Spectral Action in Noncommutative Geometry

Home , Noncommutative geometry

arXiv:1902.05306v1 [math-ph] 14 Feb 2019 ulse sSrneBif nMteaia hsc Volum Physics https://www.springer.com/la/book/9783319947877 Mathematical in SpringerBriefs as Published Geometry Noncommutative Action Spectral Iochum Bruno Eckstein, Michał in 27 e

Preface

This book is dedicated to Alain Connes, whose work has always been a fantastic source of inspiration for us.

The Least Action Principle is among the most profound laws of physics. The action — a functional of the ﬁelds relevant to a given physical system — encodes the entire dynamics. Its strength stems from its universality: The principle applies equally well in every domain of modern physics including classical mechanics, general relativity and quantum ﬁeld theory. Most notably, the action is a primary tool in model-building in particle physics and cosmology. The discovery of the Least Action Principle impelled a paradigm shift in the methodology of physics. The postulates of a theory are now formulated at the level of the action, rather than the equations of motion themselves.Whereas the success of the ‘New Method’ cannot be overestimated, it raises a big question at a higher level: “Where does the action come from?” A quick look at the current theoretical efforts in cosmology and particle physics reveals an overwhelming multitude of models determined by the actions, which are postulated basing on different assumptions, beliefs, intuitions and prejudices. Clearly, it is the empirical evidence that should ultimately select the correct theory, but one cannot help the impression that our current models are only effective and an overarching principle remains concealed. A proposal for such an encompassing postulate was formulated by Ali Chamsed- dine and Alain Connes in 1996 [5]. It reads [5, (1.8)]: The physical action should only depend upon the spectrum of D, where D is a certain unbounded operator of geometrical origin. The incarnation of the Spectral Action Principle is very simple indeed:

S(D, f ,Λ)= Tr f ( D /Λ), | | with a given energy scale Λ and a positive cut-off function f . Such a formulation provides a link with the current effective actions employed in field theoretic models and allows for a confrontation against the experimental data. The striking upshot of the spectral action is that, with a suitable choice of the operator D, it allows one to retrieve the full Standard Model of particle physics in curved (Euclidean) spacetime [4, 6, 20]. This result attracted considerable interest in both physical and mathematical communities and triggered a far-reaching outflow of theoretical research. The most recent applications include Grand Unified Theories [7], modified Einstein gravity [15] and quantum gravity [10], to name only a few.

i ii Preface

The formulation of the Least Action Principle dates back to the 18th century and the seminal works of Pierre de Maupertuis, Gottfried Leibniz and Leonhard Euler. The quest for its rigorous verbalisation sparked the development of the calculus of variations along with the Lagrangian and Hamiltonian formalisms. The modern formulation is expressed in the language of differential geometry. The Spectral Action Principle is embedded in an even more advanced domain of modern mathematics – noncommutative geometry, pioneered and strongly pushed forward by Alain Connes [8, 9]. The idea that spaces may be quantised was first ponderedby Werner Heisenberg in the 1930s (see [1] for a historical review) and the first concrete model of a ‘quantum spacetime’ was constructed by Hartland Snyder in 1949, extended by Chen-Ning Yang shortly afterwards. However, it took almost half a century for the concepts to mature and acquire the shape of a concrete mathematical structure. By now noncommutative geometry is a well-established part of mathematics. Noncommutativegeometry à la Connes sinks its roots not only in the Riemannian geometry, but also in the abstract framework of operator algebras. Its conceptual content is strongly motivated by two fundamental pillars of physics: general relativity and quantum mechanics, explaining why it has attracted both mathematicians and theoretical physicists. It offers a splendid opportunity to conceive ‘quantum spacetimes’ turning the old Heisenberg’s dream into a full-bodied concept. In this paradigm,geometry is described by a triplet (A ,H ,D), where A is a not necessarily commutative algebra, D is an operator (mimicking the Dirac operator on a spin manifold) both acting on a common scene – a Hilbert space H . Thus, by essence, this geometry is spectral. The data of a spectral triple (A ,H ,D) covers a huge variety of different geometries. The classical (i.e. commutative) case includes primarily the Riemannian manifolds, possibly tainted by boundaries or singularities, but also discrete spaces, fractals and non-Hausdorff spaces and when A is noncommutative, the resulting ‘pointless’ geometries, with the examples furnished by the duals of discrete groups, dynamical systems or quantum groups to mention but a few. At this point one should admit that the simple form of the spectral action is deceiving — an explicit computation would require the knowledge of the full spectrum of the operator D, which is hardly ever the case. Nevertheless, one can ex- tract a great deal of physically relevant information by studying the asymptotics of S(D, f ,Λ) when Λ tends to infinity. The key tool to that end is the renowned heat kernel method fruitfully employed in classical and quantum field theory, adapted here to the noncommutative setting. Beyond the (almost) commutative case the latter is still a vastly uncharted water. It is our primary intent to provide a faithful map of the mathematical aspects behind the spectral action. Whereas the physical motivation will be present in the backstage, we leave the potential applications to the Reader’s invention. To facilitate the latter, we recommend to have a glimpse into the textbooks [15, 20] and references therein. Preface iii

The plan of our guided tour presents itself as follows: In the first chapter the basics of noncommutative geometry à la Connes are laid out. Chapter 2 is designed to serve as a toolkit with several indispensable notions related to spectral functions and their functional transforms. Therein, the delicate notion of an asymptotic expansion is carefully detailed, both in the context of functions and distributions. With Chapter 3 we enter into the hard part of this book, which unveils the subtle links between the existence of asymptotic expansions of traces of heat operators and meromorphic extensions of the associated spectral zeta functions. While trying to stay as general as possible, we illustrate the concepts with friendly examples. Therein, the large-energies asymptotic expansion of the spectral action is presented in full glory. Chapter 4 is dedicated to the important concept of a fluctuation of the operator D by a ‘gauge potential’ and its impact on the action. In terms of physics, this means a passage from ‘pure gravity’ to a full theory vested with the all admissible gauge fields. In terms of mathematics, it involves rather advanced manipulations within the setting of abstract pseudodifferential operators, which we unravel step by step. We conclude in Chapter 5 with a list of open problems, which — in our personal opinion — constitute the main stumbling blocks in the quest of understanding the mathematics and physics of the Spectral Action Principle. We hope that these would inspire the Reader to have his own take on the subject. The bulk of the book is complemented with a two-part Appendix. Section A contains further auxiliary tools from the theory of pseudodifferential operators, including a detailed derivation of the celebrated heat kernel expansion. In Section B we present examples of spectral geometries of increasing complexity: spheres, tori, noncommutative tori and a quantum sphere. This book is devoted to the spectral action, which is only a small offspring in the vast domain of noncommutative geometry. Therefore, when introducing the rudiments of Connes’ theory, we are bound to be brief and focus on the specific aspects related to the spectral action. We refer the Reader to the textbooks for a complete introduction on noncommutative geometry [9, 11, 13, 14, 21]. Let us also warn the Reader that, although we have designed the book to be as self-contained as possible, some mathematical prerequisites are indispensable to grasp the presented advanced concepts. The Reader should be acquainted with the basics of functional analysis, including, in particular, the spectral theory of unbounded operators on Hilbert spaces (e.g. [2, 17, 18]) and the rudiments of operator algebras (e.g. [3, 12]). Some intuitions from global differential geometry (e.g.[16]) and the theory of pseudodifferential operators (e.g. [19]) may also prove useful. Our ultimate purpose is not only to provide a rigid first course in the spectral action, but also to charm the Reader with the marvellous interaction between mathematics and physics encapsulated in this apparently simple notion of spectral action. Let res ipsa loquitur... During the years spent in the realm of noncommutative geometry we have col- laborated with a number of our close colleagues: Driss Essouabri, Nicolas Franco, Victor Gayral, José Gracia-Bondía, Michael Heller, Cyril Levy, Thierry Masson, Tomasz Miller, Andrzej Sitarz, Jo Varilly, Dmitri Vassilevich, Raimar Wulkenhaar, Artur Zaj ˛ac. We also took benefits from discussions with Alain Connes. Moreover, iv Preface

Tomasz was a scrupulous proof reader and Thierry was a great help with the LATEX typesetting. It is our pleasure to cordially thank all of them, as without their kind support this book could not come into being. Finally, we are greatly indebted to our families for their constant support. We acknowledge the ﬁnancial support of the Copernicus Center for Interdisci- plinary Studies in Kraków Poland through the research grant “Conceptual Problems in Uniﬁcation Theories” (No. 60671) awarded by the John Templeton Foundation, and the COST Action MP1405 “Quantum Spacetime”.

Kraków and Marseille, Michał Eckstein and Bruno Iochum May 2018

References

1. Akofor, E., Balachandran, A.P., Joseph, A.: Quantum fields on the Groenewold–Moyal plane. International Journal of Modern Physics A 23(11), 1637–1677 (2008) 2. Birman, M.S., Solomjak, M.Z.: Spectral Theory of Self-adjoint Operators in Hilbert Space. D. Reidel Publishing Co., Inc. (1986) 3. Bratteli, O., Robinson, D.W.: Operator Algebras and Quantum Statistical Mechanics. Springer Science & Business Media (2012) 4. Chamseddine, A.H., Connes, A.: Universal formula for noncommutative geometry actions: Unification of gravity and the standard model. Phyical Review Letters 77, 4868–4871 (1996) 5. Chamseddine, A.H., Connes, A.: The spectral action principle. Communications in Mathe- matical Physics 186(3), 731–750 (1997) 6. Chamseddine, A.H., Connes, A., Marcolli, M.: Gravity and the standard model with neutrino mixing. Advances in Theoretical and Mathematical Physics 11(6), 991–1089 (2007) 7. Chamseddine, A.H., Connes, A., van Suijlekom, W.D.: Grand unification in the spectral Pati– Salam model. Journal of High Energy Physics 11(2015)011 8. Connes, A.: C∗-algèbres et géométrie différentielle. Comptes rendus hebdomadaires des séances de l’Académie des sciences de Paris. Série A 290(13), A599–A604 (1980) 9. Connes, A.: Noncommutative Geometry. Academic Press (1995) 10. Connes, A.: Geometry and the quantum. In: J. Kouneiher (ed.) Foundations of Mathematics and Physics one Century After Hilbert. Springer (2018) 11. Gracia-Bondía, J.M., Várilly, J.C., Figueroa, H.: Elements of Noncommutative Geometry. Springer (2001) 12. Kadison, R.V., Ringrose, J.R.: Fundamentals of the Theory of Operator Algebras. Academic Press (1986) 13. Khalkhali, M.: Basic Noncommutative Geometry. European Mathematical Society (2009) 14. Landi, G.: An Introduction to Noncommutative Spaces and Their Geometries, Lecture Notes in Physics Monographs, vol. 51. Springer Berlin (1997) 15. Marcolli, M.: Noncommutative Cosmology. World Scientific (2017) 16. Nakahara, M.: Geometry, Topology and Physics. CRC Press (2003) 17. Reed, M., Simon, B.: Methods of Modern Mathematical Physics: Vol.: 1.: Functional Analysis. Academic press (1972) 18. Rudin, W.: Functional Analysis. McGraw-Hill (1973) 19. Shubin, M.A.: Pseudodifferential Operators and Spectral Theory. Springer (2001) 20. van Suijlekom, W.D.: Noncommutative Geometry and Particle Physics. Mathematical Physics Studies. Springer (2015) 21. Várilly, J.C.: An Introduction to Noncommutative Geometry. European Mathematical Society (2006) Contents

Preface ...... i References...... iv

1 The Dwelling of the Spectral Action ...... 1 1.1 SpectralTriples ...... 1 1.2 Some Spaces Associated with D ...... 6 1.3 AbstractPseudodifferentialCalculus ...... 15 1.4 DimensionSpectrum...... 18 1.5 NoncommutativeIntegral...... 20 1.6 FluctuationsofGeometry...... 25 1.7 Intermezzo:Quasi-regularSpectralTriples...... 27 1.8 TheSpectralActionPrinciple ...... 28 References...... 33

2 The Toolkit for Computations ...... 39 2.1 SpectralFunctions...... 39 2.2 FunctionalTransforms ...... 44 2.2.1 MellinTransform...... 44 2.2.2 LaplaceTransform...... 46 2.3 Weyl’s Law and the Spectral Growth Asymptotics ...... 51 2.4 PoissonSummationFormula...... 52 2.5 AsymptoticExpansions ...... 56 2.6 ConvergentExpansions ...... 59 2.7 OntheAsymptoticsofDistributions...... 61 References...... 64

3 Analytic Properties of Spectral Functions ...... 67 3.1 FromHeatTracestoZetaFunctions ...... 68 3.2 FromZetaFunctionstoHeatTraces ...... 72 3.2.1 Finite NumberofPolesinVerticalStrips ...... 73 3.2.2 InﬁniteNumberofPoles in VerticalStrips ...... 78

v vi Contents

3.3 ConvergentExpansionsofHeatTraces...... 83 3.3.1 Convergent,Non-exact,Expansions...... 84 3.3.2 ExactExpansions...... 85 3.4 GeneralAsymptoticExpansions ...... 89 3.5 AsymptoticExpansionof the Spectral Action ...... 93 References...... 98

4 Fluctuations of the Spectral Action ...... 101 4.1 Fluctuations of the Spectral ζ-Function ...... 101 4.2 Fluctuations of NoncommutativeIntegrals...... 108 4.3 ConsequencesfortheSpectralAction...... 114 4.4 OperatorPerturbations ...... 117 References...... 120

5 Open Problems ...... 121 References...... 124

A Classical Tools from Geometry and Analysis ...... 127 A.1 About“HeatOperators”...... 127 A.2 Deﬁnition of pdos, Sobolev Spaces and a Few Spectral Properties . . 128 A.3 Complex Parameter-Dependent Symbols and Parametrix...... 131 tP A.4 About e− asapdoandAboutitsKernel ...... 133 tP A.5 The Small-t Asymptotics of e− ...... 139 A.6 Meromorphic Extensions of Certain Series and their Residues ..... 141

B Examples of Spectral Triples ...... 145 B.1 Spheres...... 145 B.2 Tori...... 145 B.3 NoncommutativeTori...... 146 B.3.1 DimensionSpectrum ...... 148 B.3.2 HeatKernelExpansion ...... 149 B.3.3 Spectral Action for NoncommutativeTori...... 150 B.4 PodlesSpheres...... ´ .. 155 References...... 159

Index ...... 161 Chapter 1 The Dwelling of the Spectral Action

Abstract The natural habitat of the spectral action is Connes’ noncommutative geometry. Therefore, it is indispensable to lay out its rudiments encoded in the notion of a spectral triple. We will, however, exclusively focus on the aspects of the structure, which are relevant for the spectral action computations. These include i.a. the abstract pseudodifferential calculus, the dimension spectrum and noncommutative integrals, based on both the Wodzicki residue and the Dixmier trace.

1.1 Spectral Triples

The basic objects of noncommutative geometry à la Connes are spectral triples. As the name itself suggests, they consist of three elements: an algebra A , a Hilbert space H and an operator D acting on H . These three constituents are tied together with a set of conditions, which could be promoted to the axioms of a new — not necessarily commutative — geometry. In the following, we shall denote successively by L (H ), B(H ), K (H ), L 1(H ) the sets of linear, bounded, compact and trace-class operators on H . As for the latter, Tr will always stand for TrH , unless stated explicitly. Definition 1.1. A spectral triple (A ,H ,D) consists of a unital involutive algebra A , with a faithful representation π : A B(H ) on a separable Hilbert space H , and D L (H ) such that: → ∈ D is a (possibly unbounded) selfadjoint operator on H , • [D,π(a)] extends to a bounded operator on H for all a A , • D has a compact resolvent – i.e. (D λ) 1 K (H ) for∈ λ / spec(D). • − − ∈ ∈ Remark that the second assumption requires that π(a)DomD DomD. It is standard to omit the symbol π of the representation when⊂ it is given once and for all. This flexible definition is tailored to encompass the largest possible spectrum of different geometries. However, in order to have workable examples one often has to

1 2 1 The Dwelling of the Spectral Action

take into account the topology of A . To this end, one can, for instance, demand that A is a dense ∗-subalgebra of a C∗-algebra or, more restrictively, a pre-C∗-algebra — if one desires to have a holomorphic functional calculus. The set of axioms adopted in Definition 1.1 is at the core of Connes’ noncommutative geometry. Shortly, we will discuss two additional properties of a spectral triple: p-summability and regularity, which are crucial for the sake of explicit spectral action computations. Before we do so, let us illustrate Definition 1.1 with the canonical example. Example 1.2. Let M be a compact Riemannian manifold without boundary and let P be any elliptic selfadjoint pseudodifferential operator (pdo) of order one on a vector bundle E over M endowed with a hermitian structure. Then, (C∞(M),L2(M,E),P) is a spectral triple. In fact, P has a purely discrete spectrum and its singular values grow to infinity [57, Lemma 1.6.3], so its resolvent is compact. Moreover, since a C∞(M) can be seen as a pdo of order zero, [P,a] is bounded as a pdo of order 0. ∈ The archetype of such situation is when M is spin: Let S be a spinor bundle over M. Let moreover A = C∞(M), H = L2(M,S ) and let D = D/ with

S D/ := iγ µ ∇ − µ – the standard Dirac operator on (M,S ) (cf. [53]). Then, (A ,H ,D) is a spectral triple. As the algebra A in the above example is commutative, the associated spectral triple is also called commutative. A natural question arises: Given a spectral triple with a commutative algebra A , can one recover the underlying manifold? The positive answer is the content of the famous Connes’ Reconstruction Theorem [33]. It requires several additional assumptions on the spectral triple (see [108, Chapter 3] for a pedagogical explanation of these). However, in the noncommutative realm, there are known examples of perfectly workable noncommutative geometries for which some of these additional assumptions are not satisfied [44, 45, 46, 49]. If M is a locally compact Riemannian spin manifold, then the natural associated C∗-algebra A = C0(M) of functions on M vanishing at infinity does not have a unit. Moreover, the Dirac operator on M does not have a compact resolvent. Thus, when A is not unital , the last item of Definition 1.1 needs to be replaced by a(D λ) 1 is compact for all a A and λ / R. • − − ∈ ∈ 2 2 1/2 Equivalently, one can require a(D + ε )− to be compact for any ε > 0, a A . Again, in practice one needs to consider the topological issues. This can be∈ done (see, for instance, [54]) by demanding that A and a preferred unitisation A of A be pre-C∗-algebras, which are faithfully represented on H in terms of bounded operators and [D,a] extends to a bounded operator for every a A . See alsof [98]. To simplify, we stick to the original Definition 1.1 and hence∈ assume from now on that the algebra A is unital, unless explicitly written. f 1.1 Spectral Triples 3

Example 1.3. A basic noncommutative spectral triple is deﬁned by

n AF = Mn(C)(complex n n-matrices), HF = C , DF = D ∗ AF . × F ∈

Since the Hilbert space HF is finite dimensional, finite direct sum of (AF ,HF ,DF )’s as in Example 1.3 are called finite spectral triples (see [76] for a classification). Taking a tensor product of a commutative spectral triple with a finite one results in an almost commutative geometry [73]. There exists an analogue of the Recon- struction Theorem for almost commutative spectral triples [11] allowing one to retrieve a smooth manifold M together with a vector bundle and a connection. Almost commutative geometries are extensively employed in building physical models of fundamental interactions [7, 20, 86, 100, 104]. The purpose of this book is, however, to study the spectral action in full generality of noncommutative geometry, beyond the almost commutative realm. We refer the Reader to the textbook [104] for a friendly introduction to almost commutative geometries and their physical applications. Example 1.4. Other illustrative examples are given by the noncommutative tori and the Podles´ spheres – see Sections B.3 and B.4 of Appendix B.

Remark 1.5. Given a spectral triple (A ,H ,D) one can always obtain a new one (A ,H ,DV ), with DV = D +V, V = V B(H ) and DomDV := DomD. Indeed, ∗ ∈ (A ,H ,DV ) is a spectral triple as [D +V,a] extends to a bounded operator for any a A whenever [D,a] does so. Furthermore, if z is in the resolvent of DV and z′ is in∈ the resolvent of D then,

1 1 1 (D +V z)− = (D z′)− 1 (V + z′ z)(D +V z)− − − − − − is compact since the ﬁrst term is compact and the second one is bounded. However, the geometry of (A ,H ,DV ) need not a priori be related to the one of (A ,H ,D). If we want to obtain a geometrywhich is in a suitable sense equivalent, the perturbation V has to acquire a precise form (see Section 1.6).

On the technical side, we need to take into account the fact that D may be non- invertible. We adopt the following convention

D := D + P0, (1.1) where P0 is the projection on KerD H . The operator P0 is a finite-rank (i.e. ⊂ dimImP0 < ∞) selfadjoint operator on H and D is an invertible operator with a compact resolvent. Thus, by the previous remark, (A ,H ,D) is also a spectral 1 triple. Moreover, notice that D = D + P0 and D − is compact. Another possibility is to define| | (see| | for e.g. [38| ])| an invertible selfadjoint operator D as the restriction of D to the Hilbert subspace (1 P0)H . Yet another option, chosen in [13], is to work with the invertible operator (−1 + D 2)1/2. The selection of a prescription to cook up an invertible operator for D is only a matter of convention (cf. [13, Section 6], [51, Remark 3.2]). However, one has to 4 1 The Dwelling of the Spectral Action stay vigilant, as different choices may affect the associated spectral functions (see, for instance, Formula (2.8)). The first vital property of a spectral triple we shall need is a ‘finite dimensional- ity’ condition: Definition 1.6. A spectral triple (A ,H ,D) is finitely summable or, more precisely, p p-summable for some p 0 if Tr D − < ∞. Note that p-summability≥ yields (|p +| ε)-summability but not (p ε)-summability for ε > 0, so we define: − The triple is said of dimension p (or p-dimensional) when

q p := inf q 0 Tr D − < ∞ < ∞. { ≥ | | | } This notion of dimension will be refined in Definition 1.23. Remark that finite spectral triples (cf. Example 1.3) are always 0-dimensional. But spectral triples with dimH = ∞ can also have dimension 0 — like, for instance, the standard Podles´ sphere (see Appendix B.4). Example 1.7. Let (A ,H ,D) be a commutative spectral triple based on a Rieman- nian manifold of dimension d, then (A ,H ,D) is d-dimensional [60, p. 489].

tD2 A slightly weaker notion is the one of θ-summability: Tre− < ∞ for t > t0 0 [29, Chapter 4, Section 8]. Every finitely summable spectral triple is θ-summable≥ with t0 = 0, but the converse is not true (cf. [27]and [29, Chapter 4]). The geometry of spectral triples which are not finitely summable is, however, too poor to accommodate various analytical notions (cf. Section 1.4), which are indispensable for the spectral action computations. On the other hand, note that we do not require p to be an integer. For instance, fractal spaces are fruitfully described via p-summable spectral triples with p being the (irrational) Hausdorff dimension of the fractal [5, 8, 22, 23, 24, 25, 36, 61, 62, 63, 74, 79, 80]. The next key property encodes the notion of smoothness: Definition 1.8. A spectral triple (A ,H ,D) is regular if

n a A a,[D,a] Domδ ′ , where δ ′ := [ D , ]. (1.2) ∀ ∈ ∈ n N | | · ∩∈

The map δ ′ is an unbounded derivation of the algebra B(H ). Recall that the notation T Domδ ′ means that T preserves Dom D and δ ′(t) = [ D ,T ] extends (uniquely)∈ to a bounded operator on H . We note| that| some authors| | refer to assumption (1.2) as smoothness [98, Deﬁnition 11] or QC∞ (Q for “quantum”) [13, Deﬁnition 2.2]. The regularity assumption allows one to equip A with a topology generated by k k the seminorms a δ ′ (a) and a δ ′ ([D,a]) . The completion of A in this topology yields a7→ Fréchet pre-C -algebra7→ A and (A ,H ,D) is again a regular ∗ δ ′ δ ′ spectral triple [98] (see also [60, p. 469], [108, Section 3.4] and [13]). 1.1 Spectral Triples 5

A commutative spectral triple (A ,H ,D) based on a Riemannian manifold is regular. Condition (1.2) assures that the functions constituting A are indeed smooth. In this case, A = C∞(M) also as topological spaces [98, Proposition 20]. δ ′ ∼ In some of the approaches (in particular, in almost commutative geometries) it is desirable to encode in the axioms of a spectral triple the fact that the classical Dirac operator is a first order differential operator [108, Section 3.3]. Such a demand was originally used to restrict admissible Dirac operators for almost commutative geometries [20, 47] and restrain the free parameters of the underlying physical models. On the other hand, in recent studies, it is argued that one does not actually need the first-order condition defined below and there exist examples of spectral triples for which it is not satisfied (see [21] and references therein). Consequently, we will not assume the first-order condition to hold throughout this book. Nevertheless, we shall state it and explain its origin. To this end we need the following notions: Definition 1.9. A spectral triple (A ,H ,D) is even if there exists a selfadjoint unitary operator γ on H such that γ2 = 1, γD = Dγ and γa = aγ for all a A . Otherwise, the spectral triple is odd. − ∈ Definition 1.10. A spectral triple (A ,H ,D) is real of KO-dimension d Z/8 if there is an antilinear isometry J : H H called the reality operator such∈ that → 2 JD = ε DJ, J = ε′, and Jγ = ε′′ γJ when the triple is even, with the following table for the signs ε,ε′,ε′′

d 01234567 ε 1-11 1 1-111 (1.3) ε′ 1 1 -1-1-1-1 1 1 ε′′ 1 -1 1 -1 and the following commutation rule (see [35]or[60, Section 9.5] for the details)

1 [a, Jb∗J− ]= 0, a,b A . (1.4) ∀ ∈ For a spectral triple based on a Riemannian manifold of dimension d the KO- dimension is just d mod 8. It encodes the fact that the Dirac operator is a square root of the Laplacian, what generates a sign problem corresponding to the choice of a spin structure (and orientation). In this context the operator J plays the role of a charge conjugation for spinors (see [60, Section 5.3] or [109]) and it encodes the nuance between spin and spinC structures. On the other hand, given a spectral triple (A ,H ,D) with a noncommutative algebra one can usually find different reality operators leading to real spectral triples with different KO-dimensions (see [76]). The operator J takes its origin in the modular theory of von Neumann algebras (see [29, Chapter 1, Section 3 & Chapter 5]) and has interesting applications in the algebraic quantum field theory [65, Chapter V]. In the context of spectral triples, with the help of J one can define a representation on H of the opposite algebra A op , which is isomorphic to A as a vector space, but 6 1 The Dwelling of the Spectral Action

op the multiplication in A is inverted, i.e. a A op b := b A a. Given a representation π : A B(H ), deﬁne the representation•πo : A op •B(H ) by → → o 1 π (a) := Ja∗J− .

1 The condition [π(a),Jπ(b∗)J− ]= 0 for a,b A means that the two representations commute [π(A ),πo(A op)] = 0, hence πo(A∈ op) is in the commutant of π(A ) in B(H ). If A is commutative, A = A op and this requirement becomes trivial. See, for instance, [108, Chapter 3] for more details. We are now ready to formulate the announced first-order condition. Definition 1.11. A real spectral triple (A ,H ,D) meets the first-order condition if

1 [[D,a],Jb∗J− ]= 0, a,b A . (1.5) ∀ ∈ As mentioned above,it encodesthe fact that for a commutative spectral triple D = D/ is a first order differential operator. The reason for the appearance of the representation of A op in (1.5) is that [D, ] is not a derivation from A to itself, but rather to the commutant of A op [21]. The· first-order condition plays an important role in the study of the spectral action fluctuations — see Section 1.6. Finally, we remark that each of the properties of being even, real, regular or p- summable can be extended to the non-unital framework [54, 98, 99].

1.2 Some Spaces Associated with D

In this section we discuss the pseudodifferential calculus suitable for noncommutative geometries. We essentially base on the classical papers[28, 37], however some of the definitions are taken from more recent works [13, 49, 67, 69]. Let us first define the following scale of spaces for a parameter s R ∈ H s := Dom D s. (1.6) | | If s 0, H s = H 0 = H and for s 0, H s are Hilbert spaces for the Sobolev norm≤ ≥

ξ 2 := ξ 2 + D sξ 2 k ks k k k| | k and H s+ε H s H 0 for s,ε 0 ⊂ ⊂ ≥ .since the injection H s+ε ֒ H s is continuous Let us note, that we do→ not need the invertibility of D since H s = Dom D s | | for s 0 as DP0 = P0D = 0 and P0 B(H ). Actually, for s 0 we also have H s =≥Dom(1+D 2)s/2 as shown in [13∈, Section 6]. On the other≥ hand, the Sobolev norms will not be the same if we swap D for D or (1 + D 2)1/2. In the second | | | | 1.2 Some Spaces Associated with D 7 case the norms will not coincide even if KerD = 0 . However, the precise form of the Sobolev norms for H s is not relevant and, as{ already} stressed, the choice of D instead of (1 + D 2)1/2 is only a matter of convention. | | We also deﬁne the domain of smoothness of D as

H ∞ := H s = H k. (1.7) s 0 k N ∩≥ ∩∈ H ∞ is dense in H and is in facta core (see [97, p. 256] for a precise definition) for D [98, Theorem 18]. Actually, it is sufficient to consider H k with k N [60, p. 467] (see also [109, Definition 6.11]). H ∞ can be equipped with a topology∈ induced by the seminorms k, which makes it a Fréchet space. With the spacesk·k H s at hand, we define the following classes of unbounded operators on H for any r R (compare [28, Section 1], [13, Definition 6.1]): ∈ opr := T L (H ) Dom T H ∞ and for any s r ∈ ⊃ ≥ H ∞ H ∞ T maps ( , s) continuously into ( , s r) . (1.8) k·k k·k − For instance r R, D r opr. Any operator in opr extends to a bounded map from H s to H s∀ r ∈s |r and| ∈ since H 0 = H , we have − ∀ ≥ op0 B(H ). ⊂ Moreover,

opr ops if r s, opr ops opr+s for all r, s R. (1.9) ⊂ ≤ · ⊂ ∈

In the previous section we met a map δ ′ on B(H ), the domain of which played a pivotal role in the axiom of regularity. For further convenience we also deﬁne:

2 1 δ( ) := [ D , ], ∇( ) := [D , ], σ( ) := D D − · | | · · · · | | · | | 1 2 E( ) := δ( ) D − E ( ) := ∇( ) D − . · · | | · · | | Let Ξ δ,∇,σ,E,E . Recall that if T DomΞ B(H ), then Ξ(T ) B(H ). Actually,∈{ one can extend} any Ξ to a map∈ on T L⊂ (H ) DomT H ∞∈ . By the habitual abuse of notation we denote the extensions{ ∈ of Ξ’s| with the⊃ same} symbols. However, one has to stay vigilant as in the following DomΞ will always mean a subset of B(H ). Furthermore, for any T L (H ) with DomT H ∞ we deﬁne the following one-parameter group of (unbounded)∈ operators: ⊃

z z σz(T ) := D T D − , for z C, (1.10) | | | | ∈ using the Cauchy integral along a curve C C (see (A.3)) to deﬁne ⊂ z 1 z/2 2 1 D − = i2π λ − (λ D )− dλ. (1.11) | | λ C − Z ∈ 8 1 The Dwelling of the Spectral Action

n Note that for T Domσ we have σ (T )= σn(T ) for all n N. ∈ ∈ n n 0 Lemma 1.12 ([37, 13]). We have Dom δ ′ = Dom δ op . n N n N ⊂ ∩∈ ∩∈ Proof. Since D = D + P0, Domδ ′ = Domδ because P0 is bounded and we get the equality of| the| lemma.| | Let now T Dom δ n. One checks that σ = id+E and, for every n N, ∈ n N ∈ ∩∈ n E n n n n E n n k k (T )= δ (T ) D − and σ (T ) = (id+ ) (T )= ∑ ( k ) δ (T ) D − . (1.12) | | k=0 | |

Hence, E n(T ) and σ n(T ) are bounded. Similarly,

n n k n k k σ − (T )= ∑ ( 1) k D − δ (T ) (1.13) k=0 − | | is bounded too. Thus, for n Z and ξ H ∞, ∈ ∈ T ξ 2 = Tξ 2 + D nT ξ 2 = T ξ 2 + σ n(T ) D n ξ 2 k kn k k k| | k k k k | | k c( ξ 2 + D n ξ 2)= c ξ 2 . ≤ k k k| | k k kn The case of an arbitrary n R follows by the interpolation theory of Banach spaces (cf. [60, Formula (10.65)]∈ and [29, Chapter 4, Appendix B]). We now introduce yet another class of operators on H (cf. [37],[13, Def. 6.6]):

0 n r r 0 OP := Dom δ , OP := T L (H ) D − T OP for r R. (1.14) n N { ∈ | | | ∈ } ∈ ∩∈ The definition of OPr is symmetric: When r N, Equations (1.12) yields ∈ r r 0 r 0 r r 0 r OP = D OP = D OP D − D OP D . (1.15) | | | | | | | | ⊂ | | When T OPr, we say that the order of T is (at most) r and write ordT := r. Since∈H ∞ is dense in H , the operatorsin opr (and a fortiori in OPr) are densely defined and we can define

r r (OP )∗ := T ∗ T OP . { | ∈ } We have r r (OP )∗ = OP , n n n 0 0 what follows from the observation that (δ (T ))∗ = ( 1) δ (T ∗), so (OP )∗ = OP for n N, and the symmetry (1.15) ofOPr. − Note∈ also that

k P0 OP− for all k R, (1.16) ∈ ∈ 0 k because P0 is trivially in OP and P0 D = P0 B(H ) for any k 1. | | ∈ ≥ 1.2 Some Spaces Associated with D 9

A OP0, [D,A ] OP0 . (1.17) ⊂ ⊂ 4 3 In that case, for instance, a D [D,b]D− OP− with a,b A . In [37] it was proved that| operators| of order∈ 0 admit another∈ characterisation: ≤ 0 ∞ OP = T t Ft (T ) C (R,B(H )) , { | 7→ ∈ } it D it D where Ft (T ) := e | |Te− | | for t R, which is reminiscent of the geodesic ﬂow [28, 35],[40, Chapter 8]. Recall that∈

Ft (T ) F (T) δ(T ) = [ D ,T]= strong-lim − 0 . | | t 0 it → Proposition 1.13. Let r, s R and z C. Then i) OPr opr. ∈ ∈ ⊂ r r ii) σz(OP ) OP . iii) OPr OPs⊂ OPr+s. iv) OPr OP⊂s when r s. v) δ(OP⊂r) OPr. ≤ r ⊂ r+1 r r 1 vi) ∇(OP ) OP and E (OP ) OP − . r ⊂s r+s 1 ⊂ vii) [ D ,OP ] OP − . | | r⊂ 1 r 1 r viii) If T OP , r 0 has an inverse with T − D B(H ), then T − OP− . ix) OP0 ∈B(H )≥and OP 1 K (H ). | | ∈ ∈ ⊂ − ⊂ Proof. i) This follows from OP0 op0 (Lemma 1.12), D r opr and (1.9). 0 k⊂ k | |0 ∈ ii) When T OP , since D − and δ (T) are also in OP for any k N, Formulae ∈ | | n 0 ∈ (1.12)and (1.13) tell us that for any n Z, σn(T )= σ (T ) OP . ∈m ∈ Let us ﬁx m N and let Fm : z C δ (σz(T )) L (H ). Since Fm(n) is bounded ∈ ∈ 7→ ∈ when n Z, a complex interpolation shows that Fm(z) is bounded for z C, so that ∈ 0 ∈ σz(T ) OP . ∈ r r 0 r r When T OP , T = D T ′ with T ′ OP , thus σz(T )= D σz(T ′) OP . ∈ r | | s ∈ r s| | ∈ 0 iii) Let T OP and T ′ OP . Then D − T and D − T ′ are in OP and by ii) s r ∈ s 0 ∈ (r+s) | | (|r+|s) s s 0 D − ( D − T) D OP . So D − T T ′ = ( D − T D )( D − T ′) OP . | | | | | | ∈ r s | | | | | |0 | | ∈ r iv) When s r, D − is a bounded operator and is in OP . Thus if T OP , s r ≥s | r| 0 ∈ D − T = D − ( D − T ) OP by iii). | |v) If T | OP| r, then| | T = ∈D r S with S OP0 and δ(T )= δ( D r)S+ D r δ(S), so the result follows∈ from δ(OP| 0|) OP0 and∈ iii). | | | | vi) Let T OPr. Since ∇(T )=⊂ δ(T ) D + D δ(T ), we get ∇(T ) OPr+1 from properties iii∈) and v). | | | | ∈ 2 r+1 2 r 1 Moreover, E (T )= ∇(T ) D − OP OP− OP − using v) and iii). | | ∈ ⊂ 10 1 The Dwelling of the Spectral Action

r 0 r 1 vii) By iii) it is sufﬁcient to prove that [ D ,OP ] OP − . This is true for r N 1 | | 1 ∈ 1 ∈ using v) and for r N using [A− ,B]= A− [A,B]A− . Finally, this also holds true for an arbitrary∈−r R by interpolation —− as in the proof of ii). ∈ 1 1 1 viii) Assume r = 0. Remark that δ(T − )= T − δ(T )T − is bounded. Thus 2 1 1 2 1 1 1 − 1 1 δ (T − )= T − δ (T )T − δ(T − )δ(T )T − T − δ(T )δ(T − ) is bounded and, − n 1 − − 1 0 by induction, δ (T − ) is bounded for any n N∗. Thus T − OP . r 0 ∈ 1 ∈ 1 r Now for r > 0, T = D S where S OP is such that S− = T − D is bounded by hypothesis, thus in OP| 0| by previous∈ argument and the claim follows| | from iii). 0 0 1 ix) By i), we have OP op B(H ). Since D − is a compact operator, the last assertion follows. ⊂ ⊂ | |

Let us stress that the op classes are, in general, strictly larger than OP’s. E.g. on the standard Podles´ sphere (cf. Appendix B.4), which is not regular, we have a op0, 0 ∈ but a / OP for any a Aq, a = c1. ∈ ∈ 6 The following, innocent-looking lemma, will prove very handy in practice: Lemma 1.14. Let T L (H ). Assume that there exist r R and 0 ε < 1 such N ∈ N r n ∈ ≤r N 1+ε that for any N , T = ∑n=0 Pn + RN, with Pn OP − and RN OP − − . ∈ r N 1 ∈ ∈ Then, actually, RN OP − − . ∈ r N 2+ε r N 1 Proof. We have RN = PN+1 + RN+1. Because RN+1 OP − − OP − − and r N 1 ∈ r N 1 ⊂ PN 1 OP − − we conclude that RN is actually in OP − − . + ∈ We will often need to integrate operators(like in Formula(1.11)) froma givenOP class and we will need a control of the order of the integral. A typical example is the ai following: For i 1,...,n , let us be given Ai(λ) OP− with ai R, commuting ∈{ bi } c ∈ ∈ with D , and Bi OP , C(λ) OP− with bi,c R, which do not necessarily | | ∈ ∈R ∈ r commute with D . For some I , we will need to show that I R(λ)dλ OP for some r R, when| | R(λ)= A (λ⊂)B A (λ)B C(λ). ∈ 1 1 n n R Let us∈ deﬁne ··· a := a1 + ... + an and b := b1 + ... + bn. a c+b 0 Assuming b a+c, we ﬁrst remark that the integrand R(λ) is in OP− − OP , ≤ a+c b ⊂ so it is bounded. We now decompose D − R(λ) in the following way: k k | | For αk = ∑ j=0 a j, βk = ∑ j=0 b j, with a0 = b0 = 0, we have

a+c b D − R(λ)= E1(λ)F1 En(λ)Fn G(λ), | | ··· where

(a+c αk 1) (b βk 1) (a+c αk)+(b βk 1) ak Ek(λ) := D − − − − − Ak (λ) D − − − − = D Ak(λ), | | | | | | (a+c αk) (b βk 1) (a+c αk)+(b βk) Fk := D − − − − Bk D − − − , | | | | G(λ) := D cC(λ). | | 0 Remark that all operators Ek(λ), Fk,G(λ) are in OP , and hence bounded. 2 1 Typically, the operators Ak(λ) would be of the form (D + λ)− , which gives 2 1 Ak(λ) OP− and Ak(λ)= ∞(λ ). So, to assure the integrability of R(λ) for ∈ O − 1.2 Some Spaces Associated with D 11

I = R+ := [0,∞) we usually have to sacriﬁce a few orders of OP. 2 2 4 r For example, with R(λ) = (D +λ)− OP− ,weget D R+ R(λ)dλ < ∞ only for r < 2. ∈ k| | k R

Theorem 1.15. Let I R. Assume that there exist r R and n N∗ such that for any λ I a given operator⊂ R(λ) L (H ) can be decomposed∈ as∈ ∈ ∈ r D R(λ)= E1(λ)F1 En(λ)Fn G(λ), (1.18) | | ··· 0 with Ek(λ),Fk,G(λ) OP and [ D ,Ek(λ)] = 0. If ∈ | | m E1(λ) En(λ) δ (G(λ)) dλ < ∞ for m N, I k k···k k·k k ∈ Z r then R(λ)dλ OP− . I ∈ Z Proof. We need to show that

r R r 0 ∞ m D := D R(λ)dλ OP = m=0 Domδ . | | | | I ∈ ∩ Z For m = 0, we have

n n r R D (∏ Fk ) ∏ Ek(λ) G(λ) dλ < ∞ k| | k≤ k k I k k·k k k=1 Z k=1 by hypothesis. Hence, D r R is bounded. An application of δ to |D|r R generates from the integrand D r R(λ) a ﬁnite linear | | | | combination of terms like those in (1.18) with either one of the Fk’s replaced by 0 δ(Fk) or G(λ) replaced by δ(G(λ)), as δ(Ek)= 0. Since Fk OP , the norm of m m r ∈ δ (Fk) is ﬁnite for any m N and the norm of δ ( D R) can be estimated along the same lines as for the case∈ m = 0. | |

As the ﬁrst application of Theorem 1.15 we derive an operator expansion (1.22), which will play a pivotal role in the construction of an abstract pseudodifferential calculus. We will need the following technical lemma (cf. [13, Lemma 6.9]):

Lemma 1.16. Let T OPr for an r R and let λ / specD2. Then, n N , N N, ∈ ∈ ∈ ∀ ∈ ∗ ∈ N 2 n j n+ j 1 j 2 ( j+n) N+1 (D λ)− T = ∑ ( 1) j− ∇ (T )(D λ)− + ( 1) RN (λ,n), (1.19) − j=0 − − − n k+N 1 2 k n 1 N+1 2 (k+N) r 2n N 1 RN (λ,n) := ∑ N− (D λ) − − ∇ (T )(D λ)− OP − − − . k=1 − − ∈ Proof. The proof follows by pure combinatorics with an induction on n and N. 1) Let us ﬁrst take n = 1 and N = 0. We have 12 1 The Dwelling of the Spectral Action

2 1 2 1 2 1 (D λ)− T = T(D λ)− + [(D λ)− ,T ] − − − 2 1 2 1 2 1 = T(D λ)− (D λ)− ∇(T )(D λ)− , (1.20) − − − − which is precisely the Formula (1.19) for n = 1, N = 0. 2) Let us now assume that Formula (1.19) holds for N = 0 and a ﬁxed n N∗, i.e. ∈ n 2 n 2 n 2 k n 1 2 k (D λ)− T = T (D λ)− ∑ (D λ) − − ∇(T )(D λ)− . (1.21) − − − k=1 − − We show that Equation (1.21) holds also for n + 1: Applying Equation (1.20) again

n 2 n 1 2 1 2 n 2 k n 1 2 k (D λ)− − T = (D λ)− T(D λ)− ∑ (D λ) − − ∇(T )(D λ)− − − − − k=1 − − 2 n 1 2 1 2 n 1 = T(D λ)− − (D λ)− ∇(T )(D λ)− − − − − − n 2 k n 2 2 k ∑ (D λ) − − ∇(T )(D λ)− . − k=1 − − 3) If (1.19) holds for any n N and a ﬁxed N N, it is sufﬁcient to show that ∈ ∗ ∈ n+N N+1 2 (N+n+1) RN (λ,n)= ∇ (T)(D λ)− RN 1(λ,n). N+1 − − + Indeed, by assumption we have

n k+N 1 2 k n 1 N+1 2 (k+N) RN(λ,n)= ∑ N− (D λ) − − ∇ (T )(D λ)− k=1 − − n k+N 1 N+1 2 (n+N+1) = ∑ N− ∇ (T )(D λ)− k=1 − n n+1 k k+N 1 − 2 j+k n 2 N+2 2 ( j+k+N) ∑ N− ∑ (D λ) − − ∇ (T)(D λ)− − k=1 j=1 − − n+N N+1 2 (N+n+1) = ∇ (T )(D λ)− N+1 − n k j+N 1 2 k n 1 N+2 2 (k+N+1) ∑ ∑ N− (D λ) − − ∇ (T)(D λ)− , − k=1 j=1 − − ℓ j+N 1 ℓ+N which is the claimed equation, since ∑ j=1 N− = N+1 and in the third equality we commuted the sums and changed the summation index j + k 1 k. − r 2n N 1 Lastly, the use of Proposition 1.13 iii), vi) shows that RN (λ,n) OP − − − . ∈ Theorem 1.17. Let T OPr for some r R. Then, for any z C and any N N, ∈ ∈ ∈ ∈ N z ℓ 2ℓ r (N+1) σ2z(T )= ∑ ℓ ∇ (T ) D − + RN(z), RN(z) OP − . (1.22) ℓ=0 | | ∈ 1.2 Some Spaces Associated with D 13

Here z := z(z 1) (z n + 1)(n!) 1 with the convention z = 1, n − ··· − − 0 Proof. For z C we decompose z = n + s, with n = ℜ(z) Z and ℜ(s) < 1 2∈z 2s 2n ⌊ ⌋ ∈ yielding D − T = D − D − T . Let us| ﬁrst| assume| | that|ℜ|(z) > 0 and ℜ(s) > 0. Hence, n N. For any N N Formula (1.19) with λ = 0 implies ∈ ∈

N 2z j n+ j 1 2s j 2( j+n) N+1 2s D − T = ∑ ( 1) j− D − ∇ (T ) D − + ( 1) D − RN(0,n). | | j=0 − | | | | − | | Now, for 0 < ℜ(s) < 1, we invoke the operator identity

∞ 2s sin(πs) 2 1 s D − = π (D + λ)− λ − dλ. | | 0 Z For any j N, Formula (1.19) with λ R+ specD2, n = 1,N j N yields ∈ − ∈− 6⊂ − − ∈ ∞ 2s j sin(πs) 2 1 j s D − ∇ (T )= π (D + λ)− ∇ (T )λ − dλ | | 0 Z N j ∞ sin(πs) − k j+k 2 k 1 s = π ∑ ( 1) ∇ (T ) (D + λ)− − λ − dλ + RN′ , j(s) − 0 k=0 Z N j sin(πs) − k Γ (1 s)Γ (s+k) j+k 2(s+k) = π ∑ ( 1) Γ−(k+1) ∇ (T ) D − + RN′ , j(s), k=0 − | | with ∞ N j+1 sin(πs) 2 1 N+1 2 N+ j 1 s RN′ , j(s) = ( 1) − π (D + λ)− ∇ (T )(D + λ)− − λ − dλ. − 0 Z Combining the formulae from above we obtain

2z D − T | | N N j − j+k n+ j 1 sin(πs) Γ (1 s)Γ (s+k) j+k 2(s+k+ j+n) R = ∑ ∑ ( 1) j− π Γ−(k+1) ∇ (T ) D − + (z) j=0 k=0 − | | N N j − j+k n+ j 1 Γ (s+k) j+k 2(k+ j+z) R = ∑ ∑ ( 1) j− Γ (k+1)Γ (s) ∇ (T ) D − + (z) j=0 k=0 − | | N ℓ ℓ n+ℓ k 1 k+s 1 ℓ 2(ℓ+z) R = ∑ ∑ ( 1) ℓ−k− k− ∇ (T ) D − + (z) ℓ=0 k=0 − − | | N N ℓ z+ℓ 1 ℓ 2(ℓ+z) R z ℓ 2(ℓ+z) R = ∑( 1) ℓ− ∇ (T ) D − + (z)= ∑ −ℓ ∇ (T ) D − + (z), ℓ=0 − | | ℓ=0 | | where along the road we employed Euler’s reﬂection formula [1, (6.1.17)] and the Chu–Vandermonde identity [1, (24.1.1)], and the global remainder reads 14 1 The Dwelling of the Spectral Action

N R j n+ j 1 2( j+n) N+1 2s (z) := ∑ ( 1) j− RN′ , j(s) D − + ( 1) D − RN (0,n). (1.23) j=0 − | | − | | N z ℓ 2ℓ R 2z We have shown that σ 2z(T )= ∑ℓ=0 −ℓ ∇ (T ) D − + (z) D , under the assumption that ℜ(z) > 0− and ℜ(s) > 0, but the result| can| be extended| | to any z C on the strength of the uniqueness of the holomorphic continuation. Thus, a swap∈ z z would complete the proof, provided we can control the remainder R(z). The− second term of (1.23) is easy to handle: From Lemma 1.16 and Proposi- 2s r 2ℜ(n+s) N 1 tion 1.13 iii) we deduce D − RN(0,n) OP − − − . The ﬁrst term of R(z)|, | ∈

N R N+1 sin(πs) n+ j 1 R 1(z) := ( 1) π ∑ j− 1, j(z) − j=0 ∞ 2 1 N+1 2 N 1+ j 2(n+ j) s with R1, j(z) := (D + λ)− ∇ (T )(D + λ)− − D − λ − dλ, 0 | | Z is more delicate. We will handle it with the aid of Theorem 1.15. For any ﬁxed j 0,1,...,N and with x = ℜ(s), we show that the operator 2x 2ε ∈{N+1 r+2n } D − R1 j(z) D − is bounded for some ε > 0: | | , | | ∞ 2x 2ε 2 1 N+1 2 N 1+ j N+1 r 2 j s D − (D + λ)− ∇ (T)(D + λ)− − D − − λ − dλ | | 0 | | Z ∞ 2 1 1 x+ε 2 1 2 x ε c1 + (D + λ)− − (D + λ)− D − ≤ 1 k k × Z N+1 N 1 r 2 1 2 N+1 j s ∇ (T ) D − − − (D + λ)− D − λ − dλ ∞ ×k | | kk k 1+x ε x c1 + c2 λ − − λ − dλ < ∞. ≤ 1 Z This computation is sound if 1 x + ε > 0 and x ε > 0. The former is always true, since x = ℜ(s) < 1, whereas in− the second one x >−0 andwe can alwaystunethe N+1 N 1 r 0 ε to be small enough. Note also that ∇ (T ) D − − − OP for any N and the convergenceof the integral of norms at 0 is automatic| | since∈ 0 < (D2 +λ) 1D2 1. − ≤ As we kept j free and R1(z) involves only a ﬁnite sum over j, we have actually 2x 2ε N+1 r+2n proven that D − R1(z) D − < ∞ for an arbitrarily small ε > 0. | | | | To show that R z OPr 2ℜ(z) (N+1)+2ε, so that R z OPr (N+1)+ε , we in- 1( ) − − N ( ) − voke Theorem 1.15 with∈ − ∈

2x 2ε 2 1 E1(λ)= D − (D + λ)− , | | 2 N 1+ j 2(N+1 j) E2(λ) = (D + λ)− − D − , N+1 N 1 r F1 = ∇ (T ) D − − − , | | s G(λ)= λ − .

Finally, to get rid of the ε we invoke the handy Lemma 1.14. 1.3 Abstract Pseudodifferential Calculus 15

This theorem was first proven in [28, 37] for T being a pseudodifferential operator (see Definition 1.18). But it only uses the fact that T OPr for r R. This formulation was first given by Nigel Higson [66] with another∈ proof (see∈ also [13]).

1.3 Abstract Pseudodifferential Calculus

In the previoussection we were only concernedwith an abstract unboundedoperator D on a Hilbert space H . We now let the algebra A enter into the game and describe the abstract pseudodifferential calculus associated with a spectral triple (A ,H ,D). We shall follow the conventions adopted in [69] (see also [49, 67]). Define first P(A ) := polynomial algebra generated by A , D, D and JA J 1 when J exists. | | − Definition 1.18. Given a regular spectral triple (A ,H ,D), one defines the set of pseudodifferential operators (pdos) as

N Ψ(A ) := T L (H ) N N P P(A ), R OP− , p N, ∈ ∀ ∈ ∃ ∈ ∈ ∈ p such that T = P D − + R . (1.24) | | In particular D r Ψ(A ) for any r R. | | ∈ ∈ N An operator T is smoothing if T OP− for all N N. Remark that any smoothing operator is automatically in Ψ(A∈ ) with P = 0 for∈ each N in Formula (1.24).

Example 1.19. Here are few other examples of smoothing abstract pdos: i) The operator P0 deﬁned in (1.1) is smoothing thanks to property (1.16). For a generalisation see (4.1). ii) If f is a Schwartz function, then f ( D ) is a smoothing pdo: We claim that for N n | | N any N N, D f ( D ) n N Dom δ , i.e. f ( D ) OP− for all N N. Indeed, the function∈ |x | R+| | x∈N ∩f (x∈) is bounded, whereas| | δ∈n( f ( D )) = 0 for∈n 1. iii) Let P be∈ a positive→ invertible operator in OPr with| | r > 0 and such≥ that 1 r tP P− D B(H ). Then, the operator A = e− is a smoothing pdo for all t > 0. To| prove| ∈ this we use Formula (A.3) to write

i tλ 1 A = e− (P λ)− dλ. 2π C − Z N N N/r N/r We ﬁrst observe that D A D P− P A < ∞ for any N N since N N/r 0 k| | k ≤ k| | kk k +∈ s tx D P− OP by Proposition 1.13 iii), viii) and the function x R x e− |is bounded| for∈ any s 0, t > 0. ∈ → Secondly, we need to≥ show that A is in Domδ n for any n N. To this end, we observe that ∈ i tλ 1 1 δ(A)= − e− (P λ)− δ(P)(P λ)− dλ 2π C − − Z converges in norm because 16 1 The Dwelling of the Spectral Action

1 1 1 δ(P)(P λ)− δ(P)P− P(P λ)− k − k≤k k·k − k 1 1 and moreover P(P λ)− λ ℑ(λ)− for ℜ(λ) 0 (or 1 if ℜ(λ) < 0) cf. k − k ≤ | | ≥ ≤ 1 (A.6), so that we can choose the curve C such that P(P λ)− is uniformly bounded for λ C . Thus the norm of δ(A) is estimatedk in− the samek way as the ∈ n tP 0 norm of A. The argument extends to any δ (A) showing that e− OP for any t > 0. ∈ To see that A is smoothing amounts to showing that the operator D N A is in OP0 for any N N but we have already checked that it is bounded. | | It is in the∈ domain of δ because

N N N N/r N/r δ( D A)= D δ(A) = [ D P− ][P δ(A)] | | | | | | and both terms are bounded: In particular for any s 0 ≥ s s s 1/r s+1/r s 1/r 1/r P δ(A) P D P− − P A + P AP P− D < ∞. k k≤k | | kk k k kk | |k Similar arguments show that D N A Domδ n. | | ∈ n N ∩∈ For T,T Ψ(A ), we deﬁne the equivalence T T if T T is smoothing. ′ ∈ ∼ ′ − ′ Lemma 1.20. The set Ψ(A ) is a Z-graded involutive algebra with

Ψ k(A ) := Ψ(A ) OPk, for k Z. (1.25) ∩ ∈ k k k Moreover, δ(Ψ (A )) Ψ (A ) and P(A ) Ψ(A ) k Z OP . ⊂ ⊂ ⊂ ∪ ∈ Proof. Only the stability under the product deserves a bit of attention. Let us take the pdos T Ψ k(A ) and T Ψ k′ (A ) with k,k Z. For any N,N N we can ∈ ′ ∈ ′ ∈ ′ ∈ p p′ write T = P D − + R and T ′ = P′ D − + R′, where p, p′ N, P,P′ P(A ) and p | k | p′ k | | N N ∈ ∈ P D − OP , P′ D − OP ′ , R OP− ,R′ OP− ′ . | | ∈ | | ∈ ∈ ∈ N We claim that, for any N′′ N, there exist p′′ N,P′′ P(A ) and R′′ OP− ′′ ∈ p′′ ∈ ∈ ∈ such that we have T T ′ = P′′ D − + R′′. Using (1.22) we get, for any|M| N, ∈ M p/2 n p p′ 2n p p′ p′ p T T ′ = P ∑ − n ∇ (P′) D − − − + PRM D − − + RP′ D − + P D − R′+ RR′ n=0 | | | | | | | | M p/2 n 2M 2n p p′ 2M p p′ 2M = P ∑ − n ∇ (P′) D − D − − − + R′′ = P′′ D − − − + R′′. n=0 | | | | | | We have P′′ P(A ) and we set p′′ = p+ p′ +2M N. Let us focus on the remainder ∈ ∈ p p′ p′ p R′′ = PRM D − − + RP′ D − + P D − R′+ RR′, | | | | | | p M 1 with RM OP ′− − . Since N,N′,M are arbitrary, we can choose N = N′′ + k′ , ∈ p′ N | | N = N + k and M = max k +N 1,0 . Thus PRM D − OP− ′′ , because we ′ ′′ | | { ′′ − } | | ∈ 1.3 Abstract Pseudodifferential Calculus 17 have k M 1 N . − − ≤− ′′ Similarly, the orders of the other terms in R′′ are successively (N′′ + k′ k′), − | |− N (N′′ + k k), (2N′′ + k + k′ ), thus less than N′′ and hence R′′ OP− ′′ proving− the| |− claim.− | | | | − ∈ It is immediate that T T Ψ k+k′ (A ). ′ ∈ Under the assumption of regularity of (A ,H ,D), the algebra Ψ(A ) can be d n seen as the set of all operators with asymptotics of the form ∑ n N Pn D − with ∈ | | Pn P(A ), i.e. ∈ A N d n N N if T Ψ( ) we have T ∑ n=0 Pn D − OP− for all N . ∈ − | | ∈ ∈ 1 If P′(A ) is the polynomial algebra generated by A , D and JA J− , when J exists, then P′(A ) P(A ) and one can construct a subalgebra Ψ ′(A ) of Ψ(A ) by taking P P (A⊂) in (1.24). In particular, D k is not necessarily in P (A ) ∈ ′ | | ′ for k odd. On the other hand, since D = D P0 P(A ), we could have defined P(A ) with D in place of D and| we| would| |− arrive∈ at the same definition ofΨ(A ). Let us remark| | that a key| point| about pdos on a manifold is that they are integral operators (see Appendix A). In the abstract framework this notion is missing even if it is reminiscent in a few specific cases, like the noncommutative torus [39].

Example 1.21. To show that the inverse of a classical pseudodifferential operator on a manifold is also a pdo is not immediate. Let us have a look at an abstract example: Let D, X Ψ 1(A ), so D2 +X Ψ(A ) and assume that D and D2 +X are invertible. We claim∈ that (D2 + X) 1 Ψ∈(A ): For any N N, we have the expansion − ∈ ∈ N 2 1 2 k 2 2 N+1 2 1 (D + X)− = ∑ ( D − X) D− + ( D − X) (D + X)− . (1.26) k=0 −| | −| |

2 The ﬁrst term is in Ψ(A ), because D − ,X Ψ(A ) and the remainder is in N 3 N 2 | 1| ∈2 2 2 OP− − OP− , provided (D + X)− OP− . But since D + X OP and 2 ⊂1 2 2 1 ∈ 2 1∈ (D + X)− D = (1 + D − X)− is bounded, because D − X OP− is compact | | | |2 ∈ 1 2 by Proposition 1.13 ix), the part viii) of the latter yields (D + X) OP− . − ∈ It is desirable to have an extension of the pseudodifferential calculus, which takes into account the complex powers of D (cf. [37, p. 239]): | | C Ψ (A ) := T D z T Ψ(A ), z C . (1.27) { | | | ∈ ∈ } By construction, T D z OPk+ℜ(z) when T Ψ k(A ). Since Formula (1.22) is true for any z C, the set| |Ψ∈C(A ) is in fact an R∈-graded algebra. The link∈ between the algebraic and geometric deﬁnitions of pseudodifferential operators is provided by the following fact:

Proposition 1.22. For a commutative spectral triple C∞(M),L2(M,S E),D/ , ⊗ E with D/ = iγ µ ∇S E (cf. Example 1.2), we have a natural inclusion E − µ ⊗ Z Ψ(C∞(M)) Ψ (M,S E), ⊂ ⊗ 18 1 The Dwelling of the Spectral Action where Ψ Z(M,S E) is the space of classical pseudodifferential operators of integer order deﬁned⊗ over a vector bundle S E. ⊗ D D 1 S This is a consequence of the observation that / E , / E Ψ (M, E) and also a Ψ 0(M,S E) for any a C∞(M). | | ∈ ⊗ For∈ the full story⊗ on classical∈ pseudodifferential operators see the references in Ap- pendix A.2 and for a noncommutative vision see [60] and[109, Section 5.1].

1.4 Dimension Spectrum

In Section 1.1 we discussed the p-summability of a spectral triple, which encodes the dimension of the underlying manifold when (A ,H ,D) is commutative. This notion does not, however, capture the whole richness of noncommutative geometry. A more adequatedescription turns out to be provided by a, possibly infinite, discrete subset of complex numbers. Definition 1.23. A regular spectral triple (A ,H ,D) of dimension p has a dimension spectrum Sd if Sd C is discrete and for any T Ψ 0(A ), the function ⊂ ∈ s ζT,D(s) := Tr T D − , defined for ℜ(s) > p, (1.28) | | extends meromorphically to C with poles located in Sd. 1 We say that the dimension spectrum is of order k N if all of the poles of ∈ ∗ functions ζT,D are of the order at most k and simple when k = 1.

Note that all functions ζT,D are well deﬁned for ℜ(s) > p, since for such s the op- s 0 s erator D − is trace-class and, as T Ψ (A ) B(H ), so is the operator T D − . By a standard| | abuse of notation we shall∈ denote⊂ the (maximal) meromorphic exten-| | sion of ζT,D with the same symbol. Observe also that if (A ,H ,D) has a dimension spectrum, then ζT,D is actually meromorphic on C for any T Ψ C(A ). Indeed, if T Ψ C(A ) then there exist S Ψ 0(A ) and z C such that∈T = S D z. Thus, for ℜ(∈s) > p+ℜ(z), the function ∈ ∈ | | ζT,D is deﬁned by

s s+z ζT,D(s) := TrT D − = TrS D − = ζS D(s z). | | | | , −

By the uniqueness of the meromorphic extension ζT,D is meromorphic on C. If, however, (A ,H ,D) is not finitely summable, then the zeta function ζ1,D is nowhere defined and the notion of a dimension spectrum does not make sense. Let us stress that the algebra does play an important role in Definition 1.23. Generally, given the meromorphic extension of the function ζ1,D, it is not clear how to get one for ζT,D, even when T A . Although in the case of a commutative ∈ spectral triple the poles of ζT,D do coincide with the poles of ζ1,D for T A , it is no longer to be expected for general noncommutative geometries. ∈

1 One could in principle also allow for essential singularities of ζT,D, as long as they are isolated. 1.4 Dimension Spectrum 19

A H D S Example 1.24. Let us consider the spectral triple ( q, q, q ) of the standard (1 q2)s S S Podles´ sphere (cf. Appendix B.4). The function ζ1 S = 4 s − (where D = D ,Dq w (1 qs)2 q q since the kernel is trivial) is regular at s = 2. On the other| | hand,− using the explicit formulae for the representation (B.17–B.25−), one can check that, for instance,

2q(1+q2) w 2 Res (s + 2)ζA,DS (s)= 2| | . s= 2 q (logq) − The definition presented above is the one of [69] (see also [67]) and it differs from the original one presented in [28, 37]. There, Sd was defined [37, Definition II.1] as the set of poles of functions ζT,D, but with T B0 – the algebra generated n ∈ by δ ′ (a) with a A and n N. This was tailored to prove the local index theorem in noncommutative∈ geometry∈ [37]. On the other hand, in the context of spectral action [35], the operator T from the definition was taken from a bigger algebra B1 n n generated by δ ′ (a) and δ ′ ([D,a]) [35, Definition 1.133]. The difference between the definition of the dimension spectrum adopted here 1 0 1 and the oneof [35] (see also [13, 99]) is that D − Ψ(A ) OP , but D − / B1. This fact led to eventual variations concerning| the| dimensi∈ on∩ spectrum of| a| commu-∈ tative spectral triple (compare for instance [36, Example 13.8] with [31, p. 22]). All these definitions of Sd satisfy Sd(disjoint sum of spaces)= Sd(spaces). There is a natural way to define the tensor product of two even∪ spectral triples (see [30, 52]) using

A := A1 A2, H := H1 H2 and D := D1 1 + γ1 D2. ⊗ ⊗ ⊗ ⊗

It would be tempting to conclude that Sd(A ,H ,D)= Sd1 +Sd2, but it is not obvious how to obtain a meromorphic extension of ζT1 T2,D given ζT1,D1 and ζT2,D2 . With Definition 1.23 the following result holds [69⊗ , Proposition A.2]: Example 1.25. Let (A ,H ,D) be the commutative spectral triple associated with a d-dimensional Riemannian manifold M and D is a first order differential operator, then Sd(A ,H ,D)= d N and it is simple. − An interesting result of Jean-Marie Lescure [82] shows that for spectral triples describing manifolds with conical singularities there appear poles of second order in the dimension spectrum. On the other hand, the dimension spectra of fractal spaces studied via noncommutative geometry [23, 24, 25, 61, 62, 63] encode the Hausdorff and spectral dimensions of the fractal along with its self-similarity structure. The latter is signalled by the appearance of complex numbers outside the real axis in Sd [74]. We see that the dimension spectrum carries much more information about the underlying geometry than a single number, for instance p-summability. However, there is no systematic procedure to compute the dimension spectrum for general spectral triples. Beyond the almost commutative geometry it is not even clear under what conditions (A ,H ,D) has a dimension spectrum. In concrete examples, one has to dutifully prove the existence of the meromorphic extensions of the whole family of 20 1 The Dwelling of the Spectral Action spectral functions and identify the poles. This was done only for few specific spectral triples like the noncommutative torus [51], quantum group SUq(2) [32, 46] and quantum Podles´ spheres [41, 44, 49, 93]. See also Problem 5) in Chapter 5. Let us note that in the definition of dimension spectrum, both the original one of [37] and the more recent one of [69], there is an (usually unspoken) assumption about the regularity of the spectral triple. Indeed, if one does not have control on the order of pseudodifferential operators, the definition of the dimension spectrum might turn out to be inconsistent with the p-summability (c.f. [49, Section 3.2]). Clearly, if a regular p-summable spectral triple has a dimension spectrum Sd, then p Sd. However, the p-summability together with regularity is not sufficient to con- ∈ clude that there is a pole of ζ1,D at p C — there might be an essential singularity, a branch cut or the boundary of analyticity,∈ in which case (A ,H ,D) does not have a dimension spectrum at all. All these pathologies certainly deserve further studies.

1.5 Noncommutative Integral

In a seminal paper [111] Mariusz Wodzicki has shown that the algebra of classical pseudodifferential operators of integer order Ψ Z(M,E) admits a single (up to normalisation) trace when M is connected and dimM > 1. It has been called the Wodz- icki residue, since it can be expressed as a residue of a certain spectral function. Alain Connes established in [26] a link between the Wodzicki residue of a compact pseudodifferential operator and its Dixmier trace (see Definition 1.29), which is an exotic trace on a certain ideal in the algebra B(H ). In this section we recollect some basic facts about the Wodzicki residue and its use for defining an integral suitable for noncommutative spaces. We start with the following definition: Definition 1.26. Let (A ,H ,D) be a regular p-dimensional spectral triple with a dimension spectrum. For any T Ψ C(A ) and any k Z define 2 ∈ ∈ [k] [1] k 1 T := Res s − ζT,D(s), T := T = Res ζT,D(s). (1.29) − s=0 − − s=0 Z Z Z If the dimension spectrum of (A ,H ,D) is of order d, then for s in an open neighbourhood of any z C we have the Laurent expansion ∈ ∞ [ k] − z k ζT,D(s)= ∑ T D − (s z) . k= d − | | − − Z The adopted notation is useful in the following result (cf. [37, Proposition II.1]):

Theorem 1.27. Let (A ,H ,D) be a regular p-dimensional spectral triple with a C dimension spectrum. Then, for any T1,T2 Ψ (A ) and any k Z we have ∈ ∈ 2 [k] k 1 Using the notation of [37] we have = 2 − τk 1 for k 1. − ≥ R 1.5 Noncommutative Integral 21

[k] [k] N n [k+ j] 1 n ( 1) j n 2n T1T2 = T2T1 + j − j T2∇ (T1) D − , − − ∑ n! ∑ 2 − | | Z Z n=1 j=0 Z n 3 with N = p+ordT1 +ordT2 and j the unsigned Stirling numbers of the ﬁrst kind. [d] C A In particular, if the dimension spectrum has order d, then is a trace on Ψ ( ). Proof. Since the triple is regular (1.22) yields, for any T R Ψ C(A ),s C,M N, ∈ ∈ ∈ M s s/2 n 2n s s ordT M 1 D − T = ∑ −n ∇ (T ) D − − + RN(s) D − , RM(s) OP − − . (1.30) | | n=0 | | | | ∈ Now, let r1 = ordT1 ,r2 = ordT2 . For ℜ(s) > p+2r1 +r2 we can use the cyclicity property of the| trace,| | |

s r1 r1 s ζT T ,D(s)= Tr T1T2 D − = Tr(T1 D − )( D T2 D − ) 1 2 | | | | | | | | r1 s r1 s = Tr( D T2 D − T1) D − = Tr T2 D − T1, | | | | | | | |

r1 s r1 s r1 since the operators D T2 D − , D T2 D − T1 are trace-class, whereas T1 D − r | | | | | | | | | | and D − 1 are bounded. Then, with the help of Formula (1.30), we obtain | | N s s/2 n 2n s s ζT1T2,D(s)= Tr T2T1 D − + ∑ −n Tr T2∇ (T1) D − − + TrT2RN (s) D − | | n=1 | | | | N s/2 s n n s h s = ζT2T1,D( )+ ∑ −n ζT2∇ (T1),D(2 + )+ ( ), (1.31) n=1 where h is holomorphic for ℜ(s) > p + ordT1 + ordT2 N 1 = 1. n C − − − For any n N, T2∇ (T1) Ψ (A ) and, since the spectral triple has a dimension ∈ ∈ n spectrum, the functions ζT2∇ (T1),D admit meromorphicextensions to the whole complex plane. Consequently, equality (1.31) holds true for ℜ(s) > 1 and, for any k Z, we have − ∈ [k] [k] N k 1 s/2 n T1T2 = T2T1 + ∑ Res s − −n ζT2∇ (T1),D(s + 2n) (1.32) − − s=0 Z Z n=1 [k] N n [k+ j] 1 n ( 1) j n 2n = T2T1 + j − j T2∇ (T1) D − . (1.33) − ∑ n! ∑ 2 − | | Z n=1 j=0 Z If the dimension spectrum of (A ,H ,D) has order d, for k = d we obtain

[d] [d] T1T2 = T2T1, − − Z Z

3 s n n s j n For any s C, n N, = ∑ j with the convention [ ]= δn 0. See [1, Sec. 24.1.3]. ∈ ∈ n j=0 n! 0 , 22 1 The Dwelling of the Spectral Action

[d+ j] C because T = 0 for any T Ψ (A ) and j N∗, whereas for j = 0, we have n = 0. ∈ ∈ j R In the case of a commutative spectral triple (as in Proposition 1.22), defined by (1.29) is related to the Wodzicki residue [111, 112, 64]. The latter is defined in R the following context: Let D Ψ(M,E) be an elliptic pseudodifferential operator of order 1. For P Ψ Z(M,E), define∈ ∈

WResP := ResζP,D(s). s=0 Mariusz Wodzicki has shown that

1 P WResP = d trσ d(x,ξ ) dµ(x,ξ ), (1.34) (2π) S Z ∗M − S 1 where ∗M := (x,ξ ) T ∗M gx− (ξ ,ξ )= 1 is the cosphere bundle over M, tr is { ∈ | P } the matrix trace of the symbol σ d(x,ξ ) and − dµ(x,ξ ) := σ(ξ ) dx , | || | dx := dx1 dxd, ∧···∧ n j 1 σ(ξ ) := ∑ ( 1) − ξ j dξ1 dξ j dξd . j=1 − ∧···∧ ∧···∧ c is a suitable Riemannian measure on S∗M (the integrand of (1.34) is a 1-density on M, so it is coordinate-independent, see [60, Section 7.3] or [108, Section 5.3]). Moreover, WRes is the unique (up to normalisation) trace on the algebra Ψ Z(M,E). D For a commutative spectral triple as in Proposition 1.22 with D = / E + P0, we simply have P = WResP for any P Ψ Z(M,S E). ∈ ⊗ Example 1.28.R For a A = C∞(M) of Example 1.2, Formula (1.34) yields ∈ d/2 d 1 d 2⌊ ⌋ Vol(S − ) a D − = d a(x)dµ(x), − | | (2π) M Z Z where dµ(x)= √gdx is the standard Riemannian measure on M, since we have d a D − d σ |d | (x,ξ )= a(x) ξ − . − k k It is worthy to store the following: In a commutative spectral triple, since ζ1,D is regular at 0 (see [57, page 108, eq. (1.12.16)]), we get

1 = 0. (1.35) − Z The noncommutative integral deﬁned in (1.29) turns out, moreover, to be related to the Dixmier trace, the construction of which we brieﬂy describe below. Given a selfadjoint operator T with purely discrete spectrum, we denote by λn(T ), with n N, its eigenvalues ordered increasingly. If T K (H ), then for ∈ ∈ 1.5 Noncommutative Integral 23

n N we deﬁne µn(T) as the (n + 1)-th singular value of T , i.e. (n + 1)-th eigen- ∈ value of T sorted in decreasing order, with the corresponding multiplicity Mn(T ). | | Note that since T is a compact operator, we have limn ∞ µn(T )= 0. Now, let us denote the partial trace →

N TrN(T ) := ∑ Mn(T ) µn(T). n=0

1 If T L (H ), limN ∞ TrN (T ) < ∞, but for a general compact operator the se- ∈ → quence TrN is unbounded. The role of a Dixmier trace is to capture the coefﬁcient of the logarithmic divergence of TrN (T ). There exists a very convenient formula [60, Proposition 7.34] for TrN (T) which uses a decomposition of T into a trace-class and a compact part:

1 TrN (T )= inf Tr R + N S R L (H ), S K (H ), T = R + S . (1.36) | | k k ∈ ∈ The partial trace, TrN (T ) may be seen as a trace of T cut-off at a scale N. But this scale does not need to be a natural number and indeed,| | Formula (1.36) still makes sense for any positive N, hence we deﬁne for any λ R+: ∈ Tr (T ) := inf Tr R + λ S R L 1(H ), S K (H ), T = R + S . (1.37) λ | | k k ∈ ∈ With the help of Formula (1.37 ) we deﬁne the subspace of compact operators, the partial trace of which diverges logarithmically

L 1,+ H K H Trλ (T ) ( ) := T ( ) T 1,+ := sup logλ < ∞ . (1.38) { ∈ | k k λ e } ≥ One has ATB A T B for every A,B B(H ) and the space k k1,+ ≤ k kk k1,+ k k ∈ L 1,+(H ) (sometimes denoted by L 1,∞(H ), see [84] for historical notes) is a C∗-ideal of B(H ) [67, Lemma 2.5]. Now, for T K (H ) and λ e, deﬁne ∈ ≥ λ 1 Tru(T ) du τλ (T ) := logλ logu u . (1.39) Ze 1,+ The functional τλ is not additive on L , but the defect is controllable [60, 1,+ Lemma 7.14], i.e. for positive operators T1,T2 L ∈ loglogλ τλ (T1 + T2) τλ (T1) τλ (T2)= λ ∞( logλ ). − − O → To obtain a useful additive functional on L 1,+, two more steps are needed. First 1,+ note that the map λ τ (T ) is in Cb([e,∞)) for T L . Deﬁne also a quotient 7→ λ ∈ C∗-algebra Q := Cb([e,∞))/C0([e,∞)) 24 1 The Dwelling of the Spectral Action and let [τ(T )] Q be the class of λ τλ (T ). Then, T [τ(T )] extends to a linear 1,∈+ 7→ 7→ map from L to Q, which is a trace, i.e. [τ(T1T2)] [τ(T2T1)] = [0]. To get a true linear functional on L 1,+, we need to apply to [τ( )]−a state ω (i.e. ω is a positive linear functional of norm 1) on Q. ·

Definition 1.29. A Dixmier trace [43] associated with a state ω on Q is defined as Trω ( ) := ω [τ( )]. · ◦ · In fact, a Dixmier trace is not only a trace, but a hypertrace on L 1,+(H ), i.e. 1,+ Trω (TS)= Trω (ST) for any T L and any S B(H ) [108, p. 45] (compare also [60, Theorem 10.20 and Corollary∈ 10.21]). ∈ The definition of a Dixmier trace, although powerful, is not completely satisfac- tory as it involves an arbitrary state on the commutative algebra Q, which is not separable (and L 1,+(H ) is also not separable). Thus, in practice, one cannot construct an explicit suitable state and there exist myriads of ‘singular traces’, different than the one of Dixmier, exploiting various notions of generalised limits [84]. Moreover, if one insists on defining the noncommutative integral via the Dixmier trace, one faces a notorious problem related to the existence of measurable (and non measurable!) sets in Lebesgue’s theory. In particular, there exists a class of operators in L 1,+(H ), for which a Dixmier trace does not depend on ω.

1,+ Deﬁnition 1.30. An operator T L (H ) is measurable if Trω (T ) does not de- ∈ pend on ω. Then one speaks about the Dixmier trace of T and denote it by TrDix(T ).

The following proposition gives a convenient characterisation of measurable operators, see [83],[84, Theorem 9.7.5] (and [107] for a link with the ζ-function)

1,+ TrN (T ) Proposition 1.31. An operatorT L is measurable if and only if limN ∞ ∈ → logN exists, in which case it is equal to TrDix(T ).

The link between Deﬁnition 1.26 of and TrDix was provided by Connes via his famous Trace Theorem [26, Theorem 1]. R Theorem 1.32. Let M be a compact Riemannian manifold of dimension d, E a vector bundle over M and T Ψ d(M,E). Then, T L 1,+, T is measurable and ∈ − ∈ 1 1 TrDix(T )= WRes(T )= T. (1.40) d d − Z Remark 1.33. Despite the nice equality (1.40), we cannot hope for proportional- ity between the Dixmier trace and the noncommutative integral for an arbitrary P Ψ(A ), even in the commutative case: Consider for instance M = S2 – the two-dimensional∈ sphere, endowed with the standard Dirac operator D/, which has singular values µn = n + 1 with multiplicities 4(n + 1) (see Appendix B.1). Now 1 0 ∞ 2 s take T = Ψ (C (S )). We have, ζ1,D/ (s)= Tr D/ − = 4ζ(s 1), which is regular at s = 0,∈ hence 1 = 0. On the other hand, 1 |is not| a compact− operator, hence its Dixmier trace does not make sense at all. R 1.6 Fluctuations of Geometry 25

We have privileged the noncommutative integral , which — as we have just explained — is more suitable in the noncommutative-geometric framework. However, R on the technical side there are possible variants: Instead of ζT,D , we could have cho- 1/2 1/2 s sen the more symmetrised Tr(T D T )− when T is positive. The not obvious links between these functions and others| | are investigated in[12].

1.6 Fluctuations of Geometry

Given a spectral triple (A ,H ,D) it is natural to ask whether there exist other spectral triples describing a noncommutativegeometry which is equivalent in some sense to the one determined by (A ,H ,D). To discuss this issue we ﬁrstly need the deﬁ- nition of noncommutative one-forms :

1 ΩD (A ) := span adb a,b A , with db := [D,b]. (1.41) { | ∈ } One can define accordingly the n-forms (modulo the so-called junk forms [60, Sec- tion 8.1]), which are building blocks of the Hochschild and cyclic homologies (see, for instance, [75, Chapter 3] for more details). On the physical side, elements of 1 ΩD (A ) are to be seen as gauge potentials of the theory. A notion of equivalence suitable for spectral triples is that of Morita equivalence [35, Chapter 1, Section 10.8] (see also [34, Section 2], [60, Section 4.5] or [75, Section 2.3]), which we now briefly describe. Recall first that a (right) module E over a unital algebra A is an Abelian group (E,+) along with the (right) action of A on E, i.e. a map E A E such that for all e, f E and any a,b A × → ∈ ∈ (e + f )a = ea + fa, e(a + b)= ea + eb, e(ab) = (ea)b, e1A = e.

A ﬁnitely generated module E over A is free if there exists N N such that E is N ∈ isomorphic to A ⊗ . A module E is called projective if there exists another (right) module F over A such that E F is a free module. The⊕ Morita equivalence of algebras can be characterised as follows (cf. [104, Proposition 6.12]).

Definition 1.34. Let A ,A ′ be two unital associative algebras. We say that A ′ is Morita equivalent to A if there exists a finitely generated projective (right) module E over A , such that A EndA E. ′ ≃ Having fixed a representation π of A on H , A acts on H := E A H . The ′ ′ ⊗ space H ′ is endowed with the scalar product:

r η,s ξ := η,π(r s)ξ , h ⊗ ⊗ i h | i 26 1 The Dwelling of the Spectral Action where ( ) is a pairing (or an A -valued inner product) E E A , which is A - linear in·|· the second variable and satisﬁes × →

(r s) = (s r)∗, (r sa) = (r s)a and (s s) > 0 for r E,0 = s E. | | | | | ∈ 6 ∈

Thus, each representation of A on H gives a representation of A ′ on H ′. For a given projective (right) module E over A , one can choose a Hermitian con- 1 nection, i.e. a linear map ∇ : E E A ΩD (A ), which satisﬁes the Leibniz rule ∇(ra) = (∇r)a + r da, for all r→ E,⊗a A . With the help of a⊗ Hermitian connection∈ ∈ one can deﬁne a selfadjoint operator D L (H ) by ′ ∈ ′

D ′(r ξ ) := r Dξ + (∇r)ξ , with r E, ξ Dom(D). ⊗ ⊗ ∈ ∈

Then, (A ′,H ′,D ′) is a spectral triple (for the compatibility with a real structure see e.g. [104, Theorem 6.16]). We say, by definition, that the spectral triple (A ′,H ′,D ′) is equivalent to the original one (A ,H ,D). This is motivated by the following observation: The algebra A is obviously Morita equivalent to itself (with E = A ), in which case 1 H ′ = H and AdD : a [D,a] ΩD (A ) is a natural hermitian connection for → ∈ 1 E. Thus, any DA = D + A with A = A Ω (A ) would provide a spectral triple ∗ ∈ D (A ,H ,DA) equivalent to (A ,H ,D). The operator DA is called a(n) (inner) fluctuation of D and A (following physicists 1 conventions) – a gauge potential. If DA = D + A with A = A∗ ΩD (A ), then for any B = B Ω 1 (A ) we have D + B = D + A with A = A∈+ B Ω 1 (A ) [35, ∗ DA A ′ ′ D Proposition 1.142].∈ In other words: “Inner fluctuations of inner fluctuations∈ are inner fluctuations”. As we want to consider real spectral triples, we should require (A ,H ,DA) to have the same KO-dimension (see p. 5) as (A ,H ,D). If DJ = εJD, then we should require DAJ = εJDA. Therefore, we define

1 1 DA := D + A, with A := A + εJAJ− , for A = A∗ ΩD (A ). (1.42) ∈ Example 1.35 (cf. [14, p. 734] and [47, p. 22]). Let M be an even dimensional spin µ manifold as in Example 1.2. For a,b A we have a[D/,b]= iγ a∂µ b. ∈ ∞ − Since A is Hermitian, Aµ := ia∂µ b C (M,R) and [J,Aµ ]= 0. On the other hand, Jγ µ = γ µ J because J commutes− with∈ D/ and anticommutes with i. Finally, − 1 1 D/ A = D/ + A = D/ + A + JAJ− = D/ + A AJJ− = D/. − Hence, there are no inner fluctuations in commutative geometry. The above considerations rely on the first-order condition (1.5). The latter can be relaxed, what induces in a more general form of fluctuations [21]. Whereas the exact shape of A does affect the physical content of the theory, the mathematics detailed 0 in Chapter 4 is not afflicted. Therefore, we shall only assume that A = A∗ Ψ (A ), which holds also for the more general form of perturbations, cf. [21, Eq.∈ (10)]. As 1.7 Intermezzo: Quasi-regular Spectral Triples 27

we will see, for such an A, (A ,H ,DA) is still a spectral triple, which inherits the regularity, p-summability and dimension spectrum properties of (A ,H ,D).

1.7 Intermezzo: Quasi-regular Spectral Triples

Before we proceed,let us come backfor a momentto the assumption of regularity of a spectral triple — recall Deﬁnition 1.8. As we have witnessed, it played an essential role in the construction of abstract pdos along with the dimension spectrum and the noncommutative integral. It turns out that at least some of these properties survive under a weaker assumption of quasi-regularity (cf. [48, Chapter 4]):

A ,[D,A ] op0 . ⊂ Clearly, every regular spectral triple is quasi-regular. An example of non-regular, though quasi-regular spectral triple, is provided by the standard Podles´ sphere (cf. Appendix B.4). 0 0 Observe that if T op , then σz(T ) op for any z C by properties (1.9). On the other hand, the∈ expansion (1.22)∈ does not hold in∈ general. Typically, it is substituted by a ‘twisted’ version (see [49, Lemma 4.3] for an illustration). Similarly to the regular case, given a quasi-regular (A ,H ,D) one can furnish the algebra A with a locally convex topology determined by the family of seminorms a σn(a) , a σn([D,a]) . The completion Aσ is a Fréchet pre-C -algebra and 7→ k k 7→ k k ∗ (Aσ ,H ,D) is again a quasi-regular spectral triple. The proof of this fact closely follows [98, Lemma 16] and the details can be found in [48, Chapter 4]. In such a context one can define the algebra of pdos Ψ(A ) as the polynomial 1 1 algebra generated by a A , D, D and D − , complemented by JA J if the ∈ | | | | − triple is real and P0 if kerD = 0 . e 6 { } In the same vein, one can consider the algebra Ψ C(A ) as being generated by the zi elements of Ψ(A ) sandwiched with D for zi C. The grading is now provided | | ∈ by the op classes, Ψ r(A ) := Ψ(A ) opr. e The dimensione spectrum of a quasi-regular∩ spectral triple can be considered naturally with T Ψ 0e(A ) in placee of Ψ 0(A ) in Definition 1.23. Being fairly involved, this notion is∈ nevertheless workable and gives reasonable results for the Podles´ sphere — cf. Theoreme B.10. Given a quasi-regular spectral triple with a dimension spectrum of order d, the noncommutative integral (1.29) still makes sense, however, the pleasant feature of [d] being a trace on the algebra of pdos is, in general, lost. Also, the study of fluctuations in the quasi-regular framework — although possible in principle — is rather R obnoxious, as the entire Chapter 4 of this book bases on Formula (1.22), hence on the regularity. 28 1 The Dwelling of the Spectral Action 1.8 The Spectral Action Principle

Having prepared the ground we are now ready to present the concept of the spectral action. In a seminal paper [14] Chamseddine and Connes put forward the following postulate: „The physical action only depends upon the spectrum of D.”

A mathematical implementation given in [14] led to the following deﬁnition.

Deﬁnition 1.36. Let (A ,H ,D) be a spectral triple and let Λ > 0. The (bosonic) spectral action associated with D reads:

S(D, f ,Λ) := Tr f ( D /Λ), (1.43) | | where f : R+ R+ is a positive function such that f ( D /Λ) exists and is a trace- class operator.→ | |

Let us emphasise (cf. [14, (1.23)]) that the operator D in Formula (1.43) should be the one dressed with gauge potentials, i.e.

D = D0 + A.

This is because (A ,H ,D0) and (A ,H ,D) cook up equivalent geometries and we must take into account all of the available degrees of freedom when constructing the action. See also Problem 1) in Chapter 5. Before we pass on to the hard mathematics hidden behind Deﬁnition 1.36, to which this book is primarily devoted, we provide a glimpse into its physical content.

The physics of the spectral action

As pointed out in [17], the spectral action has several conceptual advantages: – Simplicity: If f is the characteristic function χ[0,1], then Formula (1.43) simply counts the singular values of D smaller or equal than Λ. – Positivity: When f is positive, the action is manifestly positive — as required for a sound physical interpretation. – Invariance: The invariance group of the spectral action is the unitary group of the Hilbert space H , which is vast. The role of the parameter Λ is to provide a characteristic cut-off scale, at which (1.43) is a bare action and the theory is assumed to take a geometrical form. It 1 should have a physical dimension of length− , as the operator D does. Within the almost commutative models the value of Λ is typically taken to be in the range 1013–1017 GeV (hc¯ ) 1. · − Although f = χ[0,1] is a privileged cut-off function providing the announced simplicity, it seems that for physical applications one should stay more ﬂexible and allow it to depart from the sharp characteristic function. This is desirable in the 1.8 The Spectral Action Principle 29

almost commutative context, as the moments of f provide free parameters of the theory, which can be tuned to fit the empirical data — cf. [104, Chapter 12]. The actual physical role of f in full generality of noncommutative geometry is more obscure — see Problem 7) in Chapter 5. Let U (A ) := u A uu = u u = 1 and U := uJuJ 1, with u U (A ). Now, { ∈ | ∗ ∗ } − ∈ if the first-order condition (1.5) is satisfied, then the operator DA transforms under the action of U as follows

1 u u 1 UDAU ∗ = U(D + A + εJAJ− )U ∗ = D + A + εJA J− ,

u with A := uAu∗ +u[D,u∗]. This ensures that U (A ) is a subgroup of the symmetry group of the spectral action. In the almost commutative realm, the full symmetry group of (1.43) is the semi- 1 direct product, G⋊Diff(M), ofthegroup G := uJuJ− u U (A ) of local gauge transformations and the group of diffeomorphisms{ of the| man∈ ifold }M — see [104] for the details. If one abandons the first-order condition (1.5), then the inner fluctuations of the spectral action form a semi-group, which extends U (A ) [21]. To conclude this paragraph we exhibit the powerfulness of the spectral action: Let M be a compact Riemannian 4-manifold with a spin structure, as in Example 1.2 and let AF = C H M3(C), with H denoting the algebra of quaternions. Then, ⊕ ⊕ S(D, f ,Λ)= gd4x γ + 1 R + α C Cµνρσ + τ R⋆R⋆ + δ ∆R √ 0 2κ2 0 µνρσ 0 0 ZM 0 1 i µνi 1 a µνa 1 µν + 4 Gµν G + 4 Fµν F + 4 B Bµν (1.44) 1 2 2 2 4 1 2 1 + Dµ H µ H + λ0 H R H + ∞(Λ − ). 2 | | − 0 | | | | − 12 | | O One recognises respectively: the cosmological constant, the Einstein–Hilbert term, the modified gravity terms, the dynamical terms of gauge bosons and the Higgs 2 sector. The coefficients γ0, κ0− , α0 etc. depend on the powers of the energy scale Λ 4,Λ 2,Λ 0 and also of the moments of the cut-off function f :

∞ ∞ 3 f4 = x f (x)dx, f2 = x f (x)dx and f (0). Z0 Z0 Furthermore,they depend upon the fermionic contentof the model,which is ﬁxed by the choice of the Hilbert space HF and the matrix DF , which encodes the Yukawa couplings of elementary particles (cf. [20, 104]). To relish the full panorama of spectral physics, one supplements Formula (1.43) + 1 with the fermionic action SF := Jψ,Dψ , for ψ H = (1 + γ)H (cf. [104, h i ∈ 2 Deﬁnition 7.3] and [6, 20]). Although the physical content of SF is exciting, its mathematics is rather mundane. Therefore, shamefully, we shall ignore it in the remainder of the book. 30 1 The Dwelling of the Spectral Action

Many faces of the asymptotic expansions

How on earth the simple Formula (1.43) can yield the knotty dynamics of the full Standard Model and gravity? Actually, there is no mystery — just asymptotics. D ∞ s D /Λ Roughly (see Section 3.5 for the full story): S( , f ,Λ)= 0 Tr(e− | | )dφ(s), when f is the Laplace transform of a measure φ. But, Tr e tP is the celebrated heat − R trace associated with a (pseudo)differential operator P. The latter is known to enjoy a small-t asymptotic expansion [58, 110]: For instance, when P = D/ 2 and d = dimM

∞ t D/ 2 D2 (d k)/2 Tre− ∑ ak( / )t − . (1.45) t∼0 ↓ k=0

The alchemy is concealed in the coefficients ak known under the nickname of Seeley–deWitt coefficients. They are expressible as integrals over M of local quan- D tities polynomial in the curvature of M and, if we happen to work with / E (cf. Proposition 1.22), in the curvature of E. The Seeley–deWitt coefficients of a differential operator are local. The principle of locality lies at the core of the concept of a field, which asserts that every point of spacetime is equipped with some dynamical variables [65]. Concretely, a local quantity in quantum field theory is precisely an integral over the spacetime manifold M of some frame-independent smooth function on M, which is polynomial in the field and its derivatives. When P is a pseudodifferential operator, an asymptotic expansion similar to (1.45) is still available (cf. Appendix A.5). On the other hand, the coefficients ak(P) are, in general, nonlocal — see [59] for an explicit example. This means that pdos over classical manifolds belong already to the noncommutativeworld, which is pre- vailingly nonlocal. In the full generality of noncommutativegeometry the existence of an asymptotic expansion (1.45) is no longer guaranteed. Nevertheless, one may hope to deduce it from a meromorphic extension of the associated zeta function ζD. The bulk of Chapter 3 is devoted to this enjoyable interplay. If (A ,H ,D) is a regular spectral triple with a simple dimension spectrum one can hope to obtain the following formula (see Corollary 3.33 and Section 4.3):

∞ D α α 1 α O S( , f ,Λ)= ∑ Λ x − f (x)dx D − + f (0)ζD(0)+ ∞(1). (1.46) 0 − | | α Sd+ Z Z ∈ Astonishingly, the abstract Formula (1.46) does yield the Standard Model action (1.44), once the suitable almost commutative spectral triple has been surmised.

A careful reader have spotted the O∞(1) term in (1.46) and (s)he might wonder what does this symbol hide. Actually, if one has at one’s disposal the full asymptotic expansion (see Deﬁnition 2.33) of the form (1.45) one can expand Formula (1.46) N to the order Λ − for arbitrarily large N (vide Theorem 3.20). On the other hand, one must be aware of the fact that the explicit computation of the coefﬁcients ak(P) for k > 4 is arduous, even if P is a friendly differential operator. For a Laplace type 1.8 The Spectral Action Principle 31 operator general formulae are available up to a10 — see [4] and [110]. From the perturbative standpoint one might argue that the terms in the action, which vanish at large energies can safely be neglected. Then, however, one risks overlooking some aspects of physics. For example, it has been argued [102] that a contribution of the 2 order Λ − might affect neutrino physics and the study of cosmic topology requires the knowledge of all ak’s. To the latter end, several authors [5, 19, 17, 78, 86, 88, 89, 92, 105, 106] employed the Poisson summation formula, which we discuss in Section 2.4. The summation formulae (sometimes dubbed not quite correctly “nonperturba- ∞ tive methods”) give the spectral action modulo a reminder ∞(Λ − ), which is usually disrespected. However, the devil often sits in the details:O There exists an exten- sive catalogue of physical phenomena, which are ‘exponentially small’. An enjoyable account on this issue was produced by John P. Boyd [9].

Highlights on the research trends

The literature on the spectral action is abundant and growing fast. On top of the references already quoted, we list below some highlights on current research. The list is admittedly subjective and far from being complete — we would be pretentious to claim to possess full knowledge on the topic. To consider gravity as a low-energyeffect of quantised matter fields, rather than a fundamental force is a long-standing idea: Yakov Zeldovich considered the cosmological constant as an effect of quantum matter fluctuations and Andrei Sakharov suggested that the structure of the quantum vacuum encodes the Einstein–Hilbert action (see [91] for a short review). The heat kernel methods were successfully applied in 1960’s by Bryce S. de- Witt in order to derive a series expansion for the Feynman propagator of quantum fields on a curved background. A more recent summary of the spectral approach to quantum field theory can be found in the textbooks [10]and [50]. But the upshot of the spectral action is that it provides an exact — i.e. truly non- perturbative— formula for the bare action at the unification scale. Expression (1.43) as it stands is nonlocal, which means that it encompasses both local and global (topology, in particular) aspects of the physical world. For an approach balanced between the local and global aspects of the spectral action, which also goes beyond the weak field approximation, see [70, 71]. The impact of a boundary of a manifold on the spectral action was studied in [16, 18, 72]. The main difficulty is related to the choice of the boundary conditions for the operator D, which would guarantee a selfadjointextension, and then to define a compatible algebra A , see[68, 69]. Some aspects of spectral geometry, including the heat trace expansion, on manifolds with conical singularities were studied in [82] (cf. also [81]). The role of the torsion in the spectral context has also been explored [72, 94, 95, 103]. Surprisingly enough, the spectral action for a manifold with torsion turns out 32 1 The Dwelling of the Spectral Action to embody the Holst action, well known in the Loop Quantum Gravity approach [96]. On the physical side, a programme on the inflationary scenarios compatible with the spectral action has been launched [77, 85, 87, 88, 89, 90, 100, 101]. Also, the spectral action was approached via quantum anomalies and Higgs– dilaton interactions [2, 3].

Variations on the deﬁnition

Usually, one encounters the operator D 2 instead of D in Formula (1.43) for the spectral action. The reason for which D 2 is favoured| in| the literature is that for a commutative spectral triple D/ 2 is a differential operator (of Laplace type), whereas D is a priori only a pseudodifferential one. In the full generality of a noncommutative| | geometry, however, we are bound anyway to work with abstract operators, which are not even classical pdos and working with D allows for more flexibil- ity. Clearly, we can restore the presence of D 2 by taking| | f (x)= g(x2) (cf. Remark 3.35), but caution is needed, as the cut-off functions f and g belong to different classes (see Section 2.2.2). One could also consider the action of the form Tr f (D/Λ). If f is even, as assumed in [14], this is equivalent to (1.43). But one can take into account the asym- 1 metry of the spectrum of D: When f is odd, then f (D/Λ)= D D − g( D /Λ) for an even function g, see for instance [92]. Technically, the parity| of|f is not| | innocent and does play a role in the Poisson formula, cf. Section 2.4. In [55] a formulation of the spectral action for nonunital spectral triples has been proposed, see also [15, 56]. Alternatively, one can simply consider a compacti- fied spacetime manifold — see, for instance [19, 17], for the casus of Friedman– Lemaître–Robertson–Walker universe. It should also be recognised that the spectral action (and its entire dwelling) works under the assumption of a positive-definite metric. Hence, the action (1.43) is an Euclidean one and its physical applications require a Wick rotation [42]. A truly Lorentzian approach is a challenging programme — cf. Problem 4) in Chapter 5. We conclude with a derived notion of action: When the triple is even with a grading γ, we define the topological spectral action by

Stop(D, f ,Λ) := Tr γ f ( D /Λ). (1.47) | |

We us now show as in [104, Section 10.2.3] that Stop(D, f ,Λ)= f (0) index(D): The McKean–Singer formula

t D2 index(D)= Tr γ e− holds true for any t > 0: Indeed, let Hn be the eigenspace associated to the eigenvalue λn of D and Pn be the eigenprojection on Hn. Then, References 33

D2 2 t tλn D Tr γ e− = Tr(γ P0)+ ∑ e− Tr(Pn P n)= Tr(γ P0)= index ( ). − − λn>0

Thus, if the function x R+ f (x1/2) is a Laplace transform of a ﬁnite measure ∞ ∈t x2 7→ φ, so that f (x)= 0 e− dφ(t), then the topological spectral action is simply R ∞ ∞ t D2/Λ Stop(D, f ,Λ)= Trγ e− dφ(t)= index(D) dφ(t)= f (0) index(D). Z0 Z0

Similarly its ﬂuctuation is Stop(DA, f ,Λ)= f (0) index(DA).

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Chapter 2 The Toolkit for Computations

Abstract In this chapter we introduce a number of mathematical tools, which will prove useful in the spectral action computations. Firstly, we consider the basic properties of some spectral functions via the functional calculus and general Dirichlet series. Next, we study the interplay between these provided by the functional transforms of Mellin and Laplace. The remainder of the chapter is devoted to various notions from the theory of asymptotic behaviour of functions and distributions. Before we start off let us recall the big- and small-O notation: O Let X be a topological space and let x0 be a non-isolated point of X. Let U be a neighbourhood of x0 and V = U x0 –a punctured neighbourhood of x0. For two functions f ,g : V C we write \{ } →

f (x)= x x0 (g(x)) if limsup f (x)/g(x) < ∞, O → x x | | → 0 f (x)= Ox x (g(x)) if lim f (x)/g(x) = 0. → 0 x x → 0 | | O We use x0 (g(x)) and x0 (g(x)) when no mistake concerningthe variable can arise. We willO mostly be concerned with the cases X = R+ ∞ or X = C ∞ and ∪{ } ∪{ } x0 = 0 or x0 = ∞. ∞ ∞ k The notations 0(x ) and ∞(x− ) will stand, respectively, for x 0(x ) and k O O O → x +∞(x− ), for all k > 0, and similarly for O. O → Example 2.1. We have sinx = ∞(1), but sinx = O∞(1) and 1 = ∞(sinx). n εO ε 6 6 O For any n > 0, log x = O0(x− )= O∞(x ) for all ε > 0. For further examples and properties of and O symbols see, e.g. [17, Section 1.2]. O

2.1 Spectral Functions

The spectral action (1.43) is par excellence a spectral function, i.e. a (possibly complex valued) function on the spectrum of some operator.

39 40 2 The Toolkit for Computations

We now take a closer look at various spectral functions from the perspective of general Dirichlet series [24]. As this part only involves the properties of D and not the full force a spectral triple, we shall work with a general operator H acting on an infinite dimensional separable Hilbert space H . We shall need the following classes of positive densely defined unbounded operators, for p R+, ∈ p p ε p+ε T := H L (H ) H > 0 and ε > 0 Tr H− − < ∞, but Tr H− = ∞ . ∈ ∀ p 1 If H T for some p, then it is invertible and H− K (H ). Moreover, with L r(H )∈:= T L (H ) Tr T r < ∞ (the so-called r-th∈ Schatten ideal), we have { ∈ | | | } + 1 r p = inf r R H− L (H ) . { ∈ | ∈ } If (A ,H ,D) is a not finite (i.e. with dimH = ∞ — cf. Example 1.3) p- dimensional spectral triple then D T p and D2 T p/2 (cf. (1.1)). If (A ,H ,D) | | ∈ p ∈ 2 p/2 is a regular spectral triple, then also DA T and DA T with a suitable fluctuation A — see Section 1.6 and Chapter| | ∈4. Note, however,∈ that the requirement D T p for some p rules out the finite spectral triples, since T p L 1(H )= /0. The∈ latter situation is trivial from an analytic point of view as Tr f (∩D /Λ) is finite and explicitly computable for any bounded measurable function f and| | any Λ > 0. As the primary example of a spectral function let us consider

NH (Λ) := ∑ Mn(H), for Λ > 0, n:λn(H) Λ ≤ titled the spectral growth function. We get

N D (Λ)= Trχ[0,Λ]( D )= S(D, χ[0,1],Λ), | | | | which is the archetype of the spectral action [9]. Via the unbounded functional calculus (see, for instance [30, Chapter 13]) we deﬁne an operator

∞ f (H) := f (λ)dPλ (H)= ∑ f (λn(H))Pn(H), λ spec(H) Z ∈ n=0 with the spectral projections Pn(H) := Pλn (H), for any bounded Borel (possibly complex) function f on R+. The operator f (H) is trace-class if and only if

∞ Tr f (H)= ∑ Mn(H) f (λn(H)) < ∞. (2.1) n=0

More generally, if H T p, K is any operator in B(H ) and Tr f (H) < ∞, then ∞ ∈ s TrK f (H) = ∑n=0 Tr(Pn(H)K) f (λn(H)) < ∞. In particular, for f (x) = x− with ℜ(s) > p, we obtain the spectral zeta function 2.1 Spectral Functions 41

∞ s s ζK,H (s) := Tr KH− = ∑ Tr(Pn(H)K)λn(H)− , for ℜ(s) > p. (2.2) n=0 If H is not positive, but nethertheless H T p for some real number p, we deﬁne s | | ∈ ζK,H (s) := Tr K h − in accordance with the notation adopted in (1.28). Moreover if K =| 1| , we shall simply write

ζH := ζ1,H .

The function ζK,H will often admit a meromorphic extension to a larger region of the complex plane, in which case it is customary to denote the extension with the p same symbol. Note that H T guarantees that ζK,H , if meromorphic, has at least one pole located at s = p. ∈ Yet another propitious spectral function is the heat trace, which results from tx f (x)= e− with t > 0,

∞ t H tλn(H) Tr Ke− := ∑ Tr(Pn(H)K)e− . (2.3) n=0

t H The operator e− is called the heat operator. This nickname comes from physics: When H = ∆ is the standard Laplacian on a Riemannian manifold then e t H ψ − − solves the heat equation, ∂t φ + Hφ = 0, with the initial condition φ(0)= ψ. On the side, we note that for any positive (unbounded) operator P the function + tP t R e− B(H ) admits a holomorphic extension to the right half-plane ∈ tP7→ 1 ∈ t λ 1 C R+ via e− = i2π λ C e− (P λ)− dλ, where is a contour around . This can be proved by differentiating∈ − under the integral — see Appendix A.1. It implies, in R t H t H particular, that if e− is trace-class for any t > 0, then the function t Tr e− t H 7→ is smooth on (0,∞) and so is t Tr Ke− for any K B(H ). We will give an alternative proof of this fact in Proposition7→ 2.3. ∈ When H T p, both heat traces and spectral zeta functions (for ℜ(s) > p) are instances of ∈general Dirichlet series [24]. The latter are deﬁned as

∞ sbn ∑ an e− , (2.4) n=0 for an C, bn R with limn ∞ bn =+∞ and some s C. The region of their convergenceconstitutes∈ ∈ a half-plane→ ℜ(s) > L for some L ∈R, the latter being called the abscissa of convergence . In contradistinction to Taylor∈ series, the domain of absolute convergence of (2.4), which is also a half-plane ℜ(s) > L′, can be strictly smaller, i.e. L′ L (cf. [2, Section 11.6]). The regions of convergence of general Dirichlet series≥ can be determined from the following theorem:

∞ Theorem 2.2 ([24, Theorem 7 with the footnote]). If the series ∑n=0 an is not convergent, then the series (2.4) converges for ℜ(s) > L and diverges for ℜ(s) < L, with 42 2 The Toolkit for Computations

1 L = limsup bn− log a0 + ... + an 0. (2.5) n ∞ | |≥ → The non-negativity of L follows from the fact that for s = 0 the expression (2.4) ∞ equals to ∑n=0 an, which is not convergent. To compute the abscissa of absolute convergence one simply needs to take ai in Formula (2.5). | | For K,H > 0, we always have L′ = L in both Dirichlet series (2.3)and (2.2). For given operators K and H the associated heat trace and zeta functions are closely interrelated, which will be explored in great detail in Chapter 3. We now offer the ﬁrst glimpse into their intimate relation. tH Theorem 2.2 implies that the operator Ke− is trace-class for all t > 0 iff

N log ∑ Tr(Pn(H)K) = ON ∞(λN (H)). (2.6) n=0 →

On the other hand, ζK,H is deﬁned (i.e. (2.2) converges for ℜ(s) > p for some p 0) if and only if (cf. [12, Proposition 2]) ≥

N r ∑ Tr(Pn(H)K)= N ∞(λN (H) ), for all r > p. (2.7) n=0 O → Formula (2.7) implies (2.6), but the converse is not true. As a counterexample 2 consider K = 1 and λn(H)= log n, Mn(H)= 1. In particular, if a spectral triple (A ,H ,D) is θ-summable with t0 = 0, but not ﬁnitely summable (recall p. 4), the t D heat traces TrTe− | | exist, while the spectral zeta functions ζT,D do not. Given the abscissa of convergenceof the zeta function ζK,H we can easily deduce the behaviour of Tr Ke t H as t 0. − ↓ p t H Proposition 2.3. Let H T and K B(H ). The function t Tr Ke− is ∈ t H ∈r 7→ smooth on (0,∞) and Tr Ke = 0(t ), for all r > p. − O − α x + Proof. Let us consider the function x x e− , which is bounded on R for any 7→α x α 0. For any x > 0, we thus have x e− c(α), with some positive constant ≥ ≤ x/2 c(α). By invoking this inequality with x/2 and multiplying it by e− , we obtain α x α x/2 x e− 2 c(α)e− . Let us≤ ﬁx any r > p 0 and use xα e x c(α) again. For any r > p we have ≥ − ≤ ∞ ∞ r t H r tλn(H) r 0 t Tr e− = ∑ Mn(H)t e− c(r) ∑ Mn(H)λn(H)− = c(r)ζH (r) < ∞. ≤ n=0 ≤ n=0

t H This shows in particular that e− is trace-class for any t > 0. r t H Moreover, the function t t Tr e− is bounded for t [0,∞) for every r > p and r 7→ t H ∈ t H r the limit superior of t Tr e− as t 0 exists and is ﬁnite, giving Tr e− = 0(t− ) for all r > p. ↓ O Furthermore, for any t > 0 and any k N we have ∈ dk t H k tH k tH k t H/2 TrKe− = Tr KH e− K Tr H e− K 2 c(k) Tr e− < ∞, | d tk | | |≤k k ≤k k 2.1 Spectral Functions 43

t H which shows that the function t TrKe− is indeed smooth on (0,∞). Finally, 0 tr Tr Ke t H K →tr Tr e t H and one concludes as above. ≤ | − |≤k k − Remark 2.4. The heattrace (2.3) can also be deﬁnedwhen H is a selfadjoint operator bounded from below with a compact resolvent. In this case one writes, for any t > 0,

N ∞ t H t λn(H) t λn(H) Tr Ke− = ∑ Tr(Pn(H)K)e− + ∑ Tr(Pn(H)K)e− , n=0 n=N+1 for an N N such that λN+1(H) > 0. Since the ﬁrst ﬁnite sum is manifestly a smooth bounded∈ function on R+, Proposition 2.3 holds.

In particular, in the context of spectral triples we have (recall p. 3)

s D ζD(s)= Tr(1 P )H D− = ζD(s) dimker . − 0 − Example 2.5. Let D/ be the Dirac operator associated with the trivial spin structure on S1 equipped with the round metric (cf. Appendix B.1). Then,

∞ ∞ s s s ζD(s)= ∑ Mn( D )λn( D )− = 1− + 2 ∑ n− = 1 + 2ζ(s), n=0 | | | | n=0 where ζ is the familiar Riemann zeta function. The latter has a single simple pole at s = 1, so the triple is 1-dimensional. The corresponding heat trace reads, for t > 0,

∞ t t D nt 2e− t Tr e− | | = 1 + 2 e− = 1 + t = coth ∑ 1 e− 2 n=1 −

t D 1 and, indeed, Tr e = 0(t ) in accord with Proposition 2.3. − | | O − Remark 2.6. Let us note that Proposition 2.3 extends in a straightforward way to t Hα r Tre− = 0(t− ), for all r > p/α, for any α > 0, as ζHα (s)= ζH (αs). When appliedO to Example 2.5 it implies a non-trivial fact:

∞ nrt α ∑ e− = 0(t− ) for all α > 1/r. n=1 O We conclude this section with a sufﬁcient condition for the well-deﬁniteness of the spectral function (2.1), which is of our primary interest.

Lemma 2.7. Let H T p and let f be a positive function deﬁned on R+ such that p ε ∈ f (x)= ∞(x ) for some ε > 0. Then, f (H) is trace-class. O − − 44 2 The Toolkit for Computations

Proof. With ψn n N being the orthonormal basis on H composed of the eigen- { } ∈ ∞ ∞ vectors of H we obtain Tr f (H)= ∑ ψn, f (H)ψn = ∑ Mn(H) f (λn(H)). By n=0 h i n=0 the hypothesis on the decay of f , there exist c2,N > 0, such that for any x 0, we p ε ≥ can estimate f (x) c1 + c2 χ[N,∞)(x)x− − , with c1 = sup x

2.2 Functional Transforms

In this section we recall some basic notions and facts on two functional transforms, which will allow us to express some spectral functions in terms of others. These tools will help us to eventually establish an asymptotic expansion of the spectral action in Chapter 3.

2.2.1 Mellin Transform

We recall the following deﬁnition (see e.g. [28, Section 3.1.1] or [19, Deﬁnition 1]):

Definition 2.8. The Mellin transform of a locally (Lebesgue) integrable function f : (0,∞) (0,∞) is a complex function M[ f ] defined by → ∞ s 1 M[ f ](s)= f (t)t − dt, (2.9) Z0 1 for all s C for which the integral converges. The inverse Mellin transform M− of a meromorphic∈ function g is defined by

c+i∞ 1 1 s M− [g](t)= 2πi g(s)t− ds, (2.10) c i∞ Z − for some c R such that the integral exists for all t > 0. ∈ The domain of deﬁnition of a Mellin transform turns out to be a strip, called the fun- α β damental strip (see [19, Deﬁnition 1]). If f (t)= 0(t ) and f (t)= ∞(t ) for some α > β, then M[ f ](s) exists at least in the stripO α < ℜ(s) < βO[19, Lemma 1]. The invertibility of Mellin transform is addressed− by the following− [19, Theorem 2]:

Theorem 2.9. Let f be a continuous function. If c (0,∞) belongs to the fundamental strip of M[ f ] and the function R y M[ f∈](c + iy) is integrable, then, for any t > 0 ∋ 7→

c+i∞ 1 1 s f (t)= M− M[ f ] (t)= 2πi M[ f ](s)t− ds. c i∞ Z − 2.2 Functional Transforms 45

In the realm of spectral functions the usefulness of the Mellin transform is attested by the following result:

Proposition 2.10. Let H T p and let K B(H ), then ∈ ∈ t H M[t Tr Ke− ](s)= Γ (s)ζK,H (s), for ℜ(s) > p. (2.11) 7→ Proof. Let us pick any r such that ℜ(s) > r > p. Proposition 2.3 guarantees that t H r t H ∞ t H s 1 Tr Ke− (t)= 0(t− ) and the integral M[t Tr Ke− ](s)= 0 Tr Ke− t − dt converges (absolutely)O at 0. It also converges7→ absolutely at ∞ since R ∞ t H tλ0(H) t(λn(H) λ0(H)) Tr Ke− = e− Tr(P0(H)K)+ ∑ Tr(Pn(H)K)e− − n=1 tλ 0(H) = ∞(e− ) O and the ﬁrst eigenvalue λ0(H) is positive. tλn(H) 1 Since the map (t,n) Tr(Pn(H)K)e− is in L ((0,∞) N) we can use the Fubini theorem to swap7→ the integral and the sum in the following:×

t H M[t Tr Ke− ](s) 7→ ∞ ∞ ∞ ∞ tλn(H) s 1 tλn(H) s 1 = ∑ Tr(Pn(H)K)e− t − dt = ∑ Tr(Pn(H)K) e− t − dt Z0 n=0 n=0 Z0 ∞ ∞ s y s 1 = ∑ Tr(Pn(H)K)λn(H)− e− y − dy = ζK,H (s)Γ (s). n=0 Z0 The domain of deﬁnition is an important ingredient of the Mellin transform: Even if both functions f ,g do have Mellin transforms, M[ f + g] might not exist if DomM[ f ] DomM[g]= /0. Let us illustrate this feature by inspecting the Mellin transform of Formula∩ (2.8).

Example 2.11. Let (A ,H ,D) be a p-summable spectral triple. If D has a non- t D trivial kernel, then the Mellin transform of the heat trace Tr e− | | does not exist. t D Indeed, when µ0(D)= 0, limt ∞ Tr e− | | = dimkerD and the integral (2.10) converges at ∞ only if ℜ(s) < 0, whereas→ the convergence at 0 requires ℜ(s) > p 0. t D t D t ≥ On the other hand, Tr e− | | = Tre− | | (e− 1)dimkerD using (2.8). Notice that both functions of t on the RHS of (−2.8) do− have Mellin transforms, but with disjoint domains. In particular, we have

t M[t (e− 1)](s)= Γ (s) for ℜ(s) ( 1,0) 7→ − ∈ − (see [19, p. 13]).

The relation between the spectral functions unveiled by Proposition 2.10 can be inverted with the help of Theorem 2.9. 46 2 The Toolkit for Computations

Corollary 2.12. Let H T p and let K B(H ). Then,for anyc > pandanyt > 0, ∈ ∈ c+i∞ t H 1 s Tr Ke− = i2π Γ (s)ζK,H (s)t− ds. c i∞ Z − Proof. Recall that the function t Tr Ke t H is smooth (cf. Proposition 2.3). In 7→ − order to apply Theorem 2.9 we need the function y Γ (c + iy)ζK,H (c + iy) to be Lebesgue integrable for any c > p. This is indeed the7→ case as if c > p 0, we have c 1/2 π y /2 ≥ Γ (c + iy)= ∞ y − e (cf. Lemma 3.8), and moreover we estimate O | | − | | c iy c iy c ζK,H (c + iy) = TrKH− − K Tr H− − K TrH− . | | ≤k k ≤k k

2.2.2 Laplace Transform

The Mellin transform allows us to move back and forth between a spectral zeta function and the associated heat trace. In order to build a bridge from these to the spectral action, we will resort to the Laplace transform.

Definition 2.13. The Laplace transform of a function f L1(R+,C) is an analytic function L[ f ] : R+ C defined by ∈ → ∞ sx L[ f ](x) := f (s)e− ds. (2.12) Z0 For our purposes it is sufficient to consider x R+, although (2.12) actually defines an analytic function in the whole half-plane∈ℜ(x) 0. Also, one can consider the 1 R+ ≥ R+ Laplace transform of functions in Lloc( ) (locally integrable on ), as long as there exists M R such that the integral converges (properly) for ℜ(x) > M. More generally,∈ one can define the Laplace transform of a complex Borel measure φ on R+ via the Lebesgue integral

∞ sx L[φ](x) := e− dφ(s), for x > 0. (2.13) Z0 Since we are interested in real functions L[φ] we shall restrict ourselves to the case of Borel signed measures (see for instance [8]) on R+, i.e. σ-additive maps φ : B(R+) R ∞ ,onthe σ-algebraof all Borel subsets of R+. A signed measure may assume→ one∪{± of the} inﬁnite values ∞, but not both. The Hahn–Jordan decomposition allows± us to uniquely write any signed measure + as φ = φ φ − for two non-negative measures, at least one of which is ﬁnite. It is − + + customary to denote φ = φ + φ −, which is a non-negative measure on R called the variation of φ. | | + The support of a signed measure is suppφ := supp φ = suppφ suppφ −. Let us recall that a function f is completely monotonic| | (c.m.) if∪ f C∞((0,∞),R) and ( 1)n f (n)(x) 0 for any n N, and any x > 0. ∈ − ≥ ∈ 2.2 Functional Transforms 47

The set of such functions, denoted by , is well adapted as the range of the Laplace transform: CM Theorem 2.14 (Bernstein, see e.g. [35, p. 160] or [31, Theorem 1.4]). Given a function f we have f (x)= L[φ](x) for all x > 0 for a unique non-negative measure φ ∈on CMR+. Conversely, whenever L[φ](x) < ∞, x > 0, then L[φ] . ∀ ∈ CM The set is a convex cone which is stable under products, derivatives of even order andCM pointwise convergence. Moreover, the closure under pointwise convergence of Laplace transforms of ﬁnite measures on R+ is exactly [31, Corollary 1.6]. The Bernstein theorem naturally extends to the contextCM of signed measures:

+ Corollary 2.15. If f = f f − with c.m. functions f ±, then f (x)= L[φ](x) for all x > 0 for a unique signed measure− φ on R+. Conversely, whenever L[φ](x) < ∞ for all x > 0, then L[φ] can be uniquely written as a difference of two c.m. functions. Proof. This is a straightforward consequence of the Bernstein theorem and the Hahn–Jordan decomposition. The Laplace transform of a measure φ is well defined on the open set R+ 0 . If the measure φ is sufficiently nice, then f = L[φ] along with its derivatives can\{ be} extended to the whole R+: Proposition 2.16. Let φ be a signed measure on R+ and let f (x)= L[φ](x) for ∞ n N x > 0. If φ has a finite n-th moment, i.e. 0 s dφ(s) < ∞ for an n , then (n) n ∞ n ∈ limx 0+ f (x) exists and is equal to ( 1) 0 s dφ(s). → − R Proof. This is a consequence of the LebesgueR dominated convergence theorem: For s,x 0 we have e sx 1, thus ≥ | − |≤ ∞ ∞ sx dφ(s)= lim e− dφ(s)= lim f (x). 0 x 0+ 0 x 0+ Z → Z → The statement for n > 0 follows from a general property of the Laplace transform (cf. [37, Eq. (12)]):

L[s snφ(s)](x) = ( 1)n f (n)(x), for any n N. 7→ − ∈ Remark 2.17. The converse of Proposition 2.16 is not in general true for signed 1/x measures. A counterexample is provided by the function f (x)= 1 e− , which 1/2 1/2 − is the Laplace transform of the function φ(s)= J1(2s )s− , with Jα being the (n) Bessel function of the ﬁrst kind. Although limx 0+ f (x)= 0 for any n N∗, the function snφ(s) is not integrable on R+ for n →1. ∈ ≥ (n) If the measure φ is non-negative then limx 0+ f (x) < ∞ does imply that φ has a ﬁnite n-th moment. This is because if φ 0 one→ can use the monotoneconvergence sx≥ theorem, which implies that limx 0+ e− is integrable with respect to the measure φ (cf. [31, Proposition 1.2]). But→ for signed measures the monotone convergence fails — see [5, p. 177]. 48 2 The Toolkit for Computations

Even more generally, on can consider the Laplace transform of distributions in a suitable class (cf. [10, 17, 36]). Let be the space of smooth compactly supported functions on R endowed with the standardD topology of uniform convergence with all derivatives (cf. [10, Definition 2.1]), with ′ denoting its dual. Furthermore, let denote the space of rapidly decreasing (Schwartz)D functions on R and its dual S S ′ – the space of tempered distributions. We have ′ ′. Recall that a distribution T is said to be null onD⊂S⊂S some open set⊂ DU R if T,ϕ = 0 for all ϕ with suppϕ U. ⊂ h i The support of a distribution⊂ T is the complementof its null set (i.e. the union of all U, on which T is null) (cf. [17, Definition 2.5.2]). We shall denote by +′ the set of all right-sided distributions, i.e. distributions in D R+ ′, the support of which is contained in . Consequently, +′ = +′ ′. D Consider now a smooth function ϕ on R, which has boundedS supportD ∩ S on the left (i.e. ϕ(x)= 0 for all x < m for some m R) and equals 1 in a neighbourhood of [0,∞). Then, for any T , we define ∈ ∈ S +′ sx L[T ](x) := T,s ϕ(s)e− , for x > 0. (2.14) 7→ sx The definition is sound as for any x > 0, the function s ϕ(s)e− is in and since T is supported on R+ the choice of ϕ does not play any7→ role. S sx It is common to use the — somewhat sloppy — notation L[T ](x)= Ts,e . h − i The range of the Laplace transform of distributions in +′ is uncovered by the following result: S

Theorem 2.18 ([37, Theorem 9]). A function f is a Laplace transform of a distribution in +′ if and only if i) f is analyticS in the half-plane ℜ(s) > 0; ii) There exists a polynomial p with f (s) p( s ) for any s C with ℜ(s) > 0. | |≤ | | ∈ + ax Example 2.19. For any a R we have L[δa](x)= e− . In particular, L[δ](x)= 1. ∈ + Since δa is a legitimate measure on R , L[δa] is a c.m. function. d j N ax R+ Let now p(x)= ∑ j=0 c j x for some d and let f (x)= p(x)e− with a . Then, f is not c.m. unless d = 0. Nevertheless,∈ its inverse Laplace transform∈ exists 1 d ( j) and equals L− [ f ]= ∑ j=0 c j δa . Remark 2.20. It would be highly desirable to utilise the Laplace transform to de- tH2 tH duce the behaviour of Tre− knowing that of Tre− . Unfortunately, the Gaus- x2 sian function f (x)= e− does not satisfy the bound demanded in Theorem 2.18 as x2+y2 1 f (x + iy) = e− . Hence L− ( f ) does not exist, even as a distribution. | We also| note that if the function f has compact support in R+ — in particular if it is a smooth cut-off function as depicted in [6, Fig. 1] — then it is not complex analytic in the right half-plane, and hence cannot be a Laplace transform.

The pertinence of the Laplace transform in the context of the spectral action stands from the fact that if f = L[φ] is the Laplace transform of a measure φ, then for any positive selfadjoint (possibly unbounded) operator H, 2.2 Functional Transforms 49 ∞ sH/Λ f (H/Λ)= e− dφ(s). (2.15) Z0 The RHS of the above formula is well deﬁned in the strong operator sense (cf. [29, p. 237]). Moreover,since the trace is normal (i.e. if Hα H B(H ) strongly with H > H for α > α, then Tr H = sup Tr H ), and, if→f (H∈/Λ) is trace-class, then α′ α ′ α α ∞ sH/Λ Tr f (H/Λ)= Tre− dφ(s), (2.16) Z0 Formula (2.16) will allow us to deduce the properties of the spectral action given the corresponding heat trace. Most notably, it will enable us to establish a large energy expansion of S(D, f ,Λ) in Section 3.4. For our plan to succeed, we need to identify the suitable classes of cut-off functions. We deﬁne successively:

+ := f = f f −, f ± f (x) 0, for x > 0 , C { − ∈ CM | ∞ ≥ } n 0 := f = L[φ] n N, s d φ (s) < ∞ , C { ∈C|∀ ∈ 0 | | } Z c := f = L[φ] supp φ is compact , C { ∈ C | } p p := f f ±(x)= ∞(x− ) , for p > 0, Cp { ∈ Cp | O } 0 := 0 , for p > 0, Cp C ∩Cp := c , for p > 0. Cc C ∩C T p r Observe that, if the operator H is in then Lemma 2.7 implies that for f 0 with r > p we have Tr f (H) < ∞. The property f (0) < ∞ is not necessary for f∈C(H) to be trace-class when H is invertible, but f (0) might (and usually does) pop-up as the coefﬁcient in front of t0 in the expansion of Tr f (tH). Clearly, r p , c 0 and for r > p. CM⊂C C ⊂C C ⊂C Furthermore:

Proposition 2.21. For any p,r > 0,

p r p+r and p r p+r. C0 ·C0 ⊂C0 Cc ·Cc ⊂Cc Moreover,

∞ p m f = L[φ] 0 = for all m > p, s d φ (s) < ∞. (2.17) ∈C ⇒ − 0 | | Z Proof. Firstly, we note that since the set is closed under multiplication, so is , and hence p r p+r. Secondly, recallCM that when two signed measures φ, χ onC R+ are finiteC then·C ⊂C their convolution φ χ is defined as [4, Definition 5.4.2] ∗ ∞ ∞ ∞ + f (s)d(φ χ)(s) := f (s +t)dφ(s)dχ(t), for f Cb(R ). 0 ∗ 0 0 ∈ Z Z Z 50 2 The Toolkit for Computations

If L[φ],L[χ] 0, then φ and χ are ﬁnite and, moreover, for any n N, ∈C ∈ ∞ ∞ ∞ snd φ χ (s)= (s +t)nd φ (s)d χ (t) 0 | ∗ | 0 0 | | | | Z Z Z n ∞ ∞ n k n k = ∑ k s d φ (s) t − d χ (t) < ∞. 0 | | 0 | | k=0 Z Z From the general properties of Laplace transform [37, Theorem 10] we have L[φ χ](x)= L[φ](x)L[χ](x), hence we conclude that ∗

0 0 0 and c c c. C ·C ⊂C C ·C ⊂C Formula (2.17) for m ( p,0) follows from Fubini’s theorem: ∈ − ∞ ∞ ∞ m 1 m 1 sx s d φ (s)= Γ ( m)− x− − e− d φ (s)dx 0 | | − 0 0 | | Z Z ∞ Z 1 m 1 + = Γ ( m)− x− − f (x)+ f −(x) dx < ∞. − 0 Z On the other hand, for m 0 the statement follows from the very deﬁnition of 0: ≥ C ∞ 1 ∞ ∞ m m m m m s d φ (s) s⌊ ⌋d φ (s)+ s⌈ ⌉d φ (s) (s⌊ ⌋ + s⌈ ⌉)d φ (s). 0 | | ≤ 0 | | 1 | | ≤ 0 | | Z Z Z Z A pleasant consequence of Proposition 2.21 is that if we have at our disposal a r cut-off function f 0 for some r > 0, then we immediately obtain a cut-off function in p for any p ∈Cr by taking f n for n N, n p/r. C0 ≥ ∈ ≥ ax p Example 2.22. For any a > 0 the c.m. function f (x)= e− = L(δa)(x) is in c for ax bx p C all p > 0. Thus, when b > a > 0, the function f (x)= e− e− is in c for any p > 0, while it is not in . − C If b > a > 0 then also f (CMx)= x 1(e ax e bx)= L(χ )(x) is in p for p > 0. − − − − [a,b] Cc r Example 2.23. Let f (x) = (ax + b)− for a, b, r > 0. Then, one computes that 1 1 r r 1 bs/a r L [ f ](s)= Γ (r) a s e and hence, f , but f / c. − − − − − ∈C0 ∈C 2 Example 2.24. Let f (x)= ex [1 Erf(x)], where the error function Erf is given by 1/2 x y2 − 1 1/2 s2/4 Erf(x)= 2π− 0 e− dy. Then, f with L− [ f ](s)= π− e− . Moreover + 1 ∈CM1 f (x)= f (x)= ∞(x ), hence f . RO − ∈C0 Example 2.25. Let f (x)= e √x. Then, f since L 1[ f ](s)= 1 s 3/2e 1/4s. − ∈CM − 2√π − − On the other hand, limx 0+ f ′(x)= ∞, thus f / 0. Similarly, let f (x)= log→(a + b/x) with a 1∈C, b > 0. Then we have f with 1 1 bs/a ≥ ∈ CM L− [ f ](s)= s− (1 e− )+ log(a)δ(s), but f / 0 as limx 0+ f (x)= ∞. − ∈C → + Every signed measure on R (including ∑i δi) can be seen as a Laplace trans- formable distribution, but the space L[ +′ ] is strictly larger, as illustrated in Exam- ple 2.19. A careful reader might thus askS whether one could extend the class and C 2.3 Weyl’s Law and the Spectral Growth Asymptotics 51 allow f to be the Laplace transform of a distribution in +′ . At the operatorial level it is possible to cook up an analogue of Formula (2.16).S However, the topology of ′ obliges us to control the derivatives of the test functions and the asymptotic expansions,S which are our ultimate objective, do not behave well under differentiation. We come back to this point in Remark 3.23.

2.3 Weyl’s Law and the Spectral Growth Asymptotics

Let ∆ be the Dirichlet Laplacian ∆ on a bounded domain X Rd. In 1911 Hermann Weyl proved that ⊂

d 1 N (Λ)= Vol(S − ) Vol(X) Λ d/2 1 + O(1) as Λ ∞. ∆ (2π)d → This formula, dubbed Weyl’s law, extends to the case of X being a compact connected Riemannian manifold and ∆ the Laplace–Beltrami operator — see [25] for a nice overview. Beyond the commutative world one can deduce the asymptotic behaviour of the spectral growth function of an arbitrary positive operator H, via the Wiener–Ikehara Tauberian theorem, from the properties of the associated spectral zeta function ζH (cf. [3] and references therein, and also [13]). On the other hand, the control of NH (Λ) as Λ tends to inﬁnity yields the leading small-t behaviour of tH 1 the heat trace Tre− (cf. [12, Corollary 3] or [33] and references therein). The control of the sub-leading terms in the asymptotic behaviour of NH (Λ) is a formidable task, even in the context of classical elliptic pdos. In the latter case, (d 1)/2 the remainder is of the order ∞(Λ − ), but in full generality of noncommuta- O d/2 tive geometry one can only expect O∞(Λ ). The lower order terms, which are of great interest from the viewpoint of the spectral action, are accessible (and will be accessed in Chapter 3) provided we trade the sharp cut-off χ[0,1] for a gentler one. Let now (A ,H ,D) be a d-dimensional spectral triple with a simple dimension spectrum Sd = d N and ζD regular at the origin. Then, one deﬁnes [6, Eq. (25)] − d 1 k k ND (Λ) := Λ D − + ζD(0)+ dimkerD. (2.18) h i ∑ k − | | k=1 Z

2 1 This could be justiﬁed via Formula (1.46) using ND (Λ)= ND (Λ) + ∞(Λ − ). The computation of (2.18) is illustrated as follows [6, Propositionh i 1]:O

Proposition 2.26. Given D, assume that specD = Z∗ and that the total multiplicity d 1 j of eigenvalues n is P(n) with a polynomial P(u)= ∑ j=−0 c j u . Then, {± }

1 The cited result provides a better control than Proposition 2.3, but requires some non-trivial assumptions — see [12, Section 5] for a detailed discussion and (counter)examples. 2 Strictly speaking, Formula (1.46) requires f to be the Laplace transform of a signed measure, 1 k 1 which is not the case for the counting function. Nevertheless, naively 0 u − du = 1/k. R 52 2 The Toolkit for Computations

Λ d 1 − D ND (Λ) = P(u)du + ∑ c j ζ( j)+ dimker . h i 0 − Z j=0

1 s ∞ s Proof. The zeta function reads ζD(s)= 2 ∑n Z P(n)n− = ∑n=1 P(n)n− , thus ∈ ∗ d 1 − k ζD(s)= ∑ c j ζ(s j) and D − = Ress=k ζD(s)= ck 1. − − | | − j=0 Z

d 1 Moreover we have ζD(0)= ∑ − c j ζ( j) completing the proof with j=0 −

Λ d 1 − c j j+1 P(u)du = ∑ j+1Λ . Z0 j=0 This result will be extended in Theorem A.8 in Appendix A.6 and applied in the computation of the dimension spectrum of noncommutative tori in Section B.3.1.

2.4 Poisson Summation Formula

Given the Fourier transform of a function g L1(Rm), ∈ m i2π x.y y R F[g](y) := g(x)e− dx, ∈ 7→ Rm Z the usual Poisson formula

1 m i2πt− k.ℓ 1 Rm ∑ g(tk + ℓ)= t− ∑ e F[g](t− k), t > 0, ℓ , (2.19) k Zm k Zm ∀ ∀ ∈ ∈ ∈ is valid under mild assumptions for the decay of g and F[g] [32, VII, Corollary 2.6]. Formula (2.19) is deeply rooted in complex analysis. We have (if, e.g., g ) ∈ S ∞ ∞ ∞ s 1 ζ(s)M[g](s)= ∑ g(tk)t − dt = M ∑ g( k) (s), for ℜ(s) > 1, (2.20) 0 · Z k=1 k=1 ∞ s 1 s ∞ s 1 what can be deduced from 0 g(tk)t − dt = k− 0 g(t)t − dt. In particular, a proofof the Poisson formula(at least for m = 1) can be given by applyingthe inverse R R Mellin transform to (2.20) and invoking the Riemann functional equation [18]. The Poisson summation formulais particularly useful for the spectral action computations if one knows explicitly the singular values µn(D) along with the multiplicities. In favourable cases (which include i.a. the classical spheres and tori) when m µn(D) are indexed by Z for some m N one can rewrite the spectral action as ∈ Tr f ( D /Λ)= ∑n Mn( D ) f (µn/Λ)= ∑k Zm gΛ (k), for some function gΛ . Then, the next| | lemma shows that| | ∈ 2.4 Poisson Summation Formula 53

∞ Tr f ( D /Λ)= F[g ](0)+ ∞(Λ − ). (2.21) | | Λ O

In other words: Given a Schwartz function g, the difference between Rm g(tx) and ∑Zm g(tx) is negligible as t 0. ↓ R Proposition 2.27 (Connes). Given a function g (Rm), deﬁne ∈ S

S(g) := g(k), I(g) := g(x)dx, gt (x) := g(tx) for any t > 0. ∑ Rm k Zm ∈ Z ∞ Then, S(gt)= I(gt )+ 0(t ). O m Proof. Since t− F[g](0)= I(gt ), the Poisson formula (2.19) implies that it sufﬁces to show that for any large n N , ∈ ∗ 1 n S(gt ) I(gt)= ∑ F[g](t− k)= 0(t ). − k Zm 0 O ∈ \{ } n But F[g] is also a Schwartz function, F[g](x) c x − for x = 0 and every n N, so for n large enough, | |≤ | | 6 ∈ 1 1 n n ∑ F[g](t− k) c ∑ t− k − = c′t . k Zm 0 ≤ k Zm 0 | | ∈ \{ } ∈ \{ }

As an example, this shows that

t k 2 t x 2 ∞ m/2 m/2 ∞ e− k k = e− k k dx + 0(t )= π t− + 0(t ). (2.22) ∑ Rm k Zm O O ∈ Z The smoothness of g is indispensable in Proposition 2.27: For instance, the function g x e t x has a Fourier transform F g k 2t and ∞ e t x dx 2 , while ( )= − | | [ ]( )= 4π2k2+t2 ∞ − | | = t − ∞ ∞ R t t k e +1 2 B2n 2n 1 ∑ e− | | = et 1 = t + 2 ∑ (2n)! t − k= ∞ − n=1 − using Bernoulli numbers (cf. p. 55). Proposition 2.27 does not give a way to compute the asymptotics of I(gt ) or S(gt ). But it is hiddenly used in the computation of the heat trace asymptotics of a Laplace type operator P on a compact manifold as in Appendix A: The asymptotics tP tP ∞ t µn(P) of Tre− is not computed directly via Tre− = ∑n=0 e− but, as reminded tP in Appendix A.4, using the integral kernel of e− and an approximation of the resolvent (P λ) 1 by a pseudodifferential operator R(λ), the parametric symbols − − of which rn(x,ξ ,λ) authorise the use of integrals over x and ξ (see [20, Section 1.8]) without any reference to the discrete summation ∑n. Let us now illustrate the usefulness of the Poisson summation formula for the spectral action computations: 54 2 The Toolkit for Computations

Example 2.28. If D is the Dirac operator on the sphere Sd, then by Formula (B.1),

D d/2 +1 n+d 1 d Tr f ( /Λ)= ∑ 2⌊ ⌋ d −1 f ((n + 2 )/Λ). | | n N − ∈ Let us set d = 3,asin[6]. Assuming that f is an even Schwartz function and check- ing that 1/2 are not in the spectrum, we get, via Equation (2.19), ± Tr f ( D /Λ)= 2 ∑ (n + 1)(n + 2) f ((n + 3/2)/Λ)= ∑ k(k + 1) f ((k + 1/2)/Λ) | | n N k Z ∈ ∈ k = ∑ gΛ (k + 1/2)= ∑ ( 1) F[gΛ ](k), k Z k Z − ∈ ∈ where g (x) := (x 1/2)(x + 1/2) f (x/Λ). As already seen in Formula (2.21), we Λ − only need to compute F[gΛ ](0) and the asymptotics of the action is

3 2 1 ∞ Tr f ( D /Λ)= Λ x f (x)dx Λ f (x)dx + ∞(Λ − ). (2.23) | | R − 4 R O Z Z Example 2.29. Consider the spectral triple (C∞(Td ),L2(Td,S ),D/) and assume again that d = 3. By (B.2), the spectrum of D/ is the set of all values 2π k + ℓ where k varies in Z3 while ℓ R3 is ﬁxed and given by the chosen spin± structurek k ∈ (s1,...,sd ). There is no degeneracy, apart from zero which is a double eigenvalue. Again, to simplify we assume that f (R) and, as computed in [27], ∈ S D ∞ Tr f ( /Λ)= 2 ∑ gΛ (k)= 2 ∑ F[gΛ ](k)= 2F(g)(0)+ ∞(Λ − ), | | k Z3 k Z3 O ∈ ∈ with g (x) := f (2π x + ℓ /Λ). Since F[g ](0)= R3 f (2π x + ℓ /Λ)dx, we get Λ k k Λ k k R 1 3 ∞ Tr f ( D /Λ)= 3 Λ f ( x )dx + ∞(Λ − ). (2.24) | | 4π R3 k k O Z In Example 2.29, one can see that the spectrum of D does depend on the choice of the spin structure and so does the spectral action. But this choice disappears in (2.24), so that the dependence on the spin structure is asymptotically negligible. Is this a general fact? See the associated Problem 10) in Chapter 5. The Poisson formula has been widely employed in the computation of the spectral action, as explained in Section 1.8. It requires, however, a complete knowledge of the spectrumof D. The latter is available for a few Dirac operatorson Riemannian manifold, like the spheres, tori, Bieberbach manifolds, quotients of Lie groups, like SU(n), etc. The reader should notice that the calculations can be tricky, especially when swapping between f and g in (2.21). Although the Poisson summation formula is exact, we focused on its application for the study of asymptotics. Of course, since every summation formula is associated with an asymptotic formula [17], one can investigate the Taylor–Maclaurin or Euler– Maclaurin asymptotic formulae. For instance, the latter reads [7], forany2 m N ≥ ∈ 2.4 Poisson Summation Formula 55

N N m [g(0)+g(N)] B j ( j 1) ( j 1) ∑ g(k)= g(x)dx + 2 + ∑ j! [g − (N) g − (0)] + Rm, (2.25) 0 − k=0 Z j=2 N ( 1)m+1 (m) with Rm := − m! g (x)Bm(x x )dx, 0 −⌊ ⌋ Z where B j = 2 jζ(1 2 j) are the Bernoulli numbers, with B2 j+1 = 0 for j N∗ − 1 − 1 1 ∈ so that B2 = 6 , B4 = 30 ,B6 = 42 ,..., and B j(x) are the Bernoulli polynomials deﬁned by induction− (cf.− [1, Chap. 23]):

1 B0(x)= 1, B′j(x)= jB j 1(x), B j(x)dx = 0. − Z0 When N ∞ and 7 m N, → ≤ ∈ ∞ ∞ 2 (m) (m) Rm (2π)m ζ(m) g (x) dx g (x) dx. | |≤ 0 | | ≤ 0 | | Z Z Example 2.30. Consider the Dirac operator on the sphere S4,asin[7]. According to Appendix B.1, we have µn( D/ n N = k N and the eigenvalue k has multi- 4 3 { | | | ∈ } { ∈ } plicity 3 (k k) (thus 0, 1 are not eigenvalues).The Euler–Maclaurinformula (2.25) − D 4 ∞ 3 can be used for Tr f ( /Λ)= 3 ∑k=0 g(k), with g(x) = (x x) f (x/Λ). Assuming | | 4 m (m) 3−m 1 (k) again f ,weget Rm = ∞(Λ − ) because g (Λu)= Λ − Pm(u,Λ − , f (u)), ∈ S 1 (k) O 1 (k) where Pm(u,Λ − , f (u)) is a polynomial in u, Λ − and a ﬁnite number of f (u). Since g(2 j)(0)= 0 for all j N and ∈ (3) 1 g′(0)= f (0), g (0)= 6 f (0) 6Λ − f ′(0), − − (5) 1 2 g (0)= 120Λ − f ′(0) 60Λ − f ′′(0) − etc., we obtain

3 D 4 Tr f ( /Λ) |∞ | 3 1 1 1 2 1 2 = (x x) f (x/Λ)dx + ( 12 + 120 ) f (0) + ( 120Λ − 252Λ − ) f ′(0)+ ..., 0 − − − Z which means ∞ ∞ D 4 4 3 4 2 11 Tr f ( /Λ)= 3Λ u f (u)du 3 Λ u f (u)du + 90 f (0) | | 0 − 0 Z ∞ Z 2m (m) ∞ + ∑ cm Λ − f (0)+ ∞(Λ − ). (2.26) m=1 O

The coefﬁcients cm can be computed explicitly: c1 = 31/1890, c2 = 41/7560, c3 = 31/11880, etc. − Similarly, the Euler–Maclaurin formula can be utilised in the same way when the singular values of D are polynomials in k N with polynomial multiplicity in k. ∈ 56 2 The Toolkit for Computations

Some authors use the name “non-perturbative spectral action” for expressions ∞ like (2.26) after erasing the remainder ∞(Λ − ). We warn the Reader that the ‘ex- ∞ O ponentially small’ term ∞(Λ ) can hide the devils, as argued in Section 1.8. O −

2.5 Asymptotic Expansions

The bulk of the physical information encoded in the spectral action can be read out from its asymptotic behaviour at large energies. The leading term established in Sec- tion 2.3 provides rather scarce information. We would like to recognise the leading, subleading, etc. terms of the action and to control the remainder in a sensible way. It would be most desirable to have a Laurent expansion

∞ D D k S( , f ,Λ)= ∑ ak( f , )Λ − , (2.27) k= N − the convergence of which for Λ larger than some Λmin would guarantee that con- sidering higher and higher order terms provides a better approximation of the true, non-perturbative, action S(D, f ,Λ). Unfortunately, we are granted such a level of precision only for rather specific geometries — see Section 2.6. Typically, the best what one could hope for is a control of the order of the remainder when the series (2.27) is truncated to the first, say, M terms. This desire is formalised with the help of the notion of an asymptotic series. It firstly requires the following definition:

Definition 2.31. Let (ϕk)k be a (finite or infinite) sequence of functions on a punctured neighbourhood of x0. We call it an asymptotic scale at x0 if, for any k, ϕk(x) = 0 for x = x0 and ϕk 1(x)= Ox (ϕk(x)). 6 6 + 0 R n Example 2.32. For any x0 the sequence (x x0) n N, known from the Taylor ∈ − ∈ rn series, is an asymptotic scale at x0. More generally, one could take (x x0) for any − complex rn’s with ℜ(rn) ∞. The logarithmic termsր can also be taken into account: For instance, the sequence 1 2 1 1 2 x− log x, x− logx, x− ,log x, logx, 1,... defines an asymptotic scale at x0 = 0. { On the other hand, the sequence π , x 1}cosx, x 2 sinx, x 3 cosx,... is not an { 2 − − − } asymptotic scale at x0 = ∞, as, for instance, the limit of ϕ3(x)/ϕ2(x) for x ∞ does not exist. →

Deﬁnition 2.33. Let (ϕk)k N be an asymptotic scale at x0 and g a complex function ∈ on some punctured neighbourhood of x0. We say that g has an asymptotic expansion with respect to (ϕk)k when there exists a sequence of functions (ρk)k N such that ∈ N O N ρk(x)= x0 (ϕk(x)), g(x) ∑ ρk(x)= x0 (ϕN (x)), for any N . O − k=0 ∈ In this case, we write 2.5 Asymptotic Expansions 57

∞ g(x) ρk(x) (2.28) x x ∑ →∼ 0 k=0 ∞ and the symbol ∑k=0 ρk(x) is called an asymptotic series of g at x0. Remark 2.34. Let us warn the reader that the adopted Definition 2.33 is sometimes called in the literature an extended asymptotic expansion (cf. [17, Definition 1.3.3]). In the standard approach (see for instance [14]) one assumes that ρk(x)= ckϕk(x) with some complex constants ck. An example of a function which does not have an asymptotic expansion at infinity in this standard sense, but does have one in the x 1 extended sense is S(x)= 0 t− sint dt — see [17, Example 8]. Observe also that Definition 2.33 uses a flexible, though somewhat slippery, no- R tation – two different pairs of scale functions (ϕk)k and coefficients (ρk)k can give the same expansion (2.28) around x0 for a given function g (cf. Example 2.37). In our venture through the meanders of noncommutative geometry we will en- counter situations in which the spectral action S(D, f ,Λ) exhibits oscillations at large energies — see Example 2.39. To study its asymptotics, we will need the full force of Definition 2.33.

Let g : R R be a smooth function around x = 0. Then g always admits a Taylor → ∞ g(k)(0) k expansion g(x) x 0 ∑k=0 k! x . However, if g is not analytic, its Taylor series ∼ → ∞ s 2 2 1 is only an asymptotic one. As an example consider g(x)= 0 e− (1 + s x )− ds. It is smooth at x = 0 and g(x) ∞ ( 1)k(2k)!x2k, but this expansion is clearly x 0 ∑k=0 R not a convergent series. ∼ → − Let us also note that whereas the functions ϕk of the asymptotic scale must be non-zero in some punctured neighbourhood V of x0, the coefﬁcients of the expansion — i.e. the functions ρk — can actually vanish in the whole V. A standard 1/x example is provided by the function h(x) := e− for x > 0, h(x) := 0 for x 0. The function h, being smooth at x = 0, admits a Taylor expansion, i.e. an asymp-≤ k (k) totic expansion with respect to the asymptotic scale (x )k. But, as h (0)= 0 for ∞ every k N, we have h(x) x 0 0, i.e. h(x)= 0(x ). Note, however, that we ∈ 1/xk ∼ → O could take ϕk(x)= e− as the asymptotic scale and write a trivial expansion 1/x h(x) x 0 e− + O0(ϕN (x)) for any N N∗. The∼ casus→ of the latter function h shows∈ that an asymptotic expansion can never determine a function uniquely, even if it converges. Indeed, for any function g smooth at zero, the functions g and g + h will have the same asymptotic expansions k with respect to the scale (x )k.

Remark 2.35. Asymptotic expansions can be combined by linearity and multiplication. They can also be integrated, both with respect to the variable x, as well as to some external parameters. On the other hand, asymptotic expansions (even the standard — “non-extended”— ones) do not behave well under differentiation. This is because the only information available about the remainder is its ∞ behaviour, O which is scarce: Although f (x)= x0 (g(x)) but, in general, f ′(x) = x0 (g′(x)), as exempliﬁed by O 6 O 58 2 The Toolkit for Computations

1 1 f (x)= xsin(x− )= 0(x), f ′(x)= 0(x− ) = 0(1). O O 6 O For more details on the properties of asymptotic series see [17, Section 1.4] and references therein. Example 2.36. As in Example 1.2 let us consider an elliptic selfadjoint differential operator H of order m acting on a vector bundle E over a closed (i.e. compact without boundary) d-dimensional Riemannian manifold M and let K C∞(End(E)) be an auxiliary smooth endomorphism. If H is positive, then [21, Theorem∈ 1.3.5]

∞ t H (k d)/m TrKe− ak(K,H)t − . (2.29) t∼0 ∑ ↓ k=0

(k d)/m (k d)/m The natural asymptotic scale is ϕk(t)= t − , whereas ρk(t)= ak(K,H)t − in accordance with [20, Deﬁnition (1.8.10)]. Example 2.37. More generally, if P Ψ m(M,E) is positive, elliptic with m > 0, then ∈ ∞ ∞ tP (k d)/m ℓ Tre− ak(P)t − + bℓ(P)t logt. t∼0 ∑ ∑ ↓ k=0 ℓ=0 See Corollary A.7 in Appendix A.5 for the full story and a detailed proof. Note that this expansion can be read as the one with respect to the asymptotic scale t d/m logt, t d/m, t( d+1)/m logt, t( d+1)/m,... , with constant coefﬁcients { − − − − } c2 j+1 = a j(P) and c2 j = 0 for j < d, c2 j = b j(P) for j d. ≥ ( j d)/m ε Alternatively, one can set the asymptotic scale ϕ j(t)= t − − for some small ε > 0 and take ( j d)/m N (a j + b( j d)/m logt)t − for j d m , ρ j(t)= ( j d)/−m − ∈ a j t for j d / mN, − − ∈ to satisfy ρ j(t)= 0(ϕ j(t)). This can be furtherO generalised: If Q is a log-polyhomogeneous pdo of order q,

∞ r+1 tP k ( j q d)/m Tr Qe− ∑ ∑ a j,k(Q,P) log t t − − , t∼0 ↓ j=0 k=0 where r is the degree of log-polyhomogeneity of Q (cf. [26, Section 3]). In fact, as suggested on p. 168 of [26] (see also [23, p. 488]), this result should survive even if P is also log-polyhomogeneous rather than a classical one. Remark 2.38. Let P be an elliptic positive pseudodifferential and let Q = P + R, where R is a smoothing pdo. 1 1 1 1 Then, (P λ)− and (Q λ)− = (P λ)− [1 R(Q λ)− ] have the same parametrix.− Moreover, if P−satisﬁes Hypothesis− A.1−, so does−Q. Thus the difference tP t Q ∞ between the two pdo’s e− and e− (see Appendix A.4) is 0(t ). This means tP t Q O that Tre− and Tre− have the same asymptotics. This coincidence also follows from the Duhamel formula (4.13). 2.6 Convergent Expansions 59

Finally, let us present an example of an asymptotic expansion of the spectral action on the standard Podles´ sphere, which justiﬁes the need of the full force of Deﬁnition 2.33.

Example 2.39. Let Dq be the Dirac operator on the standard Podles´ sphere (cf. Ap- pendix B.4). For a suitable cut-off function f the spectral action admits a large- energy asymptotic expansion

∞ 2 S(Dq, f ,Λ) a 2πi (2.30) Λ +∞ ∑ ∑ ∑ 2k+ j,n ∼ k=0 n=0 j Z − logq × → ∈ n 2πi n m n n m 2k+ logq j ( 1) − f 2πi (logΛ) − Λ − . ∑ m 2k+ j,m × m=0 − − logq The numbers fz are the generalised moments of the cut-off function and the az,n’s can be computed explicitly in terms of the residues of ζDq —see[11] for the details. As in Example 2.37 the expansion (2.30) can be consider either with respect 2 2 2 2 2 to the asymptotic scale log Λ, logΛ, 1, Λ − log Λ, Λ − logΛ, Λ − ,... or with ε 2k { } respect to (Λ − )k N for any ε > 0. In both cases, the coefﬁcients ρk(Λ) of the asymptotic expansion∈ are rather involved functions of Λ given in terms of absolutely convergent Fourier series in the variable logΛ.

2.6 Convergent Expansions

Surprisingly enough, it turns out that the asymptotic series in Formula (2.30) is actually convergent for all Λ > 0 and, moreover, one could write a genuine equality in place of the asymptotic expansion symbol [11, Theorem 5.2]. The following deﬁni- tion provides a general framework for such situations.

∞ Deﬁnition 2.40. Let ∑k=0 ρk(x) be an asymptotic series of a function g at x0 with respect to an asymptotic scale (ϕk)k. We say that the asymptotic expansion of g at x0 is (absolutely / uniformly) conver- ∞ gent if the series ∑k=0 ρk(x) converges (absolutely/ uniformly) in some punctured neighbourhood V of x0. In this case, for x V , ∈ ∞ O N g(x)= ∑ ρk(x)+ R∞(x), with R∞(x)= x0 (ϕk(x)) for all k . k=0 ∈

If, moreover, R∞ = 0 then the expansion is called exact.

∞ Let us note that since ∑k=0 ρk(x) is not, in general, a Taylor series, the domain of its absolute convergence can be strictly smaller than the domain of conditional convergence — cf. page 41. In this light, the expansion presented in Example 2.39 is exact for all Λ > 0. Let us illustrate Deﬁnition 2.40 with two further examples of geometric origin: 60 2 The Toolkit for Computations

Example 2.41. Let D/ be the standard Dirac operator on S1 associated with the trivial spin structure (see Appendix B.1). Then,

∞ tD/ 2 tn2 t Tre− = ∑ e− = ϑ3(0;e− ) for t > 0 n= ∞ − where ∞ iπτ n2 2niz ϑ3(z;q = e )= ϑ3(z τ) := ∑ q e | n= ∞ − for z C, q < 1 is a Jacobi theta function(see [34, p. 464]) which enjoys the Jacobi identity∈ [34| ,| p. 475]

1/2 z2/πiτ 1 1 ϑ3(z τ) = ( iτ)− e ϑ3 zτ− τ− . (2.31) | − |− Equation (2.31) implies that, for any t > 0,

∞ t D/2 π 1/2 π2/t π 1/2 π 1/2 π2n2/t π 1/2 ∞ Tre− = ( t ) ϑ3(0;e− ) = ( t ) + 2( t ) ∑ e− = ( t ) + 0(t ). n=1 O The above asymptotic expansion is ﬁnite (thus absolutely and uniformly convergent t > 0), but it is not exact. In fact, the asymptotic expansion of the heat trace asso- ∀ciated with D/ 2 – the square of the standard Dirac operator on any odd dimensional sphere Sd is ﬁnite, but not exact — see Example 3.15 and [12, Corollary 2].

Example 2.42. Let again D/ be the standard Dirac operator on S1 associated with the trivial spin structure (see Appendix B.1). Then, for t > 0,

∞ t D/ tn t Tre− | | = 1 + 2 ∑ e− = coth 2 . (2.32) n=1 The function coth is complex analytic in the annulus s (0,π) and its Laurent expansion at s = 0 reads [1, (4.5.67)]: | | ∈

∞ 2k 1 2 B2k 2k 1 coths = s− + ∑ (2k)! s − , (2.33) k=1 where Bk are the Bernoulli numbers (cf. page 55). Hence, previous equality provides t D/ an asymptotic expansion for Tre− | | as t 0, which is convergent (absolutely and uniformly on compacts) for t (0,2π) and,↓ moreover, exact. On the side we remark that∈ (2.32) actually provides a meromorphic continuation t D/ of Tre− | | to the whole complex plane. 2.7 On the Asymptotics of Distributions 61 2.7 On the Asymptotics of Distributions

As explained in Section 2.2.2, the Laplace transform technique applied to the spectral action computations restricts the admissible cut-off function. In this section we brieﬂy sketch an alternative approach, via the asymptotics of distributions following [17] (see also [16, 15] and[22, Setion 7.4]), which works for f (R). For m N, let ∈ S ∈ ∞ n α m α n m := f C (R ,C) ∂ f (x) cα (1 + x ) −| |, mult-ind. α N K { ∈ | | |≤ k k ∀ ∈ } be equipped with the topology deﬁned by the family of seminorms

k m α f k,m := sup max(1, x − ) ∂ f (x) with α = k . k k x Rn{ k k | | | | } ∈ α In particular, f m implies ∂ f m α . ∈ K ∈ K −| | Now, let be the inductive limit of the spaces m as m tends to inﬁnity. Remark thatK all polynomials are in and the SchwartzK space is dense in so that K S K we can consider the dual ′ as a subset of tempered distributions. We remark that PropositionK 2.27 still holds true for f (Rn) and such that ∈ K Rn f (x)dx is deﬁned — see [16, Lemma 2.10]. More importantly, a distribution T (R) is in ′(R) if and only if (see [17, Theorem 6.7.1]) R ∈ D ′ K

∞ n ( 1) µn (n) n T (λx) ∑ − n+1 δ (x), with µn := T, x – the moments of T (2.34) λ∼∞ n!λ h i → n=0 in weak sense:

N For any f T λx f x µn f (n) 0 λ N 2 for any N N , ( ), ( ) = ∑ n!λ n+1 ( )+ ∞( − − ) ∗. ∈ K h i n=0 O ∈ Notice that such a moment asymptotic expansion of a distribution harmonises with the notion of the asymptotic expansion pondered in Section 2.5, as for f , 1 ∞ µn (n) n+1 ∈ K we have T (t− x), f (x) ∑n=0 f (0)t . h it ∼0 n! Formula (2.34) is directly↓ related to the Cesàro behaviour of distributions (cf. [17, Section 6.3]): Given a distribution T (R) and β R ( N ) one writes ∈ D ′ ∈ \ − ∗ β T (x) := ∞(x ) (C) O if there exist N N, a primitive TN of T of order N and a polynomial p of degree at ∈ most N 1, such that TN is locally integrable for large x and we have, in the usual − N+β β sense, TN (x)= p(x)+ ∞(x ). Similarly, one deﬁnes T(x)= O∞(x ) (C) and, O ∞ β consequently, T (x) := O∞(x− ) (C) means that T (x) := O∞(x− ) (C) for all β. ∞ It turns out that T ′ if and only if T(x)= O ∞( x − ) (C) (cf. [17, Theorem 6.7.1]), which shows∈ Kthat the asymptotic behaviour± (2.34| | ) at inﬁnity is in fact equivalent to the Cesàro behaviour. 62 2 The Toolkit for Computations

A good illustration of these notions is the following (see [17, Example 164]).

Example 2.43. A distribution S ′(R) is said to be distributionally smooth at 0 when S(n)(0) exists (in the sense∈ of S Stanisław Łojasiewicz) for n N — a property ∈ equivalent to the fact that its Fourier transform is distributionally small: F(S) ′. ∈ K Let now ξ C with ξ = 1 and assume first that ξ = 1. Let T ′ be defined ∈ n | | n k 6 ∈ K as T (x)= ∑n Z ξ δn(x). We have µk = ∑n Z ξ n = 0 (C). Thus, if S ′ is ∈ n ∈ ∞ ∈ S distributionally smooth at 0, then ∑n Z ξ S(nx)= O0(x ) in ′. ∈ D But when ξ = 1, the Dirac comb ∑n Z δn isnotin ′. Nevertheless, ∑n Z δn 1 ′ ∈ K ∈ − ∈ K and, still in ′: D 1 ∞ S(nx)= x− S(u)du + O0(x ), ∑ R n Z ∈ Z provided that R S(u)du is defined.

To justifyR this incursion within distributions, let us consider, as in [17, Sec- tion 6.16], a (possibly unbounded) selfadjoint operator on H . Its spectral decom- ∞ position reads: H = ∞ λ dPλ (H) and determines the spectral density − R dPλ (H) dH (λ) := dλ understood as a distribution from valued in B(DomH,H ). D ′ Given any function f one gets f (H)= dH (λ), f (λ) λ and it is natural to use the following notation∈ D h i dH (λ)=: δ(λ H). − ∞ n n n Observe now that, with X := Dom H , we have dH (λ), λ = H , which ∩n=1 h i means that actually δ(λ H) ′(R,B(X,H )). Hence, Formula (2.34) yields − ∈ K ∞ ( 1)n n (n) δ(λσ H) − H δ (λ) σ ∞ ∑ n!σ n+1 − ∼→ n=0 and, moreover, δ(λ H) vanishes to inﬁnite order at ∞ in the Cesàro sense, namely − ± ∞ δ(λ H)= O ∞( λ − ) (C). − ± | | R We now take a detour through [16]: Let T +′ ( ) and assume the following Cesàro asymptotic expansion: ∈ S

∞ ∞ αn j T (λ) ∑ cnλ + ∑ b jλ − (C), (2.35) λ∼∞ → n=1 j=1 where αn R N constitute a decreasing sequence. Then, by [17, Theorem 32], ∈ \−

∞ ∞ ∞ n (n) αn j j ( 1) µn δ (λ ) T(λΛ) cn (λ+Λ) + b jλ − Pf[(λΛ)− H(λ)] + − n+1 , Λ∼∞ ∑ ∑ ∑ n!Λ → n=1 j=1 n=0 2.7 On the Asymptotics of Distributions 63

∞ αn ∞ j n where µn = T (x) ∑n=1 cn x+ ∑ j=1 b j Pf [x− H(x)], x are the “generalised moments”. Here,h Pf− is the Hadamard− ﬁnite part: i

∞ Pf [h(x)], f := F. p. h(x) f (x)dx h i 0 Z and the index + means that the integral deﬁning the duality is restricted to R+. Thus,

∞ j j Pf [x− H(x)], f (x) = F. p. f (x)x− dx h i 0 Z ∞ 1 j 1 j 2 j − f (n)(0) n j − f (n)(0) = f (x)x− dx + [ f (x) ∑ n! x ]x− dx ∑ k!( j n 1) 1 0 − − Z Z n=0 n=0 − − j and, consequently, Pf [x− H(x)] is homogeneous of degree j up to a logarithmic term: −

j j j ( 1) jδ ( j 1)(λ ) Pf λΛ H λΛ Λ Pf λ H λ − − logΛ [( )− ( )] = − [ − ( )] + ( j 1)!Λ j . − Finally,

∞ ∞ αk 1 αk T (λ), f (tλ) λ ∑ ck t− − F. p. λ f (λ)dλ (2.36) h i t∼0 0 ↓ k=1 Z ∞ ∞ ( j 1) ∞ (n) b t j f (λ ) d f − (0) t µn f (0) tn + ∑ j F. p. λ j λ ( j 1)! log + ∑ n! . 0 − j=1 Z − n=0 Formula (2.36) can be utilised to obtain an asymptotic expansion of the spectral action computation, at least in the commutative case: Let P be a positive elliptic pdo of order k > 0 ona d-dimensional compact manifold M. Then, the Schwartz kernel dP(x,y;λ) of the operator δ(λ H) enjoys the following Cesàro expansion on the diagonal (cf. [16, Formula (4.5)]):−

∞ (d k n)/k dP(x,x;λ) ∑ an(x)λ − − (C), (2.37) λ∼∞ → n=0 where 1 d/k a0(x)dµ(x)= k WRes P− , ZM see (1.34). The existence of such an expansion is based on the following Ansatz: With σ denoting the total symbol,

∞ (n) σ[δ(λ P)] ∑ cn δ (λ σ(P)). − λ∼∞ − → n=0 It is important to stress that the asymptotics (2.37) cannot, in general, be integrated term by term in λ (see [16]). But we are free to integrate over x M: Given an f (R), we write ∈ ∈ S 64 2 The Toolkit for Computations

1 S(P, f ,Λ)= dP(x,x;λ), f (λΛ − ) λ dx Mh i Z and plug in Formula (2.36) with (2.37) to obtain an asymptotic expansion of S(P, f ,Λ) for large Λ. x In particular, with f (x)= e− (thus f ), one can recover as in [16, Corollary 6.1] the celebrated heat kernel expansion∈ for K an elliptic pdo, which we derive from the scratch in Corollary A.7 (with a generalisation to pdos with non-scalar symbols). Example 2.44. Let (A ,H ,D/) be as in Example 1.2 with dimM = 4. For the generalised Laplacian P = D/ 2 we get

dD2 (x,x;λ) c0 λ + c1(x) (C), / λ∼∞ → which gives

∞ ∞ 2 2 4 2 Tr f (D/ /Λ ) c0 Λ dλλ f (λ)+ c2 Λ dλ f (λ) Λ∼∞ 0 0 → Z ∞ Z n (n) 2n + ∑ ( 1) f (0)c2n+4 Λ − n=0 − (cf. [16, p. 247]). Such an expansion works for f . If we would attempt to use ∈ S f = χ[0,1] — the counting function — we would discover that the expansion is not valid after the ﬁrst term. Nevertheless, the full expansion, beyond the ﬁrst term is valid in the Cesàro sense; see also [15] for more details. It would be highly desirable to implement this approach beyond the commutative world — see Problem 9) in Chapter 5.

References

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11. Eckstein, M., Iochum, B., Sitarz, A.: Heat trace and spectral action on the standard Podles´ sphere. Communications in Mathematical Physics 332(2), 627–668 (2014) 12. Eckstein, M., Zaj ˛ac, A.: Asymptotic and exact expansions of heat traces. Mathematical Physics, Analysis and Geometry 18(1), 1–44 (2015) 13. Elizalde, E.: Ten Physical Applications of Spectral Zeta Functions, second edn. Springer (2012) 14. Erdélyi, A.: Asymptotic Expansions. Courier Dover Publications (1956) 15. Estrada, R., Fulling, S.A.: Distributional asymptotic expansions of spectral functions and of the associated Green kernels. Electronic Journal of Differential Equations 07, 1–37 (1999) 16. Estrada, R., Gracia-Bondía, J.M., Várilly, J.C.: On summability of distributions and spectral geometry. Communications in Mathematical Physics 191(1), 219–248 (1998) 17. Estrada, R., Kanwal, R.P.: A Distributional Approach to Asymptotics: Theory and Applica- tions. Springer (2002) 18. Feauveau, J.C.: A uniﬁed approach for summation formulae. ArXiv:1604.05578 [math.CV] 19. Flajolet, P., Gourdon, X., Dumas, P.: Mellin transforms and asymptotics: harmonic sums. The- oretical Computer Science 144(1), 3–58 (1995) 20. Gilkey, P.B.: Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem, second edn. Studies in Advanced Mathematics. CRC press Boca Raton (1995) 21. Gilkey, P.B.: Asymptotic Formulae in Spectral Geometry. CRC Press (2004) 22. Gracia-Bondía, J.M., Várilly, J.C., Figueroa, H.: Elements of Noncommutative Geometry. Springer (2001) 23. Grubb, G., Seeley, R.T.: Weakly parametric pseudodifferential operators and Atiyah–Patodi– Singer boundary problems. Inventiones Mathematicae 121(1), 481–529 (1995) 24. Hardy, G.H., Riesz, M.: The General Theory of Dirichlet’s Series. Courier Dover Publications (2013) 25. Ivrii, V.: 100 years of Weyl’s law. Bulletin of Mathematical Sciences 6, 379–452 (2016) 26. Lesch, M.: On the noncommutative residue for pseudodifferential operators with log- polyhomogeneous symbols. Annals of Global Analysis and Geometry 17(2), 151–187 (1999) 27. Marcolli, M., Pierpaoli, E., Teh, K.: The spectral action and cosmic topology. Communications in Mathematical Physics 304(1), 125–174 (2011) 28. Paris, R.B., Kaminski, D.: Asymptotics and Mellin–Barnes Integrals, Encyclopedia of Math- ematics and its Applications, vol. 85. Cambridge University Press (2001) 29. Reed, M., Simon, B.: Methods of Modern Mathematical Physics: Vol.: 2.: Fourier Analysis, Self-adjointness. Academic press (1972) 30. Rudin, W.: Functional Analysis. McGraw-Hill (1973) 31. Schilling, R.L., Song, R., Vondracek, Z.: Bernstein Functions: Theory and Applications, second edn. De Gruyter (2012) 32. Stein, E.M., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton Uni- versity Press (1971) 33. Usachev, A., Sukochev, F., Zanin, D.: Singular traces and residues of the ζ-function. Indiana Univ. Math. J. 66, 1107–1144 (2017) 34. Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis, fourth edn. Cambridge Univer- sity Press (1927) 35. Widder, D.V.: The Laplace Transform. Princeton University Press (1946) 36. Zemanian, A.H.: Distribution Theory and Transform Analysis: an Introduction to Generalized Functions, with Applications. Dover Publications (1965) 37. Zemanian, A.H.: The distributional Laplace and Mellin transformations. SIAM Journal on Applied Mathematics 14(1), 41–59 (1966)

Chapter 3 Analytic Properties of Spectral Functions

Abstract In the previous chapter we have witnessed the interplay between the spectral zeta functions and the associated heat traces (cf. Proposition 2.10). We have also learned, in Section 2.2.2, how to exploit the Laplace transform to compute the spectral action from a given heat trace. In this chapter we further explore the connections between the spectral functions unravelling the intimate relationship between the meromorphic continuation of a zeta function and the asymptotic expansion of the corresponding heat trace. We utilise the latter to establish the sought asymptotic expansion of the spectral action at large energies. Finally, we ponder the possibility of obtaining convergent, rather than only asymptotic, formulae for this action. Let us ﬁrst ﬁx some notations. Let f be a meromorphic function on some domain W C. For any subset V W we shall denote by P( f ,V ) the set of poles of f contained⊂ in V. If f is meromorphic⊂ on C we abbreviate P( f ) := P( f ,C). As in the previous chapter, let H T p and let K B(H ). For further convenience we adopt the following notation:∈ ∈

K,H (s) := Γ (s)ζK,H (s), for ℜ(s) > p. Z In accordance with the conventions adopter after Formula (2.2), we keep the simpli- p ﬁed notation K,H for H non-positive but enjoying H T , and set Z | | ∈

H := 1,H . Z Z

If ζK,H admits a meromorphic extension to a larger region of the complex plane, then so does K,H and we keep denoting the meromorphic extension of K,H by the same symbol.Z Z Recall also that Γ is holomorphic on C ( N) and has ﬁrst order poles at n N, n 1 \ − − ∈ with Ress= n Γ (s) = ( 1) (n!)− . Hence, P( K,H )= P(ζK,H ) ( N). − − Z ∪ −

67 68 3 Analytic Properties of Spectral Functions 3.1 From Heat Traces to Zeta Functions

As illustrated by Examples 2.36, 2.37 in the realm of pseudodifferential operators in classical differential geometry we are vested with an asymptotic expansion of the t H heat trace Tr Ke− for small t. It turns out that such an expansion can be utilised to establish a meromorphic extension of the corresponding zeta function ζK,H . As an example, we quote the following result: Theorem 3.1 ([15, Theorem 1.12.2]). Let K and H be differential operators as in Example 2.36. Then, the function K,H admits a meromorphic extension to C with (possibly removable) simple polesZ at s = (d k)/mfor k N and − ∈

Res K,H (s)= ak(K,H). s=(d k)/m Z − This agreeable interplay, which essentially relies on the Mellin transform as cap- tured by Proposition 2.10, extends to the setting of abstract operators on a sepa- t H rable Hilbert space. In this context the asymptotic expansion of Tr Ke− would, in general, be more involved including terms in logk t and/or proportional to ti — exhibiting log-periodic oscillations as t tends to 0. Theorem 3.2. Let H T p and let K B(H ). Assume that there exist d N, a ∈ ∈ ∈ sequence (rk)k N R strictly increasing to +∞ and a discrete set X C without accumulation points,∈ ⊂ such that ⊂

∞ d t H n z Tr Ke− ∑ ρk(t), with ρk(t) := ∑ ∑ az,n(K,H)log t t− , (3.1) t∼0 ↓ k=0 z Xk n=0 ∈ where Xk := z X rk+1 < ℜ(z) < rk ,X = kXk and the series deﬁning ρk(t) is absolutely{ convergent∈ |− for any t > 0 and− } any k ∪N. ∈ Then, the function ζK,H admits a meromorphic extension to the whole complex plane with the poles of order at most d + 1 and P( K,H ) X. Moreover, for any z X andn 0,1,...,d Z, ⊂ ∈ ∈{ } n n Res (s z) K,H (s) = ( 1) n! az,n(K,H). (3.2) s=z − Z − Before we prove the theorem, three comments concerning the notation are in order: Firstly, the set Xk = zi i Z comes endowed with a lexicographical order: { } ∈

zi z j iff ℑ(zi) < ℑ(z j) or ℑ(zi)= ℑ(z j) and ℜ(zi) ℜ(z j), ≤ Z what makes it isomorphic— as a poset — to . Thus, the symbol ∑z Xk f (z) should J ∈ be read as limJ ∞ ∑ j= J f (z j). Since the assumed absolute convergence of the series implies that→ the limit− does not depend on the grouping or permutation of terms, the notation ∑z Xk is unambiguous. Secondly, the∈ asymptotic expansion (3.1) should be understood with the respect r r to the asymptotic scale (t k )k, i.e. we should have ρk(t)= 0(t k ) (recall Defini- O 3.1 From Heat Traces to Zeta Functions 69 tion 2.33 and Remark 2.34). The latter is not an additional assumption and it follows from the assumed form of ρk, as we shall see in the course of the proof. Thirdly, the sequence (rk)k is needed only to fix an asymptotic scale and can be adjusted at will, provided the absolute convergence of the resulting ρk’s remains unharmed. In particular, since max ℜ(z) z X = p we can choose r0 arbitrarily close to p (see Figure 3.1). What matters{ | in∈ the} final result (3.2) are the coefficients − az,n(K,H) and not how we distribute them into ρk’s (cf. Remark 2.34 and Exam- ple 2.37) and the poles of K,H depend only on X and not on its partition by Xk’s. Z

ℑ(z)

ℜ(z)

r3 r2 r1 r0 − − − −

Fig. 3.1 Illustration explaining the notations in Theorem 3.2: The discrete set X C is partitioned ⊂ into a disjoint sum, X = Xk, with the help of the vertical lines ℜ(z)= rk. ⊔ −

1 ∞ s 1 t H Proof. For ℜ(s) > p we have ζK,H (s)= Γ (s) 0 t − Tr Ke− dt on the strength of Proposition 2.10. For such s C we can split ζK,H (s)= F0(s)+ F1(s), with ∈ R 1 ∞ 1 s 1 t H 1 s 1 t H F0(s) := Γ (s) t − Tr Ke− dt, F1(s) := Γ (s) t − Tr Ke− dt. Z0 Z1 t H t H λ (H)t Since Tr Ke− is smooth on (0,∞) and Tr Ke− = ∞(e− 0 ), the function F1 C ∞ s 1 λ0 t O s is actually holomorphic on since 0 t − e− dt = Γ (s)λ0− . p δ Note ﬁrst that Proposition 2.3 impliesR ρ0(t)= 0(t− − ) for every δ > 0, hence r O r ρ0(t)= O0(t 0 ), and we now show that actually ρk(t)= O0(t k ) for every k N. ∈ 70 3 Analytic Properties of Spectral Functions

To see this let us ﬁx the index k N and let us pick a t0 > 0. As the series over Xk is ∈ absolutely convergent for t > 0, we have, when t t0 and n 0,...,d , ≤ ∈{ } rk z ck ck t− az,n(K,H)t− az,n(K,H) t az,n(K,H) t , ∑ ≤ ∑ | | ≤ ∑ | | 0 z Xk z Xk z Xk ∈ ∈ ∈ where c : sup r ℜ z 0. k = z Xk ( k + ( )) > − ∈ 1/c Hence, for any ε > 0 there exists t0 = [ε/∑ az,n ] k > 0 such that for every t t0, rk z | | ≤ t− ∑z X az,n(K,H)t− ε, so that ∈ k ≤

rk z lim t− ∑ az,n(K,H)t− = 0. t 0 | z X | → ∈ k

Since rk < ℜ(z) for z Xk, we can ﬁnd ε′ > 0 such that rk + ε′ < ℜ(z) and − z ∈ rk+ε − ∑z X az,n(K,H)t− = O0(t ′ ). | ∈ k | n ε′ n z O rk Hence, as log t = 0(t− ) for any ε′ > 0, logt ∑z Xk az,n(K,H)t− = 0(t ) O r | | | ∈ | for n N. Thus indeed, ρk(t)= O0(t k ). Let∈ us now ﬁx N N and invoke the asymptotic expansion (3.1) to provide the ∈ meromorphic continuation of F0 to the half-plane ℜ(s) > rN . For ℜ(s) > p, − N 1 s 1 F0(s)= ∑ t − ρk(t)dt + EN(s), k=0 Z0 where

1 N s 1 t H EN(s) := t − RN(t)dt and RN (t) := Tr Ke− ∑ ρk(t). 0 − Z k=0 r But since RN(t)= O0(t N ), EN (s) is actually holomorphic for ℜ(s) > rN . On the 1 s 1 − other hand, for ℜ(s) > r0 > p, 0 t − ρk(t)dt is absolutely convergent for any k N (and so is the series− deﬁning ρ ), so that we can swap the integral with the R k series∈

1 d 1 s 1 s z 1 n t − ρk(t)dt = ∑ ∑ az,n(K,H) t − − log t dt 0 z X n=0 0 Z ∈ k Z d ( 1)nn! a K H − (3.3) = ∑ ∑ z,n( , ) (s z)n+1 . z X n=0 − ∈ k n 1 Now, we claim that, the function s ∑z X az,n(K,H)(s z)− − is holomor- 7→ ∈ k − phic on C Xk for any k N,n 0,1,...,d , and hence Formula (3.3) provides \ ∈ ∈{ } 1 s 1 a meromorphic extension of the complex function s t ρk(t)dt to the whole 7→ 0 − complex plane. Indeed, for any s / Xk there exists δ > 0 such that s z δ for all ∈ R | − |≥ z Xk, since Xk does not have accumulation points. For any such s we have ∈ 3.1 From Heat Traces to Zeta Functions 71

n 1 n 1 az,n(K,H)(s z)− − δ − − az,n(K,H) < ∞. ∑ − ≤ ∑ | | z Xk z Xk ∈ ∈

Summarising, for any ﬁxed N N we have ∈ N d 1 ( 1)nn! ζ s a K H − E s F s K,H ( )= Γ (s) ∑ ∑ ∑ z,n( , ) (s z)n+1 + N( )+ 1( ), k=0 z X n=0 − ∈ k which is a meromorphic function for ℜ(s) > rN . As N can be taken arbitrarily − large and 1/Γ is meromorphic on C, we conclude that ζK,H is also meromorphic on C. Moreover, the above reasoning shows that the function s Γ (s)ζK,H (s) can only have poles in X of order at most d + 1and (3.2) follows immediately.7→

As illustrated in Figure 3.1 (see also, for instance, [19, Figure 2.10]) the set of poles of a spectral zeta function ζK,H does not need, in general, to exhibit any symmetries. However, the geometric origin of the operators K and H can force a highly regular shape of the set X.

Example 3.3. In particular, as follows from Example 2.37, if K and H are classical pseudodifferential operators on a Riemannian manifold M then ζK,H admits a meromorphic extension to the whole complex plane with isolated simple poles located in (dimM + ordK N)/ordH R (cf. [16]). − ⊂ If K is allowed to be a polyhomogeneous pdo, then the poles of ζK,H need no longer be simple, but are still contained in the highly regular discrete subset of the real line (dimM + ordK N)/ordH — see[20]. − Example 3.4. Let us illustrate an application of Theorem 3.2 in the context of the noncommutative torus (see Appendix B.3):

Proposition 3.5. Let (AΘ ,H ,D) be the spectral triple of the noncommutative d- torus. Then, for any a AΘ , the spectral zeta function ζa,D admits a meromorphic extension to the whole∈ complex plane with a single simple pole located at s = d/2. d/2 d/2 1 Moreover, Ress=d/2 ζa,D = τ(a)2⌊ ⌋π Γ (d/2)− and ζa,D(0)= 0. Proof. In order to apply Theorem 3.2 we will show that the correspondingheat trace 2 Tr ae t D admits an asymptotic expansion as t 0. We have − ↓ d/2 2⌊ ⌋ t D2 t D2 Tr ae− = ∑ ∑ Uk e j, ae− Uk e j k Zd j=0 h ⊗ ⊗ i ∈ d/2 2⌊ ⌋ t k 2 d/2 t k 2 = ∑ ∑ e− k k Uk,aUk e j, e j = 2⌊ ⌋ ∑ τ(Uk∗aUk)e− k k k Zd j=0 h ih i k Zd ∈ ∈ d/2 t k 2 t D2 = 2⌊ ⌋ τ(a) ∑ e− k k = τ(a) Tr e− . k Zd ∈ 72 3 Analytic Properties of Spectral Functions

x 2 Invoking Proposition 2.27 with g(x)= e−k k we obtain

d/2 t D2 t1/2k 2 1/2 2−⌊ ⌋ Tr ae− = τ(a) ∑ e−k k = τ(a) ∑g(t k) k Zd k ∈ t1/2x 2 ∞ d/2 ∞ = τ(a) e−k k dx + 0(t )= τ(a)t− + 0(t ). x Rd O O Z ∈ d/2 d/2 Since dimKerD = 2⌊ ⌋ and Tr P0 a = τ(a)2⌊ ⌋ thanks to (B.7), Formula (2.8) 2 2 gives: Tr ae t D = τ(a)[Tr e t D + (e t 1)2 d/2 ] and hence − − − − ⌊ ⌋ t D2 d/2 d/2 d/2 t ∞ Tr ae− = τ(a)2⌊ ⌋ π t− + (e− 1)+ 0(t ) − O d/2 d/2 d/2 t τ(a)2⌊ ⌋π t− + τ(a)(e− 1)dimKer D. t∼0 − ↓ Since the above expansion does not contain any logt terms, Theorem 3.2 yields a meromorphic extension of ζa,D to the whole complex plane with simple poles only. To locate the poles we read out the non-zero coefﬁcients from Formula (3.1)

2 d/2 d/2 2 d/2 ( 1)k N ad/2,0(a,D )= 2⌊ ⌋π τ(a), a k,0(a,D )= 2⌊ ⌋ −k τ(a), for k ∗. − ! ∈ Thanks to Formula (3.2) we have

d/2 d/2 1 Ress=d/2 ζa,D(s)= 2⌊ ⌋π Γ (d/2)− τ(a).

2 Finally, since a0,0(a,D )= 0 and Ress=0 Γ (s)= 1, we conclude from (3.2) that ζa,D(0)= 0.

Remark that no Diophantine restriction on the matrix Θ (see Deﬁnition A.10) was needed in the previous result — in contrast with the more general Theorem B.2.

As mentioned on p. 19, the complex poles of the function ζK,H can appear, for instance, when fractal geometry is involved. We shall see an explicit example in Section 3.3. Let us now turn to the converse of Theorem 3.2.

3.2 From Zeta Functions to Heat Traces

In the context of pseudodifferential operators in classical Riemannian geometry one has at one’s disposal the powerful existence theorems about the asymptotic expansion of heat traces. From these one can construct the meromorphic continuations of the corresponding spectral zeta functions via Theorem 3.2. The original proof for classical pdos (see Appendix A) heavily relies on the symbolic calculus and integral kernels of pdos [15]. Unfortunately, these tools are not available in the noncommutative realm. On the other hand, one can hopethat given a meromorphic extension of 3.2 From Zeta Functions to Heat Traces 73 a spectral zeta function ζK,H , guaranteed, for instance, by the dimension spectrum property, one could deduce the existence (and the form) of an asymptotic expansion t H of Tr Ke− . This is indeed possible, however, one needs to control not only the local structure of ζK,H (s) around the poles, but also its asymptotic behaviour on the verticals – as ℑ(s) ∞. | | →

3.2.1 Finite Number of Poles in Vertical Strips

Let us first handle the case when ζK,H has a finite number of poles in vertical strips, i.e. for any Ua,b = z C a < ℜ(z) < b the set P(ζK,H ,Ua,b) is finite. This is always the case when{ ∈K and| H are classical} pdos, but it often occurs also beyond the classical geometry — for instance in the case of noncommutative torus (cf. Ap- pendix B.3) or the SUq(2) quantum group [8, 18].

Theorem 3.6. Let H T p and let K B(H ). Assume that: ∈ ∈ i) The function ζK,H admits a meromorphic extension to C with a ﬁnite number of poles in vertical strips and of order at most d. ii) For any ﬁxed x r0 there exists an ε(x) > 0, such that ≤ 1 ε(x) K,H (x + iy)= ∞( y − − ). (3.4) Z O | |

Fix a sequence (rk)k N R, increasing to +∞ with p < r0, giving a partition of ∈ ⊂ − P( K,H ) as P( K,H )= kXk with Z Z ⊔

Xk := z P( K,H ) rk 1 < ℜ(z) < rk . { ∈ Z |− + − } r Then, there exists an asymptotic expansion with respect to the scale (t k )k,

∞ t H Tr Ke− ∑ ρk(t), (3.5) t∼0 ↓ k=0 d n z with ρk(t) := ∑ ∑ az,n(K,H)log t t− (3.6) z X n=0 ∈ k ( 1)n n az,n(K,H) := − Res (s z) K,H (s). (3.7) n! s=z − Z

Proof. We slice the complex plane with the help of the chosen sequence (rk)k as shown on Figure 3.1. Since ζK,H (and hence K,H ) has a ﬁnite number of poles in Z vertical strips, for any k N there exists Yk > 0 such that K,H is holomorphic for ∈ Z ℑ(s) Yk. For each k let us denote | |≥ k D := rectangle s C rk 1 ℜ(s) rk, Yk ℑ(s) Yk , { ∈ |− + ≤ ≤− − ≤ ≤ } k so that Xk = P( K,H , D ). Z 74 3 Analytic Properties of Spectral Functions

Let us ﬁx the index k. By construction, the function K,H is regular at the boundary of any Dk. Hence, the residue theorem yields Z

1 s s K,H (s)t− ds = Res K,H (s)t− , (3.8) i2π k ∑ s z ∂ D Z z X = Z Z ∈ k where the contour ∂Dk is oriented counter-clockwise. Let us decompose the boundary of the rectangle using

rk+iYk + − s IV (k) := K,H (s)t− ds, r iY Z Z− k− k r +iY − k+1 k s IV−(k) := K,H (s)t− ds, r iY Z Z− k+1− k r iY − k± k s IH±(k) := K,H (s)t− ds, r iY Z Z− k+1± k so that s + + K,H (s)t− ds = IV (k) IH (k) IV−(k)+ IH−(k). ∂ Dk Z − − Z We are now concerned with the limit of Equation (3.8) as Yk ∞. Firstly, by assumption ii), we have, →

∞ 1 1 rk+1 iy Rk(t) := i2π lim IV−(k)= 2π K,H ( rk+1 + iy)t − dy < ∞, Yk ∞ ∞ Z − → Z− for any k N and any t > 0. We also have ∈ ∞ 1 + 1 r0 iy t H R 1(t) := i2π lim IV (0)= 2π K,H ( r0 + iy)t − dy = Tr Ke− − Y0 ∞ ∞ Z − → Z− on the strength of Corollary 2.12. r Moreover, for any k N, Rk(t)= O0(t k+1 ): Indeed, with F denoting the Fourier transform, we have ∈

∞ rk+1 1 iy Rk(t)t− = 2π K,H ( rk+1 + iy)t− dy ∞ Z − Z− 1 logt = F[y K,H ( rk+1 + iy)]( ) 0. (3.9) 2π 7→ Z − 2π t→0 ↓ The latter statement is a consequence of the Riemann–Lebesgue Lemma, since the 1 function y K,H ( rk + iy) is in L (R,dy) for each rk by hypothesis (3.4). Secondly,7→ hypothesis Z − (3.4) guarantees that for any k N and any t > 0 ∈ r − k x IH±(k) K,H (x iYk) t− dx ≤ r |Z ± | Z− k+1 r − k x sup K,H (x iYk) t− dx 0. Y ≤ x [ r , r ] |Z ± | rk+1 −−−→k ∞ ∈ − k+1 − k Z− → 3.2 From Zeta Functions to Heat Traces 75

Finally, let us rewrite the residues more explicitly. By assumption i) and the fact that the function Γ has only simple poles, the function K,H admits a Laurent ex- d+1 n n Z pansion K,H (s)= ∑n= ∞( 1) n!az,n(K,H)(s z)− in some open punctured disc Z − − − with the center at any z P( K,H ), in accordance with Formula (3.7). On the other hand, ∈ Z

∞ s zlogt (s z)logt z ( 1)n logn t n C t− = e− e− − = t− ∑ − n! (s z) , for any s,z , t > 0. n=0 − ∈ Therefore, d s z n Res K,H (s)t− = t− az,n(K,H)log t. s=z ∑ Z n=0 Hence, for any k N and any t > 0 Equation (3.8) yields ∈

Rk 1(t) Rk(t)= ρk(t). (3.10) − − r r Since Rk(t)= O0(t k+1 ), it follows that ρk(t)= O0(t k ) for any k N. The latter ∈ can also be easily deduced from the form of ρk given by Formula (3.6). Starting with Formula (3.10) for k = 0 and iterating it N times we obtain the t H N announced asymptotic expansion Tr Ke− = ∑k=0 ρk(t)+ RN(t).

Remark 3.7. Suppose that, for a chosen sequence (rk)k, we would require only the Lebesgue integrability of K,H on the verticals ℜ(z)= rk in place of the stronger constraint (3.4) — as weZ will do in Theorem 3.12. Then,− the theorem would still hold, but the asymptotic expansion (3.5) might get an additional contribution from the horizontal contour integrals, namely

∞ t H Tr Ke− ρk(t)+ gk(t) t∼0 ∑ ↓ k=0 with

rk 1 − iy iy x gk(t)= i2π lim K,H (x + iy)t− K,H (x iy)t t− dx, (3.11) y ∞ r Z − Z − → Z− k+1 r which is smooth for t > 0 and 0(t k ) as follows from Formula (3.8). O Let us note that even imposing a suitable decay rate of K,H on ℜ(z)= rk is not Z − sufﬁcientto getrid of the gk’s, as the Phragmén–Lindelöfinterpolation argument (cf. [24, Section 12.7]) ﬁrstly requires that K,H does not grow too fast on any vertical Z in the segments ℜ(z) [ rk 1, rk]. This is not a priori guaranteed. ∈ − + − In order to estimate the behaviour of K,H on the verticals, it is useful to recall the vertical decay rates of the Γ -function.Z 76 3 Analytic Properties of Spectral Functions

Lemma 3.8. With x,y R we have ∈ x 1/2 π y /2 ( y − e ), for x > 1/2, x iy ∞ − | | Γ ( + )= O | |π y /2 ∞(e ), for x 1/2. (O − | | ≤ Proof. The case x 1/2is standard [22, (2.1.19)].To provethe other part we invoke the Γ reﬂection formula≥

1 Γ (z)Γ (1 z)= π sin− (πz), for z C Z, (3.12) − ∈ \ along with a lower bound [6, p. 51]

1/2 Γ (x + iy) cosh(πy)− Γ (x), for x 1/2, y = 0. (3.13) | |≥ ≥ 6 For x 1/2, x / Z/2 and y = 0 we estimate: ≤ ∈ 6 1 1 Γ (x + iy) = π sin(π(1 x) iπy) − Γ (1 x iy) − | | | − − | | − − | 2 2 2 2 1/2 1 = π [sin (πx)cosh (πy)+ cos (πx)sinh (πy)] Γ (1 x iy) − | − − | π 1/2 1/2 1 [ sin(πx) cos(πx) sinh(πy) cosh(πy)]− cosh(πy) Γ (1 x)− ≤ √2 | | − π 1 1/2 1/2 Γ (1 x)− sin(πx)cos(πx) − sinh(πy) − . ≤ √2 − | | | | y π y /2 As sinh(y)= ∞(e| |), it follows that Γ (x+iy)= ∞ e− | | for x 1/2, x / Z/2 and the constraintO x / Z/2 can be dropped by continuity.O ≤ ∈ ∈ An explicit control on the growth rate of the zeta function ζK,H yields the following important corollary of Theorem 3.6:

p Corollary 3.9. Let H T , K B(H ) and let ζK,H meet the assumptions of The- ∈ ∈ orem 3.6. Moreover, assume that ζK,H is of at most polynomial growth on the verticals, i.e. for any ﬁxed x r0 there exists c(x) < ∞, such that ≤ c(x) ζK,H (x + iy)= ∞( y ). (3.14) O | | tHq Then, for any q > 0, there exists an asymptotic expansion of TrKe− as t 0 of the form (3.5). ↓

qs Proof. Observe that, for ℜ(s) > p/q, ζK,Hq (s)= TrKH− = ζK,H (qs). Then, the assumption of the polynomial growth on the verticals of ζK,H , together with Lemma 3.8, is sufﬁcient to conclude.

There is an important lesson for noncommutative geometers coming from the above result:

Remark 3.10. Let (A ,H ,D) be a regular spectral triple of dimension p. If, for a 0 given T Ψ (A ), the zeta function ζT,D has a meromorphic extension to C with a polynomial∈ growth on the verticals, then the small-t asymptotic expansions of both 3.2 From Zeta Functions to Heat Traces 77

t D tD2 TrTe− | | and TrTe− exist. Nevertheless, the two expansions can be radically different — compare Examples 3.11 and 3.15 with Example 3.18.

We indicated in Section 2.1 that the spectral zeta functions are actually complex functions deﬁned by general Dirichlet series. Estimating the growth rate of the meromorphic continuation of these is, in general, a formidable task, which sometimes relates to profound conjectures in number theory [14, p. 10]. For example, it is fairly easy to show [25, Chapter V] that

0 ∞( y ), for 1 < x, O | |(1 x)/2 ζ(x + iy)=  ∞( y − ), for 0 x 1, (3.15) O | |1/2 x ≤ ≤  ∞( y − ), for x < 0. O | |  1/4 It implies, in particular, ζ(1/2 + iy)= ∞( y ). O | | ε However, the Lindelöf Hypothesis states that actually ζ(1/2 + iy)= ∞( y ) for any ε > 0. The current best estimate of the critical exponent is 13/84[4O]. | | Although if H and K are classical pseudodifferential operators the small-t asymp- t H totic expansion of Tr Ke− is guaranteed by Theorem A.6, it is instructive to apply Theorem 3.6 in the context of the classical geometry of 2-sphere.

Example 3.11. Let D/ be the standard Dirac operator on S2 (see Appendix B.1). From Formula (B.1) we deduce

∞ 2s+1 ζD/2 (s)= 4 ∑ (n + 1)− = 4ζ(2s 1), n=0 − which is meromorphic on C with a single simple pole at s = 1. We thus have

k k ( 1) ( 1) B2k+2 Res D2 (s)= 2, and Res D2 (s)= 4 − ζ( 2k 1)= 4 − . s=1 Z / s= k Z / k! − − − k! 2k+2 −

As the partitioning sequence we can choose, for instance, rk = 3/2 + k, k N. Formula (3.15) guarantees that the assumption (3.14) is met and− Corollary 3.9∈yields

2 ∞ k t D/ 1 ( 1) B2k+2 k Tre− 2t− 4 − t . t∼0 − ∑ k! 2k+2 ↓ k=0

k+1 k This series is divergent for any t > 0 since ( 1) B2k > 2(2k)!/(2π) for k N∗ [1, (23.1.15)]. In fact, the asymptotic expansion− of the heat trace associated with∈ the square of the standard Dirac operator on any even dimensional sphere is divergent [12, Proposition 11]. 78 3 Analytic Properties of Spectral Functions 3.2.2 Inﬁnite Number of Poles in Vertical Strips

For general noncommutative geometries the spectral zeta functions might have an infinite number of poles in the vertical strips. This feature is always present when fractal geometry is in play — see, for instance, [17] — and was detected also on the standard Podles´ sphere (cf. Appendix B.4 and [11]). Theorem 3.6 carries over to this context, but the proof needs to be refined as now each term of the asymptotic expansion (3.6) can itself be an infinite series, the convergence of which is a subtle issue. Theorem 3.12. Let H T p and let K B(H ). Assume that: ∈ ∈ i) The function ζK,H admits a meromorphic extension to the whole complex plane with poles of order at most d. ii) There exists a sequence (rk)k N R strictly increasing to +∞ with p < r0, ∈ ⊂ − such that the function K,H is regular and Lebesgue integrable on the verti- Z cals ℜ(s)= rk. iii) For any k N−, t > 0 and n 0,1,...,d the series ∈ ∈{ } n z Res (s z) K,H (s)t− , (3.16) ∑ s z z X = − Z ∈ k

with Xk := z P( K,H ) rk 1 < ℜ(z) < rk , is absolutely convergent. { ∈ Z |− + − } r Then, there exists an asymptotic expansion, with respect to the scale (t k )k ,

∞ t H Tr Ke− ρk(t)+ gk(t) , (3.17) t∼0 ∑ ↓ k=0 d n z with ρk(t) := ∑ ∑ az,n(K,H)log t t− (3.18) z X n=0 ∈ k ( 1)n n az,n(K,H) := − Res (s z) K,H (s) (3.19) n! s=z − Z and the functions gk can be expressed as

rk (k) (k) 1 − (k) iym (k) iym x gk(t)= i2π lim K,H (x + iym )t− K,H (x iym )t t− dx, (3.20) m ∞ r Z − Z − → Z− k+1 (k) (k) via an (always existing) sequence (ym )m N strictly increasing to +∞ with y0 = 0 ∈ (k) and such that K,H is regular on the horizontals x iym with x [ rk 1, rk]. Z ± ∈ − + − (k) The introduction of the sequences (ym )m, illustrated in Figure 3.2, is coerced by the need to sum over the residues of an infinite number of poles in each strip. The (k) final result does not depend on the particular choice of the (ym )m’s and, actually, one could allow for curved lines instead of simple straight horizontals. Observe also that, similarly as in Theorem 3.2, the only role of the sequence (rk)k is to fix an asymptotic scale: If we choose another sequence (rk′ )k, for which 3.2 From Zeta Functions to Heat Traces 79 the assumptions ii) and iii) are met, we will obtain the same asymptotic expansion (3.17) — recall Remark 2.34.

ℑ(z)

(0) y2

(1) y2

(0) y1

(1) y1

ℜ(z) y(1) − 1

y(0) − 1 y(1) − 2 y(0) − 2

r3 r2 r1 p r0 − − − −

Fig. 3.2 An illustration of the meromorphic structure of a function ZK,H together with notations adopted in the proof of Theorem 3.12.

(k) Proof. For each k N, let us choose a sequence (ym )m N with the properties re- ∈ ∈ quired by the theorem. Such a sequence can always be found since ζK,H is meromorphic (and so is K,H ) and hence the sets Xk do not have accumulation points. Z k For each k let us denote D0 := /0and

k (k) (k) D := rectangle s C rk 1 ℜ(s) rk, ym ℑ(s) ym , for m N∗. m { ∈ |− + ≤ ≤− − ≤ ≤ } ∈ Now, let us ﬁx the indices k and m. (k) By construction of the sequences (rk)k and (ym )m, the function K,H is regular at k Z the boundary of any Dm. Hence, the residue theorem yields

1 s s K,H (s)t− ds = Res K,H (s)t− , (3.21) i2π k ∑ s z ∂ Dm Z k = Z Z z P(ZK H ,D ) ∈ , m 80 3 Analytic Properties of Spectral Functions

k where the contour ∂Dm is oriented counter-clockwise.In the above sum only a ﬁnite k number of residues is taken into account, as the region Dm is bounded. Let us decompose the boundary of the rectangle using

(k) rk+iym I+ k m : − s t s ds V ( , ) = (k) K,H ( ) − , r iym Z Z− k− (k) rk+1+iym I k m : − s t s ds V−( , ) = (k) K,H ( ) − , r iym Z Z− k+1− (k) rk iym I k m : − ± s t s ds H±( , ) = (k) K,H ( ) − , r iym Z Z− k+1± s + + so that k (s)t ds = I (k,m) I (k,m) I (k,m)+ I (k,m). ∂ Dm K,H − V H V− H− We are nowZ concerned with the limit of− Equation− (3.21) as m ∞. R Firstly, as in the proof of Theorem 3.6, we deduce from assumption→ ii) that

∞ 1 1 rk+1 iy O rk+1 Rk(t) := i2π lim IV−(k,m)= 2π K,H ( rk+1 + iy)t − dy = 0(t ), m ∞ ∞ Z − → Z− 1 + t H R 1(t) := lim IV (0,m)= Tr Ke− . − i2π m ∞ → Secondly, for any k N and any t > 0, we are allowed to write ∈ s s lim Res K,H (s)t− = Res K,H (s)t− = ρk(t), m ∞ ∑ s z ∑ s z k = Z = Z → z P(ZK H ,D ) z Xk ∈ , m ∈ as the sum over the (possibly infinite) sets Xk is absolutely convergent by assumption iii). In particular, the limit m ∞ of the sum of residues does not depend on the (k) → choice of the sequence (ym )m. Finally, for any k N and any t > 0 Equation (3.21) yields ∈ 1 + Rk 1(t) Rk(t) ρk(t)= i2π lim [IH (k,m) IH−(k,m)]. (3.22) − m ∞ − − → − Since the LHS of (3.22) is a well-defined function of k N and t (0,∞), the RHS must be so, hence, in particular, the limit m ∞ exists∈ and is finite∈ for any fixed value of k and t. We denote it by →

1 + gk(t) := lim [I (k,m) I−(k,m)], i2π m ∞ H H → − which accords with Formula (3.20). Following the same arguments as in the proof of Theorem 3.2 one can deduce r r r that ρk(t)= O0(t k ) for any k N. Since Rk(t)= O0(t k+1 ), we get gk(t)= O0(t k ) for any k N. ∈ Starting∈ with Formula (3.22) for k = 0 and iterating it N times we obtain the t H N announced asymptotic expansion: Tr Ke− = ∑k=0[ρk(t)+ gk(t)] + RN(t). 3.2 From Zeta Functions to Heat Traces 81

The assumptions in Theorem 3.12 are actually weaker than the ones adopted in Theorem 3.6 as we have only imposed Lebesgue integrability on the verticals ℜ(s)= rk, rather than demanding concrete decay rates in whole segments as in − Formula (3.4). The price for that are the possible additional terms gk in the asymptotic expansion (3.17) coming from the contour integrals (3.20) and not from the poles of K,H . This should not be a surprise in view of Remark 3.7. It is also clear Z that if K,H decays on the verticals, i.e. |Z | (k) lim K,H (x + iym ) = 0, for all k N, (3.23) m ∞ → |Z | ∈ then gk = 0 for every k. When (3.23) is assumed, the limit m ∞ of Formula (3.21) reads → s Rk 1(t) Rk(t)= lim Res K,H (s)t− . (3.24) m ∞ ∑ s z − − k = Z → z P(ZK H ,D ) ∈ , m It implies that the sum of residues converges for any k N, t > 0 and, moreover, ∈ (k) that the limit m ∞ does not depend on the chosen sequence (ym )m. Hence, one might be tempted→ to conclude that the assumption iii) is actually redundant when the constraint (3.23) is met. However, Formula (3.24) guarantees the conditional convergenceonly and we are not allowed to write the RHS of (3.24) simply as ρk(t) given by (3.16). More precisely, (3.24) says that the series of residues converges if k k we group the terms into the sets P( K,H , Dm+1 Dm) and add them subsequently with the increasing counting index mZ. Adopting a\ more stringent constraint

(k) 1 εk sup K,H (x + iym ) = ∞( m − − ), for some εk > 0, x [ r , r ] |Z | O | | ∈ − k+1 − k which looks natural when compared with assumption ii) of Theorem 3.6, does imply that ∞ s Res K,H (s)t− < ∞, for any k N, t > 0 ∑ ∑ s=z k k Z ∈ m=0 z P(ZK H ,D D ) ∈ , m+1\ m

k k (cf. [12, Proposition 4]). But the grouping of terms into P( K,H , Dm+1 Dm) might turn out indispensable. Such a situation might produce itselfZ for instance\ if the meromorphic structure of the function K,H is as on Figure 3.3. On the other hand, we are notZ able to cook up a concrete example of a function K,H , for which such a pathology occurs. So, it might well be that the geometric originZ of the operators K,H prevents such situations — see Problem 6) b). However, we will shortly witness, in Section 3.3.1, a computationof the asymptotic expansion, in which the (vertical) contour integral does give a ﬁnite, non-zero, contribution. We conclude this section with a friendly noncommutative-geometric example. D S Example 3.13. Let q be the simpliﬁed Dirac operator on the standard Podles´ sphere (cf. Appendix B.4) and let us denote u = w q(1 q2) 1 > 0. Using the | | − − 82 3 Analytic Properties of Spectral Functions

ℑ(z)

(k) y4

(k) y3

(k) y2

(k) y1

ℜ(z) y(k) − 1

y(k) − 2

y(k) − 3

y(k) − 4

rk+1 rk − −

Fig. 3.3 An illustration of a possible pathological structure of poles of a meromorphic function. The sum of residues might turn out ﬁnite if the poles are grouped by two, but inﬁnite if they are added one by one. It is unclear whether such a situation can actually produce itself for a spectral zeta function of geometrical origin.

S explicit formula for the eigenvalues of Dq we can easily compute its spectral zeta function. For ℜ(s) > 0 we have: | |

ℓ ℓ+1/2 q− s 1 s s(ℓ 1/2) ζ S (s)= w − = 2(uq ) (2ℓ + 1)q Dq ∑ ∑ ∑ 1 q2 − − ∑ − +, ℓ N+1/2 m= ℓ | | − ℓ N+1/2 { −} ∈ − ∈ ∞ 1 s sn 1 s s 2 = 4(uq− )− ∑ (n + 1)q = 4(uq− )− (1 q )− . n=0 − The latter equality, which is nothing but an elementary summation of a geometric series, provides the meromorphic extension of ζ S to the whole complex plane. It Dq has an inﬁnite number of poles of second order regularly spaced on the imaginary axis. The following meromorphic structure of the function DS emerges: Z q a third order pole at s = 0, • second order poles at s = 2πi j/(logq), with j Z∗, • ﬁrst order poles at s = k, with k N. ∈ • − ∈ S The corresponding residues az,n(1, D ), given by Formula (3.7), read | q | 3.3 Convergent Expansions of Heat Traces 83

a = 2 , a = 4 (logu + γ), 0,2 log2 q 0,1 log2 q 1 2 1 2 1 2 2 a0 0 = 2log u + π log q + 4γ logu + 2γ , , log2 q 3 − 3 4 2πi j/logq 2πi a = u− Γ ( j), for j Z∗, 2πi j/logq,1 − log2 q logq ∈ 4 2πi j/logq 2πi 2πi a = u− Γ ( j)[logu ψ( j)], for j Z∗, 2πi j/logq,0 − log2 q logq − logq ∈ where γ stands for the Euler’s constant and ψ := Γ ′/Γ is the digamma function. To apply Theorem 3.12, we can choose rk = 1/2 + k. We have on the verticals − 2 1 rk rk+iy 1 rk rk 2 ζDS ( rk + iy) = 4(uq− ) 1 q− − 4(uq− ) (1 q− )− . (3.25) | q − | − ≤ −

With the help of Lemma 3.8, we see immediately that assumptions ii) and iii) of Theorem 3.12 are fulﬁlled. 2π(m+1/2) Finally, we choose a convenient sequence ym = , m N, for which logq ∈ qiym = 1. It yields, for any x R, − ∈ 1 x x+iym 2 1 x x 2 ζDS (x + iym) = 4(uq− ) 1 q − = 4(uq− ) (1 + q )− . | q | | − | Hence, no additional contribution from the contour integral(3.20) arises. Summa summarum, Theorem 3.12 yields

S t Dq 1 2 Tre− 2 2log (ut)+ F1 log(ut) log(ut)+ F0 log(ut) | | t∼0 log q ↓ ∞ ( 1)k q k k − − ut (3.26) + ∑ (k)!(1 q k)2 ( ) , k=1 − − where F0 and F1 are periodic bounded smooth functions on R deﬁned as

2πi 2πi jx F1(x) := 4γ 4 ∑ Γ ( logq j)e , − j Z ∈ ∗ 1 2 2 2 2πi 2πi 2πi jx F0(x) := 3 (π + 6γ log q) 4 ∑ Γ ( logq j)ψ( logq j)e , − − j Z − ∈ ∗ see [11, Theorem 4.1]. Similarly as in Example 2.39, it turns out that the expansion (3.26) is actually exact for all t > 0 — see Example 3.19. We remark that these results hold also for the full Dirac operator Dq on the standard Podles´ sphere, though the estimation of the contour integrals is much more tedious — cf. [11, Section 4].

3.3 Convergent Expansions of Heat Traces

As illustrated by Example 3.11 the small-t asymptotic expansion of a heat trace is typically divergent. We have, however, witnessed the convergence of this expansion 84 3 Analytic Properties of Spectral Functions in some particular geometrical context. We shall now connect these speciﬁc situations to the behaviour of the associated zeta functions. In the context of Theorems 3.6 and 3.12 the remainder of a convergent expansion (cf. Deﬁnition 2.40) can be written explicitly as

rN +i∞ 1 − s R∞(t)= lim i2π K,H (s)t− ds. (3.27) N ∞ rN i∞ Z → Z− − Indeed, eq. (3.17) yields

N t H Tr Ke− lim [ρk(t)+ gk(t)] = lim RN(t). − N ∞ ∑ N ∞ → k=0 → Hence, we can alternatively deduce the convergence of an asymptotic expansion of the form (3.17) by inspecting the expression (3.27). Namely, if there exists T (0,∞] such that Rk(t) converges absolutely/uniformly ∈∞ on (0,T ) as k ∞, then ∑k=0(ρk(t)+ gk(t)) converges absolutely/uniformly on → t H ∞ (0,T ) and for t (0,T) we can write Tr Ke− = ∑k=0[ρk(t)+gk(t)]+R∞(t). More- ∈ ∞ over, Equation (3.9) then shows that R∞(t)= 0(t ). O

3.3.1 Convergent, Non-exact, Expansions

A particular instance of a convergent, but not exact, expansion of a heat trace pro- duces itself when the set of poles of the associated zeta function is ﬁnite. Proposition 3.14. Let H and K meet the assumptions of Theorem 3.6. If the set P( K,H ) is ﬁnite, then there exists N N such that, for all t > 0, Z ∈ ∗ N t H Tr Ke− = ∑ ρk(t)+ R∞(t), with R∞ = 0. (3.28) k=0 6

Note that since P(Γ )= N the hypothesis of Proposition 3.14 requires in par- − ticular that ζK,H ( k)= 0 for almost all k N. − ∈ Proof. This is a consequence of Theorem 3.6. If P( K,H ) is ﬁnite then, for any Z choice of the partitioning sequence (rk)k there exists an N N such that Xk = /0 ∈ ∗ for all k N, hence ρk = 0 for k N. Observe that N = 0, since ζK,H has at least one pole≥ at s = p 0 (cf. p. 41),≥ which cannot be compensated6 by Γ (s), since the function Γ has no≥ zeros. t H tλ (H) Moreover, we cannot have R∞ = 0, since Tr Ke− = ∞(e− 0 ) whereas the sum N r0 O ∑k=0 ρk(t) certainly grows faster than t at inﬁnity. As an illustration let us consider the following example: Example 3.15. Let D/ be the standard Dirac operator on S3 (see Appendix B.1). The spectral zeta function associated with D/ 2 reads 3.3 Convergent Expansions of Heat Traces 85

∞ ∞ ∞ 3 2s 3 2s+2 1 3 2s ζD/2 (s)= 2 ∑ (n + 1)(n + 2)(n + 2 )− = ∑ 2(n + 2 )− 2 ∑ (n + 2 )− n=0 n=0 − n=0 = 2ζ(2s 2,3/2) 1 ζ(2s,3/2), − − 2 where ζ(s,a) for a / N is the Hurwitz zeta function (see, for instance [2, Chap- ter 12]). For any a ∈−the latter admits a meromorphic extension to C with a single simple pole at s = 1. Moreover, it can be shown (cf. [12, Theorem 6] and [21]) that when arg(a)= 0, ζ(s,a) is of polynomial growth on the verticals. Hence, Corollary 3.9 can be applied. We have

√π √π Res D2 (s)= 2 and Res D2 (s)= 4 . s=3/2 Z / s=1/2 Z / −

With ζ(s,a + 1)= ζ(s,a) a s, we rewrite − − 1 ζ 2 (s)= 2ζ(2s 2,1/2) ζ(2s,1/2). D/ − − 2 Then, for k N, ∈ ( 1)k 1 Res D2 (s)= − [2ζ( 2k 2,1/2) ζ( 2k,1/2)]. s= k Z / k! − − − 2 − − But the properties of the Bernoulli polynomials (cf. page 55) yield, for any k N, ∈ 2k B2k+1(1/2) 1 2 ζ( 2k,1/2)= = − − B2k 1 = 0. − − 2k+1 2k+1 + Hence,

tD/ 2 √π 3/2 √π 1/2 tD/ 2 √π 3/2 √π 1/2 ∞ Tre− t− t− Tre− = t− t− + 0(t ), t∼0 2 − 4 ⇐⇒ 2 − 4 O ↓ x2 1/2 what harmonises with Formula (2.23) for f (x)= e− and Λ = t− . As remarked in Example 2.41, the asymptotic expansion of the heat trace associated with the square of the standard Dirac operator on any odd dimensional sphere is convergent for all t > 0, but is not exact.

Remark 3.16. Let us note that Proposition 3.14 can be a source of interesting iden- tities for some special functions deﬁned by general Dirichlet series: One checks (cf. 1 π 1/2 π2/t [13]) that in the context of Example 2.41 R∞(t)= 2 ( t ) [ϑ3(0;e− ) 1] and hence Formula (3.28) yields the Jacobi identity (2.31). −

3.3.2 Exact Expansions

Let us now give a sufﬁcient condition for the exactness of the small-t expansion of a heat trace, in terms of the corresponding zeta function. 86 3 Analytic Properties of Spectral Functions

Theorem 3.17. Let H and K meet the assumptions of Theorem 3.12 and let the estimate

εk y K,H ( rk + iy) cke− | | (3.29) |Z − |≤ hold for every y R and k N with some ck,εk > 0. If ∈ ∈

ck 1/rk 1 T := limsup( ε ) − > 0, (3.30) k ∞ k → ∞ then the series ∑k=0(ρk(t)+ gk(t)), with ρk and gk given by (3.18) and (3.20) re- t H spectively, converges to Tr Ke− locally uniformly on (0,T). If, moreover, logk = O∞(rk) (i.e. rk grows faster than logk), then the convergence is absolute on (0,T ).

Proof. We estimate the remainder Rk(t) as follows

∞ ∞ 1 rk iy 1 εk y rk 1 ck rk Rk(t) = 2π K,H ( rk + iy)t − dy 2π cke− | |t dy = π ε t . | | | ∞ Z − |≤ ∞ k Z− Z− Let 0 < T < T. For any t (0,T ] we have ′ ∈ ′ r limsup t k ck/εk = t/T T ′/T. k ∞ ≤ → p r Hence, for sufﬁciently large k we have t k ck/εk < a, where a (T ′/T,1) is some constant independent of t. Then, ∈ p r r ck t k a k Rk(t) < 0, (3.31) | |≤ εkπ π k→∞ ↑ so Rk(t) tends to 0 uniformly for t (0,T ′]. Since T ′ can be any number in (0,T ), the local uniform convergence is proven.∈ ∞ To check the absolute convergence we need to show that ∑k=0 ρk(t)+ gk(t) < ∞ for t (0,T ). From the recurrence relation (3.22) we obtain that,| for any k N| , ∈ ∈

ρk(t)+ gk(t) = Rk 1(t) Rk(t) Rk 1(t) + Rk(t) . | | | − − | ≤ | − | | | Now, (3.31) implies

∞ ∞ ∞ ∞ rk ckt 2 rk ρk(t)+ gk(t) 2 Rk(t) 2 < a , ∑ ∑ ∑ εkπ π ∑ k=0 | |≤ k=0 | |≤ k=0 k=0 with a (T ′/T,1) for any T ′ (0,T ). Therefore, it sufﬁces to show that the last series is∈ convergent for any a <∈1. The latter is a general Dirichlet series in variable x = loga with coefﬁcients ak = 1,bk = rk for k N. By Theorem 2.2 its abscissa − 1 ∈ of convergence equals limsupk ∞(logk)rk− , which is 0 by hypothesis. Thus, the series is convergent for x > 0, i.e.→ a < 1. 3.3 Convergent Expansions of Heat Traces 87

The absolute convergence, on top of the exactness, of a heat trace expansion is necessary to establish an exact large energies expansion of the spectral action (cf. Theorem 3.24). Given an exact expansion, the absolute convergence is fairly easy — it sufﬁces to verify whether the sequence (rk) of our choice grows faster than logk. In fact, all of the examples of exact expansions we present in this book are actually absolutely convergent in the same domain. Let us also remark that if we have an exact expansion of the heat trace for an t H open interval (0,T ), then Tr Ke− actually provides an analytic continuation of the series ∑k ρk(t)+ gk(t) to the whole half line (0,∞). In general,| Theorem| 3.17 provides only a sufﬁcient condition, so Formula (3.30) does not necessarily give the maximal range of (absolute) convergence. Neverthe- less, the following example shows that the bound (3.29) is often good enough to deduce the actual upper limit.

Example 3.18. Let D/ be the standard Dirac operator on S1 associated with the trivial spin structure (see Appendix B.1). As shown in Example 2.42, the associated heat trace can be computed explicitly for any t > 0 (recall Formula (2.32)) and developed t D/ in a Laurent series around t = 0. The latter converges to Tre− | | absolutely for t (0,2π). Let us now re-derive this result using Theorems 3.6 and 3.17: ∈The operator D/ has a kernel of dimension 1, so Formula (2.8) gives

t D/ t D Tre− | | = Tr(1 P )H e− | | + 1, for all t > 0. − 0 ∞ s C We have, ζD(s)= 2∑n=1 n− = 2ζ(s), which is meromorphic on with a single simple pole at s = 1 and

k k ( 1) ( 1) Bk+1 N Res D(s)= 2, Res D(s)= 2 − ζ( k)= 2 − , for k . s=1 Z s= k Z k! − − k! k+1 ∈ −

Since B2k 1 = 0 for k N , we actually have P( )= 1,0, 1, 3, 5,... . + ∈ ∗ ZD { − − − } As the partitioning sequence we can choose, r0 = 3/2,r1 = 1/2,r2 = 1/2 and − − rk = 2(k 2), for k 3. Then, Formula (3.15) guarantees that the assumption (3.14) is met and− Corollary≥3.9 yields

∞ t D 1 B2k 2 2k+1 Tr 1 e 2t 1 2 + t ( P0)H − | | t 0 − + ∑ (2k+2)! , − ∼ ↓ − k=0 in agreement with Formulae (2.32)and (2.33). To check that this expansion is exact we need to ﬁnd an explicit bound of the form (3.29). Recall ﬁrst the Riemann functional equation [1, Formula (23.2.6)]:

s s 1 πs ζ(s)= 2 π − sin Γ (1 s)ζ(1 s), 2 − − which yields

s 1 (s)= 2Γ (s)ζ(s) = (2π) ζ(1 s) sin[π(1 s)/2] − . ZD − − 88 3 Analytic Properties of Spectral Functions

For s = rk + iy with k 3 the denominator of the above expression equals − ≥ sin[π( 1 + k i y )] = cos[π(k i y )] = ( 1)k cosh πy . 2 − 2 − 2 − 2 Thus, for k 3 and y R we have ≥ ∈ 2k (2π)− ζ (2k+1 iy) 2k π y /2 ( rk + iy) = | − | 2(2π)− ζ(2k + 1)e− | | . |ZD − | cosh(πy/2) ≤ 2k Hence, the assumptions of Theorem 3.17 are met with ck = 2(2π)− ζ(2k + 1) and εk = π/2. Since ζ(x) 1 as x +∞, we obtain → → 1 2k+2 2k 1 1/(2k 4) 1 T − = limsup[2− π− − ζ(2k + 1)] − = 2π . k ∞ → We have recovered the radius of convergenceof the Laurent expansion of coth(t/2). Thus, as rk = ∞(k), we conclude that this expansion is absolutely convergent. O Let us now treat an exampleof an exact expansionvalid for t > 0, i.e. with T = ∞. D S Example 3.19. Let q be the simplified Dirac operator on the standard Podles´ sphere (cf. Appendix B.4) and let us denote u = w q/(1 q2) > 0. In Example | | S − t Dq 3.13 we have derived an asymptotic expansion of Tre− | | and alluded to its exactness. In fact, it is obvious that the series in (3.26) is absolutely and locally uniformly convergent for any t > 0. But there might a priori be a non-trivial contribution from the contour integral at infinity namely R∞. To show that this is not the case we will resort to Theorem 3.17. In Example 3.13 we chose rk = 1/2 + k and established an explicit estimate of ζDS on the verticals − q (cf. (3.25)). In order to proceed, we need a more precise estimate of the Gamma function on the verticals than the one derived in Lemma 3.8. The Euler reflection formula (3.12) together with inequality (3.13) and

1/2 x 1/2 x Γ (x) > (2π) x − e− for x > 0

([26, p. 253]) gives

1 π π 1 Γ ( rk + iy) = Γ (k+1/2 iy) sin[π(k 1/2+iy)] Γ (k + 1/2)− | − | | − | | − | ≤ √cosh(πy) k+1/2 k π y /2 < √πe (k + 1/2)− e− | | . (3.32)

Thus, the assumptions of Theorem 3.17 are met with

π 1 k 1/2 1/2 k 2 k εk = , ck = 4√πe(euq− ) − (1 q − )− (k + 1/2)− 2 − yielding

1 1 k 1/2 1/2 k 2 k 1/(k 1/2) lim 8π− e(euq− ) − (1 q − )− (k + 1/2)− − = 0 k ∞ − → 3.4 General Asymptotic Expansions 89 for0 < q < 1. Hence, T = ∞ and, with rk = ∞(k), the expansion(3.26) is absolutely convergent and exact for all t > 0. O

3.4 General Asymptotic Expansions

t H It turns out that the asymptotic expansion of the heat trace Tr Ke− associated with some operators K,H actually guarantees the existence of an asymptotic expansion for a larger class of spectral functions of the form TrK f (tH). This fact is especially pertinent in our quest to unravel a large-energies asymptotic expansion of the spectral action. R+ p Recall that (cf. (2.17)) a function f on belongs to 0 iff f is a Laplace transform of a signed measure φ on R+ and ∞ smd φ (s) < ∞Cfor all m > p. 0 | | − Theorem 3.20. Let H T p, K B(HR) and assume that ∈ ∈ ∞ t H rk Tr Ke− ρk(t), w.r.t. an asymptotic scale (t )k, (3.33) t∼0 ∑ ↓ k=0

∞ O rk O rk+1 N with ρk Lloc((0,∞)), ρk(t)= 0(t ) and ρk(t)= ∞(t ), forany k . ∈ r ∈ Then, for any f = L[φ] 0 with r > p, the operator Kf (tH) is trace-class for any t > 0 and there exists an∈C asymptotic expansion:

∞ ∞ rk TrK f (tH) ∑ ψk(t), with ψk(t)= ρk(st)dφ(s)= O0(t ). (3.34) t∼0 0 ↓ k=0 Z r Let us stress that the choice of the asymptotic scale (t k )k is merely a matter of convention — recall Remark 2.34. In particular, since Proposition 2.3 implies that p ε ρ0(t)= 0(t ) for all ε > 0, we can shift r0 to lie arbitrarily close to p. Hence, O − − − given any r > p, we actually have r > r0 > p > r1. − − Proof. Recall ﬁrst that, on the strength of Lemma 2.7 the operator f (tH) is trace- class for any t > 0, and hence K f (tH) is so for any K B(H ). ∈∞ stH As pointed out in Section 2.2.2 we have K f (tH)= 0 Ke− dφ(s) in the strong ∞ stH operator sense. The trace being normal, TrK f (tH)= 0 TrKe− dφ(s) for any N Rt H N t > 0. By assumption we have, for any N , Tr Ke− = ∑k=0 ρk(t)+ RN (t) and N ∞ ∈ R hence, TrK f (tH)= ∑k=0 ψk(t)+ 0 RN(st)dφ(s). r Let us ﬁrst show that ψk(t)= O0(t k ) for any k N. We have R ∈ rk rk rk ρk(t)= O0(t ) lim t− ρk(t) = 0 ε > 0 δ > 0 t δ, ρk(t) εt ⇔ t 0 | | ⇔ ∀ ∃ ∀ ≤ | |≤ → 1 rk rk ε,t > 0 δ > 0 s δt− , ρk(st) εt s . ⇔ ∀ ∃ ∀ ≤ | |≤ r r On the other hand, ρk(t)= O∞(t k+1 ) δ ′ > 0, t δ ′, ρk(t) t k+1 . + ⇒ ∃ ∀ ≥ | |≤ Since ρk is locally bounded on R , we actually have a local uniform bound: For any δ with 0 < δ δ , ′′ ′′ ≤ ′ 90 3 Analytic Properties of Spectral Functions

rk+1 t δ ′′, ρk(t) Mk(δ ′′)t with Mk(δ ′′) := supt [δ ,δ ] ρk(t) . ∀ ≥ | |≤ ∈ ′′ ′ | | 1 r r Hence, for any t > 0, s δ ′′t− , ρk(st) Mk(δ ′′)t k+1 s k+1 . We now pick any ε∀>≥0 and δ =| min |≤δ,δ . For any t > 0 we estimate ′′ { ′} 1 δ ′′t− ∞ ψk(t) ρk(st) d φ (s)+ ρk(st) d φ (s) | |≤ 0 | | | | δ t 1 | | | | Z Z ′′ − 1 δ ′′t− ∞ rk rk rk 1 rk 1 εt s d φ (s)+ Mk(δ ′′)t + s + d φ (s) 1 ≤ 0 | | δ ′′t− | | Z ∞ ∞Z rk rk rk 1 rk 1 εt s d φ (s)+ Mk(δ ′′)t + s + d φ (s). ≤ 0 | | 0 | | Z Z

Now, recall that r > r0 > p > r1 > r2 > ..., as (rk)k is strictly increasing. − − −r ∞ rk Hence, by Proposition 2.21, with f 0 both integrals 0 s d φ (s)=: c1 and ∞ rk+1 ∈ CM | | 0 s d φ (s)=: c2 are ﬁnite. | | r r r R Hence, we have t− k ψk(t) εc1 +t k+1− k c2. Since ε can be taken arbitrarily small R | |≤r we conclude that ψk = O0(t k ) for any k N. ∞ ∈ O rN N It remains to show that 0 RN (st)dφ(s) = 0(t ) for any N . This fol- r ∈ r lows by the same arguments since RN (t)= O0(t N ) and also RN(t)= O∞(t N+1 ), N R r t H λ (H)t as ∑ ρk(t)= ∞(ρN(t)) = O∞(t N+1 ), whereas Tr Ke = ∞(e 0 ). k=0 O − O − Remark 3.21. Note that if we do not insist on having a complete asymptotic expansion of Tr K f (tH), then we can relax the assumption about the heat trace to t H N O rN N the following condition: Tr Ke− = ∑k=0 ρk(t)+ 0(t ), for some N , which N O rN ∈ implies, for a suitable f , Tr K f (tH)= ∑k=0 ψk(t)+ 0(t ). Example 3.22. As an illustration of Theorem 3.20 we consider H = D/ 2 and K = 1, with D/ being the standard Dirac operator on S2 (see Appendix B.1). We have (cf. Example 3.11), ∞ 2 k t D/ 1 ( 1) B2k+2 k Tre− 2t− 2 ∑ −k! k+1 t . t∼0 − ↓ k=0 Following Example 2.23 let us take f (x) = (ax + b) r r , with some a,b,r > 0. − ∈C0 Since D/ 2 T p with p = 1 we can apply Theorem 3.20 when r > 1. From Formula (3.34) we∈ obtain

1 r k B akb k r k r b − 1 ( 1) 2k+2 − − Γ ( + ) k N ψ0(t)= 2 a(r 1) t− , ψk+1(t)= 2 −k! k+1 Γ (r) t , for k . − − ∈ Hence, we have

∞ D2 r 1 1 1 k B2k+2 r+k 1 k k Tr f (t / ) 2b− r 1 (a/b)− t− ∑ ( 1) k+1 k− (a/b) t . t∼0 − − ↓ − k=0 Having established the existence theorem for an asymptotic expansion of TrK f (tH) we can come back to the issue raised at the end of Section 2.2.2: Why we cannot allow f to be the Laplace transform of a distribution in +′ rather than of a signed measure? S 3.4 General Asymptotic Expansions 91

Remark 3.23. Observe that one could derive an analogue of the operatorial Formula sP/Λ (2.15) by regarding e− as an element of the space of operator valued test functions (R,B(X,H )) (recall Section 2.7). Along the same line, one could provethe normalityK of the trace functional (cf. (2.16)), i.e.

1 sH/Λ Tr f (H/Λ)= L− [ f ],Tre− D E 1 for L [ f ] ′ , using the convergence of the sequence of partial sums in − ∈ K ∩ S +′ K and invoking the weak- topology of ′ . ∗ K ∩ S +′ On the other hand, the topology of ′ obliges us to control the derivatives of the test 1 K functions, on which L− [ f ] act. But we do not, in general, have such a controlon the r remainder of the asymptotic series (3.33) even in the standard case of ρk(t)= ck t− k (cf. Remark 2.35).

Let us now turn to the case of convergent expansions.

Theorem 3.24. Let H T p and let K B(H ). Assume that for t (0,T ) with some T (0,+∞], ∈ ∈ ∈ ∈ ∞ ∞ t H Tr Ke− = ∑ ρk(t)+ R∞(t) and ∑ ρk(t) < ∞ (3.35) k=0 k=0 | |

∞ O rk N ∞ with ρk Lloc((0,T )) and ρk(t)= 0(t ) for any k , andR∞(t)= 0(t ). ∈ r ∈ ∞ O Then, for any f with r > r0 > p, the series ∑ ψk(t), with the functions ∈Cc − k=0 ∞ ψk(t) := ρk(st)dφ(s) Z0 is absolutely convergent on the interval (0,T/Nf ) with

1 Nf := inf N supp L− [ f ] (0,N) . { | ⊂ }

Moreover, for t (0,T /Nf ), ∈ ∞ ∞ f f ∞ TrK f (tH)= ∑ ψk(t)+ R∞(t), where R∞(t) := R∞(st)dφ(s)= 0(t ). 0 O k=0 Z Proof. On the strength of Formula (2.16) and by assumption (3.35) we have

∞ N ∞ N stH f f TrK f (tH)= TrKe− dφ(s)= ∑ ρk(st)dφ(s)+ R∞(st)dφ(s). Z0 Z0 k=0 Z0 Observe ﬁrst that the sequence of partial sums is uniformly bounded, i.e. for any N N ∞ N and any t (0,T/Nf ), ∑k=0 ρk(st) ∑k=0 ρk(st) < ∞. Via the Lebesgue ∈ ∈ ≤ | |∞ f dominated convergence theorem, we obtain TrK f (tH)= ∑k=0 ψk(t)+ R∞(t) and it f ∞ remains to show that R (t)= 0(t ). To this end, let us note that ∞ O 92 3 Analytic Properties of Spectral Functions

k k k > 0 R∞(t)= 0(t ) k > 0 M,δ > 0 t δ, R∞(t) Mt ∀ O ⇔ ∀ ∃ ∀ ≤ | |≤ 1 k k k, t > 0 M,δ > 0 s δt− , R∞(st) Mt s . ⇔ ∀ ∃ ∀ ≤ | |≤

Then, for any t (0,Nf ) and any k > 0, we have ∈ ∞ δt 1 ∞ f − R∞(t) = R∞(st)dφ(s) R∞(st) d φ (s)+ R∞(st) d φ (s) 0 ≤ 0 | | | | δt 1 | | | | Z Z Z − ∞ ∞ k k Mt s d φ (s)+ R∞(st) d φ (s). ≤ 0 | | δt 1 | | | | Z Z − ∞ k ∞ As f c, we have 0 s d φ (s) < ∞ for any k > 0 and δt 1 R∞(st) d φ (s)= 0 ∈C 1 | | − | f | | | ∞ for t < δN− . Since k can be arbitrarily large we conclude that R (t)= 0(t ). f R R ∞ O 1 In general, the compactness of the support of L− [ f ] is necessary even if the expansion of the heat trace at hand is actually convergent for all t > 0. This is because if one would like to allow for f p, rather than f p, one would need to control ∈C0 ∈Cc the behaviour of R∞(t) as t ∞. On the other hand, such a control is (trivially) → t H provided if the expansion of Tr Ke− is exact for all t > 0.

Corollary 3.25. Let K and H meet the assumptions of Theorem 3.24 with T =+∞ r and R∞ = 0. Then, for any f with r > r0 > p, and anyt > 0 we have ∈C0 − ∞ ∞ TrK f (tH)= ∑ ψk(t), with ψk(t) := ρk(st)dφ(s). k=0 Z0

f Proof. Obviously, R∞ = 0. Hence, it sufﬁces to observe that in the ﬁrst part of the proof of Theorem 3.24 we can safely allow for Nf =+∞, if the series ∑k ρk(st) is absolutely convergent for any s,t > 0.

On the other hand, we insist that the full force of condition (2.17) is necessary t H for Theorem 3.24 to hold even if the convergent expansion of Tr Ke− has a ﬁnite number of terms (recall Section 3.3.1). Let us illustrate this fact:

Example 3.26. In Example 2.41 we have for any t > 0,

t D/2 π 1/2 Tre− = ( t ) + R∞(t).

√x Let us try to apply Theorem 3.24 with f (x)= e− , which is a completely monotone function, but f / 0 (recall Example 2.25). Since, ∈C ∞ t 1 t 1/2 s 1/2 s 3/2e 1/4s ds t 1/2 ψ0( )= 2√π − − − − = 2 − , Z0 2 2 t D/ 1 f we would get Tr f (t D/ )= Tre− | | = 2t− + R∞(t). f 3 A comparison with Example 2.42 gives R∞(t) = t/6 + 0(t ). Hence, whereas ∞ f ∞ O R∞(t)= 0(t ), but R (t) = 0(t ). O ∞ 6 O 3.5 Asymptotic Expansion of the Spectral Action 93

An important question is whether one can tailor a cut-off function such that the t H asymptotic expansion of Tr Ke− turns to a convergent one of TrK f (tH). When t H ∞ k Tr Ke− t 0 ∑k=0 akt , as happens always in the context of classical differential operators (cf.∼ ↓ Example 2.36), one could seek an f with null Taylor expansion at 0 — as recognised in [7]. Unfortunately, the natural candidate – a bump function (cf. [7, Fig. 1]) is not a Laplace transform as explained in Remark 2.20. It is also clear from Proposition 2.16 that f cannot be completely monotone. Nevertheless, one can ﬁnd functions with a null Taylor expansion at 0, which are in p for some p. C0 s1/4 1/4 Example 3.27. Let φ(s)= e− sin(s ) for s > 0. This non-positive function has ∞ n N all moments vanishing, i.e. 0 s φ(s)ds = 0, n (cf. [23, Example 3.15]). More- over, its absolute moments are ﬁnite: ∀ ∈ R ∞ ∞ n n s1/4 s φ (s)ds s e− ds = 4(4n + 3)! 0 | | ≤ 0 Z Z The Laplace transform of f = L[φ] can be obtained with the help of Mathematica yielding an analytic, though rather uninviting, formula

Γ (5/4) 1 3 1 √π 3 5 1 Γ ( 5/4) 5 3 1 f (x)= 0F2 , ; 0F2 , ; − 0F2 , ; , x5/4 2 4 − 64x − 2x3/2 4 4 − 64x − 16x7/4 4 2 − 64x

∞ Γ (a)Γ (b) zk with the hypergeometric function 0F2(a,b;z) := ∑k=0 Γ (a+k)Γ (b+k) k! . One can check that f (x) > 0 for x > 0 and, since limx 0+ 0F2(b1,b2;x)= 1, we get 5/4 → f (x)= ∞(x− ). With the help of Proposition 2.21, one obtains a cut-off function f n for someO n N , which has null Taylor expansion at 0. ∈ ∗ Let us emphasise that the cut-off function exempliﬁed above would do the trick of turning an asymptotic expansion to a convergent one only if the expansion of t H ∞ k Tr Ke− has a very speciﬁc form: ∑k=0 akt . In general, the expansion(3.34) would ∞ k n involve the full shape of the cut-off function, via the integrals 0 s log (s)dφ(s). R

3.5 Asymptotic Expansion of the Spectral Action

Equipped with the general results presented in the preceding section we are at a position to formulate sufﬁcient conditions for the existence of a large-energies asymptotic expansion of the spectral action S(D, f ,Λ) associated with a given spectral triple. Recall ﬁrst that the cut-off function f 0 is smooth at 0 (Proposition 2.16). In particular, f (0) < ∞ and if the operator T∈C f (t D ) is trace-class then so is T f (t D ). D| | | | Moreover, from the spectral decomposition = 0P0 +∑λn=0 λnPn, we immediately get 6

f ( D )= f (0)P0 + f ( λn )Pn, f ( D )= f (1)P0 + f ( λn )Pn, | | ∑ | | | | ∑ | | λn=0 λn=0 6 6 94 3 Analytic Properties of Spectral Functions and

1 Tr T f ( D /Λ)= Tr T f ( D /Λ)+ f (0) f (Λ − ) TrTP0, (3.36) | | | | − 1 = Tr T f ( D /Λ)+ ∞(Λ − ). | | O Remark 3.28. From this formula we see that the kernel of D becomes irrelevant in the physical action at large energies. It also explains why we have not deﬁned the spectral action (1.43) as Tr T f ( D /Λ) despite the simpliﬁcation of working with an invertible operator. But the kernel| | pops up if we choose D instead of D:

0 Tr T f ( D /Λ) Tr T f (D/Λ)= f (0) TrTP0 = ∞(Λ ). | | − O Theorem 3.29. Let (A ,H ,D) be a p-dimensional spectral triple and T B(H ). ∈ Assume that there exists d N, a sequence (rk)k N R strictly increasing to +∞ and a discrete set X C without∈ accumulation points,∈ ⊂ such that ⊂ ∞ d t D n z Tr Te− | | ρk(t), with ρk(t)= az,n(T, D )log t t− (3.37) t∼0 ∑ ∑ ∑ | | ↓ k=0 z Xk n=0 ∈ where Xk := z X rk+1 < ℜ(z) < rk ,X = kXk and the series deﬁning ρk(t) is absolutely{ convergent∈ |− for any t > 0 and− } any k ⊔N. ∈ Then, i) The function ζT,D admits a meromorphic extension to the whole complex plane with the poles of order at most d + 1 and P( T,D) X. Moreover, for any z X andn 0,1,...,Zd , ⊂ ∈ ∈{ } d+1 [ℓ] ( 1)n z az,n(T, D )= −n! ∑ Γℓ n 1(z) T D − , (3.38) | | − − − | | ℓ=n Z with Γj(z) denoting the j-th coefﬁcient of the Laurent expansion aroundzof Γ , i.e. j 1 Γj(z) := Res(s z)− − Γ (s), for j N 1. s=z − ∈ − ii) For any f = L[φ] r with r > p, ∈C0 ∞ rk Tr T f ( D /Λ) ∑ ψk(Λ), w.r.t. the scale (Λ − )k, (3.39) | | Λ ∼+∞ → k=0 with

d d z n n m ψk(Λ)= ∑ Λ ∑ ( 1) log Λ ∑ m n az,m(T, D ) fz,m n , (3.40) z X n=0 − m=n − | | − ∈ k ∞ z n fz,n := s− log (s)dφ(s). (3.41) Z0

Let us emphasise that, as in Theorem 3.20, the onliest role of the sequence (rk)k is to ﬁx an asymptotic scale. 3.5 Asymptotic Expansion of the Spectral Action 95

Proof. i) Since the demand (3.37) exactly mimics the assumptions of Theorem 3.2 the first of part of i) follows. Then, Formula (3.38) arises from Equation (3.2) followed by some easy combinatorics: Since ζT,D is meromorphic on C we have the following Laurent expansions in some punctured neighbourhood around any z X, ∈ ∞ ∞ ℓ j ζT,D(s)= ∑ bℓ(z)(s z) , Γ (s)= ∑ Γj(z)(s z) , ℓ= d 1 − j= 1 − − − − ∞ m T,D(s)= ζT,D(s)Γ (s)= ∑ cm(z)(s z) , Z m= d 2 − − − m+1 with cm(z)= ∑ℓ= d 1 bℓ(z)Γm ℓ(z). Furthermore, − − − [ ℓ] ℓ 1 ℓ 1 s z − z bℓ(z)= Res(s z)− − ζT,D(s)= Res s− − TrT D − − = T D − . s=z − s=0 | | − | | Z Now, Formula (3.2) yields, for any z X, n 0,1,...,d , ∈ ∈{ } ( 1)n n ( 1)n az,n(T, D )= −n Res (s z) Γ (s)ζT,D(s)= −n c n 1(z) | | ! s=z − ! − − n d+1 [ℓ] ( 1)n − ( 1)n z = −n! ∑ bℓ(z)Γ n 1 ℓ(z)= −n! ∑ Γℓ n 1(z) T D − . ℓ= d 1 − − − ℓ=n − − − | | − − Z ii) Let us now turn to assertion (3.39). We have demonstrated in the course of the r proof of Theorem 3.2 that ρk defined as in (3.37) satisfies ρk(t)= O0(t k ). Using the O rk+1 ∞ same arguments, one shows that ρk(t)= ∞(t ). Moreover, ρk Lloc((0,∞)) and since D T p, Theorem 3.20 yields, with t = Λ 1, ∈ | | ∈ − ∞ ∞ Tr T f ( D /Λ) ∑ ψk(Λ), with ψk(Λ)= ρk(s/Λ)dφ(s). | | Λ ∼+∞ 0 → k=0 Z

It remains to compute the explicit form of ψk. By assumption, the series deﬁning every ρk(t) are absolutely convergent for any t > 0 so that we can invoke the Lebesgue dominated convergence theorem to commute the sum over Xk with the integral over dφ (cf. the proof of Theorem 3.24). In effect, we obtain

d ∞ z m z ψk(Λ)= ∑ Λ ∑ az,m(T, D ) log (s/Λ)s− dφ(s) z X m=0 | | 0 ∈ k Z d m z m m j 1 = ∑ Λ ∑ az,m(T, D ) ∑ j log − (Λ − ) fz, j z X m=0 | | j=0 ∈ k d d z n n m = ∑ Λ ∑ ( 1) log Λ ∑ m n az,m(T, D ) fz,m n. − | | − z Xk n=0 m=n − ∈ 96 3 Analytic Properties of Spectral Functions

[ℓ] z Formula (3.38) shows how to compute the noncommutative integrals T D − | | given the coefﬁcients az,n(T, D ): For instance, since Γ 1( k) = 0 for k N, it | | − − 6 R ∈ implies that [d+1] T D k = 0 for T Ψ 0(A )and k N. More generally: | | ∈ ∈ Corollary 3.30.R Under same hypothesis but with T Ψ 0(A ). For z = k N, ∈ − ∈ [d+1] T D k = 0, − | | Z [d] k d+k T D = ( 1) d!k!a k,d(T, D ), for d 0, − | | − − | | ≥ Z [d 1] − k d+k 1 T D = ( 1) − (d 1)!k! a k,d 1(T, D ) − | | − − − − | | Z k + ( 1) dk!Γ0( k)a k,d(T, D ) , for d 1; − − − | | ≥ whereas, for z C ( N), in which case Γ 1(z)= 0 and Γ0(z)= Γ (z), we obtain ∈ \ − − [d+1] z ( 1)dd! T D − = − az d(T, D ), for d 0, − | | Γ (z) , | | ≥ Z [d] z ( 1)d(d 1)! T D − = − 2− Γ (z)az,d 1(T, D )+Γ1(z)daz,d(T, D ) , for d 1. − | | Γ (z) − | | | | ≥ Z If d = 0, then ζT,D is regular at k N with − ∈− k ζT,D( k) = ( 1) k!a k,0(T, D ) − − − | | and we have 1 = 0 (cf. Example 1.28).

The explicitR formulae for the noncommutative integrals become rather involved when d is large, but otherwise can be computed recursively from Formula (3.38).

Remark 3.31. Let (A ,H ,D) be regular and let the assumptions of Theorem 3.29 0 be met for any T Ψ (A ) with a common d and Xk’s (i.e. there exist d,Xk such that 0 ∈ for any T Ψ (A ), d(T )= d and Xk(T )= Xk). Then, (A ,H ,D) has a dimension spectrum Sd∈ of order at most d + 1 and Sd ( N) X. Let us emphasise that this is a one-way∪ street,− ⊂ even if (A ,H ,D) is known to possess a simple dimension spectrum included in R. As we haveseen in the previous sections, the existence of a small-t asymptotic expansion of a heat trace is a stronger property than the existence of a meromorphic extension of the associated spectral zeta function. The assumptions about the growth rate of the ζD-function on the verticals are indispensable, as they allow us to control the contour integrals involved in the Mellin inversion theorem and hence manage the remainder in the asymptotic series. Its relevance was recognised by Alain Connes and Henri Moscovici in [10, p. 206] in the context of index theorems. See also Problem 6) a) in Chapter 5.

Example 3.32. For every commutative spectral triple associated with a Riemannian compact manifold M the assumptions of Theorem 3.29 are met — cf. Example 2.37. 3.5 Asymptotic Expansion of the Spectral Action 97

In this context we have d = 1, X = dimM N and Xk = dimM k . This extends naturally to almost commutative geometries,− as they are emb{ edded− in} the framework of purely classical pseudodifferential operators [3, 5].

As a straightforward corollary of Theorem 3.29 we obtain the following, slightly enhanced, known result [9, Theorem 1.145]:

Corollary 3.33. Let (A ,H ,D) be a regular p-dimensional spectral triple with a dimension spectrum Sd, denote Sd+ := z Sd ℜ(z) > 0 and let T Ψ 0(A ). Moreover, assume that { ∈ | } ∈

t D α Tr Te− | | = ∑ aα (T, D )t− + a0(T, D )+ O0(1), (3.42) α Sd+ | | | | ∈ where the, possibly inﬁnite, series over α is absolutely convergent for all t > 0. r Then, the function ζT,D is regular at 0 and for any f with r > p, ∈C0 ∞ D α α 1 α O Tr T f ( /Λ)= ∑ Λ x − f (x)dx T D − + f (0)ζT,D(0)+ ∞(1). (3.43) | | 0 − | | α Sd+ Z Z ∈ Alternatively, if in place of (3.42) one assumes

D2 t 1 2 β /2 2 O Tr Te− = 2 ∑ aβ /2(T,D )t− + a0(T,D )+ 0(1), (3.44) β Sd+ ∈ then, the expansion (3.43) holds for x f (√x) r with r > p/2. 7→ ∈C0 Proof. The existence of the expansion (3.43) follows from Theorem 3.20 enhanced by Remark 3.21, whereas the explicit form of the coefﬁcientsis providedby Formula (3.40) from Theorem 3.29. The regularity of ζT,D at 0 follows from Formula (3.38) and a0 1(T, D )= 0. Furthermore, , | | [0] 1 a0,0(T, D )= T = Res s− ζT,D(s)= ζT,D(0). | | − s=0 Z ∞ Finally, we observe that f (0)= 0 dφ(s) < ∞ (cf. Proposition 2.16) and moreover, for ℜ(α) > 0 (see the proof of Proposition 2.21), R ∞ ∞ ∞ ∞ α 1 α 1 sx 1 α 1 fα,0 = s− dφ(s)= Γ (α) x − e− dφ(s)dx = Γ (α) x − f (x)dx, Z0 Z0 Z0 Z0 whereas α aα 0(T, D )= aα (T, D )= Γ (α) T D − . , | | | | − | | Z For the alternative statement with hypothesis (3.44), we have with f (x)= g(x2), 98 3 Analytic Properties of Spectral Functions ∞ 2 2 sD2/Λ 2 Tr T f ( D /Λ)= Tr Tg(D /Λ )= Tr Te− dφg(s) | | 0 Z ∞ 1 β 2 β /2 O = 2 ∑ Λ aβ /2(T,D ) s− dφg(s)+ g(0)ζT,D2 (0)+ 0(1) 0 β Sd+ Z ∈ ∞ 1 β β /2 1 O = 2 ∑ Λ x − g(x)dx + f (0)ζT,D(0)+ 0(1) 0 β Sd+ Z ∈ ∞ β β 1 = ∑ Λ y − f (y)dy + f (0)ζT,D(0)+ O0(1). 0 β Sd+ Z ∈ Remark 3.34. Let us emphasise that the remainder in (3.43) will, in general, involve the full shape of the cut-off function f , even if (A ,H ,D) has a simple dimension spectrum. To conclude that O∞(1) depends only on the Taylor expansion of f at 0 one would need a considerably stronger assumption

∞ t D α k Tr Te− | | ∑ aα (T, D )t− + a0(T, D )+ ∑ a k(T, D )t− , t∼0 | | | | − | | ↓ α Sd+ k=0 ∈ in place of (3.42). This brings to focus the role of the cut-off function f in the dynamics of the physical system modelled via the triple (A ,H ,D). See Problem 7) in Chapter 5. Remark 3.35. As one can see, the swap between the alternative hypotheses (3.42) and (3.44) is not at all innocent. Although they both yield the same Formula (3.43), but for f in very different classes . Moreover, the existence of asymptotics of tD2 C t D Tr Te− cannot be easily deduced from the asymptotics of Tr Te− | | via the Laplace transform — see Example 2.25. The converse implication is also not guaranteed, as observed in Remark 2.20. The existence of both asymptotic expansions is, however, guaranteed if we know that the meromorphic extension of the corresponding zeta function ζT,D is of polynomial growth on the verticals — cf. Corollary 3.9. See Problem 6) e) in Chapter 5.

Theorem 3.29 allows us to unveil the large-energies asymptotic expansion of the spectral action S(D, f ,Λ) with a ﬁxed operator D. In order to understand the complete picture of dynamics, we need to fathom out how does the spectral action, and its asymptotics, behaves under the ﬂuctuations of D — recall Section 1.6. This will be our goal in the next chapter.

References

1. Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions: with Formulas, Graphs, and Mathematical Tables. Courier Dover Publications (2012) 2. Apostol, T.M.: Introduction to Analytic Number Theory. Springer, New York (1976) 3. Boeijink, J., van den Dungen, K.: On globally non-trivial almost-commutative manifolds. Journal of Mathematical Physics 55(10), 103508 (2014) References 99

4. Bourgain, J.: Decoupling, exponential sums and the Riemann zeta function. Journal of the American Mathematical Society 30(1), 205–224 (2017) 5. Ca´ ci´ c,´ B.: A reconstruction theorem for almost-commutative spectral triples. Letters in Math- ematical Physics 100(2), 181–202 (2012) 6. Carlson, B.C.: Special Functions of Applied Mathematics. Academic Press (1977) 7. Chamseddine, A.H., Connes, A.: The uncanny precision of the spectral action. Communica- tions in Mathematical Physics 293(3), 867–897 (2009) 8. Connes, A.: Cyclic cohomology, quantum group symmetries and the local index formula for SUq(2). J. Inst. Math. Jussieu 3(1), 17–68 (2004) 9. Connes, A., Marcolli, M.: Noncommutative Geometry, Quantum Fields and Motives, Collo- quium Publications, vol. 55. American Mathematical Society (2008) 10. Connes, A., Moscovici, H.: The local index formula in noncommutative geometry. Geometric And Functional Analysis GAFA 5(2), 174–243 (1995) 11. Eckstein, M., Iochum, B., Sitarz, A.: Heat trace and spectral action on the standard Podles´ sphere. Communications in Mathematical Physics 332(2), 627–668 (2014) 12. Eckstein, M., Zaj ˛ac, A.: Asymptotic and exact expansions of heat traces. Mathematical Physics, Analysis and Geometry 18(1), 1–44 (2015) 13. Elizalde, E., Leseduarte, S., Zerbini, S.: Mellin transform techniques for zeta-function resum- mations. arXiv preprint hep-th/9303126 (1993) 14. Flajolet, P., Sedgewick, R.: The average case analysis of algorithms: Mellin transform asymptotics (1996). [Research Report] RR-2956, INRIA 15. Gilkey, P.B.: Invariance Theory, the Heat Equation, and the Atiyah–Singer Index Theorem, second edn. Studies in Advanced Mathematics. CRC press Boca Raton (1995) 16. Grubb, G., Seeley, R.T.: Weakly parametric pseudodifferential operators and Atiyah–Patodi– Singer boundary problems. Inventiones Mathematicae 121(1), 481–529 (1995) 17. Guido, D., Isola, T.: New results on old spectral triples for fractals. In: D. Alpay, F. Cipriani, F. Colombo, D. Guido, I. Sabadini, J.L. Sauvageot (eds.) Noncommutative Analysis, Operator Theory and Applications, pp. 261–270. Birkhäuser (2016) 18. Iochum, B., Levy, C., Sitarz, A.: Spectral action on SUq(2). Communications in Mathematical Physics 289(1), 107–155 (2009) 19. Lapidus, M.L., van Frankenhuijsen, M.: Fractal Geometry, Complex Dimensions and Zeta Functions. Springer (2006) 20. Lesch, M.: On the noncommutative residue for pseudodifferential operators with log- polyhomogeneous symbols. Annals of Global Analysis and Geometry 17(2), 151–187 (1999) 21. Matsumoto, K., Weng, L.: Zeta-functions deﬁned by two polynomials. In: Number Theoretic Methods, eds Kanemitsu S., Jia C., pp. 233–262. Springer (2002) 22. Paris, R.B., Kaminski, D.: Asymptotics and Mellin–Barnes Integrals, Encyclopedia of Math- ematics and its Applications, vol. 85. Cambridge University Press (2001) 23. Romano, J.P., Siegel, A.F.: Counterexamples in Probability and Statistics. Wadsworth & Brooks/Cole Advanced Books & Software (1986) 24. Rudin, W.: Real and Complex Analysis. McGraw-Hill (1970) 25. Titchmarsh, E.C., Heath-Brown, D.R.: The Theory of the Riemann Zeta-function. Oxford University Press (1986) 26. Whittaker, E.T., Watson, G.N.: A Course of Modern Analysis, fourth edn. Cambridge Univer- sity Press (1927)

Chapter 4 Fluctuations of the Spectral Action

Abstract As we have learned in Section 1.6 a given spectral triple (A ,H ,D) oughtto be considered as a representativeof the entire family of triples (A ,H ,DA), which yield equivalent geometries. It is therefore of utmost importance to understand how the spectral action is affected by the ﬂuctuations of geometry. We explore the meromorphic structure of the ﬂuctuated zeta function and, for regular spectral triples with simple dimension spectra, we providea few formulae for the noncommutative integrals. Finally, we sketch the method of operator perturbations.

4.1 Fluctuations of the Spectral ζ -Function

We have seen in Chapter 3 that the spectral zeta function plays a pivotal role in the asymptotic expansion of the spectral action. In this section we study the relationship of the spectral functions ζD and ζDA . To this end, we will work with a regular s spectral triple (A ,H ,D) and provide an operator expansion of DA − for s C. 0 | | ∈ Let A be an abstract pdo such that A = A∗ Ψ (A ). This hypothesis encom- ∈ 1 passes both the standard ﬂuctuations A = A + εJAJ∗, with A = A∗ ΩD (A ) (cf. p. 27) and the more general ones considered in [2]. ∈ Recall from (1.42) that DA = D + A and deﬁne

DA := DA + PA, where PA is the projection on KerDA. We implicitly exclude the case DA = 0, which can occur only if dimH < ∞ (recall p. 40). 1 Let us remark that DA Ψ (A ) and, as already noted after Deﬁnition 1.18, ∈

VA := PA P0 is a smoothing pdo. (4.1) − k k r This follows from KerDA k 1 DomDA k 1 Dom D , the fact that D PA s ⊂ ≥ ⊂ ≥ | | | | and PA D are bounded operators for any r,s R since Im PA is ﬁnite dimensional, | | T ∈T

101 102 4 Fluctuations of the Spectral Action

k t and for any k N, t R the expression δ (PA) D , being a linear combination of ∈ ∈r s | | terms of the form D PA D , is also bounded. Thus, PA is a smoothing pdo as P0, see Example 1.19 (cf.| | [7,| Lemma| 4.1] for details). Using (1.10), deﬁne for any T Ψ(A ), n N , s C, and dt := dt1 dtn, ∈ ∈ ∗ ∈ ··· s n Kn(T,s) := ( 2 ) σ st1 (T) σ stn (T ) dt, (4.2) − 0 t tn 1 − ··· − Z ≤ 1≤···≤ ≤ α α Recall that if T OP , then σz(T ) OP for any z C (cf. Proposition 1.13) and the integrand in (4.2∈ ) is in OPαn, continuous∈ in the strong∈ topology in integration αn variables varying over a compact set. Hence, Kn(T,z) OP . We shall need the following technical operators ∈

2 2 2 X := DA D = AD + DA + A , XV := X +VA, (4.3) − 2 2 2 2 Y := log(DA) log(D )= log(D + XV ) log(D ), (4.4) − − 2 (which are well deﬁned since A preserves DomD and DA is invertible), so that 2 2 2 2 D + XV = (D + A) + PA = DA + PA = DA. We store for the sequel that XV X. In the following we use the multi-index notation n ∼ ℓ = (ℓ1, ,ℓn) N with ℓ := ℓ1 + ... + ℓn and the following complex function h ··· ∈ | |

s n st1/2 stn/2 n hn(s;ℓ) := − − dt, ℓ N , (4.5) 2 ℓ1 ℓn − 0 t tn 1 ··· ∈ Z ≤ 1≤···≤ ≤ s s(s+1) (s ℓi+1) s which is a polynomial since = ··· − for s C,ℓi N∗ and = 1: ℓi ℓi! ∈ ∈ 0 ℓ | | n+ j Nn hn(s;ℓ)= ∑ h j(ℓ)s , for ℓ . (4.6) j=0 ∈

Lemma 4.1. The operatorY has the following expansion for any N N , ∈ ∗

N N n ℓ +n+1 − ( 1)| | ℓn ℓn ℓ 2( ℓ +n) N 1 Y = − ∇ X∇ 1 (X ∇ 1 (X)) D− | | mod OP− − . (4.7) ∑ ∑ ℓ +n − ··· n=1 ℓ1,...,ℓn=0 | | So, 2 1 4 2 4 Y = XD− (∇(X)D− + X D− )+ ..., − 2 hence Y Ψ (A ) OP 1 Ψ 1(A ). ∈ ′ ∩ − ⊂ − Proof. Remark that, although Y depends on XV we could replace it with X in (4.7) as X XV . Thus, to simplify, we will, by an abuse of notation, work directly with X. ∼ ∞ 1 1 Since for x > 0, log x = 0 [(1 + λ)− (x + λ)− ]dλ, we obtain via the functional calculus − R ∞ 2 1 2 1 Y = [(D + λ)− (D + X + λ)− ]dλ. (4.8) 0 − Z 4.1 Fluctuations of the Spectral ζ-Function 103

2 2 Remark that D + X + λ = DA + λ is invertible for any λ [0,∞). Moreover, with L = L(λ) := (D2 + λ) OP2 for a given λ ∈ ∈ 1 1 1 1 1 1 (L + X)− = [(1 + XL− )L]− = L− [1 + XL− ]− , so that for a given N N , ∈ ∗ N 1 1 1 n 1 n [ + XL(λ)− ]− = ∑ ( 1) [XL(λ)− ] + RN(λ), n=0 − N+1 1 N+1 1 1 N 1 RN (λ) := ( 1) [XL(λ)− ] [1 + XL(λ)− ]− OP− − . − ∈ 1 1 2 We used the fact that X OP and Proposition 1.13 viii) to get ﬁrst L(λ)− OP− 1 ∈1 0 1 0 ∈ and then [1+XL(λ)− ]− OP , since 1+XL(λ)− OP has a discrete spectrum 1 1 ∈ 2 1 1 ∈ 1 (because XL− = (XD− )(D L− )D− is compact) and 1 + XL− is invertible. With the deﬁnition

1 1 n n 2 Bn(λ) := L(λ)− [XL(λ)− ] OP− − for n N∗, ∈ ∈ we get

∞ N n+1 1 Y = I(λ)dλ, where I(λ) := ∑ ( 1) Bn(λ)+ L(λ)− RN(λ). (4.9) 0 − Z n=1 The idea is now to move all dependence in λ in the sum to the right to facilitate an explicit integration over λ. Let us deﬁne 1 1 n Bn(T ) := L− [T L− ] for n N∗. ∈ 2 Since [D + λ,T]= ∇(T ), a recurrence gives, for any q1 N, ∈ q1 1 1 ℓ ℓ ℓ 2 B1(T )= L− T L− = ∑( 1) ∇ (T )L(λ)− − + r1,q1 (T ), ℓ=0 − q1+1 1 q1+1 q1 2 r1 q (T ) := ( 1) L− ∇ (T )L− − . , 1 − 1 For n 2 we have Bn(T )= L− TBn 1 and another recurrence gives, for any qn N, ≥ − ∈ qn q1 ℓ ℓn ℓn 1 ℓ1 ( ℓ +n+1) Bn(T )= ( 1)| |∇ T ∇ − (T ∇ (T )) L− | | + rn,q (T ), ∑ ··· ∑ − ··· n ℓn=0 ℓ1=0 qn 1 q1 − qn+ ℓ +1 1 qn+1 ℓn 1 ℓ1 (qn+ ℓ +n+1) rn,q (T )= ( 1) | | L− ∇ T∇ − (T ∇ (T )) L− | | n ∑ ··· ∑ − ··· ℓn 1=0 ℓ1=0 − 1 + L− T rn 1,qn 1 (T ). (4.10) − −

The implementation of Bn = Bn(X) in (4.9) with qi = N n for any i = 1, ,n, and − ··· sN,n(λ) := rn,N n(X) for n = 1, ,N gives − ··· 104 4 Fluctuations of the Spectral Action

N n+1 ℓ ℓn ℓn ℓ ( ℓ +n+1) I(λ)= ( 1) ( 1)| |∇ X∇ 1 (X ∇ 1 (X)) L(λ)− | | ∑ − ∑ − − ··· n=1 ℓ ℓi=0, ,N n | ··· − + RN,1(λ)+ RN,2(λ), N R 1 R n+1 with N,1 := L− RN, N,2 := ∑ ( 1) sN,n. n=1 − The integration

∞ ( ℓ +n+1) 1 2 ℓ 2n L(λ)− | | dλ = ( ℓ + n)− D− | |− 0 | | Z 2 is valid for n 1. Remark that the first term of I(λ) is XL(λ)− (with N = 1, ≥ 1 so n = 1,ℓ1 = 0) and we deduce that Y is both in Ψ ′(A ) and OP− with the announced expansion (4.7), if for all N 1 the integrals of the remainders RN (λ) are N 1 ≥ in OP− − . ∞ R N 1+ε We first claim that 0 N,1(λ)dλ OP− − for some ε (0,1) and to prove it we apply the method of Theorem 1.15∈ . ∈ R With 1 1 0 Z(λ):=[1 + XL− (λ)]− OP , ∈ 2 ε N 1 we rewrite the norm of D − RN 1 D − as | | , | | N 1 2 1 1 ε/2 ε/2 − N 1 k N+k 2 1 1 1 (D L− ) − L− ∏ [ D − − X D − ](D L− ) D − X L− Z . k 0 | | | | | | = Thus ∞ ∞ 2 ε N 1 ε/2 2 1 D − RN,1 D − dλ c L(λ)− (DA + λ)− dλ, k 0 | | | | k≤ 0 k kk k Z Z 0 2 1 since the terms in square brackets are in OP and hence bounded, D L(λ)− 1 1 2 1 k k≤ and also we have L− Z = (DA + λ)− (we use XV for the equality). This integral is finite for any ε > 0, since for λ [0,1], the integrand is smaller than 1 2 1 ∈ 1 2 1 1 D − DA− , while for λ 1 we have L− (λ) λ − and (DA + λ)− λ − . Ask| explained| kk ink Theorem 1.15≥, to provethe claimit≤ is sufficientto checkthat≤ thepre- 1 n 1 vious integral’s estimates remain valid when L− Z is swapped to δ (L− Z). Since

1 1 1 1 δ(L− Z)= L− δ(Z)= L− Z δ(X)L− Z, − this improves previous estimates in λ. ∞ R N 1+ε We also claim that 0 N,2(λ)dλ OP− − and proceed essentially in the ∈ ∞ N+1 ε same way to prove it. WeR only need to show that 0 sN,n(λ) D − dλ < ∞ for each n in 1, ,N , and we use the induction to thatk end: For| n| = 1 wek have { ··· } R 4.1 Fluctuations of the Spectral ζ-Function 105

N+1 ε N 1 N N 1 N+1 ε sN 1(λ) D − = ( 1) L− ∇ (X)L− − D − k , | | k k − | | k 1 2 2 N N+1 N 1 2N ε = (L− D )[D− ∇ (X) D − ]L− − D − k | | | | k 1 2 N ε/2 1 1+ε/2 1 1+ε/2 c L− D − L− = c′ L− , ≤ k | | k k k k k which is integrable on R+ for any ε > 0. Now we show that

∞ ∞ N+1 ε N+1 ε sN,n 1(λ) D − dλ < ∞ implies sN,n(λ) D − dλ < ∞. 0 k − | | k 0 k | | k Z Z 1 N+1 ε Using the relation (4.10), we only need to provethat L− X rn 1 D − and the − 1 qn+1 ℓn 1 ℓ1 (qn+ ℓ +n+1) N+1 |ε | operators L− ∇ X∇ − (X ∇ (X)) L− | | D − are integrable. The ﬁrst one follows directly from··· | | 1 1 2 2 2 L(λ)− X = L(λ)− D (D− X) D− X k k k k≤k k and the recurrence hypothesis. For the second one we take the decomposition

1 qn+1 ℓn ℓ (qn+ ℓ +n+1) N+1 ε L− ∇ X∇ 1 (X ∇ 1 (X)) L− | | D − − ··· | | 1 2 2 qn+1 ℓn 1 ℓ1 N ℓ +1 = L− D D− ∇ X∇ − (X ∇ (X)) D − −| | ··· | | × N+ ℓ 1 N ℓ 1 N+1 ε D | |− L− −| |− D − . × | | | | The term in the square bracket is in OP0, so that it remains to show the integrability of 2N+ ℓ ε N ℓ 1 2 1 N+ℓ ε/2 ℓ 1 1+ε/2 D | |− L− −| |− = D L− − D − L− , k| | k k k k| | kk k which follows as above. This completes the proof that

∞ N 1+ε RN := [RN,1(λ)+ RN,2(λ)]dλ OP− − . 0 ∈ Z To dispose of the ε we invoke the handy Lemma 1.14. Theorem 4.2 (see [4] and [7]). Let (A ,H ,D) be a regular spectral triple and let A = A Ψ 0(A ). Then, for any N N and any s C, ∗ ∈ ∈ ∗ ∈ N s s s (N+1) ℜ(s) DA − = D − + ∑ Kn(Y,s) D − mod OP− − , (4.11) | | | | n=1 | |

n where Kn(Y,s) Ψ (A ) and we have for each n 1, ,N ∈ − ∈{ ··· } E ℓ1 E ℓn N 1 Kn(Y,s)= ∑ hn(s;ℓ) (Y ) (Y ) mod OP− − . (4.12) ℓ 0, ,N n n ··· ∈{ ··· − } Proof. To prove Formula (4.11) we observe that

s C (s/2)Y C s 2 DA − = e − e− D − with C := ( s/2)log(D ). | | | | − 106 4 Fluctuations of the Spectral Action

We are going to use the Duhamel formula

1 P+Q P s(P+Q) sP e e− = 1 + e Qe− ds (4.13) Z0 leading to the following Volterra series (expansional formula)

∞ P+Q P 1 tP tP e e− = + ∑ Q(t1) Q(tn)dt, where Q(t) := e Qe− . 0 t tn 1 ··· n=1 Z ≤ 1≤···≤ ≤ The latter is valid for any (un)bounded selfadjoint operator P and any selfadjoint bounded operator Q in the operator norm topology (see [6, Theorem 3.5], [18, Lemma 3.32] for a more general framework). s 2 s s With P = log(D ) and Q = Y , we obtain Q(t)= σ st (Y), so that − 2 − 2 − 2 − ∞ s s s DA − = D − + ∑ Kn(Y,s) D − , (4.14) | | | | n=1 | |

n with each Kn(Y,s) in OP− , hence proving (4.11). Finally, applying the expansion (1.22) within Equation (4.2), with N n and m m 2m N 1 − E (Y )= ∇ (Y) D − , we get that Kn(Y,s) is equal, modulo OP− − , to | |

s n st /2 stn/2 ℓ ℓ ( ) − 1 − E 1 (Y ) E n (Y )dt, 2 ∑ ℓ1 ℓn − 0 t tn 1 n ··· ··· Z ≤ 1≤···≤ ≤ ℓ 0, ,N n ∈{ ··· − } which is nothing else than (4.12).

Remark that E ℓ(Y), ℓ N, are in Ψ (A ), see[7, Corollary 4.4]. ∈ ′ It turns out that, thanks to Formula (4.11), the ﬂuctuated triple (A ,H ,DA) inherits many of the properties of the original (A ,H ,D).

Corollary 4.3. Let (A ,H ,D) be regular spectral triple and let A = A Ψ 0(A ). ∗ ∈ Then, (A ,H ,DA) is a regular spectral triple and ΨA(A ) Ψ(A ), where ΨA(A ) ⊂ refers to the pseudodifferential calculus deﬁned by DA.

Proof. Recall that (A ,H ,DA) is indeed a spectral triple — cf. Remark 1.5. 0 Formula (4.11) for s = 1 yields DA = D + B(A), with B(A) Ψ (A ). In par- − 0| | | | 0 ∈ 0 ticular DomδA = Domδ since Ψ (A ) B(H ). Thus OPA = OP ; moreover, if r r 0 0 ⊂ T OPA then DA T OPA = OP , so ∈ | |− ∈ r r r r 0 D − T = ( D − DA ) DA − T OP | | | | | | | | ∈ r r 0 r as D DA OP , and T OP . By symmetry, | |− | | ∈ ∈ r r OPA = OP and by extension ΨA(A ) Ψ(A ) since we already know that DA Ψ(A ) and so ⊂ ∈ is DA by Theorem 4.2, implying that DA = DA PA is also in Ψ(A ) because | | | | | |− 4.1 Fluctuations of the Spectral ζ-Function 107

∞ k ∞ ′ k PA is smoothing (cf. p. 101). As in Lemma 1.12, k=0 DomδA = k=0 DomδA with the definition δA′ := [DA, ]. · 0 0T T Since for a A , we have a, [DA,a] OP = OPA, the regularity of the fluctuated triple is proved.∈ ∈ Consequently, we define

C z ΨA (A ) := T DA T Ψ(A ), z C . { | | | ∈ ∈ }

Theorem 4.4. Let (A ,H ,D) be p-dimensional regular spectral triple with a di- 0 mension spectrum of order d and let A = A∗ Ψ (A ). Then: s ∈ i) The function s ζD (s)= Tr DA − has a meromorphic continuation to C. 7→ A | | ii) The triple (A ,H ,DA) is also p-dimensional and has a dimension spectrum Sd(A ,H ,DA) included in Sd(A ,H ,D) and of order at most d. z C C C iii) For any z C, we have DA Ψ (A ) and ΨA (A ) Ψ (A ). iv) For any k ∈ N | | ∈ ⊂ ∈ ∗ [k] [k] p p DA − = D − . (4.15) − | | − | | Z Z Proof. i) Firstly, let us rewrite Formula (4.11), for any s C with ℜ(s) > p, as ∈ N s s s ζDA (s)= Tr DA − = Tr D − + ∑ Tr Kn(Y,s) D − + Tr RN (s). (4.16) | | | | n=1 | | s n ℜ(s) Note that it is well-deﬁned for any N 1 because Kn(Y,s) D − OP− − and N 1 ℜ(s) ≥ | | ∈ RN (s) OP− − − , so all of the involved operators are trace-class for ℜ(s) > p. ∈ Furthermore, we can rewrite ζDA using the explicit form of the Kn’s (4.12):

ζDA (s)= ζD(s)+ ∑ ∑ hn(s;ℓ)ζWn(ℓ),D(s)+ fN(s), (4.17) n=1 ℓ 0, ,N n n ∈{ ··· − } where fN (s)= Tr RN (s) and

ℓ ℓn ℓ 2ℓ1 ℓn 2ℓn Wn(ℓ) := E 1 (Y ) E (Y )= ∇ 1 (Y ) D − ∇ (Y ) D − . (4.18) ··· | | ··· | | C Since the functions ζWn(ℓ),D admit meromorphic extensions to by the dimension spectrum hypothesis on (A ,H ,D) and fN is actually a holomorphic function for ℜ(s) > p N 1, Formula (4.17) provides a meromorphic continuation of ζDA to the half-plane− −ℜ(s) > p N 1. As N can be taken arbitrarily large, we obtain − − a meromorphic extension of ζDA to the whole complex plane. Furthermore, since (A ,H ,D) is p-dimensional, ζD is singular at s = p and regular for ℜ(s) > p, and so is ζDA , hence (A ,H ,DA) is also p-dimensional. 0 ii) Secondly, let us multiply Formula (4.11) from the left by T ΨA (A ) and take the trace as in (4.16). Then, we have for N N and for ℜ(s) > p∈ ∈ ∗ 108 4 Fluctuations of the Spectral Action

ζT,DA (s)= ζT,D(s)+ ∑ ∑ hn(s;ℓ)ζTWn(ℓ),D(s)+ Tr TRN(s). (4.19) n=1 ℓ 0, ,N n n ∈{ ··· − } n ℓ 0 0 For any multi-index ℓ, Wn(ℓ) OP− −| | OP , and hence TWn(ℓ) OP . Since 0 0 ∈ ⊂ ∈ ΨA (A ) Ψ (A ), the dimension spectrum hypothesis on (A ,H ,D) then assures ⊂ C that the functions ζTWn(ℓ),D are meromorphic on and thus we establish a mero- 0 morphic continuation of ζT,DA for any T Ψ (A ). ∈ A This fact implies that (A ,H ,DA) has a dimension spectrum and, moreover, Sd(A ,H ,DA) Sd(A ,H ,D), with the former being at most of the order of the ⊂ latter, since the poles of any ζT,DA can only come from the ones of ζTWn(ℓ),D . n z C iii) Since Kn(Y,s) Ψ − (A ), Formula (4.11) shows that DA Ψ (A ). Fur- thermore, Corollary 4.3∈ implies the announced inclusion. | | ∈ iv) Using Formula (4.11) we obtain

p p s (p+s) (p+2) ( DA − D − ) D − = K1(Y, p) D − mod OP− . | | − | | | | | | (p+1+ℜ(s)) 1 This operator is in OP− , as K1(Y, p) OP− , so it is trace-class in a neigh- [k] p ∈p bourhood of s = 0. Hence, ( DA D − )= 0. | |− − | | R

4.2 Fluctuations of Noncommutative Integrals

0 C Given A = A Ψ (A ) we deﬁne, for any T ΨA (A ) and k Z, ∗ ∈ ∈ ∈ [k] k 1 T := Res s − ζT,DA (s). −A s=0 Z [k] [k] We shall now invoke Proposition 4.2 to compute A T in terms of . Proposition 4.5. Let (A ,H ,D) be a regular spectralR triple of ﬁnite dimensionR with a dimension spectrum of order d. 0 C Then, for A = A Ψ (A ), T ΨA (A ) and k Z,k < d we have: ∗ ∈ ∈ ∈

[k] [k] d k ℓ [k+n+ j ] − | | T = T + h j(ℓ) T Wn(ℓ), (4.20) −A − ∑ ∑ ∑ − Z Z n=1 ℓ 0, ,d k n n j=0 Z ∈{ ··· − − }

where h j(ℓ) are deﬁned in (4.6) and Wn(ℓ) in (4.18). C Moreover, for any T ΨA (A ), ∈ [d] [d] T = T, (4.21) −A − Z Z Proof. By exploiting Formula (4.19), we obtain, with N = d k, − 4.2 Fluctuations of Noncommutative Integrals 109

[k] [k] d k − k 1 T T = ∑ ∑ Res s − hn(s;ℓ)ζTWn(ℓ),D(s) −A − − s=0 Z Z n=1 ℓ 0, ,d k n n ∈{ ··· − − } d k ℓ − | | k+n+ j 1 = h j(ℓ)Res s − ζTW (ℓ),D(s) ∑ ∑ ∑ s=0 n n=1 ℓ 0, ,d k n n j=0 ∈{ ··· − − } d k ℓ [k+n+ j ] − | | = h j(ℓ) TWn(ℓ). ∑ ∑ ∑ − n=1 ℓ 0, ,d k n n j=0 Z ∈{ ··· − − } d+n s C Equality (4.21) results from Res s Tr T D − = 0, for T Ψ (A ), n N∗. s=0 | | ∈ ∈

C p+δ Corollary 4.6. Let T ΨA (A ) OP− for some δ < 1. Then, for any k N ,∈ we have ∩ ∈ ∗ [k] [k] T = T. −A − Z Z n ℓ Proof. Recall (cf. (4.18)) that Wn(ℓ) OP− −| |, so that the operators TWn(ℓ) appearing in Formula (4.20) are of the∈ order p ε with ε = n + ℓ δ > 0 since − − [k] | |− n 1, ℓ 0 and δ < 1, so TWn(ℓ) are trace-class and TWn(ℓ)= 0 for any k ≥N .| |≥ ∈ ∗ R Corollary 4.7. Let (A ,H ,D) be a regular spectral triple of ﬁnite dimension with 0 a simple dimension spectrum and let A = A∗ Ψ (A ). Then: C ∈ i) For any T ΨA (A ), ∈

Res ζT,DA (s)= Res ζT,D(s). (4.22) s=0 s=0

1 ii) ζDA is regular at 0 if and only if ζD is so and ζDA (0)= ζD(0) 2 Y. iii) Moreover, with p being the dimension of (A ,H ,D), − R

(p 1) (p 1) p 1 p 1 DA − − = D − − − X D − − , (4.23) − | | − | | − 2 − | | Z Z Z (p 2) (p 2) p 2 p p 2 2 p DA − − = D − − + − X D − + X D − − . (4.24) − | | − | | 2 − − | | 4 − | | Z Z Z Z Proof. i) Equation (4.22) is just a rewriting of Formula (4.21) for d = 1. 1 ii) The point i) with T = implies that ζDA is regular at 0 iff ζDA is so. Moreover, taking k = 0, d = 1,T = 1 in Formula (4.20) and recalling (4.5),(4.6) we obtain

1 1 1 Res s− ζDA (s)= Res s− ζD(s)+ h0(0) Y = ζD(0) Y. s=0 s=0 − − 2 − Z Z iii) Using respectively Formulae (4.11),(4.12) and(4.7) with N = 1 we get 110 4 Fluctuations of the Spectral Action

s s s s s s DA − = D − + K1(Y,s) D − + R1(s)= D − Y D − + R2(s) | | | | | | | | − 2 | | s s s 2 = D − X D − − + R3(s), (4.25) | | − 2 | | ℜ(s) 2 for any s C with Ri(s) OP− − . Taking s = p 1 and applying the noncom- ∈ ∈ − p 1 mutative integral yields Equation (4.23), since R3(p 1) OP− − is trace-class. The same manoeuvre with N = 2 gives − ∈

s s s2 2 2 s 3 ℜ(s) DA − = [1 Y + (∇(Y )D− +Y )] D − mod OP− − , (4.26) | | − 2 8 | | 2 1 4 2 4 3 Y = XD− (∇(X)D− + X D− ) mod OP− . (4.27) − 2 2 2 2 2 4 3 Remark that Y = XD− mod OP− and Y = X D− mod OP− . Since is a trace, R r r ∇(Y ) D − = ∇(X) D − = 0 for any r R − | | − | | ∈ Z Z and we obtain the Equation (4.24) for s = p 2—see[7, Lemma 4.10]. − Let us stress that the above corollary explicitly uses the assumption of simplicity

of the dimension spectrum of (A ,H ,D). If this is not the case, then ζDA might fail to be regular at 0, even though ζD is so, because the higher order poles of functions 1 ζT Wn(ℓ),D at 0 can render A nonzero — contrary to the commutative case, see (1.35· ). To show the next important result on the ﬂuctuation of zeta functions at zero, R we need the following lemma which closely follows [4, Lemma 2.3].

0 Lemma 4.8. Let (A ,H ,D) be a regular spectral triple and A = A∗ Ψ (A ). Then: ∈ i) For any N N, there exist B(t) Ψ(A ) such that for t > 0, we have N ∈ ∈ mod OP−

∂ 2 2 2 2 [log(D +tXV ) logD log(1 +tXV D− )] = [D +tX, B(t)], (4.28) ∂t − − 2 2 2 2 log(D + XV ) logD log(1 + XV D− ) = [D , B1] + [X, B2], (4.29) − − 1 1 A where B1 := 0 B(t)dt andB2 := 0 tB(t)dt are in Ψ( ). ii) Moreover, R R

2 2 2 Y = [log(D + XV ) logD ]= log(1 + XV D− ). (4.30) − − − − Z Z Z Proof. The operator

2 2 2 Ct := D +tXV = tDA + (1 t)D − 2 2 is selfadjoint and positive for t [0,1]. The invertibility of D and DA implies that ∈ 2 2 Ct + λ is invertible for any λ 0 and, consequently, that 1 + tXV D− = Ct D− is also invertible when t [0,1]. ≥ 2 1∈ Since XV D− OP− is compact by Proposition 1.13 ix), it has a purely discrete ∈ 2 2 2 spectrum so that the operator log(1 + XV D− )= log(DAD− ) exists. 4.2 Fluctuations of Noncommutative Integrals 111

i)Asin(4.8),

∞ 2 2 2 1 2 1 log(D +tXV ) logD = [(D + λ)− (D +tXV + λ)− ]dλ − 0 − Z always exists, so we obtain

∞ ∂ log(D2 +tX ) = 1 X 1 dλ. ∂t V Ct +λ V Ct +λ Z0 Moreover,

∞ 2 1 2 1 2 1 ∂ 1 2 XV (Ct + λ)− dλ = XVCt− = XV D− ( +tXV D− )− = ∂t log( +tXV D− ). Z0 Thus, with L(t) being the LHS of (4.28),

∞ 1 1 L(t)= (Ct + λ)− , XV (Ct + λ)− ]dλ. Z0 We now invoke Lemma 1.16 with D Ct , λ / spec Ct and n = 1 to deduce, with − ∈ ∇t (T ) := [Ct ,T ],

N 1 1 k k (k+2) [(Ct + λ)− , XV (Ct + λ)− ]= ∑ ( 1) ∇t XV (Ct + λ)− + RN(λ,t) k=1 − N+1 1 N+1 (N+2) with RN (λ,t) := ( 1) (Ct + λ)− ∇ [XV ](Ct + λ)− . (4.31) − t (k+1) Because (Ct + λ)− , being in the centraliser of ∇t , can be put inside the paren- thesis of ∇t in the ﬁrst equality, we get

N 1 1 k k 1 (k+2) [(Ct + λ)− , XV (Ct + λ)− ]= ∇t ∑ ( 1) ∇t − (XV )(Ct + λ)− + RN(λ,t) k=1 − N 4 with RN (λ,t) OP . The term in the bracket gives, after an integration over λ, ∈ − − N ( 1)k k 1 (k+1) B(t) := ∑ k−+1 ∇t − (XV )Ct− . k=1

1 Since Ct− is in Ψ(A ) (see Example 1.21), so is B(t). Thus, ∞ L(t)= ∇t [B(t)] + RN(t) where RN (t) := RN (λ,t)dλ. Z0 N We claim that RN(t) OP− . Using the method of Theorem 1.15, we show that N/2 0 ∈ RN (t)C OP . With t ∈ N+1 1 N/2 0 E = ∇ (XV )C− − OP , t t ∈ 112 4 Fluctuations of the Spectral Action we get

∞ N/2 N/2 RN(t)Ct RN(λ,t)Ct dλ k k≤ 0 k k Z ∞ 1 1+N/2 N 2 N/2 (λ +Ct )− E Ct (λ +Ct )− − Ct dλ ≤ 0 k kk kk k Z ∞ 1 N+1 1 2 E Ct (λ +Ct )− (λ +Ct )− dλ. ≤k k 0 k k k k Z 1 The latter is ﬁnite since Ct is a positive operator and Ct (λ +Ct )− 1. We now show that k k≤

N/2 N/2 N/2 δ(RN (λ,t)Ct )= δ(RN (λ,t))Ct + RN(λ,t)δ(Ct ) has a ﬁnite integral over λ R+. As for the second term, we∈ have

N/2 N/2 N/2 N/2 RN(λ,t)δ(C ) RN(λ,t)C C− δ(C ) k t k≤k t kk t t k thus yielding a finite integral. 1 1 N+2 In the first term, we expand the derivation δ on RN (λ,t)= Aλ− B(Aλ− ) as a 1 1 finite sum of expressions similar to RN (λ,t) with only one of the Aλ− = (λ +Ct )− 1 1 replaced by A− δ(Ct )A− or B replaced by δ(B). Since − λ λ 1 1 1 1 1 A− δ(Ct )A− Ct (λ +Ct )− C− δ(Ct ) (λ +Ct )− , k λ λ k≤k kk t kk k N/2 the convergence of the integral is unspoilt. Thus RN (λ,t)Ct Domδ and, with N/2 ∈n the same arguments, one deduces that RN(λ,t)C n Domδ . t ∈ ∩ The proof of (4.28) is complete because XV X, so one can swap XV and X. ∼ Finally, Equation (4.29) follows from the definition of B1 and B2. ii) The tracial property of (Theorem 1.27) applied to (4.29) gives (4.30).

Theorem 4.9 ([4]). Let (A ,HR ,D) be a p-dimensional regular spectral triple with 0 a simple dimension spectrum, ζD regular at zero and let A = A Ψ (A ). Then, ∗ ∈ p ( 1)k 1 k ζD (0) ζD(0)= − (AD− ) . (4.32) A − ∑ k − k=1 Z Proof. Thanks to Corollary 4.7 and (4.30), one has

1 1 2 ζD (0) ζD(0)= Y = log(1 + XV D− ). (4.33) A − − 2 − − 2 − Z Z We now write

N 1 2 ( 1)k+1 2 k log( + XV D− )= ∑ − k (XV D− ) + RN, k=1 4.2 Fluctuations of Noncommutative Integrals 113 where the remainder is

1 N 2 N+1 N 2 (N+1) RN = ( 1) (XV D− ) (1 t) (1 +tXV D− )− dt. − 0 − Z 2 0 We have alreadyseen in the proofof Lemma 4.8 that 1+tXV D− OP is invertible. 2 1 ∈ N 1 Thus, the integral makes sense and since XV D OP− we have RN OP− − . In − ∈ ∈ particular, RN is trace-class for N = p and

p 2 ( 1)k+1 2 k log(1 + XV D− )= − (XV D− ) , − ∑ k − Z k=1 Z where, as usual, we can safely replace XV by X. 1 1 With a = D− A and b = AD− ,

2 1 X = D(a + b + ab)D and XD− = D(a + b + ab)D− , so that, being a trace, R 2 k k (XD− ) = (a + b + ab) − − Z Z and we have to compare

N N N ( 1)k+1 k 1 ( 1)k k ( 1)k+1 k k − (a + b + ab) with − a = − (a + b ). ∑ k − − 2 ∑ k − ∑ k − k=1 Z k=1 Z k=1 Z Let us, more generally, introduce the following two formal series in x R within the free algebra generated by a and b: ∈

∞ ( 1)k+1 2 k S(x) := ∑ − k [xa + xb + x ab] , k=1 ∞ ( 1)k+1 k k T (x) := ∑ − k [(xa) + (xb) ]. k=1 . We claim that they are equal modulo commutators (denoted by =) so that, using the tracial property of , the proof of (4.32) would be complete. Since S(0)= T (0), it is sufﬁcient to compare the derivatives of S and T. Remark R n d n . n 1 ﬁrst that for the derivative of a term M(x) , we have dx M(x) = nM(x) − M′(x) k n k 1 since one can commute M′(x) to the right in M(x) M′(x)M(x) − − . Thus, 114 4 Fluctuations of the Spectral Action

∞ . k+1 2 k 1 S′(x) = ∑ ( 1) (xa + xb + x ab) − (a + b + 2xab) k=1 − 2 1 1 = (1 + xa + xb + x ab)− (a + b + 2xab) = [(1 + xa)(1 + xb)]− (a + b + 2xab) 1 1 = (1 + xb)− (1 + xa)− [(1 + xa)b + a(1 + xb)] 1 1 1 = (1 + xb)− b + (1 + xb)− (1 + xa)− a(1 + xb) . 1 1 = (1 + xb)− b + (1 + xa)− a = T ′(x).

4.3 Consequences for the Spectral Action

Theorem 4.4 assures that given a regular p-dimensional spectral triple (A ,H ,D) with a dimension spectrum, the triple (A ,H ,DA) will also be regular, p-dimensional and possessing a dimension spectrum, for any A = A Ψ 0(A ). However, to ∗ ∈ deduce an expansion of Tr f ( DA /Λ) we would need to control the behaviour of | | (the maximal meromorphic extension of) ζDA on the verticals, as explained in detail in Section 3.2. This, unfortunately, does not come for free, even if we can control 0 the behaviour of ζT,D for every T Ψ (A ). Indeed, observe that Formula (4.17) ∈ relating ζDA to ζT,D’s involves a holomorphic remainder RN(s), which is harmless when it comes to the poles and residues, but might contribute to the behaviour on the verticals. Also, there is no good reason to believe that a heat trace expansion of the form (3.37) fora given D will imply a similar one for DA, although this is indeed the case for commutative spectral triples and also for the noncommutative torus (cf. Lemma B.5). In full generality of noncommutative geometry this is a stumbling block and we list it as Problem 6) e) in Chapter 5. After revealing the blot on the landscape, let us enjoy the bright perspective. Given the heat trace asymptotic expansion of D and DA we can:

1. Deduce the large-Λ asymptotic expansion of S(D, f ,Λ) and S(DA, f ,Λ) with the help of Theorem 3.29. 2. Express the coefﬁcients of the expansion of S(DA, f ,Λ) as noncommutative integrals of operators polynomial in A, via Formulae (4.22), (4.12), (4.7),(4.3). Let us illustrate it in the case of (A ,H ,D) with a simple dimension spectrum:

Theorem 4.10. Let (A ,H ,D) be a regular p-dimensional spectral triple with a simple dimension spectrum and let A = A Ψ 0(A ). Assume, moreover, that ∗ ∈ t DA α Tr e− | | = ∑ aα ( DA )t− + a0( DA )+ O0(1), (4.34) α Sd+ | | | | ∈ where the, possibly inﬁnite, series over α is absolutely convergent for all t > 0. r Then, the function ζD is regular at 0 and, for any f with r > p, A ∈C0 4.3 Consequences for the Spectral Action 115

∞ p ℜ(α) D α α 1 ⌊ − ⌋ 1 A α Tr f ( A /Λ)= ∑ Λ x − f (x)dx ∑ Pn(α,D,D− , ) D − | | 0 − | | α Sd+ Z n=0 Z ∈ p k ( 1) A 1 k O + f (0) ζD(0)+ ∑ −k ( D− ) + ∞(1), (4.35) k 1 − h = Z i n where Pn Ψ (A ) are polynomials in all variables and of degree n in A: ∈ − 1 α 1 2 α2 2 2 P0 = 1, P1 = αAD− , P2 = (α + 2)(AD− ) + A D− ,... (4.36) − 4 4 Proof. Firstly, we apply Corollary 3.33 to obtain

α . α 2 1 2 4 α2 2 2 α DA − = [1 (XD− X D− )+ (XD− ) ] D − , | | − 2 − 2 8 | | 2 2 2 . 1 2 2 2 2 . 2 4 XD− = (AD + DA + A )D− = 2AD− + A D− , (XD− ) = X D− , 2 4 . 3 2 4 2 2 . 1 2 2 2 X D− = 2ADAD− + AD AD− + A D− = 2(AD− ) + 2A D− ,

since

3 1 2 1 2 3 1 2 3 ADAD− = (AD− ) + AD− [D ,A]D− = (AD− ) mod OP−

and similarly,

2 4 2 2 2 4 2 2 3 AD AD− = A D− + A[D ,A]D− = A D− mod OP− .

1 This gives Formulae (4.36) and makes it clear that every A is accompanied by D− , n so that Pn Ψ(A ) is a polynomial in A of order n. It is also straightforward how ∈ − to compute the higher Pn’s. Finally, we apply Theorem 4.9 to express ζDA (0) using Equation (4.32). 116 4 Fluctuations of the Spectral Action

Beyond the case of a triple (A ,H ,D) with a simple dimension spectrum we can still express the coefficients of the asymptotic expansion of S( f ,DA,Λ) in terms of the polynomials Pn, provided we have at hand the heat trace expansion (3.37) for both D and DA. Obviously, the analogue of Formula (4.35) would be considerably more involved, but the coefficients can be computed algorithmically from Formulae (4.20),(4.12),(4.7)and (4.3) — along the same lines as in Theorem 4.10. It is important to stress that if (A ,H ,D) does not have a simple dimension spectrum then the noncommutative integral is not invariant under fluctuations. In- deed, Formula (4.20) implies that A T T would, in general, involve the higher − [d] order residues. On the other hand,R the highestR residue remains insensible to fluctuations — recall Formula (4.21). More generally, we have: R Proposition 4.11. Let (A ,H ,D) be a regular p-dimensional spectral triple with a dimension spectrum of order d and let A = A Ψ 0(A ). Assume that ∗ ∈ d t D n z p+1 Tr e− | | = ∑ ∑ az,n( D )log t t− + 0(t− ), (4.37) z X n=0 | | O ∈ 0 d t DA n z p+1 Tr e− | | = az,n( DA )log t t− + 0(t− ), ∑ ∑ | | O z X0 n=0 ∈ where the, possibly infinite, series over the set X0 C are absolutely convergent for ⊂ all t > 0 and min ℜ(z) z X0 > p 1. Then, for any f { r with| r∈> p,} we have− ∈C0 p 1 Tr f ( DA /Λ) Tr f ( D /Λ)= ∞(Λ − ). (4.38) | | − | | O

Note that X0 here coincides with the one in Theorem 3.29, if we choose r1 = p 1. − − Proof. On the strength of Theorem 3.29, with Formulae (3.40)and (3.38), it is suf- [k] z [k] z ficient to check that A DA − = D − for all z X0 and k 0,1,...,d . But with ℜ(z) > p 1,| we have,| by Corollary| | 4.6 and∈ Formula∈{ (4.11) respectively,} −R R [k] [k] [k] [k] [k] z z z z DA − = DA − = D − + R(z)= D − , −A | | − | | − | | − − | | Z Z Z Z Z ℜ(z) 1 since R(z) OP− − is trace-class. ∈ In the commutative case, as well as on the noncommutative torus, Formulae t T p p+1 (4.37) are simply Tre− = ap,0(T )t− + 0(t− ), with T D , DA . In this case, Equation (4.38) is interpreted as the invarianceO of the dominant∈ {| | term| |} under the fluctuations. However, we can see that if the dimension spectrum of a given spectral triple is non-simple and/or has poles in the vertical strip p > ℜ(z) > p 1, then not only the dominant term is immune to the fluctuations. − Definition 4.12. Let (A ,H ,D) be a regularp-dimensional spectral triple with a simple real dimension spectrum.

The tadpole TadDA (α) of order α is the linear term in A in the asymptotics (4.35). 4.4 Operator Perturbations 117

We have just shown that

∞ α 1 1 + TadDA (α) = ( α x − f (x)dx) AD− if α Sd , − 0 − ∈ Z Z 1 TadD (0)= f (0) AD− . A − − Z See also [10, Proposition 3.5]. 1 This notion of a tadpole is borrowed from quantum ﬁeld theory where AD− is a one-loop graph with a fermionic internal line and only one external bosonic line A, thus looking like a tadpole. There are no tadpoles in commutative geometries on manifolds, also the ones with boundaries and torsion, or on noncommutative tori [9, 10, 13]. The vanishing of tadpoles means that a given geometry (A ,H ,D) is a critical point for the spectral action [5, p. 210]. On the other hand, the existence of tadpoles means a priori the instability of the quantum vacuum—see[8, 11].

4.4 Operator Perturbations

The spectral action (1.43) is only a part of the large programme on the differentiation of operators and on perturbation theory. There exist several strongly related approaches and we only brieﬂy sketch a few of them. The ﬁrst approach is the Lifshits formula given, for two selfadjoint, possibly unbounded, operators H0 and V acting on H , by ∞ Tr[ f (H0 +V) f (H0)] = ξ (λ) f ′(λ)dλ, (4.39) − ∞ Z− where ξ is the so-called Krein shift function (see for instance [3]). If the perturbation V is a trace-class operator and R(A) is the resolvent of A, the holomorphic function

G(z) := det 1 +VR(H0)(z)

satisﬁes 1 G− (z)G′(z)= Tr[R(H0)(z) R(H0 +V)(z)] − and we deﬁne

1 ξ (λ) := π− lim argG(λ + iε) for almost all λ. ε 0 → Then, we get

∞ 1 logG(z)= ξ (λ)(λ z)− dλ when ℑ(z) = 0, ∞ − 6 ∞ Z− ∞ ξ (λ) dλ TrV and Tr(V)= ξ (λ)dλ. ∞ | | ≤ ∞ Z− Z− 118 4 Fluctuations of the Spectral Action

Moreover, (4.39) holds true at least for functions f which are smooth and compactly supported. Remark that the function ξ is a spectral shift, because, when λ is an isolated eigenvalue of both H0 +V and H0 with respective multiplicities m and m0, then ξ (λ +0) ξ (λ 0)= m0 m and ξ has a constant integer value in any interval − − − located within the resolvents sets of both H0 +V and H0. Formula (4.39) is commonly employed in the scattering theory (see [14]) and has also been adapted to the noncommutative framework for the computation of the spectral action beyond the weak-field approximation, as in [12]. Another approach is the following: Assume that the unbounded operator H0 is ∞ selfadjoint and V is also selfadjoint, but bounded. Let λk k=1 be the eigenvalues { ∞} of H0 counted here with their multiplicities and let ψk k=1 be the corresponding normalised eigenvectors. If { } Vr,s := V ψr, ψs , h i we have the following Taylor asymptotic expansion for f CN+1(R): ∈ c Tr f (H0 +V)= (4.40) N 1 − 1 [n 1] Tr f (H0)+ ∑ n ∑ Vi1,i2 Vin 1,inVin,i1 ( f ′) − (λi1 ,...,λin )+ RH0, f ,N , − n=1 i , ,in ··· 1 ··· [m] where ( f ′) is the divided difference of order m of f ′ and, of course, the difficulty is to control the remainder RH0, f ,N. Defining N 1 R − 1 dn H0, f ,N := f (H0 +V) ∑ n! dtn t=0 f (H0 +tV), − n=0 |

d where the Gâteaux derivatives dt t=0 f (H0 + tV) are taken in some uniform topology, the strategy is to prove that |

R R N RH0, f ,N = Tr H0, f ,N and H0, f ,N = V 0( V ). (4.41) k k Ok k→ k k Recall ﬁrst that the Gâteaux derivative of a function f : X Y between two locally convex topological vector spaces X, Y , is →

1 f ′(h)(v) := lim t− [ f (h +tv) f (h)] for h, v X. t 0 − ∈ → It is linear in v if X, Y are Fréchet spaces. This approach is used in [16]or[17, Section 7.2.2] to compute the spectral action

SD (V) := Tr f (D +V) via (4.40) for a ﬁnitely summable spectral triple (A ,H ,D), setting H0 = D and under the following assumptions: ∞ tx2 1. f (x)= 0 e− dφ(t) with a positive measure dφ. ∞ α D β t(εD2 γ) 2. 0 t TrR[ e− − ]dφ(t) < ∞ for any α,β,γ > 0 and 0 ε < 1. 3. Moreover,| V| B2(H ) – as deﬁned below. ≤ R ∈ 4.4 Operator Perturbations 119

n The derivation δ ′ given in (1.2) deﬁnes a family of seminorms δ ′ (T ) n N and the vector spaces {k k} ∈

n k B (H ) := T B(H ) : δ ′ (T) < ∞ for all k n { ∈ k k ≤ } become Fréchet spaces, implying the Fréchet differentiability of the spectral action in terms of the perturbation V. 1 Remark that if the triple is regular, any selfadjoint one-form in ΩD (A ) is in B2(H ). Without entering into the details of (4.40), the main steps are an iterated use of the Duhamel formula (see (4.13))

1 t(D+V)2 t(D2+X) tD2 st(D2+X) (1 s)tD2 e− = e− = e− t e− Xe− − ds − 0 Z for the perturbation of the heat operator (the second line of the assumption guarantees the convergence of the trace of this integral), and the fact that the Taylor expansion ∞ 1 [n] SD (V)= ∑ n! SD (0)(V, ,V ) n=0 ··· has coefﬁcients which are given by

[n] SD (0)(V, ,V) = (n 1)! ∑ Vin 1,inVin,i1 f ′(λi1 ,...,λin ). − ··· − i , ,in 1 ··· Formula (4.40) has been generalised in [15] under the only assumption that the selfadjoint operator H0 has a compact resolvent (but no positivity of H0 nor summa- N+1 bility condition on its spectrum is required), V = V ∗ is bounded and f C (R) with compact support ( f is not necessarily positive or even). Even if the∈ proof is more subtle it essentially goes through the quoted steps. For instance, using the inclusion

CN+1(R) f : f (n) and F( f (n)) L1(R), n = 0, ,N , c ⊂{ ∈ ··· } the exponentials appearing in the iterated Duhamel formula are now unitaries so the second line of the assumption is not necessary. Further generalisation into the theory of multiple operator integrals, compatible with the formalism of differentiation of operator functions, is possible. It relates to the important notion of spectral ﬂow in the setting of type II von Neumann algebras — see, for instance, [1]. 120 4 Fluctuations of the Spectral Action References

1. Azamov, N., Carey, A., Dodds, P., Sukochev, F.: Operator integrals, spectral shift and spectral flow. Canadian Journal of Mathematics 61, 241–263 (2009) 2. Chamseddine, A.H., Connes, A., van Suijlekom, W.D.: Inner fluctuations in noncommutative geometry without the first order condition. Journal of Geometry and Physics 73, 222–234 (2013) 3. Chattopadhyay, A., Sinha, K.: Trace formulae in operator theory. In: T. Bhattacharyya, M. Dritschel (eds.) Operator Algebras and Mathematical Physics, pp. 1–36. Birkhäuser (2015) 4. Connes, A., Chamseddine, A.H.: Inner fluctuations of the spectral action. Journal of Geometry and Physics 57(1), 1–21 (2006) 5. Connes, A., Marcolli, M.: Noncommutative Geometry, Quantum Fields and Motives, Collo- quium Publications, vol. 55. American Mathematical Society (2008) 6. Davies, E.B.: One-Parameter Semigroups. Academic Press (1980) 7. Essouabri, D., Iochum, B., Levy, C., Sitarz, A.: Spectral action on noncommutative torus. Journal of Noncommutative Geometry 2(1), 53–123 (2008) 8. Géré, A., Wallet, J.C.: Spectral theorem in noncomutative field theories: Jacobi dynamics. Journal of Physics: Conference Series 634, 012006 (2015) 9. Iochum, B., Levy, C.: Spectral triples and manifolds with boundary. Journal of Functional Analysis 260, 117–134 (2011) 10. Iochum, B., Levy, C.: Tadpoles and commutative spectral triples. Journal of Noncommutative Geometry 5(3), 299–329 (2011) 11. Iochum, B., Levy, C., Sitarz, A.: Spectral action on SUq(2). Communications in Mathematical Physics 289(1), 107–155 (2009) 12. Iochum, B., Levy, C., Vassilevich, D.: Spectral action beyond the weak-field approximation. Communications in Mathematical Physics 316(3), 595–613 (2012) 13. Iochum, B., Levy, C., Vassilevich, D.: Spectral action for torsion with and without boundaries. Communications in Mathematical Physics 310(2), 367–382 (2012) 14. Simon, B.: Trace ideals and their applications, volume 120 of mathematical surveys and monographs. American Mathematical Society, Providence, RI (2005) 15. Skripka, A.: Asymptotic expansions for trace functionals. Journal of Functional Analysis 266(5), 2845–2866 (2014) 16. van Suijlekom, W.D.: Perturbations and operator trace functions. Journal of Functional Anal- ysis 260(8), 2483–2496 (2011) 17. van Suijlekom, W.D.: Noncommutative Geometry and Particle Physics. Mathematical Physics Studies. Springer (2015) 18. Zagrebnov, V.: Topics in the Theory of Gibbs Semigroups. Leuven University Press (2003) Chapter 5 Open Problems

Abstract As a desert, we serve a number of open problems connected with the subject matter of the book. Some of them consider the general framework of spectral triples and its possible extensions, while the other are more specific and relate to the properties of the spectral action. The problems are essentially of mathematical nature, though, at least in some cases, the conceptual skeleton strongly depends upon the input from physics. To our mind, the solution to each of these stumbling blocks would advance our understanding of the foundations and implications of the Spectral Action Principle. We therefore cordially invite the Reader to contemplate the list below, both from mathematical and physical perspectives. 1) Existence of spectral triples. Despite quite a few illustrative examples, the ter- ritory of spectral triples remains vastly uncharted. Beyond the almost-commutative enclave, the constructive procedures are available in some specific contexts: isospectral deformations [15], AF C∗-algebras [14] or crossed products [43, 46]. 1 The first road towards spectral triples is to fix a C∗-algebra A and then to find (and classify!) all possible smooth dense ∗-subalgebras A , along with the compatible operators D acting on a chosen Hilbert space H . The ultimate aim would be to understand the ‘space of the operators D’, which is the domain of the spectral action functional (1.43). This would certainly require the theory of Fredholm modules and K-homology [44], but surely much more than that. Alternatively, one can fix a sequence of real numbers seen as the spectrum of D and seek a compatible algebra A . Such a situation appears, for instance, in the context of manifolds with boundary [45]. 2) Non-unital spectral triples. We have focused exclusively on unital spectral triples, which correspond to compact manifolds in the classical case. Non-unital triples have been studied to some extent [9, 10, 57, 58]. In such a case the operator D does not have a discrete spectrum and one needs to adjust the definition of the spectral action. The easy way is to introduce a ‘spatial’ (or infrared) cut-off: Tr Φ f ( D /Λ), with a suitable Φ A , but its choice yearns for a deeper | | ∈ 1 Such a fixed C∗-algebra can come from physics — as the natural algebra of observables of a given system (see e.g. [42, 49, 61]).

121 122 5 Open Problems physical motivation. One could also promote the energy scale Λ to a dilaton field [13, 38, 39, 40]. 3) Twisted spectral triples. Motivated by the type III noncommutative geometry, where there are no finitely summable spectral triples [17], a few notions of twisted spectral triples emerged. The original one relaxes the condition [D,a] B(H ) to the demand of the boundedness of “twisted commutators”: Da σ(a)D∈ B(H ), with a given automorphism σ of the algebra A [11, 30, 54, −56]. The notions∈ of reality, pdos, regularity etc. get modified accordingly [8, 50, 52]. A related notion of modular spectral triples [12] is motivated by deformation of some physical models or quantum groups [4, 41, 46, 48, 51]. There is also an increasing interest for the ‘twist’ in particle physics [22]. However, the spectral action has not yet been systematically approached in this context, to our best knowledge. 4) The Lorentzian signature. An insistent problem, which has been swept under the carpet is that of the signature: The notion of a spectral triple generalises Riemannian manifolds with the Euclidean signature. Alas, the spacetime we are living in has a Lorentzian signature, instead. At the almost-commutative level, one can bypass the problem using the old (and somewhat murky) trick of the Wick rotation [20]. However, a rigorous approach requires a deep conceptual change: A distinctive feature of spaces with the signature ( ,+,+,...) is the existence of a causal structure. Needless to remind that the micro-causa− lity is one of the key axioms in quantum field theory. A rigorous notion of causality suitable for a noncommutative geometry has been proposed [34] and studied [25, 26, 27, 35, 36, 37]. The Dirac operators D/ on pseudo-Riemannian manifolds, formally similar to their Euclidean colleagues, are drastically different [2]. First of all, D/ is not selfadjoint in L2(M,S ). Secondly, D/ has infinite dimensional eigenspaces (in particular, dimKerD/ = ∞), hence no function f can render f ( D/ ) trace-class. Let us also point out that in the Lorentzian context| a| nonunital algebra is manda- tory, as compact spacetimes always contain causal loops, which is undesirable. The problem of compactification of spacetimes, or “attaching a causal boundary to the spacetime” is an old-standing and a hard one — both on the conceptual and on the technical side (cf. [53] for a nice abstract mathematical formulation). Yet another serious obstacle is the presence of the notorious spacetime singularities, which seem to be an inherent element of our Universe, as attested by the famous Hawking–Penrose theorems (see [64] or any other mathematically oriented textbook on general relativity). On the mathematical side, it creates problems with the incompleteness of spacetime manifolds (see, for instance, [3]). As one can see, the algebraic situation is rather dramatic already at the commutative level. Nevertheless, the programmeof “pseudo-Riemannian spectral triples” is being systematically developed [5, 21, 23, 24, 25, 26, 32, 33, 55, 62, 63]. Its central idea is to work with a Krein space [6]. The “Lorentzian spectral action” has not yet even emerged from the depth. A result, which might shed some light on it is the Lorentzian version of the index theorem [1]. 5) The dimension spectrum. The computation of the dimension spectrum of a given spectral triple is a formidable task. Beyond the almost-commutative realm it has 5 Open Problems 123 been accomplished only for a few examples listed on p. 20. Firstly, the existence of the dimension spectrum is by no means automatic — there exist spectral zeta functions admitting no meromorphic continuation (see [59] and [28, Section 5]). Secondly, even if we do have a meromorphic extension of ζD, the one of ζT,D does not come for free, even for T A . In the worked out examples, one firstly unrav- ∈ els the meromorphic extension of the basic zeta function ζD and then constructs 0 the ones for ζT,D by expressing the operators T Ψ (A ) in the eigenbasis of D. ∈ Whereas the poles of ζD and ζT,D do not coincide in general (see Example 1.24), in all known cases we actually have Sd k NP(ζ k ). So it seemsas ifthe whole D − ,D dimension spectrum is actually encoded⊂ ∪ in∈ the operator| | D. Is this a general fact or a specific property of the worked out examples? 6) Heat traces and zeta functions. In Chapter 3 we have spied into the intimate t H interplay between the small-t asymptotic expansion of Tr Ke− and the meromorphic extension of ζK,H . But on the route we only employed the properties of general Dirichlet series and the geometric origin of the operators H and K remained concealed. It would be highly desirable to understand what impact might the geometrical dwelling of H and K have on the problems we encountered. Concretely:

t H a) When is ζK,H of polynomial growth on the verticals (in which case Tr Ke− admits an asymptotic expansion with the vanishing contribution (3.11))? b) Are the pathologies illustrated on Figure 3.3 always avoided? t H c) When is the asymptotic expansion of Tr Ke− actually convergent? t D d) When is the existence of the asymptotic expansion of Tr e− | | equivalent to the tD2 existence of an expansion of Tr e− ? t D e) When does the existence of an expansion of Tre− | | imply the existence of an t DA expansion of Tre− | | for a suitable fluctuation DA = D + A? 7) The role of the cut-off function. Arguably, the smooth cut-off function f involved in the definition of the spectral action S(D, f ,Λ) is of non-geometricorigin. It might encode some physical input (such as the parameters of the Standard Model), however, one has to keep in mind that for general noncommutative geometries — and for the almost-commutative ones, but beyond the asymptotic expansion — the full shape of f enters into the game. Adopting a puritanical point of view, one should set f = χ[0,1] and create the tools to study the asymptotic expansion of the raw spectral action N D (Λ). | | 8) The coefficients of the asymptotic expansion. In the almost-commutative framework the coefficients of the large-energy asymptotic expansion of the spectral action have a pellucid geometric and physical interpretation — as pictured in formula (1.44). However, beyond the homely classical ground, the situation is more obscure. The “curvature” has been defined [16, Defintion 1.147] and computed on a con- formally rescaled noncommutative 2-torus [18, 19, 31]. However, such an interpretation remains controversial, as we have imperceptibly entered into the domain of nonminimal operators, which might also involve the torsion [47, 60]. The comprehension of geometry and physics behind the coefficients of the general asymptotic expansion of the form (3.40) is a serious challenge. 124 5 Open Problems

9) Distributional approach to the asymptotics. The framework presented in Sec- tion 2.7 is very appealing. It would be desirable to employ it beyond the realm of classical pdos. To that end, one would need to understand how the assumption (2.35) should be reformulated for general elements in ′(R,L (H )) and when it is met. K 10) Beyond the asymptotic expansion. As forewarned on p. 30, the asymptotic expansion of the spectral action (even in its full glory) might fail to capture the ‘exponentially small physical phenomena’ [7] encoded in the nonperturabtive expression Tr f ( D /Λ). The pressing question, both for mathematicians and physicists, is: What is| actually| lost? More concretely (see also Remark 2.38), assume that t D t D ∞ Tre− | | Tre− | ′| = 0(t ). What can be said about the operators D versus D ′? If D/ is the− standard DiracO operatoron the ﬂat 3-torus, then the information about the ∞ chosen spin structure S is concealed in the ∞(Λ − ) term of the spectral action expansion — see Example 2.29. Is this a generalO feature of commutative geometries? A related problem is the impact of different possible selfadjoint extensions of D on the spectral action — see [29, Theorem 7.2] for the casus of a differential operator. 11) Spectral action for the noncommutative torus. In the spectral action for the noncommutative torus (Theorem B.6), the constant term in Λ coincides with its classical value for the commutative torus after the swap τ Td . This has been demonstrated in dimensions 2 and 4. We conjecture that this↔ holds true in arbitrary R dimension. The asymptotic expansion of the spectral action for noncommutative tori relied heavily on the Diophantine hypothesis. It would be very instructive to see how does it look like (if it exists at all!) for Θ, which does not meet the Diophantine condition.

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A.1 About “Heat Operators”

Recall first that if P is an unbounded operator, then etP cannot be defined by the N tn n tP B H series -limN ∞ ∑n=0 n! P . What is needed is that G : t 0 e ( ) is a strongly-continuousk·k → contraction semigroup (i.e. G(0)= 1≥,G(→s)G(t)=∈ G(s + t), G(t) 1 and the function t 0 G(t)ψ is norm-continuous for each ψ H). Then,k k≤ a closed densely defined≥ operator→ P is the generator of this semigroup, i.e.∈ by definition G(t)= etP, if and only if R+ is contained in the resolvent set of P and 1 1 (P λ)− λ − for all λ > 0 (see [24, Chapter 14]). k − k≤tP 1 n Moreover, e = strong-limn ∞( (t/n)P)− and → − ∞ 1 λ t tP (P λ)− = e− e dt, for ℜ(λ) > 0 (A.1) − − 0 Z holds true in this generality since actually the right half-plane λ C ℜ(λ) > 0 1 1 { ∈ | } is in the resolvent set of P and (P λ)− ℜ(λ)− . The generator P is upper semibounded: ℜ( Pψ,ψ ) 0,k for all− ψ Domk≤ P. Sometimes, the generatorh i of≤G(t) is denoted∈ by P like in [33, Section X.8]. We can rephrase previous results as a constructive− way to get the exponential. Let P be an unbounded operator on the Hilbert space H such that P λ is invertible in the sector − Λ := reiφ r 0, φ θ , 0 < θ < π θ { | ≥ | |≥ } 2 and assume there exists c with

1 2 1/2 (P λ)− c(1 + λ )− , λ Λ . (A.2) − ≤ | | ∀ ∈ θ

This allows to deﬁne

tP i t λ 1 e− := e− (P λ)− dλ, (A.3) 2π C − Z

127 128 A Classical Tools from Geometry and Analysis

π iθ where C = Cr (with θ < ) is the path from ∞ along the ray re for r r0 0,θ 2 ≥ followed by a clockwise circle around zero of radius r0 and ending at infinity along iθ the ray re− . Since the two rays lie in the right-half plane, the exponential decay of t λ e− guarantees the convergence of the integral. We will need that fact for instance in (2.15) of Section 2.2.2 on Laplace transform. To close these remarks on heat operators, we recall that many functions can be defined through an integral along a curve in C. For instance, given a selfadjoint operator P, one defines (cf. [36, §10]), for any z C with ℜ(z) > 0, ∈ 2z 1 z 2 1 P− := i2π λ − (λ P )− dλ λ C − Z ∈ along the curve C as in (A.3). Typically, to control the norm-convergence of the integral, one uses

1 1 1 1 (λ P)− = sup µ λ − = dist(λ,specP)− ℑ(λ) − , (A.4) k − k µ specP | − | ≤ | | ∈ 1 1 P(λ P)− = sup µ µ λ − . (A.5) k − k µ specP | || − | ∈

Moreover, if P = P∗ is positive,

1 1 λ ℑ(λ) − if ℜ(λ) 0, P(λ P)− | || | ≥ (A.6) k − k≤ 1 if ℜ(λ) < 0, 1 1 which follows from P(λ P)− = supµ specP f (µ) with f (µ)= µ µ λ − and the computation of thek maximum− k of f . ∈ | − |

A.2 Deﬁnition of pdos, Sobolev Spaces and a Few Spectral Properties

There are several good textbooks on pdos: [19, 23, 25, 36, 37]. For the heat trace asymptotics of a pdo we closely follow [23, Section 4.2] and the nice notes [35]. See also [1, 20, 27] for the computation of heat kernel coefficients. To study pdos on Rd we need a few basic definitions: – x := (1+ x 2)1/2 and x,η := (1+ x 2 + η )1/2 for x Rn, n N , η C. h i k k h i k k | | ∈ ∈ ∗ ∈ m Rd Rd C α β m α – S := p : (x,ξ ) such that ∂ξ ∂x p(x,ξ ) cαβ ξ −| | is { ∈ × → | d |≤ h i } the set of symbols of order m R. Here α, β are in N with α = ∑αi. This yields a family of seminorms∈ on Sm defined by | |

α β m+ α p m,α,β := sup ∂ξ ∂x p(x,ξ ) ξ − | |. | | x | |h i

∞ m The set of smoothing symbols is S := m S . − ∩ A.2 Deﬁnition of pdos, Sobolev Spaces and a Few Spectral Properties 129

m ∞ m j – The symbol p S has the expansion p ∑ j=0 pm j, when pm j S − and n ∈ m n ∼ − − ∈ for each n, p ∑ j=0 pm j S − . It is named classical− if,− moreover,∈ for all j,

m j d d pm j(x,tξ )= t − p(x,ξ ), (x,ξ ) R R , ξ 1, t 1. − ∀ ∈ × k k≥ ≥ 1 m It is said to be elliptic when p(x,ξ ) is invertible and p(x,ξ )− c ξ − for all x and ξ > r for some r 0. | |≤ h i k k ≥m d A classical symbol p S is elliptic when pm(x,ξ ) is invertible for all x R and ξ = 1. ∈ ∈ kFromk now on, we assume in this appendix that all symbols are classical. – Every symbol p gives rise to a pdo acting on u in the Schwartz space (Rd ) via the inverse Fourier transform by S

1 i2π x.ξ (O(p)u)(x) := F− [(p(x, )F[u]( )](x)= e p(x,ξ )F[u](ξ )dξ . · · Rd Z This deﬁnition is compatible with the product of operators as for p Sm1 , q Sm2 ∈ ∈ there exists a symbol called the Leibniz product of p and q, denoted p q Sm1+m2 , such that O(p)O(q)= O(p q) with the expansion ◦ ∈ ◦ α ( i)| | α α (p q)(x,ξ ) ∑ −α! (∂ξ p)(x,ξ )(∂x q)(x,ξ ). ◦ ∼ α In particular, when p Sm is elliptic, there exists a symbol q called the parametrix such that p q 1 and∈q p 1are both in S ∞. ◦ − ◦ − − – The Sobolev spaces read

s d d s 2 d H (R ) := u ′(R ) ξ F(u) L (R ) { ∈ S | h i ∈ } for s R, with scalar product ∈ 2s (u,u′) := F(u)(ξ )F(v)(ξ ) ξ dξ h i Z 2 s Rd and complete for the norm u s := (u,u). For instance δy H ( ) if s < d/2. We have, H0(Rd)= L2(Rd)kandk if s > d/2 then Hs(Rd) ∈C(Rd) (Sobolev− embed- ding theorem). ⊂ When s > d/2, any bounded operator A : H s Hs is an integral operator with a − → Schwartz kernel given by kA(x,y) = (Aδx,δy). We now adapt previous deﬁnitions to a compact boundaryless Riemannian manifold M of dimension d, so we need coordinate charts (U,h) where U are open sets in M and h are diffeomorphisms from U to open sets in Rd. ∞ ∞ ∞ Let P : C (M) C (M); when φ, ψ Cc (U) (smooth functions on U with com- → ∈ ∞ pact support), the localised operator φPψ on Cc (U) is pushed-forward as h (φPψ) ∗ 130 A Classical Tools from Geometry and Analysis

∞ on Cc (h(U)). The operator P is a pseudodifferential operator of order m when each such localisation is a pdo of order m on h(U). Then, one extends Sm to symbols on M as follows:

m ∞ m S (M) := p(x,ξ ) C (T ∗M) h(φ p) S (h(U)), (U,h,φ) . { ∈ | ∈ ∀ } Of course, P is said to be elliptic or smoothing if all its local symbols have such a property and the set of classical pdos of order m is denoted by Ψ m(M) which defines m Ψ(M) := mΨ (M). For x (U∪,h) one defines the principal (or leading) symbol of P Ψ m(M) as ∈ ∈ m m 1 pm(P) := h∗ pm(h (φPφ)) S (M)/S − (M), ∗ ∈ ∞ where one chooses a φ Cc (M) equal to 1 in vicinity of x. One checks that this principal∈ symbol makes sense and is invariantly defined on T ∗M while the total symbol is quite sensitive to a change of coordinates. Moreover, m m for each p S (M), one constructs P Ψ (M) with pm(P)= p via the partition of unity. ∈ ∈ A new extension is possible when P acts on sections of a smooth vector bundle E of finite rank over M equipped with a smooth inner product. So, typically, a fiber is acted upon by a matrix. By local triviality, on can define Hs(M,E) using a partition of unity. Hence P : C∞(M,E) C∞(M,E) is called a pseudodifferential operator of order m if every localisation→ is a matrix of pdos of order m for all charts U over which E is trivial. Such operators, the symbols of which are now matrices, define Ψ m(M,E). The properties of classicality and ellipticity are generalised in a straightforward way. In particular, P has a matrix-valued kernel KP, which in local coordinates reads

i2π(x y).ξ kP(x,y)= e − p(x,ξ )dξ . (A.7) Rd Z Similarly, the Sobolev space Hs(M) is defined as the set of distributions u on s d M which, in a given local patch U, satisfy u D ′(U) with ψu H (R ) for all ∞ ∈ s t ∈ ψ Cc (U). By the Rellich theorem, the inclusion H (M) ֒ H is compact for any t <∈s and even trace-class when t + d < s. → ∞ From the beginning, classical symbols can be seen as objects defined up to S− . It has the following consequence: P Ψ(M,E) is smoothing (i.e. all of its local ∈ symbols are smoothing) if and only if P has a Schwartz kernel kP which is smooth ∞ on M M. For instance, ψPφ is smoothing if ψ,φ Cc (M) with disjoint supports. We× now recall a few classical results on pdos —∈ see loc. cit. at the beginning of this section. They provide links between a pdo and the same object, but viewed as an operator which has eventually several closed extensions on a Hilbert space. Recall that a bounded operator between Banach spaces is Fredholm if it has a finite dimensional kernel and cokernel and a closed range. A.3 Complex Parameter-Dependent Symbols and Parametrix 131

Theorem A.1. Let P Ψ m(M,E). Then: ∈ s s m i) The extension of P : H (M,E) H − (M,E) is bounded for all s R. ii) If P is elliptic, all previous extensions→ are Fredholm operators, which∈ means that there exists a Fredholm inverse which is a pdo of order m. In particular, when m > 0,P : C∞(M,E) C∞(M,E) acting− on the Hilbert space H0(M,E)= L2(M,E) has only one closed→ extension with the domains Hm(M,E) and a spectrum either equal to C or discrete without accumulation points except 0. s s m iii) When P : H (M,E) H − (M,E) is invertible for some s, then we have 1 m → P− Ψ − (M,E). iv)∈ The space Ψ 0(M,E) is an algebra. v)IfP Ψ m(M,E) with m < d, then P hasa continuouskernel andits extension PonL2(M∈,E) is trace-class with−

TrL2(M,E) P = trE kP(x,x)dx. ZM The inverse in (ii) is obtained by the construction of a local parametrix over a local chart, which, after being patched, gives rise to two pdos Q and Q of order m ′ − such that R = QP 1 and R′ = PQ′ 1 are smoothing pdos. Moreover, Q Q′ is a smoothing pdo. So,− modulo smoothing− pdos, Q is the left and right inverse− of P. We now present some details from the constructive proof of this theorem for parameter-dependent symbols.

A.3 Complex Parameter-Dependent Symbols and Parametrix

Let us be given an elliptic pdo P Ψ m(M,E) of order m > 0 with the matrix sym- ∞ ∈ 2 bol p ∑ j=0 pm j. Despite the nice unique L -extension of P provided by Theo- ∼ − tP 2 rem A.1, it is still interesting to look at e− not only as an operator on L (M,E) but as a smoothing pdo (or, similarly, to regard the complex power Ps as a pdo of order mℜ(s)). The main idea, which we expound in some detail below, is to replace the resol- 1 vent (P λ)− of P by a parameter-dependent parametrix, the symbol of which is under− control. Since we want to control the integrand of (A.3), we assume the following (uniform) parameter-ellipticity of the principal symbol of P: 1 Hypothesis A.1 The operator (P λ)− exists in the left keyhole region Vr0,θ de- C π − ﬁned by r0,θ for θ < 2 . Moreover, we have the resolvent growth condition: The matrices pm(x,ξ ) λ are − invertible for all x,ξ when λ Vr and ∈ 0,θ 1 2 2/m m/2 1/m m (pm(x,ξ ) λ)− (1 + ξ + λ )− = ξ ,λ − . − ≤ k k | | h i h For the principal symbol, let us introduce the strictly homogeneous symbol pm: 132 A Classical Tools from Geometry and Analysis

h m p (x,ξ ) := ξ pm(x,ξ / ξ ) m k k k k

(which coincides with pm for ξ 1, but is now homogeneous of degree m for all ξ = 0) and we can rephrasek thek≥ hypothesis as: ph (x,ξ ) has no eigenvalues in 6 m Vr0,θ for all ξ = 0 (see [23, Lemma 1.5.4]). Recall that pm(x,ξ ) is homogeneous of degree m only6 for ξ 1 and we have to control the integral in ξ ,asin(A.7). k k≥ Rd m Let (U,h) be a ﬁxed coordinate chart. For x U, ξ , λ = η Vr0,θ and j N, we want to generate a parametrix by an inductive∈ ∈ sequence (see [∈28]) ∈ m 1 q m(x,ξ ,η) := (pm(x,ξ ) η )− , − − j 1 α − ( i)| | α α m 1 q m j(x,ξ ,η) := − ∂ q m k(x,ξ ,η)∂x pm ℓ(x,ξ )(pm(x,ξ ) η )− − − ∑ ∑ α! ξ − k=1 α,ℓ − − − −

where the second sum is over α Nd,ℓ N such that k + ℓ + α = j. In the scalar case (i.e. the ﬁbers of∈ E are∈ one-dimensional), | |

2 j m k 1 q m j(x,ξ ,η)= p j,k(x,ξ )(pm(x,ξ ) η )− − , (A.8) − − ∑ k=1 −

where the p j,k are symbols of order mk j obtained from pm, , pm j. Using − ··· − m 1 m 1 m 1 ∂((pm η )− )= (pm η )− (∂ pm)(pm η )− , − − − − α β with ∂ = ∂x or ∂ξ , one checks that ∂ ∂x q m j is a sum of terms of the form ξ − −

m 1 α1 β1 m 1 αr βr m 1 (pm η )− ∂ ∂x pm k (pm η )− ∂ ∂x pm kr (pm η )− , (A.9) − ξ − 1 − ··· ξ − − r where ∑ℓ=1 kℓ + αℓ = j + α . Moreover, | | | |

α β m j α β ∂ ∂x q m j(x,tξ ,tη)= t− − ∂ ∂x q m j(x,ξ ,η) for ξ 1, t 1 ξ − − ξ − − k k≥ ≥ m 1 and there are at least two factors (pm(x,ξ ) η )− if either j > 0 or α + β > 0. This implies the following estimates: − | | | |

Now, deﬁning q such that q ∑ j N q m j, we get ∼ ∈ − − α β 2m m J α ∂ξ ∂x q(x,ξ ,η) ∑ q m j(x,ξ ,η) = ξ ∞( ξ ,η − ξ − −| |). (A.11) − j

We claim that: If r(x,ξ ,λ) := q(x,ξ ,λ 1/m) (p(x,ξ ) λ) id, then for any N, all N ◦ 1 − − seminorms in S− of the symbol r are ∞( λ − ). The proof is based on the decompositionO ofh i the series defining p, q in finite sums and remainders and their Leibniz products via the above estimates. Then, one gets rid of the local chart U by patching previous parametrices to get a parameter-dependent pdo Q(λ) (associated to the symbol q(x,ξ ,λ 1/m)), such that R(λ)= Q(λ)(P λ) 1 (with the symbol r(x,ξ ,λ)) is a smoothing pdo, where − − 1 the seminorms N,α,β of its symbol are ∞( λ − ) for each N. k·k− O h i Since formally [Q(λ)(P λ)] 1 = [1 + R(λ)] 1 = ∑∞ ( R(λ)) j, we deduce that − − − j=0 − ∞ 1 1 1 j (P λ)− Q(λ)= [Q(λ)(P λ)]− Q(λ)= ∑ ( R(λ)) Q(λ) − − − − j=1 − 2 1 has a norm which is 0( λ ) since Q(λ) = ∞( λ ) so that: O h i− k k O h i− 1 2 Proposition A.2. We have (P λ) Q(λ) = ∞( λ ) for all λ Λ . − − − O h i− ∈ θ Remark that we can define similarly Q (λ) such that (P λ)Q (λ) 1 = R (λ) ′ − ′ − ′ for another smoothing pdo R′(λ). Thus Q(λ) Q′(λ)= R(λ)Q′(λ) Q(λ)R′(λ) is a smoothing pdo. Moreover, the − 1 − operator (P λ)− can be seen as an elliptic pdo of order m since, by (A.10)– − 1 1/m − (A.11), the operator (P λ)− ∑ j

tP A.4 About e− as a pdo and About its Kernel

Let P Ψ m(M,E) be elliptic with m > 0 and let its principal symbol satisfy (A.1). ∈ tP Theorem A.3. For t > 0,G(t) := e− is a pdo of order zero, the symbol g(x,ξ ,t) ∞ of which has the expansion g(x,ξ ,t) ∑ j=0 g j(x,ξ ,t) with ∼ −

i tλ 1/m g j(x,ξ ,t) := e− q m j(x,ξ ,λ )dλ, for j N. (A.12) − 2π C − − ∈ Z

Moreover, g0(x,ξ ,0)= 1, while g j(x,ξ ,0)= 0 for j N∗. − ∈ Proof. We follow [23, Theorem 4.2.2]. In a coordinate chart, we have 134 A Classical Tools from Geometry and Analysis

tP i tλ 1/m i tλ 1/m e− = e− O[q(x,ξ ,λ )]dλ = O[ e− (q(x,ξ ,λ )dλ] 2π C 2π C ∞ Z Z O ∑ [g j(x,ξ ,t)]. ∼ j=0 −

tpm First, one checks that g0 = e− (by residue calculus) and the homogeneity property

m j g j(x,rξ ,r− t)= r− g j(x,ξ ,t), for ξ 1, r 1. − − k k≥ ≥ We now want to prove the following estimates: There exists c > 0 such that

α β j α 1/m a c ξ mt ∂ ∂x g j(x,ξ ,t) ξ − −| | (t ξ ) e− h i , a min(m, j + α ). (A.13) k ξ − k ≤ h i h i ∀ ≤ | | These hold true for j = 0 and for j 1 we begin with the scalar case, cf. (A.8): ≥ 2 j 2 j i tλ k 1 1 k tpm g j = ∑ p j,k 2π e− (pm λ)− − dλ = ∑ p j,k k! t e− . (A.14) − C − k=1 Z k=1 We get

k mk j k m j d k 1 m k 1 d Pj k t ξ − t = ξ − t ( ξ t) − with ( ξ t) − 1 when ξ t 1. k , k ≤ h i h i h i h i ≤ h i ≤

d k 1 tpm/2 m Moreover, ( ξ t) − e− 1 if ξ t 1. Thus, the estimates are proved when α = βk =h 0i with a = m. k≤ h i ≥ As a consequence, we cover the situation where a < m and for α, β non-zero one differentiates under the integral of (A.12) until the estimate for α = β = 0 applies. For the non-scalar case, one proceeds as in the proof of (A.10) thanks to the expansion (A.9) for the derivatives. The equality g j(x,ξ ,0)= 0 is a consequence of (A.13). We can now− conclude the proof thanks to the following argument: The symbol ∞ g(x,ξ ,t) ∑ j=0 g j(x,ξ ,t) can be chosen in such a way that ∼ − α β J α 1/µ a c ξ m t ∂ξ ∂x (g ∑ g j) ξ − −| | (t ξ ) e− h i , a min(m, j + α ). k − j

So, for any integer J,

i tλ O 1/m 1/m 2π e− [q(x,ξ ,λ ) ∑ q m j(x,ξ ,λ )]dλ C − − − Z j

tP is a pdo of order zero and the asymptotics of the symbol for G(t)= e− is fully justiﬁed.

tP But since we are interested in Tre− it is worthwhile to control the kernel of G(t) as a function of t and to give an alternative proof of the previous theorem. On tP the way, it is shown that e− is a smoothing pdo. Let G j(t) be the pdo deﬁned locally by g j(x,ξ ,t) after patching local charts. − − tP A.4 About e− as a pdo and About its Kernel 135

Lemma A.4. For any t > 0: i) The kernels of G j(t) satisfy the estimates −

d/m c′ t ( j d)/m c′ t KG0 (x,y,t) t− e− , KG j (x,y,t) t − e− , for 0 < j < m + d; k k≤ k − k≤ c′ t c′ t KG m d (x,y,t) t(1 + logt )e− , KG j (x,y,t) te− , for j > m + d. k − − k≤ | | k − k≤ h Moreover, on the diagonal we have, with c j(G,x) := Rd g j(x,ξ ,1)dξ , − ( j d)/m R KG j (x,x,t)= c j(G,x)t − + 0(t), for 0 j < m + d. − O ≤ r ii) The remainder GJ(t) := G(t) ∑ j m + d, k J k≤ | | K r x y 0 0 for J d GJ ( , , )= , > .

Proof. i) We begin with the kernel of G0:

i2π(x y).ξ tpm(x,ξ ) KG (x,y,t)= e − e− dξ . 0 Rd Z h tpm tp Since e e m = 0(t) when ξ 1, we get k − − − k O k k≤ h h tpm(x,ξ ) tpm(x,ξ ) tpm(x,ξ ) KG0 (x,y,t) c1 e− dξ + c1 e− e− dξ k k≤ Rd k k ξ 1 k − k Z Zk k≤ d/m c η m d/m c1 t− e− k k dη + c2 t c3 t− + c2 t, ≤ Rd ≤ Z and for the diagonal

h h d/m pm(x,η) tpm(x,ξ ) tpm(x,ξ ) KG0 (x,x,t)= t− e− dη + e− e− dξ Rd ξ 1 − Z Zk k≤ d/m =: c0(G,x)t− + 0(t). O h m j ct ξ m When j = 0, the estimate g (x,ξ ,t) ξ − te follows from (A.13). 6 k j k≤k k − k k Since the last function is ξ -integrable− when m j > d, we deduce that the kernel − − i2π(x y).ξ KG j (x,y,t)= e − g j(x,ξ ,t)dξ Rd − − Z ct/2 is ∞(e ), while for t > 0, O − h h KG j (x,y,t) c1 g j dξ + c1 ( g j + g j )dξ k − k≤ Rd k − k ξ 1 k − k k − k Z Zk k≤ m j ct ξ m ( j d)/m c2 t ξ − e− k k dξ + c3 t = c4 t − + c3 t. ≤ Rd k k Z 136 A Classical Tools from Geometry and Analysis

As above, still with 0 < j < d + m, one gets

h h KG j (x,x,t)= g j(x,ξ ,t)dξ + c1 (g j g j)(x,ξ ,t)dξ − Rd − ξ 1 − − − Z Zk k≤ ( j d)/m = c j(G,x)t − + 0(t). (A.15) O Moreover, using (A.13), we obtain

m j ct ξ m KG j (x,y,t) t + ξ − e− h i dξ k − k≤ ξ 1 ξ 1 h i Z Z k k≤∞ k k≥ m j d 1 ctrm t 1 + r − − − e− dr ≤ 1 Z t(c1 + c2 logt ), for j = m + d, | 1+(|j d)/m) ≤ t(c1 + c2 t ), for j > m + d. − − This completes the proof of i). ii) The remaining symbol

r 1/m 1/m qJ(x,ξ ,λ) := q(x,ξ ,λ ) ∑ q m j(x,ξ ,λ ) − j

i tλ KGr (x,y,t)= e− KO(qr )(x,y,λ)dλ, J 2π C J Z 2 with KO r (x,y,λ) = ( λ ) as in the proof of Proposition A.2. Thus, the (qJ) ∞ − k k O h i tλ ct integral over C converges uniformly for all t > 0 since e− e− for some c > 0. ct | |≤ Consequently, K r x y t is e and as such, it has a continuous extension at GJ ( , , ) ∞( − ) O i zero since K r (x,y,0)= 0, because in KO r (x,y,λ)dλ, C can be deformed GJ 2π C (qJ) into a closed contour around zero. R We already know from Section A.1 that Gr (t) C∞((0,∞),B(H )) and, for t 1, J ∈ ≤ ∞ t ct c t 1 r 1 λ r 2 λ ∂t KG (x,y,t) = 2π λe− KG (x,y,λ)dλ c1e− + c3 e− | | λ − dλ k J k C J ≤ r h i Z Z 0 ∞ c2 a 1 c4 + c5 e− a− da c6 + c7 logt . (A.16) ≤ r t ≤ | | Z 0

(See the deﬁnition of r0 after (A.3).) The Taylor series in t gives K r (x,y,t) ct(1 + logt ) when t (0,1] and hence GJ the announced estimate. | |≤ | | ∈ Technically, it is useful to employ tP A.4 About e− as a pdo and About its Kernel 137

tP i tλ k k e− = e− λ − P Q(λ)dλ, k N, 2π C ∀ ∈ Z 1 1 1 which follows from Q(λ)= λ − (λ P + P)Q(λ)= λ − + λ − PQ(λ) which k −j j 1 k k − tλ j after iteration gives Q(λ)= ∑ j=1 λ − P − +λ − P Q(λ) and C e− λ − dλ = 0. We denote: − R Q(k)(λ) := Pk Q(λ),

(k) (k) seen as a pdo of order (k 1)m with the symbol q ∑ j N q k 1 m j and, preserv- − ∼ ∈ ( ) ing the notation, − −

(k) O (k) (k) (k) (k) Q(k 1)m j := [q(k 1)m j]and QJ := Q ∑ Q(k 1)m j. − − − − − j

As an example,

1 2 2 (2) q = λ − λ − p + λ − q − − 1 2 m 1 = λ − λ − pm + + p m + ξ ∞( ξ − − ) − ··· − Ok k→ h i 2 (2) (2) 1 1/m m + λ − qm + + q m + ξ ∞( ξ − ξ ,λ − ) . ··· − Ok k→ h i h i 2 k (k 1) k (k) (ℓ) By iteration, q m j = λ − pm j λ − p(k−1)m j +λ − q(k 1)m j, where p − − − − −···− − − − − is the symbol of Pℓ. Since these symbols are independent of λ, we get

i tλ 1/m g j(x,ξ ,t)= e− q m j(x,ξ ,λ )dλ − 2π C − − Z i tλ k (k) 1/m = e− λ − q (x,ξ ,λ )dλ. (A.17) 2π C (k 1)m j Z − − In particular, for k N we rewrite the Q(k)’s as ∈ i λt k (k) G(t)= e− λ − Q (λ)dλ, 2π C Z i λt k (k) G j(t)= e− λ − Q (x,ξ ,λ)dλ, − 2π C (k 1)m j Z − − r i λt k (k) GJ(t)= e− λ − Q (λ)dλ. (A.18) 2π C J Z With the help of this representation, we can improve the key estimate (A.13) along with the estimates from Lemma A.4:

Lemma A.5. For any k N, ∈ α β k km j α c ξ m t ∂ξ ∂x ∂t g j(x,ξ ,t) c′(x) ξ − −| | e− h i . (A.19) k − k≤ h i

For the kernel of G j = O(g j) we have − − 138 A Classical Tools from Geometry and Analysis

ct k (1 + logt )e− , for j km = d, ∂t KG j (x,y,t) |( j km| d)/m ct − (A.20) k − k≤ (1 +t )e , for j km = d. − − − − 6 On the diagonal we get,

k ( j km d)/m 0 ∂t KG j (x,x,t)= c j,k(x)t − − + 0(t ), j < (k 1)m + d, (A.21) − O − with k h c j,k(x) := ∂t g j(x,ξ ,1)dξ . Rd Z − For the kernel of G j we have, with j > km + d for some k N, − ∈ k 1 − tℓ ℓ k KG j (x,y,t)= ∑ ℓ! ∂t KG j (x,y,0)+t R(x,y,t) (A.22) − ℓ=1 −

ℓ and ∂t KG j (x,y,0) and R(x,y,t) are continuous in x, y andint 0. When J >−(k + 1)m + d for some n N, the kernel of the remainder≥ Gr satisﬁes ∈ J k ct ∂ KGr (x,y,t) e− , k t J k≤ k 1 − tℓ ℓ k K r (x,y,t)= ∂ K r (x,y,0)+t R(x,y,t), (A.23) GJ ∑ ℓ! t GJ ℓ=1

ℓ where ∂ KGr (x,y,0) and R(x,y,t) are continuous in x, yandint 0. t J ≥ Proof. For ℓ N , ∈ ∗ ℓ i( 1)ℓ tλ (ℓ) 1/m ∂t g j(x,ξ ,t)= − e− q (x,ξ ,λ )dλ − 2π C (ℓ 1)m j Z − − is derived from (A.17) and the estimate (A.19) is provedin the same way as (A.13). ℓ+1 ℓ Since ∂t g j is bounded when t 0 by the above formula, the function ∂t g j − → − is continuous at t = 0. Since g j(x,ξ ,0)= 0 by Theorem A.3, we have the Taylor expansion − k 1 ℓ ℓ k g j(x,ξ ,t)= ∂t g j(x,ξ ,0)t +t r j,k(x,ξ ,t) − ∑ ℓ! − ℓ=1 − ℓ where ∂t g j(x,ξ ,0) are pdos of order ℓm j. − k O k − The operators ∂t G j(t)= (∂t g j) have kernels, for which we can repeat the same arguments used for− the proof of− Lemma A.4 i) in order to get (A.20).

If j > d, KG j (x,y,0)= 0 by Lemma A.4 and by a Taylor expansion, we get (A.22). We need− to control the remainder. For k 1, choose J > km + d and the pre- ≥ r sentation (A.18) for the kernel of K r . Since G is a pdo of order k 1 m M, GJ J ( ) r km J 1/m m − − qJ(x,ξ ,λ) ξ − ξ ,λ − , thus, after a ξ -integration, we get the estimate ofk the kernelk ≤ h i h i 1 K (k) (x,y,λ) λ − k QJ k ≤ h i tP A.5 The Small-t Asymptotics of e− 139 and i λt k (k) KGr (x,y,t)= e− λ − Q (λ)dλ J 2π C J Z because all Q(k) are holomorphic in λ. Thus, for ℓ k 1, ≤ − ℓ 1 λt ℓ k 1 λt ℓ k 1 ct ∂ K r (x,y,t) = e λ K (k) (λ)dλ e λ dλ e . t GJ 2π − − 2π − − − − k k k C QJ k≤ | C h i |≤ Z Z As a consequence we get:

k 2 − tℓ ℓ k 1 K r x y t ∂ K r x y 0 t R x y t GJ ( , , )= ∑ ℓ! t GJ ( , , )+ − ( , , ). ℓ=1

This Taylor expansion begins at 1 since K r x y 0 0 by Lemma A.4. Swap- ℓ = GJ ( , , )= ping m 1 to m completes the proof of the lemma. −

tP A.5 The Small-t Asymptotics of e−

tP The above estimates can be used to prove that e− has a smooth Schwartz kernel tP for any t > 0. Therefore, e− is a smoothing pdo, and hence is trace-class.

Theorem A.6. Let P Ψ m(M,E) be elliptic with m > 0 and let its principal symbol ∈ tP satisfy (A.1). Then, G(t)= e− is a smoothing pdo and its kernel has the following asymptotics on the diagonal:

(n d)/m (n d)/m ℓ KG(x,x,t) ∑ cn d(x,G)t − + ∑ cn d(x,G)t − logt + ∑ rℓ(x,G)t, t ∼0 − − → n N n N ℓ N n d∈/mN n d∈ mN ∈ − ∈ − ∈ ∞ where the coefﬁcient cn d(x,G) C (M) depends only on pm, , pm n, while r (x,G) C∞(M) depends− globally∈ on the operator P. ··· − ℓ ∈ Proof. The smoothness of the kernel K j(x,y,t) in x,y follows from (A.19) and it remains to control the remainder. In fact,−

γ α β ct (x y) ∂ ∂ KGr (x,y,t) e− when J > (k + 1)m γ + α + β + d k − x y J k≤ − | | | | | | which follows, as in the previous lemma, from the estimate

γ α β 1 (x y) ∂x ∂y K (k+1) (x,y,λ) λ − . k − QJ k ≤ h i Thus, G(t) is a smoothing pdo. Now, let us choose a large k N. By a successive integration of (A.21) and using (A.15) with j = d, we get, for j∈< (k 1)m + d, − 140 A Classical Tools from Geometry and Analysis

( j d)/m k Z c′j,k(x)t − + p j,k(x,t)+ 0(t ), for j d / m , KG j (x,x,t)= ( j d)/m O k − ∈ − c (x)t logt + p (x,t)+ 0(t ), for j d mZ, ( ′j,k − ′j,k O − ∈ where c′j,k(x) depends only on c j,k(x) of Lemma A.5, p j,k(x,t) and p′j,k(x,t) are polynomials of degree k in t and are continuous in x with p j,k(x,0)= p′j,k(x,0)= 0. Moreover, the remainder in (A.23) for J = (k 1)m + d > ((k 3)+ 1)m + d is − − k 4 − tℓ ℓ k 3 K r x y t ∂ K r x y 0 t GJ ( , , )= ∑ ℓ! t G J ( , , )+ 0( − ). ℓ=1 − O Thus, for the full integral kernel,

( j d)/m ℓ k 3 KG(x,x,t)= ∑ c′j,k(x)t − + ∑ c′j,k(x)t logt + pk(x,t)+ 0(t − ), 0 j

Corollary A.7. Let (M,g) be a compact Riemannian manifold of dimension d and let P Ψ m(M,E) be an elliptic pdo with m > 0, the principal symbol of which satisﬁes∈ (A.1). Then,

∞ ∞ tP (k d)/m ℓ Tre− ak(P)t − + bℓ(P)t logt, t∼0 ∑ ∑ ↓ k=0 ℓ=0 d N ak(P)= M trck d(x,P)√gd x, for k d / m , − − ∈ d N with ak(P)= RM trr(k d)/m(x,P)√gd x, for k d m , − d − ∈ b (P)= trcm (x,P)√gd x, for ℓ N. ℓ RM ℓ ∈  R It can be of interest to recall here a few links between the resolvent, complex powers and heat operators (with s C, k N, c > 0) ∈ ∈ tP k i tλ k 1 1 s s e− = t− 2π e− ∂λ (P λ)− dλ = i2π t− Γ (s)P− ds, (A.24) C − ℜ s c Z Z ( )= ∞ s 1 i k s k 1 1 s 1 tP P− = (s 1) (s k) 2π λ − ∂λ (P λ)− dλ = Γ (s) t − e− dt. C − 0 − ··· − Z Z A.6 Meromorphic Extensions of Certain Series and their Residues 141 A.6 Meromorphic Extensions of Certain Series and their Residues

We gather below some results on meromorphic extensions of certain series. These will allow us for an extension of Proposition 2.26 and provide tools for the computation of the dimension spectrum of the noncommutative torus in Section B.3.1. On the way, a ‘Diophantine condition’ will pop up guaranteeing a control on the commutation between the residues and the series. For complete proofs, see [15]. In the following, ∑′ means that we omit the division by zero in the summand. d C Theorem A.8. Let P be a polynomialP(x)= ∑ j=0 Pj(x) [x1,...,xn], where Pj is the homogeneous part of P of degree j and d is ﬁxed. ∈ Then, the function

s ′ f (s) := ∑k Zn P(k) k − ∈ k k has a meromorphic extension to C. Moreover, f is not entire if and only if

PP := j Pj(u)ds(u) = 0 = ∅ { | u Sn 1 6 } 6 Z ∈ − and f has only simple poles at j + n, j PP, with ∈

Res f (s)= Pj(u)ds(u). s= j+n u Sn 1 Z ∈ − n 1 Here ds is the Lebesgue measure on S − . The proof is based on the fact that the function s s ′ ∑k Zn P(k) k − P(x) x − dx, ∈ k k − Rn Bn k k Z \ with Bn – the unit ball in Rn, originally deﬁned for ℜ(s) > d + n, extends holomorphically to C. This result can be seen as an extension of Proposition 2.26: Let D be a selfadjoint invertible operator with only discrete spectrum equal to Zn such that each n eigenvalue k Z has multiplicity p(k) for a given polynomial P N[x1,...,xn]. ± ∈ ∈ Then, Res ζD(s)= u Sn 1 Pj(u)ds(u). s= j+n ∈ − R Example A.9. Let us consider the functions

q q s : ′ 1 n N ζq1,...,qn (s) = ∑k Zn k1 kn k − , for qi ∗. ∈ ··· k k ∈ By the symmetry k k, the functions ζq ,...,q vanish if any qi is odd. 7→ − 1 n Assume that all of qi’s are even, then ζq1,...,qn (s) is a nonzero sum of terms s P(k) k − , where P is a homogeneous polynomial of degree q1 + + qn. Theo- k k ··· C rem A.8 yields the following: ζp1,...,pn has a meromorphic extension to with a unique pole at n + q1 + + qn. This pole is simple and the residue at this pole is ··· 142 A Classical Tools from Geometry and Analysis

Γ [(q1+1)/2] Γ [(qn+1)/2] Res ζq1,...,qn (s)= 2 Γ n q ··· q 2 . (A.25) s=n+q + +qn [( + 1+ + n)/ ] 1 ··· ··· We now recall few notions from the Diophantine approximation theory.

Deﬁnition A.10. i) Let δ > 0. A vector a Rn is said to be δ-badly approximable if the Diophantine condition holds true: ∈ There exists c > 0 such that q.a m c q δ , q Zn 0 and m Z. | − |≥ | |− ∀ ∈ \{ } ∀ ∈ Let BV δ bethe set of δ-badly approximable vectors and BV := δ >0 BV δ the set of badly approximable vectors. ∪ ii) A matrix Θ Mn(R) (real n n matrices) will be called badly approximable if there exists u ∈Zn such that tΘ(u×) is a badly approximable vector of Rn. ∈ n It is known that for δ > n the Lebesgue measure of R BV δ is zero (i.e almost any element of Rn is δ-badly approximable) and, consequently,\ almost any matrix in Mn(R) is badly approximable. We store below a rather technical result [15, Theorem 2.6], omitting the proof. As compared with the previous theorem, it takes care of the possible oscillations ei2πk.a, where a is a vector in Rn, which will be allowed to vary later on.

Theorem A.11. Let P C[x1, ,xn] be a homogeneouspolynomialof degree d and let b ((Zn)q). Then,∈ ··· i)∈ For S a Rn, deﬁne ∈ s i2πk a ′ . fa(s) := ∑k Zn P(k) k − e . ∈ k k n 1.Ifa Z , then fa has a meromorphic extension to the whole space C. Moreover, ∈

fa is not entire if and only if P(u)ds(u) = 0. u Sn 1 6 Z ∈ − In that case, fa has a single simple pole at the point d + n, with

Res fa(s)= P(u)ds(u). s=d+n u Sn 1 Z ∈ − n n 2.Ifa R Z , then fa(s) extends holomorphically to C. ∈ \ q ii) Suppose that Θ Mn(R) is badly approximable. For any (εi)i 1,0,1 , the function ∈ ∈ {− } g s : b l f s ( ) = ∑l (Zn)q ( ) Θ ∑i εili ( ) ∈ extends meromorphically to C with only one possible pole at s = d + n. Moreover, if we set

q Z Zn q := l ( ) ∑ εili = 0 and V := ∑ b(l), { ∈ | i=1 } l Z ∈ then 1.If V n 1 P(u)ds(u) = 0, then s = d + n is a simple pole of g(s) and S − 6 R A.6 Meromorphic Extensions of Certain Series and their Residues 143

Res g(s)= V P(u)ds(u). s=d+n u Sn 1 Z ∈ −

2.If V n 1 P(u)ds(u)= 0, then g(s) extends holomorphically to C. S − iii) Suppose that Θ M (R) is badly approximable. For any (ε ) 1,0,1 q, R n i i the function ∈ ∈{− } g (s) := b(l) f q (s), 0 ∑l (Zn)q Z Θ ∑ εi li ∈ \ i=1 with q Z Zn q := l ( ) ∑ εili = 0 , { ∈ | i=1 } extends holomorphically to C. Is is unknown whether the Diophantine condition, which is sufﬁcient to get the results of iii), is also necessary — see nevertheless [15, Remark 2.9].

In the study of the dimension spectrum of the noncommutative toruswe will need the following result [15, Theorem 2.18 (i)]. q 1 It requires some notations: Fix q N, q 2 and r = (r1, ,rq 1) (N∗) − . ∈ ≥ ··· − ∈ When (x1,...,x2q), we set

x j := x1 + + x j + xq 1 + + xq+ j for any 1 j q ··· + ··· ≤ ≤ and we let P R[x1, ,xn] and d = degP. ∈e ··· Theorem A.12. Let 1 Θ be a badly approximable matrix, and a ((Zn)2q). Then, 2π ∈ S q 1 q − ri s ik.Θ ∑ l j s f (s) := ∑ al ∑ ′ ∏ k + li k − P(k)e 1 7→ l [(Zn)q]2 k Zn i=1 | | k k ∈ ∈ e has a meromorphic extension to C with at most simple possible poles at the points s = n + d + r1 + + rq 1 m where m N. | | ··· − − ∈

An explicit formula for the residues of f is given in [15, Theorem 2.18 (ii)].

Appendix B Examples of Spectral Triples

B.1 Spheres

A particularly illustrative example of a commutative spectral triple (recall Exam- ple 1.2) is provided by the d-dimensional unit spheres Sd. On S1 there are two possible spin structures, where the nontrivial one is associated to functions with antiperiodic boundary conditions. When d 2 there is only one spin structure available since Sd is simply connected. ≥ Let us equip Sd with the standard round metric and cook up the standard Dirac operator D/ acting on the chosen spinor bundle S . Then, (C∞(Sd),L2(Sd,S ),D/) is a d-dimensional regular spectral triple with a simple dimension spectrum d N (cf. Example 1.25), for any d 1 and any spinor bundle S . − The spectrum of D/ turns≥ out to be very simple [2, 21]: For the trivial spin 1 structure on S we have λn(D/)= n for n Z and all of the eigenspaces are one- dimensional. In particular, we have dimker∈D/ = 1. In the non-trivial case, the spectrum of the Dirac operator agrees with the general pattern for Sd and for d 1: ≥ d D d D 2 n +d 1 Z λn( /)= sign(n)(n + 2 ), Mn( /)= 2⌊ ⌋ | |d 1− , with n . (B.1) − ∈ Hence,

d D d D 2 +1 n+d 1 N µn( /)= n + 2 with Mn( / )= 2⌊ ⌋ d −1 , with n . | | − ∈

B.2 Tori

Another commutative spectral triple is given by the ﬂat tori Td = Rd/Zd and, as above, (C∞(Td),L2(Td ,S ),D/) is a d-dimensional regular spectral triple with a simple dimension spectrum d N.Thereare2d different spin structures on Td classiﬁed − d d by the twisting of each coordinates of the lattice Z : Given a basis e1, ,ed of Z , ···

145 146 B Examples of Spectral Triples this is realised by choosing s1, ,sd 0,1 , so that we have the group homomor- d s ··· ∈{ } phism: ek Z ( 1) k +1, 1 and all spin structures are given this way by ∈ → − ∈{ − } (s1, ,sd). The··· eigenvalues of the Dirac operator D/ (endowed with the induced ﬂat metric) depend on the chosen spin structures and are given by (see [2, 21])

d 1 Zd ∗ 2π k + 2 ∑ s j e∗j k . (B.2) {± k j=1 k | ∈ }

Zd where ∗ is the dual lattice and (e1∗, ,ed∗) is the dual basis of (e1, ,ed). ··· d/2 ··· The multiplicity of the eigenvalue 0 (given by s j = 0, j)is 2⌊ ⌋ while the non- d/2 1 ∀ zero eigenvalues have multiplicity 2⌊ ⌋− .

B.3 Noncommutative Tori

The noncommutative d-tori were introduced by Rieffel [34] and Connes [5] as de- d formations of T characterised a by non-zero skew-symmetric matrix Θ Md(R). ∈ ∞ Td Denote by C ( Θ ) the algebra generated by d unitaries ui, i = 1,...,d satisfying

iΘℓ j uℓ u j = e u j uℓ, (B.3) ∞ Td and with Schwartz coefﬁcients. So, a C ( Θ ) can be written as a = ∑k Zd ak Uk, Zd n∈1 nd N ∈ where ak ( ) (i.e. supk Zd k1 kd ak < ∞, ni ) and { } ∈ S ∈ | | ···| | | | ∀ ∈ i k.Θ k k1 kd d Uk := e− 2 ′ u u , k Z , 1 ··· n ∈ where Θ ′ is the restriction of Θ to its upper triangular part. Relation (B.3) reads

i k.Θq ik.Θq UkUq = e− 2 Uk+q or UkUq = e− UqUk . (B.4)

Thus, the unitary operators Uk satisfy

1 Uk∗ = U k and [Uk,Ul]= 2i sin( k.Θl)Uk+l . − − 2 ∞ Td Let τ be the trace on C ( Θ ) deﬁned by : τ ∑k Zd ak Uk = a0 ∈ H ∞ Td and τ be the GNS Hilbert space obtained by completion of C ( Θ ) with respect to the norm induced by the scalar product a,b := τ(a b). h i ∗ B.3 Noncommutative Tori 147

H 2 Zd On τ = ∑k Zd ak Uk ak k ℓ ( ) , let δµ , for µ 1,...,d , be the pair- wise commuting{ ∈ canonical|{ derivations,} ∈ given} by ∈{ }

δµ (Uk) := ikµUk . (B.5)

Deﬁne now

∞ d 2m A := C (T ) acting on H := Hτ C , with m := d/2 . Θ Θ ⊗ ⌊ ⌋

Each element of AΘ is represented on H as π(a) := L(a) 12m where L( ) (and R( )) is the left (right) regular representation of C∞(Td ). The⊗ Tomita conjugation· · Θ

J0(a) := a∗ satisﬁes J0L(a)= R(a∗)J0 and [J0,δµ ]= 0 and we deﬁne

J := J0 C0, ⊗ C2m 2 where C0 is an operator on such that C0 = 12m , depending on the parity of m. The (ﬂat) Dirac-like operator is given by ±

µ D := iδµ γ , (B.6) − ⊗ α α with hermitian Dirac matrices γ satisfying C0γ = γ C0 (see [6, 22] for details about the signs). The operator D is deﬁned and symmetric± on the dense subset of m H given by C∞(Td ) C2 and we still denote by D its selfadjoint extension. Thus Θ ⊗ µ D Uk ei = kµUk γ ei, ⊗ ⊗ 2m where (ei) is the canonical basis of C . Finally, in the even case, the chirality operator reads: γ := id ( i)mγ1 γd. The operator D is not invertible: ⊗ − ···

2m kerD = U0 C ⊗ m and this kernel has dimension 2 because if ψ = ∑k, j ck, jUk e j, then we have 2 2 ⊗ 0 = D ψ = ∑ ck j k Uk e j. Thus, k, j , k k ⊗

P0 = U0 U0 1C2m . (B.7) | ih |⊗ This yields a spectral triple:

Theorem B.1. The tuple (AΘ ,H ,D,J,γ) is a real regular spectral triple of dimension d. Its KO-dimension is also d. Most of the arguments will be revisited in the computation of the dimension spectrum — see Theorem B.2. For a complete proof see [6, 22]. We remark that the torus actions on C∗-algebras lead to interesting nonunital spectral triples , see [4, Chapter 5]. 148 B Examples of Spectral Triples B.3.1 Dimension Spectrum

1 Theorem B.2. i) If 2π Θ is badly approximable, the dimension spectrum of the triple C∞(Td ),H ,D is equal to the set d k : k N and all of the poles are simple. Θ { − ∈ } ii) ζD(0)= 0. Proof. i) Let B P(A ) and p N. Suppose that B is of the form ∈ ∈ qr 1 pr 1 q1 p1 B = arbrD − D − ar 1br 1 D D a1b1, | | − − ··· | | 1 where r N, ai A , bi JA J , qi, pi N. ∈ ∈ ∈ − ∈ We decompose ai =: ∑ Zd ai U and bi =: ∑ Zd bi U . ℓ ,ℓ ℓ ℓ′ ,ℓ′ ℓ′ ∈ µ∈1,µq µ µq With the shorthands kµ ,µ := kµ kµ and γ i := γ 1 γ i , we get 1 qi 1 ··· qi ···

q1 p1 D D a1b1 Uk e j | | ⊗ p1 µ1,µq1 = a1,ℓ b1 Uℓ UkU k + ℓ1 + ℓ′ (k + ℓ1 + ℓ′ )µ ,µq γ e j, ∑ 1 ,ℓ1′ 1 ℓ1′ | 1| 1 1 1 ⊗ ℓ1,ℓ1′ which gives, after r iterations,

r 1 − pi B U e a b U U U U U k k i ( k j)= ∑ ℓ ℓ ℓr ℓ1 k ℓ ℓr ∏ + ℓi + ℓi′ ( + ℓi + ℓi′)µ ,µi ⊗ ··· 1′ ··· ′ | | 1 qi ℓ,ℓ Zd i=1 ′∈ r 1 r 1 1 1 e e µ1− ,µq− µ1 ,µq b b b b γ r 1 γ 1 e j, ⊗ − ··· where aℓ := a1,ℓ ar,ℓ , bℓ := b1 br,ℓ , 1 ··· r ′ ,ℓ1′ ··· ′r r 1 r 1 1 1 µ µ1− ,µq− µ ,µq ℓi := ℓ1 + + ℓi, γ := γ r 1 γ 1 1 . e ··· e − ···

r 1 pi Let us denote Fµ (k,ℓ,ℓb′) := ∏i=−1 k + ℓi + ℓi′ (k + ℓi + ℓi′)µi ,µi . | | 1 qi With the shortcut c meaning equality modulo a constant function in the variable s, we have ∼ b b b b

Tr B D p s a b τ U U U U U U Fµ (k,ℓ,ℓ′) tr γ µ − − c ∑ ′ ∑ ℓ ℓ k ℓr ℓ1 k ℓ ℓr s+p ( ). | | ∼ ′ − ··· 1′ ··· ′ k k Zd ℓ,ℓ Zd k k ∈ ′∈ e e i∑r ℓ .Θk Since U U Uk = UkU U e i=1 i , we get ℓr ··· ℓ1 ℓr ··· ℓ1 − iφ(ℓ,ℓ ) i r ℓ .Θk r ′ ∑i=1 i τ U kUℓr Uℓ1UkUℓ Uℓ = δ∑ ℓ +ℓ ,0 e e− , − ··· 1′ ··· ′r i=1 i i′ where φ is a real valued function. Thus,

i r k ∑i=1 ℓi.Θ p s iφ(ℓ,ℓ′) Fµ (k,ℓ,ℓ′)e− µ Tr B D − − c ′ e δ r aℓbℓ s+p tr(γ ) | | ∼ ∑ ∑ ∑i=1 ℓi+ℓi′,0 ′ k k Zd ℓ,ℓ Zd k k ∈ ′∈ µ c fµ (s)tr(γ ). e e ∼ B.3 Noncommutative Tori 149

The function fµ (s) can be decomposed as a linear combination of zeta functions of the type described in Theorem A.12 (or, if r = 1 or if all pi’s are zero, in Theorem p s A.11). Thus, by linearity, s Tr B D − − has a meromorphicextension to C with simple poles located exclusively7→ in|Z| C. 0 ⊂ s ℜ(s) Moreover, if B Ψ (A ) and q N is such that q > d, then B D − OP− , so ∈ ∈ s | | ∈ it is trace-class around q and hence ζB,D(s)= Tr B D − is regular around q. ii) Let | | s Zd(s) := ∑ ′ k − k Zd k k ∈ 2 2 be the Epstein zeta function associated to the quadratic form q(x) := x1 + ... + xd. Then Zd enjoys the functional equation:

s d/2 1 Zd(s)= π − Γ (d/2 s/2)Γ (s/2)− Zd (d s). (B.8) − − s d/2 1 Since π − Γ (d/2 s/2)Γ (s/2)− = 0 for any negative even integer d and Zd(s) − d/2 1 is meromorphic on C with only one pole at s = d (with residue 2π Γ (d/2)− according to (A.25)), we get Zd (0)= 1. By deﬁnition, − 2m s ζD(s)= ∑ ∑ Uk e j, D − Uk e j , k Zd j=1h ⊗ | | ⊗ i ∈ so that

d/2 ζD(s)= 2⌊ ⌋(Zd(s)+ 1) (B.9) and the conclusion ζD(0)= 0 follows.

B.3.2 Heat Kernel Expansion

Proposition B.3. The heat trace asymptotics is

t D2 d/2 d/2 d/2 Tre− 2⌊ ⌋π t− . t∼0 ↓ 2 2 Proof. Since D Uk ei = k Uk ei, we know by Formula (2.22) that ⊗ k k ⊗ t D2 m t k 2 d/2 d/2 d/2 Tre− = 2 e− k k 2⌊ ⌋π t− . ∑ t∼0 k Zd ↓ ∈ This result can also be obtained (in an admittedly circuitous way) from Theo- m rem 3.6: We have D2 (s)= 2 Γ (s)(Zd (2s)+ 1) and the Epstein zeta function Zd is meromorphic on CZwith a single simple pole at s = d. Moreover, the Epstein zeta function enjoys a polynomial growth on the verticals — see the non trivial estimates demonstrated in [14, 30]. 150 B Examples of Spectral Triples

Remark that we recover the classical result: For the torus Td with the usual scalar µν Laplacian ∆ = g ∂µ ∂ν , − t ∆ d d/2 d/2 d/2 1/4t Tre− = Vol(T )(4π)− t− + 0(t− e− ) O d d d d/2 d/2 m and Vol(T ) = (2π) so that Vol(T )(4π)− = π = ad,0 (mod 2 ).

B.3.3 Spectral Action for Noncommutative Tori

We consider the d-dimensional noncommutative torus (AΘ ,H ,D) of Theorem B.1 with A = C∞(Td ), a one-form A = A Ω 1 (A ) and Θ Θ ∗ ∈ D Θ

A := A + εJAJ∗. (B.10)

0 Thus, the constraint A = A∗ Ψ (A ) is satisﬁed. Remark that A can be written∈ as

α A =: L( iAα ) γ where Aα = A∗ A , − ⊗ − α ∈ Θ so that

α DA = i[δα + L(Aα ) R(Aα )] γ . (B.11) − − ⊗ As for the commutative torus Td, we get

2 α1α2 1 α1α2 DA = δ (δα + Aα )(δα + Aα ) 12m Ωα α γ , − 1 1 2 2 ⊗ − 2 1 2 ⊗ where δ αβ is the Kronecker symbol and

α1α2 1 α1 α2 α2 α1 Aα := L(Aα ) R(Aα ), γ := (γ γ γ γ ), − 2 − Ωα α := [δα + Aα ,δα + Aα ] = L(Fα α ) R(Fα α ), 1 2 1 1 2 2 1 2 − 1 2 Fα α := δα (Aα ) δα (Aα ) + [Aα ,Aα ] A . (B.12) 1 2 1 2 − 2 1 1 2 ∈ Θ Thus,

2 α1α2 DA = δ [δα + L(Aα ) R(Aα )][δα + L(Aα ) R(Aα )] 12m − 1 1 − 1 2 2 − 2 ⊗ 1 α1α2 L(Fα α ) R(Fα α ) γ . (B.13) − 2 1 2 − 1 2 ⊗ We now prove the existence of the asymptotics of the fluctuate d heat trace on the noncommutative torus using the one for the ‘bare’ one given in Proposition B.3 (cf. also Problem 6) e) in Chapter 5). To this end, we employ the pseudodifferential d calculus introduced by Connes for C∗-dynamical system (A,R ,α) (see [5, 8, 7]) and follow the arguments given in the proof of [29, Theorem 4.2]. The idea is es- B.3 Noncommutative Tori 151 sentially to mimic the classical pdo calculus on a manifold — cf. Appendix A.3 — 2 tDA improving the Proposition 2.27 to gain control on the series defined by Tre− . We first quickly summarise this symbolic calculus — see [5, 7, 8] (and especially the complete approach by Lesch and Moscovici [29]) for details. Let Aθ be the universal C∗-algebra generated by the Uk and let Aθ consist of those ∞ d elements in A for which a αs(a) is C for each s R , with the definition θ → ∈ i2π s.k αs(Uk) := e− Uk.

A smooth map ρ : C∞(Rd ) A is named a symbol of order m Z → θ ∈ d k ℓ m ℓ if for any k,ℓ N , δ ∂ ρ(ξ ) ck (1 + ξ ) −| | for some constants ck ∈ k ξ k≤ ,ℓ k k ,ℓ ∞ Rd ∞ m and if there exists σ C ( ,Aθ ) such that lim λ − ρ(λξ )= σ(ξ ), ∈ λ ∞ → where

δ k := δ k1 δ kd for k Nd, ∂ ℓ := ∂ ℓ1 ∂ ℓd , 1 ··· d ∈ ξ ξ1 ··· ξd 1 Such a symbol is elliptic when ρ(ξ )− exists and the estimate

1 m ρ(ξ )− c(1 + ξ )− k k≤ k k holds for ξ large enough. k k d d Given a symbol ρ, let us deﬁne the pdo Pρ : (R ,A ) (R ,A ) by S θ → S θ 1 1 Pρ (u) := α x(F− [ρ](x y))u(y)dy = α x(F− [ρ](y))u(x y)dy. − − − − Z Z d d An action of the pre-C -algebra A ⋊α R on the pre-C -module (R ,A ) is ∗ θ ∗ S θ given, for a Aθ , by au(x)= α x(a)u(x) and Uy(u)(x) := u(x y). Then, ∈ − − 1 Pρ = F− [ρ](y)Uy dy Z d can be seen as an element of the multiplier space of Aθ ⋊α R . The GNS repre- d sentation of Aθ given by the trace τ can be extended to Aθ ⋊α R , with the maps 1 a Aθ a and Ux αx, and Pρ = F− [ρ](y)αy dy can be seen also as an element of∈ the multiplier→ of→A . Since θ R

1 i2π y.k Pρ Uk = F− [ρ](y)e− dyUk = ρ(k)Uk, Z we get, for any element a = ∑k Zd ak Uk, ∈ Pρ (a)= ∑ ak ρ(k)Uk. k Zd ∈ Thus, 152 B Examples of Spectral Triples

TrPρ = ∑ Uk,PρUk = ∑ τ(Uk∗ρ(k)Uk)= ∑ τ[ρ(k)] k Zd h i k Zd k Zd ∈ ∈ ∈ d 2 1/2 m is ﬁnite if m < d, because ρ(k) c0,0(1 + (∑i=1 ki ) ) . A parametric− symbol ρ(ξk,λ) withk≤λ V – a region in C is deﬁned similarly: ∈ k ℓ r m ℓ r δ ∂ ∂ ρ(ξ ,λ) ck (1 + ξ + λ ) −| |−| |. (B.14) k ξ λ k≤ ,ℓ k k | | Cf. [19, Section 1.7.1] and the (slightly different) Hypothesis A.1 of Appendix A.3. We now adapt the Proposition 2.27: m Rd ∞ Proposition B.4. If ρ S ( V,Aθ ) is a parametric symbol of order m < d, then ∈ × − ∞ ∑ ρ(k,λ)= ρ(ξ ,λ)dξ + ∞( λ − ). Rd O | | k Zd Z ∈ Proof. Using (2.19),

ρ(k,λ)= ρ(ξ ,λ)dξ + F[ρ( ,λ)](k). ∑ Rd ∑ · k Zd Z k Zd 0 ∈ ∈ \{ }

Via the Fourier transform of a derivative and (B.14), we get, for each N N∗, N d+m N ∈ d F[ρ( ,λ)] ξ − λ − and the conclusion follows from: For any q N , k · k≤k k | | ∈ N d+m N ∑ F[ρ( ,λ)](k) q cq,N ∑ k − λ − . k Zd 0 · ≤ k Zd 0 k k | | ∈ \{ } ∈ \{ }

Lemma B.5. For DA as in (B.11), we have

∞ 2 tDA (k d)/2 Tre− ak(DA)t − . t∼0 ∑ ↓ k=0 Moreover, for f (√ ) r with r > d/2, the spectral action has the asymptotics · ∈C0 d ∞ D k k 1 k 1 S( A, f ,Λ)= ∑Λ ( f (t)t − dt) DA − + f (0)ζDA (0)+ ∞(Λ − ). (B.15) 0 − | | O k=1 Z Z

Proof. The operator DA given in (B.11) can be seen as a differential multiplier of D 1 Rd B C2m B order 1. Namely, A Diffσ ( , θ ), where θ is the algebra generated by 1 ∈ ⊗ Aθ and JAθ J− , following a slight extension of [29, Deﬁnition 3.5]. D 1 α D 2 The symbol of A is σDA (ξ ) = (ξα iAα iJAα J− ) γ . Thus, A given in (B.13) has the form − − ⊗

m b ξ β where b B C2 and ξ β := ξ β1 ξ βd . ∑ β β ∈ θ ⊗ 1 ··· d β Nd, β 2 ∈ | |≤ 2 Thus DA is an elliptic differential multiplier of order 2 and as such, its symbol can be decomposed into a sum of monomials a j(ξ ) of order j = 2, 1, 0: B.3 Noncommutative Tori 153

σ 2 (ξ )= a (ξ )+ a (ξ )+ a . DA 2 1 0

If λ is the resolvent parameter for (D 2 λ) 1, in the search for the resolvent A − − parametrix Bλ , we need to solve 1 = σλ ⋆ σBλ for the parameter dependent symbol σ (ξ ) := σD2 (ξ ) λ = a′ (ξ )+ a1(ξ )+ a0 λ A − 2 with a (ξ )= a2(ξ ) λ. 2′ − Since σB = b0 +b1 +b2 + , where b j(ξ ;λ) is a symbol of order 2 j, one can λ ··· − − compute recursively the symbols b0, b1, b2,... of the parameter-dependent pseudodifferential multiplier Bλ (see [7, 8, 16, 17, 29] for examples of such computations). Then, thanks to (A.24),

2 tDA k+1 i tλ 2 k Tre− = t− e− Tr[(DA λ)− ]dλ 2π C − Z 2 k (recall that Tr[(DA λ)− ] < ∞ for k d/2 + 1), one implements the resolvent expansion into the integral− to get the asymptotics≥⌊ ⌋

∞ 2 tDA 2 (k d)/2 Tre− ak(DA)t − t∼0 ∑ ↓ k=0 exactly as in [19, Sections 1.7 and 1.8] using the Proposition B.4. Thus, 2 i λ ak(DA)= dξ e− τ[bk(ξ ,λ)]dλ. 2π Rd C Z Z Finally, we use Formula (3.43), which becomes (B.15).

While there is no Diophantine condition to get the asymptotics of the ﬂuctuated heat trace, the computation of the coefﬁcients seems to need one — this is dueto a commutation between a series and a residue.

1 Theorem B.6. Assume that 2π Θ is badly approximable. Then, the spectral action (B.15) ﬂuctuated by A as in (B.10) reads, for f (√ ) r with r > d/2 · ∈C0 4π f Λ 2 Λ 2 for d 2 D 2 + ∞( − ), = , S( A, f ,Λ)= 2 4 O4π2 µν 2 8π f4 Λ f (0)τ(Fµν F )+ ∞(Λ ), for d = 4. − 3 O − For arbitrary d,

d 1 D − d k 1 S( A, f ,Λ)= ∑ fd k cd k(A)Λ − + ∞(Λ − ), k=0 − − O where we have cd 2(A)= 0 and cd k(A)= 0 for k odd. In particular, c0(A)= 0 when d is odd. − − 154 B Examples of Spectral Triples

We do not know if, without the Diophantine condition, the above spectral action would stay the same, see [18, Appendix B]. The proof goes through several steps and the ﬁrst one is to identify the noncommutative integrals in (B.15):

1 Proposition B.7. Assume that 2π Θ is a badly approximable matrix. Then, we have

d+2 d d/2 d 1 DA − = 2⌊ 2 ⌋π Γ ( )− , − | | 2 Z d+k DA − = 0 for k odd integer, − | | Z d+2 DA − = 0. − | Z These equalities follow from the explicit computation of (4.15)–(4.24) and the d+q fact that P D − = 0 for any P Ψ ′(A ) and any odd integer q. In a similar| way,| ApD q = (εJAJ∈ )pD q = 0 and PD d+q = 0 for p 0, 1 R − ∗ − − q < d and P in the algebra generated by A , [D,A ], JA J , J[D,A ]J . All≥ these≤ R R R ∗ ∗ equalities can be proved using the deep results stored in Appendix A.6. The case p = q = 1 corresponds to the fact that there is no tadpole for the noncommutative torus — see Definition 4.12. The main point is again to be able to commute an infinite series, like those defining an element of AΘ , and a residue given by and this is where the hypothesis on 1 Θ is used. 2π R In the second step, we face explicit computations like, inter alia, the following:

α Lemma B.8. Under same hypothesis (recall that A = L( iAα ) γ ), − ⊗ α q q δq,24π τ(Aα A ) for d = 2, A D− = − π2 α α µ µ − δq 4 τ(Aα Aα ) tr(γ 1 γ 4 γ 1 γ 4 )δµ ,µ ,µ ,µ for d = 4, Z , 12 1 ··· 4 ··· ··· 1 2 3 4 where δµ1,µ2,µ3,µ4 = δµ1µ2 δµ3µ4 + δµ1µ3 δµ2µ4 + δµ1µ4 δµ2µ3 . α α 4π2 With A = L( iAα ) γ = i∑ℓ Zd aα,ℓUℓ γ and c := 3 , we have − ⊗ − ∈ ⊗

1 1 2 1 1 2 α1 α2 α1α2 2 + 2 (AD− ) = 2 (εJAJ∗D− ) = c ∑ aα1,ℓ aα2, ℓ l ℓ δ ℓ . − − − − k k Z Z ℓ Zd ∈ 1 1 3 1 1 3 ℓ1.Θℓ2 α3 3 (AD− ) = 3 (εJAJ∗D− ) = 4c ∑ aα3, ℓ1 ℓ2 aα1,ℓ2 aα1,ℓ1 sin 2 ℓ1 . − − − − d − − Z Z ℓi Z ∈ 1 1 4 1 1 4 + (AD− ) = (εJAJ∗D− ) 4 − 4 − Z Z ℓ1.Θ(ℓ2+ℓ3) ℓ2.Θℓ3 = 2c ∑ aα1, ℓ1 ℓ2 ℓ3 aα2,ℓ3 aα1,ℓ2 aα2,ℓ1 sin 2 sin 2 . d − − − ℓi Z ∈ These equalities follows from tedious computations (cf. [15] for the details). They are necessary to get ζDA (0) from (4.32), using ζD(0)= 0 obtained in Theorem B.2. B.4 Podles´ Spheres 155

µν This gives ζDA (0)= cτ(Fµν F ) in dimension d = 4, while ζDA (0)= 0 for d = 2 or when d is odd. Gathering− all these computations brings us to Theorem B.6.

B.4 Podles´ Spheres

Podles´ has introduced the eponym standard quantum spheres in [32] as homogeneous spaces under the action of the quantum deformationof the SU(2) group. They ﬁt into the picture of noncommutative geometry à la Connes and concrete spectral triples were constructed in [11, 12], see also [10]. The algebra A q introduced in [32] is a complex -algebra generated, for a pa- ∗ rameter 0 < q < 1, by A = A∗, B, B∗ subject to the relations

2 2 2 2 AB = q BA, AB∗ = q− B∗A, BB∗ = q− A(1 A), B∗B = A(1 q A). − −

As a C∗-algebra, it is isomorphic to the minimal unitisation of the algebra of compact operators on a separable Hilbert space and is an invariant subalgebra of the quantum group SUq(2) under a circle action. We will use only the polynomial algebra Aq in the above generators, which is a dense subalgebra of A q. In the following we will employ the q-numbers deﬁned as

n n n : q− q for q 0 1 and n C (B.16) [ ] = q 1− q , ( , ) . − − ∈ ∈

Observe that limq 1[n]= n for any n C. → ∈ Let H1/2 be the separable Hilbert space with an orthonormal basis l,m for 1 | i m l, l + 1,...,l and l + N. It is suitable for a representation of Aq, ∈ {− − } ∈ 2 equivariant under the action of the ∗-Hopf algebra Uq(su(2)). There exist two non- equivalent Uq(su(2))-equivariant representations of Aq on H1/2 [12]:

+ 0 π (A) l,m := Al,m, l+1,m + Al,m, l,m + Al−,m, l 1,m , (B.17) ± | i± ± | i± ± | i± ± | − i± + 0 π (B) l,m := Bl,m, l+1,m+1 + Bl,m, l,m+1 + Bl−,m, l 1,m+1 , ± | i± ± | i± ± | i± ± | − i± + 0 π (B∗) l,m := Bl,m, l+1,m 1 + Bl,m+1, l,m 1 + Bl−,m, l 1,m 1 , ± | i± ±| − i± ±| − i± ±| − − i± e e e 1 A+ := qm+l+ 2 [l m + 1][l + m + 1] α+, l,m − − l 1 0 1 p 2 0 1 A := q− 2 [l m + 1][l + m] q [l m][l + m + 1] α + , (B.18) l,m 1+q2 − − − l 1+q2 1 m l A− := q − − 2 [l m][l + m] α− , l,m − l p 156 B Examples of Spectral Triples

+ m + Bl,m := q [l + m + 1][l + m + 2] αl , B0 := qmp[l + m + 1][l m] α0 , (B.19) l,m − l m B− := q p[l m][l m 1] α− , l,m − − − l p + m 1 B := q − [l m + 2][l m + 1] α− , l,m − − l+1 0 m 1 0 B := q − p[l + m][l m + 1] α , (B.20) el,m − l m 1 + Bl−,m := q − p[l + m][l + m 1] αl 1 . e − − p2l+2 + The coefﬁcients αl read:e α− := q α and l − l 1 (q 1 )[l ][l+ 3 ]+q for π : α0 : 1 − q − 2 2 (B.21) + l = √q [2l][2l+2] , + l 2 1 α := q− − ; (B.22) l √[2l+2]([4l+4]+[2][2l+2]) 1 3 (q 1 )[l ][l+ ] q 1 0 1 − q − 2 2 − − for π : αl := , (B.23) − √q [2l][2l+2] + l 1 1 α := q− − . (B.24) l √[2l+2]([4l+4]+[2][2l+2])

Since π± are faithful, the algebra Aq is dense in A q in the operator norm. Let now Hq := H+ H , where H := H1/2, with the representation of Aq: ⊕ − ±

π+(a) 0 π(a) := 0 π (a) . (B.25) − On H we deﬁne an unbounded selfadjoint operator given by

0 w Tq 1 Dq := , Tq : l,m H [l + ] l,m H , (B.26) w Tq 0 | i ∈ 1/2 7→ 2 | i ∈ 1/2 for an arbitrary constant w C 0 . ∈ \{ } We note that kerDq = 0 and the polar decomposition of Dq reads { } T 0 D = F D , with D = w q and F := 1 0 w . (B.27) q q q 0 Tq w w 0 | | | | | | | | 1 The phase operator satisﬁes [F, π(a)] = 0 for a Aq. We have for l N + , ∈ ∈ 2 1 Dq ( l,m + l,m )= w [l + ]( l,m + l,m ), | | | i ⊕ | i− | | 2 | i ⊕ | i− which gives

µn(Dq)= λn( Dq )= w [n + 1], and Mn( Dq )= 4(n + 1), n N. (B.28) | | | | | | ∈

Let us recall (B.16) and observe that the singular values of Dq grow exponentially. B.4 Podles´ Spheres 157

In [12] it is proven that (Aq,Hq,Dq) is a spectral triple, which moreover is even 1 0 H for γ := 0 1 and real for the antiunitary operator J on q defined by − J l,m := i2m pm l, m with p R+. | i± | − i∓ ∈ 2 1 In particular, J = 1, Jγ = γJ, JaJ commutes with Aq and [Dq,J]= 0, so the − − − spectral triple is of KO-dimension 2 [9]. The operator D is the unique Uq(su(2))- equivariant operator fulfilling the first-order condition, which makes the spectral triple real [12]. In the limit q 1 one recovers the commutative spectral triple on the 2- dimensional sphere→ — cf. [32, Remark 2] and [12, p. 8]. In [32] yet another Dirac operator for the standard Podles´ sphere was introduced

0 wT S (l+1/2) D S q S H q− H q := S , Tq : l,m 1/2 1 l,m 1/2. wTq 0 | i ∈ 7→ q− q | i ∈ − which shares the property of the exponential growth of singular values with Dq. S S w q 2 S 1 The operators Dq and D commute and are related by Dq = D ( | | ) (D ) . q q − 1 q2 q − D S 1 −D S Remark that ( q )− is actually trace-class. It turns out that by taking q instead of Dq we preserve most of the properties of the spectral triple with the exception of the ﬁrst order condition. Moreover, we have

D D S w q 2 D S 1 q = q ( 1| |q2 ) q − . | | | |− − | | S Proposition B.9. The triples (Aq,Hq,Dq) and (Aq,Hq,Dq ) are 0-dimensional. Proof. This is a direct consequence of the exponential growth of singular val- S n S n ues of Dq and Dq . Indeed, µn(Dq)= ∞(q− ) and µn(Dq )= ∞(q− ), whereas S O O ε Mn( Dq )= Mn( D )= ∞(n), which means that for any ε > 0 both Tr Dq and | | | q | O | |− Tr D S ε are finite. | q |− Observe that, whereas the limit q 1 of the spectral triple (Aq,Hq,Dq) is well defined and yields the standard round→ geometry on S2, the spectral dimension jumps abruptly from 0 to 2. This phenomenon, known as the ‘dimension drop’, has also its impact on (co)homologies of A q [31] and provided inspiration for ‘twisted’ noncommutative geometries (see Problem 3) in Chapter 5). The potential physical implications of the dimension drop in the context of quantum gravity are the subject of an intensive study — see, for instance, [3]. Another drastic consequence of the exponential growth of singular values is that A H D A H D S neither of the spectral triples ( q, q, q), ( q, q, q ) is regular. In fact, already D ,[ D ,a] is an unbounded operator and more generally δ n(a) is an operator of order| | n| |1 for a generic element a A . This could be seen by computing a fixed matrix element− of the operator δ n(A∈), for instance:

n 3 1 n + l + 1, m δ (A) l,m = [l + ] [l + ] Al m . ±h | | i± 2 − 2 , 158 B Examples of Spectral Triples

+ l The behaviour of Al,m for large l is ∞(q ), which can be read from the explicit 3 O1 n nl Formula (B.18), while [l + 2 ] [l + 2 ] = ∞(q− ), and the above expression is − nO n 1 unbounded for n > 0; moreover, generally, δ (A) is in OP − . 0 1 2 Note that, in general, δ (a) and δ (a) are bounded for any a Aq, but not δ (a). ∈ Nevertheless, the spectral triple (Aq,Hq,Dq) is quasi-regular(recall Section 1.7). Within this extended framework one discovers that [13, Corollary 3.10]:

Theorem B.10. For any 0 < q < 1, the dimension spectrum of (Aq,Hq,Dq) is of the order 2 and equals to N + i 2π Z. − logq Proof (sketch). Firstly, one shows (using a simple summation of geometric series [13, Proposition 3.2]) that the basic zeta function reads

∞ s 1 q2 s Γ (s+n) q2n ζD s Tr D 4 for ℜ s 0 q ( )= q − = ( −w ) ∑ n!Γ (s) (1 qs+2n)2 , ( ) > . | | | | n=0 − This formula leads us to the realm of ‘q-zeta functions’ (see [26] and other references on [13, p. 633]). It yields a meromorphic extension of ζDq to the whole complex plane with second order poles in 2N + i 2π Z C. − logq ⊂ 0 A Secondly, one shall consider general functions ζP,Dq with P Ψ ( q) (cf. [13, Proposition 3.8]). To start, one notices that it is sufficient to consider∈ elements of p n the form P = T Dq − , with T P(Aq) op for some n N ande n p N and uses the quasi-regularity| | [13, Lemmas∈ 3.4∩ & 3.6] to commute∈ all of the≤ operators∈ Dq and Dq from T to the right. Then, one observes that the generators of Aq are | | represented via π in terms of weighted shift operators on Hq. Moreover, the weights l+m l m l are analytic functions of (bounded) variables q ,q − and q . Rewriting these in terms of infinite convergent Taylor series one arrives at a formula for ζT,Dq , which involves a (multiple) infinite series in ql. Finally, a resummation over l N + 1/2 C ∈ yields the desired meromorphic extension of ζP,Dq to — cf.[13, Eq. (30)]. D S The zeta function associated with the simplified operator q was presented in A H D S Example 3.13. Although the spectral triple ( q, q, q ) has not been studied extensively in [13], one can show along the same lines that it is quasi-regular and has a dimension spectrum of second order equal to N + i 2π Z. − logq The exponential growth of singular values of Dq has also some pros: It leads to the following spectacular result highlighted at the begining of Section 2.6: r 2πi Theorem B.11. Let f 0 for some r > 0 and denote κ := logq . Then, for any Λ > 0, ∈C

∞ 2 n D n m n n m 2k+κ j S( q, f ,Λ)= ∑ ∑ ∑ a 2k+κ j,n ∑ ( 1) − m f 2k+κ j,m (logΛ) − Λ − . k=0 j Z n=0 − m=0 − − ∈ Proof (sketch). The theoremis provenin detail in [13], via the heat trace expansion. It uses the full forceof Theorems 3.12, 3.17, 3.24 and Corollary 3.25. The estimation of the contour integrals is rather subtle and arduous — cf. [13, Proposition 4.3]. References 159

This result is remarkable for two reasons: Firstly, the formula for the spectral action contains terms proportional to log2 Λ and oscillating with Λ, which is a sign that the geometry of Podles´ sphere lies outside of the kingdom of classical pdos (recall Example 2.37). Secondly, the formula is exact for all Λ > 0 and for a fairly general class of cut-off functions. Recall that in the classical (pseudo)differential geometry one is bound to use the asymptotic expansion, which might obscure some important information (see p. 30). As for the fluctuations, these are much more tedious to control in the quasi- regular case. In [13, Theorem 5.6] it was shown that the leading term of the spectral action S(Dq + A, f ,Λ) does not depend on A, when the fluctuation is ‘small’, but a deeper understanding of the problem is missing. In particular, it is not clear whether an explicit exact formula for the fluctuated action is available at all .

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Symbols γ, 5 δ ′(T ), 4 ∇, 7 δ(T), 7 , 16 ζT D(s), 18 ∼ , , 57 ζK,H (s), 41 x ∼x0 → ζ (s), 41 , 23 H 1,+ ζ , 43 k·k , 6 D k·ks ζ(s), 43 , 20 ζ(s,a), 85 − ζ (s), 101 Z [k] DA , 20 Θ, 146 − ϑ3(z;q), 60 Z [k] ϑ3(z τ), 60 , 108 λ (T|), 22 −A n Zφ χ, 49 Λθ , 127 ∗ p q, 129 µn(T), 23 x◦,η , 128 π(a), 1 z h i σz, 7 n , 13 n , 21 ∑′, 141 j Ψ(A ), 15 [n], 155 Ψ k(A ), 16 Ψ ′(A ), 17 ΨA(A ), 106 Ψ C(A ), 17 C ΨA (A ), 107 Ψ(A ), 27 Ψ(M), 130 Ψem(M,E), 130 1 ΩD (A ), 25

161 162 Index

u A , 29 Mn(T), 23 A , 1 M[ f ], 44 A , 2 Mn(C), 3 R Aδ , 4 Mn( ), 142 op′ Af , 5 NH (Λ), 40 A O q, 155 x0 , 39 A O q, 155 x0 , 39 ∞ AΘ , 147 O∞(x− ), 39 β (A ,H ,D), 1 O∞(x ) (C), 61 B(H ), 1 O(p), 129 r Bk, 60 op , 7 C r r0,θ , 128 OP , 8 CM, 47 P0, 3 C, 49 PA, 101 C0, 49 Pn(H), 40 Cc, 49 P( f ), 67 Cp, 49 P( f ,V), 67 ∞ P A Cc (M), 129 ( ), 15 ∞ Td P A , 17 C ( Θ ), 146 ′( ) D, 1 Q, 24 + DA, 26 R =[0,∞), 11 S Dq, 156 , 2 S D Dq , 157 S( , f ,Λ), 28 D/, 2 S, 48 D/ , 17 S ′, 48 E S D, 3 +′ , 48 d DA, 101 S , 145 D, 3 Sd, 18 D, 48 Sd+, 97 Td D ′, 48 , 145 D T p +′ , 48 , 40 E, 7 TrN (T ), 23 E , 7 U (A ), 29 fz,n, 94 WRes, 22 F[g], 52 X, XV , 102 G, 29 ZK,H (s), 67 H , 1 Zd , 149 H s, 6 A H ∞, 7 H τ , 146 abscissa of convergence, 41 H 1/2, 155 almost commutative geometry, 3 H q, 156 asymptotic expansion, 56 H, 29 absolutely convergent, 59 s H , 129 exact, 59 Jα , 47 uniformly convergent, 59 K (H ), 1 asymptotic scale, 56 K, 61 asymptotic series, 56 L (H ), 1 L 1(H ), 1 B L 1,+(H ), 23 L r(H ), 40 badly approximable, 142 L[ f ] and L[φ], 46 D L[T], 48 1 R+ Lloc( ), 46 dimension spectrum, 18 Index 163

of order k, 18 O simple, 18 Diophantine condition, 142 one-form, 25 Dirichlet series, 41 order of an operator, 8 distribution P support of, 48 Dixmier trace, 24 parametrix, 129 domain of smoothness, 7 Podles´ sphere, 155 Duhamel formula, 106 Poisson formula, 52 F polynomial growth on verticals, 76 pseudodifferential operator (pdo), 15 fermionic action, 29 classical, 18 first-order condition, 6 of order k, 16 fluctuation, 26 smoothing, 15 Fourier transform, 52 punctured neighbourhood, 39 Fredholm operator, 130 Q G q-number, 155 gauge potential, 26 S H Schatten ideal, 40 heat operator, 41 Sobolev spaces, 129 heat trace, 41 spectral action, 28 Hermitian connection, 26 topological, 32 hypertrace, 24 spectral density, 62 spectral function, 39 K spectral triple, 1 KO-dimension, 5 commutative, 2 even/odd, 5 L finite, 3 finitely summable, 4 Laplace transform, 46 nonunital, 2 of a distribution, 48 p-dimensional, 4 of a measure, 46 p-summable, 4 locality, 30 quasi-regular, 27 M real, 5 regular, 4 McKean–Singer formula, 32 θ-summable, 4 measurable operator, 24 spectral zeta function, 40 measure sphere, 145 convolution of, 49 symbol moment of, 47 classical, 129 signed, 46 elliptic, 129 variation of, 46 pdo associated to, 129 Mellin transform, 44 smoothing, 128 module, 25 free, 25 T projective, 25 tadpole, 116 Morita equivalence, 25 torus, 145 N W noncommutative integral, 20 Wodzicki residue, 22 noncommutative torus, 146