/2

Γ(

n2

) .

Let us check the equality 2) for the above torus Tn and the Laplacian. Thel.h.s. is πn/2

Γ(n2 +1) the r.h.s. is

1n

(2π)−n (2π)n |Sn−1| =2n

πn/2

Γ(

n2

) .The general conclusion follows by positivity and invariance of Trω.

70

Appendix B. Spectral triple and pseudodifferen-

tial calculus

Let (A,H, D) be a spectral triple. For each s ∈ R we let Hs = Domain (|D|s)and

H∞ = ∩s≥0

Hs , H−∞ = dual of H∞ .

We obtain in this way a scale of Hilbert spaces, and for each r we define opr tobe the linear space of operators in H∞ which are continuous for every s:

opr : Hs → Hs−r .

We shall use the following smoothness condition on A: ∀a ∈ A, both a and[D, a] are in the domain of all powers of the derivation δ = [|D|, ·].

Lemma B.1. Then a, [D, a] are in op0 and

b− |D| b |D|−1 ∈ op−1 (b = a or [D, a]) .

Proof. Let us first check that |D|n b |D|−n is bounded for n ≥ 0. Withσ(·) = |D| · |D|−1, one has:

σ = id+ ε , ε(b) = δ(b) |D|−1 .

Since εk(b) is bounded, equal to δk(b) |D|−k, we get the result using σn =∑Ck

n εk.

Moreover σ−1(b) = |D|−1 b |D| = −|D|−1 δ(b) and the same argument showsthat σn(b) is bounded for n < 0. Then one uses interpolation.

For the second part one applies the above argument to δ(b); thus,

δ(b) ∈ op0 , δ(b) |D|−1 ∈ op−1 .

It is important to note that the above smoothness hypothesis can be replacedby:

a and [D, a] ∈ ∩ Dom LkRq , L(b) = |D|−1 [D2, b] , R(b) = [D2, b] |D|−1 .

Indeed, assuming the above, one has

L(b) = |D|−1 (|D| δ(b) + δ(b) |D|) ∈ op0 , R(b) ∈ op0

and the same applies to LkRq(b).

71

Corollary B.1. Under the above hypothesis one has[D2,

[D2, . . . [D2, b ]

]· · ·︸︷︷︸n

]∈ opn ∀b ∈ A , or [D,A] .

Let us now show that if b ∈ ∩ Dom LkRq then b ∈ Dom δ. The proof is moresubtle than one would expect, because the obvious argument, using

|D| = π−1

∫ ∞

0

D2

D2 + µµ−1/2 dµ ,

requires some care. Indeed, one gets from the above

[|D|, b] = π−1

∫ ∞

0

(D2 + µ)−1 [D2, b] (D2 + µ)−1 µ1/2 dµ .

We can replace [D2, b] by |D|, which has the same size, and get∫ ∞

0

(D2 + µ)−2 |D| µ1/2 dµ =∫ ∞

0

(1 + t)−2 t1/2 dt .

For this to work, we need to move [D2, b] in front of the above integral, i.e. usethe finiteness of the norm of∫ ∞

0

[(D2 + µ)−1, [D2, b]

]︸ ︷︷ ︸−(D2+µ)−1[D2,[D2,b]](D2+µ)−1

(D2 + µ)−1 µ1/2 dµ .

This finiteness follows from:1) (D2 + µ)−1 [D2, [D2, b]] bounded since b ∈ DomL2

2)∫∞0 ‖(D2 + µ)−2‖ µ1/2 dµ ≤ C

∫ 1

0 µ1/2 dµ+

∫∞1 µ−3/2 dµ <∞.

Once [D2, b] is moved in front the above calculation applies.

It follows that b ∈ Dom δ and applying the same proof to δ(b), . . . we getb ∈ ∩ Dom δk. We thus obtain:

Lemma B.2. ∩k,q

Dom LkRq = ∩n

Dom δn.

We shall define the order of operators by the following filtration:

P ∈ OPα iff |D|−α P ∈ ∩ Dom δn .

Thus OP 0 = ∩ Dom δn and we have:

OPα ⊂ opα ∀α .

72

Let us now describe the general pseudodifferential calculus.

We let ∇ be the derivation: ∇(T ) = [D2, T ], and consider the algebra generatedby the ∇n(T ), T ∈ A or [D,A].

We view this algebra D as an analogue of the algebra of differential operators.In fact by Corollary B.1 we have a natural filtration of D by the total power of∇ applied, and moreover:

(1) Dn ⊂ OPn .

We want to develop a calculus for operators of the form:

(2) A |D|z z ∈ C , A ∈ D .

We shall use the notation ∆ = D2 and begin by understanding the action of C

given by:

(3) σ2z = ∆z · ∆−z.

By construction D is stable under the derivation ∇ and:

(4) ∇(Dn) ⊂ Dn+1 .

Also for A ∈ Dn and z ∈ C one has

(5) A |D|z ∈ OPn+Re(z) .

We shall use the group σ2z to understand how to multiply operators of complexorder modulo OP−k for any k. One has: σ2 = 1 + E ,

(6) E(T ) = ∇(T ) ∆−1 .

Lemma B.3. Let T ∈ Dq then Ek(T ) ∈ OP q−k ∀k ≥ 0.

Proof.Ek(T ) = ∇k(T ) ∆−k ∈ OP q+k ∆−k ⊂ OP q−k .

We just wish to justify the formal expansion:

σ2z(T ) =(

1 + z E +z(z − 1)

2!E2 + · · ·

)(T ) .

It should give a control of σ2z(T ) modulo OP q−k−1 if we stop at Ek(T ).

73

To do this we need to control the remainder in the Taylor formula:

(1 + E)n+1−α = 1 + (n+ 1 − α)E +(n+ 1 − α)(n − α)

2!E2 + · · ·+(7)

(n+ 1 − α) · · · (n+ 1 − k − α)Ek+1

(k + 1)!+ · · · + (n+ 1 − α) · · · (2 − α)

En

n!+

En+1

∫ 1

0

(n+ 1 − α) · · · (1 − α)(1 + tE)−α (1 − t)n

n!dt .

The main lemma is the following:

Lemma B.4. Let α ∈ C, 0 < Re α < 1 and β > 0, β < a = Re α. Then thefollowing operator preserves the space OPα for any α:

Ψ = σ2β

∫ 1

0

(1 + tE)−α (1 − t)n dt .

Proof. This will be done by expressing Ψ as an integral of the form

(8) Ψ =∫σ2is dµ(s) ‖µ‖ <∞ .

One writes

(9) (1 + tE)−α =sinπαπ

∫ ∞

0

11 + tE + µ

µ−α dµ

using the standard formula:

(10) x−α =sinπαπ

∫ ∞

0

1x+ µ

µ−α dµ .

Let us then consider the resolvent of −σ2, namely

R(λ) = (λ+ σ2)−1 .

One has, with β ∈ ]0, 1[ as above,

(11) R(λ) =12

∫ ∞

−∞σ−2(β+is) λβ+is−1 ds

sinπ(β + is)

which follows from

(12)1

1 + y=

12

∫ ∞

−∞y−(β+is) ds

sinπ(β + is).

74

(With y = eu, this means that eβu

1+eu is the Fourier transform of 1sin π(β+is) , which

also follows from (10) written as πsin(π(1−β−is)) =

∫∞−∞

e(1−α)u

1+eu du , α = 1−β−is.)Thus, from (11) we get,

(13) σ2β R(λ) =12

∫ ∞

−∞σ−2is λβ−1 λis ds

sinπ(β + is)

where the measure λis dssin π(β+is) is well controled by e−|s| ds. By (9) we have:

(1 + tE)−α =sinπαπ

∫ ∞

0

1tR

(µ+ 1t

− 1)µ−α dµ ,

σ2β(1 + tE)−α =

sinπαπ

12

∫ ∞

0

1t

∫ ∞

−∞σ−2is

(µ+ 1t

− 1)β−1

µ−α λis ds

sinπ(β + is)dµ ,

with λ =(

µ+1t − 1

).

For fixed s we are thus dealing with the size:

1t

∫ ∞

0

(µ+ 1t

− 1)β−1

µ−α dµ = I .

One has 0 < t < 1 so that the behavior at µ = 0 is fine, also the integralconverges for µ→ ∞ as µ(β−α)−1, since β < α.

We get

I =1t

∫ ∞

0

(u+

1t− 1

)β−1

t−α u−α t du

=1t

∫ ∞

0

(1t− 1

)β−1

(v + 1)β−1 t−α

(1t− 1

)−α

v−α t

(1t− 1

)dv

= t−α

(1t− 1

)β−α ∫ ∞

0

(v + 1)β−1 v−α dv

= (1 − t)β−α tβ c(α, β) .

Finally, we get an equality:∫ 1

0

σ2β(1 + tE)−α (1 − t)n

n!dt =

∫ ∞

−∞σ2is dν(s)

where the total mass of the measure ν is finite.

75

Let us check this in another way, by looking directly at the L1-norm of theFourier transform of the function

u→∫ 1

0

eβu (1 + t(eu − 1))−α (1 − t)n

n!dt .

Thus, it is enough to check that the following function of u is in the Schwartzspace S(R):

ϕn(u) = (eu − 1)−(n+1)eβu(e(n+1−α)u − 1 − (n+ 1 − α)(eu − 1)

− (n+ 1 − α)(n− α)2!

(eu−1)2−· · ·− (n+ 1 − α)(n − α) · · · (2 − α)n!

(eu−1)n).

First, for u → ∞ the size is ∼ e−(n+1)u eβu e(n+1−α)u = e(β−α)u → 0. Foru → −∞ it behaves like eβu → 0. We need to know that it is smooth at u = 0but this follows from the Taylor expansion. The same argument applies to allderivatives. Thus this gives another proof of the lemma.

We are now ready to prove:

Theorem B.1. Let T ∈ Dq and n ∈ N. Then for any z ∈ C

σ2z(T )−(T + z E(T ) +

z(z − 1)2!

E2(T ) + · · · + z(z − 1) · · · (z − n+ 1)n!

En(T ))

∈ OP q−(n+1).

Proof. First for any z ∈ C and k ∈ N one has:

(14) Ek(σ2z(T )

) ∈ OP q−k .

Indeed, by (8) we know that σ2z leaves any OPn invariant; as Ek◦σ2z = σ2z◦Ek,we just use Lemma B.3.

One has, for 0 < Re α < 1, β as above and z = (n+ 1) − α:

σ2(β+(n+1)−α)(T )

−(σ2β(T ) + z σ2β E(T ) + · · · + z(z − 1) · · · (z − n+ 1)

n!σ2β En(T )

)= λ Ψ

(En+1(T )).(15)

If we apply this equality to σ2s(T ) and use (14) and Lemma B.4 we see that forany s there exists a polynomial Ps(α) of degree n in α such that:

σ2(s−α)(T ) − Ps(α) ∈ OP q−(n+1) β < Re α < 1 .

The polynomials Ps(α + s) have to agree modulo OP q−(n+1) on the overlap ofthe bands β < Re α < 1 and thus the differences between two will belong toOP q−(n+1) for all z. It follows then that there is P (z) which works for all z. Toobtain its coefficients one takes the integral values z = 0, 1, . . . , n which yieldsthe formula of Theorem B.1.

76

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