Spectral Geometry Bruno Iochum

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Bruno Iochum. Spectral Geometry. A. Cardonna, C. Neira-Jemenez, H. Ocampo, S. Paycha and A. Reyes-Lega. Spectral Geometry, Aug 2011, Villa de Leyva, Colombia. World Scientific, 2014, Geometric, Algebraic and Topological Methods for Quantum Field Theory, 978-981-4460-04-0. hal- 00947123

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Bruno Iochum

Aix-Marseille Université, CNRS UMR 7332, CPT, 13288 Marseille France Abstract The goal of these lectures is to present some fundamentals of noncommutative geometry looking around its spectral approach. Strongly motivated by physics, in particular by relativity and quantum mechanics, Chamseddine and Connes have defined an action based on spectral considerations, the so-called spectral action. The idea here is to review the necessary tools which are behind this spectral action to be able to compute it first in the case of Riemannian manifolds (Einstein–Hilbert action). Then, all primary objects defined for manifolds will be generalized to reach the level of noncommutative geometry via spectral triples, with the concrete analysis of the noncommutative torus which is a deformation of the ordinary one. The basics ingredients such as Dirac operators, heat equation asymptotics, zeta functions, noncommutative residues, pseudodifferential operators, or Dixmier traces will be presented and studied within the framework of operators on Hilbert spaces. These notions are appropriate in noncommutative geometry to tackle the case where the space is swapped with an algebra like for instance the noncommutative torus.

1 Motivations

Let us first expose few motivations from physics to study noncommutative geometry which is by essence a spectral geometry. Of course, precise mathematical definitions and results will be given in 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 electromagnetism. The notion of spectral theory is also important in functional analysis, where the spectral theorem tells us that any selfadjoint operator A can be seen as an integral over its spectral measure A = a Sp(a) adPa if Sp(A) is the spectrum of A. This is of course essential ∈ in the axiomatic formulationR of quantum mechanics, especially in the Heisenberg picture where the observables are selfadjoint operators. But this notion is also useful in geometry. In special relativity, we consider fields ψ(x) for x R4 and the electric and magnetic fields E, B Function(M = R4, R3). Einstein introduced∈ in 1915 the gravitational field and the equation∈ of motion of matter. But a problem appeared: what are the physical meaning of coordinates x and equations of fields? Assume I the general covariance of field equation. If gν (x) or the tetradfield e(x) is a solution (where I is a local inertial reference frame), then, for any diffeomorphism φ of M which is active or I ∂x I passive (i.e. change of coordinates), eν′ (x)= ∂φ(x)ν e(x) is also a solution. As a consequence, when relativity became general, the points disappeared and it remained only fields on fields in the sense that there is no fields on a given space-time. But how to practice geometry without space, given usually by a manifold M? In this later case, the spectral approach, namely the control of eigenvalues of the scalar (or spinorial) Laplacian returns important informations on M and one can address the question if they are sufficient: can one hear the shape of M?

1 There are two natural points of view on the notion of space: one is based on points (of a manifold), this is the traditional geometrical one. The other is based on algebra and this is the 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 a 2 quantum geometry with bosons satisfying the Klein-Gordon equation ( + m )ψ(x)= sb(x) and fermions satisfying (i∂/ m)ψ(x) = s (x) for sources s ,s . Here ∂/ can be seen as a − f b f square root of 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 of physics, especially of space-time. Noncommutative geometry, mainly pioneered by A. Connes (see [24, 30]), is based on a spectral triple ( , , ) where the -algebra generalizes smooth functions on space-time M (or the coordinates)A H D with pointwise∗ product,A generalizes the Hilbert space of above H quoted spinors ψ and is a selfadjoint operator on which generalizes ∂/ via a connection on a vector bundle overDM. The algebra also acts,H via a representation of -algebra, on . A ∗ 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 are 1 indistinguishable. Below the scale Λ− , a certain renormalization is necessary. Given a geometry, 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, given the data ( , , ), the appropriate notion of action was introduced by Chamseddine and A H D Connes [10] and defined as S( , Λ, f) := Tr f( /Λ) where Λ R+ plays the role of a D D ∈ cut-off and f is a positive even function. The asymptotic series in Λ yields to an effective theory. For instance, this action applied to a noncommutative model→ ∞ of space-time M F with a fine structure for fermions encoded in a finite geometry F gives rise from pure gravity× to the standard model coupled with gravity [11,20,30].

The purpose of these notes is mainly to compute this spectral action on few examples like manifolds and the noncommutative torus. In section 2, we present standard material on pseudodifferential operators over a compact Riemannian manifold. A description of the behavior of the kernel of a ΨDO near the diagonal is given with the important example of elliptic operators. Then follows the notion of Wodzicki residue and its computation. The main point being to understand why it is a residue. In section 3, the link with the Dixmier trace is shown. Different subspaces of compact 1, operators are described, and in particular, the ideal ∞( ). Its definition is on purpose because in renormalization theory, one has to control theL logarithmicH divergency of the series 1 n∞=1 n− . We will see that this “defect” of convergence of the Riemann zeta function (in Pthe sense that this 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 are the right tools which mimics this zeta function: firstly, this controls the spectral aspects of a manifold and secondly they can be generalized to any spectral triple. In section 4, we recall the basic definition of a Dirac (or Dirac-like) operator on a compact Riemannian manifold (M, g) endowed with a vector bundle E. An example is the (Clifford) bundle E = ℓ M where ℓ Tx∗M is the Clifford algebra for x M. This leads to the notion of spin structure,C spin connectionC S and Dirac operator D/∈ = ic S where c is the Clifford multiplication. A special focus∇ is put on the change of metrics− ◦ ∇g under conformal transformations.

2 In section 5 is presented the fundamentals of heat kernel theory, namely the Green function of the heat operator et∆, t R+. In particular, its expansion as t 0+ in terms of coefficients of the elliptic operator ∆,∈ with a method to compute the coeffic→ients of this expansion is explained. The idea being to replace the Laplacian ∆ by 2 later on. D In section 6, a noncommutative integration theory is developed around the notion of spectral 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 the fluctuations of the given is outlined. Section 7 concerns fewD actions in physics, such as the Einstein–Hilbert or Yang–Mills actions. The spectral action Tr f( /Λ) is justified and the link between its asymptotic D expansion in Λ and the heat kernel coefficients is given via noncommutative integrals of powers of . |D| For each section, we suggest references since this review is by no means original.

Notations: N = 1, 2,... is the set of positive integers and N0 = N 0 the set of non negative integers. On R{d, the volume} form is dx = dx1 dxd. ∪{ } Sd ∧∧ d j 1 is the sphere of radius 1 in dimension d. The induced metric dξ = j=1( 1) − ξj dξ1 d 1 | − ∧ dξ dξ restricts to the volume form on S − . ∧ j ∧∧ d| P M is a d-dimensional manifold with metric g. U, V ared open set either in M or in Rd. We denote by dvol the unique volume element such that dvol (ξ , ,ξ ) = 1 for all posi- g g 1 d tively oriented g-orthonormal basis ξ1, ,ξd of TxM for x M. Thus in a local chart √det g dx = dvol . { } ∈ x | | | g| Nd α : α1 α2 αd : d : When α is a multi-index, ∂x = ∂x1 ∂x2 ∂xd , α = i=1 αi , α! = α1α2 αd. ∈ 1/2 | | For ξ Rd, ξ := d ξ 2 is the Euclidean metric. P ∈ | | k=1 | k| is a separable Hilbert space and ( ), ( ), p( ) denote respectively the set of bounded, H P B H K H L H compact and p-Schatten-class operators, so 1( ) are trace-class operators. L H 2 Wodzicki residue and kernel near the diagonal

The aim of this section is to show that the Wodzicki’s residue WRes is a trace on the set ΨDO(M) of classical pseudodiﬀerential operators on a compact manifold M of dimension d. References for this section: Classical books are [99, 102]. For an orientation more in the spirit of noncommutative geometry since here we follow [86,87] based on [3,33], see also the excellent books [49,82,83,104,105].

2.1 A quick overview on pseudodiﬀerential operators

Deﬁnition 2.1. In the following, m C. A symbol σ(x, ξ) of order m is a C∞ function: (x, ξ) U Rd C satisfying for any∈ compact K U and any x K ∈ α ×β → (m) β ⊂ ∈ (i) ∂ ∂ σ(x, ξ) C (1 + ξ )ℜ −| |, for some constant C . | x ξ | ≤ Kαβ | | Kαβ (ii) We suppose that σ(x, ξ) j 0 σm j(x, ξ) where σk is homogeneous of degree k in ξ ≃ ≥ − where means a controlled asymptoticP behavior ≃ α β (m) N β ∂x ∂ξ σ j

m d For σ S (U R ), we get a continuous operator σ( ,D) : u C∞(U) C∞(U) given by ∈ × ∈ c → 1 ix ξ σ( ,D)(u)(x) := σ(x, D)(u) := d σ(x, ξ) u(ξ) e dξ (1) (2π) Rd Z where means the Fourier transform. This operator σ( ,D) willb be also denoted by Op(σ). For instance, if σ(x, ξ) = a (x) ξα, then σ(x, D) = a (x) Dα with D := i∂ . Re- α α α α x x − x mark that,b by transposition, there is a natural extension of σ( ,D) from the set ′ (U) of P P Dc distributions with compact support in U to the set of distributions ′(U). D By deﬁnition, the leading term for α = m is the principal symbol and the Schwartz kernel of σ(x, D) is deﬁned by | |

σ(x,D) 1 i(x y) ξ q k (x, y) := d σ(x, ξ) e − dξ = σξ y(x, x y) (2π) Rd → Z − where q is the Fourier inverse in variable ξ. Similarly, if the kernel of the operator Op(a) associated to the amplitude a is

a 1 i(x y) ξ k (x, y) := d a(x,y,ξ) e − dξ. (2) (2π) Rd Z Definition 2.2. P : C∞(U) C∞(U) (or ′(U)) is said to be smoothing if its kernel is in c → D C∞(U U) and ΨDO−∞(U) will denote the set of smoothing operators. × For m C, the set ΨDOm(U) of pseudodifferential operators of order m is the set of P such ∈ m d that P : C∞(U) C∞(U), Pu(x)= σ(x, D)+R (u) where σ S (U R ), R ΨDO−∞. c → ∈ × ∈ σ is called the symbol of P . It is important to quote that a smoothing operator is a pseudodifferential operator whose m i(x y) ξ amplitude is in A (U) for all m R: by (2), a(x,y,ξ) := e− − k(x, y) φ(ξ) where the Rd ∈ d function φ Cc∞( ) satisfies Rd φ(ξ) dξ = (2π) . The main obstruction to smoothness is ∈ σ(x,D) on the diagonal since k is RC∞ outside the diagonal. Few remarks on the duality between symbols and a subset of pseudodifferential operators: m d d m σ(x, ξ) S (U R ) k (x, y) ′(U U R ) A = Op(σ) ΨDO ∈ × ←→ σ ∈D × × ←→ ∈ A ix ξ ix ξ with the definition σ (x, ξ) := e− A(x e ) where A is properly supported, namely, → A and its adjoint map the dual of C∞(U) (distributions with compact support) into itself. Moreover,

A ( i)α α α A A 1 i(x y) ξ A σ − ∂ξ ∂y kσ (x,y,ξ) y=x , kσ (x, y)=: d e − k (x,y,ξ) dξ , α! (2π) Rd ≃ α | X Z A A A iDξ Dy A where k (x,y,ξ) is the amplitude of kσ (x, y). Actually, σ (x, ξ)= e k (x,y,ξ) y=x and | iDξ Dy 1 2 A e = 1+ iDξDy 2 (DξDy) + . Thus A = Op(σ )+ R where R is a regularizing operator on U. − There are two fundamental points about ΨDO’s: they form an algebra and this notion is stable by diﬀeomorphism justifying its extension to manifolds and then to bundles:

4 Theorem 2.3. (i) If P ΨDOm1 and P ΨDOm2 , then P P ΨDOm1+m2 with symbol 1 ∈ 2 ∈ 1 2 ∈ P P ( i)α α P α P 1 2 − 1 2 σ (x, ξ) α! ∂ξ σ (x, ξ) ∂x σ (x, ξ). ≃ α Nd X∈

P1P2 P1 P2 The principal symbol of P1P2 is σm1+m2 (x, ξ)= σm1 (x, ξ) σm2 (x, ξ). (ii) Let P ΨDOm(U) and φ Diﬀ (U, V ) where V is another open set of Rd. The ∈ ∈ 1 m operator φ P : f C∞(V ) P (f φ) φ− satisﬁes φ P ΨDO (V ) and its symbol is ∗ ∈ → ◦ ◦ ∗ ∈ φ P P 1 t ( i)α α P 1 t ∗ − σ (x, ξ)= σm φ− (x), (dφ) ξ + α! φα(x, ξ) ∂ξ σ φ− (x), (dφ) ξ α >0 |X| where φα is a polynomial of degree α in ξ. Moreover, its principal symbol is

φ∗P P 1 t σm (x, ξ)= σm φ− (x), (dφ) ξ . φ∗P P In other terms, the principal symbol is covariant by diﬀeomorphism: σ m = φ σm. ∗ While the proof of formal expressions is a direct computation, the asymptotic behavior re- quires some care, see [99,102]. P ix ξ ix ξ An interesting remark is in order: σ (x, ξ)= e− P (x e ), thus the dilation ξ tξ m itx ξ itx ξ m P m → P P →1 with t> 0 gives t− e− P e = t− σ (x, tξ) t− j 0 σm j(x, tξ)= σm(x, ξ)+o(t− ). ≃ ≥ − Thus, if P ΨDOm(U) with m 0, ∈ ≥ P P m ith(x) ith(x) σ (x, ξ) = lim t− e− P e , where h C∞(U) is (almost) deﬁned by dh(x)= ξ. m t →∞ ∈ 2.2 Case of manifolds Let M be a (compact) Riemannian manifold of dimension d. Thanks to Theorem 2.3, the following makes sense:

m Definition 2.4. ΨDO (M) is defined as the set of operators P : C∞(M) C∞(M) such c → that P (i) the kernel k C∞(M M) off the diagonal, ∈ × 1 m (ii) The map : f C∞ φ(U) P (f φ) φ− C∞ φ(U) is in ΨDO φ(U) ∈ c → ◦ ◦ ∈ for every coordinate chart (U, φ : U Rd). → Of course, this can be generalized: Given a vector bundle E over M, a linear map P : m P Γc∞(M, E) Γ∞(M, E) is in ΨDO (M, E) when k is smooth off the diagonal, and local expressions→ are ΨDO’s with matrix-valued symbols.

P The covariance formula implies that σm is independent of the chosen local chart so is globally P m defined on the bundle T ∗M M and σm is defined for every P ΨDO using overlapping charts and patching with partition→ of unity. ∈ An important class of pseudodifferential operators are those which are invertible modulo regularizing ones:

m P Deﬁnition 2.5. P ΨDO (M, E) is elliptic if σ (x, ξ) is invertible for all 0 = ξ T M ∗. ∈ m ∈ x P m This means that σ (x, ξ) c1(x) ξ for ξ c2(x), x U where c1,c2 are strictly positive continuous functions| on U|. ≥ This also| | means| | that ≥ there exists∈ a parametrix:

5 Lemma 2.6. The following are equivalent: (i) Op(σ) ΨDOm(U) is elliptic. ∈ m d (ii) There exist σ′ S− (U R ) such that σ σ′ =1 or σ′ σ =1. ∈ × ◦ ◦ (iii) Op(σ) Op(σ′)= Op(σ′) Op(σ)=1 modulo ΨDO−∞(U). m Thus Op(σ′) ΨDO− (U) is also elliptic. ∈ Remark that any P ΨDO(M, E) can be extended to a bounded operator on L2(M, E) ∈ when (m) 0 but this needs an existing scalar product for given metrics on M and E. ℜ ≤ m Theorem 2.7. (see [49]) When P ΨDO− (M, E) is elliptic with (m) > 0, its spectrum is discrete when M is compact. ∈ ℜ

We rephrase a previous remark (see [4, Proposition 2.1]): Let E be a vector bundle of m rank r over M. If P ΨDO− (M, E), then for any couple of sections s Γ∞(M, E), ∈ ∈m t∗ Γ∞(M, E∗), the operator f C∞(M) t∗,P (fs) C∞(M) is in ΨDO (M). This ∈ ∈ → ∈ means that in a local chart (U, φ), these operators are r r matrices of pseudodifferential × operators of order m. The total symbol is in C∞(T ∗U) End(E) with End(E) Mr(C). − P ⊗ ≃ The principal symbol can be globally defined: σ m(x, ξ) : Ex Ex for x M and ξ Tx∗M, − → ∈ ∈ can be seen as a smooth homomorphism homogeneous of degree m on all fibers of T ∗M. We get the simple formula which could be seen as a definition of the− principal symbol

P m ith ith σ m(x, ξ) = lim t− e− P e (x) for x M, ξ Tx∗M (3) − t ∈ ∈ →∞ where h C∞(M) is such that d h = ξ. ∈ x 2.3 Singularities of the kernel near the diagonal The question to be solved is to define a homogeneous distribution which is an extension on Rd of a given homogeneous symbol on Rd 0 . Such extension is a regularization used for instance by Epstein–Glaser in quantum field\{ theory.} d The Schwartz space on R is denoted by and the space of tempered distributions by ′. S S R d Definition 2.8. For fλ(ξ) := f(λξ), λ +∗ , define τ ′ τλ by τλ, f := λ− τ, fλ−1 ∈ ∈ S → m for all f . A distribution τ ′ is homogeneous of order m C when τ = λ τ. ∈ S ∈ S ∈ λ d Proposition 2.9. Let σ C∞(R 0 ) be a homogeneous symbol of order k Z. (i) If k > d, then σ defines∈ a homogeneous\{ } distribution. ∈ − (ii) If k = d, there is a unique obstruction to an extension of σ given by cσ = Sd−1 σ(ξ) dξ, − d namely, one can at best extend σ in τ ′ such that τ = λ− τ + c log(λ) δ . ∈ S λ σ 0 R In the following result, we are interested by the behavior near the diagonal of the kernel kP for P ΨDO. For any τ ′, we choose the decomposition as τ = φ τ +(1 φ) τ where ∈ d ∈ S ◦ − ◦ φ C∞(R ) and φ = 1 near 0. We can look at the infrared behavior of τ near the origin and ∈ c its ultraviolet behavior near infinity. Remark first that, since φ τ has a compact support, ◦ (φ τ)q ′, so the regularity of τq depends only of its ultraviolet part (1 φ) τ q. ◦ ∈ S − ◦

6 Proposition 2.10. Let P ΨDOm(U), m Z. Then, in local form near the diagonal, ∈ ∈ P k (x, y)= aj(x, x y) cP (x)log x y + (1) (m+d) j 0 − − | − | O − X≤ ≤ where a (x, y) C∞ U U x is homogeneous of order j in y and c (x) C∞(U) with j ∈ × \{ } P ∈ 1 P cP (x)= d σ d(x, ξ) dξ. (4) (2π) Sd−1 Z − We deduce readily the trace behavior of the amplitude of P :

Theorem 2.11. Let P ΨDOm(M, E), m Z. Then, for any trivializing local coordinates ∈ ∈ 0 P tr k (x, y) = aj(x, x y) cP (x)log x y + (1), j= (m+d) − − | − | O −X where aj is homogeneous of degree j in y, cP is intrinsically locally deﬁned by

1 P cP (x) := (2π)d tr σ d(x, ξ) dξ. (5) Sd−1 − Z Moreover, c (x) dx is a 1-density over M which is functorial for the diﬀeomorphisms φ: P | |

cφ∗P (x)= φ cp(x) . (6) ∗ The main diﬃculty is of course to prove that cP is well deﬁned.

2.4 Wodzicki residue The claim is that c (x) dx is a residue. For this, we embed everything in C. In the same M P | | spirit as in PropositionR 2.9, one obtains the following d Lemma 2.12. Every σ C∞ R 0 which is homogeneous of degree m C Z can be ∈ \{ } ∈ \ uniquely extended to a homogeneous distribution. Deﬁnition 2.13. Let U be an open set in Rd and Ω be a domain in C. A map σ : Ω Sm(U Rd) is said to be holomorphic when the map→ z Ω× σ(z)(x, ξ) is analytic for all x U, ξ Rd, ∈ → ∈ ∈ the order m(z) of σ(z) is analytic on Ω, the two bounds of Deﬁnition 2.1 (i) and (ii) of the asymptotics σ(z) j σm(z) j(z) ≃ − are locally uniform in z. P

This hypothesis is sufficient to get that the map z σm(z) j (z) is holomorphic from Ω − d → d to C∞ U R 0 and the map ∂ σ(z)(x, ξ) is a classical symbol on U R such that × \{ } z × ∂zσ(z)(x, ξ) j 0 ∂zσm(z) j(z)(x, ξ). ≃ ≥ − Definition 2.14.P The map P : Ω C ΨDO(U) is said to be holomorphic if it has the ⊂ → decomposition P (z) = σ(z)( ,D)+ R(z) (see definition (1)) where σ : Ω S(U Rd) and → × R : Ω C∞(U U) are holomorphic. → × 7 As a consequence, there exists a holomorphic map from Ω into ΨDO(M, E) with a holomorphic product (when M is compact). Elliptic operators: Recall that P ΨDOm(U), m C, is elliptic essentially means that P is invertible modulo smoothing operators.∈ More generally,∈ P ΨDOm(M, E) is elliptic if ∈ its local expression in each coordinate chart is elliptic. Let Q ΨDOm(M, E) with (m) > 0. We assume that M is compact and Q is elliptic. ∈ ℜ Thus Q has a discrete spectrum. We assume Spectrum(Q) R− = and the existence of a curve Γ whose interior contains the spectrum and avoid branc∩ h points∅ of λz at z = 0:

s 1 s 1 2 When (s) < 0, Q := i2π Γ λ (λ Q)− dλ makes sense as operator on L (M, E). Theℜ map s Qs is a one-parameter− group containing Q0 and Q1 which is holomorphic on → R (s) 0. We want to integrate symbols, so we will need the set Sint of integrable symbols. ℜUsing≤ same type of arguments as in Proposition 2.9 and Lemma 2.12, one proves

Z d 1 Proposition 2.15. Let L : σ S (R ) L(σ) := σq (0) = Rd σ(ξ) dξ. Then L has a ∈ int → (2π)d C Z Rd C Z unique holomorphic extension L on S \ ( ). Moreover, when σ(ξ)R j σm j(ξ), m , ≃ − ∈ \ q 1 L(σ) = σ j N τm j (0) = (2π)d Rd σ j N τm j (ξ) dξ wherePm is the order of σ, − ≤ − e − ≤ − N is an integerP with N > (m)+ d andR τm j isP the extension of σm j of Lemma 2.12. e ℜ − − Corollary 2.16. If σ : C S(Rd) is holomorphic and order σ(s) = s, then L σ(s) is → Z Z 1 meromorphic with simple poles on and for p , Res L σ(s) = d Sd−1 σ d(p)(ξ) dξ. ∈ s=p − (2π) −e e R We are now ready to get the main result of this section which is due to Wodzicki [109,110].

Definition 2.17. Let ΨDO(M, E) be an elliptic pseudodifferential operator of order 1 on a boundary-less compactD ∈ manifold M endowed with a vector bundle E. Let ΨDO (M, E) := Q ΨDOC(M, E) order(Q) < d be the class of pseudodif- int { ∈ | ℜ − } ferential operators whose symbols are in Sint, i.e. integrable in the ξ-variable. In particular, if P ΨDO (M, E), then its kernel kP (x, x) is a smooth density on the ∈ int diagonal of M M with values in End(E). For P ΨDO× Z(M, E), define ∈ s WRes P := Res Tr P − . (7) s=0 |D| This makes sense because:

Theorem 2.18. (i) Let P : Ω C ΨDOint(M, E) be a holomorphic family. Then the functional map Tr : s Ω Tr(⊂ P (s→)) C has a unique analytic extension on the family C Z ∈ → ∈ Ω ΨDO \ (M, E) still denoted by Tr. →

8 Z s (ii) If P ΨDO (M, E), the map: s C Tr P − has at most simple poles on Z ∈ ∈ → |D| and

WRes P = cP (x) dx (8) − ZM | | 1 P is independent of . Recall (see Theorem 2.11) that cP (x)= (2π)d Sd−1 tr σ d(x, ξ) dξ. D Z − (iii) WRes is a trace on the algebra ΨDO (M, E). R

s C Z Proof. The map s Tr P D − is holomorphic on C and connect P ΨDO \ (M, E) to → | | C Z ∈ the set ΨDOint(M, E) within ΨDO \ (M, E), so a analytic extension of Tr from ΨDOint to C Z ΨDO \ 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 ΨDO’s. (2) First, ﬁx s = 0. We are interested in the function Lφ(σ) := Tr φ σ(x, D) with d σ Sint(U R ) and φ C∞(U). For instance, if P = σ( ,D), ∈Tr(φP )=× φ(x) kP∈(x, x) dx = 1 φ(x) σ(x, ξ) dξ dx = φ(x) L(σ(x, )) dx , U | | (2π)d U | | U | | C Z d so one extends RL to S \ (U R ) with PropositionR 2.15 via L (σ)=R φ(x) L σ(x, ) dx . φ × φ U φ | | (3) If now σ(x, ξ) = σ(s)(x, ξ) depends holomorphically on s, we get uniform bounds in e R e x, thus we get, via Lemma 2.12 applied to L σ(s)(x, ) uniformly in x, yielding a natural φ extension to L σ(s) which is holomorphic on C Z. φ e \ Z When order(σe(s)) = s, the map Lφ σ(s) has at most simple poles on and for each Z 1 p , Res Lφ σ(s) = d U Sd−1 φ(x) σ d(p)(x, ξ) dξ dx = U φ(x) cPp(x) dx where ∈ s=p − (2π) e − | | − | | R R R we used (5)e with P = Op σp(x, ξ) . (4) In the general case, we get a unique meromorphic extension of the usual trace Tr on ΨDOZ(M, E) that we still denoted by Tr). When P : C ΨDOZ(M, E) is meromorphic with order( P (s) = s, then Tr P (s) has → at most poles on Z and Res Tr P (s) = cP (p)(x) dx for p Z. So we get the claim s=p − M | | ∈ s for the family P (s) := P − . R |D|Z C Z (iii) Let P1,P2 ΨDO (M, E). Since Tr is a trace on ΨDO \ (M, E), we get by (i), s ∈ s Tr P P − = Tr P − P . Moreover 1 2|D| 2|D| 1 s s WRes P1P2 = Res Tr P2 − P1 = Res Tr P2P1 − = WRes P2P1 s=0 |D| s=0 |D| where for the second equality we used (8) 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 ) (9) ∈ ∗ which follows from (6). The next result is due to Guillemin and Wodzicki.

Corollary 2.19. The Wodzicki residue WRes is the only trace (up to multiplication by a m constant) on the algebra ΨDO− (M, E), m N, when M is connected and d 2. ∈ ≥

9 Proof. The restriction to d 2 is used only in the part 3) below. When d = 1, T ∗M is ≥ disconnected and they are two residues. j j 1) On symbols, derivatives are commutators: [x , σ]= i∂ξj σ, [ξj, σ]= i∂x σ. P P − 2) If σ d =0, then σ (x, ξ) is a finite sum of commutators of symbols: P − P d P P If σ j σm j with m = order(P ), by Euler’s theorem, k=1 ξk ∂ξk σm j = (m j) σm j ≃ − d k P d P − − P − (this is falseP for m = j!) and k=1 [x , ξk σm j]= i k=1 ∂ξkPξk σm j = i(m j + d) σm j. So P d k − − − − σ = k=1 [ξk τ, x ]. P P Z P LetPT be another trace on ΨDO (M, E). Then T (P ) depends only on σ d because T ([ , ]) = 0. − P P 3) We have Sd−1 σ d(x, ξ) d ξ =0 if and only if σ d is sum of derivatives: − | | − The if part is directR (less than more !). P Sd 1 Only if part: σ d is orthogonal to constant functions on the sphere − and these are − P kernels of the Laplacian: ∆Sf = 0 df = 0 f = cst. Thus ∆Sd−1 h = σ d↾Sd−1 ⇐⇒ ⇐⇒ − Sd 1 ˜ d+2 ξ Rd has a solution h on − . If h(ξ) := ξ − h ξ is its extension to 0 , then we get | | | | \{ } ˜ P ξ P 1 d d 1 2 ∆Rd h(ξ) = ξ σ d ξ = σ d(ξ) because ∆Rd = r − ∂r r − ∂r)+ r− ∆Sd−1 . This means | | − − ˜ | | ˜ that h is a symbol of order d 2 and ∂ξh is a symbol of order d 1. As a consequence, P d 2 ˜ d −˜ k − σ d = k=1 ∂ξk h = i k=1[∂ξk h, x ] is a sum of commutators. − − −d P ξ | | 4) EndP of proof: theP symbol σ d(x, ξ) Vol(Sd−1) cP (x) is of order d with zero integral, − − − d Z thus is a sum of commutators by 3) and T (P )= T Op( ξ − c (x) , T ΨDO (M, E). So | | p ∀ ∈ d the map : f Cc∞(U) T Op(f ξ − ) is linear, continuous and satisfies (∂xk f)=0 ∈ d → | | because ∂ k (f) ξ − is a commutator if f has a compact support and U is homeomorphic to x | | Rd. As a consequence, is a multiple of the Lebesgue integral T (P )= c (x) = c c (x) dx = c WRes(P ). P M P | | R Example 2.20. Laplacian on a manifold M: Let M be a compact Riemannian manifold of dimension d and ∆ be the scalar Laplacian which is a differential operator of order 2. Then d/2 Sd 1 2πd/2 WRes (1+∆)− = Vol − = Γ(d/2) . −d/2 d/2 (1+∆) Proof. (1 + ∆)− ΨDO(M) has order d and its principal symbol σ d satisfies −d/2 ∈ d/2 − − (1+∆) ij − d σ d (x, ξ)= gx ξiξj = ξ x− . So (8) gives − − −|| || d/2 d Sd 1 WRes (1+∆)− = dx ξ x− dξ = dx detgx Vol − M | | Sd−1 || || M | | Z Z Z q d 1 d 1 = Vol S − dvolg = Vol S − . M | | Z

3 Dixmier trace

References for this section: [33,49,68,87,104,105]. The trace Tr on the operators on a Hilbert space has an interesting property, it is normal. Recall that Tr acting on ( ) is a particularH case of a weight ω acting on a von B H + Neumann algebra : it is a homogeneous additive map from := aa∗ a to [0, ]. M M { | ∈ M} ∞ 10 A state is a weight ω ∗ (so ω(a) < , a ) such that ω(1) = 1. ∈M ∞ ∀ ∈M A trace is a weight such that ω(aa∗)= ω(a∗a) for all a . ∈M + A weight ω is normal if ω(sup aα) = sup ω(aα) whenever (aα) is an increasing α α ⊂ M bounded net. This is equivalent to say that ω is lower semi-continuous with respect to the σ-weak topology. In particular, the usual trace Tr is normal on ( ). Remark that the net (a ) converges B H α α in ( ) and this property looks innocent since a trace preserves positivity. NeverthelessB H it is natural to address the question: are all traces (in particular on an arbitrary von Neumann algebra) normal? In 1966, Dixmier answered by the negative [34] by exhibiting non-normal, say singular, traces. Actually, his motivation was to answer the following related question: is any trace ω on ( ) proportional to the usual trace on the set where ω is finite? B H The aim of this section is first to define this Dixmier trace, which essentially means 1 N TrDix(T ) “ = ” limN n=0 n(T ), where the n(T ) are the singular values of T ordered →∞ log N in decreasing order and thenP to relate this to the Wodzicki trace. It is a non-normal trace on some set that we have to identify. Naturally, the reader can feel the link with the Wodzicki d trace via Proposition 2.10. We will see that if P ΨDO− (M) where M is a compact Riemannian manifold of dimension d, then, ∈ 1 1 P TrDix(P )= d WRes(P )= d M S∗M σ d(x, ξ) dξ dx − | | where S∗M is the cosphere bundle on M. R R 1 The physical motivation is quite essential: we know how n N∗ n diverges and this is ∈ 1 related to the fact the electromagnetic or Newton gravitationalP potentials are in r which has 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 Riemann C 1 zeta function ζ : s n N∗ ns . This is also why we have already seen a logarithmic ∈ → ∈ obstruction in Theorem 2.11P and define a zeta function associated to a pseudodifferential s operator P by ζ (s) = Tr P − in (7). P |D| We now have a quick review on the main properties of singular values of an operator.

3.1 Singular values of compact operators In noncommutative geometry, infinitesimals correspond to compact operators: for T ( ) ∈K H (compact operators), let (T ) := inf T ⊥ E subspace of with dim(E)= n , n N. n { ↾E | H } ∈ This could looks strange but actually, by mini-max principle, n(T ) is nothing else than the (n + 1)th of eigenvalues of T sorted in decreasing order. Since limn n(T ) = 0, for any | | →∞ ⊥ ǫ> 0, there exists a finite-dimensional subspace Eǫ such that T↾Eǫ < ǫ and this property being equivalent to T compact, T deserves the name of infinitesimal.

Moreover, we have following properties: n(T )= n(T ∗)= n( T ). 1 | | T ( ) (meaning T 1 := Tr( T ) < ) n N n(T ) < . ∈L(A T BH) A (T ) B when| A,| B ∞ ⇐⇒( ). ∈ ∞ n ≤ n ∈ B H P N (U T U ∗)= N (T ) when U is a unitary. Deﬁnition 3.1. For T ( ), the partial trace of order N N is σ (T ) := N (T ). ∈K H ∈ N n=0 n P

11 Remark that T σ (T ) N T which implies σ on ( ). Then ≤ N ≤ n ≃ K H σ (T + T ) σ (T )+ σ (T ), N 1 2 ≤ N 1 N 2 σ (T )+ σ (T ) σ (T + T ) when T , T 0. N1 1 N2 2 ≤ N1+N2 1 2 1 2 ≥ 1 This norm σN splits: σN (T ) = inf x 1 + N y T = x + y with x ( ), y ( ) . This justifies a continuous approach{ with the | ∈L H ∈K H } Definition 3.2. The partial trace of T of order λ R+ is ∈ σ (T ) := inf x + λ y T = x + y with x 1( ), y ( ) . λ { 1 | ∈L H ∈K H } R+ As before, σλ1 (T1)+ σλ2 (T2)= σλ1+λ2 (T1 + T2), for λ1,λ2 , 0 T1, T2 ( ). We define a real interpolate space between 1( ) and ( ) by ∈ ≤ ∈K H L H K H 1, σλ(T ) ∞ := T ( ) T 1, := sup log λ < . L { ∈K H | ∞ λ e ∞} ≥ p p 1 1/p If ( ) is the ideal of operators T such that Tr T < , so σ (T )= (λ − ), we have L H | | ∞ λ O 1 1, p ( ) ∞ ( ) for p> 1, T T 1, T 1 . (10) L H ⊂L ⊂L H ≤ ∞ ≤ 1, Lemma 3.3. ∞ is a C∗-ideal of ( ) for the norm 1, . Moreover, it is equal to the ∞ L1,+ B H σN (T ) Macaev ideal := T ( ) T 1,+ := supN 2 log(N) < . L { ∈K H | ≥ ∞} 3.2 Dixmier trace

σρ(T ) R We begin with a Cesàro mean of log ρ with respect of the Haar measure of the group +∗ : Deﬁnition 3.4. For λ e and T ( ), let τ (T ) := 1 λ σρ(T ) dρ . ≥ ∈K H λ log λ e log ρ ρ R Clearly, σρ(T ) log ρ T 1, and τλ(T ) T 1, , thus the map: λ τλ(T ) is in Cb([e, ]). ≤ 1, ∞ ≤ ∞ → ∞ It is not additive on ∞ but this defect is under control: L log (log λ) 1, τλ(T1 + T2) τλ(T1) τλ(T2) , when 0 T1, T2 ∞ : − − λ ≃ O log λ ≤ ∈L →∞ Lemma 3.5. In fact, we have

log 2(2+log log λ) 1, τλ(T1 + T2) τλ(T1) τλ(T2) T1 + T2 1, , when T1, T2 +∞. | − − | ≤ log λ ∞ ∈L The Dixmier’s idea was to force additivity: since the map λ τλ(T ) is in Cb([e, ]) and log 2(2+log log λ) → ∞ λ is in C ([e, [), consider the C∗-algebra := C ([e, ])/C ([e, [). → log λ 0 ∞ A b ∞ 0 ∞ If [τ(T )] is the class of the map λ τλ(T ) in , previous lemma shows that [τ] : T [τ(T )] → A1, → is additive and positive homogeneous from +∞ into satisfying [τ(UT U ∗)] = [τ(T )] for any unitary U. L A Now let ω be a state on , namely a positive linear form on with ω(1) = 1. Then, A 1, 1, A ω [τ( )] is a tracial weight on +∞ . Since ∞ is a C∗-ideal of ( ), each of its element is◦ generated by (at most) fourL positive elements,L and this map canB H be extended to a map 1, 1, ω [τ( )] : T ∞ ω([τ(T )]) C such that ω([τ(T1T2)]) = ω([τ(T2T1)]) for T1, T2 ∞. This◦ leads to∈L the following→ deﬁnition∈ and result: ∈L

12 Definition 3.6. The Dixmier trace Tr associated to a state ω on is Tr ( ) := ω [τ( )]. ω A ω ◦ 1, Theorem 3.7. Trω is a trace on ∞ which depends only on the locally convex topology of , not on its scalar product. L H 1) Note that Tr (T )=0if T 1( ) and all Dixmier traces vanish on the closure for the ω ∈ L H norm . 1, of the ideal of finite rank operators. So Dixmier traces are not normal. ∞ 2) The C∗-algebra is not separable, so it is impossible to exhibit any state ω! Despite A p 1, (10) and the fact that the ( ) are separable ideals for p 1, ∞ is not a separable. L H ≥ L Moreover, as for Lebesgue integral, there are sets which are not measurable. For instance, a function f Cb([e, ]) has a limit ℓ = limλ f(λ) if and only if ℓ = ω(f) for all state ω. ∈ ∞ →∞ 1, Definition 3.8. The operator T ∞ is said to be measurable if Tr (T ) is independent of ∈L ω ω. In this case, Trω is denoted TrDix.

1, Lemma 3.9. The operator T ∞ is measurable and Tr (T ) = ℓ if and only if the map ∈ L ω λ R+ τ (T ) converges at inﬁnity to ℓ. ∈ → λ ∈A After Dixmier, singular (i.e. non normal) traces have been deeply investigated, see for instance [72,74,75], but we quote only the following characterization of measurability: 1 N If T +( ), then T is measurable if and only if limN n=0 n(T ) exists. ∈K H →∞ log N Example 3.10. Computation of the Dixmier trace of the inverse LaplacianP on the torus:

Td Rd Zd d 2 Let = /2π be the d-dimensional torus and ∆ = i=1 ∂xi be the scalar Laplacian 2 d − p seen as unbounded operator on = L (T ). We want to compute Tr (1 + ∆)− for H P ω d N 2 p ∗. We use 1 + ∆ to avoid the kernel problem with the inverse. As the following proof≤ shows,∈ 1 can be replaced by any ǫ> 0 and the result does not depends on ǫ. 1 ik x Td Zd Td Notice that the functions ek(x) := 2π e with x , k =( )∗ form a basis of ∈ ∈ Hd 2 R t t(1+∆) t k 2 tk2 of eigenvectors: ∆ ek = k ek. For t +∗ , e Tr e− = k Zd e− | | = k Z e− . | | ∈ ∈ ∈ tx2 tk2 π We know that ∞ e− dx k Z e− 1, and since theP ﬁrst integral isP t , we get | −∞ − ∈ | ≤ d/2 t t(1+∆) R π Pd/2 q e Tr e− =: α t− . t 0+ t ↓≃ d/2 1 1 We will use a Tauberian theorem: n (1+∆)− ) (α , see [49,54]. Thus n ≃ Γ(d/2+1) n 1 N d/2 →∞ α πd/2 limN n=0 n (1+∆)− = = . →∞ log N Γ(d/2+1) Γ(d/2+1) d/2 d/2 d/2 πd/2 So (1+∆)− is measurable andP TrDix (1+∆)− ) = Trω (1+∆)− ) = Γ(d/2+1) . Since p d p (1+∆)− is traceable for p> 2 , TrDix (1+∆)− = 0. This result has been generalized in Connes’ trace theorem [23]: since WRes and TrDix are m traces on ΨDO− (M, E), m N, we get the following ∈ Theorem 3.11. Let M be a d-dimensional compact Riemannian manifold, E a vector bundle d 1, 1 over M and P ΨDO− (M, E). Then, P ∞, is measurable and Tr (P )= WRes(P ). ∈ ∈L Dix d

13 4 Dirac operator

There are several ways to define a Dirac-like operator. The best one is to define Clifford algebras, their representations, the notion of Clifford modules, spinc structures on orientable manifolds M defined by Morita equivalence between the C∗-algebras C(M) and Γ( ℓ M). Then the notion of spin structure and finally, with the notion of spin and Clifford connection,C we reach the definition of a (generalized) Dirac operator. Here we try to bypass this approach to save time. References: a classical book is [70], but I suggest [46]. Here we follow [86], see also [49].

4.1 Definition and main properties Let (M, g) be a compact Riemannian manifold with metric g, of dimension d and E be a vector bundle over M. An example is the (Clifford) bundle E = ℓ T ∗M where the fiber C ℓ T ∗M is the Clifford algebra of the real vector space T ∗M for x M endowed with the C x x ∈ nondegenerate quadratic form g. Given a connection on E, a differential operator P of order m N on E is an element m ∇ ∈ of Diff (M, E) :=Γ M,End(E) V ect X Γ(M,TM), j m . { ∇X1 ∇Xj | j ∈ ≤ } In particular, Diff m(M, E) is a subalgebra of End Γ(M, E) and the operator P has a prin- P cipal symbol σm in Γ T ∗M, π∗End(E) where π : T ∗M M is the canonical submersion P → and σm(x, ξ) is given by (3). Example: Let E = T ∗M. The exterior product and the contraction given on ω,ωj E by m j 1 ∈ ǫ(ω ) ω := ω ω , ι(ω)(ω ω ) := ( 1) − g(ω,ω ) ω ω ω 1 2 1 ∧ 2 V 1 ∧∧ m j=1 − j 1 ∧∧ j ∧∧ m suggest the following definition c(ω) := ǫ(ω)+ ιP(ω) and one checks that c

c(ω1) c(ω2)+ c(ω2)c(ω1)=2g(ω1,ω2) idE. (11)

E has a natural scalar product: if e , , e is an orthonormal basis of T ∗M, then the scalar 1 d x product is chosen such that ei1 eip for i1 < < ip is an orthonormal basis. 1 ∧∧ If d Diﬀ is the exterior derivative and d∗ is its adjoint for the deduced scalar product on Γ(M,∈ E), then their principal symbols are

∗ σd(ω)= iǫ(ω), σd (ω)= iι(ω). (12) 1 1 − d 1 ith(x) ith(x 1 This follows from σ1 (x, ξ) = limt e− de (x) = limt it dxh = idxh = iξ →∞ t →∞ t d d∗ where h is such that dxh = ξ, so σ1 (x, ξ)= iξ and similarly for σ1 . m P 1 m More generally, if P Diff (M), then σm(dh) = imm! (ad h) (P ) with ad h = [ , h] and P ∗ P ∈ σm (ω)= σm(ω)∗ where the adjoint P ∗ is for the scalar product on Γ(M, E) associated to an hermitean metric on E: ψ, ψ′ := ψ(x), ψ′(x) dx is a scalar product on Γ(M, E). M x | | Definition 4.1. The operator P R Diff 2(M, E) is called a generalized Laplacian when its P 2 ∈ symbol satisfies σ (x, ξ)= ξ id for x M, ξ T ∗M. 2 | |x Ex ∈ ∈ x ij j This is equivalent to say that, in local coordinates, P = i,j g (x)∂xi ∂xj + b (x)∂xj + c(x) j − where the b are smooth and c is in Γ M,End(E) . P

14 + Deﬁnition 4.2. Assume that E = E E− is a Z2-graded vector bundle. 1 0 D⊕+ When D Diﬀ (M, E) and D = − (D is odd) where D± : Γ(M, E∓) Γ(M, E±), ∈ D 0 → 2 D−D+ 0 D is called a Dirac operator if D = 0 D+D− is a generalized Laplacian. even odd A good example is given by E = T ∗M = T ∗M T ∗M and the de Rham operator 2 ⊕ D := d + d∗. It is a Dirac operatorV since D V= dd∗ + d∗dVis a generalized Laplacian according to (12). D2 is also called the Laplace–Beltrami operator.

Definition 4.3. Define ℓ M as the vector bundle over M whose fiber in x M is the C ∈♭ Clifford algebra ℓ T ∗M (or ℓ T M using the musical isomorphism X T M X T ∗M). C x C x ∈ ↔ ∈ A bundle E is called a Clifford bundle over M when there exists a Z2-graduate action c : Γ(M, ℓ M) End Γ(M, E) . C → The main idea which drives this definition is that Clifford actions correspond to principal symbols of Dirac operators:

Proposition 4.4. If E is a Clifford module, every odd D Diff 1 such that [D, f]= ic(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]. − even odd Consider previous example: E = T ∗M = T ∗M T ∗M is a Clifford module for ⊕ c := i(ǫ + ι) coming from the DiracV operatorVD = d + d∗:V by (12) d d∗ i[D, f]= i[d + d∗, f]= i iσ (df) iσ (df) = i(ǫ + ι)(df). 1 − 1 − Definition 4.5. Let E be a Clifford module over M. A connection on E is a Clifford connection if for a Γ(M, ℓ M) and X Γ(M,TM), [ ,c(a)] = c(∇ LC a) where LC is ∈ C ∈ ∇X ∇X ∇X the Levi-Civita connection after its extension to the bundle ℓ M (here ℓ M is the bundle C C with fiber ℓ TxM). A Dirac operator D is associated to a Clifford connection : C ∇ c 1 ∇ D := ic , Γ(M, E) ∇ Γ(M, T ∗M E) ⊗ Γ(M, E), ∇ where we use c for c −1. ◦ ∇ −→ ⊗ −−→ ⊗ d j Thus if in local coordinates, = j=1 dx ∂j , the associated Dirac operator is given j ∇ ⊗ ∇ by D = i j c(dx ) ∂i . In particular,P for f C∞(M), ∇ − ∇ d i ∈ d j [D , fidE]= i i=1 c(dx ) [ ∂j , f]= j=1 ic(dx ) ∂jf = ic(df). P∇ By Proposition 4.4, D deserves− the name∇ of Dirac operator!− − ∇ P P Examples: 1) For the previous example E = T ∗M, the Levi-Civita connection is indeed a Clifford connection whose associated Dirac operatorV coincides with the de Rham operator D = d+d∗. 2) The spinor bundle: Recall that the spin group Spind is the non-trivial two-fold covering Z ξ of SOd, so we have 0 2 Spind SOd 1. Let SO(T M) M−→be the−→ SO -principal−→ bundle−→ of positively oriented orthonormal frames → d on 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 Spind- η principal bundle Spin(T M) π M with a two-fold covering map Spin(T M) SO(T M) such that the following diagram−→ commutes: −→

15 / Spin(T M) Spind Spin(T M) XXπX × XXXXX X3+ η ξ η ffff3 M × fffff SO(T M) SO / SO(T M) π × d where the horizontal maps are the actions of Spind and SOd on the principal fiber bundles Spin(T M) and SO(T M). A spin manifold is an oriented Riemannian manifold admitting a spin structure. c In above definition, one can replace Spind by the group Spind which is a central extension of T T c ξ SOd by : 0 Spind SOd 1. An oriented−→ Riemannian−→ manifold−→ (M,g−→) is spin if and only if the second Stiefel–Whitney class of its tangent bundle vanishes. Thus a manifold is spin if and only both its first and second Stiefel–Whitney classes vanish. In this case, the set of spin structures on (M,g) stands 1 in one-to-one correspondence with H (M, Z2). In particular the existence of a spin structure does not depend on the metric or the orientation of a given manifold. d 2⌊d/2⌋ Let ρ be an irreducible representation of ℓ C EndC(Σ ) with Σ C as set of C → d d ≃ complex spinors. Of course, ℓ Cd is endowed with its canonical complex bilinear form. The spinor bundle S of MCis the complex vector bundle associated to the principal bundle Spin(T M) with the spinor representation, namely S := Spin(T M) Σ . Here ρ is a ×ρd d d representation of Spind on Aut(Σd) which is the restriction of ρ. + More precisely, if d =2m is even, ρd = ρ +ρ− where ρ± are two nonequivalent irreducible + complex representations of Spin and Σ = Σ Σ− , while for d = 2m + 1 odd, the 2m 2m 2m ⊕ 2m spinor representation ρd is irreducible. + In practice, M is a spin manifold means that there exists a Clifford bundle S = S S− ⊕ such that S T ∗M. Due to the dimension of M, the Clifford bundle has fiber ≃ V M2m (C) when d =2m is even, ℓxM = C M2m (C) M2m (C) when d =2m +1. ( ⊕ Locally, the spinor bundle satisfies S M Cd/2. S ≃ × A spin connection : Γ∞(M,S) Γ∞(M,S) Γ∞(M, T ∗M) is any connection which is ∇ → ⊗ compatible 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 4.6. The Dirac (called Atiyah–Singer) operator given by the spin structure is D/ := ic S. (13) − ◦ ∇ In coordinates, D/ = ic(dxj ) ∂ ω (x) where ω is the spin connection part which can be − j − j j 1 k α β i l computed in the coordinate basis ωj = 4 Γji gkl ∂i(hj )δαβhl c(dx ) c(dx ) and the matrix α t − H := [hj ] is such that H H = [gij] (we use Latin letters for coordinate basis indices and Greek letters for orthonormal basis indices). D j This gives σ1 (x, ξ)= c(ξ)+ ic(dx ) ωj(x). Thus in normal coordinates around x0, j j D j c(dx )(x0)= γ , σ1 (x0,ξ)= c(ξ)= γ ξj where the γ’s are constant hermitean matrices. The Hilbert space of spinors is

2 = L (M,g),S) := ψ Γ∞(M,S) ψ, ψ x dvolg(x) < (14) H { ∈ | M ∞} Z 16 where we have a scalar product which is C∞(M)-valued. On its domain Γ∞(M,S), D/ is symmetric: ψ, D/φ = D/ψ, φ . Moreover, it has a selfadjoint closure (which is D/ ∗∗): Theorem 4.7. (see [49,70,104,105]) Let (M,g) be an oriented compact Riemannian spin manifold without boundary. By extension to , D/ is essentially selfadjoint on its original H domain Γ∞(M,S). It is a diﬀerential (unbounded) operator of order one which is elliptic. There is a nice formula which relates the Dirac operator D/ to the spinor Laplacian S S S ∆ := Trg( ): Γ∞(M,S) Γ∞(M,S). − ∇ ◦ ∇ → 2 Before to give it, we need to ﬁx few notations: let R Γ∞ M, T ∗M End(T M) be ∈ ⊗ the Riemann curvature tensor with components Rijkl := g(∂i, R(∂Vk, ∂l)∂j), the Ricci tensor ik jl components are Rjl := g Tijkl and the scalar curvature is s := g Rjl. Proposition 4.8. Schrödinger–Lichnerowicz formula: if s is the scalar curvature of M, 2 S 1 D/ = ∆ + 4 s. The proof is just a lengthy computation (see for instance [49]). 1 We already know via Theorems 2.7 and 4.7 that D/ − is compact so has a discrete spectrum. For T +( ), we denote by λn(T ) n N its spectrum sorted in decreasing order including ∈K H { } ∈ 1 multiplicity (and in increasing order for an unbounded positive operator T such that T − is compact) and by N (λ) := # λ (T ) λ λ its counting function. T { n | n ≤ } Theorem 4.9. With same hypothesis, the asymptotics of the Dirac operator counting func- 2dVol(Sd−1) d tion is N D/ (λ) d Vol(M) λ where Vol(M)= M dvol. | | λ d(2π) →∞∼ R We already encounter such computation in Example 3.10.

4.2 Dirac operators and change of metrics

Recall that the spinor bundle Sg and square integrable spinors g defined in (14) depends H 2 on the chosen metric g, so we note Mg instead of M and g := L (Mg,Sg) and a natural question is: what happens to a Dirac operator when the metricH changes? Let g′ be another Riemannian metric on M. Since the space of d-forms is one-dimensional, + there exists a positive function f ′ : M R such that dvol ′ = f ′ dvol . g,g → g g ,g g Let I ′ (x) : S S ′ the natural injection on the spinors spaces above point x M which g,g g → g ∈ is a pointwise linear isometry: Ig,g′ (x) ψ(x) g′ = ψ(x) g. Let us first see its construction: | | | | d/2 there always exists a g-symmetric automorphism Hg,g′ of the 2⌊ ⌋- dimensional vector space 1/2 T M such that g′(X,Y ) = g(H ′ X,Y ) for X,Y T M so define ι ′ X := H− ′ X. Note g,g ∈ g,g g,g that ιg,g′ commutes with right action of the orthogonal group Od and can be lifted up to a diffeomorphism P ind-equivariant on the spin structures associated to g and g′ and this lift is denoted by Ig,g′ (see [6]). This isometry is extended as operator on the Hilbert spaces Ig,g′ : g g′ with (Ig,g′ ψ)(x) := Ig,g′ (x) ψ(x). NowH define→ H U ′ := f ′ I ′ : ′ . g ,g g,g g ,g Hg → Hg Then by construction, U ′ is a unitaryq operator from ′ onto : for ψ′ ′ , g ,g Hg Hg ∈ Hg 2 2 2 ′ ′ ′ ′ ′ ′ Ug ,gψ′, Ug ,gψ′ g = Ug,g ψ′ g dvolg = Ig ,gψ′ g fg ,g dvolg = ψ′ g′ dvolg H ZM | | ZM | | ZM | | = ψg′ , ψg′ ′ . Hg

17 So we can realize D/ ′ as an operator D ′ acting on with g g Hg 1 D ′ : ,D ′ := U − ′ D/ ′ U ′ . (15) g Hg → Hg g g,g g g,g

This is an unbounded operator on g which has the same eigenvalues as D/ g′ . H k In the same vein, the k-th Sobolev space H (Mg,Sg) (which is the completion of the space 2 k j 2 Γ∞(M ,S ) under the norm ψ = ψ(x) dx; be careful, applied to ψ is g g k j=0 M |∇ | ∇ ∇ the tensor product connection on T ∗MPg SRg etc, see Theorem 2.7) can be transported: the k k ⊗ map U ′ : H (M ,S ) H (M ′ ,S ′ ) is an isomorphism, see [96]. In particular, (after the g,g g g → g g transport map U), the domain of Dg′ and D/ g′ are the same. A nice example of this situation is when g′ is in the conformal class of g where we can compute explicitly D/ g′ and Dg′ [2,6,46,56].

2h Theorem 4.10. Let g′ = e g with h C∞(M, R). Then there exists an isometry I ′ ∈ g,g between the spinor bundle S and S ′ such that for ψ Γ∞(M,S ), g g ∈ g h d 1 D/ ′ I ′ ψ = e− I ′ D/ ψ i − c (grad h) ψ , g g,g g,g g − 2 g d+1 d 1 h h 2 1 −2 D/ g′ = e− Ig,g′ D/ g Ig,g− ′ e , h/2 h/2 Dg′ = e− D/ g e− .

Note that Dg′ is not a Dirac operator as defined in (13) since its principal symbol has an Dg′ h(x) x-dependence: σ (x, ξ)= e− cg(ξ). The principal symbols of D/ g′ and D/ g are related by D/ g′ h(x)/2 1 D/ g h(x)/2 σd (x, ξ)= e− Ug−′,g(x) σd (x, ξ) Ug′,g(x) e− , ξ Tx∗M. h(x) 1 ∈ Thus cg′ (ξ) = e− Ug−′,g(x) cg(ξ) Ug′,g(x), ξ Tx∗M. This formula gives a verification of 2h ∈ g′(ξ, η)= e− g(ξ, η) using cg(ξ)cg(η)+ cg(η)cg(ξ)=2g(ξ, η) idSg . It is also natural to look at the changes on a Dirac operator when the metric g is modified by a diffeomorphism α which preserves the spin structure. The diffeomorphism α can be lifted to a diffeomorphism Od-equivariant on the Od-principal bundle of g-orthonormal frames with 1/2 α˜ := Hα−∗g,g T α, and this lift also exists on Sg when α preserves both the orientation and the spin structure. However, the last lift is defined up to a Z2-action which disappears if α is connected to the identity. The pull-back g′ := α∗g of the metric g is defined by (α∗g)x(ξ, η) = gα(x)(α (ξ), α η), ∗ ∗ x M, where α is the push-forward map : TxM Tα(x)M. Of course, the metric g′ and g ∈ ∗ → are different but the geodesic distances are the same and one checks that dg′ = α∗dg. D The principal symbol of a Dirac operator D is σd (x, ξ) = cg(ξ) so gives the metric g by (11). This information will be used in the definition of a spectral triple. A commutative spectral triple associated to a manifold generates the so-called Connes’ distance which is nothing else but the metric distance; see the remark after (24). The link between dα∗g and dg is explained by (15), since the unitary induces an automorphism of the C∗-algebra C∞(M).

5 Heat kernel expansion

References for this section: [4,43,44] and especially [107].

18 The heat kernel is a Green function of the heat operator et∆ (recall that ∆ is a positive − operator) which measures the temperature evolution in a domain whose boundary has a given temperature. For instance, the heat kernel of the Euclidean space Rd is

1 x y 2/4t k (x, y)= e−| − | for x = y (16) t (4πt)d/2 and it solves the heat equation d ∂tkt(x, y) = ∆xkt(x, y), t> 0, x,y R , initial condition: limt 0 kt(x, y)= δ(x y). 1 ∀ ts2 is(x y) ∈ ↓ − Actually, kt(x, y)= 2π ∞ e− e − ds when d = 1. d−∞ Note that for f (R ), we have limt 0 Rd kt(x, y) f(y) dy = f(x). ∈D R ↓ For a connected domain (or manifold withR boundary with vector bundle V ) U, let λn be the eigenvalues for the Dirichlet problem of minus the Laplacian ∆φ = λψ in U, ψ =0 on ∂U. 2 − 1 If ψ L (U) are the normalized eigenfunctions, the inverse Dirichlet Laplacian ∆− is a n ∈ selfadjoint compact operator, 0 λ1 λ2 λ3 ,λn . ≤ ≤ ≤ ≤ →∞tx The interest for the heat kernel is that, if f(x)= 0∞ dt e − φ(x) is the Laplace transform t ∆ of φ, then Tr f( ∆) = ∞ dtφ(t) Tr e (if everything makes sense) is controlled by − 0 R t∆ t∆ t λn Tr e = M dvol(x)trVx ktR(x, x) since Tr e = n∞=1 e and t ∆ t ∆ t λ k (x, y)=R x, e y = ∞ x, ψ ψ , e ψP ψ ,y = ∞ ψ (x) ψ (y) e n . t n,m=1 m m n n n,m=1 n n So it is useful to know theP asymptotics of the heat kernel kt Pon the diagonal of M M especially near t = 0. ×

5.1 The asymptotics of heat kernel Let now M be a smooth compact Riemannian manifold without boundary, V be a vector bundle over M and P ΨDOm(M,V ) be a positive elliptic operator of order m > 0. If ∈ tP kt(x, y) is the kernel of the heat operator e− , then the following asymptotics exits on the diagonal: ∞ ( d+k)/m kt(x, x) ak(x) t − t ∼0+ ↓ kX=0 ( d+k)/m n which means that kt(x, x) k k(n) ak(x) t − < cn t for 0 0, see [43, sectionP 1.6, 1.7]). tP More generally, we will use k(t,f,P ) := Tr f e− where f is a smooth function. We have similarly

∞ ( d+k)/m k(t,f,P ) ak(f,P ) t − . (17) t ∼0+ ↓ kX=0

The utility of function f will appear later for the computation of coefficients ak. The following points are of importance: 1) The existence of this asymptotics is non-trivial [43,44]. 2) The coefficients a2k(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 derivatives of a finite number of terms in the asymptotic expansion of the total symbol of P .

19 In noncommutative geometry, local generally means that it is concentrated at inﬁnity in momentum space. 3) The odd coeﬃcients are zero: a2k+1(f,P )=0. For instance, let us assume from now on that P is a Laplace type operator of the form

P = (gν ∂ ∂ + A∂ + B) (18) − ν ν where (g )1 ,ν d is the inverse matrix associated to the metric g on M, and A and B are ≤ ≤ smooth L(V )-sections on M (endomorphisms) (see also Definition 4.1). Then (see [44, Lemma 1.2.1]) there is a unique connection on V and a unique endomorphism E such that 2 ∇ 2 P = (Trg + E), (X,Y ) := [ X , Y ] LC Y , − ∇ ∇ ∇ ∇ − ∇∇X X,Y are vector fields on M and LC is the Levi-Civita connection on M. Locally ∇ 2 ν ρ Trg := g ( ν Γν ρ) ρ ∇ ∇LC∇ − ∇ where Γν are the Christoffel coefficients of . Moreover (with local frames of T ∗M and V ), = dx (∂ + ω ) and E are related to∇ gν, A and B through ∇ ⊗ 1 A σε ων = 2 gν( + g Γσε idV ) , E = B gν(∂ ω + ω ω ω Γσ ) . − ν ν − σ ν i In this case, the coefficients ak(f,P )= M dvolg trE f(x) ak(P )(x) and the ak(P )= ci αk(P ) i are linear combination with constants Rci of all possible independent invariants αk(P ) of dimension 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 only independent invariants. Point 3) follow since there is no odd-dimension invariant. If s is the scalar curvature and ‘;’ denote multiple covariant derivative with respect to Levi-Civita connection on M, one finds, using variational methods

d/2 a0(f,P )=(4π)− dvolg trV (f) , ZM (4π)−d/2 a2(f,P )= 6 dvolg trV (f(6E + s) , (19) M Z h i (4π)−d/2 2 2 a4(f,P )= 360 dvolg trV f(60E;kk + 60Es + 180E + 12R;kk +5s M Z h 2R R +2R R + 30Ω Ω ) . − ij ij ijkl ijkl ij ij i The coefficient a6 was computed by Gilkey, a8 by Amsterdamski, Berkin and O’Connor and a10 in 1998 by van de Ven [108]. Some higher coefficients are known in flat spaces.

5.2 Wodzicki residue and heat expansion

1 Wodzicki has proved that, in (17), ak(P )(x) = m cP (k−d)/m (x) is true not only for k = 0 as 1 seen in Theorem 3.11 (where P P − ), but for all k N. In this section, we will prove this ↔ ∈ result when P is is the inverse of a Dirac operator and this will be generalized in the next section. Let M be a compact Riemannian manifold of dimension d even, E a Clifford module over M and D be the Dirac operator (definition 4.2) given by a Clifford connection on E. By Theorem 4.7, D is a selfadjoint (unbounded) operator on := L2(M,S). H 20 tD2 2 We are going to use the heat operator e− since D is related to the Laplacian via the Schrödinger–Lichnerowicz formula (4.2) and since the asymptotics of the heat kernel of this Laplacian is known. tD2 1 tD2 2 (d+1)/2 tD2 2 (d+1)/2 For t > 0, e− : follows from e− = (1+ D ) e− (1 + D )− , since 2 (d+1)/2 1 ∈ L 2 (d+1)/2 tλ2 tD2 tλ2 (1+D )− and λ (1+λ ) e− is bounded. So Tr e− = e− n < . ∈L → n ∞ tD2 Moreover, the operator e− has a smooth kernel since it is regularizing (P [70]) and the 2 1 j dg (x,y) /4t asymptotics of its kernel is, see (16): kt(x, y) d/2 √detgx j 0 kj(x, y) t e− t 0+ (4πt) ≥ ↓∼ where k is a smooth section on E∗ E. Thus P j ⊗ tD2 (j d)/2 2 Tr e− t − aj(D ) (20) t 0+ ↓∼ j 0 X≥ with for j N, a (D2) := 1 tr k (x, x) √detg dx , a (D2)=0. ∈ 2j (4π)d/2 M j x | | 2j+1 R p p Theorem 5.1. For any integer p, 0 p d, D− ΨDO− (M, E) and ≤ ≤ ∈ p 2 2 2 WRes D− = Γ(p/2) ad p(D )= (4π)d/2Γ(p/2) tr k(d p)/2(x, x) dvolg(x). − M − Z Few remarks are in order: d 2 2 2 Rank(E) 1) If p = d, WRes D− = Γ(p/2) a0(D )= Γ(p/2) (4π)d/2 Vol(M). tD2 2 d/2 Since Tr(e− ) a0(D ) t− , the Tauberian theorem used in Example 3.10 implies that t ∼0+ d 2 d/2 ↓ D− =(D− ) is measurable and we obtain Connes’ trace Theorem 3.11 2 d d a0(D ) 1 d TrDix(D− ) = Trω(D− )= Γ(d/2+1) = d WRes(D− ). 2 2) When D = D/ and E is the spinor bundle, the Seeley-deWit coefficient a2( ) (see (19) with f = 1) can be easily computed (see [43,49]): if s is the scalar curvature, D

2 1 a2(D/ )= 12(4π)d/2 s(x) dvolg(x). (21) − ZM d+2 2 2 − So WRes D/ = Γ(d/2 1) a2(D/ ) = c M s(x) dvolg(x). This is a quite important result − since this last integral is nothing else butR the Einstein–Hilbert action. In dimension 4, this is an example of invariant by diﬀeomorphisms, see (9).

6 Noncommutative integration

The Wodzicki residue is a trace so can be viewed as an integral. But of course, it is quite natural to relate this integral to zeta functions used in (7): with notations of Section 2.4, let P ΨDOZ(M, E) and D ΨDO1(M, E) which is elliptic. The deﬁnition of zeta function P ∈ s ∈ P ζ (s) := Tr P D − has been useful to prove that WRes P = Res ζ (s)= cP (x) dx . D | | s=0 D − M | | The aim now is to extend this notion to noncommutative spaces encoded inR the notion of spectral triple. References: [24,30,33,36,49].

6.1 Notion of spectral triple The main properties of a compact spin Riemannian manifold M can be recaptured using 2 1 d the following triple ( = C∞(M), = L (M,S), D/ ). The coordinates x = (x , , x ) are A H 21 exchanged with the algebra C∞(M), the Dirac operator gives the dimension d as seen in D Theorem 4.9, but also the metric of M via Connes formula and more generally generates a quantized calculus. The idea is to forget about the commutativity of the algebra and to impose axioms on a triplet ( , , ) to generalize the above one in order to be able to A H D obtain appropriate definitions of other notions: pseudodifferential operators, measure and integration theory, KO-theory, orientability, Poincaré duality, Hochschild (co)homology etc. Definition 6.1. A spectral triple ( , , ) is the data of an involutive (unital) algebra with a faithful representation π on aA HilbertH D space and a selfadjoint operator with compactA H D resolvent (thus with discrete spectrum) such that [ , π(a)] is bounded for any a . D ∈A

We could impose the existence of a C∗-algebra A where := a A [ , π(a)] is bounded A { ∈ | D } is norm dense in A so is a pre-C∗-algebra stable by holomorphic calculus. A When there is no confusion, we will write a instead of π(a). We now give useful deﬁnitions: Deﬁnition 6.2. Let ( , , ) be a spectral triple. A H D It is even if there is a grading operator χ s.t. χ = χ∗, [χ, π(a)]=0, a , χ = χ . It is real of KO-dimension d Z/8 if there is an antilinear isometry∀ J∈A: D −suchD 2 ∈ H → H that J = ǫ J, J = ǫ′, Jχ = ǫ′′ χJ with the following table for the signs ǫ, ǫ′, ǫ′′ D D d 01234567 ǫ 1-1 1 1 1 -1 1 1 (22) ǫ′ 1 1 -1 -1 -1 -1 1 1 ǫ′′ 1 -1 1 -1 and the following commutation rules

[π(a), π(b)◦)=0, [ , π(a)], π(b)◦ =0, a, b (23) D ∀ ∈A h i 1 where π(a)◦ := Jπ(a∗)J − is a representation of the opposite algebra ◦. It is d-summable (or has metric dimension d) if the singular valuesA of behave like 1 1/d D n( − )= (n− ). DIt is regularO if and [ , ] are in the domain of δn for all n N where δ(T ) := [ , T ]. A D A ∈ |D|k It satisfies the finiteness condition if the space of smooth vectors ∞ := k Dom is a finitely projective left -module. H D A T It satisfies the orientation condition if there is a Hochschild cycle c Z ( , ◦) such ∈ d A A⊗A that π (c) = χ, where π (a b◦) a1 ad := π(a)π(b)◦[ , π(a1)] [ , π(ad)] and D D ⊗ ⊗ ⊗⊗ D D d is its metric dimension. An interesting example of noncommutative space of non-zero KO-dimension is given by the finite part of the noncommutative standard model [20,27,30]. Moreover, the reality (or charge conjugation in the commutative case) operator J is related to Tomita theory [101]. A reconstruction of the manifold is possible, starting only with a spectral triple where the algebra is commutative (see [28] for a more precise formulation, and also [92]): Theorem 6.3. [28] Given a commutative spectral triple ( , , ) satisfying the above ax- c A H D ioms, then there exists a compact spin manifold M such that C∞(M) and is a Dirac operator. A ≃ D

22 The manifold is known as a set, M = Sp( ) = Sp(A). Notice that is known only via its A D principal symbol, so is not unique. J encodes the nuance between spin and spinc structures. The spectral action selects the Levi-Civita connection so the Dirac operator D/ . The way, the operator recaptures the original Riemannian metric g of M is via the D Connes’ distance: the map

d(φ ,φ ) := sup φ (a) φ (a) [ , π(a)] 1, a (24) 1 2 {| 1 − 2 || D ≤ ∈A} defines a distance (eventually infinite) between two states φ1,φ2 on the C∗-algebra A. The role of is non only to provide a metric by (24), but its homotopy class represents D the K-homology fundamental class of the noncommutative space . It is known that one cannot hear the shape of a drum since the knA owledge of the spectrum of a Laplacian does not determine the metric of the manifold, even if its conformal class is given [7]. But Theorem 6.3 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 operator and the volume form, via its cohomological content, is sufficient to recapture the metric and spin structure. See however the more precise refinement made in [29].

6.2 Notion of pseudodifferential operators Definition 6.4. Let ( , , ) be a spectral triple. A H D it it For t R define the map F : T ( ) e |D|T e− |D| and for α R ∈ t ∈ B H → ∈ 0 OP := T t F (T ) C∞ R, ( ) is the set of operators or order 0, { | → t ∈ B H } ≤ α α 0 OP := T T − OP is the set of operators of order α. { | |D| ∈ } ≤ Moreover, we set δ(T ) := [ , T ], (T ) := [ 2, T ]. |D| ∇ D 0 For instance, C∞(M) = OP L∞(M) and L∞(M) is the von Neumann algebra gener- α ated by = C∞(M). The spaces OP have the expected properties: A T 0 k Proposition 6.5. Assume that ( , , ) is regular so OP = k 0 Dom δ ( ). Then, for any α, β R, A H D A ⊂ ≥ ⊂ B H ∈ T OP α OP β OP α+β, OP α OP β if α β, δ(OP α) OP α, (OP α) OP α+1. ⊂ ⊂ ≤ ⊂ ∇ ⊂ 3 As an example, let us compute the order of X = a [ , b] − : since the order of a is 0, of 3 |D| D2 D is 1, of [ , b] is 0 and of − is -3, we get X OP − . |D| D D ∈ Definition 6.6. Let ( , , ) be a spectral triple and ( ) be the polynomial algebra gen- A H D D A erated by , ◦, and . Define the set of pseudodifferential operators as A A D |D| N p Ψ( ) := T N N, P ( ), R OP − , p N such that T = P − + R A { | ∀ ∈ ∃ ∈D A ∈ ∈ |D| }

The idea behind this definition is that we want to work modulo the set OP −∞ of smoothing operators. This explains the presence of the arbitrary N and R. In the commutative case 1 where Diff (M, E), we get the natural inclusion Ψ C∞(M) ΨDO(M, E). D ∈ ⊂ The reader should be aware that Definition 6.6 is not exactly the same as in [30,33,49] since it pays attention to the reality operator J when it is present.

23 6.3 Zeta-functions and dimension spectrum

Definition 6.7. For P Ψ∗( ), we define the zeta-function associated to P (and ) by ∈ A D P s ζ : s C Tr P − (25) D ∈ → |D| s 1 which makes sense since for (s) 1, P − ( ). The dimension spectrum Sdℜ of≫( , ,|D|) is∈L the setH poles of ζP (s) P Ψ( ) OP 0 . A H D { D | ∈ A ∩ } It is said simple if it contains poles of order at most one. The noncommutative integral of P is defined by P := Res ζP (s). − s=0 D R In (25), we assume invertible since otherwise, one can replace by the invertible operator + P , P being theD projection on Ker . This change does notD modify the computation of D D the integrals − which follow since − X = 0 when X is a trace-class operator. R R 6.4 One-forms and fluctuations of D The unitary group ( ) of gives rise to the automorphism αu : a uau∗ . This defines the innerU A automorphismsA group Inn( ) which is a normal∈A→ subgroup of∈ the A automorphisms Aut( ) := α Aut(A) α( ) A . For instance, in case of a gauge A { ∈ | A ⊂ A} theory, the algebra = C∞ M, M (C) C∞(M) M (C) is typically used. Then, A n ≃ ⊗ n Inn( ) is locally isomorphic to = C∞ M,PSU(n) . Since Aut C∞(M) Diff(M), we A G ≃ get a complete parallel analogy between following two exact sequences: 1 Inn( ) Aut( ) Aut( )/Inn( ) 1, −→ A −→ A −→ A A −→ 1 ⋊ Diff(M) Diff(M) 1. Thus the internal−→ symmetries G −→ of physics G have to be−→ replaced by the inner automorphisms.−→ The appropriate framework for inner fluctuations of a spectral triple ( , , ) is Morita equivalence, see [24,49]. A H D Definition 6.8. Let ( , , ) be a spectral triple. One-forms are definedA as ΩH1 (D ) := span adb a, b , db := [ , b] which is a -bimodule. D A { | ∈A} D A The Morita equivalence which does not change neither the algebra nor the Hilbert space A 1 , gives a natural hermitean fluctuation of : A := + A with A = A∗ Ω ( ). H 1D D→D D ∈ D A For instance, in commutative geometries, Ω C∞(M) = c(da) a C∞(M) . D/ { | ∈ } When a reality operator J exists, we also want J = ǫJ , so we choose DA DA 1 := + A, A := A + ǫJAJ − , A = A∗. (26) DA D The next two results show that,e withe the same algebra and Hilbert space , a fluctu- e A H ation of still give rise to a spectral triple ( , , ) or ( , , ). D A H DA A H DA Lemma 6.9. Let ( , , ) be a spectral triple with a reality operator J and chirality χ. If 1 A H D e A Ω , the fluctuated Dirac operator A or A is an operator with compact resolvent, and in∈ particularD its kernel is a finite dimensionalD D space. This space is invariant by J and χ. e Note that ( ) acts on by = u u∗ leaving invariant the spectrum of . Since U A D D→Du D D u = + u[ ,u∗] and in a C∗-algebra, any element a is a linear combination of at most four Dunitaries,D DefinitionD 6.8 is quite natural.

24 The inner automorphisms of a spectral triple correspond to inner fluctuation of the metric defined by (24). One checks directly that a fluctuation of a fluctuation is a fluctuation and that the unitary group ( ) is gauge compatible for the adjoint representation but to be an inner fluctuation U A is not a symmetric relation. It can append that = 0 with = 0. DA D Lemma 6.10. Let ( , , ) be a spectral triple and X Ψ( ). Then − X∗ = − X. If the A D H 1 ∈ A 1 spectral triple is real, then, for X Ψ( ), JXJ − Ψ( ) and − JXJ − R= − X∗ =R − X. ∈ A ∈ A k N l k R1 l R k R l k If A = A∗, then for k, l , the integrals − A − , − A − , − A − , − χA − , l k ∈ D D |D| |D| A − are real valued. R R R R − D|D| R We remark that the fluctuations leave invariant the first term of the spectral action (33). This is a generalization of the fact that in the commutative case, the noncommutative integral depends only on the principal symbol of the Dirac operator and this symbol is stable by adding a gauge potential like in + A. Note however that theD symmetrized gauge potential 1 D A + ǫJAJ − is always zero in this commutative case for any selfadjoint one-form A. Theorem 6.11. Let ( , , ) be a regular spectral triple which is simple and of dimension d. 1 A H D d ( 1)q 1 q Let A Ω ( ) be a selfadjoint gauge potential. Then, ζD (0) = ζD(0)+ q=1 −q (AD− ) . ∈ D A A − P R The proof of this result, necessary for spectral action computation, needs few preliminaries.e e Definition 6.12. For an operator T , define the one-parameter group and notation

z z 2 2 σ (T ) := D T D − , z C. ǫ(T ) := (T )D− , (recall that (T ) = [ , T ]). z | | | | ∈ ∇ ∇ D The expansion of the one-parameter group σ gives for T OP q z ∈ N r N 1+q σ (T ) g(z,r) ε (T ) mod OP − − (27) z ∼ rX=0 where g(z,r) := 1 ( z ) ( z (r 1)) = z/2 with the convention g(z, 0) := 1. r! 2 2 − − r We fix a regular spectral triple ( , , ) of dimension d and a self-adjoint 1-form A. Despite previous remark before LemmaA H 6.10,D we pay attention here to the kernel of since DA this operator can be non-invertible even if is, so we define D 1 := + A where A := A + εJAJ − ,D := + P (28) DA D A DA A where P is the projection one Ker .e Remark that A ( ) OP 0 and ( ) OP 1. A DA ∈D A ∩ DA ∈D A ∩ We note VA := PA P0 . and as the following lemma shows, VA is a smoothing operator: − e k k Lemma 6.13. (i) k 1 Dom( A) k 1 Dom D . ≥ Dk ⊆ ≥ | | (ii) Ker A Tk 1 Dom D . T (iii) ForD any⊆α, β≥ R, D| β|P D α is bounded. T ∈ | | A| | (iv) P OP −∞. A ∈ 2 2 2 1 Let us define X := A = A + A + A , XV := X + VA, thus X 1( ) OP D −D D D 2 ∈2 D A ∩ and by Lemma 6.13, XV X mod OP −∞. Now let Y := log(DA) log(D ) which makes 2 2 ∼ e e e − sense since DA = A + PA is invertible for any A. By definition of XD, we get Y = log(D2 + X ) log(D2). V V − 25 1 Lemma 6.14. Y is a pseudodifferential operator in OP − with the expansion for any N N ∈ N N p − |k| +p+1 ( 1) 1 kp kp−1 k1 2( k 1+p) N 1 Y − (X ( X (X) ))D− | | mod OP − − . k 1+p ∼ | | ∇ ∇ ∇ p=1 k1, ,kp=0 X X For any N N and s C, ∈ ∈ N s s s N 1 (s) D − D − + K (Y,s) D − mod OP − − −ℜ (29) | A| ∼| | p | | pX=1

p r1 rp with K (Y,s) OP − . For any p N and r , ,r N , ε (Y ) ε (Y ) Ψ( ). p ∈ ∈ 1 p ∈ 0 ∈ A Proof of Theorem 6.11. See [12]. Since the spectral triple is simple, equation (29) entails that s 1 ζD (0) ζD(0) = Tr(K1(Y,s) D − ) s=0 . Thus, with (27), we get ζD (0) ζD(0) = Y . A − | | | A − − 2 − Now the conclusion follows from − log (1 + S)(1 + T ) = − log(1 + S)+ − log(1 + TR ) for 1 ( 1)n+1 n 1 1 R − R R S, T Ψ( ) OP − (since log(1 + S) = n∞=1 n S ) with S = D− A and T = AD− ; ∈ A ∩ 2 1 1 d ( 1)q 1 q so log(1 + XD− )=2 log(1 + AD− ) and Y = − (AD− ) . − − P − 2 − q=1 q − R R R P R Lemma 6.15. For any k N , e ∈ 0 k k p − r1 rp s Res ζD (s) = Res ζD(s)+ Res h(s,r,p) Tr ε (Y ) ε (Y ) D − , s=d k A s=d k s=d k | | − − p=1 r1, ,rp=0 − X X : p where h(s,r,p) =( s/2) 0 t1 tp 1 g( st1,r1) g( stp,rp) dt . − ≤ ≤≤ ≤ − − R Our operators D k are pseudodiﬀerential operators: for any k Z, D k Ψk( ). | A| ∈ | A| ∈ A The following result is quite important since it shows that one can use − for D or DA:

s R Proposition 6.16. If the spectral triple is simple, Res Tr P DA − = P for any pseudod- s=0 | | − (d k) ifferential operator P . In particular, for any k N0 DA − − = ResR ζD (s). Moreover, − s=d k A ∈ | | − R d d DA − = D − , (30) Z− | | Z− | | (d 1) (d 1) d 1 d 1 DA − − = D − − ( −2 ) X D − − , Z− | | Z− | | − Z− | | (d 2) (d 2) d 2 d d 2 2 d D − − = D − − + − X D − + X D − − . − | A| − | | 2 − − | | 4 − | | Z Z Z Z 6.5 Tadpole In [30], the following definition is introduced inspired by the quantum field theory.

Deﬁnition 6.17. In ( , , ), the tadpole T ad +A(k) of order k, for k d l : l N A H D 1 k D ∈{ − ∈ } is the term linear in A = A∗ Ω , in the Λ term of (33) where + A. ∈ D D→D If moreover, the triple ( , , , J) is real, the tadpole T ad +A˜(k) is the term linear in A H D D A, in the Λk term of (33) where + A. D→D e 26 Proposition 6.18. Let ( , , ) be a spectral triple of dimension d with simple dimension A H D spectrum. Then (d k) 2 1 Tad +A(d k)= (d k) A − − − , k = d, Tad +A(0) = A − . D − − − − D|D| ∀ D −− D Moreover, if the triple is real, TadR +A = 2 Tad +A. R D D 1 Corollary 6.19. In a real spectral triple ( , , ), if A = A∗ Ω ( ) is such that A =0, e A H D ∈ D A then TadD+A(k)=0 for any k Z, k d. ∈ ≤ e The vanishing tadpole of order 0 has the following equivalence (see [12]) 1 1 1 A − =0, A Ω ( ) ab = aα(b), a, b , α(b) := b − . − D ∀ ∈ D A ⇐⇒ − − ∀ ∈A D D TheR existence of tadpoles is importantR since,R for instance, A = 0 is not necessarily a stable solution of the classical ﬁeld equation deduced from spectral action expansion, [50].

6.6 Commutative geometry Definition 6.20. Consider a commutative spectral triple given by a compact Riemannian spin manifold M of dimension d without boundary and its Dirac operator D/ associated to 2 the Levi–Civita connection. This means := C∞(M), := L (M,S), D/ where S is A H the spinor bundle over M. This triple is real since, due to the existence of a spin structure, the charge conjugation operator generates an anti-linear isometry J on such that 1 H JaJ − = a∗, a , and when d is even, the grading is given by the chirality matrix ∀ ∈ A χ := ( i)d/2 γ1γ2 γd. Such triple is said to be a commutative geometry. − In the polynomial algebra ( ) of Definition 6.6, we added ◦. In the commutative case, 1 D A 1 1A ◦ J J − which also gives JAJ − = ǫ A∗, A Ω ( ) or A = 0 when A = A∗. A ≃ A ≃A − ∀ ∈ D A As noticed by Wodzicki, − P is equal to 2 times the coefficient in log t of the asymptotics t D/ 2 − e of Tr(P e− ) as t 0. It isR remarkable that this coefficient is independent of D/ as seen in Theorem 2.18 and this→ gives a close relation between the ζ function and heat kernel expansion with WRes. Actually, by [47, Theorem 2.7] t D/ 2 (k ord(P ) d)/2 k Tr(P e− ) k∞=0 ak t − − + k∞=0( ak′ log t + bk) t , t 0+ ↓∼ − P P so − P =2a0′ . Since −, WRes are traces on Ψ C∞(M) , Corollary 2.19 gives − P = c WRes P. 2s 1 s 1 t D/ 2 BecauseR Tr(P D/ − )=R ∞ t − Tr(P e− ) dt, the non-zero coefficient a′R, k = 0 creates a Γ(s) 0 k 2s 1 s 1 k ( 1)kk! 1 − R − pole of Tr(P D/ ) of order k +2 as we get 0 t − log(t) dt = sk+1 and Γ(s)= s +γ +sg(s) where γ is the Euler constant and the functionR g is also holomorphic around zero. Proposition 6.21. Let Sp(M) be the dimension spectrum of a commutative geometry of dimension d. Then Sp(M) is simple and Sp(M)= d k k N . { − | ∈ } Remark 6.22. When the dimension spectrum is not simple, the analog of WRes is no longer a trace. −d −d A The equation (30) can be obtained via (8) as σd|D | = σd|D| . In dimension d =4, the computation in (19) of coefficient a (1, 2 ) gives 4 DA 2 ν νρσ ν ζ (0) = c1 (5R 8Rνr 7RνρσR dvol + c2 tr(Fν F ) dvol, DA ZM − − ZM 2 see Corollary 7.4 to see precise correspondence between ak(1, A) and ζ A (0). One recognizes D D the Yang–Mills action which will be generalized in Section 7.1.3 to arbitrary spectral triples. According to Corollary 6.19, a commutative geometry has no tadpoles [58].

27 6.7 Scalar curvature What could be the scalar curvature of a spectral triple ( , , )? Of course, we consider first the case of a commutative geometry of dimension d =4:A H if sDis the scalar curvature and d+2 f C∞(M), we know that f(x)D/ − = f(x) s(x) dvol(x). This suggests the ∈ − M Definition 6.23. Let ( , R, ) be a spectralR triple of dimension d. The scalar curvature is A H D d+2 the map : a C defined by (a) := a − . R ∈A→ R − D In the commutative case, is a trace on theR algebra. More generally R Proposition 6.24. d+1 [30, Proposition 1.153] If is a trace on and the tadpoles − A − 1 R A D are zero for all A Ω , is invariant by inner fluctuations + A. R ∈ D R D→D 6.8 Tensor product of spectral triples There is a natural notion of tensor for spectral triples which corresponds to direct product of manifolds in the commutative case. Let ( , , ), i = 1, 2, be two spectral triples of Ai Di Hi dimension di with simple dimension spectrum. Assume the first to be of even dimension, with grading χ . The spectral triple ( , , ) associated to the tensor product is defined by 1 A D H := 1 2, := 1 1+ χ1 2, := 1 2. A A ⊗A D D ⊗ 2 ⊗D2 H H2 ⊗ H The interest of χ1 is to guarantee additivity: = 1 1+1 2. 2 D D ⊗ ⊗D2 t d1/2 t d2/2 Lemma 6.25. Assume that Tr(e− D1 ) t 0 a1 t− and Tr(e− D2 ) t 0 a2 t− . ∼ → ∼ → The triple ( , , ) has dimension d = d1 + d2. A D H s Moreover, the function ζ (s) = Tr( D − ) has a simple pole at s = d1 + d2 with D | | 1 Γ(d1/2)Γ(d2 /2) Ress=d1+d2 ζ (s) = Ress=d1 ζ 1 (s) Ress=d2 ζ 2 (s) . D 2 Γ(d/2) D D s 2 2 s 2 2 − Proof. We get ζ (2s)= n∞=0 n(D1 1+1 2)− = n,m∞ =0 n( 1)+ m( 2) . Since D ⊗ ⊗D D D (c1+c2)/2 2 2 − P c1 c2 P n( 1) + m( 2) n( 1)− m( 2)− , ζ (c1 + c2) ζ 1 (c1)ζ 2 (c2) if ci >di, D D D D D + ≤ D D ≤ so d := inf c R : ζ (c) < d1 + d2. We claim that d = d1 + d2: recall first that { ∈ D ∞ } ≤ 1 ai := Res Γ(s)ζ i (2s) = Γ(di/2) Res ζ i (2s) = 2 Γ(di/2) Res ζ i (s) . If s=di/2 D s=di/2 D s=di D 1 2 1 2 2 t s 1 t 1 t 2 s 1 f(s) := Γ(s) ζD(2s), f(s)= 0 Tr e− D t − dt+g(s)= 0 Tr e− D Tr e− D t − dt+g(s) R tx2 x 1 where g is holomorphic sinceR the map x 1∞ e− Rt − dt is in Schwartz space. 2 2 t t ∈ (d→1+d2)/2 Since Tr e− D1 Tr e− D2 t 0 a1a2 t− R , the function f(s) has a simple pole at ∼ → s =(d1 + d2)/2. We conclude that ζ (s) has a simple pole at s = d1 + d2. Moreover, using 1 1 D 1 ai, 2 Γ((d1 + d2)/2)Res ζ (s) = 2 Γ(d1/2)Res ζ 1 (s) 2 Γ(d2/2)Res ζ 2 (s) . s=d D s=d1 D s=d2 D 7 Spectral action

7.1 On the search for a good action functional We would like to obtain a good action for any spectral triple and for this it is useful to look at some examples in physics. In any physical theory based on geometry, the interest of an action functional is, by a minimization process, to exhibit a particular geometry, for instance, trying to distinguish between diﬀerent metrics. This is the case in general relativity with the Einstein–Hilbert action (with its Riemannian signature).

28 7.1.1 Einstein–Hilbert action

This action is SEH (g) := M sg(x) dvolg(x) where s is the scalar curvature (chosen positive − 2 − for the sphere). Up to a constant,R this is − D/ in dimension 4 as quoted after (21). This action is interesting for the followingR reason: Let 1 be the set of Riemannian metrics g on M such that dvol = 1. By a theorem of HilbertM [5], g is a critical point of M g ∈ M1 SEH(g) restricted toR 1 if and only if (M,g) is an Einstein manifold (the Ricci curvature R of g is proportionalM by a constant to g: R = cg). Taking the trace, this means that sg = c dim(M) and such manifold have a constant scalar curvature. But in the search for invariants under diﬀeomorphisms, they are more quantities than the

Einstein–Hilbert action, a trivial example being M f sg(x) dvolg(x) and they are others [42]. In this desire to implement gravity in noncommutativeR geome try, the eigenvalues of the Dirac operator look as natural variables [69]. However we are looking for observables which add up under disjoint unions of diﬀerent geometries.

7.1.2 Quantum approach and spectral action

1 In a way, a spectral triple fits quantum field theory since − can be seen as the propagator (or line element ds) for (Euclidean) fermions and we canD compute Feynman graphs with fermionic internal lines. As glimpsed in section 6.4, the gauge bosons are only derived objects obtained from internal fluctuations via Morita equivalence given by a choice of a connection which is associated to a one-form in Ω1 ( ). Thus, the guiding principle followed by Connes D A and Chamseddine is to use a theory which is pure gravity with a functional action based on the spectral triple, namely which depends on the spectrum of [10]. They proposed the D Definition 7.1. The spectral action of ( , , ) is defined by ( , f, Λ) := Tr f( 2/Λ2) A H D S D D where Λ R+ plays the role of a cut-off and f is any positive function (such that f( 2/Λ2) is a trace-class∈ operator). D

Remark 7.2. We can also define ( , f, Λ) = Tr f( /Λ) when f is positive and even. S D D With this second definition, S( , g, Λ) = Tr f( 2/Λ2) withg(x) := f(x2). D D For f, one can think of the characteristic function of [ 1, 1], thus f( /Λ) is nothing else but the number of eigenvalues of within [ Λ, Λ]. − D When this action has an asymptoticD series− in Λ , we deal with an effective theory. → ∞ Naturally, has to be replaced by A which is a just a decoration. To this bosonic part of the D D1 action, one adds a fermionic term 2 Jψ, ψ for ψ to get a full action. In the standard model of particle physics, this latter correspondsD to∈ the H integration of the Lagrangian part for the coupling between gauge bosons and Higgs bosons with fermions. Actually, the finite dimension part of the noncommutative standard model is of KO-dimension 6, thus ψ, ψ D has to be replaced by 1 Jψ, ψ for ψ = χψ , see [30]. 2 D ∈ H 7.1.3 Yang–Mills action This action plays an important role in physics so recall first the classical situation: let G be a compact Lie group with its Lie algebra g and let A Ω1(M, g) be a connection. If ∈ F := da + 1 [A, A] Ω2(M, g) is the curvature (or field strength) of A, then the Yang-Mills 2 ∈

29 action is S (A)= tr(F ⋆F ) dvol . In the abelian case G = U(1), it is the Maxwell action YM M ∧ g and its quantum versionR is the quantum electrodynamics (QED) since the un-gauged U(1) of electric charge conservation can be gauged and its gauging produces electromagnetism [97]. It is conformally invariant when dim(M)=4. The study of its minima and its critical values can also been made for a spectral triple ( , , ) of dimension d [23, 24]: let A Ω1 ( ) and curvature θ = dA + A2; then it is A H D 2 ∈d D A natural to consider I(A) := TrDix(θ − ) since it coincides (up to a constant) with the |D| 2 d previous Yang-Mills action in the commutative case: if P = θ − , then Theorems 2.18 |D| and 3.11 give the claim since for the principal symbol, tr σP (x, ξ) = c tr(F ⋆F )(x). ∧ There is nevertheless a problem with the definition of dA: if A = j π(aj)[ , π(bj)], then dA = [ , π(a )][ , π(b )] can be non-zero while A = 0. This ambiguityD means j D j D j P that, to get a graded differential algebra Ω∗ ( ), one must divide by a junk, for instance P D A Ω2 π(Ω2/π δ(Ker(π) Ω1) where Ωk( ) is the set of universal k-forms over given by D ≃ ∩ A A the set of a0δa1 δak (before representation on : π(a0δa1 δak) := a0[ , a1] [ , ak]). Hk D D Let k be the Hilbert space completion of π(Ω ( )) with the scalar product defined by H d k A A1, A2 k := TrDix(A2∗A1 − ) for Aj π(Ω ( )). The Yang–Mills action on|D| Ω1( ) is S∈ (V ) :A= δV + V 2,δV + V 2 . It is positive, quartic A YM and gauge invariant under V π(u)V π(u∗)+ π(u)[ , π(u∗)] when u ( ). Moreover, 1 → D ∈ U A SYM (V ) = inf I(ω) ω Ω ( ), π(ω) = V since the above ambiguity disappears when taking the infimum.{ | ∈ A } This Yang–Mills action can be extended to the equivalent of Hermitean vector bundles on M, namely finitely projective modules over . A The spectral action is more conceptual than the Yang–Mills action since it gives no fundamental role to the distinction between gravity and matter in the artificial decomposition = + A. For instance, for the minimally coupled standard model, the Yang–Mills action DA D for the vector potential is part of the spectral action, as far as the Einstein–Hilbert action for the Riemannian metric [11]. As quoted in [16], 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 action Tr f( /Λ) 0 has the correct sign for a Euclidean action: the positivity of the function f D ≥ will insure that the actions for gravity, Yang-Mills, Higgs couplings are all positive and the Higgs mass term is negative. - Invariance: the spectral action has a much stronger invariance group than the usual diffeomorphism group as for the gravitational action; this is the unitary group of the Hilbert space . However,H this action is not local but becomes local when replaced by its asymptotic expansion:

7.2 Asymptotic expansion for Λ →∞ The heat kernel method already used in previous sections will give a control of spectral action S( , f, Λ) when Λ goes to inﬁnity. D Theorem 7.3. Let ( , , ) be a spectral triple with a simple dimension spectrum Sd. A H D

30 We assume that

t 2 α Tr e− D aα t with aα =0. (31) t 0 ∼↓ α Sd X∈ Then, for the zeta function ζ deﬁned in (25) D 1 aα = Ress= 2α Γ(s/2)ζ (s) . (32) 2 − D 1 (i) If α< 0, ζ has a pole at 2α with aα = Γ( α) Res ζ (s). D 2 s= 2α D − − − (ii) For α =0, we get a0 = ζ (0) + dim Ker . D D (iii) If α> 0, aα = ζ( 2α) Res Γ(s). − s= α (iv) The spectral action has the− asymptotic expansion over the positive part Sd+of Sd:

β β Tr f( /Λ) fβ Λ + f(0) ζ (0) + (33) Λ + D D →∼ ∞ β Sd+ Z− |D| ∈X β 1 where the dependence of the even function f is f := ∞ f(x) x − dx and involves the β 0 full Taylor expansion of f at 0. R

s t 2 s/2 1 1 t 2 s/2 1 Proof. (i) Since Γ(s/2) − = 0∞ e− D t − dt = 0 e− D t − dt + f(s), where the func- |D| tx2 s/2 1 tion f is holomorphic (since theR map x 1∞ e− Rx − dt is in the Schwartz space), the 2 t α → 1 α+s/2 1 2αa swap of Tr e− D with a sum of aα t and aRα 0 t − dt = s+2α yields (32). 1 (ii) The regularity of Γ(s/2)− s/2 aroundR zero implies that only the pole part at s∼0 2 → t s/2 1 1 s/2 1 2a0 s =0of 0∞ Tr e− D t − dt contributes to ζ (0). This contribution is a0 0 t − dt = . D s (iii) followsR from (32). R 2 sx (iv) Assume f(x) = g(x ) where g is a Laplace transform: g(x) := 0∞ e− φ(s) ds. We will see in Section 7.3 how to relax this hypothesis. R 2 st 2 2 α α Since g(t ) = 0∞ e− D φ(s) ds, Tr g(t ) α Sp+ aα t 0∞ s g(s) ds. When α < 0, t 0 ∈ D D ∼↓ α 1 R sy α 1 α 1 R α 1 s = Γ( α)− 0∞ e− y− − dy and 0∞ s φ(s) ds =P Γ( α)− 0∞ g(y) y− − dy. Thus − 2 1 − α 1 α R Tr g(t ) αR Sp− 2 Res ζ (s) 0∞ g(yR) y− − dy t . t 0 ∈ s= 2α D D ∼↓ − P 1 h β/2 1 R β 1 i Thus (33) follows from (i), (ii) and 2 0∞ g(y) y − dy = 0∞ f(x) x − dx. R R It can be useful to make a connection with (20) of Section 5.2: Corollary 7.4. Assume that the spectral triple ( , , ) has dimension d. If A H D t 2 (k d)/2 2 Tr e− D t − ak( )+ , (34) t 0 ∼↓ k 0, ,d D ∈{X } k 2 2 then ( , f, Λ) k 1, ,d fk Λ ad k( )+ f(0) ad( )+ with t 0 ∈{ } − S D ∼↓ D D 1 k/P2 1 fk := Γ(k/2) 0∞ f(s)s − ds. Moreover, R 2 1 d k d+k 2 ak( )= 2 Γ( −2 ) − for k =0, ,d 1, ad( ) = dim Ker + ζ 2 (0). (35) D Z− |D| − D D D The asymptotics (33) uses the value of ζ (0) in the constant term Λ0, so it is fundamental D to look at its variation under a gauge ﬂuctuation + A as we saw in Theorem 6.11. D→D 31 7.3 Remark on the use of Laplace transform

d n 2 The spectral action asymptotic behavior S( , f, Λ) n∞=0 cn Λ − an( ) has been Λ + D →∼ ∞ D proved for a smooth function f which is a Laplace transform forP an arbitrary spectral triple (with simple dimension spectrum) satisfying (31). However, this hypothesis is too restrictive x since it does not cover the heat kernel case where f(x)= e− . When the triple is commutative and 2 is a generalized Laplacian on sections of a vector bundle over a manifold of dimension 4,D Estrada–Gracia-Bondía–Várilly proved in [37] that previous asymptotics is

2 2 1 ∞ 4 2 ∞ 2 Tr f( /Λ ) (4π)2 rk(E) xf(x) dx Λ + b2( ) f(x) dx Λ D ∼ 0 D 0 Z Z ∞ m (m) 2 2m + ( 1) f (0) b ( )Λ− , Λ − 2m+4 D →∞ mX=0

m 2 (4π)2 2 where ( 1) b2m+4( )= m! m( ) are suitably normalized, integrated moment terms of the spectral− densityD of 2. D D The main point is that this asymptotics makes sense in the Cesàro sense (see [37] for deﬁnition) for f in ′(R), which is the dual of (R). This latter is the space of smooth K (k) K a k functions φ such that for some a R, φ (x) = ( x − ) as x , for each k N. In ∈ O | | | |→∞ ∈ particular, the Schwartz functions are in (R) (and even dense). K Of course, the counting function is not smooth but is in ′(R), so the given asymptotic K behavior is wrong beyond the ﬁrst term, but is correct in the Cesàro sense. Actually there are more derivatives of f at 0 as explained on examples in [37, p. 243].

7.4 About convergence and divergence, local and global aspects of the asymptotic expansion The asymptotic expansion series (34) of the spectral action may or may not converge. It is 1 known that each function g(Λ− ) deﬁnes at most a unique expansion series when Λ → ∞ but the converse is not true since several functions have the same asymptotic series. We give here examples of convergent and divergent series of this kind. Td ν t∆ (4π)−d/2 Vol(Td) When M = as in Example 3.10 with ∆ = δ ∂∂ν, Tr(e ) = td/2 + d/2 1/4t t∆ (4π)−d/2 Vol(T d) (t− e− ), so the asymptotic series Tr(e ) td/2 , t 0, has only one term. O In the opposite direction, let now M be the unit≃ four-sphere S4 and→ D/ be the usual Dirac operator. By Proposition 6.21, equation (31) yields (see [19]):

n 2 k tD/ 1 2 2 k+2 n+3 ( 1) 4 B2k+2 B2k+4 Tr(e− )= + t + a t + (t ) , a := − t2 3 3 k O k 3 k! 2k+2 − 2k+4 kX=0

2 tD/ 2 2 2 k+2 with Bernoulli numbers B . Thus t Tr(e− ) + t + ∞ a t when t 0 and this 2k ≃ 3 3 k=0 k → series is a not convergent but only asymptotic: P 2k+4 4 B2k+4 (2k+4)! k+2 a > | | > 0 and B =2 ζ(2k+4) 4 π(k + 2) if k . k 3 k! 2k+4 | 2k+4| (2π)2k+4 ≃ πe →∞ →∞ More generally, in the commutative case consideredq above and when is a differential D t 2 operator—like a Dirac operator, the coefficients of the asymptotic series of Tr(e− D ) are locally defined by the symbol of 2 at point x M but this is not true in general: in [45] D ∈ 32 is given a positive elliptic pseudodifferential such that non-locally computable coefficients especially appear in (34) when 2k>d. Nevertheless, all coefficients are local for 2k 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 derivatives of a finite number of terms in the asymptotic expansion of the total symbol of 2. For instance, some nonlocal information contained in the ultraviolet asymptotics can be recoveredD t√ ∆ 1 1 if one looks at the (integral) kernel of e− − : in T , with Vol(T )=2π, we get [38]

∞ 2 4 t√ ∆ sinh(t) t 2 B2k 2k 2 t t 6 Tr(e− − )= cosh(t) 1 = coth( 2 )= t (2k)! t = t [1 + 12 720 + (t )] − − O kX=0

B k+1 2 ζ(2k) B2k 2 so the series converges when t< 2π as 2k =( 1) , thus | | when k . (2k)! − (2π)2k (2k)! ≃ (2π)2k →∞ Thus we have an example where t cannot be used with the asymptotic series. Thus the spectral action of Corollary→∞ 7.4 precisely encodes these local and nonlocal behavior which appear or not in its asymptotics for different f. The coefficient of the action for the positive part (at least) of the dimension spectrum correspond to renormalized traces, namely the noncommutative integrals of (35). In conclusion, the asymptotics of spectral action may or may not have nonlocal coefficients. For the flat torus Td, the difference between Tr(et∆) and its asymptotic series is an term which is related to periodic orbits of the geodesic flow on Td. Similarly, the counting function N(λ) (number of eigenvalues including multiplicities of ∆ less than λ) obeys Weyl’s law: (4π)−d/2 Vol(Td) d/2 d/2 N(λ) = Γ(d/2+1) λ + o(λ ) — see [1] for a nice historical review on these fundamental points. The relationship between the asymptotic expansion of the heat kernel and the formal expansion of the spectral measure is clear: the small-t asymptotics of heat kernel is determined by the large-λ asymptotics of the density of eigenvalues (and eigenvectors). However, the latter is defined modulo some average: Cesàro sense as reminded in Section 7.3, or Riesz mean of the measure which washes out ultraviolet oscillations, but also gives informations on intermediate values of λ [38]. In [16, 76] are examples of spectral actions on commutative geometries of dimension 4 whose asymptotics have only two terms. In the quantum group SUq(2), the spectral action itself has only 4 terms, independently of the choice of function f. See [62] for more examples.

7.5 On the physical meaning of the asymptotics of spectral action The spectral action is non-local. Its localization does not cover all situations: consider for instance the commutative geometry of a spin manifold M of dimension 4. One adds a gauge connection A Γ∞ M,End(S) to D/ such that = iγ (∂ +A), thus with a field strength ∈ D 2 Fν = ∂Aν ∂ν A+ [A, Aν]. We apply (18) with P = and find the coefficients ai(1,P ) of (19) with−i =0, 2, 4. The asymptotic expansion correspondsD to a weak field expansion. Moreover a commutative geometry times a finite one where the finite one is algebra is a sum of matrices has been deeply and intensively investigated for the noncommutative approach to standard model of particle physics, see [20, 30]. This approach offers a lot of interesting perspectives, for instance, the possibility to compute the Higgs representations and mass (for each noncommutative model) is particularly instructive [10,15,17,57,63,64,71,79]. Of course, since the first term in (33) is a cosmological term, one may be worried by its large value (for instance in the noncommutative standard model where the cutoff is, roughly

33 speaking the Planck scale). At the classical level, one can work with unimodular gravity where the metric (so the Dirac operator) varies within the set 1 of metrics which preserve the volume as in Section 7.1.1. Thus itD remains only (!) to controM l the inflaton: see [13]. The spectral action has been computed in [60] for the quantum group SUq(2) which is not a deformation of SU(2) of the type considered on the Moyal plane. It is quite peculiar since (33) has only a finite number of terms. Due to the difficulties to deal with non-compact manifolds, the case of spheres S4 or S3 S1 has been investigated in [16,19] for instance in the case of Robertson–Walker metrics. ×All the machinery of spectral geometry as been recently applied to cosmology, computing the spectral action in few cosmological models related to inflation, see [66,76–78,81,95]. Spectral triples associated to manifolds with boundary have been considered in [14,18,18, 58,59,61]. 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 . This is probably a must to obtain a result in a noncommutative Hamiltonian theory in dimensionA 1+3. The case of manifolds with torsion has also been studied in [53, 84, 85], and even with boundary in [61]. These works show that the Holst action appears in spectral actions and that torsion could be detected in a noncommutative world.

8 The noncommutative torus

The aim of this section is to compute the spectral action of the noncommutative torus. Due to a fundamental appearance of small divisors, the number theory is involved via a Diophantine condition. As a consequence, the result which essentially says that the spectral action of the noncommutative torus coincide with the action of the ordinary torus (up few constants) is awfully technical and this shows how life can be hard in noncommutative geometry! Reference: [36].

8.1 Definition of the nc-torus Tn Let C∞( Θ) be the smooth noncommutative n-torus associated to a non-zero skew-symmetric deformation matrix Θ Mn(R). It was introduced by Rieffel [93] and Connes [21] to gen- Tn ∈ Tn eralize the n-torus . This means that C∞( Θ) is the algebra generated by n unitaries ui, i Θlj i = 1,...,n subject to the relations ul uj = e uj ul, and with Schwartz coefficients: an Tn Zn element a C∞( Θ) can be written as a = k Zn ak Uk, where ak ( ) with the Weyl ∈ i k ∈ { } ∈ S 2 k.χk 1 kn Zn elements defined by Uk := e− u1 un ,P k , the constraint relation reads i k.Θq ∈ ik.Θq UkUq = e− 2 Uk+q, and UkUq = e− UqUk where χ is the matrix restriction of Θ to its upper triangular part. Thus unitary operators 1 Uk satisfy Uk∗ = U k and [Uk, Ul]= 2i sin( k.Θl) Uk+l. − − 2 Tn Let τ be the trace on C∞( Θ) defined by τ k Zn ak Uk := a0 and τ be the GNS Tn ∈ H Hilbert space obtained by completion of C∞( Θ)P with respect of the norm induced by the scalar product a, b := τ(a∗b). 2 Zn On τ = k Zn ak Uk ak k l ( ) , we consider the left and right regular repre- H { T∈n | { } ∈ } sentations of C∞P( Θ) by bounded operators, that we denote respectively by L(.) and R(.). Let also δ , 1,...,n , be the n (pairwise commuting) canonical derivations, defined ∈{ } by δ(Uk) := ikUk.

34 Tn C2m We need to fix notations: let Θ := C∞( Θ) acting on := τ with n = 2m or n A n H H ⊗ n = 2m + 1 (i.e., m = 2 is the integer part of 2 ), the square integrable sections of the ⌊ n ⌋ trivial spin bundle over T and each element of is represented on as L(a) 1 m . The AΘ H ⊗ 2 Tomita conjugation J0(a) := a∗ satisfies [J0, δ] = 0 and we define J := J0 C0 where C0 is m ⊗ an operator on C2 . The Dirac-like operator is given by := i δ γ, D − ⊗ where we use hermitian Dirac matrices γ. It is defined and symmetric on the dense subset Tn C2m α α of given by C∞( Θ) and denotes its selfadjoint extension. Thus C0γ = εγ C0, H ⊗ D 2m 2− and U e = k U γ e , where (e ) is the canonical basis of C . Moreover, C = 1 m D k ⊗ i k ⊗ i i 0 ± 2 depending on the parity of m. Finally, the chirality in the even case is χ := id ( i)mγ1 γn. This yields a spectral triple: ⊗ −

Theorem 8.1. [24,49] The 5-tuple ( Θ, , , J, χ) is a real regular spectral triple of dimension n. It satisfies the finitenessA andH orientabilityD conditions of Definition 6.2. It is n-summable and its KO-dimension is also n.

For every unitary u , uu∗ = u∗u = U0, the perturbed operator Vu Vu∗ by the unitary ∈ A 1 D Vu := L(u) 12m J L(u) 12m J − , must satisfy condition J = ǫ J. The yields the ⊗ ⊗ D D 1 necessity of a symmetrized covariant Dirac operator := +A+ǫJAJ − since V V ∗ = DA D u D u ∗ L(u) 12m [ ,L(u ) 12m ]. Moreover, we get the gauge transformation Vu AVu∗ = γu(A) where D ⊗ D ⊗ D D the gauged transform one-form of A is γ (A) := u[ ,u∗]+ uAu∗, with the shorthand L(u) u D ⊗ m 12 u. So the spectral action is gauge invariant: ( A, f, Λ) = ( γu(A), f, Λ). −→ 1 S D S Dα Any selfadjoint one-form A Ω ( ), is written as A = L( iAα) γ , Aα = Aα∗ Θ, ∈ D A α − ⊗ − ∈A thus A = i δα + L(Aα) R(Aα) γ . Deﬁning A˜α := L(Aα) R(Aα), we get 2 D α α− − ⊗ 1 α α − = g 1 2 (δ + A˜ )(δ + A˜ ) 1 m Ω γ 1 2 where DA − α1 α1 α2 α2 ⊗ 2 − 2 α1α2 ⊗ γα1α2 := 1 (γα1 γα2 γα2 γα1 ), 2 − Ω := [δ + A˜ , δ + A˜ ] = L(F ) R(F ) α1α2 α1 α1 α2 α2 α1α2 − α1α2 F := δ (A ) δ (A ) + [A , A ]. α1α2 α1 α2 − α2 α1 α1 α2 8.2 Kernels and dimension spectrum Since the dimension of the kernel appears in the coeﬃcients (35) of the spectral action, we now compute the kernel of the perturbed Dirac operator:

2m m Proposition 8.2. Ker = U0 C , so dim Ker =2 . For any selfadjoint one-form A, Ker Ker and orD any unitary⊗ u , Ker D = V Ker . D ⊆ DA ∈A Dγu(A) u DA One shows that Ker = Ker in the following cases: DA D (i) A = Au := L(u) 12m [ , L(u∗) 12m ] when u is a unitary in . 1 ⊗ D ⊗ A (ii) A < 2 . || || 1 (iii) The matrix 2π Θ has only integral coefficients. Conjecture: Ker = Ker A at least for generic Θ’s. We will use freelyD the notationD (28) about the difference between and D. D Since we will have to control small divisors, we give first some Diophantine condition:

35 Deﬁnition 8.3. Let δ > 0. A vector a Rn is said to be δ-badly approximable if there exists δ ∈ n c> 0 such that q.a m c q − , q Z 0 and m Z. | − | ≥ | | ∀ ∈ \{ } ∀ ∈ We note (δ) the set of δ-badly approximable vectors and := δ>0 (δ) the set of badly approximableBV vectors. BV ∪ BV A matrix Θ n(R) (real n n matrices) will be said to be badly approximable if there exists u Zn such∈M that tΘ(u) is a× badly approximable vector of Rn. ∈ Remark. A result from Diophantine approximation asserts that for δ > n, the Lebesgue measure of Rn (δ) is zero (i.e almost any element of Rn is δ badly approximable.) \ BV − n Let Θ n(R). If its row of index i is a badly approximable vector of R (i.e. if ∈ Mt Li ) then Θ(ei) and thus Θ is a badly approximable matrix. It follows that almost∈ BV any matrix of ∈( BVR) Rn2 is badly approximable. Mn ≈ 1 Proposition 8.4. When 2π Θ is badly approximable, the spectrum dimension of the spectral n triple C∞(T ), , is equal to the set n k : k N and all these poles are simple. Θ H D { − ∈ 0 } Moreover ζD(0) = 0.

To get this result, we must compute the residues of inﬁnite series of functions on C and the commutation between residues and series works under the suﬃcient Diophantine condition.

We can compute ζD(0) easily but the main diﬃculty is precisely to calculate ζDA (0).

8.3 The spectral action We ﬁx a self-adjoint one-form A on the noncommutative torus of dimension n.

1 Proposition 8.5. If 2π Θ is badly approximable, then the ﬁrst elements of the spectral action expansion (33) are given by n n m+1 n/2 n 1 n+k n+2 D − = D − =2 π Γ( )− , D − =0 for k odd, D − =0. − | A| − | | 2 − | A| − | A| R R R R Here is the main result of this section.

n Theorem 8.6. Consider the noncommutative torus C∞(T ), , of dimension n N Θ H D ∈ 1 where 2π Θ is a real n n real skew-symmetric badly approximable matrix, and a selfadjoint × α 1 one-form A = L( iAα) γ . Then, the full spectral action of A = + A + ǫJAJ − is − ⊗ 2 2 D D (i) for n =2, ( A, f, Λ)=4π f2 Λ + (Λ− ), S D 2 O4 4π2 ν 2 (ii) for n =4, ( A, f, Λ)=8π f4 Λ 3 f(0) τ(FνF )+ (Λ− ), S D − n n kO 1 (iii) More generally, in ( A, f, Λ) = k=0 fn k cn k(A)Λ − + (Λ− ), cn 2(A)=0, S D − − O − cn k(A)=0 for k odd. In particular, c0(A)=0 when n is odd. − P This result (for n = 4) has also been obtained in [41] using the heat kernel method. It is however interesting to get the result via direct computations of (33) since it shows how this formula is eﬃcient.

Remark 8.7. Note that all terms must be gauge invariants, namely invariant by A α −→ γu(Aα)= uAαu∗ + uδα(u∗). A particular case is u = Uk where Ukδα(Uk∗)= ikαU0. In the same way, note that there is no contradiction with the commutative− case where, for any selfadjoint one-form A, = (so A is equivalent to 0!), since we assume in Theorem DA D 8.6 that Θ is badly approximable, so cannot be commutative. A 36 Conjecture 8.8. The constant term of the spectral action of on the noncommutative n- DA torus is proportional to the constant term of the spectral action of + A on the commutative n-torus. D

Remark 8.9. The appearance of a Diophantine condition for Θ has been characterized in 0 1 R dimension 2 by Connes [22, Prop. 49] where in this case, Θ = θ 1 0 with θ . In j − ∈ fact, the Hochschild cohomology H( Θ, Θ∗) satisﬁes dim H ( Θ, Θ∗)=2 (or 1) for j =1 (or j = 2) if and only if the irrationalA A number θ satisﬁesA a DiophantineA condition like i2πnθ 1 k 1 e − = (n ) for some k. |The− result| of TheoremO 8.6 without this Diophantine condition is unknown. Recall that when the matrix Θ is quite irrational (the lattice generated by its columns is dense n after translation by Z , see [49, Def. 12.8]), then the C∗-algebra generated by Θ is simple. It is possible to go beyond the Diophantine condition: see [41]. A

Acknowledgments

I would like to thank Driss Essouabri, Victor Gayral, José Gracia-Bondía, Cyril Levy, Pierre Martinetti, Thierry Masson, Thomas Schücker, Andrzej Sitarz, Jo Várilly and Dmitri Vas- silevich, for our discussions during our collaborations along years. Few results presented here are directly extracted from these collaborations. I also took beneﬁts from the questions of participants (professors and students) of the school during the lectures, to clarify few points. I would like to emphasize the huge kindness of the organizers of this school, Alexander Cardona, Hermán Ocampo, Sylvie Paycha and Andrés Reyes which seems to be a natural colombian quality. I strongly encourage all students or professors to participate to this school in a wonderful country which deserves to be visited despite diﬃculties.

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