Spectral Geometry
Bruno
Iochum
Centre de Physique ThéoriqueAix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France
Notes of lectures delivered at the Summer School“Geometric, Algebraic and Topological Methods for Quantum Field Theory 2011”Villa de Leyva (Columbia), July 4–22
Abstract
The goal of these lectures is to present the few fundamentals of noncommu-tative geometry looking around its spectral approach. Strongly motivatedby physics, in particular by relativity and quantum mechanics, Chamsed-dine and Connes have defined an action based on spectral considerations,the so-called spectral action.The idea is to review the necessary tools which are behind this spectralaction to be able to compute it first in the case of Riemannian manifolds(Einstein–Hilbert action). Then, all primary objects defined for manifoldswill be generalized to reach the level of noncommutative geometry via spec-tral triples, with the concrete analysis of the noncommutative torus whichis a deformation of the ordinary one.The basics of different ingredients will be presented and studied like, Diracoperators, heat equation asymptotics, zeta functions and then, how to getwithin the framework of operators on Hilbert spaces, the notion of noncom-mutative residue, Dixmier trace, pseudodifferential operators etc. Thesenotions are appropriate in noncommutative geometry to tackle the casewhere the space is swapped with an algebra like for instance the noncom-mutative torus. Its non-compact generalization, namely the Moyal plane,is also investigated.Update: 2017, December 15 a r X i v : . [ m a t h - ph ] D ec otivations: Let us first expose few motivations from physics to study noncommutative geometry whichis by essence a spectral geometry. Of course, precise mathematical definitions and resultswill be given in the other sections.The notion of spectrum is quite important in physics, for instance in classical mechanics,the Fourier spectrum is essential to understand vibrations or the light spectrum in electro-magnetism. The notion of spectral theory is also important in functional analysis, where thespectral theorem tells us that any selfadjoint operator A can be seen as an integral over itsspectral measure A = R a ∈ Sp( a ) a dP a if Sp( A ) is the spectrum of A . This is of course essentialin the axiomatic formulation of quantum mechanics, especially in the Heisenberg picturewhere the tools are the observables namely are selfadjoint operators.But this notion is also useful in geometry. In special relativity, we consider fields ψ ( ~x ) for ~x ∈ R and the electric and magnetic fields E, B ∈ Function( M = R , R ). Einstein intro-duced in 1915 the gravitational field and the equation of motion of matter. But a problemappeared: what are the physical meaning of coordinates x µ and equations of fields? Assumethe general covariance of field equation. If g µν ( x ) or the tetradfield e Iµ ( x ) is a solution (where I is a local inertial reference frame), then, for any diffeomorphism φ of M which is active orpassive (i.e. change of coordinates), e Iν ( x ) = ∂x µ ∂φ ( x ) ν e Iµ ( x ) is also a solution. As a consequence,when relativity became general, the points disappeared and it remained only fields on fields inthe sense that there is no fields on a given space-time. But how to practice geometry withoutspace, given usually by a manifold M ? In this later case, the spectral approach, namely thecontrol of eigenvalues of the scalar (or spinorial) Laplacian return important informations on M and one can even address the question if they are sufficient: can one hear the shape of M ?There are two natural points of view on the notion of space: one is based on points (of amanifold), this is the traditional geometrical one. The other is based on algebra and this isthe spectral one. So the idea is to use algebra of the dual spectral quantities.This is of course more in the spirit of quantum mechanics but it remains to know what is aquantum geometry with bosons satisfying the Klein-Gordon equation ( (cid:3) + m ) ψ ( ~x ) = s b ( ~x )and fermions satisfying ( i∂/ − m ) ψ ( ~x ) = s f ( ~x ) for sources s b , s f . Here ∂/ can be seen as asquare root of (cid:3) and the Dirac operator will play a key role in noncommutative geometry.In some sense, quantum forces and general relativity drive us to a spectral approach ofphysics, especially of space-time.Noncommutative geometry, mainly pioneered by A. Connes (see [25, 31]), is based on aspectral triple ( A , H , D ) where the ∗ -algebra A generalizes smooth functions on space-time M (or the coordinates) with pointwise product, H generalizes the Hilbert space of abovequoted spinors ψ and D is a selfadjoint operator on H which generalizes ∂/ via a connectionon a vector bundle over M . The algebra A also acts, via a representation of ∗ -algebra, on H .Noncommutative geometry treats space-time as quantum physics does for the phase-space since it gives a uncertainty principle: under a certain scale, phase-space points areindistinguishable. Below the scale Λ − , a certain renormalization is necessary. Given ageometry, the notion of action plays an essential role in physics, for instance, the Einstein–Hilbert action in gravity or the Yang–Mills–Higgs action in particle physics. So here, giventhe data ( A , H , D ), the appropriate notion of action was introduced by Chamseddine andConnes [11] and defined as S ( D , Λ , f ) := Tr (cid:16) f ( D / Λ) (cid:17) ∈ R + plays the role of a cut-off and f is a positive even function. The asymp-totic series in Λ → ∞ yields to an effective theory. For instance, this action applied toa noncommutative model of space-time M × F with a fine structure for fermions encodedin a finite geometry F gives rise from pure gravity to the standard model coupled withgravity [12, 21, 31].The purpose of these notes is mainly to compute this spectral action on few examples likemanifolds and the noncommutative torus.In section 1, we present standard material on pseudodifferential operators over a compactRiemannian manifold. A description of the behavior of the kernel of a ΨDO near the diagonalis given with the important example of elliptic operators. Then follows the notion of Wodzickiresidue and its computation. The main point being to understand why it is a residue.In section 2, the link with the Dixmier trace is shown. Different subspaces of compact op-erators are described in particular, the ideal L , ∞ ( H ). Its definition is on purpose because inrenormalization theory, one has to control the logarithmic divergency of the series P ∞ n =1 n − .We will see that this “defect” of convergence of the Riemann zeta function (in the sense thatthis generates a lot of complications of convergence in physics) is in fact an “advantage”because it is precisely the Dixmier trace and more generally the Wodzicki residue which arethe right tools which mimics this zeta function: firstly, this controls the spectral aspects of amanifold and secondly they can be generalized to any spectral triple.In section 3, we recall the basic definition of a Dirac (or Dirac-like) operator on a compactRiemannian manifold ( M, g ) endowed with a vector bundle E . An example is the (Clifford)bundle E = C ‘ M where C ‘ T ∗ x M is the Clifford algebra for x ∈ M . This leads to the notionof spin structure, spin connection ∇ S and Dirac operator D/ = − ic ◦ ∇ S where c is theClifford multiplication. A special focus is put on the change of metrics g under conformaltransformations.In section 4 is presented the fundamentals of heat kernel theory, namely the Green functionof the heat operator e t ∆ , t ∈ R + . In particular, its expansion as t → + in terms of coefficientsof the elliptic operator ∆, with a method to compute the coefficients of this expansion isexplained. The idea being to replace the Laplacian ∆ by D later on.In section 5, a noncommutative integration theory is developed around the notion ofspectral triple. This means to understand the notion of differential (or pseudodifferential)operators in this context. Within differential calculus, the link between the one-form and thefluctuations of the given D is outlined.Section 6 concerns few actions in physics, such that the Einstein–Hilbert and Yang–Millsactions. The spectral action Tr (cid:16) f ( D / Λ) (cid:17) is justified and the link between its asymptoticexpansion in Λ and the heat kernel coefficients is given via the noncommutative integrals ofpowers of |D| .Section 7 gathers several results on the computation of a residue of a series of holomorphicfunctions, a real difficulty since one cannot commute residue and infinite sums. The notionof Diophantine condition appears and allows nevertheless this commutation for meromorphicextension of a class of zeta functions.Section 8 is devoted to the computation of the spectral action on the noncommutativetorus. After the very definitions, it is shows how to calculate with the noncommutativeintegral. The main technical difficulty stems from a Diophantine condition which seemsnecessary (but is sufficient) since any element of the smooth algebra of the torus is a series of3ts generators, so the previous section is fully used. All proofs are not given, but the readershould be aware of all the main steps.Section 9 is an approach of non-compact spectral triples. This is mandatory for physicssince, a priori, the space-time is not compact. After a quick review on the difficulties whichoccur when M = R d due to the fact that the Dirac operator has a continuous spectrum,the example of the Moyal plane is analyzed. This plane is a non-compact version of thenoncom-mutative torus. Thus, no Diophantine condition appears, but the price to pay isthat functional analysis is deeply used.For each section, we suggest references since this review is by no means original.4 ontents D . . . . . . . . . . . . . . . . . . . . . . . . . . 455.5 Tadpole . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.6 Commutative geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 555.7 Scalar curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 575.8 Tensor product of spectral triples . . . . . . . . . . . . . . . . . . . . . . . . . 57 → ∞ . . . . . . . . . . . . . . . . . . . . . . . . . 616.3 Remark on the use of Laplace transform . . . . . . . . . . . . . . . . . . . . . 636.4 About convergence and divergence, local and global aspects of the asymptoticexpansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 636.5 About the physical meaning of the spectral action via its asymptotics . . . . . 65 i = 1: . . . . . . . . . . . . . . . . . . . . . . . 705.2.2 Proof of Lemma 7.6 for i = 0: . . . . . . . . . . . . . . . . . . . . . . . 717.2.3 Proof of item ( i.
2) of Theorem 7.5: . . . . . . . . . . . . . . . . . . . . 727.2.4 Proof of item ( iii ) of Theorem 7.5: . . . . . . . . . . . . . . . . . . . . 727.2.5 Commutation between sum and residue . . . . . . . . . . . . . . . . . . 747.3 Computation of residues of zeta functions . . . . . . . . . . . . . . . . . . . . . 767.4 Meromorphic continuation of a class of zeta functions . . . . . . . . . . . . . . 777.4.1 A family of polynomials . . . . . . . . . . . . . . . . . . . . . . . . . . 787.4.2 Residues of a class of zeta functions . . . . . . . . . . . . . . . . . . . . 78 − R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 928.5.1 Even dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . 948.5.2 Odd dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . . 978.6 Proof of the main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 978.7 Beyond Diophantine equation . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Notations: N = { , , . . . } is the set of positive integers and N = N ∪ { } the set of non negative integers.On R d , the volume form is dx = dx ∧ · · · ∧ dx d . S d is the sphere of radius one in dimension d . The induced metric: dξ = | d X j =1 ( − j − ξ j dξ ∧ · · · ∧ d dξ j ∧ · · · ∧ dξ d | restricts to the volume form on S d − . M is a d -dimensional manifold with metric g . U, V are open set either in M or in R d .We denote by dvol g the unique volume element such that dvol g ( ξ , · · · , ξ d ) = 1 for all posi-tively oriented g -orthonormal basis { ξ , · · · , ξ d } of T x M for x ∈ M . Thus in a local chart √ det g x | dx | = | dvol g | .When α ∈ N d is a multi-index, we define ∂ αx := ∂ α x ∂ α x · · · ∂ α d x d , | α | := d X i =1 α i , α ! := α a · · · α d . ξ ∈ R d , | ξ | := (cid:16) P dk =1 | ξ k | (cid:17) / is the Euclidean metric. H is a separable Hilbert space and B ( H ) , K ( H ) , L p ( H ) denote respectively the set of bounded,compact and p -Schatten-class operators, so L ( H ) are trace-class operators. The aim of this section is to show that the Wodzicki’s residue
WRes is a trace on the setΨ DO ( M ) of classical pseudodifferential operators on a compact manifold M of dimension d .Let us first describe the steps:- Define WRes ( P ) = 2 Res s =0 ζ ( s ) for P ∈ Ψ DO m of order m and ζ : s ∈ C → Tr( P ∆ − s ),which is holomorphic when < ( s ) ≥ ( d + m ).- If k P ( x, y ) is the kernel of P , then its trace can be developed homogeneously as thefollowing : tr (cid:16) k P ( x, y ) (cid:17) = P j = − ( m + d ) a j ( x, x − y ) − c P ( x ) log | x − y | + · · · where a j is homoge-neous of degree j in y and c P is a density on M defined by c P ( x ) := π ) d R S d − tr (cid:16) σ P − d ( x, ξ ) (cid:17) dξ ;here, σ P − d is the symbol of P of order − d .The Wodzicki’s residue has a simple computational form, namely WRes P = R M c P ( x ) | dx | .Then, the trace property follows.References for this section: Classical books are [101, 104]. For an orientation more in thespirit of noncommutative geometry since here we follow [88, 89] based on [3, 34], see also theexcellent books [50, 84, 85, 106, 107]. In the following, m ∈ C . Definition 1.1.
A symbol σ ( x, ξ ) of order m is a C ∞ function: ( x, ξ ) ∈ U × R d → C satisfying for any compact K ⊂ U and any x ∈ K (i) | ∂ αx ∂ βξ σ ( x, ξ ) | ≤ C Kαβ (1 + | ξ | ) < ( m ) −| β | , for some constant C Kαβ .(ii) We suppose that σ ( x, ξ ) ’ P j ≥ σ m − j ( x, ξ ) where σ k is homogeneous of degree k in ξ where ’ means a controlled asymptotic behavior | ∂ αx ∂ βξ (cid:16) σ − P j It is important to quote that a smoothing operator is a pseudodifferential op-erator whose amplitude is in A m ( U ) for all m ∈ R : by (3) , a ( x, y, ξ ) := e − i ( x − y ) · ξ k ( x, y ) φ ( ξ ) where the function φ ∈ C ∞ c ( R d ) satisfies R R d φ ( ξ ) dξ = (2 π ) d . Clearly, the main obstruction to smoothness is on the diagonal since Lemma 1.4. k σ ( x,D ) is C ∞ outside the diagonal.Proof. q σ is smooth since it is given for y = 0 by the oscillatory integral Z R d σ ( x, ξ ) e iy · ξ dξ = Z R d ( P ky σ ) e iy · ξ dξ where k is an integer such that k > < ( m ) + n and moreover P y = P ( y, D ξ ) is chosen with P y ( e iy · ξ ) = e iy · ξ ; for instance P y = | y | P j y j ∂∂ξ j . The last integral is absolutely converging.Few remarks on the duality between symbols and a subset of pseudodifferential operators: σ ( x, ξ ) ∈ S m ( U × R d ) ←→ k σ ( x, y ) ∈ C ∞ c ( U × U × R d ) ←→ A = Op ( σ ) ∈ Ψ DO m with the definition σ A ( x, ξ ) := e − ix · ξ A ( x → e ix · ξ )8here A is properly supported (since x → e ixξ has not a compact support) namely, A and itsadjoint map the dual of C ∞ ( U ) (distributions with compact support) into itself. Moreover, σ A ’ X α ( − i ) α α ! ∂ αξ ∂ αy k Aσ ( x, y, ξ ) | y = x , k Aσ ( x, y ) =: π ) d Z R d e i ( x − y ) · ξ k A ( x, y, ξ ) dξ , where k A ( x, y, ξ ) is the amplitude of k Aσ ( x, y ). Actually, σ A ( x, ξ ) = e iD ξ D y k A ( x, y, ξ ) | y = x and e iD ξ D y = 1 + iD ξ D y − ( D ξ D y ) + · · · . Thus A = Op ( σ A ) + R where R is a regularizingoperator on U .A technical point is the following: a pseudodifferential operator P is properly supportedwhen both P and P ∗ maps C ∞ c ( U ) not only in C ∞ ( U ) but in C ∞ c ( U ). Any Ψ DO is the sumof a properly supported Ψ DO and a smothing one.A point of interest is that differential operators are local: if f = 0 on U c (complementaryset of U ) then P f = 0 on U c . While pseudodifferential operators are pseudo-local: P f issmooth on U when f is smooth.There are two fundamental points about Ψ DO ’s: they form an algebra and this notion isstable by diffeomorphism justifying its extension to manifolds and then to bundles: Theorem 1.5. (i) If P ∈ Ψ DO m and P ∈ Ψ DO m are properly supported Ψ DO ’s, then P P ∈ Ψ DO m + m is properly supported with symbol σ P P ( x, ξ ) ’ X α ∈ N d ( − i ) α α ! ∂ αξ σ P ( x, ξ ) ∂ α × σ P ( x, ξ ) . The principal symbol of P P is σ P P m + m ( x, ξ ) = σ P m ( x, ξ ) σ P m ( x, ξ ) . (ii) Let P ∈ Ψ DO m ( U ) and φ ∈ Diff ( U, V ) where V is another open set of R d . The operator φ ∗ P : f ∈ C ∞ ( V ) → P ( f ◦ φ ) ◦ φ − satisfies φ ∗ P ∈ Ψ DO m ( V ) and its symbol is σ φ ∗ P ( x, ξ ) = σ Pm (cid:16) φ − ( x ) , ( dφ ) t ξ (cid:17) + X | α | > − i ) α α ! φ α ( x, ξ ) ∂ αξ σ P (cid:16) φ − ( x ) , ( dφ ) t ξ (cid:17) where φ α is a polynomial of degree α in ξ . Moreover, its principal symbol is σ φ ∗ Pm ( x, ξ ) = σ Pm (cid:16) φ − ( x ) , ( dφ ) t ξ (cid:17) . In other terms, the principal symbol is covariant by diffeomorphism: σ φ ∗ P m = φ ∗ σ Pm .(iii) If P ∈ Ψ DO m is properly supported, then P ∗ ∈ Ψ DO m and σ P ∗ ( x, ξ ) ’ X α ( − i ) α α ! ∂ αξ ∂ αx σ ( x, ξ ) . While the proof of formal expressions is a direct computation, the asymptotic behavior re-quires some care, see [101, 104].An interesting remark is in order: σ P ( x, ξ ) = e − ix · ξ P ( x → e ix · ξ ), thus the dilation ξ → tξ with t > t − m e − itx · ξ P e itx · ξ = t − m σ P ( x, tξ ) ’ t − m X j ≥ σ Pm − j ( x, tξ ) = σ Pm ( x, ξ ) + o ( t − ) . Thus, if P ∈ Ψ DO m ( U ) with m ≥ σ Pm ( x, ξ ) = lim t →∞ t − m e − ith ( x ) P e ith ( x ) , where h ∈ C ∞ ( U ) is (almost) defined by dh ( x ) = ξ. .2 Case of manifolds Let M be a (compact) Riemannian manifold of dimension d . Thanks to Theorem 1.5, thefollowing makes sense: Definition 1.6. Ψ DO m ( M ) is defined as the set of operators P : C ∞ c ( M ) → C ∞ ( M ) suchthat (i) the kernel k P ∈ C ∞ ( M × M ) off the diagonal,(ii) the map : f ∈ C ∞ c (cid:16) φ ( U ) (cid:17) → P ( f ◦ φ ) ◦ φ − ∈ C ∞ (cid:16) φ ( U ) (cid:17) is in Ψ DO m (cid:16) φ ( U ) (cid:17) for every coordinate chart ( U, φ : U → R d ) . Of course, this can be generalized: Definition 1.7. Given a vector bundle E over M , a linear map P : Γ ∞ c ( M, E ) → Γ ∞ ( M, E ) is in Ψ DO m ( M, E ) when k P is smooth off the diagonal, and local expressions are Ψ DO ’swith matrix-valued symbols. The covariance formula implies that σ Pm is independent of the chosen local chart so is globallydefined on the bundle T ∗ M → M and σ Pm is defined for every P ∈ Ψ DO m using overlappingcharts and patching with partition of unity.An important class of pseudodifferential operators are those which are invertible moduloregularizing ones: Definition 1.8. P ∈ Ψ DO m ( M, E ) is elliptic if σ Pm ( x, ξ ) is invertible for all ξ ∈ T M ∗ x , ξ = 0 . This means that | σ P ( x, ξ ) | ≥ c ( x ) | ξ | m for | ξ | ≥ c ( x ) , x ∈ U where c , c are strictly positivecontinuous functions on U .This also means that there exists a parametrix: Lemma 1.9. The following are equivalent:(i) Op ( σ ) ∈ Ψ DO m ( U ) is elliptic.(ii) There exist σ ∈ S − m ( U × R d ) such that σ ◦ σ = 1 or σ ◦ σ = 1 .(iii) Op ( σ ) Op ( σ ) = Op ( σ ) Op ( σ ) = 1 modulo Ψ DO −∞ ( U ) .Thus Op ( σ ) ∈ Ψ DO − m ( U ) is also elliptic. At this point, it is useful to remark that any P ∈ Ψ DO m ( M, E ) can be extended to a boundedoperator on L ( M, E ) when < ( m ) ≤ 0. Of course, this needs an existing scalar product forgiven metrics on M and E . Theorem 1.10. Assume M is compact. If P ∈ Ψ DO − m ( M, E ) is elliptic with < ( m ) > ,then P is compact, so its spectrum is discrete.Proof. We need to get the result first for an open set U , for a manifold M and then for abundle E over M .For any s ∈ R , the usual Sobolev spaces H s ( R d ) (with H ( R d ) = L ( R d )) and H sc ( U )(defined as the union of all H s ( K ) over compact subsets K ⊂ U ) or H sloc ( U ) (defined as theset of distributions u ∈ D ( U ) such that φu ∈ H s ( R d ) for all φ ∈ C ∞ c ( U )) can be extendedfor any manifold M to the Sobolev spaces H sc ( M ) (obvious definition) and H sloc ( M ): if10 U, φ : U → R d ) is a local chart and χ ∈ C ∞ c (cid:16) φ ( U ) (cid:17) , we say that a distribution u ∈ D ( M )is in H sloc ( M ) when (cid:16) φ − (cid:17) ∗ ( ξ u ) ∈ H s ( R d ). When M is compact, H sloc ( M ) = H sc ( M ) (thusdenoted H s ( M ). Using Rellich’s theorem, the inclusion H sc ( U ) , → H tc ( U ) for s < t is compact.Since P : H sc ( M ) → H s −< ( m ) loc ( M ) is a continuous linear map for a (non-necessarily compact)manifold M , both results yield that P is compact. Finally, the extended operator on a bundleis P : L ( M, E ) → H −< ( m ) ( M, E ) , → L ( M, E ) where the second map is the continuousinclusion, so P being compact as an L operator has a discrete spectrum.We rephrase a previous remark (see [4, Proposition 2.1]):Let E be a vector bundle of rank r over M . If P ∈ Ψ DO − m ( M, E ), then for any couple ofsections s ∈ Γ ∞ ( M, E ), t ∗ ∈ Γ ∞ ( M, E ∗ ), the operator f ∈ C ∞ ( M ) → h t ∗ , P ( f s ) i ∈ C ∞ ( M )is in Ψ DO m ( M ). This means that in a local chart ( U, φ ), these operators are r × r matricesof pseudodifferential operators of order − m . The total symbol is in C ∞ ( T ∗ U ) ⊗ End ( E ) with End ( E ) ’ M r ( C ). The principal symbol can be globally defined: σ P − m ( x, ξ ) : E x → E x for x ∈ M and ξ ∈ T ∗ x M , can be seen as a smooth homomorphism homogeneous of degree − m on all fibers of T ∗ M . Moreover, we get the simple formula which could be seen as a definitionof the principal symbol (as already noticed at the end of previous section) σ P − m ( x, ξ ) = lim t →∞ t − m (cid:16) e − ith · P · e ith (cid:17) ( x ) for x ∈ M, ξ ∈ T ∗ x M (4)where h ∈ C ∞ ( M ) is such that d x h = ξ . The question to be solved is to define a homogeneous distribution which is an extension on R d of a given homogeneous symbol on R d \{ } . Such extension is a regularization used forinstance by Epstein–Glaser in quantum field theory.The Schwartz space on R d is denoted by S and the space of tempered distributions by S . Definition 1.11. For f λ ( ξ ) := f ( λξ ) , λ ∈ R ∗ + , define τ ∈ S → τ λ by h τ λ , f i := λ − d h τ, f λ − i for all f ∈ S .A distribution τ ∈ S is homogeneous of order m ∈ C when τ λ = λ m τ . Proposition 1.12. Let σ ∈ C ∞ ( R d \{ } ) be a homogeneous symbol of order k ∈ Z .(i) If k > − d , then σ defines a homogeneous distribution.(ii) If k = − d , there exists a unique obstruction to the extension of σ given by c σ = Z S d − σ ( ξ ) dξ, namely, one can at best extend σ in τ ∈ S such that τ λ = λ − d (cid:16) τ + c σ log( λ ) δ (cid:17) . (5) Proof. ( i ) For k > − d , σ is integrable near zero, increases slowly at ∞ , so defines by extensiona unique distribution τ ∈ S which will be homogeneous of order k .( ii )Assume k = − d . Then σ extends to a continuous linear form L σ ( f ) := R R d f ( ξ ) σ ( ξ ) dξ on S := { f ∈ S | f (0) = 0 } . By Hahn–Banach theorem, L σ extends to S and L σ ∈ E where E := { τ ∈ S | τ |S = L σ } is given by the direction δ .11his affine space E is stable by the endomorphism τ → λ d τ λ : actually if f ∈ S , f λ − ∈ S and λ d h τ λ , f i = h τ, f λ − i = L σ ( f λ − ) = Z R d f ( λ − ξ ) σ ( ξ ) dξ = Z R d f ( ξ ) σ ( ξ ) dξ = L σ ( f ) , thus λ d τ λ = L σ on S .Moreover, λ d ( δ ) λ = δ ; thus there exists c ( λ ) ∈ C such that τ λ = λ − d τ + c ( λ ) λ − d δ (6)for all τ ∈ E . The computation of c ( λ for a specific example in E gives c ( λ ) = c σ log( λ ): forinstance, choose g ∈ C ∞ c ([0 , ∞ ]) which is 1 near 0 and define τ ∈ S by h τ, f i := L σ (cid:16) f − f (0) g ( | · | ) (cid:17) = Z R d (cid:16) f ( ξ ) − f (0) g ( | ξ | ) (cid:17) σ ( ξ ) dξ, ∀ f ∈ S . Thus if f (0) = 1, we get c ( λ ) λ − d h δ , f i = c ( λ ) λ − d , so by (6) c ( λ ) λ − d = h τ, f λ − i − λ − d h τ, f i = Z R d (cid:16) f ( λ − ξ ) − g ( | ξ | ) (cid:17) σ ( ξ ) dξ − λ − d Z R d (cid:16) f ( ξ ) − g ( | ξ | ) σ ( ξ ) dξ = − λ − d Z R d (cid:16) g ( λ | ξ | ) − g ( | ξ | ) σ ( ξ ) dξ = − λ − d c σ Z ∞ (cid:16) g ( λ | ξ | ) − g ( | ξ | ) d | ξ || ξ | with c σ := R S d − σ ( ξ ) d d − ξ . Since λ ddλ Z ∞ (cid:16) g ( λ | ξ | ) − g ( | ξ | ) (cid:17) d | ξ || ξ | = λ Z ∞ g ( λ | ξ | ) d | ξ | = − g (0) = − , we get c ( λ ) = c σ log( λ ). Thus, when c σ = 0, every element of E is a homogeneous distributionon R d which extends the symbol σ .Conversely, let τ ∈ S be a homogeneous distribution extending σ and let ˜ τ ∈ E . Since τ − ˜ τ is supported at the origin, we can write τ = ˜ τ + P | α |≤ N a α ∂ α δ where a α ∈ C and0 = τ λ − λ − d τ = c σ λ − d log( λ ) δ + X ≤| α |≤ N a α λ − d (cid:16) λ | α | − (cid:17) ∂ α δ . The linear independence of ( ∂ α δ ) gives a α = 0 , ∀ a α . So c σ = 0 and τ ∈ E . The condition c σ = 0 is so necessary and sufficient to extend σ in a homogeneous distribution. And in thegeneral case, one can at best extend it in a distribution satisfying (5), but it is only possiblewith elements of E .In the following result, we are interested by the behavior near the diagonal of the kernel k P for P ∈ Ψ DO . For any τ ∈ S , we choose the decomposition as τ = φ ◦ τ + (1 − φ ) ◦ τ where φ ∈ C ∞ c ( R d ) and φ = 1 near 0. We can look at the infrared behavior of τ near theorigin and its ultraviolet behavior near infinity. Remark first that, since φ ◦ τ has a compactsupport, ( φ ◦ τ ) q ∈ S , so the regularity of τ q depends only of its ultraviolet part (cid:16) (1 − φ ) ◦ τ (cid:17) q .12 roposition 1.13. Let P ∈ Ψ DO m ( U ) , m ∈ Z . Then, in local form near the diagonal, k P ( x, y ) = X − ( m + d ) ≤ j ≤ a j ( x, x − y ) − c P ( x ) log | x − y | + O (1) where a j ( x, y ) ∈ C ∞ (cid:16) U × U \{ x } (cid:17) is homogeneous of order j in y and c P ( x ) ∈ C ∞ ( U ) isgiven by c P ( x ) = π ) d Z S d − σ P − d ( x, ξ ) dξ. (7) Proof. We know that σ P ( x, ξ ) ’ P j ≤ m σ Pj ( x, ξ ) and by (2), k P ( x, y ) = ~ σ ξ → y ( x, x − y ) so weneed to control ~ σ ξ → y ( x, x − y ) when y → Assume first that σ P ( x, ξ ) is independent of x : For − d < j ≤ m , σ j ( ξ ) extends to τ j ∈ S . For j > − d , this extension is homogeneous (ofdegree j ) and unique.For j = − d , we may assume that τ − d satisfies (5). Thus τ := σ P − P mj = − d τ j ∈ S behaves inthe ultraviolet as a integrable symbol. In particular τ q is continuous near 0 and we get | σ P ( y ) = m X j = − d q τ j ( y ) + O (1) . (8)Note that the inverse Fourier transform of the infrared part of τ j is in C ∞ ( R d ) while thoseof its ultraviolet part is in C ∞ (cid:16) R d \{ } (cid:17) , so q τ j is smooth near 0.Moreover, for j > − d , q τ j is homogeneous of degree − ( d + j ) while for j = − d , | τ − d ( λy ) = λ − d [ (cid:16) τ − d ) λ − ] q ( y ) = [ τ − d − c σ − d log( λ ) δ ] q ( y ) = | τ − d ( y ) − π ) d c σ − d log λ. For λ = | y | − , we get | τ − d (cid:16) y | y | (cid:17) = | τ − d (cid:16) y | y | (cid:17) − π ) d c σ − d log | y | . Summation over j in (8) yields the result. Assume now that σ P ( x, ξ ) is dependent of x : We do the same with families { τ x } x ∈ U and { τ j,x } x ∈ U . Their ultraviolet behaviors are thoseof smooth symbols on U × R d , so given by smooth functions on U × R d \{ } and for τ x by acontinuous function on U × R d . For the infrared part, we get smooth maps from U to E ( R d ) (distributions with compact support), thus applying inverse Fourier transform, we end upwith smooth functions on U × R d . Actually, for τ j,x with j > − d , this follows from the factthat it is the extension of σ j ( x, · ) which is integrable near the origin: let f ∈ S , h φ ◦ τ j,x , f i = h τ j,x , φ ◦ f i = Z R d φ ( ξ ) f ( ξ ) σ j ( x, ξ ) dξ. While for j = − d , h φ ◦ τ − d,x , f i = h τ − d,x , φ ◦ f i = Z R d φ ( ξ ) (cid:16) f ( ξ ) − f (0) (cid:17) σ − d ( x, ξ ) dξ, x → φ ◦ τ − d,x is smooth from U to E ( R d ) . In conclusion, q σ ξ → y ( x, y ) = X − ( m + d ) ≤ j ≤ a j ( x, y ) − c P ( x ) log | y | + R ( x, y )where a j ( x, y ) is a smooth function on U × R d \{ } , is homogeneous of degree j in y , c P is given by (7) and R ( x, y ) is a function, continuous on U × R d . So we get the claimedasymptotic behavior. Theorem 1.14. Let P ∈ Ψ DO m ( M, E ) with m ∈ Z . Then, for any trivializing local coordi-nates tr (cid:16) k P ( x, y ) (cid:17) = X j = − ( m + d ) a j ( x, x − y ) − c P ( x ) log | x − y | + O (1) , where a j is homogeneous of degree j in y , c P is intrinsically locally defined by c P ( x ) := π ) d Z S d − tr (cid:16) σ P − d ( x, ξ ) (cid:17) dξ. (9) Moreover, c P ( x ) | dx | is a 1-density over M which is functorial with respect to diffeomorphisms φ : c φ ∗ P ( x ) = φ ∗ (cid:16) c p ( x ) (cid:17) . (10) Proof. The asymptotic behavior follows from Proposition 1.13 but we first have to understandwhy c P is well defined: Assume first that E is a trivial line bundle and P is a scalar Ψ DO . Define a change of coordinates by y := φ − ( x ). Thus k P ( x, x ) φ ∗ −→ k φ ∗ P ( y, y ) with k φ ∗ P ( y, y ) = | J φ ( y ) | k P (cid:16) φ ( y ) , φ ( y ) (cid:17) = X j = − ( m + d ) | J φ ( y ) | h a j (cid:16) φ ( y ) , φ ( y ) − φ ( y ) (cid:17) − c P (cid:16) φ ( y ) (cid:17) log | φ ( y ) − φ ( y ) | i + O (1) . A Taylor expansion around (cid:16) φ ( y ) , φ ( y ) · ( y − y ) (cid:17) of a j gives a j (cid:16) φ ( y ) , φ ( y ) − φ ( y ) (cid:17) ’ | y − y | j a j (cid:16) φ ( y ) , φ ( y ) · y − y | y − y | (cid:17) + · · · , since a j (cid:16) φ ( y ) , · (cid:17) is smooth outside 0, so we get only homogeneous and continuous terms.Moreover the only contribution to the log-term is | J φ ( y ) | c P (cid:16) φ ( y ) (cid:17) log | φ ( y ) − φ ( y ) | ’ | J φ ( y ) | c P (cid:16) φ ( y ) (cid:17) log | φ ( y ) − φ ( y ) | + O (1)and we get c φ ∗ P ( y ) = | J φ ( y ) | c P (cid:16) φ ( y ) (cid:17) . In particular c P ( x ) | dx ∧ · · · ∧ dx d | can be globally defined on M as a 1-density. (Recallthat a α -density on a vector space E of dimension n is any application f : V n E → R such14hat for any λ ∈ R , f ( λx ) = | λ | α f ( x ) and the set of these densities is denoted | V | α E ∗ ; thisis generalized to a vector bundle E over M where each fiber is | V | α E ∗ x . The interest of thebundle of 1-densities is to give a class of objects directly integrable on M . In particular, weget here something intrinsically defined, even when the manifold is not oriented). General case: P acts on section of a bundle. By a change of trivialization, the action of P is conjugateon each fiber by a smooth matrix-valued map A ( x ), so k P ( x, x ) → A ( x ) − k P ( x, x ) A ( x ).We are looking for the logarithmic term: only the 0-order term in A ( x ) will contributeand tr (cid:16) A ( x ) − k P ( x, x ) A ( x ) (cid:17) has the same logarithmic singularity than the similar termtr (cid:16) A ( x ) − k P ( x, x ) A ( x ) (cid:17) = tr (cid:16) k P ( x, x ) (cid:17) near the diagonal. Thus c P ( x ) is independent of achosen trivialization.Similarly, if P is not a scalar but End ( E )-valued, the above proof can be generalized (thespace of C ∞ (cid:16) ( M, | V | ( M ) ⊗ End ( E ) (cid:17) of End ( E )-valued densities is a sheaf).Remark that, when M is Riemannian with metric g and d g ( x, y ) is the geodesic distance,then tr (cid:16) k P ( x, y ) (cid:17) = X j = − ( m + d ) a j ( x, x − y ) − c P ( x ) log (cid:16) d g ( x, y ) (cid:17) + O (1) , since there exists c > c − | x − y | ≤ d g ( x, y ) ≤ c | x − y | . The claim is that R M c P ( x ) | dx | is a residue.For this, we embed everything in C . In the same spirit as in Proposition 1.12, one obtainsthe following Lemma 1.15. Every σ ∈ C ∞ (cid:16) R d \{ } (cid:17) which is homogeneous of degree m ∈ C \ Z can beuniquely extended to a homogeneous distribution. Definition 1.16. Let U be an open set in R d and Ω be a domain in C .A map σ : Ω → S m ( U × R d ) is said to be holomorphic whenthe map: z ∈ Ω → σ ( z )( x, ξ ) is analytic for all x ∈ U , ξ ∈ R d ,the order m ( z ) of σ ( z ) is analytic on Ω ,the two bounds of Definition 1.1 ( i ) and ( ii ) of the asymptotics σ ( z ) ’ P j σ m ( z ) − j ( z ) are locally uniform in z . This hypothesis is sufficient to get:The map: z → σ m ( z ) − j ( z ) is holomorphic from Ω to C ∞ (cid:16) U × R d \{ } (cid:17) .The map ∂ z σ ( z )( x, ξ ) is a classical symbol on U × R d and one obtains: ∂ z σ ( z )( x, ξ ) ’ X j ≥ ∂ z σ m ( z ) − j ( z )( x, ξ ) . Definition 1.17. The map P : Ω ⊂ C → Ψ DO ( U ) is said to be holomorphic if it has thedecomposition P ( z ) = σ ( z )( · , D ) + R ( z ) (see definition (1) ) where σ : Ω → S ( U × R d ) and R : Ω → C ∞ ( U × U ) are holomorphic. 15s a consequence, there exists a holomorphic map from Ω into Ψ DO ( M, E ) with a holo-morphic product (when M is compact). Example 1.18. Elliptic operators: Recall that P ∈ Ψ DO m ( U ), m ∈ C , is elliptic if there exist strictly positive continuousfunctions c and C on U such that | σ P ( x, ξ ) | ≥ c ( x ) | ξ | m for ξ | ≥ C ( x ) , x ∈ U . This essentiallymeans that P is invertible modulo smoothing operators. More generally, P ∈ Ψ DO m ( M, E )is elliptic if its local expression in each coordinate chart is elliptic.Let Q ∈ Ψ DO m ( M, E ) with < ( m ) > 0. We assume that M is compact and Q is elliptic.Thus Q has a discrete spectrum and we suppose Spectrum( Q ) ∩ R − = ∅ . Since we want tointegrate in C , we assume that there exists a curve Γ coming from + ∞ along the real axisin the upper half plane, turns around the origin and goes back to infinity in the lower halfplane whose interior contains the spectrum of Q . The curve Γ must avoid branch points of λ s at s = 0 (so the branch of λ s defined in the right half-plane is such that 1 s = 1).Γ O When < ( s ) < Q s := i π R Γ λ s ( λ − Q ) − dλ makes sense as operator on L ( M, E ) (when < ( s ) ≥ 0, define Q s := Q s − k Q k for k ∈ N large enough so < ( s ) − k < 0; there is nounambiguity because Q s Q = Q s +1 for < ( s ) < − 1. Note that if < ( m ) < 0, we can define Q s = ( Q − ) − s and apply previous definition).Actually, Q s ∈ Ψ DO ms ( M, E ) and ( λ − Q ) − = σ ( λ )( · , D ) + R ( λ ) where R ( λ ) is aregularizing operator and σ ( λ )( · , D ) has a symbol smooth in λ such that σ ( λ )( x, ξ ) ’ P j ≥ a − m − j ( λ, x, ξ ) with a n ( λ, x, ξ ) homogeneous of degree n in ( λ /m , ξ ).The map s → Q s is a one-parameter group containing Q = 1 and Q = Q which isholomorphic on < ( s ) ≤ S int of integrable symbols. Usingsame type of arguments as in Proposition 1.12 and Lemma 1.15, one proves Proposition 1.19. Let L : σ ∈ S Z int ( R d ) → L ( σ ) := q σ (0) = π ) d Z R d σ ( ξ ) dξ. Then L has a unique holomorphic extension e L on S C \ Z ( R d ) .Moreover, when σ ( ξ ) ’ P j σ m − j ( ξ ) , m ∈ C \ Z , e L ( σ ) = (cid:16) σ − X j ≤ N τ m − j (cid:17) q (0) = π ) d Z R d (cid:16) σ − X j ≤ N τ m − j (cid:17) ( ξ ) dξ here m is the order of σ , N is an integer with N > < ( m ) + d and τ m − j is the extension of σ m − j of Lemma 1.15. e L is holomorphic extension of L on S C \ Z ( R d ) which is unique since every element of S C \ Z ( R d )is arcwise connected to S int ( R d ) via a holomorphic path within S C \ Z ( R d ).This result has an important consequence here: Corollary 1.20. If σ : C → S ( R d ) is holomorphic and order (cid:16) σ ( s ) (cid:17) = s , then e L (cid:16) σ ( s ) (cid:17) ismeromorphic with at most simple poles on Z and for p ∈ Z , Res s = p e L (cid:16) σ ( s ) (cid:17) = − π ) d Z S d − σ − d ( p )( ξ ) dξ. Proof. Using Lemma 1.15, one proves that if m ( s ) is holomorphic near m ( s ) = p , then e L (cid:16) σ ( s ) (cid:17) is meromorphic near p .Now we look at the singularity near p ∈ Z . In the half plane { < ( s ) < p } , only the infraredpart of τ m − j ( s ) is a problem since its ultraviolet part is holomorphic. For 0 ≤ j ≤ p + m and < ( s ) < p , σ s − j ( s )( ξ ) is integrable near 0 thus defines its unique extension τ s − j ( s ). So, theonly possible singularity near s = p could come from − π ) d Z | ξ |≤ σ s − j ( s )( ξ ) dξ = − π ) d Z t s − j + d − dt Z | ξ |≤ σ s − j ( s )( ξ | ξ | ) d ( ξ | ξ | )= − π ) d s − j + d Z S d − σ s − j ( s )( ξ ) dξ. where we used for the first equality σ s − j ( s )( ξ ) = | ξ | s − j σ s − j ( s )( ξ | ξ | ). Thus, e L (cid:16) σ ( s ) (cid:17) has atmost only simple pole at s = − d + j .We are now ready to get the main result of this section which is due to Wodzicki [111, 112]. Definition 1.21. Let D ∈ Ψ DO ( M, E ) be an elliptic pseudodifferential operator of order 1on a boundary-less compact manifold M endowed with a vector bundle E .Let Ψ DO int ( M, E ) := { Q ∈ Ψ DO C ( M, E ) | < (cid:16) order ( Q ) (cid:17) < − d } be the class of pseudodif-ferential operators whose symbols are in S int , i.e. integrable in the ξ -variable.In particular, if P ∈ Ψ DO int ( M, E ) , then its kernel k P ( x, x ) is a smooth density on thediagonal of M × M with values in End ( E ) .For P ∈ Ψ DO Z ( M, E ) , defineWRes P := Res s =0 Tr (cid:16) P |D| − s (cid:17) . (11)This makes sense because: Theorem 1.22. (i) Let P : Ω ⊂ C → Ψ DO int ( M, E ) be a holomorphic family. Then thefunctional map Tr : s ∈ Ω → Tr( P ( s )) ∈ C has a unique analytic extension on the family Ω → Ψ DO C \ Z ( M, E ) still denoted by Tr .(ii) If P ∈ Ψ DO Z ( M, E ) , the map: s ∈ C → Tr (cid:16) P |D| − s (cid:17) has at most simple poles on Z and WRes P = Z M c P ( x ) | dx | (12) is independent of D . Recall (see Theorem 1.14) that c P ( x ) = π ) d R S d − tr (cid:16) σ P − d ( x, ξ ) (cid:17) dξ. (iii) WRes is a trace on the algebra Ψ DO Z ( M, E ) . roof. ( i ) The map s → Tr (cid:16) P | D | − s (cid:17) is holomorphic on C and connect P ∈ Ψ DO C \ Z ( M, E )to the set Ψ DO int ( M, E ) within Ψ DO C \ Z ( M, E ), so an analytic extension of Tr from Ψ DO int to Ψ DO C \ Z is necessarily unique.( ii ) one apply the above machinery:(1) Notice that Tr is holomorphic on smoothing operator, so, using a partition of unity,we can reduce to a local study of scalar Ψ DO ’s.(2) First, fix s = 0. We are interested in the function L φ ( σ ) := Tr (cid:16) φ σ ( x, D ) (cid:17) with σ ∈ S int ( U × R d ) and φ ∈ C ∞ ( U ). For instance, if P = σ ( · , D ),Tr( φ P ) = Z U φ ( x ) k P ( x, x ) | dx | = π ) d Z U φ ( x ) σ ( x, ξ ) dξ | dx | = Z U φ ( x ) L ( σ ( x, · )) | dx | , so one extends L φ to S C \ Z ( U × R d ) with Proposition 1.19 via e L φ ( σ ) = R U φ ( x ) e L φ (cid:16) σ ( x, · ) (cid:17) | dx | .(3) If now σ ( x, ξ ) = σ ( s )( x, ξ ) depends holomorphically on s , we get uniform bounds in x , thus we get, via Lemma 1.15 applied to e L φ (cid:16) σ ( s )( x, · ) (cid:17) uniformly in x , yielding a naturalextension to e L φ (cid:16) σ ( s ) (cid:17) which is holomorphic on C \ Z .When order( σ ( s )) = s , the map e L φ (cid:16) σ ( s ) (cid:17) has at most simple poles on Z and for each p ∈ Z , Res s = p e L φ (cid:16) σ ( s ) (cid:17) = − π ) d R U R S d − φ ( x ) σ − d ( p )( x, ξ ) dξ | dx | = − R U φ ( x ) c P p ( x ) | dx | wherewe used (9) with P = Op (cid:16) σ p ( x, ξ ) (cid:17) .(4) In the general case, we get a unique meromorphic extension of the usual trace Tr onΨ DO Z ( M, E ) that we still denoted by Tr).When P : C → Ψ DO Z ( M, E ) is meromorphic with order( (cid:16) P ( s ) (cid:17) = s , then Tr (cid:16) P ( s ) (cid:17) has atmost poles on Z and Res s = p Tr (cid:16) P ( s ) (cid:17) = − R M c P ( p ) ( x ) | dx | for p ∈ Z . So we get the claim forthe family P ( s ) := P |D| − s . ( iii ) Let P , P ∈ Ψ DO Z ( M, E ). Since Tr is a trace on Ψ DO C \ Z ( M, E ), we get by ( i ),Tr (cid:16) P P |D| − s (cid:17) = Tr (cid:16) P |D| − s P (cid:17) . Moreover WRes (cid:16) P P (cid:17) = Res s =0 Tr (cid:16) P |D| − s P (cid:17) = Res s =0 Tr (cid:16) P P |D| − s (cid:17) = WRes (cid:16) P P (cid:17) where for the second equality we used (12) so the residue depends only of the value of P ( s )at s = 0.Note that WRes is invariant by diffeomorphism:if φ ∈ Diff( M ) , WRes ( P ) = WRes ( φ ∗ P ) (13)which follows from (10). The next result is due to Guillemin and Wodzicki. Corollary 1.23. The Wodzicki residue WRes is the only trace (up to multiplication by aconstant) on the algebra Ψ DO − N ( M, E ) , when M is connected and d ≥ .Proof. The restriction to d ≥ below. When d = 1, T ∗ M isdisconnected and they are two residues. 18 ) On symbols, derivatives are commutators: [ x j , σ ] = i∂ ξ j σ, [ ξ j , σ ] = − i∂ x j σ. 2) If σ P − d = 0 , then σ P ( x, ξ ) is a finite sum of commutators of symbols: When σ P ’ P j σ Pm − j with m = order ( P ), by Euler’s theorem, d X k =1 ξ k ∂ ξ k σ Pm − j = ( m − j ) σ Pm − j (this is false for m = j !) and d X k =1 [ x k , ξ k σ Pm − j ] = i d X k =1 ∂ ξ k ξ k σ Pm − j = i ( m − j + d ) σ Pm − j . So σ P = P dk =1 [ ξ k τ, x k ] (where τ ’ i P j ≥ m − j + d σ Pm − j and here we need for m − j = − d that σ Pd = 0!).Let T be another trace on Ψ DO Z ( M, E ). Then T ( P ) depends only on σ P − d because T ([ · , · ]) = 0. 3) We have R S d − σ P − d ( x, ξ ) d | ξ | = 0 if and only if σ P − d is sum of derivatives: The if part is direct (less than more !).Only if part: σ P − d is orthogonal to constant functions on the sphere S d − and these arekernels of the Laplacian: ∆ S f = 0 ⇐⇒ df = 0 ⇐⇒ f = cst . Thus ∆ S d − h = σ P − d (cid:22) S d − has a solution h on S d − . If ˜ h ( ξ ) := | ξ | − d +2 h (cid:16) ξ | ξ | (cid:17) is its extension to R d \{ } , then we get∆ R d ˜ h ( ξ ) = | ξ | σ P − d (cid:16) ξ | ξ | (cid:17) = σ P − d ( ξ ) because ∆ R d = r − d ∂ r (cid:16) r d − ∂ r ) + r − ∆ S d − . This meansthat ˜ h is a symbol of order d − ∂ ξ ˜ h is a symbol of order d − 1. As a consequence, σ P − d = P dk =1 ∂ ξ k ˜ h = − i P dk =1 [ ∂ ξ k ˜ h, x k ] is a sum of commutators. 4) End of proof: σ P − d ( x, ξ ) − | ξ | − d Vol( S d − ) c P ( x ) is a symbol of order − d with zero integral, thus is a sum of com-mutators by and T ( P ) = T (cid:16) Op ( | ξ | − d c p ( x ) (cid:17) for all T ∈ Ψ DO Z ( M, E ). In other words,the map µ : f ∈ C ∞ c ( U ) → T (cid:16) Op ( f | ξ | − d ) (cid:17) is linear, continuous and satisfies µ ( ∂ x k f ) = 0because ∂ x k ( f ) | ξ | − d is a commutator if f has a compact support and U is homeomorphic to R d . As a consequence, µ is a multiple of the Lebesgue integral: T ( P ) = µ (cid:16) c P ( x ) (cid:17) = c Z M c P ( x ) | dx | = c WRes ( P ) . Example 1.24. Laplacian on a manifold M : Let M be a compact Riemannian manifold ofdimension d and ∆ be the scalar Laplacian which is a differential operator of order 2. ThenWRes (cid:16) (1 + ∆) − d/ (cid:17) = Vol (cid:16) S d − (cid:17) = π d/ Γ( d/ . Proof. (1 + ∆) − d/ ∈ Ψ DO ( M ) has order − d and its principal symbol σ (1+∆) − d/ − d satisfies σ (1+∆) − d/ − d ( x, ξ ) = (cid:16) g ijx ξ i ξ j (cid:17) − d/ = || ξ || − dx . 19o (12) gives WRes (cid:16) (1 + ∆) − d/ (cid:17) = Z M | dx | Z S d − || ξ || − dx dξ = Z M | dx | q det g x Vol (cid:16) S d − (cid:17) = Vol (cid:16) S d − (cid:17) Z M | dvol g | = Vol (cid:16) S d − (cid:17) . Dixmier trace References for this section: [34, 50, 69, 89, 106, 107].The trace on the operators on a Hilbert space H has an interesting property, it is normal .Recall first that Tr acting on B ( H ) is a particular case of a weight ω acting on a von Neumannalgebra M : it is a homogeneous additive map from positive elements M + := { aa ∗ | a ∈ M } to [0 , ∞ ].A state is a weight ω ∈ M ∗ (so ω ( a ) < ∞ , ∀ a ∈ M ) such that ω (1) = 1.A trace is a weight such that ω ( aa ∗ ) = ω ( a ∗ a ) for all a ∈ M . Definition 2.1. A weight ω is normal if ω (sup α a α ) = sup α ω ( a α ) whenever ( a α ) ⊂ M + isan increasing bounded net. This is equivalent to say that ω is lower semi-continuous with respect to the σ -weak topology. Lemma 2.2. The usual trace Tr is normal on B ( H ) . Remark that the net ( a α ) α converges in B ( H ) and this property looks innocent since a tracepreserves positivity.Nevertheless it is natural to address the question: are all traces (in particular on an arbitraryvon Neumann algebra) normal? In 1966, Dixmier answered by the negative [35] by exhibitingnon-normal, say singular, traces. Actually, his motivation was to answer the following relatedquestion: is any trace ω on B ( H ) proportional to the usual trace on the set where ω is finite?The aim of this section is first to define this Dixmier trace, which essentially meansTr Dix ( T ) “ = ” lim N →∞ N P Nn =0 µ n ( T ), where the µ n ( T ) are the singular values of T orderedin decreasing order and then to relate this to the Wodzicki trace. It is a non-normal trace onsome set that we have to identify. Naturally, the reader can feel the link with the Wodzickitrace via Proposition 1.13. We will see that if P ∈ Ψ DO − d ( M ) where M is a compactRiemannian manifold of dimension d , then,Tr Dix ( P ) = d WRes ( P ) = d Z M Z S ∗ M σ P − d ( x, ξ ) dξ | dx | where S ∗ M is the cosphere bundle on M .The physical motivation is quite essential: we know how P n ∈ N ∗ n diverges and this isrelated to the fact the electromagnetic or Newton gravitational potentials are in r whichhas the same singularity (in one-dimension as previous series). Actually, this (logarithmic-type) divergence appears everywhere in physics and explains the widely use of the Riemannzeta function ζ : s ∈ C → P n ∈ N ∗ n s . This is also why we have already seen a logarithmicobstruction in Theorem 1.14 and define a zeta function associated to a pseudodifferentialoperator P by ζ P ( s ) = Tr (cid:16) P |D| − s (cid:17) in (11).We now have a quick review on the main properties of singular values of an operator. In noncommutative geometry, infinitesimals correspond to compact operators: for T ∈ K ( H )(compact operators), define for n ∈ N µ n ( T ) := inf { k T (cid:22) E ⊥ k | E subspace of H with dim( E ) = n } . µ n ( T ) is nothing else than the( n + 1)th of eigenvalues of | T | sorted in decreasing order. Since lim n →∞ µ n ( T ) = 0, for any (cid:15) > 0, there exists a finite-dimensional subspace E (cid:15) such that (cid:13)(cid:13)(cid:13) T (cid:22) E ⊥ (cid:15) (cid:13)(cid:13)(cid:13) < (cid:15) and this propertybeing equivalent to T compact, T deserves the name of infinitesimal.Moreover, we have following properties: µ n ( T ) = µ n ( T ∗ ) = µ n ( | T | ). T ∈ L ( H ) (meaning k T k := Tr( | T | ) < ∞ ) ⇐⇒ P n ∈ N µ n ( T ) < ∞ . µ n ( A T B ) ≤ k A k µ n ( T ) k B k when A, B ∈ B ( H ). µ N ( U T U ∗ ) = µ N ( T ) when U is a unitary. Definition 2.3. For T ∈ K ( H ) , the partial trace of order N ∈ N is σ N ( T ) := P Nn =0 µ n ( T ) . Remark that k T k ≤ σ N ( T ) ≤ N k T k which implies σ n ’ k·k on K ( H ). Then σ N ( T + T ) ≤ σ N ( T ) + σ N ( T ) ,σ N ( T ) + σ N ( T ) ≤ σ N + N ( T + T ) when T , T ≥ . (14)The proof of the sub-additivity is based on the fact that σ N is a norm on K ( H ). Moreover T ≥ ⇒ σ N ( T ) = sup { Tr( T E ) | E subspace of H with dim( E ) = n } . which implies σ N ( T ) = sup { Tr( k T E k | dim( E ) = n } and gives the second inequality.The norm σ N can be decomposed: σ N ( T ) = inf { k x k + N k y k | T = x + y with x ∈ L ( H ) , y ∈ K ( H ) } . In fact if ˜ σ N is the right hand-side, then the sub-additivity gives ˜ σ N ≥ σ N ( T ). To getequality, let ξ n ∈ H be such that | T | ξ n = µ n ( T ) ξ n and define x N := (cid:16) | T | − µ N ( T ) (cid:17) E N , y N := µ N ( T ) E N + | T | (1 − E N ) where E N := P n Definition 2.4. The partial trace of T of order λ ∈ R + is σ λ ( T ) := inf { k x k + λ k y k | T = x + y with x ∈ L ( H ) , y ∈ K ( H ) } . It interpolates between two consecutive integers since the map: λ → σ λ ( T ) is concave for T ∈ K ( H ) and moreover, it is affine between N and N + 1 because σ λ ( T ) = σ N ( T ) + ( λ − N ) σ N ( T ) , where N = [ λ ] . (15)Thus, as before, σ λ ( T ) + σ λ ( T ) = σ λ + λ ( T + T ) , for λ , λ ∈ R + , ≤ T , T ∈ K ( H ) . We define a real interpolate space between L ( H ) and K ( H ) by L , ∞ := { T ∈ K ( H ) | k T k , ∞ := sup λ ≥ e σ λ ( T )log λ < ∞ } . L p ( H ) is the ideal of operators T such that Tr (cid:16) | T | p (cid:17) < ∞ , so σ λ ( T ) = O ( λ − /p ), we havenaturally L ( H ) ⊂ L , ∞ ⊂ L p ( H ) for p > , (16) k T k ≤ k T k , ∞ ≤ k T k . Lemma 2.5. L , ∞ is a C ∗ -ideal of B ( H ) for the norm k·k , ∞ .Moreover, it is equal to the Macaev ideal L , + := { T ∈ K ( H ) | k T k , + := sup N ≥ σ N ( T )log( N ) < ∞ } . Proof. k·k , ∞ is a norm as supremum of norms. By (15),sup ρ ≥ e σ ρ ( T )log ρ ≤ sup N ≥ sup ≤ α ≤ P N − n =0 µ N ( T ) + αµ N ( T )log( N + α )and L , + ∞ is a left and right ideal of B ( H ) since k A T B k , ∞ ≤ k A k k T k , ∞ k B k for every A, B ∈ B ( H ), T ∈ L , ∞ , and moreover k T k , ∞ = k T ∗ k , ∞ = k | T | k , ∞ .This ideal L , ∞ is closed for k·k , ∞ : this follows from a 3- (cid:15) argument since Cauchy sequencesfor k·k , ∞ are Cauchy sequences for each norm σ λ which are equivalent to k·k .Despite this result, the reader should notice that k·k , ∞ = k·k , + since the norms are onlyequivalent. We begin with a Cesàro mean of σ ρ ( T )log ρ with respect of the Haar measure of the group R ∗ + : Definition 2.6. For λ ≥ e and T ∈ K ( H ) , let τ λ ( T ) := λ Z λe σ ρ ( T )log ρ dρρ . Clearly, σ ρ ( T ) ≤ log ρ k T k , ∞ and τ λ ( T ) ≤ k T k , ∞ , thus the map: λ → τ λ ( T ) is in C b ([ e, ∞ ]).It is not additive on L , ∞ but this defect is under control: τ λ ( T + T ) − τ λ ( T ) − τ λ ( T ) ’ λ →∞ O (cid:16) log (log λ )log λ (cid:17) , when 0 ≤ T , T ∈ L , ∞ . More precisely, using previous results, one get Lemma 2.7. | τ λ ( T + T ) − τ λ ( T ) − τ λ ( T ) | ≤ (cid:16) log 2(2+log log λ )log λ (cid:17) k T + T k , ∞ , when T , T ∈ L , ∞ + . Proof. By the sub-additivity of σ ρ , τ λ ( T + T ≤ τ λ ( T ) + τ λ ( T ) and thanks to (14), we get σ ρ ( T ) + σ ρ ( T ) ≤ σ ρ ( T + T ). Thus τ λ ( T ) + τ λ ( T ) ≤ λ Z λe σ ρ ( T + T )log ρ dρρ ≤ λ Z λ e σ ρ ( T + T )log ρ/ dρρ λ ) | τ λ ( T + T ) − τ λ ( T ) − τ λ ( T ) | ≤ (cid:15) + (cid:15) with (cid:15) := Z λe σ ρ ( T + T )log ρ dρρ − Z λ e σ ρ ( T + T )log ρ/ dρρ ,(cid:15) := Z λ e σ ρ ( T + T ) (cid:16) ρ/ − ρ (cid:17) dρρ . By triangular inequality and the fact that σ ρ ( T + T ) ≤ log ρ k T + T k , ∞ when ρ ≥ e , (cid:15) ≤ Z ee σ ρ ( T + T )log ρ dρρ + Z λλ σ ρ ( T + T )log ρ dρρ ≤ k T + T k , ∞ (cid:16) Z ee dρρ + Z λλ dρρ (cid:17) ≤ k T + T k , ∞ . Moreover, (cid:15) ≤ k T + T k , ∞ Z λ e log ρ (cid:16) ρ/ − ρ (cid:17) dρρ ≤ k T + T k , ∞ Z λ e log ρ/ dρρ ≤ k T + T k , ∞ log(2) log (log λ ) . The Dixmier’s idea was to force additivity: since the map λ → τ λ ( T ) is in C b ([ e, ∞ ]) and λ → (cid:16) log 2(2+log log λ )log λ (cid:17) is in C ([ e, ∞ [), let us consider the C ∗ -algebra A := C b ([ e, ∞ ]) /C ([ e, ∞ [) . If [ τ ( T )] is the class of the map λ → τ λ ( T ) in A , previous lemma shows that [ τ ] : T → [ τ ( T )]is additive and positive homogeneous from L , ∞ + into A satisfying [ τ ( U T U ∗ )] = [ τ ( T )] forany unitary U .Now let ω be a state on A , namely a positive linear form on A with ω (1) = 1.Then, ω ◦ [ τ ( · )] is a tracial weight on L , ∞ + (a map from L , ∞ + to R + which is additive,homogeneous and invariant under T → U T U ∗ ). Since L , ∞ is a C ∗ -ideal of B ( H ), each ofits element is generated by (at most) four positive elements, and this map can be extendedto a map ω ◦ [ τ ( · )] : T ∈ L , ∞ → ω ([ τ ( T )]) ∈ C such that ω ([ τ ( T T )]) = ω ([ τ ( T T )]) for T , T ∈ L , ∞ . This leads to the following Definition 2.8. The Dixmier trace Tr ω associated to a state ω on A := C b ([ e, ∞ ]) /C ([ e, ∞ [) is Tr ω ( · ) := ω ◦ [ τ ( · )] . Theorem 2.9. Tr ω is a trace on L , ∞ which depends only on the locally convex topology of H , not of its scalar product.Proof. We already know that Tr ω is a trace.If h· , ·i is another scalar product on H giving the same topology as h· , ·i , then there exist aninvertible U ∈ B ( H ) with h· , ·i = h U · , U ·i . Let H be the Hilbert space for h· , ·i and Tr ω bethe associated Dixmier trace to a given state ω . Since the singular value of U − T U ∈ K + ( H )are the same of T ∈ K + ( H ), we get L , ∞ ( H ) = L , ∞ ( H ) andTr ω ( T ) = Tr ω ( U − T U ) = Tr ω ( T ) for T ∈ L , ∞ + . wo important points: 1) Note that Tr ω ( T ) = 0 if T ∈ L ( H ) and more generally all Dixmier traces vanish onthe closure for the norm k . k , ∞ of the ideal of finite rank operators. In particular, Dixmiertraces are not normal.2) The C ∗ -algebra A is not separable, so it is impossible to exhibit any state ω ! Despitethe inclusions (16) and the fact that the L p ( H ) are separable ideals for p ≥ L , ∞ is not aseparable.Moreover, as for Lebesgue integral, there are sets which are not measurable. For instance, afunction f ∈ C b ([ e, ∞ ]) has a limit ‘ = lim λ →∞ f ( λ ) if and only if ‘ = ω ( f ) for all state ω . Definition 2.10. The operator T ∈ L , ∞ is said to be measurable if Tr ω ( T ) is independentof ω . In this case, Tr ω is denoted Tr Dix . Lemma 2.11. The operator T ∈ L , ∞ is measurable and Tr ω ( T ) = ‘ if and only if the map λ ∈ R + → τ λ ( T ) ∈ A converges at infinity to ‘ .Proof. If lim τ →∞ τ λ ( T ) = ‘ , then Tr ω ( T ) = ω (cid:16) τ ( T ) (cid:17) = ω ( ‘ ) = ‘ ω (1) = ‘ .Conversely, assume T is measurable and ‘ = Tr ω ( T ) for any state ω . Then we get, ω (cid:16) τ ( T ) − ‘ (cid:17) = Tr ω ( T ) − ‘ = 0. Since the set of states separate the points of A , τ ( T ) = ‘ and lim τ →∞ τ λ ( T ) = ‘ .After Dixmier, the singular (i.e. non normal) traces have been deeply investigated, seefor instance the recent [73, 75, 76], but we do not enter into this framework and technically,we just make the following characterization of measurability: Remark 2.12. If T ∈ K + ( H ) , then T is measurable if and only if lim N →∞ N P Nn =0 µ n ( T ) exists. Actually, if ‘ = lim N →∞ N P Nn =0 µ n ( T ) since Tr ω ( T ) = ‘ for any ω , so Tr Dix = ‘ and theconverse is proved in [74]. Example 2.13. Computation of the Dixmier trace of the inverse Laplacian on the torus: Let T d = R d / π Z d be the d-dimensional torus and ∆ = − P di =1 ∂ x i be the scalar Laplacianseen as unbounded operator on H = L ( T d ). We want to compute Tr ω (cid:16) (1 + ∆) − p (cid:17) for d ≤ p ∈ N ∗ . We use 1 + ∆ to avoid the kernel problem with the inverse. As the followingproof shows, 1 can be replaced by any (cid:15) > (cid:15) .Notice that the functions e k ( x ) := √ π e ik · x with x ∈ T d , k ∈ Z d = ( T d ) ∗ form a basis of H of eigenvectors: ∆ e k = | k | e k . Moreover, for t ∈ R ∗ + , e t Tr (cid:16) e − t (1+∆) (cid:17) = X k ∈ Z d e − t | k | = (cid:16) X k ∈ Z e − tk (cid:17) d . We know that | R ∞−∞ e − tx dx − P k ∈ Z e − tk | ≤ t > 0, and since the first integral is q πt , we get e t Tr (cid:16) e − t (1+∆) (cid:17) ’ t ↓ + (cid:16) πt (cid:17) d/ =: α t − d/ .We will use a Tauberian theorem: µ n (cid:16) (1 + ∆) − d/ (cid:17) ’ n →∞ [ α d/ ] n , see [55] (one needs toestimates the cardinality of the set { k ∈ Z d | | k | ≤ n } , see [50]). Thuslim N →∞ N N X n =0 µ n (cid:16) (1 + ∆) − d/ (cid:17) = α Γ( d/ = π d/ Γ( d/ . 25o (1 + ∆) − d/ is measurable andTr Dix (cid:16) (1 + ∆) − d/ (cid:17) = Tr ω (cid:16) (1 + ∆) − d/ (cid:17) = π d/ Γ( d/ . Since (1 + ∆) − p is traceable for p > d , Tr Dix (cid:16) (1 + ∆) − p (cid:17) = 0.This result has been generalized in Connes’ trace theorem [24]: Theorem 2.14. Let M be a compact Riemannian manifold of dimension d , E a vectorbundle over M and P ∈ Ψ DO − d ( M, E ) . Then, P ∈ L , ∞ , is measurable and Tr Dix ( P ) = d WRes ( P ) . Proof. Since WRes and Tr Dix are traces on Ψ DO − m ( M, E ), m ∈ N , Tr Dix = c WRes for someconstant c using Corollary 1.23. Above example, when compare with Example 1.24, showsthat the c = d . 26 Dirac operator There are several ways to define a Dirac-like operator. The best one is to define Cliffordalgebras, their representations, the notion of Clifford modules, spin c structures on orientablemanifolds M defined by Morita equivalence between the C ∗ -algebras C ( M ) and Γ( C ‘ M )(this approach is more of the spirit of noncommutative geometry). Then the notion of spinstructure and finally, with the notion of spin and Clifford connection, we reach the definitionof a (generalized) Dirac operator.Here we try to bypass this approach to save time.References: a classical book is [71], but I recommend [47]. Here, we follow [88], but seealso [50]. Let ( M, g ) be a compact Riemannian manifold with metric g , of dimension d and E be avector bundle over M . An example is the (Clifford) bundle E = C ‘ T ∗ M where the fiber C ‘ T ∗ x M is the Clifford algebra of the real vector space T ∗ x M for x ∈ M endowed with thenondegenerate quadratic form g .Given a connection ∇ on E , recall that a differential operator P of order m on E is anelement ofDiff m ( M, E ) := Γ (cid:16) M, End ( E ) (cid:17) · V ect { ∇ X · · · ∇ X j | X j ∈ Γ( M, T M ) , j ≤ m } . In particular, Diff m ( M, E ) is a subalgebra of End (cid:16) Γ( M, E ) (cid:17) and the operator P has a prin-cipal symbol σ Pm in Γ (cid:16) T ∗ M, π ∗ End ( E ) (cid:17) where π : T ∗ M → M is the canonical submersionand σ Pm ( x, ξ ) is given by (4). An example: Let E = V T ∗ M . The exterior product and the contraction given on ω, ω j ∈ E by (cid:15) ( ω ) ω := ω ∧ ω ,ι ( ω ) ( ω ∧ · · · ∧ ω m ) := m X j =1 ( − j − g ( ω, ω j ) ω ∧ · · · ∧ c ω j ∧ · · · ∧ ω m suggest the following definition c ( ω ) := (cid:15) ( ω ) + ι ( ω ) and one checks that c ( ω ) c ( ω ) + c ( ω ) c ( ω ) = 2 g ( ω , ω ) id E . (17) E has a natural scalar product: if e , · · · , e d is an orthonormal basis of T ∗ x M , then the scalarproduct is chosen such that e i ∧ · · · ∧ e i p for i < · · · < i p is an orthonormal basis.If d ∈ Diff is the exterior derivative and d ∗ is its adjoint for the deduced scalar product onΓ( M, E ), then their principal symbols are σ d ( ω ) = i(cid:15) ( ω ) , (18) σ d ∗ ( ω ) = − iι ( ω ) . (19)This follows from σ d ( x, ξ ) = lim t →∞ t (cid:16) e − ith ( x ) de ith ( x (cid:17) ( x ) = lim t →∞ t it d x h = i d x h = i ξ where h is such that d x h = ξ , so σ d ( x, ξ ) = i ξ and similarly for σ d ∗ .27ore generally, if P ∈ Diff m ( M ), σ Pm ( dh ) = i m m ! ( ad h ) m ( P ) with ad h = [ · , h ] and σ P ∗ m ( ω ) = σ Pm ( ω ) ∗ where the adjoint P ∗ is for the scalar product on Γ( M, E ) associated toan hermitean metric on E : h ψ, ψ i := R M h ψ ( x ) , ψ ( x ) i x | dx | is a scalar product on the spaceΓ( M, E ). Definition 3.1. The operator P ∈ Diff ( M, E ) is called a generalized Laplacian when itssymbol satisfies σ P ( x, ξ ) = | ξ | x id E x for x ∈ M, ξ ∈ T ∗ x M (note that | ξ | x depends on themetric g ). This is equivalent to say that, in local coordinates, P = − P i,j g ij ( x ) ∂ x i ∂ x j + b j ( x ) ∂ x j + c ( x )where the b j are smooth and c is in Γ (cid:16) M, End ( E ) (cid:17) . Definition 3.2. Assume that E = E + ⊕ E − is a Z -graded vector bundle.When D ∈ Diff ( M, E ) and D = (cid:16) D + D − (cid:17) ( D is odd) where D ± : Γ( M, E ∓ ) → Γ( M, E ± ) , D is called a Dirac operator if D = (cid:16) D − D + D + D − (cid:17) is a generalized Laplacian. A good example is given by E = V T ∗ M = V even T ∗ M ⊕ V odd T ∗ M and the de Rham operator D := d + d ∗ . It is a Dirac operator since D = dd ∗ + d ∗ d is a generalized Laplacian accordingto (18) (19). D is also called the Laplace–Beltrami operator. Definition 3.3. Define C ‘ M as the vector bundle over M whose fiber in x ∈ M is theClifford algebra C ‘ T ∗ x M (or C ‘ T x M using the musical isomorphism X ∈ T M ↔ X [ ∈ T ∗ M ).A bundle E is called a Clifford bundle over M when there exists a Z -graduate action c : Γ( M, C ‘ M ) → End (cid:16) Γ( M, E ) (cid:17) . The main idea which drives this definition is that Clifford actions correspond to principalsymbols of Dirac operators: Proposition 3.4. If E is a Clifford module, every odd D ∈ Diff such that [ D, f ] = i c ( df ) for f ∈ C ∞ ( M ) is a Dirac operator.Conversely, if D is a Dirac operator, there exists a Clifford action c with c ( df ) = − i [ D, f ] .Proof. Let x ∈ M , ξ ∈ T ∗ x M and f ∈ C ∞ ( M ) such that d x f = ξ . Then σ D ( df )( x ) = (cid:16) i ad f ) D = − i [ D, f ] = c ( df ) , so, thanks to Theorem 1.5, σ D ( x, ξ ) = (cid:16) σ D ( x, ξ ) (cid:17) = | ξ | x id E x and D is a generalizedLaplacian.Conversely, if D is a Dirac operator, then we can define c ( df ) := i [ D, f ]. This makessense since D ∈ Diff and for f ∈ C ∞ ( M ), x ∈ M , [ D, f ]( x ) = iσ D ( df )( x ) = iσ D ( x, d x f ) isan endomorphism of E x depending only on d x f . So c can be extended to the whole T ∗ M with c ( x, ξ ) := c ( dh )( x ) = iσ D ( x, ξ ) where h ∈ C ∞ ( M ) is chosen such that ξ = d x h . Themap ξ → c ( x, ξ ) is linear from T ∗ x M to End ( E x ) and c ( x, ξ ) = σ D ( x, ξ ) = σ D ( x, ξ ) = | ξ | x for each ξ ∈ T ∗ x M . Thus c can be extended to an morphism of algebras from C ‘ ( T ∗ x M ) in End ( E x ). This gives a Clifford action on the bundle E .Consider previous example: E = V T ∗ M = V even T ∗ M ⊕ V odd T ∗ M is a Clifford module for c := i ( (cid:15) + ι ) coming from the Dirac operator D = d + d ∗ : by (18) and (19) i [ D, f ] = i [ d + d ∗ , f ] = i (cid:16) iσ d ( df ) − iσ d ∗ ( df ) (cid:17) = − i ( (cid:15) + ι )( df ) . efinition 3.5. Let E be a Clifford module over M . A connection ∇ on E is a Cliffordconnection if for a ∈ Γ( M, C ‘ M ) and X ∈ Γ( M, T M ) , [ ∇ X , c ( a )] = c ( ∇ LCX a ) where ∇ LCX isthe Levi-Civita connection after its extension to the bundle C ‘ M (here C ‘ M is the bundlewith fiber C ‘ T x M ).A Dirac operator D ∇ is associated to a Clifford connection ∇ in the following way: D ∇ := − i c ◦ ∇ , Γ( M, E ) ∇ −→ Γ( M, T ∗ M ⊗ E ) c ⊗ −−→ Γ( M, E ) . where we use c for c ⊗ . Thus if in local coordinates, ∇ = P dj =1 dx j ⊗ ∇ ∂ j , the associated Dirac operator is givenby D ∇ = − i P j c ( dx j ) ∇ ∂ i . In particular, for f ∈ C ∞ ( M ),[ D ∇ , f id E ] = − i d X i =1 c ( dx i ) [ ∇ ∂ j , f ] = d X j =1 − i c ( dx j ) ∂ j f = − ic ( df ) . By Proposition 3.4, D ∇ deserves the name of Dirac operator! Examples: 1) For the previous example E = V T ∗ M , the Levi-Civita connection is indeed a Cliffordconnection whose associated Dirac operator coincides with the de Rham operator D = d + d ∗ .2) The spinor bundle : Recall that the spin group Spin d is the non-trivial two-fold coveringof SO d , so we have 0 −→ Z −→ Spin d ξ −→ SO d −→ . Let SO( T M ) → M be the SO d -principal bundle of positively oriented orthonormal frameson T M of an oriented Riemannian manifold M of dimension d .A spin structure on an oriented d-dimensional Riemannian manifold ( M, g ) is a Spin d -principal bundle Spin( T M ) π −→ M with a two-fold covering map Spin( T M ) η −→ SO( T M )such that the following diagram commutes:Spin( T M ) × Spin d (cid:47) (cid:47) η × ξ (cid:15) (cid:15) Spin( T M ) η (cid:15) (cid:15) π (cid:43) (cid:43) M SO( T M ) × SO d (cid:47) (cid:47) SO( T M ) π (cid:51) (cid:51) where the horizontal maps are the actions of Spin d and SO d on the principal fiber bundlesSpin( T M ) and SO( T M ).A spin manifold is an oriented Riemannian manifold admitting a spin structure.In above definition, one can replace Spin d by the group Spin cd which is a central extension of SO d by T : 0 −→ T −→ Spin cd ξ −→ SO d −→ . An oriented Riemannian manifold ( M, g ) is spin if and only if the second Stiefel–Whitneyclass of its tangent bundle vanishes. Thus a manifold is spin if and only both its first andsecond Stiefel–Whitney classes vanish (the vanishing of the first one being equivalent to theorientability of the manifold). In this case, the set of spin structures on ( M, g ) stands inone-to-one correspondence with H ( M, Z ). In particular the existence of a spin structuredoes not depend on the metric or the orientation of a given manifold. Note that all manifolds29f dimension d ≤ c structures but C P is a 4-dimensional (complex) manifoldwithout spin structures.Let ρ be an irreducible representation of C ‘ C d → End C (Σ d ) with Σ d ’ C b d/ c as set ofcomplex spinors. Of course, C ‘ C d is endowed with its canonical complex bilinear form.The spinor bundle S of M is the complex vector bundle associated to the principal bundleSpin( T M ) with the spinor representation, namely S := Spin( T M ) × ρ d Σ d . Here ρ d is arepresentation of Spin d on Aut(Σ d ) which is the restriction of ρ .More precisely, if d = 2 m is even, ρ d = ρ + + ρ − where ρ ± are two nonequivalent irreduciblecomplex representations of Spin m and Σ m = Σ +2 m ⊕ Σ − m , while for d = 2 m + 1 odd, thespinor representation ρ d is irreducible.In practice, M is a spin manifold means that there exists a Clifford bundle S = S + ⊕ S − such that S ’ V T ∗ M . Due to the dimension of M , the Clifford bundle has fiber C ‘ x M = ( M m ( C ) when d = 2 m is even, M m ( C ) ⊕ M m ( C ) when d = 2 m + 1 . Locally, the spinor bundle satisfies S ’ M × C d/ . A spin connection ∇ S : Γ ∞ ( M, S ) → Γ ∞ ( M, S ) ⊗ Γ ∞ ( M, T ∗ M ) is any connection which iscompatible with Clifford action: [ ∇ S , c ( · )] = c ( ∇ LC · ) . It is uniquely determined by the choice of a spin structure on M (once an orientation of M is chosen). Definition 3.6. The Dirac (also called Atiyah–Singer) operator given by the spin structureis D/ := − i c ◦ ∇ S . (20)In coordinates, D/ = − ic ( dx j ) (cid:16) ∂ j − ω j ( x ) (cid:17) (21)where ω j is the spin connection part which can be computed in the coordinate basis ω j = (cid:16) Γ kji g kl − ∂ i ( h αj ) δ αβ h βl (cid:17) c ( dx i ) c ( dx l )where the matrix H := [ h αj ] is such that H t H = [ g ij ] (we use Latin letters for coordinatebasis indices and Greek letters for orthonormal basis indices).This gives σ D ( x, ξ ) = c ( ξ ) + ic ( dx j ) ω j ( x ). Thus in normal coordinates around x , c ( dx j )( x ) = γ j ,σ D ( x , ξ ) = c ( ξ ) = γ j ξ j (22)where the γ ’s are constant hermitean matrices.A fundamental result concerning a Dirac operator (definition 3.2) is its unique continua-tion property: if ψ satisfies Dψ = 0 and ψ vanishes on an open subset of the smooth manifold30 (with or without boundary), then ψ also vanishes on the whole connected component of M . The Hilbert space of spinors is H = L (cid:16) ( M, g ) , S ) (cid:17) := { ψ ∈ Γ ∞ ( M, S ) | Z M h ψ, ψ i x dvol g ( x ) < ∞ } (23)where we have a scalar product which is C ∞ ( M )-valued. On its domain Γ ∞ ( M, S ), the Diracoperator is symmetric: h ψ, D/ φ i = h D/ ψ, φ i . Moreover, it has a selfadjoint closure (which is D/ ∗∗ ): Theorem 3.7. Let ( M, g ) be an oriented compact Riemannian spin manifold without bound-ary. By extension to H , D/ is essentially selfadjoint on its original domain Γ ∞ ( M, S ) . It isa differential (unbounded) operator of order one which is elliptic. See [50, 71, 106, 107] for a proof.There is a nice formula which relates the Dirac operator D/ to the spinor Laplacian∆ S := − Tr g ( ∇ S ◦ ∇ S ) : Γ ∞ ( M, S ) → Γ ∞ ( M, S ) . Before to give it, we need to fix few notations: let R ∈ Γ ∞ (cid:16) M, V T ∗ M ⊗ End ( T M ) (cid:17) bethe Riemann curvature tensor with components R ijkl := g ( ∂ i , R ( ∂ k , ∂ l ) ∂ j ), the Ricci tensor components are R jl := g ik T ijkl and the scalar curvature is s := g jl R jl . Proposition 3.8. Schrödinger–Lichnerowicz formula: with same hypothesis, D/ = ∆ S + s (24) where s is the scalar curvature of M . The proof is just a lengthy computation (see for instance [50]).We already know via Theorems 1.10 and 3.7 that D/ − is compact so has a discrete spectrum.For T ∈ K + ( H ), we denote by { λ n ( T ) } n ∈ N its spectrum sorted in decreasing order includingmultiplicity (and in increasing order for an unbounded positive operator T such that T − iscompact) and by N T ( λ ) := { λ n ( T ) | λ n ≤ λ } its counting function. Theorem 3.9. With same hypothesis, the asymptotics of the Dirac operator counting func-tion is N | D/ | ( λ ) ∼ λ →∞ d Vol ( S d − ) d (2 π ) d Vol ( M ) λ d where Vol ( M ) = R M dvol .Proof. By Weyl’s theorem, we know the asymptotics of N ∆ ( λ ) for the the scalar Laplacian∆ := − Tr g (cid:16) ∇ T ∗ M ⊗ T ∗ M ◦ ∇ T ∗ M (cid:17) which in coordinates is ∆ = − g ij ( ∂ i ∂ j − Γ kij ∂ k ). It is givenby: N ∆ ( λ ) ∼ λ →∞ Vol( S d − ) d (2 π ) d Vol( M ) λ d/ . For the spinor Laplacian, we get the same formula with an extra factor of Tr(1 S ) = 2 d andProposition 3.8 shows that N D/ ( λ ) has the same asymptotics than N ∆ ( λ ) since s gives riseto a bounded operator.We already encounter such computation in Example 2.13.31 .2 Dirac operators and change of metrics Recall that the spinor bundle S g and square integrable spinors H g defined in (23) dependson the chosen metric g , so we note M g instead of M and H g := L ( M g , S g ) and a naturalquestion is: what happens to a Dirac operator when the metric changes?Let g be another Riemannian metric on M . Since the space of d -forms is one-dimensional,there exists a positive function f g,g : M → R + such that dvol g = f g ,g dvol g .Let I g,g ( x ) : S g → S g the natural injection on the spinors spaces above point x ∈ M whichis a pointwise linear isometry: | I g,g ( x ) ψ ( x ) | g = | ψ ( x ) | g . Let us first see its construction:there always exists a g -symmetric automorphism H g,g of the 2 b d/ c - dimensional vector space T M such that g ( X, Y ) = g ( H g,g X, Y ) for X, Y ∈ T M so define ι g,g X := H − / g,g X . Notethat ι g,g commutes with right action of the orthogonal group O d and can be lifted up to adiffeomorphism P in d -equivariant on the spin structures associated to g and g and this liftis denoted by I g,g (see [6]). This isometry is extended as operator on the Hilbert spaces I g,g : H g → H g with ( I g,g ψ )( x ) := I g,g ( x ) ψ ( x ).Now define U g ,g := q f g,g I g ,g : H g → H g . (25)Then by construction, U g ,g is a unitary operator from H g onto H g : for ψ ∈ H g , h U g ,g ψ , U g ,g ψ i H g = Z M | U g,g ψ | g dvol g = Z M | I g ,g ψ | g f g ,g dvol g = Z M | ψ | g dvol g = h ψ g , ψ g i H g . So we can realize D/ g as an operator D g acting on H g with D g : H g → H g , D g := U − g,g D/ g U g,g . (26)This is an unbounded operator on H g which has the same eigenvalues as D/ g .In the same vein, the k -th Sobolev space H k ( M g , S g ) (which is the completion of the spaceΓ ∞ ( M g , S g ) under the norm k ψ k k = P kj =0 R M |∇ j ψ ( x ) | dx ; be careful, ∇ applied to ∇ ψ isthe tensor product connection on T ∗ M g ⊗ S g etc, see Theorem 1.10) can be transported: themap U g,g : H k ( M g , S g ) → H k ( M g , S g ) is an isomorphism, see [98]. In particular, (after thetransport map U ), the domain of D g and D/ g are the same.A nice example of this situation is when g is in the conformal class of g where we cancompute explicitly D/ g and D g [2, 6, 47, 57]. Theorem 3.10. Let g = e h g be a conformal transformation of g with h ∈ C ∞ ( M, R ) . Thenthere exists an isometry I g,g between the spinor bundle S g and S g such that D/ g I g,g ψ = e − h I g,g (cid:16) D/ g ψ − i d − c g ( grad h ) ψ (cid:17) ,D/ g = e − d +12 h I g,g D/ g I − g,g e d − h ,D g = e − h/ D/ g e − h/ . for ψ ∈ Γ ∞ ( M, S g ) . roof. The isometry X → X := e − h X from ( T M, g ) onto ( T M, g ) defines a principal bundleisomorphism SO g ( T M ) → SO g ( T M ) lifting to the spin level. More precisely, it induces avector-bundle isomorphism I g,g : S g → S g , preserving the pointwise hermitean inner product(i.e. H g,g = e h ), such that e − h c g ( X ) I g,g ψ = c g ( X ) I g,g ψ = I g,g c g ( X ) ψ .For a connection f ∇ compatible with a metric k and without torsion, we have for X, Y, Z in Γ ∞ ( T M ) 2 k ( f ∇ X Y, Z ) = k ([ X, Y ] , Z ) + k ([ Z, Y ] , X ) + k ([ Z, X ] , Y )+ X · k ( Y, Z ) + Y · k ( X, Z ) − Z · k ( X, Y ) (27)which is obtained via k ( f ∇ X Y, Z ) + k ( Y, f ∇ X Z ) = X · k ( Y, Z ) minus two cyclic permutations.The set { e j := e − h e j | ≤ j ≤ d } is a local g -orthonormal basis of T U for g if and onlyif { e j } is a local g -orthonormal basis of T U where U is a trivializing open subset of M .Applying (27) to ∇ LC , we get2 g ( ∇ LCX e i , e j ) = e h g ([ X, e − h e i ] , e − h e j ) + e h g ([ e − h e j , e − h e i ] , X ) + e h g ([ e − h e j , X ] , e − h e i )+ X · e h g ( e − h e i , e − h e j ) + e − h e i · e h g ( X, e − h e j ) − e − h e j · e h g ( X, e − h e i )= 2 g ( ∇ LCX e i , e j ) + 2( e i · h ) g ( e j , X ) − e j · h ) g ( e i , X ) . Since ∇ S g X ψ = − g ( ∇ LCX e i , e j ) c g ( e i ) c g ( e j ) ψ , for ψ ∈ Γ ∞ ( U, S g ), ∇ S g e k I g,g ψ = I g,g h ∇ S g e k + c g ( e k ) c g (grad h ) − e k ( h ) i ψ. (28)Hence D/ g I g,g ψ = − ic g ( e k ) ∇ S g e k I g,g ψ = − ie − h c g ( e k ) ∇ S g e k I g,g ψ = − ie − h c g ( e k ) I g,g h ∇ S g e k + c g ( e k ) c g (grad h ) − e k ( h ) i ψ = − ie − h I g,g c g ( e k ) h ∇ S g e k + c g ( e k ) c g (grad h ) − e k ( h ) i ψ = e − h I g,g h D/ g ψ − i d − c g (grad h ) i ψ. So, using [ D/ g , f ] = − ic g (grad f ) for f = e − d − h , D/ g e − d − h I g,g ψ = e − h I g,g h D/ g e − d − h ψ − i d − e − d − h c g (grad h ) ψ i = e − h I g,g h e − d − h D/ g ψ + [ D/ g , e − d − h ] ψ − i d − e − d − h c g (grad h ) ψ i = e − h I g,g h e − d − h D/ g ψ − d − e − d − h ( − i ) c g (grad h ) ψ − i d − e − d − h c g (grad h ) ψ i = e − d +12 h I g,g D/ g ψ. Thus D/ g = e − d +12 h I g,g D/ g I − g,g e d − h and since dvol g = e dh dvol g , using (25) D/ g = e − h/ U − g ,g D/ g U g ,g e − h/ . Finally, (26) yields D g = e − h/ D/ g e − h/ . 33ote that D g is not a Dirac operator as defined in (20) since its principal symbol has an x -dependence: σ D g ( x, ξ ) = e − h ( x ) c g ( ξ ).The principal symbols of D/ g and D/ g are related by σ D/ g d ( x, ξ ) = e − h ( x ) / U − g ,g ( x ) σ D/ g d ( x, ξ ) U g ,g ( x ) e − h ( x ) / , ξ ∈ T ∗ x M. Thus c g ( ξ ) = e − h ( x ) U − g ,g ( x ) c g ( ξ ) U g ,g ( x ) , ξ ∈ T ∗ x M. (29)Using c g ( ξ ) c g ( η ) + c g ( η ) c g ( ξ ) = 2 g ( ξ, η ) id S g , formula (29) gives a verification of the formula g ( ξ, η ) = e − h g ( ξ, η ).Note that two volume forms µ, µ on a compact connected manifold M are related byan orientation preserving diffeomorphism α of M in the following sense [81]: there existsa constant c = ( R M µ ) − R M µ such that µ = c α ∗ µ where α ∗ µ is the pull-back of µ (i.e. R α ( S ) α ∗ µ = R S µ for any set S ⊂ M ). The proof is based on the construction of an orientationpreserving automorphism homotopic to the identity.It is also natural to look at the changes on a Dirac operator when the metric g is modifiedby a diffeomorphism α which preserves the spin structure. The diffeomorphism α can be liftedto a diffeomorphism O d -equivariant on the O d -principal bundle of g -orthonormal frames with˜ α := H − / α ∗ g,g T α , and this lift also exists on S g when α preserves both the orientation and thespin structure. However, the last lift is defined up to a Z -action which disappears if α isconnected to the identity.The pull-back g := α ∗ g of the metric g is defined by ( α ∗ g ) x ( ξ, η ) = g α ( x ) ( α ∗ ( ξ ) , α ∗ η ), x ∈ M , where α ∗ is the push-forward map : T x M → T α ( x ) M . Of course, the metric g and g are different but the geodesic distances are the same. Let us check that d g = α ∗ d g :In local coordinates, we note ∂ µ := ∂/∂x µ and ∂ µ := ∂/∂ ( α ( x )) µ . Thus ∂ = (Λ − T ) ∂ where Λ µ µ := ∂ ( α ( x )) µ /∂x µ . The dependence in the metric g of Cristoffel symbols isΓ ρµν = g ρβ ( ∂ µ g βν + ∂ ν g µβ − ∂ β g µν ). Thus the same symbols Γ associated to g areΓ ρ µ ν = Λ ρ ρ (Λ − T ) µ µ (Λ − T ) ν ν Γ ρµν + Λ ρ ρ (Λ − T ) µ µ ∂ µ (Λ − T ) ν ν . (30)The geodesic equation is ¨ x ρ + Γ ρµν ˙ x µ ˙ x ν = 0 for all ρ (note that neither x µ nor ¨ x µ are 4-vectorsin the sense that they are not transformed like v µ = Λ µ µ v µ , while ˙ x µ is a 4-vector; in fact¨ α ( x ) ρ = Λ ρ ρ ¨ x ρ + ∂ µ Λ ρ ρ ˙ x µ ˙ x ρ . This relation and (30) give the invariance of the geodesicequation and the same for the distance since for any path γ joining points x = γ (0) , y = γ (1) Z r ( α ∗ g ) γ ( t ) (cid:16) γ ( t ) , γ ( t ) (cid:17) dt = Z r g α ◦ γ ( t ) (cid:16) ( α ◦ γ ) ( t ) , ( α ◦ γ ) ( t ) (cid:17) dt and ( α ◦ γ )(0) = α ( x ) , ( α ◦ γ )(1) = α ( y ). Note that α is an isometry only if α ∗ d g = d g .Recall that the principal symbol of a Dirac operator D is σ Dd ( x, ξ ) = c g ( ξ ) so gives themetric g by (17) as we checked above. This information will be used later in the definitionof a spectral triple. A commutative spectral triple associated to a manifold generates theso-called Connes’ distance which is nothing else but the metric distance; see the remark after(44). Again, the link between d α ∗ g and d g is explained by (26), since the unitary induces anautomorphism of the C ∗ -algebra C ∞ ( M ). 34 Heat kernel expansion References for this section: [4, 44, 45] and especially [109].Recall that the heat kernel is a Green function of the heat operator e t ∆ (recall that − ∆ isa positive operator) which measures the temperature evolution in a domain whose boundaryhas a given temperature. For instance, the heat kernel of the Euclidean space R d is k t ( x, y ) = πt ) d/ e −| x − y | / t for x = y (31)and it solves the heat equation ( ∂ t k t ( x, y ) = ∆ x k t ( x, y ) , ∀ t > , x, y ∈ R d initial condition: lim t ↓ k t ( x, y ) = δ ( x − y ) . Actually, k t ( x, y ) = π R ∞−∞ e − ts e is ( x − y ) ds when d = 1.Note that for f ∈ D ( R d ), we have lim t ↓ R R d k t ( x, y ) f ( y ) dy = f ( x ).For a connected domain (or manifold with boundary with vector bundle V ) U , let λ n be theeigenvalues for the Dirichlet problem of minus the Laplacian ( − ∆ φ = λψ in Uψ = 0 on ∂U. If ψ n ∈ L ( U ) are the normalized eigenfunctions, the inverse Dirichlet Laplacian ∆ − is aselfadjoint compact operator, 0 ≤ λ ≤ λ ≤ λ ≤ · · · , λ n → ∞ .The interest for the heat kernel is that, if f ( x ) = R ∞ dt e − tx φ ( x ) is the Laplace transformof φ , then Tr (cid:16) f ( − ∆) (cid:17) = R ∞ dt φ ( t ) Tr (cid:16) e t ∆ (cid:17) (if everything makes sense) is controlled byTr (cid:16) e t ∆ (cid:17) = R M dvol ( x ) tr V x k t ( x, x ) since Tr (cid:16) e t ∆ (cid:17) = P ∞ n =1 e t λ n and k t ( x, y ) = h x, e t ∆ y i = ∞ X n,m =1 h x, ψ m i h ψ m , e t ∆ ψ n i h ψ n , y i = ∞ X n =1 ψ n ( x ) ψ n ( y ) e t λ n . So it is useful to know the asymptotics of the heat kernel k t on the diagonal of M × M especially near t = 0. Let now M be a smooth compact Riemannian manifold without boundary, V be a vectorbundle over M and P ∈ Ψ DO m ( M, V ) be a positive elliptic operator of order m > 0. If k t ( x, y ) is the kernel of the heat operator e − tP , then the following asymptotics exits on thediagonal: k t ( x, x ) ∼ t ↓ + ∞ X k =0 a k ( x ) t ( − d + k ) /m which means that (cid:12)(cid:12)(cid:12) k t ( x, x ) − n X k =0 a k ( x ) t ( − d + k ) /m (cid:12)(cid:12)(cid:12) ∞ ,n < c n t n for 0 < t < | f | ∞ ,n := sup x ∈ M P | α |≤ n | ∂ αx f | (since P is elliptic, k t ( x, y ) is a smooth function of( t, x, y ) for t > 0, see [44, section 1.6, 1.7]).More generally, we will use k ( t, f, P ) := Tr (cid:16) f e − tP (cid:17) where f is a smooth function. We have similarly k ( t, f, P ) ∼ t ↓ + ∞ X k =0 a k ( f, P ) t ( − d + k ) /m . (32)The utility of function f will appear later for the computation of coefficients a k .The following points are of importance:1) The existence of this asymptotics is non-trivial [44, 45].2) The coefficients a k ( f, P ) can be computed locally as integral of local invariants:Recall that a locally computable quantity is the integral on the manifold of a local frame-independent smooth function of one variable, depending only on a finite number of derivativesof a finite number of terms in the asymptotic expansion of the total symbol of P .In noncommutative geometry, local generally means that it is concentrated at infinity inmomentum space.3) The odd coefficients are zero: a k +1 ( f, P ) = 0.For instance, let us assume from now on that P is a Laplace type operator of the form P = − ( g µν ∂ µ ∂ ν + A µ ∂ µ + B ) (33)where ( g µν ) ≤ µ,ν ≤ d is the inverse matrix associated to the metric g on M , and A µ and B aresmooth L ( V )-sections on M (endomorphisms) (see also Definition 3.1). Then (see [45, Lemma1.2.1]) there is a unique connection ∇ on V and a unique endomorphism E such that P = − (Tr g ∇ + E ) , ∇ ( X, Y ) := [ ∇ X , ∇ Y ] − ∇ ∇ LCX Y ,X, Y are vector fields on M and ∇ LC is the Levi-Civita connection on M . LocallyTr g ∇ := g µν ( ∇ µ ∇ ν − Γ ρµν ∇ ρ )where Γ ρµν are the Christoffel coefficients of ∇ LC . Moreover (with local frames of T ∗ M and V ), ∇ = dx µ ⊗ ( ∂ µ + ω µ ) and E are related to g µν , A µ and B through ω ν = g νµ ( A µ + g σε Γ µσε id V ) , (34) E = B − g νµ ( ∂ ν ω µ + ω ν ω µ − ω σ Γ σνµ ) . (35)In this case, the coefficients a k ( f, P ) = R M dvol g tr V (cid:16) f ( x ) a k ( P )( x ) (cid:17) and the a k ( P ) = c i α ik ( P )are linear combination with constants c i of all possible independent invariants α ik ( P ) of di-mension k constructed from E, Ω , R and their derivatives (Ω is the curvature of the connection ω , and R is the Riemann curvature tensor). As an example, for k = 2, E and s are the onlyindependent invariants.Point 3) follow since there is no odd-dimension invariant.36 .2 Computations of heat kernel coefficients The computation of coefficients a k ( f, P ) is made by induction using first a variational method:for any smooth functions f, h one has dd(cid:15) | (cid:15) =0 a k (1 , e − (cid:15)f P ) = ( d − k ) a k ( f, P ) , (36) dd(cid:15) | (cid:15) =0 a k (1 , P − (cid:15)h ) = a k − ( h, P ) , (37) dd(cid:15) | (cid:15) =0 a d − ( e − (cid:15)f h, e − (cid:15)f P ) = 0 . (38)The first equation follows from dd(cid:15) | (cid:15) =0 Tr (cid:16) e − e − (cid:15)f tP (cid:17) = − t ddt Tr (cid:16) f e − tP (cid:17) with an expansion in power series in t . Same method for (37).For the proof of (38), we use P ( (cid:15), δ ) := e − f ( P − δh ); with (36) for k = d ,0 = dd(cid:15) | (cid:15) =0 a d (cid:16) , P ( (cid:15), δ ) (cid:17) , thus after a variation of δ ,0 = ddδ | δ =0 dd(cid:15) | (cid:15) =0 a d (cid:16) , P ( (cid:15), δ ) (cid:17) = dd(cid:15) | (cid:15) =0 ddδ | δ =0 a d (cid:16) , P ( (cid:15), δ ) (cid:17) , we derive (38) from (37).The idea behind equations (36), (37) and (38) is that (37) shows dependence of coefficients a k on E , while the two others describe their behaviors under local scale transformations.Then, the a k ( P ) = c i α ik ( P ) are computed with arbitrary constants c i (they are dependentonly of the dimension d ) and these constants are inductively calculated using (36), (37) and(38). If s is the scalar curvature and ‘;’ denote multiple covariant derivative with respect toLevi-Civita connection on M , one finds, with rescaled α ’s, a ( f, P ) = (4 π ) − d/ Z M dvol g tr V ( α f ) ,a ( f, P ) = (4 π ) − d/ Z M dvol g tr V h ( f ( α E + α s ) i , (39) a ( f, P ) = (4 π ) − d/ Z M dvol g tr V h f ( α E ; kk + α Es + α E + α R ; kk + α s + α R ij R ij + α R ijkl R ijkl + α Ω ij Ω ij ) i . In a , they are no other invariants: for instance, R ij ; ij is proportional to R ; ij .Using the scalar Laplacian on the circle, one finds α = 1.Using (37) with k = 2, under the change P → P − (cid:15)h , E becomes E + (cid:15)h , so Z M dvol g tr V ( α h ) = Z M dvol g tr V ( h )yielding α = 6. For k = 4, it gives now: Z M dvol g tr V ( α hs + 2 α hE ) = Z M dvol g tr V ( α hE + α hs ) , α = 180 and α = 60 α .To go further, one considers the scale transformation on P given in (36) and (38). In (36), P is transformed covariantly, the metric g is changed into e − (cid:15)f g implying conformal trans-formation of the Riemann tensor, Ricci tensor and scalar curvature giving the modificationson ω and E via (34), (35). This gives (we collect here all terms appearing in a and only fewterms appearing in a ) dd(cid:15) | (cid:15) =0 dvol g = d f dvol g , dd(cid:15) | (cid:15) =0 E = − f E + ( d − f ; ii , dd(cid:15) | (cid:15) =0 s = − f s − d − f ; ii , dd(cid:15) | (cid:15) =0 Es = − f Es + ( d − s f ; ii − d − f ; ii E , dd(cid:15) | (cid:15) =0 E = − f E + ( d − f ; ii E , dd(cid:15) | (cid:15) =0 s = − f s − d − f ; ii s , dd(cid:15) | (cid:15) =0 R ijkl = − f R ijkl + δ jl f ; ik + δ ; ik f ; jl − δ il f ; jk − δ jk f ; il , dd(cid:15) | (cid:15) =0 Ω ij Ω ij = − f Ω ij Ω ij , · · · Applying (38) with d = 4, we get dd(cid:15) | (cid:15) =0 a ( e − (cid:15)f h, e − (cid:15)f P ) = 0 . Picking terms with R M dvol g tr V ( hf ; ii ), we find α = 6 α , so α = 1 and α = 60. Thus a ( f, P ) has been determined.Similar method gives a ( f, P ), but only after lengthy computation despite the use of Gauss–Bonnet theorem for the determination of α ! One finds: α = 60 , α = 180 , α = 12 , α = 5 , α = − , α = 2 , α = 30 . The coefficient a was computed by Gilkey, a by Amsterdamski, Berkin and O’Connorand a in 1998 by van de Ven [110]. Some higher coefficients are known in flat spaces. Wodzicki has proved that, in (32), a k ( P )( x ) = m c P ( k − d ) /m ( x ) is true not only for k = 0 asseen in Theorem 2.14 (where P ↔ P − ), but for all k ∈ N . In this section, we will prove thisresult when P is is the inverse of a Dirac operator and this will be generalized in the nextsection.Let M be a compact Riemannian manifold of dimension d even, E a Clifford module over M and D be the Dirac operator (definition 3.2) given by a Clifford connection on E . ByTheorem 3.7, D is a selfadjoint (unbounded) operator on H := L ( M, S ).We are going to use the heat operator e − tD since D is related to the Laplacian via theSchrödinger–Lichnerowicz formula (3.2) and since the asymptotics of the heat kernel of thisLaplacian is known. 38or t > 0, we have e − tD ∈ L : the result follows from the decomposition e − tD = (1 + D ) ( d +1) / e − tD (1 + D ) − ( d +1) / , since (1 + D ) − ( d +1) / ∈ L and the function: λ → (1 + λ ) ( d +1) / e − tλ is bounded.Thus Tr (cid:16) e − tD (cid:17) = P n e − tλ n < ∞ .Another argument is the following: (1 + D ) − d/ maps L ( M, S ) into the Sobolev space H k ( M, S ) (see Theorem 1.10) and the injection H k ( M, S ) , → L ( M, S ) is Hilbert–Schmidtoperator for k > d . Thus t → e − tD is a semigroup of Hilbert–Schmidt operators for t > e − tD has a smooth kernel since it is regularizing, see Remark 1.3(or [71, Theorem 6.2]) and the asymptotics of its kernel is (recall (31)), see [4, Theorem 2.30]: k t ( x, y ) ∼ t ↓ + πt ) d/ q det g x X j ≥ k j ( x, y ) t j e − d g ( x,y ) / t where k j is a smooth section on E ∗ ⊗ E . ThusTr (cid:16) e − tD (cid:17) ∼ t ↓ + X j ≥ t ( j − d ) / a j ( D ) (40)with for j ∈ N , ( a j ( D ) := π ) d/ R M tr (cid:16) k j ( x, x ) (cid:17) √ det g x | dx | ,a j +1 ( D ) = 0 . The aim now is to compute WRes (cid:16) D − p (cid:17) for an integer p such that 0 ≤ p ≤ d : Theorem 4.1. For any integer p , ≤ p ≤ d , D − p ∈ Ψ DO − p ( M, E ) andWRes (cid:16) D − p (cid:17) = 0 , for odd p, WRes (cid:16) D − p (cid:17) = p/ a d − p ( D ) = π ) d/ Γ( p/ Z M tr (cid:16) k ( d − p ) / ( x, x ) (cid:17) dvol g ( x ) , for even p. Proof. Assume D is invertible, otherwise swap D for the invertible operator D + P where P is the projection on the kernel of D . Since the kernel is finite dimensional, P has a finiterank and generates a smoothing operator.Since the trace of γ -matrices in (22) is zero, WRes (cid:16) D − p (cid:17) = 0 for p odd.Assuming now that p is an even integer, by spectral theory, D − p = | D | − p = p/ Z ∞ t p/ e − t D t − dt = p/ ( Z (cid:15) + Z ∞ (cid:15) ) t p/ e − t D t − dt The second integral is a smooth operator since the map x → R ∞ (cid:15) t p/ e − tx t − dt is in theSchwartz space S .Define the first integral as the operator D − p(cid:15) and choose (cid:15) small enough such that for 0 < t ≤ (cid:15) and x and y close enough, | k t ( x, y ) − πt ) d/ ( d − p ) / X j =0 t j q det g x k j ( x, y ) e − d g ( x,y ) / t | ≤ c t p/ e − d g ( x,y ) / t . d ) tr (cid:16) k D − p(cid:15) ( x, y ) (cid:17) = Z ∞ t p/ tr (cid:16) k t ( x, y ) (cid:17) t − dt = √ det g x (4 π ) d/ ( d − p ) / X j =0 tr (cid:16) k j ( x, y ) (cid:17) Z (cid:15) t j − ( p − d ) / e − d g ( x,y ) / t dt + O (cid:16) Z (cid:15) e − d g ( x,y ) / t dt (cid:17) . For m integer and µ > 0, we get after a change of variable t → t − , Z (cid:15) t m e − µ/t t − dt = µ m Z ∞ µ(cid:15) t − m e − t t − dt = Polynomial in µ + O (1) for m < , − log µ + O (1) for m = 0 , O (1) for m > . Thus, the logarithmic behavior of Γ( d ) tr (cid:16) k D − p(cid:15) ( x, y ) (cid:17) comes from √ det g x (4 π ) d/ tr (cid:16) k ( d − p ) / ( x, y ) (cid:17) Z (cid:15) e − d g ( x,y ) / t t − dt = √ det g x (4 π ) d/ tr (cid:16) k ( d − p ) / ( x, y ) (cid:17) (cid:16) − log (cid:16) d g ( x, y ) / (cid:17) + O (1) (cid:17) = √ det g x (4 π ) d/ tr (cid:16) k ( d − p ) / ( x, y ) (cid:17) (cid:16) − (cid:16) d g ( x, y ) (cid:17) + O (1) (cid:17) . Thus (see Theorem 1.14 for the sign) WRes ( (cid:16) D − p (cid:17) = WRes (cid:16) D − p(cid:15) (cid:17) = Z M c D − p(cid:15) ( x ) | dx | = π ) d/ Z M tr (cid:16) k ( d − p ) / ( x, x ) (cid:17)q det g x | dx | , which is, by definition, p/ a d − p ( D ).Few remarks are in order:1) If p = d is even, WRes (cid:16) D − d (cid:17) = p/ a ( D ) = p/ Rank ( E )(4 π ) d/ Vol( M ).Since Tr( e − tD ) ∼ t ↓ + a ( D ) t − d/ , the Tauberian theorem used in Example 2.13 implies that D − d = ( D − ) d/ is measurable and we obtain Connes’ trace theorem 2.14Tr Dix ( D − d ) = Tr ω ( D − d ) = a ( D )Γ( d/ = d WRes ( D − d ) . 2) When D = D/ and E is the spinor bundle, the Seeley-deWit coefficient a ( D/ ) (see(39) with f = 1) can be easily computed (see [44, 50]): if s is the scalar curvature, a ( D/ ) = − π ) d/ Z M s ( x ) dvol g ( x ) . (41)So WRes (cid:16) D/ − d +2 (cid:17) = d/ − a ( D/ ) = c R M s ( x ) dvol g ( x ). This is a quite important resultsince this last integral is nothing else but the Einstein–Hilbert action (71). In dimension 4,this is an example of invariant by diffeomorphisms, see (13).40 Noncommutative integration We already saw that the Wodzicki residue is a trace and, as such, can be viewed as an integral.But of course, it is quite natural to relate this integral to zeta functions used in (11): withnotations of Section 1.4, let P ∈ Ψ DO Z ( M, E ) and D ∈ Ψ DO ( M, E ) which is elliptic. Thedefinition of zeta function ζ PD ( s ) := Tr (cid:16) P | D | − s (cid:17) has been useful to prove that WRes P = Res s =0 ζ PD ( s ) = R M c P ( x ) | dx | .The aim now is to extend this notion to noncommutative spaces encoded in the notion ofspectral triple.References: [25, 31, 34, 37, 50]. The main properties of a compact spin Riemannian manifold M can be recaptured usingthe following triple ( A = C ∞ ( M ) , H = L ( M, S ) , D/ ). The coordinates x = ( x , · · · , x d ) areexchanged with the algebra C ∞ ( M ), the Dirac operator D gives the dimension d as we sawin Theorem 3.9, but also the metric of M via Connes formula and more generally generatesa quantized calculus. The idea of noncommutative geometry is to forget about the commu-tativity of the algebra and to impose axioms on a triplet ( A , H , D ) to generalize the aboveone in order to be able to obtain appropriate definitions of important notions: pseudodiffer-ential operators, measure and integration theory, KO -theory, orientability, Poincaré duality,Hochschild (co)homology etc.An important remark, probably due to Atiyah, is that the commutator of a pseudodiffer-ential operator of order 1 (resp. order 0) with the multiplication by a function is a boundedoperator (resp. compact). This is at the origin of the notion of Fredholm module (or K-cycle)with its K-homology class and via duality to its K-theory culminating with the Kasparov KK-theory. Thus, it is quite natural to define (unbounded) Fredholm module since for instance D/ is unbounded: Definition 5.1. A spectral triple ( A , H , D ) is the data of an involutive (unital) algebra A with a faithful representation π on a Hilbert space H and a selfadjoint operator D with compactresolvent (thus with discrete spectrum) such that [ D , π ( a )] extends to a bounded operator forany a ∈ A . We could impose the existence of a C ∗ -algebra A such that A := { a ∈ A | [ D , π ( a )] is bounded } is norm dense in A so A is a pre- C ∗ -algebra stable by holomorphic calculus. Such A is alwaysa ∗ -subalgebra of A .When there is no confusion, we will write a instead of π ( a ).We now give useful definitions: Definition 5.2. Let ( A , H , D ) be a spectral triple.It is even if there is a grading operator χ such that χ = χ ∗ , [ χ, π ( a )] = 0 , ∀ a ∈ A and D χ = − χ D . t is real of KO-dimension d ∈ Z / if there is an antilinear isometry J : H → H suchthat J D = (cid:15) D J, J = (cid:15) , J χ = (cid:15) χJ with the following table for the signs (cid:15), (cid:15) , (cid:15) d 0 1 2 3 4 5 6 7 (cid:15) (cid:15) (cid:15) (42) and the following commutation rules [ π ( a ) , π ( b ) ◦ ) = 0 , h [ D , π ( a )] , π ( b ) ◦ i = 0 , ∀ a, b ∈ A (43) where π ( a ) ◦ := J π ( a ∗ ) J − is a representation of the opposite algebra A ◦ .It is d -summable (or has metric dimension d ) if the singular values of D behave like µ n ( D − ) = O ( n − /d ) .It is regular if A and [ D , A ] are in the domain of δ n for all n ∈ N where δ ( T ) := [ |D| , T ] . (Recall that the domain of the unbounded derivation δ is the set of all bounded operators T on H which map Dom( |D| ) ⊂ H into itself and [ |D| , T ] can be (uniquely) extended to a boundedoperator.)It satisfies the finiteness condition if the space of smooth vectors H ∞ := T k Dom D k is afinitely projective left A -module.It satisfies the orientation condition if there is a Hochschild cycle c ∈ Z d ( A , A ⊗ A ◦ ) suchthat π D ( c ) = χ , where π D (cid:16) ( a ⊗ b ◦ ) ⊗ a ⊗ · · · ⊗ a d (cid:17) := π ( a ) π ( b ) ◦ [ D , π ( a )] · · · [ D , π ( a d )] and d is its metric dimension. The above definition of KO -dimension comes from the fact that a Dirac operator is asquare root a Laplacian. This generates a sign problem which corresponds to a choice of aspin structure (or orientation). Up to some subtleties, the choice of a manifold of a chosenhomotopy needs a Poincaré duality between homology and cohomology and the necessaryrefinement yields to the KO -homology introduced by Atiyah and Singer.An interesting example of noncommutative space of non-zero KO -dimension is given bythe finite part of the noncommutative standard model [21, 28, 31].Moreover, the reality (or charge conjugation in the commutative case) operator J is relatedto the problem of the adjoint: If M is a von Neumann algebra acting on the Hilbert space H with a cyclic and separating vector ξ ∈ H (which means M ξ is dense in H and aξ = 0 implies a = 0, for a ∈ M ), then the closure S of the map: aξ → a ∗ ξ has an unbounded extensionto H with a polar decomposition S = J ∆ / where ∆ := S ∗ S is a positive operator and J isantilinear operator such that J M J − = M , see Tomita theory in [103]. This explains thecommutation relations (43). Moreover ∆ it M ∆ − it = M , a point related to Definition 5.5.A fundamental point is that a reconstruction of the manifold is possible, starting only witha spectral triple where the algebra is commutative (see [29] for a more precise formulation,and also [94]): 42 heorem 5.3. [29] Given a commutative spectral triple ( A , H , D ) satisfying the above ax-ioms, then there exists a compact spin c manifold M such that A ’ C ∞ ( M ) and D is a Diracoperator. The manifold is known as a set, M = Sp( A ) = Sp( A ). Notice that D is known only via itsprincipal symbol, so is not unique. J encodes the nuance between spin and spin c structures.The spectral action selects the Levi-Civita connection so the Dirac operator D/ .The way, the operator D recaptures the original Riemannian metric g of M is via theConnes’ distance: Definition 5.4. Given a spectral triple ( A , H , D ) , d ( φ , φ ) := sup { | φ ( a ) − φ ( a ) | | k [ D , π ( a )] k ≤ , a ∈ A } (44) defines a distance (eventually infinite) between two states φ , φ on the C ∗ -algebra A . In a commutative geometry, any point x ∈ M defines a state via φ x : a ∈ C ∞ ( M ) → a ( x ) ∈ C .Since the geodesic distance is also given by d g ( x, y ) = sup { | a ( x ) − a ( y ) | | a ∈ C ∞ ( M ) , k grad a k ∞ ≤ } , we get d ( x, y ) = d g ( x, y ) because k c ( da ) k = k grad a k ∞ . Recall that g is uniquely determinedby its distance function by Myers–Steenrod theorem: if α : ( M, g ) → ( M , g ) is a bijectionsuch that d g (cid:16) α ( x ) , α ( y ) (cid:17) = d g ( x, y ) for x, y ∈ M , then g = α ∗ g .The role of D is non only to provide a metric by (44), but its homotopy class representsthe K -homology fundamental class of the noncommutative space A .It is known that one cannot hear the shape of a drum since the knowledge of the spec-trum of a Laplacian does not determine the metric of the manifold, even if its conformalclass is given [7]. But Theorem 5.3 shows that one can hear the shape of a spinorial drum(or better say, of a spectral triple) since the knowledge of the spectrum of the Dirac op-erator and the volume form, via its cohomological content, is sufficient to recapture themetric and spin structure. See however the more precise refinement made in [30]: for in-stance, if ( M, g ) is a compact oriented smooth Riemannian manifold, the spectral triple (cid:16) L ∞ ( M ) , L ( M, V T ∗ M ) , D (cid:17) where D = d + d ∗ is the signature operator (see example afterdefinition 3.2) uniquely determines the manifold M . Definition 5.5. Let ( A , H , D ) be a spectral triple.For t ∈ R define the map F t : T ∈ B ( H ) → e it |D| T e − it |D| and for α ∈ R OP := { T | t → F t ( T ) ∈ C ∞ (cid:16) R , B ( H ) (cid:17) } is the set of operators or order ≤ ,OP α := { T | T |D| − α ∈ OP } is the set of operators of order ≤ α. Moreover, we set δ ( T ) := [ |D| , T ] , ∇ ( T ) := [ D , T ] . For instance, C ∞ ( M ) = OP T L ∞ ( M ) and L ∞ ( M ) is the von Neumann algebra gener-ated by A = C ∞ ( M ). 43 roposition 5.6. Assume that ( A , H , D ) is regular so A ⊂ OP = T k ≥ Dom δ k ⊂ B ( H ) .Then, for any α, β ∈ R , OP α OP β ⊂ OP α + β , OP α ⊂ OP β if α ≤ β, δ ( OP α ) ⊂ OP α , ∇ ( OP α ) ⊂ OP α +1 . As an example, let us compute the order of X = a |D| [ D , b ] D − : since the order of a is 0, of |D| is 1, of [ D , b ] is 0 and of D − is -3, we get X ∈ OP − . Definition 5.7. Let ( A , H , D ) be a spectral triple and D ( A ) be the polynomial algebra gen-erated by A , A ◦ , D and |D| .Define the set of pseudodifferential operators as Ψ( A ) := { T | ∀ N ∈ N , ∃ P ∈ D ( A ) , R ∈ OP − N , p ∈ N such that T = P |D| − p + R } The idea behind this definition is that we want to work modulo the set OP −∞ of smoothingoperators . This explains the presence of the arbitrary N and R . In the commutative caseof a manifold M with spectral triple (cid:16) C ∞ ( M ) , L ( M, E ) , D (cid:17) where D ∈ Diff ( M, E ), we getthe natural inclusion Ψ (cid:16) C ∞ ( M ) (cid:17) ⊂ Ψ DO ( M, E ).The reader should be aware that Definition 5.7 is not exactly the same as in [31, 34, 50] sinceit pays attention to the reality operator J when it is present. Definition 5.8. For P ∈ Ψ ∗ ( A ) , we define the zeta-function associated to P (and D ) by ζ P D : s ∈ C → Tr (cid:16) P |D| − s (cid:17) (45) which makes sense since for < ( s ) (cid:29) , P |D| − s ∈ L ( H ) .The dimension spectrum Sd ( A , H , D ) of ( A , H , D ) is the set of all poles of ζ P D ( s ) suchthat P ∈ Ψ( A ) ∩ OP . It is said simple if it contains poles of order at most one.The noncommutative integral of P is defined by − Z P := Res s =0 ζ P D ( s ) . (46)In (45), we assume D invertible since otherwise, one can replace D by the invertible operator D + P , P being the projection on Ker D . This change does not modify the computation ofthe integrals − R which follow since − R X = 0 when X is a trace-class operator.The notion of dimension spectrum contains more informations than the usual dimensioneven for a manifold as we will see in Proposition 5.34. Remark 5.9. If Sp ( A , H , D ) denotes the set of all poles of the functions s Tr (cid:16) P | D | − s (cid:17) where P is any pseudodifferential operator, then, Sd ( A , H , D ) ⊆ Sp ( A , H , D ) .When Sp ( A , H , D ) = Z , Sd ( A , H , D ) = { n − k : k ∈ N } : indeed, if P is a pseudodiffer-ential operator in OP , and q ∈ N is such that q > n , P | D | − s is in OP −< ( s ) so is trace-classfor s in a neighborhood of q ; as a consequence, q cannot be a pole of s Tr (cid:16) P | D | − s (cid:17) . Due to the little difference of behavior between scalar and nonscalar pseudodifferentialoperators (i.e. when coefficients like [ D , a ], a ∈ A appears in P of Definition 5.7), it isconvenient to also introduce 44 efinition 5.10. Let D ( A ) be the algebra generated by A , J A J − and D , and Ψ ( A ) be theset of pseudodifferential operators constructed as before with D ( A ) instead of D ( A ) . Notethat Ψ ( A ) is subalgebra of Ψ( A ) . Remark that Ψ ( A ) does not necessarily contain operators such as | D | k where k ∈ Z isodd. This algebra is similar to the one defined in [13]. D The unitary group U ( A ) of A gives rise to the automorphism α u : a ∈ A → uau ∗ ∈ A .This defines the inner automorphisms group Inn ( A ) which is a normal subgroup of theautomorphisms Aut ( A ) := { α ∈ Aut ( A ) | α ( A ) ⊂ A } . For instance, in case of a gaugetheory, the algebra A = C ∞ (cid:16) M, M n ( C ) (cid:17) ’ C ∞ ( M ) ⊗ M n ( C ) is typically used. Then, Inn ( A ) is locally isomorphic to G = C ∞ (cid:16) M, P SU ( n ) (cid:17) . Since Aut (cid:16) C ∞ ( M ) (cid:17) ’ Diff( M ), weget a complete parallel analogy between following two exact sequences:1 −→ Inn ( A ) −→ Aut ( A ) −→ Aut ( A ) /Inn ( A ) −→ −→ G −→ G (cid:111) Diff( M ) −→ Diff( M ) −→ A , H , D ) and ( A , H , D ) giving riseto the same geometry. Of course, we could use unitary equivalence: there exists a unitary U : H → H such that D = U D U ∗ , U π ( a ) U ∗ := π (cid:16) α ( a ) (cid:17) for some α ∈ Aut ( A ), and in theeven real case [ U, χ ] = [ U, J ] = 0. But this is not useful since it does not change the metric(44). So we need to vary not only D but the algebra and its representation. The appropriate framework for inner fluctuations of a spectral triple ( A , H , D ) is Moritaequivalence that we describe now: A is Morita equivalent to B if there is a finite projective right A -module E such that B ’ End A ( E ). Thus B acts on H = E ⊗ A H and H is endowed with scalar product h r ⊗ η, s ⊗ ξ i := h η, π ( r | s ) ξ i where ( ·|· ) is a pairing E × E → A that is A -linear in the secondvariable and satisfies ( r | s ) = ( s | r ) ∗ , ( r | sa ) = ( r | s ) a and ( s | s ) > = s ∈ E (this can beseen as a A -valued inner product).A natural operator D associated to B and H is a linear map D ( r ⊗ η ) = r ⊗ D η + ( ∇ r ) η where ∇ : E → E ⊗ A Ω D ( A ) is a linear map obeying to Leibniz rule ∇ ( ra ) = ( ∇ r ) a + r ⊗ [ D , a ]for r ∈ E , a ∈ A where we took the following Definition 5.11. Let ( A , H , D ) be a spectral triple. The set of one-forms is defined as Ω D ( A ) := span { a db | a, b ∈ A } , db := [ D , b ] . It is a A -bimodule. Such ∇ is called a connection on E and by a result of Cuntz–Quillen, only projective modulesadmit (universal) connections (see [50][Proposition 8.3]). Since we want D selfadjoint, ∇ must be hermitean with respect to D which means: π (cid:16) ( r |∇ s ) − ( ∇ r | s ) (cid:17) = [ D , π ( r | s )].In particular, when E = A (any algebra is Morita equivalent to itself) and A is regardedas a right A -module, E has a natural hermitean connection with respect to D given by Ad D : a ∈ A → [ D , a ] ∈ Ω D ( A ) and using the Leibniz rule, any another hermitean connection45 must verify: ∇ a = Ad D a + A a where A = A ∗ ∈ Ω D ( A ). So this process, which does notchange neither the algebra A nor the Hilbert space H , gives a natural hermitean fluctuationof D : D → D A := D + A with A = A ∗ ∈ Ω D ( A ) . In conclusion, the Morita equivalent geometries for ( A , H , D ) keeping fixed A and H is anaffine space modelled on the selfadjoint part of Ω D ( A ).For instance, in commutative geometries, Ω D/ (cid:16) C ∞ ( M ) (cid:17) = { c ( da ) | a ∈ C ∞ ( M ) } .When a reality operator J exists, we also want D A J = (cid:15) J D A , so we choose D e A := D + e A, e A := A + (cid:15)J AJ − , A = A ∗ . (47)The next two results show that, with the same algebra A and Hilbert space H , a fluctu-ation of D still give rise to a spectral triple ( A , H , D A ) or ( A , H , D e A ). Lemma 5.12. Let ( A , H , D ) be a spectral triple with a reality operator J and chirality χ . If A ∈ Ω D is a one-form, the fluctuated Dirac operator D A or D e A is an operator with compactresolvent, and in particular its kernel is a finite dimensional space. This space is invariantby J and χ .Proof. Let T be a bounded operator and let z be in the resolvent of D + T and z be in theresolvent of D . Then( D + T − z ) − = ( D − z ) − [1 − ( T + z − z )( D + T − z ) − ] . Since ( D − z ) − is compact by hypothesis and since the term in bracket is bounded, D + T has a compact resolvent. Applying this to T = A + (cid:15)J AJ − , D A has a finite dimensionalkernel (see for instance [66, Theorem 6.29]).Since according to the dimension, J = ± J commutes or anticommutes with χ , χ commutes with the elements in the algebra A and D χ = − χ D , see (42), we get D A χ = − χ D A and D A J = ± J D A which gives the result.Note that U ( A ) acts on D by D → D u = u D u ∗ leaving invariant the spectrum of D . Since D u = D + u [ D , u ∗ ] and in a C ∗ -algebra, any element a is a linear combination of at most fourunitaries, Definition 5.11 is quite natural.The inner automorphisms of a spectral triple correspond to inner fluctuation of the metricdefined by (44).One checks directly that a fluctuation of a fluctuation is a fluctuation and that the unitarygroup U ( A ) is gauge compatible for the adjoint representation: Lemma 5.13. Let ( A , H , D ) be a spectral triple (which is eventually real) and A ∈ Ω D ( A ) , A = A ∗ .(i) If B ∈ Ω D A ( A ) ( or B ∈ Ω D e A ( A ) ), D B = D C (or D e B = D e C ) with C := A + B .(ii) Let u ∈ U ( A ) . Then U u := uJ uJ − is a unitary of H such that U u D e A U u ∗ = D (cid:94) γ u ( A ) , where γ u ( A ) := u [ D , u ∗ ] + uAu ∗ . Remark 5.14. To be an inner fluctuation is not a symmetric relation. It can append that D A = 0 with D 6 = 0 . emma 5.15. Let ( A , D , H ) be a spectral triple and X ∈ Ψ( A ) . Then − Z X ∗ = − Z X. If the spectral triple is real, then, for X ∈ Ψ( A ) , J XJ − ∈ Ψ( A ) and − Z J XJ − = − Z X ∗ = − Z X. Proof. The first result follows from (for < s large enough, so the operators are traceable)Tr( X ∗ |D| − s ) = Tr (cid:16) ( |D| − ¯ s ) X ) ∗ (cid:17) = Tr( |D| − ¯ s X ) = Tr( X |D| − ¯ s ) . The second result is due to the anti-linearity of J , Tr( J Y J − ) = Tr( Y ), and J |D| = |D| J ,so Tr( X |D| − s ) = Tr( J X |D| − s J − ) = Tr( J XJ − |D| − ¯ s ) . Corollary 5.16. For any one-form A = A ∗ , and for k, l ∈ N , − Z A l D − k ∈ R , − Z (cid:16) A D − (cid:17) k ∈ R , − Z A l |D| − k ∈ R , − Z χA l |D| − k ∈ R , − Z A l D |D| − k ∈ R . We remark that the fluctuations leave invariant the first term of the spectral action (75).This is a generalization of the fact that in the commutative case, the noncommutative integraldepends only on the principal symbol of the Dirac operator D and this symbol is stable byadding a gauge potential like in D + A . Note however that the symmetrized gauge potential A + (cid:15)J AJ − is always zero in this case for any selfadjoint one-form A , see (65). Theorem 5.17. Let ( A , H , D ) be a regular spectral triple which is simple and of dimension d . Let A ∈ Ω D ( A ) be a selfadjoint gauge potential. Then, ζ D e A (0) = ζ D (0) + d X q =1 ( − q q − Z ( e AD − ) q . (48)The proof needs few preliminaries. Definition 5.18. For an operator T , define the one-parameter group and notation σ z ( T ) := | D | z T | D | − z , z ∈ C .(cid:15) ( T ) := ∇ ( T ) D − , (recall that ∇ ( T ) = [ D , T ]) . The expansion of the one-parameter group σ z gives for T ∈ OP q σ z ( T ) ∼ N X r =0 g ( z, r ) ε r ( T ) mod OP − N − q (49)where g ( z, r ) := r ! ( z ) · · · ( z − ( r − (cid:16) z/ r (cid:17) with the convention g ( z, 0) := 1.47e fix a regular spectral triple ( A , H , D ) of dimension d and a self-adjoint 1-form A .Despite previous remark before Lemma 5.15, we pay attention here to the kernel of D A sincethis operator can be non-invertible even if D is, so we define D A := D + e A where e A := A + εJ AJ − ,D A := D A + P A (50)where P A is the projection on Ker D A . Remark that e A ∈ D ( A ) ∩ OP and D A ∈ D ( A ) ∩ OP .We note V A := P A − P . As the following lemma shows, V A is a smoothing operator: Lemma 5.19. (i) T k ≥ Dom( D A ) k ⊆ T k ≥ Dom | D | k .(ii) Ker D A ⊆ T k ≥ Dom | D | k .(iii) For any α, β ∈ R , | D | β P A | D | α is bounded.(iv) P A ∈ OP −∞ .Proof. ( i ) Let us define for any p ∈ N , R p := ( D A ) p − D p , so R p ∈ OP p − and moreover R p (cid:16) Dom | D | p (cid:17) ⊆ Dom | D | .Let us fix k ∈ N , k ≥ 2. Since Dom D A = Dom D = Dom | D | , we haveDom( D A ) k = { φ ∈ Dom | D | : ( D j + R j ) φ ∈ Dom | D | , ∀ j ≤ j ≤ k − } . Let φ ∈ Dom( D A ) k . We prove by recurrence that for any j ∈ { , · · · , k − } , φ ∈ Dom | D | j +1 :We have φ ∈ Dom | D | and ( D + R ) φ ∈ Dom | D | . Thus, since R φ ∈ Dom | D | , we have D φ ∈ Dom | D | , which proves that φ ∈ Dom | D | . Hence, case j = 1 is done.Suppose now φ ∈ Dom | D | j +1 for a j ∈ { , · · · , k − } . Since ( D j +1 + R j +1 ) φ ∈ Dom | D | ,and R j +1 φ ∈ Dom | D | , we get D j +1 φ ∈ Dom | D | , which proves that φ ∈ Dom | D | j +2 .Finally, if we set j = k − 1, we get φ ∈ Dom | D | k , so Dom( D A ) k ⊆ Dom | D | k .( ii ) follows from Ker D A ⊆ T k ≥ Dom( D A ) k and ( i ).( iii ) Let us first check that | D | α P A is bounded. We define D as the operator withdomain Dom D = Im P A ∩ Dom | D | α and such that D φ = | D | α φ. Since Dom D is finitedimensional, D extends as a bounded operator on H with finite rank. We havesup φ ∈ Dom | D | α P A , k φ k≤ k| D | α P A φ k ≤ sup φ ∈ Dom D , k φ k≤ k| D | α φ k = k D k < ∞ so | D | α P A is bounded. We can remark that by ( ii ), Dom D = Im P A and Dom | D | α P A = H .Let us prove now that P A | D | α is bounded: Let φ ∈ Dom P A | D | α = Dom | D | α . By ( ii ),we have Im P A ⊆ Dom | D | α so we get k P A | D | α φ k ≤ sup ψ ∈ Im P A , k ψ k≤ | < ψ, | D | α φ > | ≤ sup ψ ∈ Im P A , k ψ k≤ | < | D | α ψ, φ > |≤ sup ψ ∈ Im P A , k ψ k≤ k| D | α ψ k k φ k = k D k k φ k . ( iv ) For any k ∈ N and t ∈ R , δ k ( P A ) | D | t is a linear combination of terms of the form | D | β P A | D | α , so the result follows from ( iii ). 48 emark 5.20. We will see later on the noncommutative torus example how important is thedifference between D A and D + A . In particular, the inclusion Ker D ⊆ Ker D + A is notsatisfied since A does not preserve Ker D contrarily to e A . Let us define X := D A − D = e A D + D e A + e A ,X V := X + V A , thus X ∈ D ( A ) ∩ OP and by Lemma 5.19, X V ∼ X mod OP −∞ . (51)We will use Y := log( D A ) − log( D )which makes sense since D A = D A + P A is invertible for any A . By definition of X V , we get Y = log( D + X V ) − log( D ) . Lemma 5.21. (i) Y is a pseudodifferential operator in OP − with the following expansionfor any N ∈ N Y ∼ N X p =1 N − p X k , ··· ,k p =0 ( − | k | p +1 | k | + p ∇ k p ( X ∇ k p − ( · · · X ∇ k ( X ) · · · )) D − | k | + p ) mod OP − N − . (ii) For any N ∈ N and s ∈ C , | D A | − s ∼ | D | − s + N X p =1 K p ( Y, s ) | D | − s mod OP − N − −< ( s ) (52) with K p ( Y, s ) ∈ OP − p .Proof. ( i ) We follow [13, Lemma 2.2]. By functional calculus, Y = R ∞ I ( λ ) dλ , where I ( λ ) ∼ N X p =1 ( − p +1 (cid:16) ( D + λ ) − X V (cid:17) p ( D + λ ) − mod OP − N − . By (51), (cid:16) ( D + λ ) − X V (cid:17) p ∼ (cid:16) ( D + λ ) − X (cid:17) p mod OP −∞ and we get I ( λ ) ∼ N X p =1 ( − p +1 (cid:16) ( D + λ ) − X (cid:17) p ( D + λ ) − mod OP − N − . We set A p ( X ) := (cid:16) ( D + λ ) − X (cid:17) p ( D + λ ) − and L := ( D + λ ) − ∈ OP − for a fixed λ .Since [ D + λ, X ] ∼ ∇ ( X ) mod OP −∞ , a recurrence proves that if T is an operator in OP r ,then, for q ∈ N , A ( T ) = LT L ∼ q X k =0 ( − k ∇ k ( T ) L k +2 mod OP r − q − . A p ( X ) = LXA p − ( X ), another recurrence gives, for any q ∈ N , A p ( X ) ∼ q X k , ··· ,k p =0 ( − | k | ∇ k p ( X ∇ k p − ( · · · X ∇ k ( X ) · · · )) L | k | + p +1 mod OP − q − p − , which entails that I ( λ ) ∼ N X p =1 ( − p +1 N − p X k , ··· ,k p =0 ( − | k | ∇ k p ( X ∇ k p − ( · · · X ∇ k ( X ) · · · )) L | k | + p +1 mod OP − N − . With R ∞ ( D + λ ) − ( | k | + p +1) dλ = | k | + p D − | k | + p ) , we get the result provided we controlthe remainders. Such a control is given in [13, (2.27)].( ii ) Applied to | D A | − s = e B − ( s/ Y e − B | D | − s where B := ( − s/ 2) log( D ), the Duhamel’sexpansion formula e U + V e − U = ∞ X n =0 Z ≤ t ≤ t ≤···≤ t n ≤ V ( t ) · · · V ( t n ) dt · · · dt n with V ( t ) := e tU V e − tU gives | D A | − s = | D | − s + ∞ X p =1 K p ( Y, s ) | D | − s . (53)and each K p ( Y, s ) is in OP − p . Corollary 5.22. For any p ∈ N and r , · · · , r p ∈ N , ε r ( Y ) · · · ε r p ( Y ) ∈ Ψ ( A ) .Proof. If for any q ∈ N and k = ( k , · · · , k q ) ∈ N q ,Γ kq ( X ) := ( − | k | q +1 | k | + q ∇ k q ( X ∇ k q − ( · · · X ∇ k ( X ) · · · )) , then, Γ kq ( X ) ∈ OP | k | + q . For any N ∈ N , Y ∼ N X q =1 N − q X k , ··· ,k q =0 Γ kq ( X ) D − | k | + q ) mod OP − N − . (54)Since the Γ kq ( X ) are in D ( A ), this proves with (54) that Y and thus ε r ( Y ) = ∇ r ( Y ) D − r ,are also in Ψ ( A ). Proof of Theorem 5.17. Again, we follow [13]. Since the spectral triple is simple, equation(53) entails that ζ D A (0) − ζ D (0) = Tr( K ( Y, s ) | D | − s ) | s =0 . Thus, with (49), we get ζ D A (0) − ζ D (0) = − − R Y .Now the conclusion follows from − R log (cid:16) (1 + S )(1 + T ) (cid:17) = − R log(1 + S ) + − R log(1 + T ) for S, T ∈ Ψ( A ) ∩ OP − (since log(1 + S ) = P ∞ n =1 ( − n +1 n S n ) with S = AD − and T = DAD − ;so − R log(1 + XD − ) = 2 − R log(1 + AD − ) and − − Z Y = d X q =1 ( − q q − Z ( e AD − ) q . emma 5.23. For any k ∈ N , Res s = d − k ζ D A ( s ) = Res s = d − k ζ D ( s ) + k X p =1 k − p X r , ··· ,r p =0 Res s = d − k h ( s, r, p ) Tr (cid:16) ε r ( Y ) · · · ε r p ( Y ) | D | − s (cid:17) , where h ( s, r, p ) := ( − s/ p Z ≤ t ≤···≤ t p ≤ g ( − st , r ) · · · g ( − st p , r p ) dt . Proof. By Lemma 5.21 ( ii ), | D A | − s ∼ | D | − s + P kp =1 K p ( Y, s ) | D | − s mod OP − ( k +1) −< ( s ) , wherethe convention P ∅ = 0 is used. Thus, we get for s in a neighborhood of d − k , | D A | − s − | D | − s − k X p =1 K p ( Y, s ) | D | − s ∈ OP − ( k +1) −< ( s ) ⊆ L ( H )which gives Res s = d − k ζ D A ( s ) = Res s = d − k ζ D ( s ) + k X p =1 Res s = d − k Tr (cid:16) K p ( Y, s ) | D | − s (cid:17) . (55)Let us fix 1 ≤ p ≤ k and N ∈ N . By (49) we get K p ( Y, s ) ∼ ( − s ) p Z ≤ t ≤··· t p ≤ N X r , ··· ,r p =0 g ( − st , r ) · · · g ( − st p , r p ) ε r ( Y ) · · · ε r p ( Y ) dt mod OP − N − p − . (56)If we now take N = k − p , we get for s in a neighborhood of d − kK p ( Y, s ) | D | − s − k − p X r , ··· ,r p =0 h ( s, r, p ) ε r ( Y ) · · · ε r p ( Y ) | D | − s ∈ OP − k − −< ( s ) ⊆ L ( H )so (55) gives the result.Our operators | D A | k are pseudodifferential operators: Lemma 5.24. For any k ∈ Z , | D A | k ∈ Ψ k ( A ) .Proof. Using (56), we see that K p ( Y, s ) is a pseudodifferential operator in OP − p , so (52)proves that | D A | k is a pseudodifferential operator in OP k .The following result is quite important since it shows that one can use − R for D or D A : Proposition 5.25. If the spectral triple is simple, Res s =0 Tr (cid:16) P | D A | − s (cid:17) = − R P for any pseu-dodifferential operator P . In particular, for any k ∈ N − Z | D A | − ( d − k ) = Res s = d − k ζ D A ( s ) . roof. Suppose P ∈ OP k with k ∈ Z and let us fix p ≥ 1. With (56), we see that for any N ∈ N , P K p ( Y, s ) | D | − s ∼ N X r , ··· ,r p =0 h ( s, r, p ) P ε r ( Y ) · · · ε r p ( Y ) | D | − s mod OP − N − p − k −< ( s ) . Thus if we take N = d − p + k , we getRes s =0 Tr (cid:16) P K p ( Y, s ) | D | − s (cid:17) = n − p + k X r , ··· ,r p =0 Res s =0 h ( s, r, p ) Tr (cid:16) P ε r ( Y ) · · · ε r p ( Y ) | D | − s (cid:17) . Since s = 0 is a zero of the analytic function s h ( s, r, p ) and s Tr P ε r ( Y ) · · · ε r p ( Y ) | D | − s has only simple poles by hypothesis, we get Res s =0 h ( s, r, p ) Tr (cid:16) P ε r ( Y ) · · · ε r p ( Y ) | D | − s (cid:17) = 0and Res s =0 Tr (cid:16) P K p ( Y, s ) | D | − s (cid:17) = 0 . (57)Using (52), P | D A | − s ∼ P | D | − s + P k + dp =1 P K p ( Y, s ) | D | − s mod OP − d − −< ( s ) and thus,Res s =0 Tr( P | D A | − s ) = − Z P + k + d X p =1 Res s =0 Tr (cid:16) P K p ( Y, s ) | D | − s (cid:17) . (58)The result now follows from (57) and (58). To get the last equality, one uses the pseudodif-ferential operator | D A | − ( d − k ) . Proposition 5.26. If the spectral triple is simple, then − Z | D A | − d = − Z | D | − d . (59) Proof. Lemma 5.23 and previous proposition for k = 0. Lemma 5.27. If the spectral triple is simple, ( i ) − Z | D A | − ( d − = − Z | D | − ( d − − ( d − ) − Z X | D | − d − . ( ii ) − Z | D A | − ( d − = − Z | D | − ( d − + d − (cid:16) − − Z X | D | − d + d − Z X | D | − − d (cid:17) . Proof. ( i ) By (52),Res s = d − ζ D A ( s ) − ζ D ( s ) = Res s = d − ( − s/ 2) Tr (cid:16) Y | D | − s (cid:17) = − d − Res s =0 Tr (cid:16) Y | D | − ( d − | D | − s (cid:17) where for the last equality we use the simple dimension spectrum hypothesis. Lemma 5.21 ( i )yields Y ∼ XD − mod OP − and Y | D | − ( d − ∼ X | D | − d − mod OP − d − ⊆ L ( H ). Thus,Res s =0 Tr (cid:16) Y | D | − ( d − | D | − s (cid:17) = Res s =0 Tr (cid:16) X | D | − d − | D | − s (cid:17) = − Z X | D | − d − . ( ii ) Lemma 5.23 ( ii ) givesRes s = d − ζ D A ( s ) = Res s = d − ζ D ( s ) + Res s = d − X r =0 h ( s, r, 1) Tr (cid:16) ε r ( Y ) | D | − s (cid:17) + h ( s, , 2) Tr (cid:16) Y | D | − s (cid:17) . 52e have h ( s, , 1) = − s , h ( s, , 1) = ( s ) and h ( s, , 2) = ( s ) . Using again Lemma 5.21( i ), Y ∼ XD − − ∇ ( X ) D − − X D − mod OP − . Thus, Res s = d − Tr (cid:16) Y | D | − s (cid:17) = − Z X | D | − d − − Z ( ∇ ( X ) + X ) | D | − − d . Moreover, using − R ∇ ( X ) | D | − k = 0 for any k ≥ − R is a trace,Res s = d − Tr (cid:16) ε ( Y ) | D | − s (cid:17) = Res s = d − Tr (cid:16) ∇ ( X ) D − | D | − s (cid:17) = − Z ∇ ( X ) | D | − − d = 0 . Similarly, since Y ∼ XD − mod OP − and Y ∼ X D − mod OP − , we getRes s = d − Tr (cid:16) Y | D | − s (cid:17) = Res s = d − Tr (cid:16) X D − | D | − s (cid:17) = − Z X | D | − − d . Thus, Res s = d − ζ D A ( s ) = Res s = d − ζ D ( s )+( − d − )( − Z X | D | − d − − Z ( ∇ ( X ) + X ) | D | − − d )+ ( d − ) − Z ∇ ( X ) | D | − − n + ( d − ) − Z X | D | − − d . Finally,Res s = d − ζ D A ( s ) = Res s = d − ζ D ( s ) + ( − d − ) (cid:16) − Z X | D | − d − − Z X | D | − − d (cid:17) + ( d − ) − Z X | D | − − d and the result follows from Proposition 5.25. Corollary 5.28. If the spectral triple satisfies − R | D | − ( d − = − R e A D| D | − d = − R D e A | D | − d = 0 ,then − Z | D A | − ( d − = d ( d − (cid:16) − Z e A D e A D| D | − d − + d − d − Z e A | D | − d (cid:17) . Proof. By previous lemma,Res s = d − ζ D A ( s ) = d − (cid:16) − − Z e A | D | − d + d − Z ( e A D e A D + D e A D e A + e A D e A + D e A D ) | D | − d − (cid:17) . Since ∇ ( e A ) ∈ OP , the trace property of − R yields the result. In [31], the following definition is introduced: Definition 5.29. In ( A , H , D ) , the tadpole T ad D + A ( k ) of order k , for k ∈ { d − l : l ∈ N } is the term linear in A = A ∗ ∈ Ω D , in the Λ k term of (75) (considered as an infinite series)where D → D + A .If moreover, the triple ( A , H , D , J ) is real, the tadpole T ad D + ˜ A ( k ) is the term linear in A , in the Λ k term of (75) where D → D + e A . roposition 5.30. Let ( A , H , D ) be a spectral triple of dimension d with simple dimensionspectrum. Then Tad D + A ( d − k ) = − ( d − k ) − Z A D|D| − ( d − k ) − , ∀ k = d, (60)Tad D + A (0) = −− Z A D − . (61) Moreover, if the triple is real, Tad D + e A = 2 Tad D + A .Proof. We already proved the following formula, for any k ∈ N , − Z |D A | − ( d − k ) = − Z |D| − ( d − k ) + k X p =1 k − p X r , ··· ,r p =0 Res s = d − k h ( s, r, p ) Tr (cid:16) ε r ( Y ) · · · ε r p ( Y ) |D| − s (cid:17) , with here X := e A D + D e A + e A , e A := A + (cid:15)J AJ − .As a consequence, for k = n , only the terms with p = 1 contribute to the linear part:Tad D + e A ( d − k ) = Lin A ( − Z |D A | − ( d − k ) ) = k − X r =0 Res s = d − k h ( s, r, 1) Tr (cid:16) ε r (Lin A ( Y )) |D| − s (cid:17) . We check that for any N ∈ N ∗ ,Lin A ( Y ) ∼ N − X l =0 Γ l ( e A D + D e A ) D − l +1) mod OP − N − . Since Γ l ( e A D + D e A ) = ( − l l +1 ∇ l ( e A D + D e A ) = ( − l l +1 {∇ l ( e A ) , D} , we get, assuming the dimensionspectrum to be simpleTad D + e A ( d − k ) = k − X r =0 Res s = d − k h ( s, r, p ) Tr (cid:16) ε r (Lin A ( Y )) |D| − s (cid:17) = k − X r =0 h ( n − k, r, k − − r X l =0 ( − l l +1 Res s = d − k Tr (cid:16) ε r ( {∇ l ( e A ) , D} ) |D| − s − l +1) (cid:17) = 2 k − X r =0 h ( d − k, r, k − − r X l =0 ( − l l +1 − Z ∇ r + l ( e A ) D|D| − ( d − k +2( r + l )) − = − ( n − k ) − Z e A D|D| − ( d − k ) − , because in the last sum it remains only the case r + l = 0, so r = l = 0.Formula (61) is a direct application of Theorem 5.17.The link between Tad D + e A and Tad D + A follows from J D = (cid:15) D J and Lemma 5.15. Corollary 5.31. In a real spectral triple ( A , H , D ) , if A = A ∗ ∈ Ω D ( A ) is such that e A = 0 ,then Tad D + A ( k ) = 0 for any k ∈ Z , k ≤ d . The vanishing tadpole of order 0 has the following equivalence (see [13]) − Z A D − = 0 , ∀ A ∈ Ω D ( A ) ⇐⇒ − Z ab = − Z aα ( b ) , ∀ a, b ∈ A , (62)where α ( b ) := D b D − .The existence of tadpoles is important since, for instance, A = 0 is not necessarily a stablesolution of the classical field equation deduced from spectral action expansion, [51].54 .6 Commutative geometry Definition 5.32. Consider a commutative spectral triple given by a compact Riemannianspin manifold M of dimension d without boundary and its Dirac operator D/ associated tothe Levi–Civita connection. This means (cid:16) A := C ∞ ( M ) , H := L ( M, S ) , D/ (cid:17) where S is thespinor bundle over M . This triple is real since, due to the existence of a spin structure, thecharge conjugation operator generates an anti-linear isometry J on H such that J aJ − = a ∗ , ∀ a ∈ A , (63) and when d is even, the grading is given by the chirality matrix χ := ( − i ) d/ γ γ · · · γ d . (64) Such triple is said to be a commutative geometry. In the polynomial algebra D ( A ) of Definition 5.7, we added A ◦ . In the commutative case, A ◦ ’ J A J − ’ A as indicated by (63) which also gives J AJ − = − (cid:15) A ∗ , ∀ A ∈ Ω D ( A ) or e A = 0 when A = A ∗ . (65)As noticed by Wodzicki, − R P is equal to − t of the asymptoticsof Tr( P e − t D/ ) as t → 0. It is remarkable that this coefficient is independent of D/ as seen inTheorem 1.22 and this gives a close relation between the ζ function and heat kernel expansionwith WRes . Actually, by [48, Theorem 2.7]Tr( P e − t D/ ) ∼ t ↓ + ∞ X k =0 a k t ( k − ord ( P ) − d ) / + ∞ X k =0 ( − a k log t + b k ) t k , (66)so − Z P = 2 a . Remark that − R , WRes are traces on Ψ (cid:16) C ∞ ( M ) (cid:17) , thus Corollary 1.23 implies − Z P = c WRes P. (67)Since, via Mellin transform, Tr( P D/ − s ) = s ) R ∞ t s − Tr( P e − t D/ ) dt , the non-zero coeffi-cient a k , k = 0 creates a pole of Tr( P D/ − s ) of order k + 2 because we get R t s − log( t ) k dt = ( − k k ! s k +1 and Γ( s ) = 1 s + γ + s g ( s ) (68)where γ is the Euler constant and the function g is also holomorphic around zero.We have − R WRes ( P ) = 0 for all zero-order pseudodifferentialprojections [112].As the following remark shows, being a commutative geometry is more than just havinga commutative algebra: 55 emark 5.33. Since J π ( a ) J − = π ( a ∗ ) for all a ∈ A and ˜ A = 0 for all A = A ∗ ∈ Ω D when A is commutative by (65) , one can only use D A = D + A to get fluctuation of D : Itis amazing to see that in the context of noncommutative geometry, to get an abelian gaugefield, we need to go outside of abelian algebras. In particular, as pointed out in [107], acommutative manifold could support relativity but not electromagnetism.However, we can have A commutative and J π ( a ) J − = π ( a ∗ ) for some a ∈ A [27, 68]:Let A = C ⊕ C represented on H = C with, for some complex number m = 0 , π ( a ) := b b 00 0 b , f or a = ( b , b ) ∈ A , and D := m m ¯ m m , χ := − − , J := ◦ cc where cc is the complex conjugation. Then ( A , H , D ) is a commutative real spectral tripleof dimension d = 0 with non zero one-forms and such that J π ( a ) J − = π ( a ∗ ) only if a = ( b , b ) .Take now a commutative geometry (cid:16) A = C ∞ ( M ) , H = L ( M, S ) , D , χ , J (cid:17) defined in5.32 where d = dimM is even, and then take the tensor product of the two spectral triples,see Section 5.8, namely A = A ⊗ A , H = H ⊗ H , π = π ⊗ π , D = D ⊗ χ + 1 ⊗ D , χ = χ ⊗ χ and J is either χ J ⊗ J when d ∈ { , } mod 8 or J ⊗ J in the other cases,see [27, 105].Then ( A , H , D ) is a real commutative spectral triple of dimension d such that ˜ A = 0 forsome selfadjoint one-forms A , so is not exactly like in Definition 5.32. Proposition 5.34. Let Sp ( M ) be the dimension spectrum of a commutative geometry ofdimension d . Then Sp ( M ) is simple and Sp ( M ) = { d − k | k ∈ N } .Proof. Let a ∈ A = C ∞ ( M ) such that its trace norm || a || L is non zero and for k ∈ N , let P k := a | D | − k . Then P k ∈ OP − k ⊂ OP and its associated zeta-function has a pole at d − k :Res s = d − k ζ P D ( s ) = Res s =0 ζ P D ( s + d − k ) = Res s =0 Tr (cid:16) a |D| − k |D| − ( s + d − k ) (cid:17) = − Z a |D| − d = Z M a ( x ) Z S ∗ x M Tr (cid:16) ( σ |D| ) − d ( x, ξ ) (cid:17) | dξ | | dx | = c Z M a ( x ) Z S ∗ x M || ξ || − d | dξ | | dx | = c Z M a ( x ) dvol g ( x ) = c || a || L = 0 . Conversely, since Ψ ( A ) is contained in the algebra of all pseudodifferential operators of orderless or equal to 0, it is known [52,111,112] that Sp ( M ) ⊂ { d − k : k ∈ N } as seen in Theorem4.1.All poles are simple since D being differential and M being without boundary, a k = 0, forall k ∈ N ∗ in (66). 56 emark 5.35. Due to our efforts to mimic the commutative case, we get as in Theorem1.22 that the noncommutative integral is a trace on Ψ ∗ ( A ) . However, when the dimensionspectrum is not simple, the analog of WRes is no longer a trace.The equation (59) can be obtained via (12) and (67) since σ |D A | − d d = σ |D| − d d .In dimension d = 4 , the computation in (39) of coefficient a (1 , D A ) gives ζ D A (0) = c Z M (5 R − Rµνr µν − R µνρσ R µνρσ dvol + c Z M tr ( F µν F µν ) dvol, see Corollary 6.4 to see precise correspondence between a k (1 , D A ) and ζ D A (0) . One recognizesthe Yang–Mills action which will be generalized in Section 6.1.3 to arbitrary spectral triples.According to Corollary 5.31, a commutative geometry has no tadpoles. What could be the scalar curvature of a spectral triple ( A , H , D )? Of course, we need toconsider first the case of a commutative geometry ( C ∞ ( M ) , L ( M, S ) , D/ ) of dimension d = 4:We know that − R f ( x ) D/ − d +2 = R M f ( x ) s ( x ) dvol ( x ) where s is the scalar curvature for any f ∈ C ∞ ( M ). This suggests the following Definition 5.36. Let ( A , H , D ) be a spectral triple of dimension d . The scalar curvature isthe map R : a ∈ A → C defined by R ( a ) := − Z a D − d +2 . In the commutative case, R is a trace on the algebra. More generally Proposition 5.37. If R is a trace on A and the tadpoles − R A D − d +1 are zero for all A ∈ Ω D , R is invariant by inner fluctuations D → D + A . See [31, Proposition 1.153] for a proof. There is a natural notion of tensor for spectral triples which corresponds to direct productof manifolds in the commutative case. Let ( A i , D i , H i ), i = 1 , 2, be two spectral triples ofdimension d i with simple dimension spectrum. Assume the first to be of even dimension,with grading χ .The spectral triple ( A , D , H ) associated to the tensor product is defined by A := A ⊗ A , D := D ⊗ χ ⊗ D , H := H ⊗ H . The interest of χ is to guarantee additivity: D = D ⊗ ⊗ D .We assume that Tr( e − t D ) ∼ t → a t − d / , Tr( e − t D ) ∼ t → a t − d / . (69) Lemma 5.38. The triple ( A , D , H ) has dimension d = d + d .Moreover, the function ζ D ( s ) = Tr( | D | − s ) has a simple pole at s = d + d with Res s = d + d (cid:16) ζ D ( s ) (cid:17) = 12 Γ( d / d / d/ Res s = d (cid:16) ζ D ( s ) (cid:17) Res s = d (cid:16) ζ D ( s ) (cid:17) . roof. If ( µ n ( A )) are the singular values of A , ζ D (2 s ) = ∞ X n =0 µ n ( D ⊗ ⊗ D ) − s = ∞ X n,m =0 (cid:16) µ n ( D ) + µ m ( D ) (cid:17) − s . Since (cid:16) µ n ( D ) + µ m ( D ) (cid:17) − ( c + c ) / ≤ µ n ( D ) − c µ m ( D ) − c , this shows in particular that ζ D ( c + c ) ≤ ζ D ( c ) ζ D ( c ) if c i > d i , and in particular that d := inf { c ∈ R + : ζ D ( c ) < ∞ } ≤ d + d . We claim that d = d + d : recall first that in (69) a i := Res s = d i / (cid:16) Γ( s ) ζ D i (2 s ) (cid:17) = Γ( d i / 2) Res s = d i / (cid:16) ζ D i (2 s ) (cid:17) = Γ( d i / 2) Res s = d i (cid:16) ζ D i ( s ) (cid:17) . (70)If f ( s ) := Γ( s ) ζ D (2 s ), f ( s ) = Γ( s ) Tr (cid:16) D − s (cid:17) = Tr( Z ∞ e − t D t s − dt (cid:17) = Z Tr (cid:16) e − t D (cid:17) t s − dt + g ( s )= Z Tr (cid:16) e − t D (cid:17) Tr (cid:16) e − t D (cid:17) t s − dt + g ( s )where g is a holomorphic function since the map x ∈ R → R ∞ e − tx t x − dt is in Schwartzspace.Since Tr (cid:16) e − t D (cid:17) Tr (cid:16) e − t D (cid:17) ∼ t → a a t − ( d + d ) / , we get that the function f ( s ) has a simplepole at s = ( d + d ) / 2. We conclude that ζ D ( s ) has a simple pole at s = d + d .Moreover, thanks to (70), Γ(( d + d ) / 2) Res s = d (cid:16) ζ D ( s ) (cid:17) = Γ( d / 2) Res s = d (cid:16) ζ D ( s ) (cid:17) Γ( d / 2) Res s = d (cid:16) ζ D ( s ) (cid:17) . Remark that we can apply the last lemma to our Remark 5.33 in the commutative case,but we could also use A = M n ( C ) acting on C n and D any selfadjoint n × n -matrix. Thisis the way the internal degrees of freedom for fermions are implemented in particle physics,see for instance references in [31]. 58 Spectral action We would like to obtain a good action for any spectral triple and for this it is useful to lookat some examples in physics.In any physical theory based on geometry, the interest of an action functional is, by aminimization process, to exhibit a particular geometry, for instance, trying to distinguishbetween different metrics. This is the case in general relativity with the Einstein–Hilbertaction (with its Riemannian signature). This action is S EH ( g ) := − Z M s g ( x ) dvol g ( x ) (71)where s is the scalar curvature (chosen positive for the sphere). This is nothing else (up toa constant, in dimension 4) than − R D/ − as quoted after (41).This action is interesting for the following reason: Let M be the set of Riemannian metrics g on M such that R M dvol g = 1. By a theorem of Hilbert [5], g ∈ M is a critical point of S EH ( g ) restricted to M if and only if ( M, g ) is an Einstein manifold (the Ricci curvature R of g is proportional by a constant to g : R = c g ). Taking the trace, this means that s g = c dim( M ) and such manifold have a constant scalar curvature.But in the search for invariants under diffeomorphisms, they are more quantities than theEinstein–Hilbert action, a trivial example being R M f (cid:16) s g ( x ) (cid:17) dvol g ( x ) and they are others [43].In this desire to implement gravity in noncommutative geometry, the eigenvalues of the Diracoperator look as natural variables [70]. However we are looking for observables which add upunder disjoint unions of different geometries. In a way, a spectral triple fits quantum field theory since D − can be seen as the propagator(or line element ds ) for (Euclidean) fermions and we can compute Feynman graphs withfermionic internal lines. As glimpsed in section 5.4, the gauge bosons are only derived objectsobtained from internal fluctuations via Morita equivalence given by a choice of a connectionwhich is associated to a one-form in Ω D ( A ). Thus, the guiding principle followed by Connesand Chamseddine is to use a theory which is pure gravity with a functional action based onthe spectral triple, namely which depends on the spectrum of D [11]. They proposed thefollowing Definition 6.1. The spectral action of a spectral triple ( A , H , D ) is defined by S ( D , f, Λ) := Tr (cid:16) f ( D / Λ ) (cid:17) where Λ ∈ R + plays the role of a cut-off and f is any positive function (such that f ( D / Λ ) is a trace-class operator). emark 6.2. We can also define S ( D , f, Λ) = Tr (cid:16) f ( D / Λ) (cid:17) when f is positive and even.With this second definition, S ( D , g, Λ) = Tr (cid:16) f ( D / Λ ) (cid:17) with g ( x ) := f ( x ) . For f , one can think of the characteristic function of [ − , f ( D / Λ) is nothing elsebut the number of eigenvalues of D within [ − Λ , Λ].When this action has an asymptotic series in Λ → ∞ , we deal with an effective theory.Naturally, D has to be replaced by D A which is a just a decoration. To this bosonic part of theaction, one adds a fermionic term h J ψ, D ψ i for ψ ∈ H to get a full action. In the standardmodel of particle physics, this latter corresponds to the integration of the Lagrangian partfor the coupling between gauge bosons and Higgs bosons with fermions. Actually, the finitedimension part of the noncommutative standard model is of KO -dimension 6, thus h ψ, D ψ i has to be replaced by h J ψ, D ψ i for ψ = χψ ∈ H , see [31]. This action plays an important role in physics so it is natural to consider it in the noncommu-tative framework. Recall first the classical situation: let G be a compact Lie group with its Liealgebra g and let A ∈ Ω ( M, g ) be a connection. If F := da + [ A, A ] ∈ Ω ( M, g ) is the cur-vature (or field strength) of A , then the Yang-Mills action is S Y M ( A ) = R M tr( F ∧ ?F ) dvol g .In the abelian case G = U (1), it is the Maxwell action and its quantum version is the quan-tum electrodynamics (QED) since the un-gauged U (1) of electric charge conservation can begauged and its gauging produces electromagnetism [99]. It is conformally invariant when thedimension of M is d = 4.The study of its minima and its critical values can also been made for a spectral triple( A , H , D ) of dimension d [24, 25]: let A ∈ Ω D ( A ) and curvature θ = dA + A ; then it isnatural to consider I ( A ) := Tr Dix ( θ |D| − d )since it coincides (up to a constant) with the previous Yang-Mills action in the commutativecase: if P = θ |D| − d , then Theorems 1.22 and 2.14 give the claim since for the principalsymbol, tr (cid:16) σ P ( x, ξ ) (cid:17) = c tr( F ∧ ?F )( x ).There is nevertheless a problem with the definition of dA : if A = P j π ( a j )[ D , π ( b j )],then dA = P j [ D , π ( a j )][ D , π ( b j )] can be non-zero while A = 0. This ambiguity meansthat, to get a graded differential algebra Ω ∗D ( A ), one must divide by a junk, for instanceΩ D ’ π (Ω /π (cid:16) δ (Ker( π ) ∩ Ω ) (cid:17) where Ω k ( A ) is the set of universal k -forms over A given bythe set of a δa · · · δa k (before representation on H : π ( a δa · · · δa k ) := a [ D , a ] · · · [ D , a k ]).Let H k be the Hilbert space completion of π (Ω k ( A )) with the scalar product defined by h A , A i k := Tr Dix ( A ∗ A |D| − d ) for A j ∈ π (Ω k ( A )).The Yang–Mills action on Ω ( A ) is S Y M ( V ) := h δV + V , δV + V i . (72)It is positive, quartic and gauge invariant under V → π ( u ) V π ( u ∗ ) + π ( u )[ D , π ( u ∗ )] when u ∈ U ( A ). Moreover, S Y M ( V ) = inf { I ( ω ) | ω ∈ Ω ( A ) , π ( ω ) = V } since the above ambiguity disappears when taking the infimum.60his Yang–Mills action can be extended to the equivalent of Hermitean vector bundleson M , namely finitely projective modules over A .The spectral action is more conceptual than the Yang–Mills action since it gives nofundamental role to the distinction between gravity and matter in the artificial decomposition D A = D + A . For instance, for the minimally coupled standard model, the Yang–Mills actionfor the vector potential is part of the spectral action, as far as the Einstein–Hilbert actionfor the Riemannian metric [12].As quoted in [17], the spectral action has conceptual advantages:- Simplicity: when f is a cutoff function, the spectral action is just the counting function.- Positivity: when f is positive (which is the case for a cutoff function), the actionTr (cid:16) f ( D / Λ) (cid:17) ≥ f will insure that the actions for gravity, Yang-Mills, Higgs couplings are all positive and theHiggs mass term is negative.- Invariance: the spectral action has a much stronger invariance group than the usualdiffeomorphism group as for the gravitational action; this is the unitary group of the Hilbertspace H .However, this action is not local. It only becomes so when it is replaced by the asymptoticexpansion: Λ → ∞ The heat kernel method already used in previous sections will give a control of spectral action S ( D , f, Λ) when Λ goes to infinity. Theorem 6.3. Let ( A , H , D ) be a spectral triple with a simple dimension spectrum Sd .We assume that Tr (cid:16) e − t D (cid:17) ∼ t ↓ X α ∈ Sd a α t α with a α = 0 . (73) Then, for the zeta function ζ D defined in (45) a α = Res s = − α (cid:16) Γ( s/ ζ D ( s ) (cid:17) . (74) (i) If α < , ζ D has a pole at − α with a α = Γ( − α ) Res s = − α ζ D ( s ) .(ii) For α = 0 , we get a = ζ D (0) + dim Ker D .(iii) If α > , a α = ζ ( − α ) Res s = − α Γ( s ) .(iv) The spectral action has the asymptotic expansion over the positive part Sd + of Sd : Tr (cid:16) f ( D / Λ) (cid:17) ∼ Λ → + ∞ X β ∈ Sd + f β Λ β − Z |D| − β + f (0) ζ D (0) + · · · (75) where the dependence of the even function f is f β := R ∞ f ( x ) x β − dx and · · · involves thefull Taylor expansion of f at 0.Proof. ( i ) Since Γ( s/ |D| − s = R ∞ e − t D t s/ − dt = R e − t D t s/ − dt + f ( s ), where the func-tion f is holomorphic (since the map s → R ∞ e − tx t s/ − dt is in the Schwartz space), theswap of Tr (cid:16) e − t D (cid:17) with a sum of a α t α and a α R t α + s/ − dt = α a s +2 α yields (74).61 ii ) The regularity of Γ( s/ − ∼ s → s/ s = 0 of R ∞ Tr (cid:16) e − t D (cid:17) t s/ − dt contributes to ζ D (0). This contribution is a R t s/ − dt = a s .( iii ) follows from (74).( iv ) Assume f ( x ) = g ( x ) where g is a Laplace transform: g ( x ) := R ∞ e − sx φ ( s ) ds . Wewill see in Section 6.3 how to relax this hypothesis.Since g ( t D ) = R ∞ e − st D φ ( s ) ds , Tr (cid:16) g ( t D ) (cid:17) ∼ t ↓ P α ∈ Sp a α t α R ∞ s α φ ( s ) ds . When α < s α = Γ( − α ) − R ∞ e − sy y − α − dy and R ∞ s α φ ( s ) ds = Γ( − α ) − R ∞ g ( y ) y − α − dy . ThusTr (cid:16) g ( t D ) (cid:17) ∼ t ↓ X α ∈ Sp − h Res s = − α ζ D ( s ) Z ∞ g ( y ) y − α − dy i t α . Finally (75) follows from ( i ) , ( ii ) and R ∞ g ( y ) y β/ − dy = R ∞ f ( x ) x β − dx .It can be useful to make a connection with (40) of Section 4.3: Corollary 6.4. Assume that the spectral triple ( A , H , D ) has dimension d . If Tr (cid:16) e − t D (cid:17) ∼ t ↓ X k ∈{ , ··· ,d } t ( k − d ) / a k ( D ) + · · · , (76) then S ( D , f, Λ) ∼ Λ →∞ X k ∈{ , ··· ,d } f k Λ k a d − k ( D ) + f (0) a d ( D ) + · · · with f k := k/ R ∞ f ( s ) s k/ − ds .Moreover, a k ( D ) = Γ( d − k ) − Z |D| − d + k for k = 0 , · · · , d − , (77) a d ( D ) = dim Ker D + ζ D (0) . Proof. We rewrite the hypothesis on Tr (cid:16) e − t D (cid:17) asTr (cid:16) e − t D (cid:17) ∼ t ↓ X α ∈{ − d/ , ··· , − / } A α t α + A = X k ∈{ , ··· ,d } t ( k − d ) / a k ( D ) + a d ( D )with a k ( D ) := A ( k − d ) / .For α < 0, we repeat the above proof: S ( D , f, Λ) ∼ Λ →∞ X α ∈{ − d/ , ··· , − / } A α Λ − α − α ) Z ∞ f ( s ) s − α − ds + A = X l ∈{ , ··· ,d } A ( l − d ) / Λ d − l d − l ) / Z ∞ f ( s ) s ( d − l ) / − ds + A f (0)= X l ∈{ , ··· ,d } a l Λ d − l d − l ) / Z ∞ f ( s ) s ( d − l ) / − ds + a d f (0)= X k ∈{ , ··· ,d } a d − k Λ k k/ Z ∞ f ( s ) s k/ − ds + a d f (0) . α < A α = Γ( − α ) Res s = − α Tr (cid:16) |D| − s (cid:17) = Γ( − α ) Res s =0 Tr (cid:16) |D| − ( s − α ) (cid:17) = Γ( − α ) Res s =0 Tr (cid:16) |D| α |D| − s (cid:17) = Γ( − α ) − Z |D| α . Thus, for α = k − d < k = 0 , · · · , d − a k ( D ) = A ( k − d ) / = Γ( d − k ) − Z |D| − d + k . The asymptotics (75) uses the value of ζ D (0) in the constant term Λ , so it is fundamentalto look at its variation under a gauge fluctuation D → D + A as we saw in Theorem 5.17. The spectral action asymptotic behavior S ( D , f, Λ) ∼ Λ → + ∞ ∞ X n =0 c n Λ d − n a n ( D ) (78)has been proved for a smooth function f which is a Laplace transform for an arbitraryspectral triple (with simple dimension spectrum) satisfying (73). However, this hypothesis istoo restrictive since it does not cover the heat kernel case where f ( x ) = e − x .When the triple is commutative and D is a generalized Laplacian on sections of a vectorbundle over a manifold of dimension 4, Estrada–Gracia-Bondía–Várilly proved in [38] thatprevious asymptotics isTr (cid:16) f ( D / Λ ) (cid:17) ∼ π ) (cid:20) rk(E) Z ∞ xf ( x ) dx Λ + b ( D ) Z ∞ f ( x ) dx Λ + ∞ X m =0 ( − m f ( m ) (0) b m +4 ( D ) Λ − m (cid:21) , Λ → ∞ where ( − m b m +4 ( D ) = (4 π ) m ! µ m ( D ) are suitably normalized, integrated moment terms ofthe spectral density of D .The main point is that this asymptotics makes sense in the Cesàro sense (see [38] fordefinition) for f in K ( R ), which is the dual of K ( R ). This latter is the space of smoothfunctions φ such that for some a ∈ R , φ ( k ) ( x ) = O ( | x | a − k ) as | x | → ∞ , for each k ∈ N . Inparticular, the Schwartz functions are in K ( R ) (and even dense).Of course, the counting function is not smooth but is in K ( R ), so such behavior (78) iswrong beyond the first term, but is correct in the Cesàro sense. Actually there are morederivatives of f at 0 as explained on examples in [38, p. 243]. See also Section 9.4. The asymptotic expansion series (76) of the spectral action may or may not converge. It isknown that each function g (Λ − ) defines at most a unique expansion series when Λ → ∞ M is the torus T d as in Example 2.13 with ∆ = δ µν ∂ µ ∂ ν ,Tr( e t ∆ ) = (4 π ) − d/ Vol( T d ) t d/ + O ( t − d/ e − / t ) , thus the asymptotic series Tr( e t ∆ ) ’ (4 π ) − d/ Vol( T d ) t d/ , t → 0, has only one term.In the opposite direction, let now M be the unit four-sphere S and D/ be the usual Diracoperator. By Propostion 5.34, equation (73) yields (see [20]):Tr( e − tD/ ) = t (cid:16) + t + n X k =0 a k t k +2 + O ( t n +3 ) (cid:17) ,a k := ( − k k ! (cid:16) B k +2 k +2 − B k +4 k +4 (cid:17) with Bernoulli numbers B k . Thus t Tr( e − tD/ ) ’ + t + P ∞ k =0 a k t k +2 when t → a k > k ! | B k +4 | k +4 > | B k +4 | = 2 (2 k +4)!(2 π ) k +4 ζ (2 k +4) ’ q π ( k + 2) (cid:16) k +2 πe (cid:17) k +4 → ∞ if k → ∞ .More generally, in the commutative case considered above and when D is a differentialoperator—like a Dirac operator, the coefficients of the asymptotic series of Tr( e − t D ) arelocally defined by the symbol of D at point x ∈ M but this is not true in general: in [46]is given a positive elliptic pseudodifferential such that non-locally computable coefficientsespecially appear in (76) when 2 k > d . Nevertheless, all coefficients are local for 2 k ≤ d .Recall that a locally computable quantity is the integral on the manifold of a local frame-independent smooth function of one variable, depending only on a finite number of derivativesof a finite number of terms in the asymptotic expansion of the total symbol of D . Forinstance, some nonlocal information contained in the ultraviolet asymptotics can be recoveredif one looks at the (integral) kernel of e − t √− ∆ : in T , with Vol( T ) = 2 π , we get [39]Tr( e − t √− ∆ ) = 2 ∞ X n =1 e − tn + 1 = sinh( t )cosh( t ) − = coth( t ) = t ∞ X k =0 B k (2 k )! t k = t [1 + t − t + O ( t )]and the series converges when t < π , since B k (2 k )! = ( − k +1 2 ζ (2 k )(2 π ) k , thus | B k | (2 k )! ’ π ) k when k → ∞ .Thus we have an example where t → ∞ cannot be used with the asymptotic series.Thus the spectral action of Corollary 6.4 precisely encodes these local and nonlocal be-havior which appear or not in its asymptotics for different f . The coefficient of the actionfor the positive part (at least) of the dimension spectrum correspond to renormalized traces,namely the noncommutative integrals of (77). In conclusion, the asymptotics (78) of spectralaction may or may not have nonlocal coefficients.For the flat torus T d , the difference between Tr( e t ∆ ) and its asymptotic series is an termwhich is related to periodic orbits of the geodesic flow on T d . Similarly, the counting func-tion N ( λ ) (number of eigenvalues including multiplicities of ∆ less than λ ) obeys Weyl’s law: N ( λ ) = (4 π ) − d/ Vol( T d )Γ( d/ λ d/ + o ( λ d/ ) — see [1] for a nice historical review on these funda-mental points. The relationship between the asymptotic expansion of the heat kernel andthe formal expansion of the spectral measure is clear: the small- t asymptotics of heat kernelis determined by the large- λ asymptotics of the density of eigenvalues (and eigenvectors).64owever, the latter is defined modulo some average: Cesàro sense as reminded in Section6.3, or Riesz mean of the measure which washes out ultraviolet oscillations, but also givesinformations on intermediate values of λ [39].In [17, 77] are given examples of spectral actions on (compact) commutative geometriesof dimension 4 whose asymptotics have only two terms. In the quantum group SU q (2), thespectral action itself has only 4 terms, independently of the choice of function f .See [63] for more examples. As explained before, the spectral action is non-local. Its localization does not cover allsituations: consider for instance the commutative geometry of a spin manifold M (see Section5.6) of dimension 4. One adds a gauge connection A ∈ Γ ∞ (cid:16) M, End ( S ) (cid:17) to the Dirac operator D/ such that D = iγ µ ( ∂ µ + A µ ), thus with a field strength F µν = ∂ µ A ν − ∂ ν A µ + [ A µ , A ν ]. Wecan apply (33) with P = D and find the coefficients a i (1 , P ) of (39) with i = 0 , , 4. Theexpansion (78) corresponds to a weak field expansion.Moreover a commutative geometry times a finite one where the finite one is algebrais a sum of matrices (like in Remark 5.33) has been deeply and intensively investigatedfor the noncommutative approach to standard model of particle physics, see [21, 31]. Thisapproach offers a lot of interesting perspectives, for instance, the possibility to compute theHiggs representations and mass (for each noncommutative model) is particularly instructive[11, 16, 18, 58, 64, 65, 72, 80]. Of course, since the first term in (75) is a cosmological term, onemay be worried by its large value (for instance in the noncommutative standard model wherethe cutoff is, roughly speaking the Planck scale). At the classical level, one can work withunimodular gravity where the metric (so the Dirac operator) D varies within the set M ofmetrics which preserve the volume as in Section 6.1.1. Thus it remains only (!) to controlthe inflaton: see [14].The spectral action has been computed in [61] for the quantum group SU q (2) which isnot a deformation of SU (2) of the type considered in Section 9.5 on the Moyal plane. It isquite peculiar since (75) has only a finite number of terms.Due to the difficulties to deal with non-compact manifolds (see nevertheless Section 9),the case of spheres S or S × S has been investigated in [17, 20] for instance in the case ofRobertson–Walker metrics.All the machinery of spectral geometry as been recently applied to cosmology, computingthe spectral action in few cosmological models related to inflation, see [67, 77–79, 83, 97].Spectral triples associated to manifolds with boundary have been considered in [15,19,19,59, 60, 62]. The main difficulty is precisely to put nice boundary conditions to the operator D to still get a selfadjoint operator and then, to define a compatible algebra A . This is probablya must to obtain a result in a noncommutative Hamiltonian theory in dimension 1+3.The case of manifolds with torsion has also been studied in [54, 86, 87], and even withboundary in [62]. These works show that the Holst action appears in spectral actions andthat torsion could be detected in a noncommutative world.65 Residues of series and integral, holomorphic contin-uation, etc The aim of this section is to control the holomorphy of series of holomorphic functions.The necessity of a Diophantine condition appears quite naturally. This section has its owninterest, but will be fully applied in the next one devoted to the noncommutative torus. Themain idea is to get a condition which guarantee the commutation of a residue and a series.This section is quite technical, but with only non-difficult notions. Nevertheless, the devilis hidden into the details and I recommend to the reader to have a look at the proofs despitetheir lengths.Reference: [37].Notations:In the following, the prime in P means that we omit terms with division by zero in thesummand. B n (resp. S n − ) is the closed ball (resp. the sphere) of R n with center 0 andradius 1 and the Lebesgue measure on S n − will be noted dS .For any x = ( x , . . . , x n ) ∈ R n we denote by | x | = q x + · · · + x n the Euclidean normand | x | := | x | + · · · + | x n | .By f ( x, y ) (cid:28) y g ( x ) uniformly in x , we mean that | f ( x, y ) | ≤ a ( y ) | g ( x ) | for all x and y for some a ( y ) > In order to be able to compute later the residues of certain series, we prove here the following Theorem 7.1. Let P ( X ) = P dj =0 P j ( X ) ∈ C [ X , · · · , X n ] be a polynomial function where P j is the homogeneous part of P of degree j . The function ζ P ( s ) := X k ∈ Z n P ( k ) | k | s , s ∈ C has a meromorphic continuation to the whole complex plane C .Moreover ζ P ( s ) is not entire if and only if P P := { j | R u ∈ S n − P j ( u ) dS ( u ) = 0 } 6 = ∅ . Inthat case, ζ P has only simple poles at the points j + n , j ∈ P P , with Res s = j + n ζ P ( s ) = Z u ∈ S n − P j ( u ) dS ( u ) . The proof of this theorem is based on the following lemmas. Lemma 7.2. For any polynomial P ∈ C [ X , . . . , X n ] of total degree δ ( P ) := P ni =1 deg X i P and any α ∈ N n , we have ∂ α (cid:16) P ( x ) | x | − s (cid:17) (cid:28) P,α,n (1 + | s | ) | α | | x | − σ −| α | + δ ( P ) uniformly in x ∈ R n , | x | ≥ , where σ = < ( s ) .Proof. By linearity, we may assume without loss of generality that P ( X ) = X γ is a monomial.It is easy to prove (for example by induction on | α | ) that for all α ∈ N n and x ∈ R n \ { } : ∂ α (cid:16) | x | − s (cid:17) = α ! X β,µ ∈ N n β +2 µ = α (cid:16) − s/ | β | + | µ | (cid:17) ( | β | + | µ | )! β ! µ ! x β | x | σ +2( | β | | µ | . 66t follows that for all α ∈ N n , we have uniformly in x ∈ R n , | x | ≥ ∂ α (cid:16) | x | − s (cid:17) (cid:28) α,n (1 + | s | ) | α | | x | − σ −| α | . (79)By Leibniz formula and (79), we have uniformly in x ∈ R n , | x | ≥ ∂ α (cid:16) x γ | x | − s (cid:17) = X β ≤ α (cid:16) αβ (cid:17) ∂ β ( x γ ) ∂ α − β (cid:16) | x | − s (cid:17) (cid:28) γ,α,n X β ≤ α ; β ≤ γ x γ − β (1 + | s | ) | α | −| β | | x | − σ −| α | + | β | (cid:28) γ,α,n (1 + | s | ) | α | | x | − σ −| α | + | γ | . Lemma 7.3. Let P ∈ C [ X , . . . , X n ] be a polynomial of degree d . Then, the difference ∆ P ( s ) := X k ∈ Z n P ( k ) | k | s − Z R n \ B n P ( x ) | x | s dx which is defined for < ( s ) > d + n , extends holomorphically on the whole complex plane C .Proof. We fix in the sequel a function ψ ∈ C ∞ ( R n , R ) such that for all x ∈ R n ≤ ψ ( x ) ≤ , ψ ( x ) = 1 if | x | ≥ ψ ( x ) = 0 if | x | ≤ / . The function f ( x, s ) := ψ ( x ) P ( x ) | x | − s , x ∈ R n and s ∈ C , is in C ∞ ( R n × C ) and dependsholomorphically on s .Lemma 7.2 above shows that f is a “gauged symbol” in the terminology of [53, p. 4].Thus [53, Theorem 2.1] implies that ∆ P ( s ) extends holomorphically on the whole complexplane C . However, to be complete, we will give here a short proof of Lemma 7.3:It follows from the classical Euler–Maclaurin formula that for any function h : R → C ofclass C N +1 satisfying lim | t |→ + ∞ h ( k ) ( t ) = 0 and R R | h ( k ) ( t ) | dt < + ∞ for any k = 0 . . . , N + 1,that we have X k ∈ Z h ( k ) = Z R h ( t ) + ( − N ( N +1)! Z R B N +1 ( t ) h ( N +1) ( t ) dt where B N +1 is the Bernoulli function of order N + 1 (it is a bounded periodic function.)Fix m ∈ Z n − and s ∈ C . Applying to the function h ( t ) := ψ ( m , t ) P ( m , t ) | ( m , t ) | − s (we use Lemma 7.2 to verify hypothesis), we obtain that for any N ∈ N : X m n ∈ Z ψ ( m , m n ) P ( m , m n ) | ( m , m n ) | − s = Z R ψ ( m , t ) P ( m , t ) | ( m , t ) | − s dt + R N ( m ; s ) (80)where R N ( m ; s ) := ( − N ( N +1)! R R B N +1 ( t ) ∂ N +1 ∂x nN +1 ( ψ ( m , t ) P ( m , t ) | ( m , t ) | − s ) dt .By Lemma 7.2, Z R (cid:12)(cid:12)(cid:12)(cid:12) B N +1 ( t ) ∂ N +1 ∂x nN +1 (cid:16) ψ ( m , t ) P ( m , t ) | ( m , t ) | − s (cid:17) (cid:12)(cid:12)(cid:12)(cid:12) dt (cid:28) P,n,N (1 + | s | ) N +1 ( | m | + 1) − σ − N + δ ( P ) . Thus P m ∈ Z n − R N ( m ; s ) converges absolutely and define a holomorphic function in the halfplane { σ = < ( s ) > δ ( P ) + n − N } . 67ince N is an arbitrary integer, by letting N → ∞ and using (80) above, we concludethat: s X ( m ,m n ) ∈ Z n − × Z ψ ( m , m n ) P ( m , m n ) | ( m , m n ) | − s − X m ∈ Z n − Z R ψ ( m , t ) P ( m , t ) | ( m , t ) | − s dt has a holomorphic continuation to the whole complex plane C .After n iterations, we obtain that s X m ∈ Z n ψ ( m ) P ( m ) | m | − s − Z R n ψ ( x ) P ( x ) | x | − s dx has a holomorphic continuation to the whole C .To finish the proof of Lemma 7.3, it is enough to notice that: • ψ (0) = 0 and ψ ( m ) = 1, ∀ m ∈ Z n \ { } ; • s R B n ψ ( x ) P ( x ) | x | − s dx = R { x ∈ R n :1 / ≤| x |≤ } ψ ( x ) P ( x ) | x | − s dx is a holomorphicfunction on C . Proof of Theorem 7.1. Using the polar decomposition of the volume form dx = ρ n − dρ dS in R n , we get for < ( s ) > d + n , Z R n \ B n P j ( x ) | x | s dx = Z ∞ ρ j + n − ρ s Z S n − P j ( u ) dS ( u ) = j + n − s Z S n − P j ( u ) dS ( u ) . Lemma 7.3 now gives the result. Before stating the main result of this section, we give first in the following some preliminariesfrom Diophantine approximation theory: Definition 7.4. (i) Let δ > . A vector a ∈ R n is said to be δ -badly approximable if thereexists c > such that | q.a − m | ≥ c | q | − δ , ∀ q ∈ Z n \ { } and ∀ m ∈ Z .We note BV ( δ ) the set of δ -badly approximable vectors and BV := ∪ δ> BV ( δ ) the set of badlyapproximable vectors.(ii) A matrix Θ ∈ M n ( R ) (real n × n matrices) will be said to be badly approximable ifthere exists u ∈ Z n such that t Θ( u ) is a badly approximable vector of R n . Remark. A classical result from Diophantine approximation asserts that for δ > n , theLebesgue measure of R n \ BV ( δ ) is zero (i.e almost any element of R n is δ − badly approx-imable.)Let Θ ∈ M n ( R ). If its row of index i is a badly approximable vector of R n (i.e. if L i ∈ BV ) then t Θ( e i ) ∈ BV and thus Θ is a badly approximable matrix. It follows thatalmost any matrix of M n ( R ) ≈ R n is badly approximable.The goal of this section is to show the following Theorem 7.5. Let P ∈ C [ X , · · · , X n ] be a homogeneous polynomial of degree d and let b be in S ( Z n × · · · × Z n ) ( q times, q ∈ N ). Then,(i) Let a ∈ R n . We define f a ( s ) := P k ∈ Z n P ( k ) | k | s e πik.a . . If a ∈ Z n , then f a has a meromorphic continuation to the whole complex plane C .Moreover if S is the unit sphere and dS its Lebesgue measure, then f a is not entire if andonly if R u ∈ S n − P ( u ) dS ( u ) = 0 . In that case, f a has only a simple pole at the point d + n ,with Res s = d + n f a ( s ) = R u ∈ S n − P ( u ) dS ( u ) .2. If a ∈ R n \ Z n , then f a ( s ) extends holomorphically to the whole complex plane C .(ii) Suppose that Θ ∈ M n ( R ) is badly approximable. For any ( ε i ) i ∈ {− , , } q , thefunction g ( s ) := X l ∈ ( Z n ) q b ( l ) f Θ P i ε i l i ( s ) extends meromorphically to the whole complex plane C with only one possible pole on s = d + n .Moreover, if we set Z := { l ∈ ( Z n ) q | P qi =1 ε i l i = 0 } and V := P l ∈Z b ( l ) , then1. If V R S n − P ( u ) dS ( u ) = 0 , then s = d + n is a simple pole of g ( s ) and Res s = d + n g ( s ) = V Z u ∈ S n − P ( u ) dS ( u ) . 2. If V R S n − P ( u ) dS ( u ) = 0 , then g ( s ) extends holomorphically to the whole complexplane C .(iii) Suppose that Θ ∈ M n ( R ) is badly approximable. For any ( ε i ) i ∈ {− , , } q , thefunction g ( s ) := X l ∈ ( Z n ) q \Z b ( l ) f Θ P qi =1 ε i l i ( s ) where Z := { l ∈ ( Z n ) q | P qi =1 ε i l i = 0 } extends holomorphically to the whole complex plane C . Proof of Theorem 7.5: First we remark thatIf a ∈ Z n then f a ( s ) = P k ∈ Z n P ( k ) | k | s . So, the point ( i. 1) follows from Theorem 7.1; g ( s ) := P l ∈ ( Z n ) q \Z b ( l ) f Θ P i ε i l i ( s ) + ( P l ∈Z b ( l )) P k ∈ Z n P ( k ) | k | s . Thus, the point ( ii )rises easily from ( iii ) and Theorem 7.1.So, to complete the proof, it remains to prove the items ( i. 2) and ( iii ).The direct proof of ( i. 2) is easy but is not sufficient to deduce ( iii ) of which the proofis more delicate and requires a more precise (i.e. more effective) version of ( i. F := { P ( X )( X + ··· + X n +1) r/ | P ( X ) ∈ C [ X , . . . , X n ] and r ∈ N } . We set g =deg( G ) =deg( P ) − r ∈ Z , the degree of G = P ( X )( X + ··· + X n +1) r/ ∈ F .By convention, we set deg(0) = −∞ . Lemma 7.6. Let a ∈ R n . We assume that d ( a.u, Z ) := inf m ∈ Z | a.u − m | > for some u ∈ Z n . For all G ∈ F , we define formally, F ( G ; a ; s ) := X k ∈ Z n G ( k ) | k | s e πi k.a and F ( G ; a ; s ) := X k ∈ Z n G ( k )( | k | +1) s/ e πi k.a . Then for all N ∈ N , G ∈ F and i ∈ { , } , there exist positive constants C i := C i ( G, N, u ) , B i := B i ( G, N, u ) and A i := A i ( G, N, u ) such that s F i ( G ; α ; s ) extends holomorphicallyto the half-plane {< ( s ) > − N } and verifies in it: F i ( G ; a ; s ) ≤ C i (1 + | s | ) B i (cid:16) d ( a.u, Z ) (cid:17) − A i . emark 7.7. The important point here is that we obtain an explicit bound of F i ( G ; α ; s ) in {< ( s ) > − N } which depends on the vector a only through d ( a.u, Z ) , so depends on u andindirectly on a (in the sequel, a will vary.) In particular the constants C i := C i ( G, N, u ) , B i = B i ( G, N ) and A i := A i ( G, N ) do not depend on the vector a but only on u . This iscrucial for the proof of items ( ii ) and ( iii ) of Theorem 7.5! i = 1 : Let N ∈ N be a fixed integer, and set g := n + N + 1.We will prove Lemma 7.6 by induction on g =deg( G ) ∈ Z . More precisely, in order to provecase i = 1, it suffices to prove that:Lemma 7.6 is true for all G ∈ F with deg( G ) ≤ − g .Let g ∈ Z with g ≥ − g + 1. If Lemma 7.6 is true for all G ∈ F such thatdeg( G ) ≤ g − G ∈ F satisfying deg( G ) = g . • Step 1: Checking Lemma 7.6 for deg( G ) ≤ − g := − ( n + N + 1).Let G ( X ) = P ( X )( X + ··· + X n +1) r/ ∈ F with deg( G ) ≤ − g . It is easy to see that we have uniformlyin s = σ + iτ ∈ C and in k ∈ Z n : | G ( k ) e πi k.a | ( | k | +1) σ/ = | P ( k ) | ( | k | +1) ( r + σ ) / (cid:28) G | k | +1) ( r + σ − deg ( P )) / (cid:28) G | k | +1) ( σ − deg ( G )) / (cid:28) G | k | +1) ( σ + g / . It follows that F ( G ; a ; s ) = P k ∈ Z n G ( k )( | k | +1) s/ e πi k.a converges absolutely and defines a holo-morphic function in the half plane { σ > − N } . Therefore, we have for any s ∈ {< ( s ) > − N } : | F ( G ; a ; s ) | (cid:28) G X k ∈ Z n | k | +1) ( − N + g / (cid:28) G X k ∈ Z n | k | +1) ( n +1) / (cid:28) G . Thus, Lemma 7.6 is true when deg( G ) ≤ − g . • Step 2: Induction.Now let g ∈ Z satisfying g ≥ − g + 1 and suppose that Lemma 7.6 is valid for all G ∈ F withdeg( G ) ≤ g − 1. Let G ∈ F with deg( G ) = g . We will prove that G also verifies conclusionsof Lemma 7.6:There exist P ∈ C [ X , . . . , X n ] of degree d ≥ r ∈ N such that G ( X ) = P ( X )( X + ··· + X n +1) r/ and g =deg( G ) = d − r .Since G ( k ) (cid:28) ( | k | + 1) g/ uniformly in k ∈ Z n , we deduce that F ( G ; a ; s ) converges abso-lutely in { σ = < ( s ) > n + g } .Since k k + u is a bijection from Z n into Z n , it follows that we also have for < ( s ) > n + gF ( G ; a ; s ) = X k ∈ Z n P ( k )( | k | +1) ( s + r ) / e πi k.a = X k ∈ Z n P ( k + u )( | k + u | +1) ( s + r ) / e πi ( k + u ) .a = e πi u.a X k ∈ Z n P ( k + u )( | k | +2 k.u + | u | +1) ( s + r ) / e πi k.a = e πi u.a X α ∈ N n ; | α | = α + ··· + α n ≤ d u α α ! X k ∈ Z n ∂ α P ( k )( | k | +2 k.u + | u | +1) ( s + r ) / e πi k.a = e πi u.a X | α | ≤ d u α α ! X k ∈ Z n ∂ α P ( k )( | k | +1) ( s + r ) / (cid:16) k.u + | u | ( | k | +1) (cid:17) − ( s + r ) / e πi k.a . M := sup( N + n + g, ∈ N . We have uniformly in k ∈ Z n (cid:16) k.u + | u | ( | k | +1) (cid:17) − ( s + r ) / = M X j =0 (cid:16) − ( s + r ) / j (cid:17) ( k.u + | u | ) j ( | k | +1) j + O M,u (cid:16) (1+ | s | ) M +1 ( | k | +1) ( M +1) / (cid:17) . Thus, for σ = < ( s ) > n + d , F ( G ; a ; s ) = e πi u.a X | α | ≤ d u α α ! X k ∈ Z n ∂ α P ( k )( | k | +1) ( s + r ) / (cid:16) k.u + | u | ( | k | +1) (cid:17) − ( s + r ) / e πi k.a = e πi u.a X | α | ≤ d M X j =0 u α α ! (cid:16) − ( s + r ) / j (cid:17) X k ∈ Z n ∂ α P ( k ) ( k.u + | u | ) j ( | k | +1) ( s + r +2 j ) / e πi k.a + O G,M,u (cid:16) (1 + | s | ) M +1 X k ∈ Z n | k | +1) ( σ + M +1 − g ) / (cid:17) . (81)Set I := { ( α, j ) ∈ N n × { , . . . , M } | | α | ≤ d } and I ∗ := I \ { (0 , } .Set also G ( α,j ); u ( X ) := ∂ α P ( X ) ( X.u + | u | ) j ( | X | +1) ( r +2 j ) / ∈ F for all ( α, j ) ∈ I ∗ .Since M ≥ N + n + g , it follows from (81) that(1 − e πi u.a ) F ( G ; a ; s ) = e πi u.a X ( α,j ) ∈ I ∗ u α α ! (cid:16) − ( s + r ) / j (cid:17) F (cid:16) G ( α,j ); u ; α ; s (cid:17) + R N ( G ; a ; u ; s ) (82)where s R N ( G ; a ; u ; s ) is a holomorphic function in the half plane { σ = < ( s ) > − N } , inwhich it satisfies the bound R N ( G ; a ; u ; s ) (cid:28) G,N,u α, j ) ∈ I ∗ ,deg (cid:16) G ( α,j ); u (cid:17) = deg( ∂ α P ) + j − ( r + 2 j ) ≤ d − | α | + j − ( r + 2 j ) = g − | α | − j ≤ g − . Relation (82) and the induction hypothesis imply then that(1 − e πi u.a ) F ( G ; a ; s ) verifies the conclusions of Lemma 7.6 . (83)Since | − e πi u.a | = 2 | sin( πu.a ) | ≥ d ( u.a, Z ), then (83) implies that F ( G ; a ; s ) satisfiesconclusions of Lemma 7.6. This completes the induction and the proof for i = 1. i = 0 : Let N ∈ N be a fixed integer. Let G ( X ) = P ( X )( X + ··· + X n +1) r/ ∈ F and g = deg( G ) = d − r where d ≥ P . Set also M := sup( N + g + n, ∈ N .Since P ( k ) (cid:28) | k | d for k ∈ Z n \ { } , it follows that F ( G ; a ; s ) and F ( G ; a ; s ) convergeabsolutely in the half plane { σ = < ( s ) > n + g } .71oreover, we have for s = σ + iτ ∈ C with σ > n + g : F ( G ; a ; s ) = X k ∈ Z n \{ } G ( k )( | k | +1 − s/ e πi k.a = X k ∈ Z n G ( k )( | k | +1) s/ (cid:16) − | k | +1 (cid:17) − s/ e πi k.a = X k ∈ Z n M X j =0 (cid:16) − s/ j (cid:17) ( − j G ( k )( | k | +1) ( s +2 j ) / e πi k.a + O M (cid:16) (1 + | s | ) M +1 X k ∈ Z n G ( k ) | ( | k | +1) ( σ +2 M +2) / (cid:17) = M X j =0 (cid:16) − s/ j (cid:17) ( − j F ( G ; a ; s + 2 j )+ O M h (1 + | s | ) M +1 (cid:16) X k ∈ Z n G ( k ) | ( | k | +1) ( σ +2 M +2) / (cid:17)i . (84)In addition we have uniformly in s = σ + iτ ∈ C with σ > − N , X k ∈ Z n G ( k ) | ( | k | +1) ( σ +2 M +2) / (cid:28) X k ∈ Z n k | g ( | k | +1) ( − N +2 M +2) / (cid:28) X k ∈ Z n | k | n +1 < + ∞ . So (84) and Lemma 7.6 for i = 1 imply that Lemma 7.6 is also true for i = 0. This completesthe proof of Lemma 7.6. ( i. of Theorem 7.5: Since a ∈ R n \ Z n , there exists i ∈ { , . . . , n } with a i Z . So d ( a.e i , Z ) = d ( a i , Z ) > a satisfies the assumption of Lemma 7.6 with u = e i . Thus, for all N ∈ N , s f a ( s ) = F ( P ; a ; s ) has a holomorphic continuation to the half-plane {< ( s ) > − N } . Itfollows, by letting N → ∞ , that s f a ( s ) has a holomorphic continuation to the wholecomplex plane C . ( iii ) of Theorem 7.5: Let Θ ∈ M n ( R ), ( ε i ) i ∈ {− , , } q and b ∈ S ( Z n × Z n ). We assume that Θ is a badly ap-proximable matrix. Set Z := { l = ( l , . . . , l q ) ∈ ( Z n ) q | P i ε i l i = 0 } and P ∈ C [ X , . . . , X n ]of degree d ≥ σ > n + d : X l ∈ ( Z n ) q \Z | b ( l ) | X k ∈ Z n P ( k ) || k | σ | e πi k. Θ P i ε i l i | (cid:28) P X l ∈ ( Z n ) q \Z | b ( l ) | X k ∈ Z n | k | σ − d (cid:28) P,σ X l ∈ ( Z n ) q \Z | b ( l ) | < + ∞ . So g ( s ) := X l ∈ ( Z n ) q \Z b ( l ) f Θ P i ε i l i ( s ) = X l ∈ ( Z n ) q \Z b ( l ) X k ∈ Z n P ( k ) | k | s e πi k. Θ P i ε i l i converges absolutely in the half plane {< ( s ) > n + d } .Moreover with the notations of Lemma 7.6, we have for all s = σ + iτ ∈ C with σ > n + d : g ( s ) = X l ∈ ( Z n ) q \Z b ( l ) f Θ P i ε i l i ( s ) = X l ∈ ( Z n ) q \Z b ( l ) F ( P ; Θ X i ε i l i ; s ) (85)72ut Θ is badly approximable, so there exists u ∈ Z n and δ, c > | q. t Θ u − m | ≥ c (1 + | q | ) − δ , ∀ q ∈ Z n \ { } , ∀ m ∈ Z . We deduce that ∀ l ∈ ( Z n ) q \ Z , | (cid:16) Θ X i ε i l i (cid:17) .u − m | = | (cid:16)X i ε i l i (cid:17) . t Θ u − m | ≥ c (cid:16) | X i ε i l i | (cid:17) − δ ≥ c (1 + | l | ) − δ . It follows that there exists u ∈ Z n , δ > c > ∀ l ∈ ( Z n ) q \ Z , d (cid:16) (Θ X i ε i l i ) .u ; Z (cid:17) ≥ c (1 + | l | ) − δ . (86) Therefore, for any l ∈ ( Z n ) q \ Z , the vector a = Θ P i ε i l i verifies the assumption of Lemma7.6 with the same u . Moreover δ and c in (86) are also independent on l .We fix now N ∈ N . Lemma 7.6 implies that there exist positive constants C := C ( P, N, u ), B := B i ( P, N, u ) and A := A ( P, N, u ) such that for all l ∈ ( Z n ) q \Z , s F ( P ; Θ P i ε i l i ; s )extends holomorphically to the half plane {< ( s ) > − N } and verifies in it the bound F ( P ; Θ X i ε i l i ; s ) ≤ C (1 + | s | ) B d (cid:16) (Θ X i ε i l i ) .u ; Z (cid:17) − A . This and (86) imply that for any compact set K included in the half plane {< ( s ) > − N } ,there exist two constants C := C ( P, N, c, δ, u, K ) and D := D ( P, N, c, δ, u ) (independent on l ∈ ( Z n ) q \ Z ) such that ∀ s ∈ K and ∀ l ∈ ( Z n ) q \ Z , F ( P ; Θ X i ε i l i ; s ) ≤ C (1 + | l | ) D . (87)It follows that s P l ∈ ( Z n ) q \Z b ( l ) F ( P ; Θ P i ε i l i ; s ) has a holomorphic continuation to thehalf plane {< ( s ) > − N } .This and ( 85) imply that s g ( s ) = P l ∈ ( Z n ) q \Z b ( l ) f Θ P i ε i l i ( s ) has a holomorphic contin-uation to {< ( s ) > − N } . Since N is an arbitrary integer, by letting N → ∞ , it follows that s g ( s ) has a holomorphic continuation to the whole complex plane C which completesthe proof of the theorem. Remark 7.8. By equation (83), we see that a Diophantine condition is sufficient to getLemma 7.6. Our Diophantine condition appears also (in equivalent form) in Connes [23,Prop. 49] (see Remark 4.2 below). The following heuristic argument shows that our conditionseems to be necessary in order to get the result of Theorem 7.5:For simplicity we assume n = 1 (but the argument extends easily to any n ).Let θ ∈ R \ Q . We know that for any l ∈ Z \ { } , g θl ( s ) := X k ∈ Z e πiθlk | k | s = π s − / Γ( 1 − s Γ( s ) h θl (1 − s ) where h θl ( s ) := X k ∈ Z | θl + k | s . So, for any ( a l ) ∈ S ( Z ) , the existence of meromorphic continuation of g ( s ) := P l ∈ Z a l g θl ( s ) is equivalent to the existence of meromorphic continuation of h ( s ) := X l ∈ Z a l h θl ( s ) = X l ∈ Z a l X k ∈ Z | θl + k | s . So, for at least one σ ∈ R , we must have | a l || θl + k | σ = O (1) uniformly in k, l ∈ Z ∗ . It follows that for any ( a l ) ∈ S ( Z ) , | θl + k | (cid:29) | a l | /σ uniformly in k, l ∈ Z ∗ . Therefore,our Diophantine condition seems to be necessary. .2.5 Commutation between sum and residue Let p ∈ N . Recall that S (( Z n ) p ) is the set of the Schwartz sequences on ( Z n ) p . In other words, b ∈ S (( Z n ) p ) if and only if for all r ∈ N , (1 + | l | + · · · | l p | ) r | b ( l , · · · , l p ) | is bounded on( Z n ) p . We note that if Q ∈ R [ X , · · · , X np ] is a polynomial, ( a j ) ∈ S ( Z n ) p , b ∈ S ( Z n ) and φ a real-valued function, then l := ( l , · · · , l p ) e a ( l ) b ( − b l p ) Q ( l ) e iφ ( l ) is a Schwartz sequenceon ( Z n ) p , where e a ( l ) := a ( l ) · · · a p ( l p ) , b l i := l + . . . + l i . In the following, we will use several times the fact that for any ( k, l ) ∈ ( Z n ) such that k = 0 and k = − l , we have 1 | k + l | = 1 | k | − k.l + | l | | k | | k + l | . (88) Lemma 7.9. There exists a polynomial P ∈ R [ X , · · · , X p ] of degree p and with positivecoefficients such that for any k ∈ Z n , and l := ( l , · · · , l p ) ∈ ( Z n ) p such that k = 0 and k = − b l i for all ≤ i ≤ p , the following holds: | k + b l | . . . | k + b l p | ≤ | k | p P ( | l | , · · · , | l p | ) . Proof. Let’s fix i such that 1 ≤ i ≤ p . Using two times (88), Cauchy–Schwarz inequality andthe fact that | k + b l i | ≥ 1, we get | k + b l i | ≤ | k | + | k || b l i | + | b l i | | k | + (2 | k || b l i | + | b l i | ) | k | | k + b l i | ≤ | k | + | k | | b l i | + (cid:16) | k | + | k | (cid:17) | b l i | + | k | | b l i | + | k | | b l i | . Since | k | ≥ 1, and | b l i | j ≤ | b l i | if 1 ≤ j ≤ 4, we find | k + b l i | ≤ | k | X j =0 | b l i | j ≤ | k | (cid:16) | b l i | (cid:17) ≤ | k | (cid:16) X pj =1 | l j | ) (cid:17) , | k + b l | ... | k + b l p | ≤ p | k | p (cid:16) X pj =1 | l j | ) (cid:17) p . Taking P ( X , · · · , X p ) := 5 p (cid:16) P pj =1 X j ) (cid:17) p now gives the result. Lemma 7.10. Let b ∈ S (( Z n ) p ) , p ∈ N , P j ∈ R [ X , · · · , X n ] be a homogeneous polynomialfunction of degree j , k ∈ Z n , l := ( l , · · · , l p ) ∈ ( Z n ) p , r ∈ N , φ be a real-valued function on Z n × ( Z n ) p and h ( s, k, l ) := b ( l ) P j ( k ) e iφ ( k,l ) | k | s + r | k + b l | · · · | k + b l p | , with h ( s, k, l ) := 0 if, for k = 0 , one of the denominators is zero.For all s ∈ C such that < ( s ) > n + j − r − p , the series H ( s ) := X ( k,l ) ∈ ( Z n ) p +1 h ( s, k, l ) is absolutely summable. In particular, X k ∈ Z n X l ∈ ( Z n ) p h ( s, k, l ) = X l ∈ ( Z n ) p X k ∈ Z n h ( s, k, l ) . roof. Let s = σ + iτ ∈ C such that σ > n + j − r − p . By Lemma 7.9 we get, for k = 0, | h ( s, k, l ) | ≤ | b ( l ) P j ( k ) | | k | − r − σ − p P ( l ) , where P ( l ) := P ( | l | , · · · , | l p | ) and P is a polynomial of degree 4 p with positive coefficients.Thus, | h ( s, k, l ) | ≤ F ( l ) G ( k ) where F ( l ) := | b ( l ) | P ( l ) and G ( k ) := | P j ( k ) || k | − r − σ − p . Thesummability of P l ∈ ( Z n ) p F ( l ) is implied by the fact that b ∈ S (( Z n ) p ). The summability of P k ∈ Z n G ( k ) is a consequence of the fact that σ > n + j − r − p . Finally, as a product of twosummable series, P k,l F ( l ) G ( k ) is a summable series, which proves that P k,l h ( s, k, l ) is alsoabsolutely summable. Definition 7.11. Let f be a function on D × ( Z n ) p where D is an open neighborhood of in C .We say that f satisfies (H1) if and only if there exists ρ > such that(i) for any l , s f ( s, l ) extends as a holomorphic function on U ρ , where U ρ is theopen disk of center 0 and radius ρ ,(ii) if k H ( · , l ) k ∞ ,ρ := sup s ∈ U ρ | H ( s, l ) | , the series P l ∈ ( Z n ) p k H ( · , l ) k ∞ ,ρ is summable.We say that f satisfies (H2) if and only if there exists ρ > such that(i) for any l , s f ( s, l ) extends as a holomorphic function on U ρ − { } ,(ii) for any δ such that < δ < ρ , the series P l ∈ ( Z n ) p k H ( · , l ) k ∞ ,δ,ρ is summable,where k H ( · , l ) k ∞ ,δ,ρ := sup δ< | s | <ρ | H ( s, l ) | . Remark 7.12. Note that (H1) implies (H2). Moreover, if f satisfies (H1) (resp. (H2)for ρ > , then it is straightforward to check that f : s P l ∈ ( Z n ) p f ( s, l ) extends as anholomorphic function on U ρ (resp. on U ρ \ { } ). Corollary 7.13. With the same notations of Lemma 7.10, suppose that r + 2 p − j > n , then,the function H ( s, l ) := P k ∈ Z n h ( s, k, l ) satisfies (H1).Proof. ( i ) Let’s fix ρ > ρ < r + 2 p − j − n . Since r + 2 p − j > n , U ρ is insidethe half-plane of absolute convergence of the series defined by H ( s, l ). Thus, s H ( s, l ) isholomorphic on U ρ .( ii ) Since (cid:12)(cid:12)(cid:12) | k | − s (cid:12)(cid:12)(cid:12) ≤ | k | ρ for all s ∈ U ρ and k ∈ Z n \ { } , we get as in the above proof | h ( s, k, l ) | ≤ | b ( l ) P j ( k ) | | k | − r + ρ − p P ( | l | , · · · , | l p | ) . Since ρ < r + 2 p − j − n , the series P k ∈ Z n | P j ( k ) || k | − r + ρ − p is summable.Thus, k H ( · , l ) k ∞ ,ρ ≤ K F ( l ) where K := P k | P j ( k ) || k | − r + ρ − p < ∞ . We have alreadyseen that the series P l F ( l ) is summable, so we get the result.We note that if f and g both satisfy (H1) (or (H2)), then so does f + g . In the following,we will use the equivalence relation f ∼ g ⇐⇒ f − g satisfies (H1) . Lemma 7.14. Let f and g be two functions on D × ( Z n ) p where D is an open neighborhoodof in C , such that f ∼ g and such that g satisfies (H2). Then Res s =0 X l ∈ ( Z n ) p f ( s, l ) = X l ∈ ( Z n ) p Res s =0 g ( s, l ) . roof. Since f ∼ g , f satisfies (H2) for a certain ρ > 0. Let’s fix η such that 0 < η < ρ anddefine C η as the circle of center 0 and radius η . We haveRes s =0 g ( s, l ) = Res s =0 f ( s, l ) = πi I C η f ( s, l ) ds = Z I u ( t, l ) dt . where I = [0 , π ] and u ( t, l ) := π ηe it f ( η e it , l ). The fact that f satisfies (H2) entails that theseries P l ∈ ( Z n ) p k f ( · , l ) k ∞ ,C η is summable. Thus, since k u ( · , l ) k ∞ = π η k f ( · , l ) k ∞ ,C η , the series P l ∈ ( Z n ) p k u ( · , l ) k ∞ is summable, so, R I P l ∈ ( Z n ) p u ( t, l ) dt = P l ∈ ( Z n ) p R I u ( t, l ) dt which gives theresult. Since, we will have to compute residues of series, let us introduce the following Definition 7.15. ζ ( s ) := ∞ X n =1 n − s ,Z n ( s ) := X k ∈ Z n | k | − s ,ζ p ,...,p n ( s ) := X k ∈ Z n k p · · · k p n n | k | s , for p i ∈ N , where ζ ( s ) is the Riemann zeta function (see [56] or [36]).By the symmetry k → − k , it is clear that these functions ζ p ,...,p n all vanish for odd valuesof p i .Let us now compute ζ , ··· , , i , ··· , , j , ··· , ( s ) in terms of Z n ( s ):Since ζ , ··· , , i , ··· , , j , ··· , ( s ) = A i ( s ) δ ij , exchanging the components k i and k j , we get ζ , ··· , , i , ··· , , j , ··· , ( s ) = δ ij n Z n ( s − . Similarly, X Z n k k | k | s +8 = n ( n − Z n ( s + 4) − n − X Z n k | k | s +8 but it is difficult to write explicitly ζ p ,...,p n ( s ) in terms of Z n ( s − 4) and other Z n ( s − m )when at least four indices p i are non zero.When all p i are even, ζ p ,...,p n ( s ) is a nonzero series of fractions P ( k ) | k | s where P is a homo-geneous polynomial of degree p + · · · + p n . Theorem 7.1 now gives us the following Proposition 7.16. ζ p ,...,p n has a meromorphic extension to the whole plane with a uniquepole at n + p + · · · + p n . This pole is simple and the residue at this pole is Res s = n + p + ··· + p n ζ p ,...,p n ( s ) = 2 Γ( p +12 ) ··· Γ( p n +12 )Γ( n + p + ··· + p n (89) when all p i are even or this residue is zero otherwise.In particular, for n = 2 , Res s =0 X k ∈ Z k i k j | k | s +4 = δ ij π , (90)76 nd for n = 4 , Res s =0 X k ∈ Z k i k j | k | s +6 = δ ij π , Res s =0 X k ∈ Z k i k j k l k m | k | s +8 = ( δ ij δ lm + δ il δ jm + δ im δ jl ) π . (91) Proof. Equation (89) follows from Theorem (7.1)Res s = n + p + ··· + p n ζ p ,...,p n ( s ) = Z k ∈ S n − k p · · · k p n n dS ( k )and standard formulae (see for instance [100, VIII,1;22]). Equation (90) is a straightforwardconsequence of Equation (89). Equation (91) can be checked for the cases i = j = l = m and i = j = l = m .Remark that Z n ( s ) is an Epstein zeta-function which is associated to the quadratic form q ( x ) := x + ... + x n , so Z n satisfies the following functional equation Z n ( s ) = π s − n/ Γ( n/ − s/ s/ − Z n ( n − s ) . Since π s − n/ Γ( n/ − s/ 2) Γ( s/ − = 0 for any negative even integer n and Z n ( s ) is mero-morphic on C with only one pole at s = n with residue 2 π n/ Γ( n/ − according to previousproposition, so we get Z n (0) = − 1. We have proved thatRes s =0 Z n ( s + n ) = 2 π n/ Γ( n/ − , (92) Z n (0) = − . (93)There are many applications of Proposition 7.16 for instance in ζ -regularization, multiplica-tive anomalies or Casimir effect, see for instance [36]. Let n, q ∈ N , q ≥ 2, and p = ( p , . . . , p q − ) ∈ N q − .Set I := { i | p i = 0 } and assume that I = ∅ and I := { α = ( α i ) i ∈ I | ∀ i ∈ I α i = ( α i, , . . . , α i,p i ) ∈ N p i } = Y i ∈ I N p i . We will use in the sequel also the following notations:- for x = ( x , . . . , x t ) ∈ R t recall that | x | = | x | + · · · + | x t | and | x | = q x + · · · + x t ;- for all α = ( α i ) i ∈ I ∈ I = Q i ∈ I N p i , | α | = X i ∈ I | α i | = X i ∈ I p i X j =1 | α i,j | and (cid:16) / α (cid:17) = Y i ∈ I (cid:16) / α i (cid:17) = Y i ∈ I p i Y j =1 (cid:16) / α i,j (cid:17) . .4.1 A family of polynomials In this paragraph we define a family of polynomials which plays an important role later.Consider first the variables:- for X , . . . , X n we set X = ( X , . . . , X n );- for any i = 1 , . . . , q , we consider the variables Y i, , . . . , Y i,n and set Y i := ( Y i, , . . . , Y i,n )and Y := ( Y , . . . , Y q );- for Y = ( Y , . . . , Y q ), we set for any 1 ≤ j ≤ q , e Y j := Y + · · · + Y j + Y q +1 + · · · + Y q + j .We define for all α = ( α i ) i ∈ I ∈ I = Q i ∈ I N p i the polynomial P α ( X, Y ) := Y i ∈ I p i Y j =1 (2 h X, e Y i i + | e Y i | ) α i,j . (94)It is clear that P α ( X, Y ) ∈ Z [ X, Y ], deg X P α ≤ | α | and deg Y P α ≤ | α | .Let us fix a polynomial Q ∈ R [ X , · · · , X n ] and note d := deg Q . For α ∈ I , we want toexpand P α ( X, Y ) Q ( X ) in homogeneous polynomials in X and Y so defining L ( α ) := { β ∈ N (2 q +1) n | | β | − d β ≤ | α | and d β ≤ | α | + d } where d β := P n β i , we set (cid:16) / α (cid:17) P α ( X, Y ) Q ( X ) =: X β ∈ L ( α ) c α,β X β Y β where c α,β ∈ R , X β := X β · · · X β n n and Y β := Y β n +1 , · · · Y β q +1) n q,n . By definition, X β is ahomogeneous polynomial of degree in X equals to d β . We note M α,β ( Y ) := c α,β Y β . In this section we will prove the following result, used in Proposition 8.5 for the computationof the spectrum dimension of the noncommutative torus: Theorem 7.17. (i) Let π Θ be a badly approximable matrix, and e a ∈ S (cid:16) ( Z n ) q (cid:17) . Then s f ( s ) := X l ∈ [( Z n ) q ] e a l X k ∈ Z n q − Y i =1 | k + e l i | p i | k | − s Q ( k ) e ik. Θ P q l j has a meromorphic continuation to the whole complex plane C with at most simple possiblepoles at the points s = n + d + | p | − m where m ∈ N .(ii) Let m ∈ N and set I ( m ) := { ( α, β ) ∈ I × N (2 q +1) n | β ∈ L ( α ) where we have taken m = 2 | α | − d β + d } . Then I ( m ) is a finite set and s = n + d + | p | − m is a pole of f if andonly if C ( f, m ) := X l ∈ Z e a l X ( α,β ) ∈ I ( m ) M α,β ( l ) Z u ∈ S n − u β dS ( u ) = 0 , with Z := { l | P q l j = 0 } and the convention P ∅ = 0 . In that case s = n + d + | p | − m is asimple pole of residue Res s = n + d + | p | − m f ( s ) = C ( f, m ) . 78n order to prove the theorem above we need the following Lemma 7.18. For all N ∈ N we have q − Y i =1 | k + e l i | p i = X α =( α i ) i ∈ I ∈ Q i ∈ I { ,...,N } pi (cid:16) / α (cid:17) P α ( k,l ) | k | | α | −| p | + O N ( | k | | p | − ( N +1) / ) uniformly in k ∈ Z n and l ∈ ( Z n ) q such that | k | > U ( l ) := 36 ( P q − i =1 , i = q | l i | ) .Proof. For i = 1 , . . . , q − 1, we have uniformly in k ∈ Z n and l ∈ ( Z n ) q with | k | > U ( l ), (cid:12)(cid:12)(cid:12) h k, e l i i + | e l i | (cid:12)(cid:12)(cid:12) | k | ≤ √ U ( l )2 | k | < √ | k | . (95)In that case, | k + e l i | = (cid:16) | k | + 2 h k, e l i i + | e l i | (cid:17) / = | k | (cid:16) h k, e l i i + | e l i | | k | (cid:17) / = ∞ X u =0 (cid:16) / u (cid:17) | k | u − P iu ( k, l )where for all i = 1 , . . . , q − u ∈ N , P iu ( k, l ) := (cid:16) h k, e l i i + | e l i | (cid:17) u , with the convention P i ( k, l ) := 1.In particular P iu ( k, l ) ∈ Z [ k, l ], deg k P iu ≤ u and deg l P iu ≤ u . Inequality (95) impliesthat for all i = 1 , . . . , q − u ∈ N , | k | u | P iu ( k, l ) | ≤ (cid:16) q | k | (cid:17) − u uniformly in k ∈ Z n and l ∈ ( Z n ) q such that | k | > U ( l ).Let N ∈ N . We deduce from the previous that for any k ∈ Z n and l ∈ ( Z n ) q with | k | > U ( l ) and for all i = 1 , . . . , q − 1, we have | k + e l i | = N X u =0 (cid:16) / u (cid:17) | k | u − P iu ( k, l ) + O (cid:16) X u>N | k | | (cid:16) / u (cid:17) | (2 q | k | ) − u (cid:17) = N X u =0 (cid:16) / u (cid:17) | k | u − P iu ( k, l ) + O N (cid:16) | k | ( N − / (cid:17) . It follows that for any N ∈ N , we have uniformly in k ∈ Z n and l ∈ ( Z n ) q with | k | > U ( l )and for all i ∈ I , | k + e l i | p i = X α i ∈{ ,...,N } pi (cid:16) / α i (cid:17) | k | | αi | − pi P iα i ( k, l ) + O N (cid:16) | k | ( N +1) / − pi (cid:17) where P iα i ( k, l ) = Q p i j =1 P iα i,j ( k, l ) for all α i = ( α i, , . . . , α i,p i ) ∈ { , . . . , N } p i and Y i ∈ I | k + e l i | p i = X α =( α i ) ∈ Q i ∈ I { ,...,N } pi (cid:16) / α (cid:17) | k | | α | −| p | P α ( k, l ) + O N (cid:16) | k | ( N +1) / −| p | (cid:17) where P α ( k, l ) = Q i ∈ I P iα i ( k, l ) = Q i ∈ I Q p i j =1 P iα i,j ( k, l ).79 roof of Theorem 7.17. ( i ) All n , q , p = ( p , . . . , p q − ) and e a ∈ S (( Z n ) q ) are fixed as aboveand we define formally for any l ∈ ( Z n ) q F ( l, s ) := X k ∈ Z n q − Y i =1 | k + e l i | p i Q ( k ) e ik. Θ P q l j | k | − s . (96)Thus, still formally, f ( s ) := X l ∈ ( Z n ) q e a l F ( l, s ) . (97)It is clear that F ( l, s ) converges absolutely in the half plane { σ = < ( s ) > n + d + | p | } where d = deg Q .Let N ∈ N . Lemma 7.18 implies that for any l ∈ ( Z n ) q and for s ∈ C such that σ > n + | p | + d , F ( l, s ) = X | k |≤ U ( l ) q − Y i =1 | k + e l i | p i Q ( k ) e ik. Θ P q l j | k | − s + X α =( α i ) i ∈ I ∈ Q i ∈ I { ,...,N } pi (cid:16) / α (cid:17) X | k | >U ( l ) 1 | k | s +2 | α | −| p | P α ( k, l ) Q ( k ) e ik. Θ P q l j + G N ( l, s ) . where s G N ( l, s ) is a holomorphic function in the half-plane D N := { σ > n + d + | p | − N +12 } and verifies in it the bound G N ( l, s ) (cid:28) N,σ l .It follows that F ( l, s ) = X α =( α i ) i ∈ I ∈ Q i ∈ I { ,...,N } pi H α ( l, s ) + R N ( l, s ) , (98)where H α ( l, s ) := X k ∈ Z n (cid:16) / α (cid:17) | k | s +2 | α | −| p | P α ( k, l ) Q ( k ) e ik. Θ P q l j ,R N ( l, s ) := X | k |≤ U ( l ) q − Y i =1 | k + e l i | p i Q ( k ) e ik. Θ P q l j | k | − s − X | k |≤ U ( l ) X α =( α i ) i ∈ I ∈ Q i ∈ I { ,...,N } pi (cid:16) / α (cid:17) P α ( k,l ) | k | s +2 | α | −| p | Q ( k ) e ik. Θ P q l j + G N ( l, s ) . In particular there exists A ( N ) > s R N ( l, s ) extends holomorphically to thehalf-plane D N and verifies in it the bound R N ( l, s ) (cid:28) N,σ | l | A ( N ) uniformly in l .Let us note formally h α ( s ) := X l e a l H α ( l, s ) . Equation (98) and R N ( l, s ) (cid:28) N,σ | l | A ( N ) imply that f ( s ) ∼ N X α =( α i ) i ∈ I ∈ Q i ∈ I { ,...,N } pi h α ( s ) , (99)where ∼ N means modulo a holomorphic function in D N .80ecall the decomposition (cid:16) / α (cid:17) P α ( k, l ) Q ( k ) = P β ∈ L ( α ) M α,β ( l ) k β and we decompose sim-ilarly h α ( s ) = P β ∈ L ( α ) h α,β ( s ) . Theorem 7.5 now implies that for all α = ( α i ) i ∈ I ∈ Q i ∈ I { , . . . , N } p i and β ∈ L ( α ),- the map s h α,β ( s ) has a meromorphic continuation to the whole complex plane C with only one simple possible pole at s = n + | p | − | α | + d β ,- the residue at this point is equal toRes s = n + | p | − | α | + d β h α,β ( s ) = X l ∈Z e a l M α,β ( l ) Z u ∈ S n − u β dS ( u ) (100)where Z := { l ∈ ( Z ) n ) q : P q l j = 0 } . If the right hand side is zero, h α,β ( s ) is holomorphicon C .By (99), we deduce therefore that f ( s ) has a meromorphic continuation on the halfplane D N , with only simple possible poles in the set { n + | p | + k : − N | p | ≤ k ≤ d } . Takingnow N → ∞ yields the result.( ii ) Let m ∈ N and set I ( m ) := { ( α, β ) ∈ I × N (2 q +1) n | β ∈ L ( α ) and m = 2 | α | − d β + d } .If ( α, β ) ∈ I ( m ), then | α | ≤ m and | β | ≤ m + d , so I ( m ) is finite.With a chosen N such that 2 N | p | + d > m , we get by (99) and (100)Res s = n + d + | p | − m f ( s ) = X l ∈Z e a l X ( α,β ) ∈ I ( m ) M α,β ( l ) Z u ∈ S n − u β dS ( u ) = C ( f, m )with the convention P ∅ = 0. Thus, n + d + | p | − m is a pole of f if and only if C ( f, m ) = 0.81 The noncommutative torus The aim of this section is to compute the spectral action of the noncommutative torus.After the basic definitions, the result is presented in Theorem 8.13. Due to a fundamentalappearance of small divisors, the number theory is involved via a Diophantine condition. As aconsequence, the result which essentially says that the spectral action of the noncommutativetorus coincide with the action of the ordinary torus (up few constants) is awfully technicaland use the machinery of Section 7. A bunch of proofs are not given, but the essential lemmasare here: they show to the reader how life can be hard in noncommutative geometry!Reference: [37]. Let C ∞ ( T n Θ ) be the smooth noncommutative n -torus associated to a non-zero skew-symmetricdeformation matrix Θ ∈ M n ( R ). It was introduced by Rieffel [95] and Connes [22] to gener-alize the n -torus T n .This means that C ∞ ( T n Θ ) is the algebra generated by n unitaries u i , i = 1 , . . . , n subject tothe relations u l u j = e i Θ lj u j u l , (101)and with Schwartz coefficients: an element a ∈ C ∞ ( T n Θ ) can be written as a = P k ∈ Z n a k U k ,where { a k } ∈ S ( Z n ) with the Weyl elements defined by U k := e − i k.χk u k · · · u k n n ,k ∈ Z n , relation (101) reads U k U q = e − i k. Θ q U k + q , and U k U q = e − ik. Θ q U q U k (102)where χ is the matrix restriction of Θ to its upper triangular part. Thus unitary operators U k satisfy U ∗ k = U − k and [ U k , U l ] = − i sin( k. Θ l ) U k + l . Let τ be the trace on C ∞ ( T n Θ ) defined by τ (cid:16) X k ∈ Z n a k U k (cid:17) := a and H τ be the GNS Hilbert space obtained by completion of C ∞ ( T n Θ ) with respect of thenorm induced by the scalar product h a, b i := τ ( a ∗ b ) . On H τ = { P k ∈ Z n a k U k | { a k } k ∈ l ( Z n ) } , we consider the left and right regular repre-sentations of C ∞ ( T n Θ ) by bounded operators, that we denote respectively by L ( . ) and R ( . ).Let also δ µ , µ ∈ { , . . . , n } , be the n (pairwise commuting) canonical derivations, definedby δ µ ( U k ) := ik µ U k . (103)82e need to fix notations: let A Θ := C ∞ ( T n Θ ) acting on H := H τ ⊗ C m with n = 2 m or n = 2 m + 1 (i.e., m = b n c is the integer part of n ), the square integrablesections of the trivial spin bundle over T n .Each element of A Θ is represented on H as L ( a ) ⊗ m . The Tomita conjugation J ( a ) := a ∗ satisfies [ J , δ µ ] = 0 and we define J := J ⊗ C where C is an operator on C m . The Dirac-like operator is given by D := − i δ µ ⊗ γ µ , (104)where we use hermitian Dirac matrices γ . It is defined and symmetric on the dense subsetof H given by C ∞ ( T n Θ ) ⊗ C m . We still note D its selfadjoint extension. This implies C γ α = − εγ α C , (105)and D U k ⊗ e i = k µ U k ⊗ γ µ e i , where ( e i ) is the canonical basis of C m . Moreover, C = ± m depending on the parity of m . Finally, one introduces the chirality, which in the even case is χ := id ⊗ ( − i ) m γ · · · γ n . This yields a spectral triple: Theorem 8.1. The 5-tuple ( A Θ , H , D , J, χ ) is a real regular spectral triple of dimension n .It satisfies the finiteness and orientability conditions of Definition 5.2. It is n -summable andits KO -dimension is also n . We do not give a proof since most of its arguments will be emphasized in this section; seehowever [25, 50] for a specific proof.For instance, we prove in Proposition 8.5 that this triple has simple dimension spectrumwhen Θ is badly approximable (see Definition 7.4).The perturbed Dirac operator V u D V ∗ u by the unitary V u := (cid:16) L ( u ) ⊗ m (cid:17) J (cid:16) L ( u ) ⊗ m (cid:17) J − , defined for every unitary u ∈ A , uu ∗ = u ∗ u = U , must satisfy condition J D = (cid:15) D J (which isequivalent to H being endowed with a structure of A Θ -bimodule). This yields the necessityof a symmetrized covariant Dirac operator D A := D + A + (cid:15)J A J − V u D V ∗ u = D L ( u ) ⊗ m [ D ,L ( u ∗ ) ⊗ m ] : in fact, for a ∈ A Θ , using J L ( a ) J − = R ( a ∗ ), we get (cid:15)J (cid:16) L ( a ) ⊗ γ α (cid:17) J − = − R ( a ∗ ) ⊗ γ α and that the representation L and the anti-representation R are C -linear, commute andsatisfy [ δ α , L ( a )] = L ( δ α a ) , [ δ α , R ( a )] = R ( δ α a ) . This induces some covariance property for the Dirac operator: one checks that for all k ∈ Z n , L ( U k ) ⊗ m [ D , L ( U ∗ k ) ⊗ m ] = 1 ⊗ ( − k µ γ µ ) , (106)so with (105), we get U k [ D , U ∗ k ] + (cid:15)J U k [ D , U ∗ k ] J − = 0 and V U k D V ∗ U k = D = D L ( U k ) ⊗ m [ D ,L ( U ∗ k ) ⊗ m ] . (107)Moreover, we get the gauge transformation (see Lemma 5.13): V u D A V ∗ u = D γ u ( A ) (108)where the gauged transform one-form of A is γ u ( A ) := u [ D , u ∗ ] + uAu ∗ , (109)with the shorthand L ( u ) ⊗ m −→ u . As a consequence, the spectral action is gauge invariant: S ( D A , f, Λ) = S ( D γ u ( A ) , f, Λ) . An arbitrary selfadjoint one-form A ∈ Ω D ( A ), can be written as A = L ( − iA α ) ⊗ γ α , A α = − A ∗ α ∈ A Θ , (110)thus D A = − i (cid:16) δ α + L ( A α ) − R ( A α ) (cid:17) ⊗ γ α . (111)Defining ˜ A α := L ( A α ) − R ( A α ) , we get D A = − g α α ( δ α + ˜ A α )( δ α + ˜ A α ) ⊗ m − Ω α α ⊗ γ α α where γ α α := ( γ α γ α − γ α γ α ) , Ω α α := [ δ α + ˜ A α , δ α + ˜ A α ] = L ( F α α ) − R ( F α α )with F α α := δ α ( A α ) − δ α ( A α ) + [ A α , A α ] . (112)In summary, D A = − δ α α (cid:18) δ α + L ( A α ) − R ( A α ) (cid:19)(cid:18) δ α + L ( A α ) − R ( A α ) (cid:19) ⊗ m − (cid:16) L ( F α α ) − R ( F α α ) (cid:17) ⊗ γ α α . (113)84 .2 Kernels and dimension spectrum We now compute the kernel of the perturbed Dirac operator: Proposition 8.2. (i) Ker D = U ⊗ C m , so dim Ker D = 2 m .(ii) For any selfadjoint one-form A , Ker D ⊆ Ker D A .(iii) For any unitary u ∈ A , Ker D γ u ( A ) = V u Ker D A .Proof. ( i ) Let ψ = P k,j c k,j U k ⊗ e j ∈ Ker D . Thus, 0 = D ψ = P k,i c k,j | k | U k ⊗ e j whichentails that c k,j | k | = 0 for any k ∈ Z n and 1 ≤ j ≤ m . The result follows.( ii ) Let ψ ∈ Ker D . So, ψ = U ⊗ v with v ∈ C m and from (111), we get D A ψ = D ψ + ( A + (cid:15)J AJ − ) ψ = ( A + (cid:15)J AJ − ) ψ = − i [ A α , U ] ⊗ γ α v = 0since U is the unit of the algebra, which proves that ψ ∈ Ker D A .( iii ) This is a direct consequence of (108). Corollary 8.3. Let A be a selfadjoint one-form. Then Ker D A = Ker D in the followingcases:(i) A = A u := L ( u ) ⊗ m [ D , L ( u ∗ ) ⊗ m ] when u is a unitary in A .(ii) || A || < .(iii) The matrix π Θ has only integral coefficients.Proof. ( i ) This follows from previous result because V u ( U ⊗ v ) = U ⊗ v for any v ∈ C m .( ii ) Let ψ = P k,j c k,j U k ⊗ e j be in Ker D A (so P k,j | c k,j | < ∞ ) and φ := P j c ,j U ⊗ e j .Thus ψ := ψ − φ ∈ Ker D A since φ ∈ Ker D ⊆ Ker D A and || X = k ∈ Z n , j c k,j k α U k ⊗ γ α e j || = ||D ψ || = || − ( A + (cid:15)J AJ − ) ψ || ≤ || A || || ψ || < || ψ || . Defining X k := P α k α γ α , X k = P α | k α | m is invertible and the vectors { U k ⊗ X k e j } = k ∈ Z n , j are orthogonal in H , so X = k ∈ Z n , j (cid:16) X α | k α | (cid:17) | c k,j | < X = k ∈ Z n , j | c k,j | which is possible only if c k,j = 0 , ∀ k, j that is ψ = 0 and ψ = φ ∈ Ker D .( iii ) This is a consequence of the fact that the algebra is commutative, thus the argumentsof (65) apply and e A = 0.Note that if e A u := A u + (cid:15)J A u J − , then by (106), e A U k = 0 for all k ∈ Z n and k A U k k = | k | ,but for an arbitrary unitary u ∈ A , e A u = 0 so D A u = D .Naturally the above result is also a direct consequence of the fact that the eigenspace ofan isolated eigenvalue of an operator is not modified by small perturbations. However, it isinteresting to compute the last result directly to emphasize the difficulty of the general case:Let ψ = P l ∈ Z n , ≤ j ≤ m c l,j U l ⊗ e j ∈ Ker D A , so P l ∈ Z n , ≤ j ≤ m | c l,j | < ∞ . We have to showthat ψ ∈ Ker D that is c l,j = 0 when l = 0.Taking the scalar product of h U k ⊗ e i | with0 = D A ψ = X l, α, j c l, j ( l α U l − i [ A α , U l ]) ⊗ γ α e j , 85e obtain 0 = X l, α, j c l, j (cid:16) l α δ k,l − i h U k , [ A α , U l ] i (cid:17) h e i , γ α e j i . If A α = P α,l a α,l U l ⊗ γ α with { a α,l } l ∈ S ( Z n ), note that [ U l , U m ] = − i sin( l. Θ m ) U l + m and h U k , [ A α , U l ] i = X l ∈ Z n a α,l ( − i sin( l . Θ l ) h U k , U l + l i = − i a α,k − l sin( k. Θ l ) . Thus0 = X l ∈ Z n n X α =1 2 m X j =1 c l, j (cid:16) l α δ k,l − a α,k − l sin( k. Θ l ) (cid:17) h e i , γ α e j i , ∀ k ∈ Z n , ∀ i, ≤ i ≤ m . (114) We conjecture that Ker D = Ker D A at least for generic Θ ’s : the constraints (114) shouldimply c l,j = 0 for all j and all l = 0 meaning ψ ∈ Ker D . When π Θ has only integercoefficients, the sin part of these constraints disappears giving the result.We will use freely the notation (50) about the difference between D and D . Lemma 8.4. If π Θ is badly approximable (see Definition 7.4), Sp (cid:16) C ∞ ( T n Θ ) , H , D (cid:17) = Z andall these poles are simple.Proof. Let B ∈ D ( A ) and p ∈ N . Suppose that B is of the form B = a r b r D q r − |D| p r − a r − b r − · · · D q |D| p a b where r ∈ N , a i ∈ A , b i ∈ J A J − , q i , p i ∈ N . We note a i =: P l a i,l U l and b i =: P i b i,l U l .With the shorthand k µ ,µ qi := k µ · · · k µ qi and γ µ ,µ qi = γ µ · · · γ µ qi , we get D q |D| p a b U k ⊗ e j = X l , l a ,l b ,l U l U k U l | k + l + l | p ( k + l + l ) µ ,µ q ⊗ γ µ ,µ q e j which gives, after r iterations, BU k ⊗ e j = X l,l e a l e b l U l r · · · U l U k U l · · · U l r r − Y i =1 | k + b l i + b l i | p i ( k + b l i + b l i ) µ i ,µ iqi ⊗ γ µ r − ,µ r − qr − · · · γ µ ,µ q e j where e a l := a ,l · · · a r,l r and e b l := b ,l · · · b r,l r .Let us note F µ ( k, l, l ) := Q r − i =1 | k + b l i + b l i | p i ( k + b l i + b l i ) µ i ,µ iqi and γ µ := γ µ r − ,µ r − qr − · · · γ µ ,µ q .Thus, with the shortcut ∼ c meaning modulo a constant function towards the variable s, Tr (cid:16) B | D | − p − s (cid:17) ∼ c X k X l,l e a l e b l τ (cid:16) U − k U l r · · · U l U k U l · · · U l r (cid:17) F µ ( k,l,l ) | k | s +2 p Tr( γ µ ) . Since U l r · · · U l U k = U k U l r · · · U l e − i P r l i . Θ k we get τ (cid:16) U − k U l r · · · U l U k U l · · · U l r (cid:17) = δ P r l i + l i , e iφ ( l,l ) e − i P r l i . Θ k φ is a real valued function. Thus,Tr (cid:16) B | D | − p − s (cid:17) ∼ c X k X l,l e iφ ( l,l ) δ P r l i + l i , e a l e b l F µ ( k,l,l ) e − i P r li. Θ k | k | s +2 p Tr( γ µ ) ∼ c f µ ( s ) Tr( γ µ ) . The function f µ ( s ) can be decomposed as a linear combination of zeta function of typedescribed in Theorem 7.17 (or, if r = 1 or all the p i are zero, in Theorem 7.5). Thus, s Tr (cid:16) B | D | − p − s (cid:17) has only poles in Z and each pole is simple. Finally, by linearity, we getthe result.The dimension spectrum of the noncommutative torus is simple: Proposition 8.5. (i) If π Θ is badly approximable, the spectrum dimension of the spectraltriple (cid:16) C ∞ ( T n Θ ) , H , D (cid:17) is equal to the set { n − k : k ∈ N } and all these poles are simple.(ii) ζ D (0) = 0 . Proof. ( i ) Lemma 8.4 and Remark 5.9.( ii ) ζ D ( s ) = P k ∈ Z n P ≤ j ≤ m h U k ⊗ e j , | D | − s U k ⊗ e j i = 2 m ( P k ∈ Z n | k | s + 1) = 2 m ( Z n ( s ) + 1) . By (93), we get the result.We have computed ζ D (0) relatively easily but the main difficulty of the present Sectionis precisely to calculate ζ D A (0). We fix a self-adjoint one-form A on the noncommutative torus of dimension n . Proposition 8.6. If π Θ is badly approximable, then the first elements of the spectral actionexpansion (75) are given by − Z | D A | − n = − Z | D | − n = 2 m +1 π n/ Γ( n ) − . − Z | D A | − n + k = 0 for k odd . − Z | D A | − n +2 = 0 . We need a few technical lemmas: Lemma 8.7. On the noncommutative torus, for any t ∈ R , − Z e A D| D | − t = − Z D e A | D | − t = 0 . Proof. Using notations of (110), we haveTr( e A D| D | − s ) ∼ c X j X k h U k ⊗ e j , − ik µ | k | − s [ A α , U k ] ⊗ γ α γ µ e j i∼ c − i Tr( γ α γ µ ) X k k µ | k | − s h U k , [ A α , U k ] i = 0since h U k , [ A α , U k ] i = 0. SimilarlyTr( D e A | D | − s ) ∼ c X j X k h U k ⊗ e j , | k | − s X l a α,l k. Θ l ( l + k ) µ U l + k ⊗ γ µ γ α e j i∼ c γ µ γ α ) X k X l a α,l sin k. Θ l ( l + k ) µ | k | − s h U k , U l + k i = 0 . h in the algebra generated by A and [ D , A ] can be written as a linearcombination of terms of the form a p · · · a np r where a i are elements of A or [ D , A ]. Such aterm can be written as a series b := P a ,α ,l · · · a q,α q ,l q U l · · · U l q ⊗ γ α · · · γ α q where a i,α i areSchwartz sequences and when a i =: P l a l U l ∈ A , we set a i,α,l = a i,l with γ α = 1. We define L ( b ) := τ (cid:16)X l a ,α ,l · · · a q,α q ,l q U l · · · U l q (cid:17) Tr( γ α · · · γ α q ) . By linearity, L is defined as a linear form on the whole algebra generated by A and [ D , A ]. Lemma 8.8. If h is an element of the algebra generated by A and [ D , A ] , Tr (cid:16) h | D | − s (cid:17) ∼ c L ( h ) Z n ( s ) . In particular, Tr (cid:16) h | D | − s (cid:17) has at most one pole at s = n .Proof. We get with b of the form P a ,α ,l · · · a q,α q ,l q U l · · · U l q ⊗ γ α · · · γ α q ,Tr (cid:16) b | D | − s (cid:17) ∼ c X k ∈ Z n h U k , X l a ,α ,l · · · a q,α q ,l q U l · · · U l q U k i Tr( γ α · · · γ α q ) | k | − s ∼ c τ ( X l a ,α ,l · · · a q,α q ,l q U l · · · U l q ) Tr( γ α · · · γ α q ) Z n ( s ) = L ( b ) Z n ( s ) . The results follows now from linearity of the trace. Lemma 8.9. If π Θ is badly approximable, the function s Tr (cid:16) εJ AJ − A | D | − s (cid:17) extendsmeromorphically on the whole plane with only one possible pole at s = n . Moreover, this poleis simple and Res s = n Tr (cid:16) εJ AJ − A | D | − s (cid:17) = a α, a α m +1 π n/ Γ( n/ − . Proof. With A = L ( − iA α ) ⊗ γ α , we get (cid:15)J AJ − = R ( iA α ) ⊗ γ α , and by multiplication εJ AJ − A = R ( A β ) L ( A α ) ⊗ γ β γ α . Thus,Tr (cid:16) εJ AJ − A | D | − s (cid:17) ∼ c X k ∈ Z n h U k , A α U k A β i | k | − s Tr( γ β γ α ) ∼ c X k ∈ Z n X l a α,l a β, − l e ik. Θ l | k | − s Tr( γ β γ α ) ∼ c m X k ∈ Z n X l a α,l a α − l e ik. Θ l | k | − s . Theorem 7.5 ( ii ) entails that P k ∈ Z n P l a α,l a α − l e ik. Θ l | k | − s extends meromorphically on thewhole plane C with only one possible pole at s = n . Moreover, this pole is simple and wehave Res s = n X k ∈ Z n X l a α,l a α − l e ik. Θ l | k | − s = a α, a α Res s = n Z n ( s ) . Equation (92) now gives the result. Lemma 8.10. If π Θ is badly approximable, then for any t ∈ R , − Z X | D | − t = δ t,n m +1 (cid:16) − X l a α,l a α − l + a α, a α (cid:17) π n/ Γ( n/ − . where X = e A D + D e A + e A and A =: − i P l a α,l U l ⊗ γ α . roof. By Lemma 8.7, we get − R X | D | − t = Res s =0 Tr( e A | D | − s − t ). Since A and εJ AJ − com-mute, we have e A = A + J A J − + 2 εJ AJ − A . Thus,Tr( e A | D | − s − t ) = Tr( A | D | − s − t ) + Tr( J A J − | D | − s − t ) + 2 Tr( εJ AJ − A | D | − s − t ) . Since | D | and J commute, we have with Lemma 8.8,Tr (cid:16) e A | D | − s − t (cid:17) ∼ c L ( A ) Z n ( s + t ) + 2 Tr (cid:16) εJ AJ − A | D | − s − t (cid:17) . Thus Lemma 8.9 entails that Tr( e A | D | − s − t ) is holomorphic at 0 if t = n . When t = n ,Res s =0 Tr (cid:16) e A | D | − s − t (cid:17) = 2 m +1 (cid:16) − X l a α,l a α − l + a α, a α (cid:17) π n/ Γ( n/ − , (115)which gives the result. Lemma 8.11. If π Θ is badly approximable, then − Z e A D e A D| D | − − n = − n − n − Z e A | D | − n . Proof. With D J = εJ D , we get − Z e A D e A D| D | − − n = 2 − Z A D A D| D | − − n + 2 − Z εJ AJ − D A D| D | − − n . Let us first compute − R A D A D| D | − − n . We have, with A =: − iL ( A α ) ⊗ γ α =: − i P l a α,l U l ⊗ γ α ,Tr (cid:16) A D A D| D | − s − − n (cid:17) ∼ c − X k X l ,l a α ,l a α ,l τ ( U − k U l U l U k ) k µ ( k + l ) µ | k | s +2+ n Tr( γ α,µ )where γ α,µ := γ α γ µ γ α γ µ . Thus, − Z A D A D| D | − − n = − X l a α , − l a α ,l Res s =0 (cid:16)X k k µ k µ | k | s +2+ n (cid:17) Tr( γ α,µ ) . We have also, with εJ AJ − = iR ( A α ) ⊗ γ a ,Tr (cid:16) εJ AJ − D A D| D | − s − − n (cid:17) ∼ c X k X l ,l a α ,l a α ,l τ ( U − k U l U k U l ) k µ ( k + l ) µ | k | s +2+ n Tr( γ α,µ ) . which gives − Z εJ AJ − D A D| D | − − n = a α , a α , Res s =0 (cid:16)X k k µ k µ | k | s +2+ n (cid:17) Tr( γ α,µ ) . Thus, − Z e A D e A D| D | − − n = (cid:16) a α , a α , − X l a α , − l a α ,l (cid:17) Res s =0 (cid:16)X k k µ k µ | k | s +2+ n (cid:17) Tr( γ α,µ ) . With P k k µ k µ | k | s +2+ n = δ µ µ n Z n ( s + n ) and C n := Res s =0 Z n ( s + n ) = 2 π n/ Γ( n/ − we obtain − Z e A D e A D| D | − − n = (cid:16) a α , a α , − X l a α , − l a α ,l (cid:17) C n n Tr( γ α γ µ γ α γ µ ) . Since Tr( γ α γ µ γ α γ µ ) = 2 m (2 − n ) δ α ,α , we get − Z e A D e A D| D | − − n = 2 m (cid:16) − a α, a α + X l a α, − l a αl (cid:17) C n ( n − n . Equation (115) now proves the lemma. 89 emma 8.12. If π Θ is badly approximable, then for any P ∈ Ψ ( A ) and q ∈ N , q odd, − Z P | D | − ( n − q ) = 0 . Proof. There exist B ∈ D ( A ) and p ∈ N such that P = BD − p + R where R is in OP − q − .As a consequence, − R P | D | − ( n − q ) = − R B | D | − n − p + q . Assume B = a r b r D q r − a r − b r − · · · D q a b where r ∈ N , a i ∈ A , b i ∈ J A J − , q i ∈ N . If we prove that − R B | D | − n − p + q = 0, then thegeneral case will follow by linearity. We note a i =: P l a i,l U l and b i =: P l b i,l U l . With theshorthand k µ ,µ qi := k µ · · · k µ qi and γ µ ,µ qi = γ µ · · · γ µ qi , we get D q a b U k ⊗ e j = X l ,l a ,l b ,l U l U k U l ( k + l + l ) µ ,µ q ⊗ γ µ ,µ q e j which gives, after iteration, B U k ⊗ e j = X l,l e a l e b l U l r · · · U l U k U l · · · U l r r − Y i =1 ( k + b l i + b l i ) µ i ,µ iqi ⊗ γ µ r − ,µ r − qr − · · · γ µ ,µ q e j where e a l := a ,l · · · a r,l r and e b l := b ,l · · · b r,l r . Let’s note Q µ ( k, l, l ) := Q r − i =1 ( k + b l i + b l i ) µ i ,µ iqi and γ µ := γ µ r − ,µ r − qr − · · · γ µ ,µ q . Thus, − Z B | D | − n − p + q = Res s =0 X k X l,l e a l e b l τ (cid:16) U − k U l r · · · U l U k U l · · · U l r (cid:17) Q µ ( k,l,l ) | k | s +2 p + n − q Tr( γ µ ) . Since U l r · · · U l U k = U k U l r · · · U l e − i P r l i . Θ k , we get τ (cid:16) U − k U l r · · · U l U k U l · · · U l r (cid:17) = δ P r l i + l i , e iφ ( l,l ) e − i P r l i . Θ k where φ is a real valued function. Thus, − Z B | D | − n − p + q = Res s =0 X k X l,l e iφ ( l,l ) δ P r l i + l i , e a l e b l Q µ ( k,l,l ) e − i P r li. Θ k | k | s +2 p + n − q Tr( γ µ )=: Res s =0 f µ ( s ) Tr( γ µ ) . We decompose Q µ ( k, l, l ) as a sum P rh =0 M h,µ ( l, l ) Q h,µ ( k ) where Q h,µ is a homogeneouspolynomial in ( k , · · · , k n ) and M h,µ ( l, l ) is a polynomial in (cid:16) ( l ) , · · · , ( l r ) n , ( l ) , · · · , ( l r ) n (cid:17) .Similarly, we decompose f µ ( s ) as P rh =0 f h,µ ( s ). Theorem 7.5 ( ii ) entails that f h,µ ( s )extends meromorphically to the whole complex plane C with only one possible pole for s + 2 p + n − q = n + d where d := deg Q h,µ . In other words, if d + q − p = 0, f h,µ ( s ) isholomorphic at s = 0. Suppose now d + q − p = 0 (note that this implies that d is odd,since q is odd by hypothesis), then, by Theorem 7.5 ( ii )Res s =0 f h,µ ( s ) = V Z u ∈ S n − Q h,µ ( u ) dS ( u )where V := P l,l ∈ Z M h,µ ( l, l ) e iφ ( l,l ) δ P r l i + l i , e a l e b l and Z := { l, l : P ri =1 l i = 0 } . Since d isodd, Q h,µ ( − u ) = − Q h,µ ( u ) and R u ∈ S n − Q h,µ ( u ) dS ( u ) = 0. Thus, Res s =0 f h,µ ( s ) = 0 in any case,which gives the result. 90s we have seen, the crucial point of the preceding lemma is the decomposition of thenumerator of the series f µ ( s ) as polynomials in k . This has been possible because we restrictedour pseudodifferential operators to Ψ ( A ). Proof of Proposition 8.6. The top element follows from Proposition 5.26 and according to(92), − Z | D | − n = Res s =0 Tr (cid:16) | D | − s − n (cid:17) = 2 m Res s =0 Z n ( s + n ) = m +1 π n/ Γ( n/ . For the second equality, we get from Lemmas 8.8 and 5.23Res s = n − k ζ D A ( s ) = k X p =1 k − p X r , ··· ,r p =0 h ( n − k, r, p ) − Z ε r ( Y ) · · · ε r p ( Y ) | D | − ( n − k ) . Corollary 5.22 and Lemma 8.12 imply that − R ε r ( Y ) · · · ε r p ( Y ) | D | − ( n − k ) = 0, which gives theresult.Last equality follows from Lemma 8.11 and Corollary 5.28. Here is the main result of this section. Theorem 8.13. Consider the noncommutative torus (cid:16) C ∞ ( T n Θ ) , H , D (cid:17) of dimension n ∈ N where π Θ is a real n × n real skew-symmetric badly approximable matrix, and a selfadjointone-form A = L ( − iA α ) ⊗ γ α . Then, the full spectral action of D A = D + A + (cid:15)J AJ − is(i) for n = 2 , S ( D A , f, Λ) = 4 π f Λ + O (Λ − ) , (ii) for n = 4 , S ( D A , f, Λ) = 8 π f Λ − π f (0) τ ( F µν F µν ) + O (Λ − ) , (iii) More generally, in S ( D A , f, Λ) = n X k =0 f n − k c n − k ( A ) Λ n − k + O (Λ − ) ,c n − ( A ) = 0 , c n − k ( A ) = 0 for k odd. In particular, c ( A ) = 0 when n is odd. This result (for n = 4) has also been obtained in [42] using the heat kernel method.It is however interesting to get the result via direct computations of (75) since it showshow this formula is efficient. As we will see, the computation of all the noncommutativeintegrals require a lot of technical steps. One of the main points, namely to isolate where theDiophantine condition on Θ is assumed, is outlined here.91 emark 8.14. Note that all terms must be gauge invariants, namely, according to (109),invariant by A α −→ γ u ( A α ) = uA α u ∗ + uδ α ( u ∗ ) . A particular case is u = U k where U k δ α ( U ∗ k ) = − ik α U .In the same way, note that there is no contradiction with the commutative case where, forany selfadjoint one-form A , D A = D (so A is equivalent to 0!), since we assume in Theorem8.13 that Θ is badly approximable, so A cannot be commutative. Conjecture 8.15. The constant term of the spectral action of D A on the noncommutativen-torus is proportional to the constant term of the spectral action of D + A on the commutativen-torus. Remark 8.16. The appearance of a Diophantine condition for Θ has been characterizedin dimension 2 by Connes [23, Prop. 49] where in this case, Θ = θ (cid:16) − (cid:17) with θ ∈ R .In fact, the Hochschild cohomology H ( A Θ , A Θ ∗ ) satisfies dim H j ( A Θ , A Θ ∗ ) = 2 (or ) for j = 1 (or j = 2 ) if and only if the irrational number θ satisfies a Diophantine condition like | − e i πnθ | − = O ( n k ) for some k .Recall that when the matrix Θ is quite irrational (the lattice generated by its columns isdense after translation by Z n , see [50, Def. 12.8]), then the C ∗ -algebra generated by A Θ issimple. Remark 8.17. It is possible to generalize above theorem to the case D = − i g µν δ µ ⊗ γ ν instead of (104) when g is a positive definite constant matrix. The formulae in Theorem 8.13are still valid, up to obvious modifications due to volume variation. − R In order to get this theorem, let us prove a few technical lemmas.We suppose from now on that Θ is a skew-symmetric matrix in M n ( R ). No other hy-pothesis is assumed for Θ, except when it is explicitly stated.When A is a selfadjoint one-form, we define for n ∈ N , q ∈ N , 2 ≤ q ≤ n and σ ∈ {− , + } q A + := A D D − , A − := (cid:15)J AJ − D D − , A σ := A σ q · · · A σ . Lemma 8.18. We have for any q ∈ N , − Z ( e AD − ) q = − Z ( e A D D − ) q = X σ ∈{ + , − } q − Z A σ . Proof. Since P ∈ OP −∞ , D − = D D − mod OP −∞ and − R ( e AD − ) q = − R ( e A D D − ) q . Lemma 8.19. Let A be a selfadjoint one-form, n ∈ N and q ∈ N with ≤ q ≤ n and σ ∈ {− , + } q . Then − Z A σ = − Z A − σ . efinition 8.20. In [13] has been introduced the vanishing tadpole hypothesis: − Z AD − = 0 , for all A ∈ Ω D ( A ) . (116)By the following lemma, this condition is satisfied for the noncommutative torus. Lemma 8.21. Let n ∈ N , A = L ( − iA α ) ⊗ γ α = − i P l ∈ Z n a α,l U l ⊗ γ α , A α ∈ A Θ , where { a α,l } l ∈ S ( Z n ) , be a hermitian one-form. Then,(i) − R A p D − q = − R ( (cid:15)J AJ − ) p D − q = 0 for p ≥ and ≤ q < n (case p = q = 1 is tadpolehypothesis.)(ii) If π Θ is badly-approximable, then − R BD − q = 0 for ≤ q < n and any B in the algebragenerated by A , [ D , A ] , J A J − and J [ D , A ] J − .Proof. ( i ) Let us compute − Z A p ( (cid:15)J AJ − ) p D − q . With A = L ( − iA α ) ⊗ γ α and (cid:15)J AJ − = R ( iA α ) ⊗ γ α , we get A p = L ( − iA α ) · · · L ( − iA α p ) ⊗ γ α · · · γ α p and ( (cid:15)J AJ − ) p = R ( iA α ) · · · R ( iA α p ) ⊗ γ α · · · γ α p . We note e a α,l := a α ,l · · · a α p ,l p . Since L ( − iA α ) · · · L ( − iA α p ) R ( iA α ) · · · R ( iA α p ) U k = ( − i ) p i p X l,l e a α,l e a α ,l U l · · · U l p U k U l p · · · U l , and U l · · · U l p U k = U k U l · · · U l p e − i ( P i l i ) . Θ k , we get, with U l,l := U l · · · U l p U l p · · · U l ,g µ,α,α ( s, k, l, l ) := e ik. Θ P j l j k µ ...k µq | k | s +2 q e a α,l e a α ,l ,γ α,α ,µ := γ α · · · γ α p γ α · · · γ α p γ µ · · · γ µ q ,A p ( (cid:15)J AJ − ) p D − q | D | − s U k ⊗ e i ∼ c ( − i ) p i p X l,l g µ,α,α ( s, k, l, l ) U k U l,l ⊗ γ α,α ,µ e i . Thus, − R A p ( (cid:15)J AJ − ) p D − q = Res s =0 f ( s ) where f ( s ) : = Tr (cid:16) A p ( (cid:15)J AJ − ) p D − q | D | − s (cid:17) ∼ c ( − i ) p i p X k ∈ Z n h U k ⊗ e i , X l,l g µ,α,α ( s, k, l, l ) U k U l,l ⊗ γ α,α ,µ e i i∼ c ( − i ) p i p X k ∈ Z n τ (cid:16) X l,l g µ,α,α ( s, k, l, l ) U l,l (cid:17) Tr( γ µ,α,α ) ∼ c ( − i ) p i p X k ∈ Z n X l,l g µ,α,α ( s, k, l, l ) τ (cid:16) U l,l (cid:17) Tr( γ µ,α,α ) . 93t is straightforward to check that the series P k,l,l g µ,α,α ( s, k, l, l ) τ (cid:16) U l,l (cid:17) is absolutely sum-mable if < ( s ) > R for a R > 0. Thus, we can exchange the summation on k and l, l , whichgives f ( s ) ∼ c ( − i ) p i p X l,l X k ∈ Z n g µ,α,α ( s, k, l, l ) τ (cid:16) U l,l (cid:17) Tr( γ µ,α,α ) . If we suppose now that p = 0, we see that, f ( s ) ∼ c ( − i ) p X l X k ∈ Z n k µ ...k µq | k | s +2 q e a α,l δ P pi =1 l i , Tr( γ µ,α,α )which is, by Proposition 7.16, analytic at 0. In particular, for p = q = 1, we see that − R AD − = 0, i.e. the vanishing tadpole hypothesis is satisfied. Similarly, if we suppose p = 0,we get f ( s ) ∼ c ( − i ) p X l X k ∈ Z n k µ ...k µq | k | s +2 q e a α,l δ P p i =1 l i , Tr( γ µ,α,α )which is holomorphic at 0.( ii ) Adapting the proof of Lemma 8.12 to our setting (taking q i = 0, and adding gammamatrices components), we see that − Z B D − q = Res s =0 X k X l,l e iφ ( l,l ) δ P r l i + l i , e a α,l e b β,l k µ ··· k µq e − i P r li. Θ k | k | s +2 q Tr( γ ( µ,α,β ) )where γ ( µ,α,β ) is a complicated product of gamma matrices. By Theorem 7.5 ( ii ), since wesuppose here that π Θ is badly approximable, this residue is 0. Same hypothesis as in Lemma 8.21.(i) Case n = 2 : − Z A q D − q = − δ q, π τ (cid:16) A α A α (cid:17) . (ii) Case n = 4 : with the shorthand δ µ ,...,µ := δ µ µ δ µ µ + δ µ µ δ µ µ + δ µ µ δ µ µ , − Z A q D − q = δ q, π τ (cid:16) A α · · · A α (cid:17) Tr( γ α · · · γ α γ µ · · · γ µ ) δ µ ,...,µ . Proof. ( i, ii ) The same computation as in Lemma 8.21 ( i ) (with p = 0, p = q = n ) gives − Z A n D − n = Res s =0 ( − i ) n (cid:16) X k ∈ Z n k µ ...k µn | k | s +2 n (cid:17) τ (cid:16) X l ∈ ( Z n ) n e a α,l U l · · · U l n (cid:17) Tr( γ α · · · γ α n γ µ · · · γ µ n )and the result follows from Proposition 7.16.We will use few notations: 94or n ∈ N , q ≥ l := ( l , · · · , l q − ) ∈ ( Z n ) q − , α := ( α , · · · , α q ) ∈ { , · · · , n } q , k ∈ Z n \{ } , σ ∈ {− , + } q , ( a i ) ≤ i ≤ n ∈ ( S ( Z n )) n , l q := − X ≤ j ≤ q − l j , λ σ := ( − i ) q Y j =1 ...q σ j , e a α,l := a α ,l . . . a α q ,l q ,φ σ ( k, l ) := X ≤ j ≤ q − ( σ j − σ q ) k. Θ l j + X ≤ j ≤ q − σ j ( l + . . . + l j − ) . Θ l j ,g µ ( s, k, l ) := k µ ( k + l ) µ ... ( k + l + ... + l q − ) µq | k | s +2 | k + l | ... | k + l + ... + l q − | , with the convention P ≤ j ≤ q − = 0 when q = 2, and g µ ( s, k, l ) = 0 whenever b l i = − k for a1 ≤ i ≤ q − Lemma 8.23. Let A = L ( − iA α ) ⊗ γ α = − i P l ∈ Z n a α,l U l ⊗ γ α where A α = − A ∗ α ∈ A Θ and { a α,l } l ∈ S ( Z n ) , with n ∈ N , be a hermitian one-form, and let ≤ q ≤ n , σ ∈ {− , + } q .Then, − R A σ = Res s =0 f ( s ) where f ( s ) := X l ∈ ( Z n ) q − X k ∈ Z n λ σ e i φ σ ( k,l ) g µ ( s, k, l ) e a α,l Tr( γ α q γ µ q · · · γ α γ µ ) . In the following, we will use the shorthand c := π . Lemma 8.24. Suppose n = 4 . Then, with the same hypothesis of Lemma 8.23,(i) − Z ( A + ) = − Z ( A − ) = c X l ∈ Z a α ,l a α , − l (cid:16) l α l α − δ α α | l | (cid:17) . (ii) − − Z ( A + ) = − − Z ( A − ) = 4 c X l i ∈ Z a α , − l − l a α l a α ,l sin l . Θ l l α . (iii) − Z ( A + ) = − Z ( A − ) = 2 c X l i ∈ Z a α , − l − l − l a α ,l a α l a α l sin l . Θ( l + l )2 sin l . Θ l . (iv) Suppose π Θ badly approximable. Then the crossed terms in − R ( A + + A − ) q vanish: if C is the set of all σ ∈ {− , + } q with ≤ q ≤ , such that there exist i, j satisfying σ i = σ j ,we have P σ ∈ C − R A σ = 0 . Lemma 8.25. Suppose n = 4 and π Θ badly approximable. For any self-adjoint one-form A , ζ D A (0) − ζ D (0) = − c τ ( F α ,α F α α ) . Proof. By (48) and Lemma 8.18 we get ζ D A (0) − ζ D (0) = n X q =1 ( − q q X σ ∈{ + , − } q − Z A σ . By Lemma 8.24 ( iv ), we see that the crossed terms all vanish. Thus, with Lemma 8.19, weget ζ D A (0) − ζ D (0) = 2 n X q =1 ( − q q − Z ( A + ) q . (117)95y definition, F α α = i X k (cid:16) a α ,k k α − a α ,k k α (cid:17) U k + X k, l a α ,k a α ,l [ U k , U l ]= i X k h ( a α ,k k α − a α ,k k α ) − X l a α ,k − l a α ,l sin( k. Θ l ) i U k . Thus τ ( F α α F α α ) = m X α , α =1 X k ∈ Z h ( a α ,k k α − a α ,k k α ) − X l ∈ Z a α ,k − l a α ,l sin( k. Θ l ) ih ( a α , − k k α − a α , − k k α ) − X l ” ∈ Z a α , − k − l ” a α ,l ” sin( k. Θ l ”2 ) i . One checks that the term in a q of τ ( F α α F α α ) corresponds to the term − R ( A + ) q given byLemma 8.24. For q = 2, this is − X l ∈ Z , α , α a α ,l a α , − l (cid:16) l α l α − δ α α | l | (cid:17) . For q = 3, we compute the crossed terms: i X k,k ,l ( a α ,k k α − a α ,k k α ) a α k a α l (cid:16) U k [ U k , l ] + [ U k , U l ] U k (cid:17) , which gives the following a -term in τ ( F α α F α α ) − X l i a α , − l − l a α l a α ,l sin l . Θ l l α . For q = 4, this is − X l i a α , − l − l − l a α ,l a α l a α l sin l . Θ( l + l )2 sin l . Θ l which corresponds to the term − R ( A + ) . We get finally, n X q =1 ( − q q − Z ( A + ) q = − c τ ( F α ,α F α α ) . (118)Equations (117) and (118) yield the result. Lemma 8.26. Suppose n = 2 . Then, with the same hypothesis as in Lemma 8.23,(i) − Z ( A + ) = − Z ( A − ) = 0 . (ii) Suppose π Θ badly approximable. Then − Z A + A − = − Z A − A + = 0 . Lemma 8.27. Suppose n = 2 and π Θ badly approximable. Then, for any self-adjoint one-form A , ζ D A (0) − ζ D (0) = 0 . Proof. As in Lemma 8.25, we use (48) and Lemma 8.18 so the result follows from Lemma8.26. 96 .5.2 Odd dimensional caseLemma 8.28. Suppose n odd and π Θ badly approximable. Then for any self-adjoint one-form A and σ ∈ {− , + } q with ≤ q ≤ n , − Z A σ = 0 . Proof. Since A σ ∈ Ψ ( A ), Lemma 8.12 with k = n gives the result. Corollary 8.29. With the same hypothesis of Lemma 8.28, ζ D A (0) − ζ D (0) = 0 . Proof. As in Lemma 8.25, we use (48) and Lemma 8.18 so the result follows from Lemma8.28. Proof of Theorem 8.13.. ( i ) By (75) and Proposition 8.6, we get S ( D A , f, Λ) = 4 f Λ + f (0) ζ D A (0) + O (Λ − ) , where f = R ∞ f ( t ) dt . By Lemma 8.27, ζ D A (0) − ζ D (0) = 0 and from Proposition 8.5, ζ D (0) = 0, so we get the result.( ii ) Similarly, S ( D A , f, Λ) = 8 π f Λ + f (0) ζ D A (0) + O (Λ − ) with f = R ∞ f ( t ) t dt .Lemma 8.25 implies that ζ D A (0) − ζ D (0) = − c τ ( F µν F µν ) and Proposition 8.5 yields theequality ζ D A (0) = − c τ ( F µν F µν ) and the result.( iii ) is a direct consequence of (75), Propositions 8.5, 8.6, and Corollary 8.29. This section is an attempt to understand what happens if Θ is ‘in between’ rational numbersand “Diophantine numbers”. Consider the simplest case: T withΘ = θ − 11 0 ! , To proceed, we need some results from number theory [8]: Definition 8.30. Let f : R ≥ → R > be a continuous function such that x → x f ( x ) isnon-increasing. Consider the set F ( f ) := { θ ∈ R : | θq − p | < q f ( q ) for infinitely many rational numbers pq } . The elements of F ( f ) are termed f -approximable. Note that we cannot expect the above estimate to be valid for all rational numbers pq sincefor all irrational numbers θ , the set of fractional values of ( θq ) q ≥ is dense in [0 , Theorem 8.31. There exists an uncountable family of real numbers θ/ (2 π ) which are f -approximable but not cf -approximable for any < c < . f ( x ) = (2 πx ) − e − x , and fix a constant c < 1. Let us pick a θ which is f -approximable, but not cf -approximable.Consider now g ( t ) := Tr (cid:16) aJ bJ − e − t D (cid:17) . It is shown in [42] that, by tuning a, b ∈ A Θ , it ispossible make the difference g ( t ) − g ( t ) Dioph (of g ( t ) and its value if we suppose that θ is aDiophantine number) of arbitrary order in t .This shows how subtle can be the computation of spectral action!98 The non-compact case When a Riemannian spin manifold M is non-compact, the Dirac operator, which exists asa selfadjoint extension when M is (geodesically) complete has no more a compact resolvent:its spectrum is not discrete but is R ( [47, Theorem 7.2.1] and similar results for hyperbolicspaces [47, p. 106].)To see what happens, let us consider for instance the flat space M = R d and the Hilbertspace H = L ( R d ). Then the operator f ( x ) g ( − i ∇ ) is formally given on ψ, φ in appropriatedomains by h ψ, f ( x ) g ( − i ∇ ) φ i = Z R d ¯ f ( x ) φ ( x ) ( g b ψ ) q ( x ) dx. For k ∈ Z d , let χ k be the characteristic function of the unit cube in R d with center at k anddefine for p, q > ‘ q (cid:16) L p ( R d ) (cid:17) := { f | k f k p,q := (cid:16) X k k f χ k k qp (cid:17) /q < ∞ } where k g k p := (cid:16) R R d | g ( x ) | p dx (cid:17) /p is the usual norm of L p ( R d ). Theorem 9.1. Birman–Solomjak.(i) If f, g ∈ ‘ p (cid:16) L ( R d ) (cid:17) for ≤ p ≤ , then f ( x ) g ( − i ∇ ) is in the Schatten class L p and k f ( x ) g ( − i ∇ ) k p ≤ c p k f k ,p k g k ,p .(i) If f, g are non zero, then f ( x ) g ( − i ∇ ) ∈ L ( H ) if and only f and g are in ‘ (cid:16) H ) (cid:17) . For a proof, see [102, Chapter 4].This shows that even if g ( x ) = e − tx , the heat kernel e − t ∆ is never trace-class since f = 1 isnot in ‘ (cid:16) L ( R d ) (cid:17) .Thus, to cover at least the non-compact manifold case, Definition 5.1 has to be improved: Definition 9.2. A non-compact spectral triple ( A , H , D ) is the data of an involutive algebra A with a faithful representation π on a Hilbert space H , a preferred unitization e A of A anda selfadjoint operator D such that- a ( D − λ ) − is compact for all a ∈ A and λ / ∈ Sp D .- [ D , π ( a )] is bounded for any a ∈ e A . All definitions of regularity, finiteness and orientation have to be modified with e A instead of A , see also [9].In the first constraint of this definition we recover a certain discreteness which, with a = 1,is the compact case (the algebra can have a unit). This matter is not only technical sincenow there is a deeper intertwining of the choice of the algebra A and the operator D to geta spectral triple. Moreover, a tentative of modification of D is quite often forbidden by thesecond constraint.The case of non-compact spin manifold has been considered by Rennie [91–93]. This hasbeen improved in [40] which studied the Moyal plane. Actually, a compactification of thisplane is the noncommutative torus! 99 .2 The Moyal product Reference: [40].For any finite dimension k , let Θ be a real skewsymmetric k × k matrix, let s · t denotethe usual scalar product on Euclidean R k and let S ( R k ) be the space of complex Schwartzfunctions on R k . One defines, for f, h ∈ S ( R k ), the corresponding Moyal or twisted product: f ? Θ h ( x ) := (2 π ) − k ZZ f ( x − Θ u ) h ( x + t ) e − iu · t d k u d k t. (119)In Euclidean field theory, the entries of Θ have the dimensions of an area. Because Θ isskewsymmetric, complex conjugation reverses the product: ( f ? Θ h ) ∗ = h ∗ ? Θ f ∗ .Assume Θ to be nondegenerate, that is to say, σ ( s, t ) := s · Θ t to be symplectic. Thisimplies even dimension, k = 2 N . We note that Θ − is also skewsymmetric; let θ > θ N := det Θ. Then formula (119) may be rewritten as f ? Θ h ( x ) = ( πθ ) − N ZZ f ( x + s ) h ( x + t ) e − is · Θ − t d N s d N t. (120)This form is very familiar from phase-space quantum mechanics, where R N is parametri-zed by N conjugate pairs of position and momentum variables, and the entries of Θ havethe dimensions of an action; one then selects Θ = (cid:126) S := (cid:126) (cid:16) N − N (cid:17) Indeed, the product ? (or rather, its commutator) was introduced in that context by Moyal [82], using a seriesdevelopment in powers of (cid:126) whose first nontrivial term gives the Poisson bracket; later, itwas rewritten in the above integral form. These are actually oscillatory integrals, of whichMoyal’s series development, f ? (cid:126) g ( x ) = X α ∈ N N (cid:16) i (cid:126) (cid:17) | α | α ! ∂f∂x α ( x ) ∂g∂ ( Sx ) α ( x ) , (121)is an asymptotic expansion. The first integral form (119) of the Moyal product was exploitedby Rieffel in a remarkable monograph [96], who made it the starting point for a more generaldeformation theory of C ∗ -algebras.With the choice Θ = θS made, the Moyal product can also be written f ? θ g ( x ) := ( πθ ) − N ZZ f ( y ) g ( z ) e iθ ( x − y ) · S ( x − z ) d N y d N z. (122)Of course, our definitions make sense only under certain hypotheses on f and g [49, 108]. Lemma 9.3. [49] Let f, g ∈ S ( R N ) . Then(i) f ? θ g ∈ S ( R N ) .(ii) ? θ is a bilinear associative product on S ( R N ) . Moreover, complex conjugation of func-tions f f ∗ is an involution for ? θ .(iii) Let j = 1 , , . . . , N . The Leibniz rule is satisfied: ∂∂x j ( f ? θ g ) = ∂f∂x j ? θ g + f ? θ ∂g∂x j . (123)100 iv) Pointwise multiplication by any coordinate x j obeys x j ( f ? θ g ) = f ? θ ( x j g ) + i θ ∂f∂ ( Sx ) j ? θ g = ( x j f ) ? θ g − i θ f ? θ ∂g∂ ( Sx ) j . (124) (v) The product has the tracial property: h f , g i := πθ ) N Z f ? θ g ( x ) d N x = πθ ) N Z g? θ f ( x ) d N x = πθ ) N Z f ( x ) g ( x ) d N x. (vi) Let L θf ≡ L θ ( f ) be the left multiplication g f ? θ g . Then lim θ ↓ L θf g ( x ) = f ( x ) g ( x ) ,for x ∈ R N . Property (vi) is a consequence of the distributional identitylim ε ↓ ε − k e ia · b/ε = (2 π ) k δ ( a ) δ ( b ) , for a, b ∈ R k ; convergence takes place in the standard topology [100] of S ( R N ). To sim-plify notation, we put S := S ( R N ) and let S := S ( R N ) be the dual space of tempereddistributions. In view of (vi), we may denote by L f the pointwise product by f . Theorem 9.4. [49] A θ := ( S , ? θ ) is a nonunital associative, involutive Fréchet algebra witha jointly continuous product and a distinguished faithful trace. Definition 9.5. The algebra A θ has a natural basis of eigenvectors f mn of the harmonicoscillator, indexed by m, n ∈ N N . If H l := ( x l + x l + N ) for l = 1 , . . . , N and H := H + H + · · · + H N , then the f mn diagonalize these harmonic oscillator Hamiltonians: H l ? θ f mn = θ ( m l + ) f mn ,f mn ? θ H l = θ ( n l + ) f mn . (125) They may be defined by f mn := √ θ | m | + | n | m ! n ! ( a ∗ ) m ? θ f ? θ a n , (126) where f is the Gaussian function f ( x ) := 2 N e − H/θ , and the annihilation and creationfunctions respectively are a l := √ ( x l + ix l + N ) and a ∗ l := √ ( x l − ix l + N ) . (127) One finds that a n := a n . . . a n N N = a ? θ n ? θ · · · ? θ a ? θ n N N . Proposition 9.6. [49, p. 877] The algebra ( S , ? θ ) has the (nonunique) factorization prop-erty: for all h ∈ S there exist f, g ∈ S such that h = f ? θ g . Lemma 9.7. [49, 108] Let f, g ∈ L ( R N ) . Then(i) For θ = 0 , f ? θ g lies in L ( R N ) . Moreover, f ? θ g is uniformly continuous. ii) ? θ is a bilinear associative product on L ( R N ) . The complex conjugation of functions f f ∗ is an involution for ? θ .(iii) The linear functional f R f ( x ) dx on S extends to R − ( R N ) := L ( R N ) ? θ L ( R N ) ,and the product has the tracial property: h f , g i := ( πθ ) − N Z f ? θ g ( x ) d N x = ( πθ ) − N Z g? θ f ( x ) d N x = ( πθ ) − N Z f ( x ) g ( x ) d N x. (iv) lim θ ↓ L θf g ( x ) = f ( x ) g ( x ) almost everywhere on R N . Definition 9.8. Let A θ := { T ∈ S : T ? θ g ∈ L ( R N ) for all g ∈ L ( R N ) } , provided withthe operator norm k L θ ( T ) k op := sup { k T ? θ g k / k g k : 0 = g ∈ L ( R N ) } .Obviously A θ = S , → A θ . But A θ is not dense in A θ . Note that G ⊂ A θ . This is clear from the following estimate. Lemma 9.9. [49] If f, g ∈ L ( R N ) , then f ? θ g ∈ L ( R N ) and k L θf k op ≤ (2 πθ ) − N/ k f k .Proof. Expand f = P m,n c mn α mn and g = P m,n d mn α mn with respect to the orthonormalbasis { α nm } := (2 πθ ) − N/ { f nm } of L ( R N ). Then k f ? θ g k = (2 πθ ) − N (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)X m,l (cid:18)X n c mn d nl (cid:19) f ml (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) = (2 πθ ) − N X m,l (cid:12)(cid:12)(cid:12)(cid:12)X n c mn d nl (cid:12)(cid:12)(cid:12)(cid:12) ≤ (2 πθ ) − N X m,j | c mj | X k,l | d kl | = (2 πθ ) − N k f k k g k , on applying the Cauchy–Schwarz inequality. Proposition 9.10. [108] ( A θ , k . k op ) is a unital C ∗ -algebra of operators on L ( R N ) , iso-morphic to L ( L ( R N )) and including L ( R N ) . Moreover, there is a continuous injection of ∗ -algebras A θ , → A θ , but A θ is not dense in A θ . Proposition 9.11. A θ is a (nonunital) Fréchet pre- C ∗ -algebra.Proof. We adapt the argument for the commutative case in [50, p. 135]. To show that A θ is stable under the holomorphic functional calculus, we need only check that if f ∈ A θ and 1 + f is invertible in A θ with inverse 1 + g , then the quasi-inverse g of f must lie in A θ . From f + g + f ? θ g = 0, we obtain f ? θ f + g? θ f + f ? θ g? θ f = 0, and it is enough toshow that f ? θ g? θ f ∈ A θ , since the previous relation then implies g? θ f ∈ A θ , and then g = − f − g? θ f ∈ A θ also.Now, A θ ⊂ G − r, for any r > N [108, p. 886]. Since f ∈ G s,p + r ∩ G qt , for s, t arbitraryand p, q positive, we conclude that f ? θ g? θ f ∈ G s,p + r ? θ G − r, ? θ G qt ⊂ G st ; as S = T s,t ∈ R G st , theproof is complete. Lemma 9.12. If f ∈ S , then L θf is a regularizing ΨDO .Proof. From (119), one at once sees that left Moyal multiplication by f is the pseudodiffer-ential operator on R N with symbol f ( x − θ Sξ ). Clearly L θf extends to a continuous linearmap from C ∞ ( R N ) , → S to C ∞ ( R N ). The lemma also follows from the inequality | ∂ αx ∂ βξ f ( x − θ Sξ ) | ≤ C Kαβ (1 + | ξ | ) ( d −| β | ) / , valid for all α, β ∈ N N , any compact K ⊂ R N , and any d ∈ R , since f ∈ S . Remark 9.13. Unlike for the case of a compact manifold, regularizing ΨDO s are not nec-essarily compact operators! .3 The preferred unitization of the Schwartz Moyal algebra Definition 9.14. Following Schwartz, we denote B := O , the space of smooth functionsbounded together with all derivatives. A unitization of A θ is given by the algebra e A θ := ( B , ? θ ). The inclusion of A θ in B is notdense, but this is not needed. e A θ contains the constant functions and the plane waves, butno nonconstant polynomials and no imaginary-quadratic exponentials, such as e iax x in thecase N = 1 (we will see later the pertinence of this).Since B is a unital ∗ -algebra with the Moyal product, Proposition 9.15. e A θ is a unital Fréchet pre- C ∗ -algebra. An advantage of e A θ is that the covering relation of the noncommutative plane to the NCtorus is made transparent. To wit, the smooth noncommutative torus algebra C ∞ ( T N Θ ) seenin Section 8.1 can be embedded in B as periodic functions (with a fixed period parallelogram).This is in fact a Hopf algebra homomorphism: recall that C ∞ ( T N Θ ) is a cotriangular Hopfalgebra by exploiting the integral form (119) of (a periodic version of) the Moyal product.We finally note the main reason for suitability of e A θ , namely, that each [ D/ , L θ ( f ) ⊗ N ]lies in A θ ⊗ M N ( C ), for f ∈ e A θ and D/ the Dirac operator on R N . When Θ = 0 the Moyal product is the ordinary product.Let A be some appropriate subalgebra of C ∞ ( M ) and D/ be the Dirac operator, with k equal to the ordinary dimension of the spin manifold M = R k . Let H be the spaceof square-integrable spinors. Then [ D/ , f ] = D/ ( f ), just as in the unital case, and so theboundedness of [ D, A ] is unproblematic. In order to check whether ( A , H , D/ , χ ) is a spectraltriple, one first needs to determine whether products of the form f ( | D/ | + ε ) − k are compactoperators of Dixmier trace class, whose Dixmier trace is (a standard multiple of) R f ( x ) d k x .This compactness condition is guaranteed in the flat space case (taking A = S ( R k ), say) bycelebrated estimates in scattering theory [102].The summability condition is a bit tougher. The Cesàro summability theory of [38]establishes that, for a positive pseudodifferential operator H of order d , acting on spinors,the spectral density asymptotically behaves as d H ( x, x ; λ ) ∼ b k/ c d (2 π ) k (cid:16) WRes H − k/d ( λ ) ( k − d ) /d + · · · (cid:17) , in the Cesàro sense. (If the operator is not positive, one uses the “four parts” argument.) Inour case, H = a ( | D/ | + ε ) − k is pseudodifferential of order − k , so d H ( x, x ; λ ) ∼ − b k/ c Ω k a ( x ) k (2 π ) k ( λ + · · · ) , as λ → ∞ in the Cesàro sense; here Ω k is the hyper-area of the unit sphere in R k . Weindependently know that H is compact, so on integrating the spectral density over x andover 0 ≤ λ ≤ λ , we get that the number of eigenvalues of H less than λ is N H ( λ ) ∼ b k/ c Ω k R a ( x ) d k xk (2 π ) k λ − as λ → ∞ . λ m ( H ) ∼ b k/ c Ω k R a ( x ) d k xk (2 π ) k m − as m → ∞ , (128)and the Dixmier traceability of a ( | D/ | + ε ) − k , plus the value of its trace, follow at once.The rest is a long but almost trivial verification. For instance, J is the charge conjugationoperator on spinors; the algebra ( B , ? ) is a suitable compactification; the domain H ∞ consistsof the smooth spinors; and so on. Thus, we get the following Theorem 9.16. The triple ( S ( R k ) , L ( R k ) ⊗ C b k/ c , D/ ) on R k defines a noncompact com-mutative geometry of spectral dimension k . What about the non-flat case (of a spin manifold such that D/ is selfadjoint)? Mainlybecause the previous Cesàro summability argument is purely local, everything carries over,if we choose for A the algebra of smooth and compactly supported functions. Of course, insome contexts it may be useful to demand that M also has conic exits. Let A = ( S ( R N ) , ? θ ), with preferred unitization e A := ( B ( R N ) , ? θ ). The Hilbert space willbe H := L ( R N ) ⊗ C N of ordinary square-integrable spinors. The representation of A isgiven by π θ : A → L ( H ) : f L θf ⊗ N , where L θf acts on the “reduced” Hilbert space H r := L ( R N ). In other words, if a ∈ A and Ψ ∈ H , to obtain π θ ( a )Ψ we just left Moyalmultiply Ψ by a componentwise.This operator π θ ( f ) is bounded, since it acts diagonally on H and k L θf k ≤ (2 πθ ) − N/ k f k was proved in Lemma 9.9. Under this action, the elements of H get the lofty name of Moyalspinors .The selfadjoint Dirac operator is not “deformed”: it will be the ordinary Euclidean Diracoperator D/ := − i γ µ ∂ µ , where the hermitian matrices γ , . . . , γ N satisfying { γ µ , γ ν } = +2 δ µν irreducibly represent the Clifford algebra C ‘ R N associated to ( R N , η ), with η the standardEuclidean metric.As a grading operator χ we take the usual chirality associated to the Clifford algebra: χ := γ N +1 := 1 H r ⊗ ( − i ) N γ γ . . . γ N . The notation γ N +1 is a nod to physicists’ γ . Thus χ = ( − N ( γ . . . γ N ) = ( − N = 1and χγ µ = − γ µ χ .The real structure J is chosen to be the usual charge conjugation operator for spinors on R N endowed with an Euclidean metric. Here, we only assume that J = ± J (1 H r ⊗ γ µ ) J − = − H r ⊗ γ µ which guarantees the other requirements of the table. In general, in a given representation,it can be written as J := CK, (129)104here C denotes a suitable 2 N × N unitary matrix and K means complex conjugation. Animportant property of J is J ( L θ ( f ∗ ) ⊗ N ) J − = R θ ( f ) ⊗ N , (130)where R θ ( f ) ≡ R θf is the right Moyal multiplication by f ; this follows from the antilinearityof J and the reversal of the twisted product under complex conjugation.Lemma 9.3(iii) implies that [ D/ , π θ ( f )] = − iL θ ( ∂ µ f ) ⊗ γ µ =: π θ ( D/ ( f )) and this is boundedfor f ∈ e A θ = B ( R N ) just as in the commutative case. In this subsection and the next, the main tools are techniques developed some time ago forscattering theory problems, as summarized in Simon’s booklet [102, Chap. 4]. We adopt theconvention that L ∞ ( H ) := K ( H ), with k A k ∞ := k A k op .Let g ∈ L ∞ ( R N ). We define the operator g ( − i ∇ ) on H r as g ( − i ∇ ) ψ := F − ( g F ψ ) , where F is the ordinary Fourier transform. More in detail, for ψ in the correct domain, g ( − i ∇ ) ψ ( x ) = (2 π ) − N ZZ e iξ · ( x − y ) g ( ξ ) ψ ( y ) d N ξ d N y. The inequality k g ( − i ∇ ) ψ k = kF − g F ψ k ≤ k g k ∞ k ψ k entails that k g ( − i ∇ ) k ∞ ≤ k g k ∞ . Theorem 9.17. Let f ∈ A and λ / ∈ sp D/ . Then, if R D/ ( λ ) is the resolvent operator of D/ ,then π θ ( f ) R D/ ( λ ) is compact. Thanks to the first resolvent equation, R D/ ( λ ) = R D/ ( λ ) + ( λ − λ ) R D/ ( λ ) R D/ ( λ ), we mayassume that λ = iµ with µ ∈ R ∗ . The theorem will follow from a series of lemmas interestingin themselves. Lemma 9.18. If f ∈ S and = µ ∈ R , then π θ ( f ) R D/ ( iµ ) ∈ K ( H ) ⇐⇒ π θ ( f ) | R D/ ( iµ ) | ∈ K ( H ) . Proof. We know that L θ ( f ) ∗ = L θ ( f ∗ ). The “only if” part is obvious since R D/ ( iµ ) is abounded normal operator. Conversely, if π θ ( f ) | R D/ ( iµ ) | is compact, then the operator π θ ( f ) | R D/ ( iµ ) | π θ ( f ∗ ) is compact. Since an operator T is compact if and only if T T ∗ iscompact, the proof is complete.The usefulness of this lemma stems from the diagonal nature of the action of the operator π θ ( f ) | R D/ ( iµ ) | on H = H r ⊗ C N ; so in our arguments it is feasible to replace H by H r , π θ ( f ) by L θf , and to use the scalar Laplacian − ∆ := − P Nµ =1 ∂ µ instead of the square of theDirac operator D/ . Lemma 9.19. When f, g ∈ H r , L θf g ( − i ∇ ) is a Hilbert–Schmidt operator such that, for allreal θ , k L θf g ( − i ∇ ) k = k L f g ( − i ∇ ) k = (2 π ) − N k f k k g k . roof. To prove that an operator A with integral kernel K A is Hilbert–Schmidt, it suffices tocheck that R | K A ( x, y ) | dx dy is finite, and this will be equal to k A k [102, Thm. 2.11]. Sowe compute K L θ ( f ) g ( − i ∇ ) . In view of Lemma 9.12,[ L θ ( f ) g ( − i ∇ ) ψ ]( x ) = π ) N ZZ f ( x − θ Sξ ) g ( ξ ) ψ ( y ) e iξ · ( x − y ) d N ξ d N y. Thus K L θ ( f ) g ( − i ∇ ) ( x, y ) = π ) N Z f ( x − θ Sξ ) g ( ξ ) e iξ · ( x − y ) d N ξ, and R | K L θ ( f ) g ( − i ∇ ) ( x, y ) | dx dy is given by π ) N Z · · · Z ¯ f ( x − θ Sξ ) ¯ g ( ξ ) f ( x − θ Sζ ) g ( ζ ) e i ( x − y ) · ( ζ − ξ ) d N x d N y d N ζ d N ξ = π ) N ZZ | f ( x − θ Sξ ) | | g ( ξ ) | d N x d N ξ = (2 π ) − N k f k k g k < ∞ . Remark 9.20. As a consequence, we get k . k - lim θ → L θf g ( − i ∇ ) = L f g ( − i ∇ ) . Lemma 9.21. If f ∈ H r and g ∈ L p ( R N ) with ≤ p < ∞ , then L θf g ( − i ∇ ) ∈ L p ( H r ) and k L θf g ( − i ∇ ) k p ≤ (2 π ) − N (1 / /p ) θ − N (1 / − /p ) k f k k g k p . Proof. The case p = 2 (with equality) is just the previous lemma. For p = ∞ , we esti-mate k L θf g ( − i ∇ ) k ∞ ≤ (2 πθ ) − N/ k f k k g k ∞ : since k L θf g ( − i ∇ ) k ∞ ≤ k L θf k ∞ k g ( − i ∇ ) k ∞ , thisfollows from Lemma 9.9 and a previous remark.Now use complex interpolation for 2 < p < ∞ . For that, we first note that we maysuppose g ≥ 0: defining the function a with | a | = 1 and g = a | g | , we see that k L θf g ( − i ∇ ) k = Tr( | L θf g ( − i ∇ ) | ) = Tr(¯ g ( − i ∇ ) L θf ∗ L θf g ( − i ∇ ))= Tr( | g | ( − i ∇ ) ¯ a ( − i ∇ ) L θf ∗ L θf a ( − i ∇ ) | g | ( − i ∇ ))= Tr(¯ a ( − i ∇ ) | g | ( − i ∇ ) L θf ∗ L θf | g | ( − i ∇ ) a ( − i ∇ ))= Tr( | L θf | g | ( − i ∇ ) | ) = k L θf | g | ( − i ∇ ) k , and k L θf g ( − i ∇ ) k ∞ = k L θf a ( − i ∇ ) | g | ( − i ∇ ) k ∞ = k L θf | g | ( − i ∇ ) a ( − i ∇ ) k ∞ ≤ k L θf | g | ( − i ∇ ) k ∞ k a ( − i ∇ ) k ∞ = k L θf | g | ( − i ∇ ) k ∞ . Secondly, for any positive, bounded function g with compact support, we define the maps: F p : z L θf g zp ( − i ∇ ) : S = { z ∈ C | ≤ < z ≤ } → L ( H r ) . For all y ∈ R , F p ( iy ) = L θf g iyp ( − i ∇ ) ∈ L ∞ ( H r ) by Lemma 9.19 since g , being compactlysupported, lies in H r . Moreover, k F p ( iy ) k ∞ ≤ (2 πθ ) − N/ k f k .106lso, by Lemma 9.19, F p ( + iy ) ∈ L ( H r ) and k F p ( + iy ) k = (2 π ) − N k f k k g p/ k .Then complex interpolation (see [90, Chap. 9] and [102]) yields F ( z ) ∈ L / < z ( H r ), for all z in the strip S . Moreover, k F p ( z ) k / < z ≤ k F (0) k − < z ∞ k F ( ) k < z = k f k (2 πθ ) − N (1 − < z ) (2 π ) − N < z k g p/ k < z , and applying this result at z = 1 /p , we get for such g : k L θf g ( − i ∇ ) k p = k F (1 /p ) k p ≤ (2 π ) − N (1 / /p ) θ − N (1 / − /p ) k f k k g k p . We finish by using the density of compactly supported bounded functions in L p ( R N ). Lemma 9.22. If f ∈ S and = µ ∈ R , then π θ ( f ) | R D/ ( iµ ) | ∈ L p for p > N .Proof. We see that π θ ( f ) | R D/ ( iµ ) | = ( L θf ⊗ N ) ( D/ − iµ ) − ( D/ + iµ ) − = L θf ( − ∂ ν ∂ ν + µ ) − ⊗ N . So this operator acts diagonally on H r ⊗ C N and Lemma 9.21 implies that (cid:13)(cid:13)(cid:13) L θf ( − ∂ ν ∂ ν + µ ) − (cid:13)(cid:13)(cid:13) p ≤ (2 π ) − N (1 / /p ) θ − N (1 / − /p ) k f k Z d N ξ ( ξ ν ξ ν + µ ) p ! /p , which is finite for p > N . Proof of Theorem 9.17. By Lemma 9.18, it was enough to prove that π θ ( f ) | R D/ ( iµ ) | is com-pact for a nonzero real µ . The spectral dimension of the Moyal N -plane spectral triple is N . We shall first establish existence properties.Thanks to Lemma 9.21 and because [ D/ , π θ ( f )] = − iL θ ( ∂ µ f ) ⊗ γ µ , we see that π θ ( f )( D/ + ε ) − l and [ D/ , π θ ( f )] ( D/ + ε ) − l lie in L p ( H ) whenever p > N/l (we always assume ε > | D/ | , π θ ( f )] ( D/ + ε ) − l has the same property of summability; thiswill become our main technical instrument for the subsection. Lemma 9.24. If f ∈ S and ≤ l ≤ N , then [ | D/ | , π θ ( f )] ( D/ + ε ) − l ∈ L p ( H ) for p > N/l .Proof. We use the following spectral identity for a positive operator A : A = π Z ∞ A A + µ dµ √ µ , and another identity for any operators A , B and λ / ∈ sp A :[ B, ( A − λ ) − ] = ( A − λ ) − [ A, B ]( A − λ ) − . (131)107ence, for any ρ > | D/ | , π θ ( f )] = [ | D/ | + ρ, π θ ( f )] = π Z ∞ " ( | D/ | + ρ ) ( | D/ | + ρ ) + µ , π θ ( f ) dµ √ µ = π Z ∞ − ( | D/ | + ρ ) ( | D/ | + ρ ) + µ !h ( | D/ | + ρ ) , π θ ( f ) i | D/ | + ρ ) + µ dµ √ µ = π Z ∞ | D/ | + ρ ) + µ h ( | D/ | + ρ ) , π θ ( f ) i | D/ | + ρ ) + µ √ µ dµ (132)= π Z ∞ | D/ | + ρ ) + µ − π θ ( ∂ µ ∂ µ f ) − i ( L θ ( ∂ µ f ) ⊗ γ µ ) D/ + 2 ρ h | D/ | , π θ ( f ) i! × | D/ | + ρ ) + µ √ µ dµ. This implies that (cid:13)(cid:13)(cid:13) [ | D/ | , π θ ( f )] ( D/ + ε ) − l (cid:13)(cid:13)(cid:13) p ≤ π Z ∞ (cid:13)(cid:13)(cid:13) | D/ | + ρ ) + µ (cid:18) − π θ ( ∂ µ ∂ µ f ) − i ( L θ ( ∂ µ f ) ⊗ γ µ ) D/ + 2 ρ h | D/ | , π θ ( f ) i(cid:19) | D/ | + ρ ) + µ ( D/ + ε ) − l (cid:13)(cid:13)(cid:13) p √ µ dµ. Thus, the proof reduces to show that for any f ∈ S , π Z ∞ (cid:13)(cid:13)(cid:13) | D/ | + ρ ) + µ π θ ( f ) D/ | D/ | + ρ ) + µ ( D/ + ε ) − l (cid:13)(cid:13)(cid:13) p √ µ dµ < ∞ . (133)Since the Schatten p -norm is a symmetric norm, and since, as in the proof of Theorem 9.17,only the reduced Hilbert space is affected, expression (133) is bounded by by π Z ∞ (cid:13)(cid:13)(cid:13) | D/ | + ρ ) + µ (cid:13)(cid:13)(cid:13) / (cid:13)(cid:13)(cid:13) D/ ( D/ + ε ) / (cid:13)(cid:13)(cid:13) (cid:13)(cid:13)(cid:13) π θ ( f ) D/ + ε ) l − / | D/ | + ρ ) + µ ) / (cid:13)(cid:13)(cid:13) p √ µ dµ ≤ π Z ∞ (cid:13)(cid:13)(cid:13) π θ ( f ) ( D/ + ε ) − l +1 / (( | D/ | + ρ ) + µ ) − / (cid:13)(cid:13)(cid:13) p √ µ dµ ( µ + ρ ) / . Thanks to Lemma 9.21, we can estimate the µ -dependence of the last p -norm: (cid:13)(cid:13)(cid:13) π θ ( f )(( | D/ | + ρ ) + µ ) − / ( D/ + ε ) − l +1 / (cid:13)(cid:13)(cid:13) p ≤ (2 π ) − N (1 / /p ) θ − N (1 / − /p ) k f k (cid:13)(cid:13)(cid:13) (( | ξ | + ρ ) + µ ) − / ( | ξ | + ε ) − l +1 / (cid:13)(cid:13)(cid:13) p ≤ C ( p, θ ) (cid:13)(cid:13)(cid:13) (( | ξ | + ρ ) + µ ) − / (cid:13)(cid:13)(cid:13) q (cid:13)(cid:13)(cid:13) ( | ξ | + ε ) − l +1 / (cid:13)(cid:13)(cid:13) r ;with p − = q − + r − appropriately chosen, these integrals are finite for all q > N and r > N/ (2 l − l = , take r = ∞ and q = p . For such values, (cid:13)(cid:13)(cid:13) π θ ( f )(( | D/ | + ρ ) + µ ) − / ( D/ + ε ) − l +1 / (cid:13)(cid:13)(cid:13) p ≤ C ( p, θ, N ; f ) k ( | ξ | + ε ) − l +1 / k r Ω /q N Z ∞ R N − (( R + ρ ) + µ ) q/ dR ! /q = C ( p, θ, N ; f ) k ( | ξ | + ε ) − l +1 / k r π N/q Γ /q ( q − N )Γ /q ( q µ − / N/q =: C ( p, q, θ, N ; f ) µ − / N/q . C ( p, q, θ, N ; f ) Z ∞ µ N/q ( µ + ρ ) / dµ, which is finite for q > N and p > N/l . This concludes the proof. Lemma 9.25. If f ∈ S , then π θ ( f ) ( | D/ | + ε ) − π θ ( f ∗ ) ∈ L N + ( H ) .Proof. This is an extension to the Moyal context of the renowned inequality by Cwikel [102].As remarked before, it is possible to replace D/ by − ∆, π θ ( f ) by L θf and H by H r . Consider g ( − i ∇ ) := ( √− ∆ + ε ) − . Since g is positive, it can be decomposed as g = P n ∈ Z g n where g n ( x ) := g ( x ) if 2 n − < g ( x ) ≤ n , . For each n ∈ Z , let A n and B n be the two operators A n := X k ≤ n L θf g k ( − i ∇ ) L θf ∗ , B n := X k>n L θf g k ( − i ∇ ) L θf ∗ . We estimate the uniform norm of the first part: k A n k ∞ ≤ k L θf k (cid:13)(cid:13)(cid:13)X k ≤ n g k ( − i ∇ ) (cid:13)(cid:13)(cid:13) ∞ ≤ (2 πθ ) − N k f k (cid:13)(cid:13)(cid:13)X k ≤ n g k (cid:13)(cid:13)(cid:13) ∞ ≤ (2 πθ ) − N k f k n =: 2 n c ( θ, N ; f ) . The trace norm of B n can be computed using Lemma 9.19: k B n k = (cid:13)(cid:13)(cid:13)(cid:18)X k>n g k ( − i ∇ ) (cid:19) / L θf ∗ (cid:13)(cid:13)(cid:13) = (cid:13)(cid:13)(cid:13) L θf (cid:18)X k>n g k ( − i ∇ ) (cid:19) / (cid:13)(cid:13)(cid:13) = (2 π ) − N k f k (cid:13)(cid:13)(cid:13)(cid:18)X k>n g k (cid:19) / (cid:13)(cid:13)(cid:13) = (2 π ) − N k f k (cid:13)(cid:13)(cid:13)X k>n g k (cid:13)(cid:13)(cid:13) = (2 π ) − N k f k X k>n k g k k ≤ (2 π ) − N k f k X k>n k g k k ∞ ν { supp( g k ) } , where ν is the Lebesgue measure on R N . By definition, k g k k ∞ ≤ k and ν { supp( g k ) } = ν { x ∈ R N : 2 k − < g ( x ) ≤ k } ≤ ν { x ∈ R N | ( | x | + ε ) − ≥ k − }≤ N (1 − k ) c . Therefore k B n k ≤ (2 π ) − N k f k N c X k>n k (1 − N ) < π − N c k f k n (1 − N ) =: 2 n (1 − N ) c ( N ; f ) , where the second inequality follows because N > .We can now estimate the m th singular value µ m of B n (arranged in decreasing orderwith multiplicity): k B n k = P ∞ k =0 µ k ( B n ). Note that, for m = 1 , , , · · · , we get that109 B n k ≥ P m − k =0 µ k ( B n ) ≥ m µ m ( B n ). Thus, µ m ( B n ) ≤ k B n k m − ≤ n (1 − N ) c m − . NowFan’s inequality [102, Thm. 1.7] yields µ m ( L θf g ( − i ∇ ) L θf ∗ ) = µ m ( A n + B n ) ≤ µ ( A n ) + µ m ( B n ) ≤ k A n k + k B n k m − ≤ n c + 2 n (1 − N ) c m − . Given m , choose n ∈ Z so that 2 n ≤ m − / N < n +1 . Then µ m ( L θf g ( − i ∇ ) L θf ∗ ) ≤ c m − / N + c m − (1 − N ) / N m − =: c ( θ, N ; f ) m − / N . Therefore L θf ( √− ∆ + ε ) − L θf ∗ ∈ L N + ( H r ), and the statement of the lemma follows. Corollary 9.26. If f, g ∈ S , then π θ ( f ) ( | D/ | + ε ) − π θ ( g ) ∈ L N + ( H ) .Proof. Consider π θ ( f ± g ∗ ) ( | D/ | + ε ) − π θ ( f ∗ ± g ) and π θ ( f ± ig ∗ ) ( | D/ | + ε ) − π θ ( f ∗ ∓ ig ). Corollary 9.27. If h ∈ S , then π θ ( h ) ( | D/ | + ε ) − ∈ L N + ( H ) .Proof. Let h = f ? θ g . Then π θ ( h ) ( | D/ | + ε ) − = π θ ( f ) ( | D/ | + ε ) − π θ ( g ) + π θ ( f ) [ π θ ( g ) , ( | D/ | + ε ) − ] , and we obtain from the identity (131) that π θ ( h ) ( | D/ | + ε ) − = π θ ( f ) ( | D/ | + ε ) − π θ ( g ) + π θ ( f ) ( | D/ | + ε ) − [ | D/ | , π θ ( g )] ( | D/ | + ε ) − . By arguments similar to those of lemmata 9.21 and 9.24, the last term belongs to L p for p > N , and thus to L N + .Boundedness of ( | D/ | + ε )( D/ + ε ) − / follows from elementary Fourier analysis. And sothe last corollary means that the spectral triple is “2 N + -summable”. We have taken care ofthe first assertion of the theorem. The next lemma is the last property of existence that weneed. Lemma 9.28. If f ∈ S , then π θ ( f )( | D/ | + ε ) − N and π θ ( f )( D/ + ε ) − N are in L ( H ) .Proof. It suffices to prove that π θ ( f )( | D/ | + ε ) − N ∈ L ( H ). We factorize f ∈ S accordingto Proposition 9.6, with the following notation: f = f ? θ f = f ? θ f ? θ f = f ? θ f ? θ f ? θ f = · · · = f ? θ f ? θ f ? θ · · · ? θ f ··· ? θ f ··· . Therefore, π θ ( f ) ( | D/ | + ε ) − N = π θ ( f ) ( | D/ | + ε ) − π θ ( f ) ( | D/ | + ε ) − N +1 + π θ ( f ) ( | D/ | + ε ) − [ | D/ | , π θ ( f )] ( | D/ | + ε ) − N . (134)By Lemma 9.21, π θ ( f )( | D/ | + ε ) − ∈ L p ( H ) whenever p > N ; and by Lemma 9.24, theterm [ | D/ | , π θ ( f )]( | D/ | + ε ) − N lies in L q ( H ) for q > 1. Hence, the last term on the right handside of equation (134) lies in L ( H ). We may write the following equivalence relation: π θ ( f )( | D/ | + ε ) − N ∼ π θ ( f )( | D/ | + ε ) − π θ ( f )( | D/ | + ε ) − N +1 , A ∼ B for A, B ∈ K ( H ) means that A − B is trace-class. Thus, π θ ( f )( | D/ | + ε ) − N ∼ π θ ( f )( | D/ | + ε ) − π θ ( f )( | D/ | + ε ) − N +1 = π θ ( f )( | D/ | + ε ) − π θ ( f )( | D/ | + ε ) − π θ ( f )( | D/ | + ε ) − N +2 + π θ ( f )( | D/ | + ε ) − π θ ( f )( | D/ | + ε ) − [ | D/ | , π θ ( f )] ( | D/ | + ε ) − N +1 ∼ π θ ( f )( | D/ | + ε ) − π θ ( f )( | D/ | + ε ) − π θ ( f )( | D/ | + ε ) − N +2 ∼ · · ·∼ π θ ( f )( | D/ | + ε ) − π θ ( f )( | D/ | + ε ) − π θ ( f )( | D/ | + ε ) − . . . π θ ( f ··· )( | D/ | + ε ) − . The second equivalence relation holds because π θ ( f )( | D/ | + ε ) − π θ ( f )( | D/ | + ε ) − ∈ L p ( H )for p > N by Lemma 9.21, and [ | D/ | , π θ ( f )]( | D/ | + ε ) − N +1 ∈ L q ( H ) for q > N/ (2 N − L ( H ) ⊂ L ( H ) finally yieldthe result.Now we go for the computation of the Dixmier trace. Using the regularized trace for aΨDO: Tr Λ ( A ) := (2 π ) − N ZZ | ξ |≤ Λ σ [ A ]( x, ξ ) d N ξ d N x, the result can be conjectured because lim Λ →∞ Tr Λ ( · ) / log(Λ N ) is heuristically linked withthe Dixmier trace, and the following computation:lim Λ →∞ N log Λ Tr Λ (cid:16) π θ ( f )( D/ + ε ) − N (cid:17) = lim Λ →∞ t N N (2 π ) N log Λ ZZ | ξ |≤ Λ f ( x − θ Sξ ) ( | ξ | + ε ) − N d N ξ d N x = N Ω N N (2 π ) N Z f ( x ) d N x. This is precisely the same result of (128), in the commutative case, for k = 2 N . However,to establish it rigorously in the Moyal context requires a subtler strategy. We shall computethe Dixmier trace of π θ ( f ) ( D/ + ε ) − N as the residue of the ordinary trace of a relatedmeromorphic family of operators. In turn we are allowed to introduce the explicit symbolformula that will establish measurability [25, 50], too.We seek first to verify that A θ has analytical dimension equal to 2 N ; that is, for f ∈ A θ the operator π θ ( f ) ( D/ + ε ) − z/ is trace-class if < z > N . Lemma 9.29. If f ∈ S , then L θf ( D/ + ε ) − z/ is trace-class for < z > N , and Tr[ L θf ( D/ + ε ) − z/ ] = (2 π ) − N ZZ f ( x ) ( | ξ | + ε ) − z/ d N ξ d N x. Proof. If a ( x, ξ ) ∈ K p ( R k ), for p < − k , is the symbol of a pseudodifferential operator A ,then the operator is trace-class and moreoverTr A = (2 π ) − k ZZ a ( x, ξ ) d k x d k ξ. This is easily proved by taking a ∈ S ( R k ) first and extending the resulting formula bycontinuity. 111n our case, the symbol formula for a product of ΨDOs yields, for p > N , σ h L θf ( − ∆ + ε ) − p i ( x, ξ ) = X α ∈ N N ( − i ) | α | α ! ∂ αξ σ [ L θf ]( x, ξ ) ∂ αx σ h ( − ∆ + ε ) − p i ( x, ξ )= σ [ L θf ]( x, ξ ) σ h ( − ∆ + ε ) − p i ( x, ξ )= f ( x − θ Sξ ) ( | ξ | + ε ) − p . Therefore, for p > N ,Tr (cid:16) L θf ( − ∆ + ε ) − p (cid:17) = (2 π ) − N ZZ f ( x − θ Sξ ) ( | ξ | + ε ) − p d N ξ d N x = (2 π ) − N ZZ f ( x ) ( | ξ | + ε ) − p d N ξ d N x. We continue with a technical lemma, in the spirit of [93]. Consider the approximate unit { e K } K ∈ N ⊂ A c where e K := P ≤| n |≤ K f nn . These e K are projectors with a natural ordering: e K ? θ e L = e L ? θ e K = e K for K ≤ L , and they are local units for A c . Lemma 9.30. Let f ∈ A c,K . Then π θ ( f ) ( D/ + ε ) − N − π θ ( f ) (cid:16) π θ ( e K )( D/ + ε ) − π θ ( e K ) (cid:17) N ∈ L ( H ) . Proof. For simplicity we use the notation e := e K and e n := e K + n . By the boundedness of π θ ( f ), we may assume that f = e ∈ A c,K .Because e n ? θ e = e? θ e n = e , it is clear that π θ ( e )( D/ + λ ) − (cid:16) − π θ ( e n ) (cid:17) = π θ ( e ) ( D/ + λ ) − [ D/ , π θ ( e n )] ( D/ + λ ) − . (135)Also, π θ ( e ) [ D/ , π θ ( e n )] = [ D/ , π θ ( e? θ e n )] − [ D/ , π θ ( e )] π θ ( e n ) = 0 because we have the relation[ D/ , π θ ( e )] π θ ( e n ) = [ D/ , π θ ( e )] for n ≥ 1. We obtain A n := π θ ( e )( D/ + λ ) − [ D/ , π θ ( e n )]( D/ + λ ) − = π θ ( e )( D/ + λ ) − [ D/ , π θ ( e )]( D/ + λ ) − [ D/ , π θ ( e n )]( D/ + λ ) − = π θ ( e )( D/ + λ ) − [ D/ , π θ ( e )] π θ ( e )( D/ + λ ) − [ D/ , π θ ( e n )]( D/ + λ ) − = · · · = (cid:16) π θ ( e )( D/ + λ ) − (cid:17)(cid:16) [ D/ , π θ ( e )]( D/ + λ ) − (cid:17)(cid:16) [ D/ , π θ ( e )]( D/ + λ ) − (cid:17) · · ·· · · (cid:16) [ D/ , π θ ( e n )]( D/ + λ ) − (cid:17) . Taking n = 2 N here, A N appears as a product of 2 N + 1 terms in parentheses, each in L N +1 ( H ) by Lemma 9.21. Hence, by Hölder’s inequality, A N is trace-class and therefore π θ ( e )( D/ + λ ) − (1 − π θ ( e N )) ∈ L ( H ). Thus, π θ ( e ) ( D/ + ε ) − (cid:16) − π θ ( e N ) (cid:17) = π θ ( e )( D/ − iε ) − (cid:16) − π θ ( e N ) + π θ ( e N ) (cid:17) ( D/ + iε ) − (cid:16) − π θ ( e N ) (cid:17) = π θ ( e )( D/ − iε ) − (cid:16) − π θ ( e N ) (cid:17) ( D/ + iε ) − (cid:16) − π θ ( e N ) (cid:17) + π θ ( e )( D/ − iε ) − π θ ( e N )( D/ + iε ) − (cid:16) − π θ ( e N ) (cid:17) ∈ L ( H ) . (136)112his is to say π θ ( e )( D/ + ε ) − ∼ π θ ( e )( D/ + ε ) − π θ ( e N ). Shifting this property, we get π θ ( e )( D/ + ε ) − N ∼ π θ ( e )( D/ + ε ) − π θ ( e N )( D/ + ε ) − N +1 ∼ π θ ( e )( D/ + ε ) − π θ ( e N )( D/ + ε ) − π θ ( e N )( D/ + ε ) − N +2 ∼ · · ·∼ π θ ( e )( D/ + ε ) − π θ ( e N )( D/ + ε ) − π θ ( e N ) · · · ( D/ + ε ) − π θ ( e N ) . By identity (131), the last term on the right equals π θ ( e )( D/ + iε ) − π θ ( e )( D/ − iε ) − π θ ( e N )( D/ + ε ) − π θ ( e N ) · · · ( D/ + ε ) − π θ ( e N )+ π θ ( e )( D/ + iε ) − [ D/ , π θ ( e )]( D/ + ε ) − π θ ( e N )( D/ + ε ) − π θ ( e N ) · · ( D/ + ε ) − π θ ( e N ) . The last term is trace-class because it is a product of N terms in L p ( H ) for p > N andone term in L q ( H ) for q > N , by Lemma 9.21. Removing the second π θ ( e ) once again, bythe ordering property of the local units e K yields π θ ( e )( D/ + iε ) − π θ ( e )( D/ − iε ) − π θ ( e N )( D/ + ε ) − π θ ( e N ) · · · ( D/ + ε ) − π θ ( e N )= π θ ( e )( D/ + ε ) − π θ ( e )( D/ + ε ) − π θ ( e N ) · · · ( D/ + ε ) − π θ ( e N )+ π θ ( e )( D/ + ε ) − [ D/ , π θ ( e )]( D/ − iε ) − π θ ( e N )( D/ + ε ) − π θ ( e N ) · · ( D/ + ε ) − π θ ( e N ) . The last term is still trace-class, hence π θ ( e )( D/ + ε ) − N ∼ π θ ( e )( D/ + ε ) − π θ ( e )( D/ + ε ) − π θ ( e N ) · · · ( D/ + ε ) − π θ ( e N ) . This algorithm, applied another ( N − 1) times, yields the result: π θ ( e )( D/ + ε ) − N ∼ (cid:16) π θ ( e )( D/ + ε ) − π θ ( e ) (cid:17) N . We retain the following consequence. Corollary 9.31. Tr + (cid:16) π θ ( g ) [ π θ ( f ) , ( D/ + ε ) − N ] (cid:17) = 0 for any g ∈ S and any projector f ∈ A c .Proof. This follows from Lemma 9.30 applied to π θ ( f ) ( D/ + ε ) − N and its adjoint.Now we are finally ready to evaluate the Dixmier traces. Proposition 9.32. For f ∈ S , any Dixmier trace Tr + of π θ ( f ) ( D/ + ε ) − N is independentof ε , and Tr + (cid:16) π θ ( f ) ( D/ + ε ) − N (cid:17) = N Ω N N (2 π ) N Z f ( x ) d N x = N ! (2 π ) N Z f ( x ) d N x. Proof. We will first prove it for f ∈ A c . Choose e a unit for f , that is, e? θ f = f ? θ e = f . ByLemmata 9.28 and 9.30, and because L ( H ) lies inside the kernel of the Dixmier trace, weobtain Tr + ( π θ ( f ) ( D/ + ε ) − N ) = Tr + (cid:16) π θ ( f ) ( π θ ( e )( D/ + ε ) − π θ ( e )) N (cid:17) . f = e implies that (cid:16) π θ ( e )( D/ + ε ) − π θ ( e ) (cid:17) N is a positive operatorin L ( H ), since it is equal to π θ ( e )( D/ + ε ) − N plus a term in L ( H ). Thus, [10, Thm. 5.6]yields (since the limit converges, any Dixmier trace will give the same result):Tr + (cid:16) π θ ( f ) ( D/ + ε ) − N (cid:17) = lim s ↓ ( s − 1) Tr h π θ ( f ) ( π θ ( e )( D/ + ε ) − π θ ( e )) Ns i = lim s ↓ ( s − 1) Tr (cid:16) π θ ( f ) π θ ( e )( D/ + ε ) − Ns π θ ( e ) + E Ns (cid:17) , (137)where E Ns := π θ ( f ) (cid:16) π θ ( e )( D/ + ε ) − π θ ( e ) (cid:17) Ns − π θ ( f ) π θ ( e )( D/ + ε ) − Ns π θ ( e ) . Lemma 9.30 again shows that E N ∈ L ( H ).Now for s > 1, the first term π θ ( f ) (cid:16) π θ ( e )( D/ + ε ) − π θ ( e ) (cid:17) Ns of E Ns is in L ( H ). Ineffect, using Lemma 9.21 and since π θ ( e )( D/ + ε ) − ∈ L p ( H ) for p > N , we get that π θ ( e )( D/ + ε ) − π θ ( e ) ∈ L Ns ( H ). This operator being positive, one concludes (cid:16) π θ ( e )( D/ + ε ) − π θ ( e ) (cid:17) Ns ∈ L ( H ) . The second term π θ ( f ) π θ ( e )( D/ + ε ) − Ns π θ ( e ) lies in L ( H ) too, because k π θ ( e )( D/ + ε ) − Ns π θ ( e ) k = k ( D/ + ε ) − Ns/ π θ ( e ) k = k π θ ( e )( D/ + ε ) − Ns/ k is finite by Lemma 9.19. So E Ns ∈ L ( H ) for s ≥ 1, and (137) impliesTr + (cid:16) π θ ( f ) ( D/ + ε ) − N (cid:17) = lim s ↓ ( s − 1) Tr (cid:16) π θ ( f ) π θ ( e )( D/ + ε ) − Ns π θ ( e ) (cid:17) = lim s ↓ ( s − 1) Tr (cid:16) π θ ( f )( D/ + ε ) − Ns (cid:17) . Applying now Lemma 9.29, we obtainTr + (cid:16) π θ ( f ) ( D/ + ε ) − N (cid:17) = lim s ↓ ( s − 1) Tr(1 N ) Tr (cid:16) L θf ( − ∆ + ε ) − Ns (cid:17) = 2 N (2 π ) − N lim s ↓ ( s − ZZ f ( x ) ( | ξ | + ε ) − Ns d N ξ d N x = N ! (2 π ) N Z f ( x ) d N x, where the identity Z ( | ξ | + ε ) − Ns d N ξ = π N Γ( N ( s − Ns ) ε N ( s − , and Γ( N α ) ∼ /N α as α ↓ f ∈ A c .Finally, take f arbitrary in S , and recall that { e K } is an approximate unit for A θ . Since f = g? θ h for some g, h ∈ S , Corollary 9.31 implies (cid:12)(cid:12)(cid:12) Tr + (cid:16) ( π θ ( f ) − π θ ( e K ? θ f ? θ e K ))( D/ + ε ) − N (cid:17)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Tr + (cid:16) ( π θ ( f ) − π θ ( e K ? θ f )) ( D/ + ε ) − N (cid:17)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) Tr + (cid:16) ( π θ ( g ) − π θ ( e K ? θ g )) π θ ( h )( D/ + ε ) − N (cid:17)(cid:12)(cid:12)(cid:12) ≤ k π θ ( g ) − π θ ( e K ? θ g ) k ∞ Tr + (cid:12)(cid:12)(cid:12) π θ ( h ) ( D/ + ε ) − N (cid:12)(cid:12)(cid:12) . k π θ ( g ) − π θ ( e K ? θ g ) k ∞ ≤ (2 πθ ) − N/ k g − e K ? θ g k tends to zero when K increases, theproof is complete because e K ? θ f ? θ e K lies in A c and Z [ e K ? θ f ? θ e K ]( x ) d N x → Z f ( x ) d N x as K ↑ ∞ . Remark 9.33. Similar arguments to those of this section (or a simple comparison argument)show that for f ∈ S , Tr + (cid:16) π θ ( f ) ( | D/ | + ε ) − N (cid:17) = Tr + (cid:16) π θ ( f ) ( D/ + ε ) − N (cid:17) . In conclusion: the analytical and spectral dimension of Moyal planes coincide. AndLemma 9.28, Proposition 9.32 and the previous remark have concluded the proof of Theo-rem 9.23.The conclusion is that ( A , e A , H , D/ , χ, J ) defines a non-compact spectral triple; recall thatwe already know that both A and its preferred compactification e A are pre- C ∗ -algebras. Theorem 9.34. The Moyal planes ( A , e A , H , D/ , J, χ ) are connected real non-compact spectraltriples of spectral dimension N . One can compute the Yang–Mills action (72) of this triple: Theorem 9.35. Let ω = − ω ∗ ∈ Ω A θ . Then the Yang–Mills action Y M ( V ) of the universalconnection δ + ω , with V = ˜ π θ ( ω ) , is equal to S Y M ( V ) = − g Z F µν ? θ F µν ( x ) d N x = − g Z F µν ( x ) F µν ( x ) d N x, where F µν := ( ∂ µ A ν − ∂ ν A µ + [ A µ , A ν ] ? θ ) and A µ is defined by V = L θ ( A µ ) ⊗ γ µ . The spectral action has been computed in [41, 42]. As one can expected, it is the same,up to few universal coefficients, to the one of Theorem 8.13. Acknowledgments I would like to thank Driss Essouabri, Victor Gayral, José Gracia-Bondía, Cyril Levy, PierreMartinetti, Thierry Masson, Thomas Schücker, Andrzej Sitarz, Jo Várilly and Dmitri Vassile-vich, for our discussions during our collaborations along years. Some of the results presentedhere are directly extracted from these collaborations. I also took benefits from the questionsof participants (professors and students) of the school during the lectures, to clarify fewpoints. 115 eferences [1] W. Arendt, R. Nittka, W. Peter and F. 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