Noncommutative Geometry, the spectral standpoint

Alain Connes

October 24, 2019

In memory of John Roe, and in recognition of his pioneering achievementsin coarse geometry and index theory.

Abstract

We update our Year 2000 account of Noncommutative Geometry in [68]. There, we de-scribed the following basic features of the subject:I The natural “time evolution" that makes noncommutative spaces dynamical from their

measure theory.I The new calculus which is based on operators in Hilbert space, the Dixmier trace and

the Wodzicki residue.I The spectral geometric paradigm which extends the Riemannian paradigm to the non-

commutative world and gives a new tool to understand the forces of nature as pure gravityon a more involved geometric structure mixture of continuum with the discrete.I The key examples such as duals of discrete groups, leaf spaces of foliations and de-

formations of ordinary spaces, which showed, very early on, that the relevance of this newparadigm was going far beyond the framework of Riemannian geometry.

Here, we shall report1 on the following highlights from among the many discoveries madesince 2000:

1. The interplay of the geometry with the modular theory for noncommutative tori.

2. Great advances on the Baum-Connes conjecture, on coarse geometry and on higher in-dex theory.

3. The geometrization of the pseudo-differential calculi using smooth groupoids.

4. The development of Hopf cyclic cohomology.

5. The increasing role of topological cyclic homology in number theory, and of the lambdaoperations in archimedean cohomology.

6. The understanding of the renormalization group as a motivic Galois group.

7. The development of quantum field theory on noncommutative spaces.

1this report makes no claim to be exhaustive and we refer the reader to the collection of surveys [35, 90, 104, 114,117, 124, 134, 195, 199, 272, 276, 281], published in Advances in Noncommutative Geometry, Springer, ISBN 978-3-030-29596-7, (2019), for a more complete account and bibliography.

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8. The discovery of a simple equation whose irreducible representations correspond to 4dimensional spin geometries with quantized volume and give an explanation of theLagrangian of the standard model coupled to gravity.

9. The discovery that very natural toposes such as the scaling site provide the missingalgebro-geometric structure on the noncommutative adele class space underlying thespectral realization of zeros of L-functions.

Contents

1 Introduction, two key examples 3

2 New paradigm: spectral geometry 82.1 Geometry and the modular theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.2 Inner fluctuations of the metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.3 The spectral action and standard model coupled to gravity . . . . . . . . . . . . . . 142.4 Dimension 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.5 Relation with quantum gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 New tool: Quantized calculus 183.1 Quantum formalism and variables, the dictionary . . . . . . . . . . . . . . . . . . . 183.2 The principle of locality in NCG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

4 Cyclic cohomology 234.1 Transverse elliptic theory and cyclic cohomology of Hopf algebras . . . . . . . . . 244.2 Cyclic cohomology and archimedean cohomology . . . . . . . . . . . . . . . . . . . 254.3 Topological cyclic homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

5 Overall picture, interaction with other fields 255.1 Index theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255.2 Smooth groupoids and their C∗-algebras in geometry . . . . . . . . . . . . . . . . . 295.3 Novikov conjecture for hyperbolic groups . . . . . . . . . . . . . . . . . . . . . . . . 295.4 The Baum-Connes conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305.5 Coarse geometry and the coarse BC-conjecture . . . . . . . . . . . . . . . . . . . . . 315.6 Comparison with toposes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325.7 Quantum field theory on noncommutative spaces . . . . . . . . . . . . . . . . . . . 335.8 Noncommutative geometry and solid state physics . . . . . . . . . . . . . . . . . . 34

6 Spectral realization of zeros of zeta and the scaling site 346.1 Quantum statistical mechanics and number theory . . . . . . . . . . . . . . . . . . . 346.2 Anosov foliation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 366.3 The adele class space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 376.4 The scaling site . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

2

1 Introduction, two key examples

Noncommutative geometry has its roots both in quantum physics and in pure mathematics.One of its main themes is to explore and investigate “new spaces" which lie beyond the scopeof standard tools. In physics the discovery of such new spaces goes back to Heisenberg’s matrixmechanics which revealed the non-commutative nature of the phase space of quantum systems.The appearance of a whole class of “new spaces" in mathematics and the need to treat themwith new tools came later. It is worth explaining in simple terms when one can recognize thata given space such as the space of leaves of a foliation, the space of irreducible representationsof a discrete group, the space of Penrose tilings, the space of lines2 in a negatively curved com-pact Riemann surface, the space of rank one subgroups of R, etc, etc, is “noncommutative" inthe sense that classical tools are inoperative. The simple characteristic of such spaces is visibleeven at the level of the underlying “sets": for such spaces their cardinality is the same as thecontinuum but nevertheless it is not possible to put them constructively in bijection with thecontinuum. More precisely any explicitly constructed map from such a set to the real line fails tobe injective! It should then be pretty obvious to the reader that there is a real problem when onewants to treat such spaces in the usual manner. One may of course dismiss them as “patholog-ical" but this is ignoring their abundance since they appear typically when one defines a spaceas an inductive limit. Moreover extremely simple toposes admit such spaces as their spaces ofpoints and discarding them is an act of blindness. The reason to call such spaces “noncommu-tative" is that if one accepts to use noncommuting coordinates to encode them, and one extendsthe traditional tools to this larger framework, everything falls in place. The basic principle is thatone should take advantage of the presentation of the space as a quotient of an ordinary spaceby an equivalence relation and instead of effecting the quotient in one stroke, one should asso-ciate to the equivalence relation its convolution algebra over the complex numbers, thus keepingtrack of the equivalence relation itself. The noncommutativity of the algebra betrays the identi-fication of points3 of the ordinary space and when the quotient happens to exist as an ordinaryspace, the algebra is Morita equivalent to the commutative algebra of complex valued functionson the quotient. One needs of course to refine the perception of the space one is handling accord-ing to the natural hierarchy of geometric points of view, going from measure theory, topology,differential geometry to metric geometry.It is important to stress from the start that noncommutative geometry is not just a “generaliza-tion" of ordinary geometry, inasmuch as while it covers as a special case the known spaces, ithandles them from a completely different perspective, which is inspired by quantum mechanicsrather than by its classical limit. Moreover there are many surprises and the new spaces ad-mit features which have no counterpart in the ordinary case, such as their time evolution whichappears already from their measure theory and turns them into thermodynamical objects.The obvious question then, from a conservative standpoint, is why one needs to explore suchnew mathematical entities, rather than staying in the well paved classical realm ? The simpleanswer is that noncommutative geometry allows one to comprehend in a completely new waythe following two spaces whose relevance can hardly be questioned :

2infinite geodesics3The groupoid identifying two points gives the noncommutative algebra of matrices M2(C)

3

Space-Time and NCG

The simplest reason why noncommutative geometry is relevant for the understanding ofthe geometry of space-time is the key role of the non-abelian gauge theories in the StandardModel of particles and forces. The gauge theories enhance the symmetry group of grav-ity, i.e. the group of diffeomorphisms Diff(M) of space-time, to a larger group which is asemi-direct product with the group of gauge transformations of second kind. Searching fora geometric interpretation of the larger group as a group of diffeomorphisms of an higherdimensional manifold is the essence of the Kaluza-Klein idea. Noncommutative geometrygives another track. Indeed the group of diffeomorphisms Diff(M) is the group of automor-phisms of the star algebra A of smooth functions on M, and if one replaces A by the (Moritaequivalent) noncommutative algebra Mn(A) of matrices over A one enhances the group ofautomorphisms of the algebra in exactly the way required by the non-abelian gauge theorywith gauge group SU(n). The Riemannian geometric paradigm is extended to the noncom-mutative world in an operator theoretic and spectral manner. A geometric space is encodedby its algebra of coordinatesA and its “line element" which specifies the metric. The new ge-ometric paradigm of spectral triples (see §2) encodes the discrete and the continuum on thesame stage which is Hilbert space. The Yukawa coupling matrix DF of the Standard Modelprovides the inverse line element for the finite geometry (AF,HF, DF) which displays thefine structure of space-time detected by the particles and forces discovered so far. For a longtime the structure of the finite geometry was introduced “by hand" following the trip inwardbound [228] in our understanding of matter and forces as foreseen by Newton:

The Attractions of Gravity, Magnetism and Electricity, reach to very sensible distances,and so have been observed by vulgar Eyes, but there may be others which reach to so smalldistances as hitherto escape Observation.

and adapting by a bottom up process the finite geometry to the particles and forces, withthe perfect fitting of the Higgs phenomenon and the see-saw mechanism with the geometricinterpretation. There was however no sign of an ending in this quest, nor any sensible justi-fication for the presence of the noncommutative finite structure. This state of affairs changedrecently [31,32] with the simultaneous quantization of the fundamental class in K-homologyand in K-theory. The K-homology fundamental class is represented by the Dirac operator.Representing the K-theory fundamental class, by requiring the use of the Feynman slash ofthe coordinates, explains the slight amount of non-commutativity of the finite algebra AFfrom Clifford algebras. From a purely geometric problem emerged the very same finite al-gebra AF = M2(H)⊕M4(C) which was the outcome of the bottom-up approach.One big plus of the noncommutative presentation is the economy in encoding. This economyis familiar in the written language where the order of letters is so important. Apart fromthe finite algebra AF the algebra of coordinates is generated by adjoining a “punctuationsymbol" Y of K-theoretic nature, which is a unitary with Y4 = 1. It is the noncommutativityof the obtained system which gets one outside the finite dimensional algebraic frameworkand generates the continuum. It allows one to write a higher analogue of the commutationrelations which quantizes the volume and encodes all 4-dimensional spin geometries.

4

Zeta and NCG

Let us explain in a simple manner why the problem of locating the zeros of the Riemannzeta function is intimately related to the adèle class space. In fact what matters for the zerosis not the Riemann zeta function itself but rather the ideal it generates among holomorphicfunctions of a suitable class. A key role as a generator of this ideal is played by the operationon functions f (u) of a real positive variable u which is defined by

f 7→ E( f ), E( f )(v) := ∑N×

f (nv) (1)

Indeed this operation is, in the variable log v, a sum of translations log v 7→ log v + log nby log n and thus after a suitable Fourier transform it becomes a product by the Fouriertransform of the sum of the Dirac masses δlog n, i.e. by the Riemann zeta function ∑ e−is log n =ζ(is). It is thus by no means mysterious that one obtains a spectral realization by consideringthe cokernel of the map E. But what is interesting is the geometric meaning of this fact andthe link with the explicit formulas. Thanks to the use of adèles in the thesis of Tate the abovesummation over N× relates to a summation over the group Q∗ of non-zero rational numbers.The natural adelic space on which the group Q∗ is acting is simply the space AQ of adèlesover Q and in order to focus on the Riemann zeta function one needs to restrict oneself tofunctions invariant under the maximal compact subgroup Z∗ of the group of idèle classeswhich acts canonically on the noncommutative space of adèle classes: Q∗\AQ. In fact letY be the quotient A f /Z∗ of the locally compact space of finite adèles by Z∗ and y ∈ Y theelement whose all components are 1. The closure F of the orbit N×y is compact and one hasfor q ∈ Q∗ the equivalence qy ∈ F ⇐⇒ q ∈ ±N×. It is this property which allows oneto replace the sum over N× involved in (1) by the summation over the associated group Q∗,simply by considering the function 1F ⊗ f on the product Y ×R. The space of adèle classesX = Q∗\AQ/Z∗ = Q∗\(Y×R) is noncommutative because the action of Q∗ on AQ is ergodicfor the Haar measure. This geometrization gives a trace formula interpretation of the explicitformulas of Riemann-Weil (see §6 below).The same space X appears naturally from a totally different perspective which is that ofGrothendieck toposes. Indeed the above action of N× by multiplication on the half line [0, ∞)naturally gives rise to a Grothendieck topos and it turns out that X is the space of points ofthis topos as shown in our recent work with C. Consani, [86,87]. Moreover the trace formulainterpretation of the explicit formulas of Riemann-Weil allows one to rewrite the Riemannzeta function in the Hasse-Weil form where the role of the frobenius is played by its analoguein characteristic one. The space X is the space of points of the arithmetic site defined over thesemifield Rmax

+ , the Galois group of Rmax+ over the Boolean semifield B is the multiplicative

group R∗+ which acts by the automorphisms x 7→ xλ and its action on X replaces the action ofthe frobenius on the points over Fq in finite characteristic. This topos theoretic interpretationyields a natural structure sheaf of tropical nature on the above scaling site which becomesa curve in a tropical sense and the two sides : (topos theoretic, tropical) on one hand and(adelic, noncommutative) on the other hand, are in the same relative position as the twosides of the class field theory isomorphism.

5

The theory of Grothendieck toposes is another crucial extension of the notion of space whichplays a major role in algebraic geometry and is liable to treat quotient spaces in a successful man-ner. The main difference of point of view between noncommutative geometry and the theory oftoposes is that noncommutative geometry makes fundamental use of the complex numbers ascoefficients and thus has a very close relation with the formalism of quantum mechanics. It takescomplex numbers as the preferred coefficient field and Hilbert space as the main stage. The birthof noncommutative geometry can be traced back to this night of June 1925 when, around threein the morning, W. Heisenberg while working alone in the Island of Helgoland in the northsea, discovered matrix mechanics and the noncommutativity of the phase space of microscopicmechanical systems. After the formulation of Heisenberg’s discovery as matrix mechanics, vonNeumann reformulated quantum mechanics using Hilbert space operators and went much fur-ther with Murray in identifying “subsystems" of a quantum system as “factorizations" of theunderlying Hilbert space H. They discovered unexpected factorizations which did not corre-spond to tensor product decompositions H = H1 ⊗ H2 and developed the theory of factorswhich they classified into three types. In my thesis I showed that factorsM admit a canonicaltime evolution, i.e. a canonical homomorphism

R δ−→ Out(M) = Aut(M)/Int(M)

by showing that the class of the modular automorphism σϕt in Out(M) does not depend on thechoice of the faithful normal stateϕ. This of course relied crucially on the Tomita-Takesaki theory[262] which constructs the map ϕ 7→ σϕt . The above uniqueness of the class of the modularautomorphism [41] drastically changed the status of the two invariants which I had previouslyintroduced in [39], [40] by making them computable. The kernel of δ, T(M) = Kerδ formsa subgroup of R, the periods of M, and many non-trivial non-closed subgroups appear in thisway. The fundamental invariant of factors is the modular spectrum S(M) of [39]. Its intersectionS(M) ∩ R∗+ is a closed subgroup of R∗+, [44], and this gave the subdivision of type III into typeIIIλ ⇐⇒ S(M) ∩ R∗+ = λZ. I showed moreover that the classification of factors of type IIIλ,λ ∈ [0, 1) is reduced to that of type II and automorphisms

M = N oθ Z, N type II∞, θ ∈ Aut(N)

In the case IIIλ, λ ∈ (0, 1), N is a factor and the automorphism θ ∈ Aut(N) is of module λi.e. it scales the trace by the factor λ. In the III0 case, N has a non-trivial center and, using therestriction of θ to the center, this gave a very rich invariant, a flow, and I used it in 1972, [42],to show the existence of hyperfinite non ITPFI factor. Only the case III1 remained open in mythesis [43] and was solved later by Takesaki [261] using crossed product by R∗+.At that point one could fear that the theory of factors would remain foreign from main streammathematics. This is not the case thanks to the following construction [46] of the von Neumannalgebra W(V, F) canonically associated to a foliated manifold V with foliation F. It is the vonNeumann algebra of random operators, i.e. of bounded families (T`) parameterized by leaves` and where each (T`) acts in the Hilbert space L2(`) of square integrable half-densities on theleaf. Modulo equality almost everywhere one gets a von Neumann algebra for the algebraicoperations given by pointwise addition and multiplication:

W(V, F) := (T`) | acting on L2(`)

6

The geometric examples lead naturally to the most exotic factors, the prototype being the Anosovfoliation of the sphere bundle of Riemann surfaces (see §6.2) whose von Neumann algebra is thehyperfinite factor of type III1.The above construction of the von Neumann algebra of a foliation only captures the “measuretheory" of the space of leaves. But such spaces inherit a much richer structure from the topo-logical and differential geometric structure of the ambient manifold V. This fact served as afundamental motivation for the development of noncommutative geometry and Figure 1 givesan overall picture of how the refined properties of leaf spaces and more general noncommutativespaces are treated in noncommutative geometry. One of the striking new features is the richnessof the measure theory and how it interacts with a priori foreign features such as the secondarycharacteristic classes of foliations. Cyclic cohomology plays a central role in noncommutative ge-ometry as the replacement of de-Rham cohomology [52,53]. One of its early striking applicationsobtained in [51] (following previous work of S. Hurder) using as a key tool the time derivativeof the cyclic cocycle which is the transverse fundamental class, is that the flow of weights4 of thevon Neumann algebra of a foliation with non-vanishing Godbillon-Vey class, admits an invari-ant probability measure. This implies immediately that the von Neumann algebra is of type IIIand exhibits the deep interplay between characteristic classes and ergodic theory of foliations.

Figure 1: Aspects of a noncommutative space

4The fundamental invariant of [43, 45]

7

2 New paradigm: spectral geometry

In the name “Noncommutative Geometry" the adjective “noncommutative" has an obvious al-gebraic meaning if one recalls that “commutative" is one of the leitmotivs of algebraic geometry.As mentioned earlier reducing the new theory to the extension of previously known notions be-yond the commutative case misses the point. In fact we shall now justify the term “Geometry"given to the field. The new paradigm for a geometric space is spectral. A spectral geometry isgiven by a triple (A,H, D) where A is an algebra of operators in the Hilbert spaceH and whereD is a self-adjoint operator, generally unbounded, acting in the same Hilbert spaceH. Intuitivelythe algebra A represents the functions on the geometric space and the operator D plays the roleof the inverse line element. It specifies how to measure distances by the formula

d(a, b) = Sup | f (a)− f (b)|, f such that ‖[D, f ]‖ ≤ 1. (2)

Given a spin compact Riemannian manifold M the associated spectral triple (A,H, D) is givenby the algebra A of functions on M acting in the Hilbert space H of L2 spinors and the Diracoperator D. The formula (2) gives back the geodesic distance in the usual Riemannian sense butmeasures distances in a Kantorovich dual manner. The word “spectral" in “spectral triple" (or“spectral geometry") has two origins, the straightforward one is the well-known Gelfand dual-ity between a (locally compact) space and its algebra of complex valued continuous functions(vanishing at infinity). The deeper one is the full reconstruction of the geometry from the spec-trum of the Dirac operator (the D in spectral triples) and the relative position in Hilbert space(the H of the spectral triple) of the two von Neumann algebras given on one hand by the weakclosure (double commutant) of the algebraA (theA of the spectral triple) and on the other handthe functions f (D) of the operator D. We refer to [76] for this point5. The great advantage ofthe new paradigm is that (2) continues to make sense for spaces which are no longer arcwiseconnected and works equally well for discrete or fractal spaces. Moreover the paradigm doesnot make use of the commutativity of A. When A is noncommutative the evaluation f (a) off ∈ A at a point a no longer makes sense but the formula (2) still makes sense and measures thedistance between states6 on A, i.e. positive linear forms φ : A → C, normalized by φ(1) = 1.The formula measures the distance between states as follows :

d(φ,ψ) = Sup |φ( f )−ψ( f )|, f such that ‖[D, f ]‖ ≤ 1. (3)

The new spectral paradigm of geometry extends the Riemannian paradigm and we explainedin [95] to which extent it allows one to go further in the query of Riemann on the geometricparadigm of space in the infinitely small. Here “space" is the space in which we live and is, afterall, the one which truly deserves the name of “geometry" independently of all the later devel-opments which somehow usurp this name. The new paradigm ensures that the line element

5One recovers the algebra of smooth functions from the von Neumann algebra of measurable ones as thosemeasurable functions which preserve the intersection of domains of powers of D. One then gets the topologicalspace as the Gelfand spectrum of the norm closure of the algebra of smooth functions, one also gets its smoothstructure, its Riemannian metric etc..

6The pure states are the extreme points of the space of states on A. In quantum mechanics they play the roleof the points of the classical phase space. They are endowed with the natural equivalence relation defined by theunitary equivalence of the associated irreducible representations. The space of equivalence classes of pure states istypically a non-commutative space except in the easy type I situation.

8

does encapsulate all the binding forces known so far and is flexible enough to follow the inwardbound quest of the geometric structure of space for microscopic distances [228]. The words ofRiemann are so carefully chosen and his vision so far-seeing that the reader will hopefully excuseour repeated use of this quotation:

Nun scheinen aber die empirischen Begriffe, in welchen die räumlichen Massbestimmungengegründet sind, der Begriff des festen Körpers und des Lichtstrahls, im Unendlichkleinen ihreGültigkeit zu verlieren; es ist also sehr wohl denkbar, dass die Massverhältnisse des Raumesim Unendlichkleinen den Voraussetzungen der Geometrie nicht gemäss sind, und dies würdeman in der That annehmen müssen, sobald sich dadurch die Erscheinungen auf einfachereWeise erklären liessen. Es muss also entweder das dem Raume zu Grunde liegende Wirk-liche eine discrete Mannigfaltigkeit bilden, oder der Grund der Massverhältnisse ausserhalb,in darauf wirkenden bindenen Kräften, gesucht werden.7

We also explained in [95] the deep roots of the notion of spectral geometry in pure mathematicsfrom the conceptual understanding of the notion of (smooth, compact, oriented and spin) man-ifold. The fundamental property of a “manifold" is not Poincaré duality in ordinary homologybut is Poincaré duality in the finer theory called KO-homology. The key result behind this, dueto D. Sullivan (see [209], epilogue), is that a PL-bundle is the same thing 8 as a spherical fibrationtogether with a KO-orientation. The fundamental class in KO-homology contains all the infor-mation about the Pontrjagin classes of the manifold and these are not at all determined by itshomotopy type: in the simply connected case only the signature class is fixed by the homotopytype.The second key step towards the notion of spectral geometry is the link with the Hilbert spaceand operator formalism which came as a byproduct of the work of Atiyah and Singer on theindex theorem. They understood that operators in Hilbert space provide the right realizationfor KO-homology cycles [2,253]. Their original idea was developed by Brown-Douglas-Fillmore[15], Voiculescu [271], Mishchenko [211] and acquired its definitive form in the work of Kasparovat the end of the 1970’s. The great new tool is bivariant Kasparov theory [173,174], and as far as K-homology cycles are concerned the right notion is already in Atiyah’s paper [2]: A K-homologycycle on a compact space X is given by a representation of the algebra C(X) (of continuousfunctions on X) in a Hilbert space H, together with a Fredholm operator F acting in the sameHilbert space fulfilling some simple compatibility condition (of commutation modulo compactoperators) with the action of C(X). One striking feature of this representation of K-homologycycles is that the definition does not make any use of the commutativity of the algebra C(X).At the beginning of the 1980’s, motivated by numerous examples of noncommutative spacesarising naturally in geometry from foliations and in particular the noncommutative torus T2

θ de-scribed in §2.1, I realized that specifying an unbounded representative of the Fredholm operator

7Now it seems that the empirical notions on which the metric determinations of space are founded, the notionof a solid body and of a ray of light, cease to be valid in the infinitely small. It is therefore quite conceivable thatthe metric relations of space in the infinitely small do not conform to the hypotheses of geometry; and we ought infact to assume this, if we can thereby obtain a simpler explanation of phenomena. Either therefore the reality whichunderlies space must be discrete, or we must seek the foundation of its metric relations outside it, in binding forceswhich act upon it.

8Modulo the usual “small-print" qualifications at the prime 2, [251]

9

gave the right framework for spectral geometry. The corresponding K-homology cycle9 onlyretains the stable information and is insensitive to deformations while the unbounded represen-tative encodes the metric aspect. These are the deep mathematical reasons which are the rootsof the notion of spectral triple. Since the year 2000 ( [68]) one important progress was obtainedin the reconstruction theorem [79] which gives an abstract characterization of the spectral triplesassociated to ordinary spin geometries. Moreover a whole subject has developed concerningnoncommutative manifolds and for lack of space we shall not cover it here but refer to the pa-pers [69–72, 99, 111, 112, 189]. We apologise for the brevity of our account of the topics coveredin the present survey and the reader will find a more complete treatment and bibliography inthe collection of surveys [35, 90, 104, 114, 117, 124, 134, 195, 199, 272, 276, 281] to be published bySpringer.

2.1 Geometry and the modular theory

Foliated manifolds provide a very rich source of examples of noncommutative spaces and oneis quite far from a full understanding of these spaces. In this section we discuss in an examplethe interaction of the new geometric paradigm with the modular theory (type III). The exampleis familiar from [68], it is the noncommutative torus T2

θ whose K-theory was computed in thebreakthrough paper of Pimsner and Voiculescu [232]. We first briefly recall the geometric pictureunderlying the noncommutative torus T2

θ and then describe the new development since year2000 which displays a highly non-trivial interaction between the new paradigm of geometryand the modular type III theory. The underlying geometric picture is the Kronecker foliationdy = θdx of the unit torus V = R2/Z2 as shown in Figure 2. If one considers the restrictionof the foliation to a neighborhood of a transversal as shown in Figure 2 the structure is that ofthe product of a circle S1 by an interval and the leaf space is S1. But for the whole foliationof the torus and irrational θ the leaf space is a non-commutative space. It corresponds to theidentification of those points on S1 which are on the same orbit of the irrational rotation of angleθ. At the algebraic level the foliation algebra is the same10 as the crossed product C(T2

θ) :=C(S1)oθ Z of the transversal by the irrational rotation of angle θ.The differential geometry of the noncommutative torus T2

θ was first defined and investigated inC.R. Acad. Sc. Paris Note of 1980 (cf. [48]) “C∗ algèbres et géométrie différentielle”. While at theC∗-algebra level the presentation of the algebra C(T2

θ) of continuous functions on T2θ is by means

of two unitaries U, V such thatVU = e2π iθ UV,

the smooth functions on T2θ are given by the dense subalgebra C∞(T2

θ) ⊂ C(T2θ) of all series

x = ∑m,n∈Z

am,n UmVn,

such that the sequence of complex coefficients (am,n) is rapidly decaying in the sense that

supm,n∈Z

|am,n|(1 + |n|+ |m|)k < ∞,

9In [4], S. Baaj and P. Julg have extended the unbounded construction to the bivariant case. The most efficientaxioms have been found by M. Hilsum in [167] which allows to handle the case of symmetric non self-adjointoperators.

10up to Morita equivalence, the algebraic notion has been adapted to C∗-algebras by M. Rieffel in [242]

10

Figure 2: Kronecker foliation dy = θdx

for any non-negative integer k. Thus one has a simple explicit description of the generic elementof C∞(T2

θ) (while there is no such explicit description for C(T2θ)). At the geometric level one

has the smooth groupoid G = S1 oθ Z which is the reduction of the holonomy groupoid of thefoliation to the transversal. The elements of G are described equivalently as pairs of points ofS1 which are on the same orbit of the irrational rotation or as pairs (u, n) ∈ S1 × Z. The algebrais the convolution algebra of G. Any other transversal L gives canonically a module C∞(T2

θ , L)over the convolution algebra of G and the main discovery of [48] can be summarized as follows:

1. The Schwartz space S(R) is a finite projective module over C∞(T2θ) for the action of the

generators U, V on Schwartz functionsξ(s) is given by translation of the variable s 7→ s−θand multiplication by e2π is.

2. The Murray-von Neumann dimension of S(R) is θ.

3. The module S(R) is naturally endowed with a connection of constant curvature and theproduct of the constant curvature by the Murray-von Neumann dimension is an integer.

It is this integrality of the product of the constant curvature by the (irrational) Murray-vonNeumann dimension which was the starting point of noncommutative differential geometry(cf. [52, 53]). It holds irrespective of the transversal and the description of the general finite pro-jective modules thus obtained was done in [48]11. The differential geometry as well as pseudo-differential operator calculus of the noncommutative two torus T2

θ were first developed in [48].As in the case of an ordinary torus the conformal structure is specified by a complex modu-lus τ ∈ C with =(τ) > 0. The flat spectral geometry (A,H, D0) is constructed using the twoderivations δ1, δ2 which generate the action by rotation of the generators U, V of the algebraA = C∞(T2

θ). To obtain a curved geometry (A,H, D) from the flat one (A,H, D0), one in-troduces (cf. [37], [78]) a noncommutative Weyl conformal factor (or dilaton), which changes

11and was often misattributed as in [202]

11

the metric by modifying the noncommutative volume form while keeping the same conformalstructure. Both notions of volume form and of conformal structure are well understood in thegeneral case (cf. [60, §VI]). The new and crucial ingredient which has no classical analogue is themodular operator ∆ of the non-tracial weightϕ(a) =ϕ0(ae−h) associated to the dilaton h.

In noncommutative geometry the local geometric invariants such as the Riemannian curvatureare extracted from the spectral triple using the functionals defined by the coefficients of heatkernel expansion

Tr(ae−tD2) ∼t0 ∑

n≥0an(a, D2)t

−d+n2 , a ∈ A,

where d is the dimension of the geometry. Equivalently, one may consider special values ofthe corresponding zeta functions. Thus, the local curvature of the geometry is detected by thehigh frequency behavior of the spectrum of D coupled with the action of the algebra A in H.In the case of T2

θ the computation of the value at s = 0 of the zeta function Tr(a|D|−2s) for the2-dimensional curved geometry associated to the dilaton h, or equivalently of the coefficienta2(a, D2) of the heat expansion was started in the late 1980’s (cf. [37]), and the proof of the ana-logue of the Gauss–Bonnet formula was published in [78]. It was subsequently extended in [119]to the case of arbitrary values of the complex modulus τ (one had τ = i in [78]). In these papersonly the total integral of the curvature was needed, and this allowed one to make simplifica-tions under the trace which are no longer possible when one wants to fully compute the localexpression for the functional a ∈ A 7→ a2(a, D2).

The complete calculation of a2(a, D2) was actually performed in our joint work with H. Moscoviciin [80], and independently by F. Fathizadeh and M. Khalkhali in [120]. The spectral geometry ofnoncommutative tori is a key testground for new ideas and basic contributions have been donein [78,80,121,122,124,126,127,168,177,193,194,197,235], with beautiful surveys by M. Lesch andH. Moscovici [195] and by F. Fathizadeh and M. Khalkhali [124] to which we refer for a morecomplete account and bibliography. In this topic the hard direct calculations alternate with theirconceptual understanding and the main result of [80] is the conceptual explanation for the com-plicated formulas. The explanation is obtained by expressing in terms of a closed formula theRay-Singer log-determinant of D2. The gradient of the log-determinant functional yields in turna local curvature formula, which arises as a sum of two terms, each involving a function in themodular operator, of one and respectively two variables. Computing the gradient in two dif-ferent ways leads to the proof of a deep internal consistency relation between these two distinctconstituents. This relation was checked successfully and gave confidence in the pertinence ofthe whole framework. Moreover it elucidates the meaning of the intricate two operator-variablefunction which we now briefly describe. As in the case of the standard torus viewed as a com-plex curve, the total Laplacian associated to such a spectral triple splits into two components,one 4ϕ on functions and the other 4(0,1)

ϕ on (0, 1)-forms, the two operators being isospectraloutside zero. The corresponding curvature formulas involve second order (outer) derivatives ofthe Weyl factor. For the Laplacian4ϕ the result is of the form

a2(a,4ϕ) = −π

2τ2ϕ0(a

(K0(∇)(4(h)) +

12

H0(∇1,∇2)(<(h))

, (4)

12

where ∇ = log ∆ is the inner derivation implemented by −h,

4(h) = δ21(h) + 2<(τ)δ1δ2(h) + |τ |2δ2

2(h),

< is the Dirichlet quadratic form

<(`) := (δ1(`))2 +<(τ) (δ1(`)δ2(`) + δ2(`)δ1(`)) + |τ |2(δ2(`))

2 ,

and∇i, i = 1, 2, signifies that∇ is acting on the ith factor. The operators K0(∇) and H0(∇1,∇2)are new ingredients, whose occurrence has no classical analogue and is a vivid manifestation ofthe non-unimodular nature of the curved geometry of the noncommutative 2-torus with Weylfactor. The functions K0(u) and H0(u, v) by which the modular derivatives act seem at first verycomplicated, and of course beg for a conceptual understanding. This was obtained in [80] wheredenoting K0(s) = 4 sinh(s/2)

s K0(s) and H0(s, t) = 4 sinh((s+t)/2)s+t H0(s, t), we give an abstract proof

of the functional relation

−12

H0(s1, s2) =K0(s2)− K0(s1)

s1 + s2+

K0(s1 + s2)− K0(s2)

s1− K0(s1 + s2)− K0(s1)

s2(5)

which determines the whole structure in terms of the function K0 which turns out to be (up to thefactor 1

8 ) the generating function of the Bernoulli numbers, i.e. one has 18 K0(u) = ∑

∞1

B2n(2n)! u

2n−2 .Among the recent developments are the full understanding of the interplay of the geometry withMorita equivalence (a purely noncommutative feature) in [193] and a new hard computation in[92]. This latter result gives the formulas for the a4 term for the noncommutative torus and, as aconsequence, allows one to compute this term for the noncommutative 4-tori which are productsof two 2-tori. We managed in [92] to derive abstractly and check the analogues of the abovefunctional relations, but the formulas remain very complicated and mysterious, thus beggingfor a better understanding. One remarkable fact is that the noncommutative 4-tori which areproducts of two 2-tori, are no longer conformally flat so that the computation of the a4 term forthem involves a more complicated modular structure which is given by a two parameter groupof automorphisms. This provides a first hint in the following program:

The long range goal in exploring the spectral geometry of noncommutative tori is to arrive atthe formulation of the full fledged modular theory involving the full Jacobian matrix ratherthan its determinant, as suggested long ago by the transverse geometry of foliations in [51]where the reduction from type III to type II was done for the full jacobian. The hypoelliptictheory was used in [65] to obtain the transverse geometry at the type II level. The first step ofthe general theory is to adapt the twist defined in [77] to a notion involving the full Jacobian.

2.2 Inner fluctuations of the metric

In the same way as inner automorphisms enhance the group of diffeomorphisms, they providespecial deformations of the metric for a spectral geometry (A,H, D) which correspond to thegauge fields of non-abelian gauge theories. In our joint work with A. Chamseddine and W.

13

van Suijlekom [30], we obtained a conceptual understanding of the role of the gauge bosons inphysics as the inner fluctuations of the metric. I will describe this result here in a non-technicalmanner.Ignoring the important nuance coming from the real structure J, the inner fluctuations of themetric were first defined as the transformation

D 7→ D + A, A = ∑ a j[D, b j], a j, b j ∈ A, A = A∗

which imitates the way classical gauge bosons appear as matrix-valued one-forms in the usualframework. The really important facts were that the spectral action applied to D + A deliversthe Einstein-Yang-Mills action which combines gravity with matter in a natural manner, and thatthe gauge invariance becomes transparent at this level since an inner fluctuation coming froma gauge potential of the form A = u[D, u∗] where u is a unitary element (i.e. uu∗ = u∗u = 1)simply results in a unitary conjugation D 7→ uDu∗ which does not change the spectral action.What we discovered in our joint work with A. Chamseddine and W. van Suijlekom [30] is thatthe inner fluctuations arise in fact from the action on metrics (i.e. the D) of a canonical semigroupPert(A) which only depends upon the algebraA and extends the unitary group. The semigroupis defined as the self-conjugate elements:

Pert(A) := A = ∑ a j ⊗ bopj ∈ A⊗A

op |∑ a jb j = 1, θ(A) = A

where θ is the antilinear automorphism of the algebra A⊗Aop given by

θ : ∑ a j ⊗ bopj 7→∑ b∗j ⊗ a∗op

j .

The composition law in Pert(A) is the product in the algebra A⊗Aop. The action of this semi-group Pert(A) on the metrics is given, for A = ∑ a j ⊗ bop

j by

D 7→ D′ =AD = ∑ a jDb j.

Moreover, the transitivity of inner fluctuations which is a key feature since one wants that aninner fluctuation of an inner fluctuation is itself an inner fluctuation now results from the semi-group structure according to the equality: A′(AD) =(A′A) D. We refer to [30] for the full treatment.Inner fluctuations have been adapted in the twisted case in [187, 188].

2.3 The spectral action and standard model coupled to gravity

The new spectral paradigm of geometry, because of its flexibility, provides the needed tool torefine our understanding of the structure of physical space in the small and to “seek the foundationof its metric relations outside it, in binding forces which act upon it". The main idea, described indetails in [95], is that the line element now embodies not only the force of gravity but all theknown forces, electroweak and strong, appear from the spectral action and the inner fluctuationsof the metric. This provides a completely new perspective on the geometric interpretation of thedetailed structure of the Standard model and of the Brout-Englert-Higgs mechanism. One getsthe following simple mental picture for the appearance of the scalar field : imagine that thespace under consideration is two sided like a sheet S of paper in two dimensions. Then when

14

differentiating a function on such a space one may restrict the function to either side S± of thesheet and thus obtain two spin one fields. But one may also take the finite difference f (s+)−f (s−) of the function at the related points of the two sides. The corresponding field is clearlyinsensitive to local rotations and is a scalar spin zero field. This, in a nutshell, is how the Brout-Englert-Higgs field appears geometrically once one accepts that there is a “fine structure" whichis revealed by the detailed structure of the standard model of matter and forces. This allows oneto uncover the geometric meaning of the Lagrangian of gravity coupled to the standard model.This extremely complicated Lagrangian is obtained from the spectral action developed in ourjoint work with A. Chamseddine [19] which is the only natural additive spectral invariant ofa noncommutative geometry. In order to comply with Riemann’s requirement that the inverseline element D embodies the forces of nature, it is evidently important that we do not separateartificially the gravitational part from the gauge part, and that D encapsulates both forces in aunified manner. In the traditional geometrization of physics, the gravitational part specifies themetric while the gauge part corresponds to a connection on a principal bundle. In the NCGframework, D encapsulates both forces in a unified manner and the gauge bosons appear asinner fluctuations of the metric but form an inseparable part of the latter. The noncommutativegeometry dictated by physics is the product of the ordinary 4-dimensional continuum by a finitenoncommutative geometry (AF,HF, DF) which appears naturally from the classification of finitegeometries of KO-dimension equal to 6 modulo 8 (cf. [21,23]). The finite dimensional algebraAFwhich appeared is of the form

AF = C+ ⊕ C−, C+ = M2(H), C− = M4(C).

The agreement of the mathematical formalism of spectral geometry and all its subtleties such asthe periodicity of period 8 of the KO-theory, might still be accidental but I personally got con-vinced of the pertinence of this approach when recovering [21] the see-saw mechanism (whichwas dictated by the pure math calculation of the model) while I was unaware of its key physicsrole to provide very small non-zero masses to the neutrinos, and of how it is “put by hand" inthe standard model. The low Higgs mass then came in 2012 as a possible flaw of the model, butin [28] we proved the compatibility of the model with the measured value of the Higgs mass,due to the role in the renormalization of the scalar field which was already present in [25] buthad been ignored thinking that it would not affect the running of the self-coupling of the Higgs.In all the previous developments we had followed the “bottom-up" approach, i.e. we uncoveredthe details of the finite noncommutative geometry (AF,HF, DF) from the experimental informa-tion contained in the standard model coupled to gravitation. In 2014, in collaboration with A.Chamseddine and S. Mukhanov [31, 32] we were investigating the purely geometric problem ofencoding 4-manifolds in the most economical manner in the spectral formalism. Our problemhad no a priori link with the standard model of particle and forces and the idea was to treatthe coordinates in the same way as the momenta are assembled together in a single operatorusing the gamma matrices. The great surprise was that this investigation gave the conceptualexplanation of the finite noncommutative geometry from Clifford algebras! This is described indetails in [95] to which we refer. What we obtained is a higher form of the Heisenberg commu-tation relations between p and q, whose irreducible Hilbert space representations correspond to4-dimensional spin geometries. The role of p is played by the Dirac operator and the role of q bythe Feynman slash of coordinates using Clifford algebras. The proof that all spin geometries are

15

obtained relies on deep results of immersion theory and ramified coverings of the sphere. Thevolume of the 4-dimensional geometry is automatically quantized by the index theorem and thespectral model, taking into account the inner automorphisms due to the noncommutative natureof the Clifford algebras, gives Einstein gravity coupled with the slight extension of the standardmodel which is a Pati-Salam model. This model was shown in [29, 33] to yield unification ofcoupling constants. We refer to the survey [35] by A. Chamseddine and W. van Suijlekom for anexcellent account of the whole story of the evolution of this theory from the early days to now.The dictionary between the physics terminology and the fine structure of the geometry is of thefollowing form:

Standard Model Spectral Model

Higgs Boson Inner metric(0,1)

Gauge bosons Inner metric(1,0)

Fermion masses u,ν Dirac(0,1) in ↑

CKM matrix, Masses down Dirac(0,1) in (↓ 3)

Lepton mixing, Masses leptons e Dirac(0,1) in (↓ 1)

Majorana mass matrix Dirac(0,1) on ER ⊕ JFER

Gauge couplings Fixed at unification

Higgs scattering parameter Fixed at unification

Tadpole constant −µ20 |H|2

2.4 Dimension 4

In our work with A. Chamseddine and S. Mukhanov [31,32], the dimension 4 plays a special rolefor the following reason. In order to encode a manifold M one needs to construct a pair of mapsφ,ψ from M to the sphere (of the same dimension d) in such a way that the sum of the pullbacksof the volume form of the (round) sphere vanishes nowhere on M. This problem is easy to solvein dimension 2 and 3 because one first writes M as a ramified cover φ : M → Sd of the sphereand one pre-composes φ with a diffeomorphism f of M such that Σ ∩ f (Σ) = ∅ where Σ is thesubset whereφ is ramified. This subset is of codimension 2 in M and there is no difficulty to findf because (d− 2) + (d− 2) < d for d < 4. It is worth mentioning that the 2 for the codimension

16

of Σ is easy to understand from complex analysis: For an arbitrary smooth map φ : M → Sd

the Jacobian will vanish on a codimension 1 subset, but in one dimensional complex analysis theJacobian is a sum of squares and its vanishing means the vanishing of the derivative which givestwo conditions rather than one. Note also in this respect that quaternions do not help.12 Thusdimension d = 4 is the critical dimension for the above existence problem of the pairφ,ψ. Sucha pair does not always exist13 but as shown in [31, 32], it always exist for spin manifolds whichis the relevant case for us.The higher form of the Heisenberg commutation relation mentioned above involves in dimen-sion d the power d of the commutator [D, Z] of the Dirac operator with the operator Z whichis constructed (using the real structure J, see [95]) from the coordinates. We shall now explainbriefly how this fits perfectly with the framework of D. Sullivan on Sobolev manifolds, i.e. ofmanifolds of dimension d where the pseudo-group underlying the atlas preserves continuousfunctions with one derivative in Ld. He discovered the intriguing special role of dimension 4in this respect. He showed in [260] that topological manifolds in dimensions > 5 admit bi-lipschitz coordinates and these are unique up to small perturbations, moreover existence anduniqueness also holds for Sobolev structures: one derivative in Ld. A stronger result was knownclassically for dimensions 1,2 and 3. There the topology controls the smooth structure up tosmall deformation. In dimension 4, he proved with S. Donaldson in [109] that for manifoldswith coordinate atlases related by the pseudo-group preserving continuous functions with onederivative in L4 it is possible to develop the SU(2) gauge theory and the famous Donaldsoninvariants. Thus in dimension 4 the Sobolev manifolds behave like the smooth ones as opposedto Freedman’s abundant topological manifolds.14 The obvious question then is to which extentthe higher Heisenberg equation of [31,32] singles out the Sobolev manifolds as the relevant onesfor the functional integral involving the spectral action.

2.5 Relation with quantum gravity

Our approach on the geometry of space-time is not concerned with quantum gravity but it ad-dresses a more basic related question which is to understand “why gravity coupled to the stan-dard model", which we view as a preliminary. The starting point of our approach is an extensionof Riemannian geometry beyond its classical domain which provides the needed flexibility inorder to answer the querry of Riemann in his inaugural lecture. The modification of the ge-ometric paradigm comes from the confluence of the abstract understanding of the notion ofmanifold from its fundamental class in KO-homology with the advent of the formalism of quan-tum mechanics. This confluence comes from the realization of cycles in KO-homology fromrepresentations in Hilbert space. The final touch on the understanding of the geometric rea-son behind gravity coupled to the standard model, came from the simultaneous quantization ofthe fundamental class in KO-homology and its dual in KO-theory which gave rise to the higherHeisenberg relation. So far we remain at the level of first quantization and the problem of secondquantization is open. A reason why this issue cannot be ignored is that the quantum corrections

12As an exercice one can compute the Jacobian of the power map q 7→ qn, q ∈ H, and show that it vanishes on acodimension 1 subset. For instance for n = 3 the Jacobian is given by 9

(3a2 − b2 − c2 − d2)2 (a2 + b2 + c2 + d2)2.

13It does not exist for M = P2(C).14for which any modulus of continuity whatsoever is not known.

17

to the line element as explained in Section 3.2.3 below (and shown in Figure 3) are ony the tip ofthe iceberg since the dressing also occurs for all the n-point functions for Fermions while Figure3 only involves the two point function. One possible strategy to pass to this second quantizedhigher level of geometry is to try to give substance to the proposal we did in [95]:

The duality between KO-homology and KO-theory is the origin of the higher Heisenbergrelation. As already mentioned in [55], algebraic K-theory, which is a vast refinement oftopological K-theory, is begging for the development of a dual theory and one should expectprofound relations between this dual theory and the theory of interacting quanta of geometry.As a concrete point of departure, note that the deepest results on the topology of diffeomor-phism groups of manifolds are given by the Waldhausen algebraic K-theory of spaces and werefer to [113] for a unifying picture of algebraic K-theory.

The role of the second quantization of Fermions in “ Entropy and the spectral action" [34]which interprets the spectral action as an entropy is a first step towards building from quan-tum field theory a second quantized version of spectral geometry.

Note also that an analogous result to [34] has been obtained in [110] for the bosonic case.

3 New tool: Quantized calculus

A key tool of noncommutative geometry is the quantized calculus. Its origin is the notion of realvariable provided by the formalism of quantum mechanics in terms of self-adjoint operators inHilbert space. One basic very striking fact is that if φ : R → R is an arbitrary Borel functionand H a self-adjoint operator in Hilbert space, thenφ(H) makes sense even thoughφ can have acompletely different definition in various parts of R. It is this fact which reveals in which sensethe formalism of quantum mechanics is compatible with the naive notion of real variable. It issuperior to the classical notion of random variable because it models discrete variables with fi-nite multiplicity as coexisting with continuous variables while a discrete random variable withfinite multiplicity only exists on a countable sample space. The price one pays for this coex-istence is non-commutativity since the commutant of a self-adjoint operator H with countablespectrum of multiplicity 1 is the algebra of Borel functions φ(H) and they all have countablerange and atomic spectral measure. This implies in particular that the infinitesimal variablesof the theory, which are modeled by compact operators, cannot commute with the continuousvariables. We refer to [95] for a detailed discussion of this point. We now display the dictionarywhich translates the classical geometric notions into their noncommutative analogue.

3.1 Quantum formalism and variables, the dictionary

We refer to Chapter IV of [60] for a detailed description of the calculus and many concrete exam-ples.

18

Real variable f : X → R Self-adjoint operator H in Hilbert spaceRange f (X) ⊂ R of the variable Spectrum of the operator H

Compositionφ f ,φ measurable Measurable functionsφ(H) of self-adjoint operatorsBounded complex variable Z Bounded operator A in Hilbert space

Infinitesimal variable dx Compact operator TInfinitesimal of orderα > 0 Characteristic values µn(T) = O(n−α) for n→ ∞

Algebraic operations on functions Algebra of operators in Hilbert space

Integral of function∫

f (x)dx∫−T = coefficient of log(Λ) in TrΛ(T)

Line element ds2 = gµνdxµdxν ds = •−−−−−• : Fermion propagator D−1

d(a, b) = Inf∫γ

√gµ ν dxµ dxν d(µ,ν) = Sup |µ(A)− ν(A)|, | ‖[D, A]‖ ≤ 1.

Riemannian geometry (X, ds2) Spectral geometry (A,H, D)Curvature invariants Asymptotic expansion of spectral action

Gauge theory Inner fluctuations of the metricWeyl factor perturbation D 7→ ρDρ

Conformal Geometry Fredholm module (A,H, F), F2 = 1.Perturbation by Beltrami differential F 7→ (aF + b)(bF + a)−1, a = (1−µ2)−1/2, b = µa

Distributional derivative Quantized differential dZ := [F, Z]

Measure of conformal weight p f 7→∫− f (Z)|dZ|p

While the first lines of the dictionary are standard in quantum mechanics, they become moreand more involved as one goes down the list. We refer to the survey of Dan Voiculescu [272] forhis deep work on commutants modulo normed ideals and the modulus of quasicentral approxi-mation for an n-tuple of operators with respect to a normed ideal which provides a fundamentaltool, used crucially in the reconstruction theorem [79]. We did include the inner fluctuationsin the dictionary as corresponding to the gauge theories and the precise relation is explainedin §2.2. Note also in this respect that the alteration of the metric by a Weyl factor is of a simi-lar nature as the inner fluctuations. One understands from [60], how the Beltrami differentialsact on conformal structures when the latter are encoded in bounded form, but the problem ofencoding a general change of metric is yet open. After listing below the various entries of thedictionary we shall concentrate on the highly non-trivial part which is integration in the form ofthe Dixmier trace and the Wodzicki residue.

3.2 The principle of locality in NCG

It is worth describing a fundamental principle which has emerged over the years. It concerns thenotion of “locality" in noncommutative geometry. The notion of locality is built in for topologicalspaces or more generally for toposes. It is also an essential notion in modern physics.Locality also plays a key role in noncommutative geometry and makes sense in the above frame-work of the quantized calculus, but it acquires another meaning, more subtle than the straight-forward topological one. The idea comes from the way the Fourier transform translates the local

19

behavior of functions on a space in terms of the decay at ∞ of their Fourier coefficients. Therelevant properties of the “local functionals" which makes them “local" is translated in Fourierby their dependance on the asymptotic behavior of the Fourier coefficients.This is familiar in particle physics where it is the ultraviolet behavior which betrays the finelocal features, in particular through the ultraviolet divergencies. More specifically, in QuantumField Theory, the counterterms which need to be added along the way in the renormalizationprocess as corrections to the initial Lagrangian appear from the ultraviolet divergencies and areautomatically “local". One merit of the divergencies is to be insensitive to bounded perturbationsof the momentum space behavior. The fundamental ones appear either as the coefficient of alogarithmic divergency when a cutoff parameter Λ (with the dimension of an energy) is movedto ∞ or, in the Dim-Reg regularization scheme, as a residue.In noncommutative geometry the above two ways of isolating the relevant term in divergencieshave been turned into two key tools which allow one to construct “local functionals". These are

I The Dixmier trace

I The Wodzicki residue

These tools are functionals on suitable classes of operators, and their role is to filter out theirrelevant details. In some sense these filters wipe out the quantum details and give a semi-classical picture.

3.2.1 Dixmier trace and fractals

The integration in noncommutative geometry needs to combine together two properties whichseem contradictory: being a positive trace (in order to allow for cyclic permutations under thefunctional) and vanishing on infinitesimals of high enough order (order > 1). This latter prop-erty together with the approximation from below of positive compact operators by finite rankones entails that the integration cannot commute with increasing limits and is non-normal asa functional. Such a functional was discovered by J. Dixmier in 1966, [108]. One might beskeptical at first, on the use of such functionals for the reason which was invoked in the in-troduction concerning the non-constructive aspect of some proofs for the cardinality of naturalnon-commutative spaces. Fortunately there is a perfect answer to this objection both at the ab-stract general level as well as in practice. The ingredient needed to construct the non-normalfunctional is a limiting process limω on bounded sequences of real numbers. It is required to belinear and positive (i.e. limω((an)) ≥ 0 if an ≥ 0 ∀n) and to vanish for sequences which tendto 0 at infinity. The book of Hardy [149] on divergent series shows the role of iterated Cesaromeans and of such limiting processes in many questions of number theory related to Tauberiantheorems. If one would require the limiting process to be multiplicative it would correspond toan ultrafilter and would easily be shown to break the rule of measurability which we imposedfrom the start in the introduction. At this point a truly remarkable result of G. Mokobodzki [207]saves the day. He managed to show, using the continuum hypothesis, the existence of a limiting

20

process which besides the above conditions is universally measurable and fulfills

limω

(∫a(α)dµ(α)

)=∫

limω

(a(α))dµ(α) (6)

for any bounded measurable family a(α) of sequences of real numbers. In other words if oneuses this medial limit of Mokobodski one no longer needs to worry about the measurabilityproblems of the limit and one can permute it freely with integrals! One might object that themedial limit is not explicitly constructed since its existence relies on the continuum hypothesisbut the medial limit is to be used as a tool and there are very few places in mathematics whereone can use the power of subtle independent axioms such as the continuum hypothesis. We referto the article of Paul Cohen [38] for the interplay between axioms of set theory and mathematicalproof. He explains how certain proofs might require the use of stronger axioms even though theoriginal question is formulated in simple number theoretic language. But the objection that theDixmier trace is “non-constructively defined" does not apply in practice, the main point therebeing that this issue does not arise when there is convergence. Since year 2000 a lot of progesshas been done and for instance the construction of Dixmier traces and the proof of its importantproperties have been extended to the type II situation in [10]. Moreover a very powerful schoolwith great analytical skill has developed around F. Sukochev and D. Zanin and their book withS. Lord [200]. With them we have undertaken the task of giving complete proofs of severalimportant results of [60] which were only briefly sketched there. In [93] we give complete proofsof Theorem 17 of Chapter IV of [60] and in [94], with E. McDonald, the proof of the result onJulia sets announced in [60] page 23. At the technical level an important hypothesis plays a roleconcerning the invariance of the limiting process under rescaling. It is simpler to state it for alimiting process ω on bounded continuous functions f (t) of t ∈ [0, ∞], where limω f neglectsall functions which tend to 0 at ∞. One letsα > 0 and defines the notation

limωα

f := limω

g, g(u) := f (uα) , ∀u ≥ 0 (7)

Then the relevant additional requirement is power invariance, i.e. the equality of limωα withlimω. It can always be achieved due to the amenability of solvable groups and is technicallyvery useful as shown in §7 of [93]. We refer to [199] for a survey of the many key developmentsaround the singular traces whose prototype examples were uncovered by Dixmier.

3.2.2 Wodzicki residue

The Wodzicki residue [275] is a remarkable discovery which considerably enhances the powerof the quantized calculus because it allows one, under suitable assumptions, to give a soundmeaning to the integral of infinitesimals whose order is strictly less than one, while it agrees(up to normalization) with the Dixmier trace for infinitesimals of order ≥ 1. Just as the Dixmiertrace, it vanishes for infinitesimals of order > 1 but the extension of its domain is a considerableimprovement which has no classical meaning. A striking example is given by the following newline in the dictionary (up to normalization):

21

Scalar curvature of manifold of dimension 4 Functional A 3 f 7→∫− f ds2 =

∫−R f ds4

The Wodzicki residue was extended to the framework of spectral triples with simple dimensionspectrum in our joint work [61] with Henri Moscovici where we obtained a local expression (lo-cal in the above sense) for the cyclic cocycle which computes the index pairing (see §4). There is arevealing example [73] where the computation of this local formula displays well the simplifica-tions which occur when on manipulates infinitesimals (in the operator theoretic sense) modulothose of high order. The explicit form of the operators provided by the quantum group is quiteinvolved but when one works modulo higher order, all sorts of irrelevant details are wiped out.The noncommutative residue was extended by R. Ponge to Heisenberg manifolds in [233] whoobtained new invariants of CR-geometry in a series of papers [234].

3.2.3 Locality and spectral action

Another striking result obtained since the year 2000, was obtained in our joint work with AliChamseddine [20]. It gives a general local formula for the scale independent terms ζD+A(0)−ζD(0) in the spectral action, under the perturbation of D given by the addition of a gauge poten-tial A. Here (A,H, D) is a spectral triple with simple dimension spectrum consisting of positiveintegers and the general formula has finitely many terms and gives

ζD+A(0)−ζD(0) = −∫− log(1 + A D−1) = ∑

n

(−1)n

n

∫− (A D−1)n. (8)

In dimension 4 this formula reduces to

ζD+A(0)−ζD(0) = −∫−AD−1 +

12

∫−(AD−1)2

− 13

∫−(AD−1)3 +

14

∫−(AD−1)4.

(9)

and under the hypothesis of vanishing of the tadpole term, the variation (9) is the sum of a Yang–Mills action and a Chern–Simons action relative to a cyclic 3-cocycle on the algebra A. Thisresult should be seen as a first step in the long term program of performing the renormalizationtechnique entirely in the framework of spectral triples. We refer to [256–258] for fundamentalsteps in this direction. After the initial breakthrough discovery in our joint work with D. Kreimerof the conceptual meaning of perturbative renormalization as the Birkhoff decomposition, weunveiled in our joint work with M. Marcolli [75] Theorem 1.106 and Corollary 1.107, an affinegroup scheme which acts on the coupling constants of physical theories as the fundamentalambiguity inherent to the renormalization process.At the conceptual level one can expect that since the local functionals filter out the quantum de-tails, and give a semiclassical picture of the quantum reality, the knowledge of this semiclassicalpicture only allows to guess the quantum reality. In fact the interpretation of the line element

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ds as the fermion propagator (in the Euclidean signature) already gives a hint of the modifica-tions of geometry due to the quantum corrections. Indeed these corrections “dress" the Fermionpropagator as shown in Figure 3 :

ℏ

Figure 3: Dressed Propagator

This dressing of the propagator should be interpreted physically as quantum corrections to thegeometry, i.e. the replacement of the initial “classical" spectral triple by a new one in whichthe inverse line element undergoes the same dressing. In perturbative renormalization, thiscan only be expressed in the form of an asymptotic series in powers of h which modifies D bymultiplication by such a series of functions of the operator D2/Λ2. This suggests that in realitythe geometry of space-time is far more interrelated to the quantum and should not be fixed andfrozen to its semi-classical limit when one performs renormalization. The geometric paradigmof noncommutative geometry is well suited for this interaction with the quantum world. Werefer to the books [259] of W. van Suijlekom and [115] of M. Eckstein and B. Iochum for completeaccounts of the above topics on the spectral action.

4 Cyclic cohomology

The quantized calculus briefly presented in the dictionary replaces the differential df of a func-tion by the commutator [F, f ] which is an operator in Hilbert space where the self-adjoint opera-tor F acts and fulfills F2 = 1. One can then develop the calculus using products of k one forms asforms of degree k and the graded commutator with F as the differential. One has d2 = 0 becauseF2 = 1. Moreover under good summability conditions the higher forms will be trace class oper-ators and puting things together one will get an abstract cycle of degree n on the algebra A i.e. agraded differential algebra Ω which has A in degree zero and is equipped with a closed gradedtrace τ : Ωn → C. The integer n is only relevant by its parity and by being large enough so thatall n forms are trace class operators. In 1981 (see the Oberwolfach 1981 report [49] on “Spectralsequence and homology of currents for operator algebras”) I discovered cyclic cohomology, theSBI sequence and the related spectral sequence from the above framework. In particular the pe-riodicity operator S in cyclic cohomology was imposed by the ability to replace n by n + 2 in thisframework. Since then cyclic cohomology plays a central role in noncommutative geometry asthe replacement of de Rham cohomology. Moreover and parallel to what happens in differentialgeometry the integral pairing between K-homology and K-theory is computed by the pairing ofthe Chern characters in cyclic theory according to the diagram:

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K− Theory

Ch∗

oo // K−Homology

Ch∗

HC∗ oo // HC∗

(10)

The Chern character in K-homology is defined as explained above from the quantized calcu-lus associated to a Fredholm module. However the obtained formula is non-local and difficultto compute explicitly. In [60] Theorem 8 of Chapter IV, I stated an explicit formula using theDixmier trace for the Hochschild class of the Chern character, i.e. its image under the forgetfulmap I from cyclic to Hochschild cohomology. Following my unpublished notes a proof wasincluded in the book [125]. The result was then refined and improved in [17].The Hochschild class of the character only gives partial information on the Chern character anda complete local formula, involving the Wodzicki residue in place of the Dixmier trace was ob-tained in 1995 in our joint work with H. Moscovici [61]. In fact index theory is an essentialapplication of the cyclic theory and we shall briefly describe its role in §5.1 below.

De-Rham homology of manifold Periodic Cyclic cohomology

Atiyah-Singer Index Theorem Local index formula in NCG [61]

Characteristic classes and Lie algebra cohomology Cyclic cohomology of Hopf algebras

4.1 Transverse elliptic theory and cyclic cohomology of Hopf algebras

In our work with H. Moscovici on the transverse elliptic operators for foliations we were led todevelop the theory of characteristic classes in the framework of noncommutative geometry as areplacement of its classical differential geometric ancestor. We also developed another key in-gredient of the transverse geometry which is the geometric analogue of the reduction from typeIII to type II and automorphisms and we used the theory of hypoelliptic operators to obtain aspectral triple describing the type II geometry. In fact the explicit computations involved in thelocal index formula for this spectral triple dictated two essential new ingredients: first a HopfalgebraHn which governs transverse geometry in codimension n, second a general constructionof the cyclic cohomology of Hopf algebras as a far reaching generalization of Lie algebra coho-mology [65–67] and [1,147,148]. This theory has been vigorously developed by H. Moscovici andB. Rangipour [217–220], the DGA version of the Hopf cyclic cohomology and the characteristicmap from Hopf cyclic to cyclic were investigated by A. Gorokhovsky in [136].

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4.2 Cyclic cohomology and archimedean cohomology

In [83] with C. Consani, we showed that cyclic homology which was invented for the needs ofnoncommutative geometry is the right theory to obtain, using the λ-operations, Serre’s archimedeanlocal factors of the L-function of an arithmetic variety as regularized determinants. Cyclic homol-ogy provides a conceptual general construction of the sought for “archimedean cohomology”.This resurgence of cyclic homology in the area of number theory was totally unexpected and isa witness of the coherence of the general line of thoughts. In fact the recent work [96, 97] of G.Cortinas, J. Cuntz, R. Meyer and G. Tamme relates rigid cohomology to cyclic homology.

4.3 Topological cyclic homology

We refer to the book [113] for a complete view of the relation of topological Hochschild and cyclictheories with algebraic K-theory. At the conceptual level one may think of topological cyclic ho-mology as cyclic homology performed over the smallest possible ground ring which in algebraictopology is not the familiar ring Z of integers but the far more absolute sphere spectrum. Thegreat advantage of working over this base is that the analogue of the tensor product becomesmuch more manageable as far as the action of permutations on the factors as well as their fixedpoints are concerned. This fact gives rise to new operations which do not exist in the generalframework. These operations include an analogue of the Frobenius operator and in his workwith I. Madsen [152, 153] and then in [151, 154, 155], L. Hesselholt has shown that the local andglobal Witt construction, the de Rham-Witt complex and the Fontaine theory all emerge natu-rally in this framework. All the more since he showed recently how topological periodic cyclichomology with its inverse Frobenius operator may be used to give a cohomological interpreta-tion of the Hasse-Weil zeta function of a scheme smooth and proper over a finite field in the formentirely similar to [83]. Finally, the paper [88] shows that topological cyclic homology is cyclichomology when applied to the Gamma-rings of G. Segal and that this framework, based on thesphere spectrum, subsumes all previous attempts of developing “absolute algebra". We referto the work of B. Dundas in [114] for a survey of the topological cyclic theory and an attempttowards extending the construction in order to incorporate the unstable information.

5 Overall picture, interaction with other fields

We describe in this section a few of the interactions of noncommutative geometry with otherfields with no claim of being exhaustive.

5.1 Index theory

5.1.1 Type II index theory

:Atiyah’s index theorem for covering spaces [3] gave a striking application of the type II indextheory which he used in order to define the index of an elliptic operator acting on the non-compact Galois covering of a smooth compact manifold. In Atiyah’s theorem the type II index

25

Figure 4: Interactions with noncommutative geometry

26

takes the same value as on the compact quotient and is hence integer valued. In [47] the typeII index theorem for foliations displayed the role of the Ruelle-Sullivan current associated to atransverse measure and the index as well as the dimensions of the solution spaces are real val-ued and give striking examples of real dimensions as discovered by Murray and von-Neumann.These real dimensions are best understood at the intuitive level in the same way as densities ofinfinite sets along the leaves relate to cardinalities of finite sets and this point of view is exten-sively developed in [47].

5.1.2 Higher index theory

The need for a higher index theory is already visible starting with the longitudinal index theo-rem for foliations which made use of a transverse measure as explained in Section 5.1.1. Suchtransverse measures do not always exist and the theory would seem to be limited to the type IIsituation as opposed to the type III which often occurs in geometry. Fortunately, the need of atransverse measure for the formulation of the theorem can be overcome when the result is for-mulated in terms of K-theory as was done in our joint work with G. Skandalis [54]. But in orderto obtain numeric indices one still needs to define natural numerical invariants of K-theory. Thisis obtained using the pairing of K-theory with cyclic cohomology and the early applications ofthe theory go back to the work on the transverse fundamental class [51]. The strategy can bedivided in several steps

1. Prove analytic invariance or vanishing properties of the analytic index as an element ofthe K-theory of a suitable algebra (most often a C∗-algebra associated to the geometricsituation).

2. Compute the pairing of the analytic index with cyclic cocycles by explicit geometric for-mulae, in the spirit of the original result of Atiyah and Singer.

3. Show that the algebraic pairing with the cyclic cocycle in fact makes sense at the level ofthe K-theory of the C∗-algebra.

Note that steps 1 and 3 are of hard analytic nature while the step 2 is of a more algebraic and com-putational nature. The third step is usually the most difficult. The early applications of the higherindex theory included the vanishing of A-genus for a foliated compact manifolds with leafwisepositive scalar curvature [51] and [9,284], the Novikov conjecture for Gelfand-Fuchs classes [51]and the Novikov conjecture for hyperbolic groups ([58] see §5.3). The proof of the Novikov con-jecture for Gelfand-Fuchs classes illustrates the power of cyclic cohomology theory and there isno other approach available to prove this result which inspired the recent proof by S. Gong, J. Wuand G. Yu of the Novikov conjecture for discrete subgroups of the group of volume preservingdiffeomorphisms in [135]. M. Puschnigg has made key contributions [239–241] to step 3 and wasable to use it to prove the Kadison-Kaplansky conjecture (which states that the reduced groupC∗-algebra does not contain non-trivial idempotents) for hyperbolic groups. Moreover R. Meyerfully developed in [206] the cyclic theory in the context of bornological algebras. A basic newidea is the notion of analytic nilpotence for bornological algebras which allows him to use theCuntz-Quillen approach to cyclic theories. In [215, 216], H. Moriyoshi and P. Piazza proved an

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higher index analogue of the Atiyah-Patodi-Singer formula for foliated manifolds using the pair-ing with the Godbillon-Vey cocycle. In [159–161] N. Higson and J. Roe construct an analyticalanalogue of surgery theory and connect this theory with the usual surgery theory for manifolds.They define an analytic analogue of algebraic Poincaré complexes and show that such a complexhas a signature, lying in the K-theory of the appropriate C∗-algebra, and that this signature is in-variant under bordism and homotopy. They construct a natural commutative diagram of exactsequences, sending the surgery exact sequence of Wall to an analytic surgery sequence improv-ing on previous results of J. Rosenberg [249]. The above gives only a brief glance at the primaryhigher index theory and we refer to the survey [117] of E. van Erp and A. Gorokhovsky for anexcellent introduction to the subject as well as a detailed account of the local index formula inNCG.We now turn to the recent developments [136,137,215,231,277,279,281] in the secondary higherindex theory. When the primary index vanishes as in the case of positive scalar curvature, orin the relative situation of homotopic manifolds for the signature, the secondary theory comesinto play, and one obtains a secondary invariant called the rho-invariant. This theory was firstintroduced by Higson and Roe [161, 163]. We refer to the splendid survey by Zhizhang Xie andGuoliang Yu [281] for the remarkable recent developments in this domain. We shall not attempt(for lack of space) to do justice to these considerable progress but briefly mention the papers [231]of P. Piazza and T. Schick and [277, 279, 281] of G. Yu and Z. Xie who develop sophisticatedsecondary invariants involving the group C∗-algebra of Galois covers to analyse the connectedcomponents of the space of metrics of positive scalar curvature. They proved in full generalitythe connection of the higher index of the Dirac operator on a spin manifold with boundary to thehigher rho invariant of the Dirac operator on the boundary, where the boundary is endowed witha positive scalar curvature metric. In [274] with S. Weinberger, G. Yu and Z. Xie settled a longstanding question by showing that the higher rho invariant is a group homomorphism. Theythen apply this result to study non-rigidity of topological manifolds by giving a lower bound forthe size of the reduced structure group of a closed oriented topological manifold, by the numberof torsion elements in the fundamental group of the manifold. Note that step 3 above comesinto play when one uses cyclic cohomology to obtain numerical invariants by pairing with therho invariant. In [36] the pairing of delocalized cyclic cocycles with the higher rho invariants iscomputed in terms of the delocalized eta invariants which were shown in [280] to be algebraic ifthe Baum-Connes conjecture holds, thus giving a new test of the conjecture. We refer to [281] foran account of this theory.

5.1.3 Algebraic index theory

The algebraic index theory is a vast subject which has been successfully developed by R. Nestand B. Tsygan (see [221–225]) and with P. Bressler and A. Gorokhovsky in [14]. It is an approachto the Atiyah-Singer index formula using deformation quantizations. It gives a powerful way ofunderstanding conceptually and proving the algebraic index formulas which play for instance akey role in step 2 of the higher index theory (§5.1.2). The algebra of pseudo-differential operatorsprovides a deformation quantization of the algebra of smooth functions on the cotangent spaceof a manifold. The index map is formulated using the trace density, which gives a map from theHochschild or negative cyclic homology of the deformation quantization to the de Rham coho-

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mology of the manifold. We refer to [14,221] for this line of development as well as [229,230] forindex theory in the framework of Lie groupoids and to [212–214, 226] for the analysis on singu-lar spaces consisting of a smooth interior part and a tower of foliated structures at the boundary,including corner and cone points and edges of all possible dimensions. This leads us to the nexttopic which is the geometrization of the pseudodifferential calculus and of deformations.

5.2 Smooth groupoids and their C∗-algebras in geometry

In [60] the construction of the tangent groupoid of a manifold was presented as a geometricframework for the theory of pseudo-differential operators. The tangent groupoid embodies in ageometric manner the deformation from symbols to operators. The main tool there is the functo-rial association of a C∗-algebra C∗(G) to a smooth groupoid G. As in the case of discrete groups,it admits a reduced and unreduced version, a distinction which disappears in the amenable case.This geometrization of operators allows one to give a completely geometric proof of the Atiyah-Singer index theorem as sketched in [60] and shown in [165]. In a remarkable series of papersC. Debord and G. Skandalis [100–102] have considerably extended this idea and advanced ina general program of geometrization of the existing pseudo-differential calculi. In her work C.Debord had shown how to construct the holonomy groupoid for a singular foliation defined by aLie algebroid when the fiber dimension is the same as the leaf dimension. The general construc-tion of the holonomy groupoid for a singular foliation was done by Androulidakis and Skandalisand this was the starting point of the index theory in this framework. In [103], C.Debord andG. Skandalis construct Lie groupoids by using the classical procedures of blowups and of de-formations to the normal cone. They show that the blowup of a manifold sitting in a transverseway in the space of objects of a Lie groupoid leads to a calculus similar to the Boutet de Monvelcalculus for manifolds with boundary. Their geometric constructions gives rise to extensionsof C∗-algebras and they compute the corresponding K-theory maps and the associated index.Smooth groupoids and the associated C∗-algebras thus provide a powerful tool in differentialgeometry. We refer to [104] for an excellent survey of this theory. This tool also applies success-fully in representation theory as shown by N. Higson in [162] who gave substance to the idea ofG. Mackey on the analogy between complex semisimple groups and their Cartan motion groups.

5.3 Novikov conjecture for hyperbolic groups

In the C. R. Acad. Sc. Paris Note of 1988 “Conjecture de Novikov et groupes hyperboliques”[57] and in [58], Henri Moscovici and myself proved the Novikov conjecture for hyperbolicgroups using cyclic cohomology combined with analytic techniques. The basic observationwhich was part of the origin of cyclic cohomology, is that, given a discrete group Γ , a groupcocycle c ∈ Zn(Γ ,C) gives a cyclic cocycle τc on the noncommutative group ring C[Γ]. Moreoveran index formula [57] [58] shows that the pairing of the cyclic cocycle gives the higher signature.The main difficulty is to pass from the very small domain of definition of the cocycle provided byC[Γ] to a larger domain having the same K-theory as the C∗-algebra C∗r (Γ) of the group, in whichthe signature is known to be homotopy invariant. The key technical tool which we used is theHaagerup property which provides a good notion of Schwartz rapid decay elements in C∗r (Γ)for hyperbolic groups. This together with the existence of bounded group cocycles representing

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cohomology classes of dimension > 1 (which is dual to the non-vanishing of the Gromov norm)gave the solution of step 3 of the higher index technique of §5.1.2 and proved the Novikov conjec-ture for hyperbolic groups [58]. Subsequently, and together with M. Gromov and H. Moscoviciwe produced another proof based on geometry. The original proof based on analysis and sub-algebras of C∗ algebras stable under holomorphic functional calculus is still a main tool for theBaum-Connes conjecture as in the work of V. Lafforgue who proved the latter conjecture for allhyperbolic groups (see §5.4).

5.4 The Baum-Connes conjecture

In this section we give a short overview of the Baum-Connes conjecture and refer to [134] for aremarkable survey of this topic. At the beginning of 1980, I had shown that closed transversalsof foliations provide idempotents in the C∗-algebra of the foliation. This geometric constructionplayed a key role in the explicit construction of the finite projective modules for the noncommu-tative torus [48] as described in §2.1. It was however quite clear that this geometric constructioncould not describe all of the K-theory of C∗(V, F) even in the simplest case of a fibration. I metPaul Baum at the Kingston conference of 1980 and he explained his work with Ron Douglasdescribing the K-theory of a manifold W using geometric cycles given by triples (M, E, f ) whereM is a compact manifold, E a vector bundle over M and f : M→ W is a continuous map, whileone makes suitable K-orientation hypothesis. This construction immediately gave the clue onhow to extend the construction of K-theory classes from transversals to a much more generaland flexible one; the principle at work there is a general principle of noncommutative geometrywhich is worth emphasizing because of its potential use in other circumstances, it can be statedas follows:

While it is in general difficult to map a noncommutative space X (such as a leaf space) to anordinary space, it is quite easy to construct maps the other way, and use such maps to testhomological properties of X.

In fact, in the case of foliations (V, F), in order to get a map f : M → V/F from a manifoldto a leaf space it is enough to cover M by open sets and maps f j : Ω j → V and to provide aone-cocycle ensuring that the maps match in the leaf space. The Baum-Connes conjecture wasformulated by P. Baum and myself in 1981 and had a great impact on the K-theory of operatoralgebras (see [134]). It was first stated for discrete groups and foliations, for Lie groups (asthe Connes-Kasparov conjecture) and then generalized (see Baum–Connes–Higson [6], and J-L.Tu [266]) to the framework of crossed products by locally compact groupoids. It has been a majorapplication of the Kasparov bivariant theory. By its relevance for representation theory of realor p-adic Lie groups, the analysis of discrete groups, quantum groups, group actions etc, it isone of the most powerful and unifying principles emerging as a link between algebraic topologyand analysis. It holds in full generality for Lie groups and in the amenable framework, andvery powerful results have been obtained (see e.g. [156]) but its limits were found in [164] wherecounterexamples were obtained.

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One of the most remarkable achievements since 2000 is the breakthrough contributions of Vin-cent Lafforgue who developed the analogue of the Kasparov bivariant theory in the context ofBanach algebras [185] and successfully applied this new tool combined with his work on thecomparison of K-theory in the Banach and C∗-contexts. He was able to cross the obstructiongiven, for discrete groups, by the property T of Kazhdan which had from the beginning in the1980’s forced to use the reduced C∗-algebra of the group as the natural one for the general for-mulation, but seemed an insurmountable obstacle in general. In [184] V. Lafforgue establishesthe Baum-Connes conjecture for many groups with property T. In [210] I. Mineyev and G. Yushowed that hyperbolic groups are strongly bolic in the sense of V. Lafforgue and therefore sat-isfy the Baum-Connes conjecture. The results of Lafforgue cover not only reductive real andp–adic Lie groups but also discrete cocompact subgroups of real Lie groups of real rank oneand discrete cocompact subgroups of real Lie groups SL3 over a local field. The conjecture withcoefficients was proved by P. Julg for the property T group Sp(n, 1) in [170]. Finally in [186] V.Lafforgue proved the Baum-Connes conjecture with coefficients for hyperbolic groups, a stun-ning positive result.

5.5 Coarse geometry and the coarse BC-conjecture

Coarse geometry is the study of spaces from a “large-scale" point of view. Its link with C∗-algebras and index theory is due to John Roe [244–246,248]. The book [246] gives a self-containedand excellent introduction to the topic. In his thesis John Roe managed to develop the index the-ory for non-compact spaces such as the leaves of a foliation of a compact manifold without theneed of the ambient compact manifold (which was needed for the L2 index theorem for folia-tions) and he associated a C∗-algebra to a coarse geometry thus allowing to apply the operatortechnique in this new context. His construction is obtained by abstracting the properties whichin the foliation context hold for operators along the leaves coming from the holonomy groupoid.Let (X, d) be a metric space in which every closed ball is compact. Let H be a separable Hilbertspace equipped with a faithful and non-degenerate representation of C0(X) whose range con-tains no nonzero compact operator. The Roe algebra associated to a coarse geometry is the C∗-algebra C∗(X, d) which is the norm closure of operators T inH which are locally compact15 andof finite propagation16, two notions which have a clear geometric meaning using the metric d.This C∗-algebra is independent of the choice of the representation of C0(X). The BC-conjecture isadapted to the coarse context under the hypothesis that (X, d) has bounded geometry and usinga discrete model.In [282] Guoliang Yu proved the Novikov conjecture for groups with finite asymptotic dimen-sion, and in [283] he proved the coarse BC-conjecture for metric spaces which admit a uniformembedding into Hilbert space, and as a corollary the Novikov conjecture for discrete groupswhose coarse geometry is embeddable in Hilbert space. Moreover he introduced Property A formetric spaces and showed that it implies uniform embedding into Hilbert space. We refer to thetextbook [227] for large scale geometry and the many other applications of property A.At the conceptual level, the coarse point of view is best expressed in Gromov’s words

15i.e. such that f T and T f are compact operators for any f ∈ C0(X)16i.e. their support stays at a finite distance from the diagonal. A pair (x, y) ∈ X × X is not in the support of T iff

there are f , g ∈ C0(X), f (x) 6= 0, g(y) 6= 0, f Tg = 0.

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To regain the geometric perspective one has to change one’s position and move the observa-tion point far away from X. Then the metric in X seen from the distance δ becomes the originaldistance divided by δ and as δ tends to infinity the points in X coalesce into a connected con-tinuous solid unity which occupies the visual horizon without any gaps or holes and fills ourgeometers heart with joy....

This point of view is clearly fitting with the principle of locality in noncommutative geometryexplained in §3.2 provided one works “in Fourier" i.e. one deals with momentum space. Thelarge scale geometry of momentum space corresponds to the local structure of space. Thus themomentum space of a spectral geometry (A,H, D) should be viewed as a coarse space in asuitable manner. A first attempt is to consider the distance d(A, A′) := ‖A− A′‖ defined by theoperator norm on the gauge potentials, i.e. the self-adjoint elements of theA-bimodule of formalone forms Ω1 := ∑ a jdb j | a j, b j ∈ A which one endows with

d(A, A′) = ‖A− A′‖D, where ‖a jdb j‖D := ‖∑ a j[D, b j]‖ (11)

The A-bimodule structure should be combined with the metric structure to investigate thiscoarse space. Note that the “zooming out" for the coarse metric corresponds to the rescalingD 7→ D/Λ for large Λ, which governs the spectral action. Under this rescaling D 7→ D/Λ, themetric d(φ,ψ) on states of (3) is multiplied by Λ, i.e. one zooms in, which corresponds to the du-ality between the two metric structures, the first given by (3) on the state space and the secondgiven by (11) on gauge potentials.

5.6 Comparison with toposes

Both noncommutative geometry and the theory of toposes provide a solution to the problem ofthe coexistence of the continuous and the discrete. The first by the use of the quantum formalismfor real variables as self-adjoint operators and the second by a far reaching generalization of thenotion of space which embodies usual topological spaces on the same footing as combinatoricaldatas such as small categories. The relation between noncommutative geometry and the theoryof toposes belongs to the same principle as the relation established by the Langlands correspon-dence between analysis of complex representations on one hand and Galois representations onthe other.The functorial construction of the C∗-algebra associated to a smooth groupoid (§5.2) is the proto-type of the correspondence between noncommutative geometry and the theory of toposes. Therole of the operator algebra side is to obtain global results such as index theory, homotopy in-variance of the signature or vanishing theorems of the index of Dirac operators in the presenceof positive scalar curvature. Such results would be inaccessible in a purely local theory.A systematic relation between the two theories has been developed by S. Henry in [150] and isbased on the use of groupoids on both sides.But as explained in [16] the theory of Grothendieck toposes goes very far beyond its geometricrole as a generalization of the notion of space. The new input comes from its relation with logicsand the key notion of the classifying topos for a geometric theory of first order. We refer to [16]for an introduction to this fundamental aspect of the theory and the role of the duality betweena topos and its presentation from a specific site.

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From my own point of view, the great surprise was that very simple toposes provide, throughtheir sets of points, very natural examples of noncommutative spaces. The origin of this find-ing came from realizing that the classifying space of a small category C gets much improvedwhen it is replaced by the topos C of contravariant functors from C to the category of sets. Inthe C.R. Acad. Sc. Paris Note of 1983 “Cohomologie cyclique et foncteurs Extn” (cf. [50]) I in-troduced the cyclic category Λ and showed that cyclic cohomology is a derived ext-functor afterembedding the non additive category of algebras in the abelian category of Λ-modules. I alsoshowed that the classifying space (in Quillen’s sense) of the small category Λ is the same as forthe compact group U(1) thus exhibiting the strong relation of Λ with the circle. But since theclassifying space of the ordinal category (of totally ordered finite sets and non-decreasing maps)is contractible, the information captured by the classifying space is only partial. This loss of in-formation is completely repaired by the use of the topos C associated to a small category C. Forinstance the topos thus associated to the ordinal category classifies the abstract intervals (totallyordered sets with a minimal and a maximal element) and encodes Hochschild cohomology asa derived functor. The lambda operations in cyclic homology [198] enrich the cyclic category Λ

to the epicyclic category and the latter fibers over the category (?,N×) with a single object andmorphisms given by the semigroup N× of non-zero positive integers under multiplication. Thetopos N× associated as above to this category underlies the arithmetic site (whose structure alsoinvolves a structure sheaf) and it was a great surprise to realize [84] that the points of this toposform a noncommutative space intimately related to the adele class space of §6.One is surely quite far from a complete understanding of the inter-relations between topos theoryand noncommutative geometry, and as another instance of an unexpected relation we mentionthat the above topos N× underlying the arithmetic site qualifies for representing the point innoncommutative geometry. More precisely noncommutative geometry works with (separable)C∗-algebras up to Morita equivalence and in each class there is a unique (up to isomorphism)A which is stable i.e. such that A ' A ⊗ K where K is the algebra of compact operators. Thekey fact then ( [85], §8) is that the algebra K of compact operators is naturally a C∗-algebra inthe above topos, i.e. it admits a natural action (unique up to inner) of the semi-group N× byendomorphisms which is hence inherited by any stable C∗-algebra. Moreover the classificationof matroids by J. Dixmier [107] is given by the same space as the space of points of the topos andthe corresponding algebras are obtained as stalks of the sheaf associated to K.

5.7 Quantum field theory on noncommutative spaces

Noncommutative tori appear naturally in compactifications of string theory as shown in [140](based on [141]), where the need for developing quantum field theory in such noncommutativespaces became apparent. The simplest case to consider is the Moyal deformation of Euclideanspace. What became quickly discovered is that there is a mixing between ultraviolet and infrareddivergencies so that the lack of compactness of the resolvent of the Laplacian becomes a realproblem. In a remarkable series of papers H. Grosse and R. Wulkenhaar [139] together with V.Rivasseau and his collaborators [243] managed to overcome this difficulty and to renormalize thetheory by adding a quadratic term which replaces the Laplacian by the harmonic oscillator. Theythen developped Euclideanφ4-quantum field theory on four-dimensional Moyal space with har-monic propagation even further than its commutative analogue. For instance they showed that

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in contrast with the commutative case there is no Landau ghost in this theory: it is asymptoticallysafe [106], leading to the tantalizing prospect of a full non-perturbative construction, includingsectors with Feynman graphs of all genera.Grosse and Wulkenhaar went on to extensively study and fully solve the planar sector of thetheory [140] (certainly its most interesting part since it contains all the ultraviolet divergencies).This sector also identifies with an infinite volume limit of the model in a certain sense. Very sur-prisingly both Euclidean invariance and translation invariance are restored in this limit. Buildingon the identities introduced in [106] they were able to fully solve all correlation functions of amodel in terms of a single sector of the two-point function They went even further and gavestrong numerical evidence [141] (and a recent proof in the case of the two point function of theφ3 theory, in collaboration with A. Sako [142]) that Osterwalder-Schrader positivity also holdsfor such models. In particular, given the specific problems (such as Gribov ambiguities) whichhave plagued and delayed the construction of ordinary non-Abelian gauge theories, the pla-nar sector of the Grosse-Wulkenhaar theory has now become our best candidate for a rigorousreconstruction of a non-trivial Wightman theory in four dimensions.The study of the Grosse-Wulkenhaar model has been also inspirational for promising generaliza-tions of matrix models and non-commutative field theories, namely the tensor models and tensorfield theories discovered and ceveloped by R. Gurau, V. Rivasseau and collaborators [145]. Werefer to [276] for an extensive survey of the remarkable developments of quantum field theoryon noncommutative spaces.

5.8 Noncommutative geometry and solid state physics

Since the pioneering work of Jean Bellissard on the quantum Hall effect and the deep relationwhich he unveiled as the noncommutative nature of the Brillouin zone (see [60] Chapter IV,6) the relation between noncommutative geometry and solid state physics has been vigorouslypursued. We refer to the book [236] and papers [237, 238] for the recent progress as well as tothe papers of Terry Loring on the finite dimensional manifestations of the K-theory obstructionsand of C. Bourne, A. Carey, A. Rennie [12] for the bulk-edge correspondence.

6 Spectral realization of zeros of zeta and the scaling site

We describe in this section the main steps from quantum statistical mechanical models arising innumber theory, the adele class space as a noncommutative space, to the scaling site, an object ofalgebraic geometry underlying the spectral realization of the zeros of the Riemann zeta function.

6.1 Quantum statistical mechanics and number theory

Discrete groups Γ provide very non-trivial examples of factors of type II1 and their left regularrepresentation given by the left action of Γ in the Hilbert space `2(Γ) of square integrable func-tions on Γ generates a type II1 factor R(Γ) as long as the non-trivial conjugacy classes of elementsof Γ are infinite. It was shown by Atiyah that the type II index theory relative to R(Γ) can be used

34

very successfully for Galois coverings of compact manifolds. To obtain type III factors from dis-crete groups, one considers the relative situation of pairs (Γ , Γ0), where Γ0 ⊂ Γ is a subgroupwhich is almost normal inasmuch as the left action of Γ0 on the coset space Γ/Γ0 only has finiteorbits. The left action of Γ in the Hilbert space `2(Γ/Γ0) then generates a von Neumann alge-bra which is no longer of type II in general and whose modular theory depends on the integervalued function L(γ) := cardinality of the orbit Γ0(/γΓ0) in the coset space Γ/Γ0. In fact thecommutant von Neumann algebra is generated by the Hecke algebraH(Γ , Γ0) of convolution offunctions f of double cosets which have finite support in Γ0\Γ/Γ0. The modular automorphismof the canonical state is given by [13]

σt( f ) =(

L(γ)R(γ)

)it

f (γ), R(γ) := L(γ−1).

Almost normal subgroups Γ0 ⊂ Γ arise naturally by considering the inclusion of points over Zinside points over Q for algebraic groups as in the construction of Hecke algebras. The BC systemarises from the “ax + b” algebraic group P. By construction P+

Z ⊂ P+Q is an inclusion Γ0 ⊂ Γ of

countable groups, where P+Z and P+

Q denote the restrictions to a > 0. This inclusion fulfills theabove commensurability condition and the associated Hecke algebra with its dynamics is theBC system. Its main interest is that it exhibits spontaneous symmetry breaking [13] when it iscooled down using the thermodynamics of noncommutative spaces (see [75] for this generalnotion) and that its partition function is the Riemann zeta function. Considerable progress hasbeen done since 2000 on extending the BC-system to number fields. For lack of space we shallnot describe these developments but refer to [75] and to [182,183]. The paper [182] elucidates thefunctoriality of the construction in Theorem 4.4, where they show that the construction of Bost-Connes type systems extends to a functor which to an inclusion of number fields K ⊂ L assigns aC∗-correspondence which is equivariant with respect to their suitably rescaled natural dynamics.In the paper [183] M. Laca, N. Larsen and S. Neshveyev succeeded to extend the properties offabulous states of the BC system to arbitrary number fields. We refer to the introduction oftheir paper for an historical account of the various steps that led to the solution after the initialsteps of Ha and Paugam. They consider the Hecke pair consisting of the group P+

K of affinetransformations of a number field K that preserve the orientation in every real embedding andthe subgroup P+

O of transformations with algebraic integer coefficients. They then proceed andshow that the Hecke algebra H associated to the pair (P+

K , P+O ) fulfills perfect analogues of the

properties of the BC–system including phase transitions and Galois action. More precisely theyobtain an arithmetic subalgebra (through the isomorphism of H to a corner in the larger systemestablished in [182]) on which ground states exhibit the “fabulous states" property with respectto an action of the Galois group Gal(Kab : H+(K)), where H+(K) is the narrow Hilbert classfield, and Kab is the maximal abelian extension of K.We refer to [98] for a survey of the important other line of developments coming from the ex-ploration by Joachim Cuntz and his collaborators of the C∗-algebra associated to the action ofthe multiplicative semigroup of a Dedekind ring on its additive group. Representations of suchactions give rise to particularly intriguing problems and the study of the corresponding C∗-algebras has motivated many of the new methods and general results obtained in this area.Another very interesting broad generalization of the BC-system has been developed by M. Mar-colli and G. Tabuada in [203]. Using the Tannakian formalism, they categorify the algebraic

35

data used in constructing the system such as roots of unity, algebraic numbers, and Weil num-bers. They study the partition function, low temperature Gibbs states, and Galois action onzero-temperature states for the associated quantum statistical system and they show that in theparticular case of the Weil numbers the partition function and the low temperature Gibbs statescan be described as series of polylogarithms.

6.2 Anosov foliation

The factor of type III1 generated by the regular representation of the BC-system is the same(unique injective factor of type III1, [146]) as the factor associated to the Anosov foliation of thesphere bundle of a Riemann surface of genus > 1. This fact suggested an analogy between thegeometry of the continuous decomposition (as a crossed product: type II∞ oR∗+) in both caseswhich led me in [63], based on the paper of Guillemin [144], to the adele class space as a naturalcandidate for the trace formula interpretation of the explicit formulas of Riemann-Weil. It isworth explaining briefly the framework17 of [144]. Let Γ ⊂ SL(2,R) be a discrete cocompactsubgroup, and V = SL(2,R)/Γ . Let F be the foliation of V whose leaves are the orbits of theaction on the left of the subgroup P ⊂ SL(2,R) of upper triangular matrices. At the Lie algebralevel P is generated by the elements

E+ =

(0 10 0

), H =

12

(1 00 −1

)One denotes by η and ξ the corresponding vector fields. The associated flows are the horocycleand geodesic flows. They correspond to the one-parameter subgroups of P ⊂ SL(2,R) given by

n(a) =(

1 a0 1

), g(t) =

(et/2 00 e−

t2

)

The horocycle flow is normalized by the geodesic flow, more precisely one has g(t)n(a)g(−t) =n(aet). The von Neumann algebra of the foliation, M = W(V, F) is a factor of type III1 and itscontinuous decomposition is visible at the geometric level: the associated factor of type II∞ isthe von Neumann algebra N = W(V, η) of the horocycle foliation. The one parameter group ofautomorphismsθλ scaling the trace is given by the action of the geodesic flow by automorphismsof N = W(V, η) which has clear geometric meaning by the naturality of the construction ofthe von Neumann algebra of a foliation. The trace on N = W(V, η) corresponds to the RuelleSullivan current which is the contraction of the SL(2,R)-invariant volume form of M by thevector field η. The rescaling of the trace by the automorphisms θλ follows from the rescalingof η by the geodesic flow. In [144] a heuristic proof of the Selberg trace formula is given usingthe action of the geodesic flow on the horocycle foliation. Our interpretation of the explicitformulas as a trace formula involves in a similar manner the action of R∗+ on the geometricspace associated to the type II∞ factor of the continuous decomposition for the BC-system. Thisgeometric space is the adele class space (for the precise formulation we refer to [75]).

17The same paper [144] was used two years later by Deninger in [105] to motivate his search for an hypotheticalcohomology theory using foliations.

36

6.3 The adele class space

In [64] we gave a spectral realization of the zeros of the Riemann zeta function and of L-functionsbased on the above action of the Idele class group on the noncommutative space of Adèle classeswhich is the type II space associated to the BC system by the reduction from type III to typeII and automorphisms. This result determined a geometric interpretation of the Riemann-Weilexplicit formulas as a Lefschetz formula and also a reformulation of the Riemann Hypothesisin terms of the validity of a trace formula. The understanding of the geometric side of the traceformula is simpler when one works with the full adèle class space i.e. one does not further divideby Z∗. The local contribution from a place v of a global field K is obtained as the distributionaltrace of an integral

∫f (λ−1)Tλd∗λ of operators of the form, with λ ∈ K∗v ,

(Tλξ)(x) := ξ(λx) , ∀x ∈ Kv

where Kv is the local completion of K at the place v. The distributional trace of Tλ is the integral∫k(x, x)dx on the diagonal of the Schwartz kernel which represents Tλ i.e. by

(Tλξ)(x) =∫

k(x, y)ξ(y)dy⇒ k(x, y) = δ(y− λx)

This gives for the trace the formula∫

k(x, x)dx =∫δ(x− λx)dx = |1− λ|−1 using the change of

variables u = (1− λ)x and the local definition of the module which implies dx = |1− λ|−1du.For the convolution operator

∫f (λ−1)Tλd∗λ this delivers the local contribution to the Riemann-

Weil explicit formula as∫

K∗vf (λ−1)|1− λ|−1d∗λ. The above computation is formal but as shown

in [64] the more precise calculation even delivers the subtle finite parts involved in the Riemann-Weil explicit formula. In [205], R. Meyer showed how, by relaxing the Sobolev condition, onecan effectively reprove the explicit formulas as a trace formula on the adèle class space.

6.4 The scaling site

6.4.1 Hasse-Weil form of the Riemann zeta function

After a few years of direct attack using only analysis I came to the conclusion that a much betterunderstanding of the geometry underlying the adèle class space was required in order to make avigorous advance toward the solution of this problem. This work was undertaken in the ongoingcollaboration with C. Consani. Among the major results obtained in these past years is a geo-metric framework in which one can transpose to the case of RH many of the ingredients of theWeil proof for function fields as reformulated by Mattuck, Tate and Grothendieck. The startingpoint [81, 82] is the determination and interpretation as intersection number using the geometryof the adèle class space, of the real counting function N(q) which gives the complete Riemannzeta function by a Hasse-Weil formula in the limit q → 1, in the line of the limit geometry on Fqfor q = 1 proposed by J.Tits and C.Soulé.

6.4.2 Characteristic one

The limit q → 1 is taken analytically in the above development, and this begs for an algebraicunderstanding of the meaning of “characteristic one". Characteristic p is defined by the congru-ence p ' 0. While the congruence 1 ' 0 is fruitless, its variant given by 1+ 1 ' 1 opens the door

37

to a whole field which we discuss in [91] as the “world of characteristic one" whose historicalorigin goes quite far back. It has strong connections with the field of optimization [129,130], withtropical geometry, with lattice theory and very importantly it appeared independently with theschool of “dequantization", of V. P. Maslov and his collaborators [179, 196]. They developed asatisfactory algebraic framework which encodes the semiclassical limit of quantum mechanicsand called it idempotent analysis. The starting observation is that one can encode the limit h→ 0by simply conjugating the addition of numbers by the power operation x 7→ xε and passing tothe limit when ε→ 0. The new addition of positive real numbers is

limε→0

(x

1ε + y

1ε

)ε= maxx, y = x ∨ y

When endowed with this operation as addition and with the usual multiplication, the positivereal numbers become a semifield Rmax

+ . It is of characteristic 1, i.e. 1 ∨ 1 = 1 and contains thesmallest semifield of characteristic 1, namely the Boolean semifield B = 0, 1. Moreover, Rmax

+admits non-trivial automorphisms and one has

GalB(Rmax+ ) := AutB(Rmax

+ ) = R∗+, Frλ(x) = xλ , ∀x ∈ Rmax+ , λ ∈ R∗+

thus providing a first glimpse of an answer to Weil’s query in [273] of an algebraic frameworkin which the connected component of the idele class group would appear as a Galois group.The most striking discovery of this school of Maslov, Kolokolstov and Litvinov [179, 196] is thatthe Legendre transform which plays a fundamental role in all of physics and in particular inthermodynamics in the nineteenth century, is simply the Fourier transform in the framework ofidempotent analysis.

6.4.3 Tropical geometry and Riemann-Roch theorems

The tropical semiring Nmin = N ∪ ∞ with the operations min and + was introduced by ImreSimon in [252] to solve a decidability problem in rational language theory. His work is at theorigin of the term “tropical" which was coined by Marco Schutzenberger. Tropical geometry is avast subject, see e.g. [131,201,208]. We refer to [269] for an excellent introduction starting from thesixteenth Hilbert problem. In its simplest form ([128]) a tropical curve is given by a graph with ausual line metric on its edges. The natural structure sheaf on the graph is the sheaf of real valuedfunctions which are continuous, convex, piecewise affine with integral slopes. The operationson such functions are given by ( f ∨ g)(x) = f (x)∨ g(x) for all x ∈ Γ and the product is given bypoint-wise addition. One also adjoins the constant −∞ which plays the role of the zero elementin the semirings of sections. One proceeds as in the classical case with the construction of thesheaf of semifields of quotients and finds the same type of functions as above but no longerconvex. Cartier divisors make sense and one finds that the order of a section f at a point x ∈ Γ

is given by the sum of the (integer valued) outgoing slopes. The conceptual explanation of whythe discontinuities of the derivative should be interpreted as zeros or poles is due to Viro, [270]who showed that it follows automatically if one understands that, as seen when dealing withvaluations the sum x ∨ x of two equal terms should be viewed as ambiguous with all values inthe interval [−∞, x] on equal footing. In their work Baker and Norine [5] proved in the discreteset-up of graphs an analogue of the Riemann-Roch theorem whose essence is that the inequality

38

Deg(D) ≥ g (where g is the genus of the graph) for a divisor implies that the divisor is equivalentto an effective divisor. Once translated in the language of the chip firing game (op.cit.), this factis equivalent to the existence of a winning strategy if one assumes that the total sum of dollarsattributed to the vertices of the graph is ≥ g. We refer to [128] for the tropical curve version ofthe Baker and Norine theorem.

6.4.4 The topos, its points and structure sheaf

The major discovery [84–86] is that of the scaling site, the topos of N×-equivariant sheaves onthe Euclidean half-line, which is obtained by extension of scalars from the arithmetic site. Thepoints of this topos form exactly the sector of the adèle class space involved in RH, and theaction of R∗+ on the adele class space corresponds to the action of the Frobenius. This providesin full the missing geometric structure on the adèle class space which becomes a tropical curvesince the topos inherits, from its construction by extension of scalars, a natural sheaf of regularfunctions as piecewise affine convex functions. This structure is central in the well known resultson the localization of zeros of analytic functions which involve Newton polygons in the non-archimedean case and the Jensen’s formula in the complex case. The new feature given by theaction of N× corresponds to the transformation f (z) 7→ f (zn) on analytic functions. The Newtonpolygons play a key role in the construction of the analogue of the Frobenius correspondencesas a one parameter semigroup of correspondences already defined at the level of the arithmeticsite. Finally the restriction to the periodic orbit of the scaling flow associated to each prime pgives a quasi-tropical structure which turns this orbit into a variant Cp = R∗+/pZ of the classicalJacobi description C∗/qZ of an elliptic curve. On Cp, [87] develops the theory of Cartier divisors,determines the structure of the quotient of the abelian group of divisors by the subgroup ofprincipal divisors, develops the theory of theta functions, and proves the Riemann-Roch formulawhich involves real valued dimensions, as in the type II index theory. The current situationconcerning the evolution of this strategy towards RH is summarized in the essay [91] in thevolume on open problems in mathematics, initiated by John Nash and recently published bySpringer. After developing homological algebra in characteristic one in [89] we understood in[90] how to go back and forth to the complex situation using the above Jensen formula so thatthe fundamental structure takes place over C and fits well with noncommutative geometry.

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