An Introduction to Noncommutative Geometry

AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY Joseph C. Varilly Departamento de Matematicas, Universidad de Costa Rica, 2060 San Jose, Costa Rica Introduction These are lecture notes for a course given at the Summer School on Noncommuta- tive Geometry and Applications, sponsored by the European Mathematical Society, at Monsaraz, Portugal and at Lisboa, from the 1st to the 10th of September, 1997. Noncommutative geometry, which already occupies an extensive and wide-ranging area of mathematics, has come under increasing scrutiny from physicists interested in what it has to say about fundamental problems of Nature. This course sought to address a mixed audience of students and young researchers, both mathematicians and physicists, and to provide a gateway to some of its more recent developments. Many approaches can be taken to introducing noncommutative geometry. I decided to focus on the geometry of Riemannian spin manifolds and their noncommutative cousins, which are geometries determined by a suitable generalization of the Dirac operator. These geometries underlie the NCG approach to phenomenological particle models and recent attempts to place gravity and matter elds on the same geometrical footing. The rst two lectures are devoted to commutative geometry; we set up the general framework and then compute a simple example, the two-sphere, in noncommutative terms. The general denition of a geometry is then laid out and exemplied with the noncom- mutative torus. Enough details are given so that one can see clearly that NCG is just ordinary geometry, extended by discarding the commutativity assumption on the coordi- nate algebra. Classication up to equivalence is dealt with brie y in lecture 7. Other lectures explore some of the tools of the trade: the noncommutative integral, the r^ole of quantization, and the spectral action functional. Physical models are not treated directly, since these were the subject of other lectures at the Summer School, but most of the mathematical issues needed for their understanding are dealt with here. I wish to thank several people who contributed in no small way to assembling these lectures. Jose M. Gracia-Bonda gave decisive help at many points; he and Alejandro Rivero provided constructive criticism throughout. I thank Daniel Kastler, Bruno Iochum, Thomas Schucker and Daniel Testard for the opportunity to visit the Centre de Physique Theorique of the CNRS at Marseille, and the pleasure of learning and practising noncom- mutative geometry at the source. I am grateful for enlightening discussions with Alain Connes, Robert Coquereaux, Ricardo Estrada, Hector Figueroa, Thomas Krajewski, Gio- vanni Landi, Fedele Lizzi, Carmelo Martn, William Ugalde and Mark Villarino. Thanks also to Jesus Clemente, Stephan de Bievre and Markus Walze who provided indispens- able references. Several improvements to the original draft notes were suggested by Eli Hawkins, Thomas Schucker and Georges Skandalis. Last but by no means least, I want to discharge a particular debt of gratitude to Paulo Almeida for his energy and foresight in organizing this Summer School in the right place at the right time. 1 2 AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY Contents 1. Commutative Geometry from the Noncommutative Point of View 4 The Gelfand{Namark cofunctors The functor Hermitian metrics and spinc structures The Dirac operator and the distance formula 2. Spectral Triples on the Riemann Sphere 13 Line bundles and the spinor bundle The Dirac operator on the sphere Spinor harmonics and the spectrum of D= Twisted spinor modules A reducible spectral triple 3. Real Spectral Triples: the Axiomatic Foundation 22 The data set Innitesimals and dimension The order-one condition Smoothness of the algebra Hochschild cycles and orientation Finiteness of the K -cycle Poincare duality and K -theory The real structure 4. Geometries on the Noncommutative Torus 32 Algebras of Weyl operators The algebra of the noncommutative torus The skeleton of the noncommutative torus A family of geometries on the torus 5. The Noncommutative Integral 42 The Dixmier trace on innitesimals Pseudodierential operators The Wodzicki residue The trace theorem Integrals and zeta residues 6. Quantization and the Tangent Groupoid 51 Moyal quantizers and the Heisenberg deformation Groupoids The tangent groupoid Moyal quantization as a continuity condition The hexagon and the analytical index Remarks on quantization and the index theorem  CONTENTS 3 7. Equivalence of Geometries 61 Unitary equivalence of geometries Morita equivalence and Hermitian connections Vector bundles over the noncommutative torus Morita-equivalent toral geometries Gauge potentials 8. Action Functionals 70 Automorphisms of the algebra The fermionic action The spectral action principle Spectral densities and asymptotics References 79 1. Commutative Geometry from the Noncommutative Point of View The traditional arena of geometry and topology is a set of points with some particular structure that, for want of a better name, we call a space . Thus, for instance, one studies curves and surfaces as subsets of an ambient Euclidean space. It was recognized early on, however, that even such a fundamental geometrical object as an elliptic curve is best studied not as a set of points (a torus) but rather by examining functions on this set, specically the doubly periodic meromorphic functions. Weierstrass opened up a new approach to geometry by studying directly the collection of complex functions that satisfy an algebraic addition theorem, and derived the point set as a consequence. In probability theory, the set of outcomes of an experiment forms a measure space, and one may regard events as subsets of outcomes; but most of the information is obtained from \random variables", i.e., measurable functions on the space of outcomes. In noncommutative geometry, under the in uence of quantum physics, this general idea of replacing sets of points by classes of functions is taken further. In many cases the set is completely determined by an algebra of functions, so one forgets about the set and obtains all information from the functions alone. Also, in many geometrical situations the associated set is very pathological, and a direct examination yields no useful information. The set of orbits of a group action, such as the rotation of a circle by multiples of an irrational angle, is of this type. In such cases, when we examine the matter from the algebraic point of view, we often obtain a perfectly good operator algebra that holds the information we need; however, this algebra is generally not commutative. Thus, we proceed by rst discovering how function algebras determine the structure of point sets, and then learning which relevant properties of function algebras do not depend on commutativity. In a famous paper [52] that has become a cornerstone of noncommutative geometry, Gelfand and Namark in 1943 characterized the involutive algebras of operators by just dropping commutativity from the most natural axiomatization for the algebra of continuous functions on a locally compact Hausdor space. The starting point for noncommutative geometry that we shall adopt here is to study ordinary \commutative" spaces via their algebras of functions, omitting wherever possible any reference to the commutativity of these algebras. The Gelfand{Namark cofunctors The Gelfand{Namark theorem can be thought of as the construction of two con- travariant functors (cofunctors for short) from the category of locally compact Hausdor spaces to the category of C  -algebras. The rst cofunctor C takes a compact space X to the C  -algebra C (X ) of continuous complex-valued functions on X , and takes a continuous map f : X ! Y to its transpose Cf : h 7! h  f : C (Y ) ! C (X ). If X is only a locally compact space, the corresponding C  -algebra is C0 (X ) whose elements are continuous functions vanishing at innity, and we require that the continuous maps f : X ! Y be proper (the preimage of a compact set is compact) in order that h 7! h  f take C0 (Y ) into C0 (X ). The other cofunctor M goes the other way: it takes a C  -algebra A onto its space of characters , that is, nonzero homomorphisms : A ! C . If A is unital, M (A) is closed in 4  1. COMMUTATIVE GEOMETRY FROM THE NONCOMMUTATIVE POINT OF VIEW 5 the weak* topology of the unit ball of the dual space A and hence is compact. If : A ! B is a unital -homomorphism, the cofunctor M takes  to its transpose M:  7!    : M (B ) ! M (A). Write X + := X ]f1g for the space X with a point at innity adjoined (whether X is compact or not), and write A+ := C  A for the C  -algebra A with an identity adjoined via the rule (; a)(; b) := (; b+a+ab), whether A is unital or not; then C (X + ) ' C0 (X )+ as unital C  -algebras. If 0: A+ ! C : (; a) 7! , then M (A) = M (A+) n f0g is locally compact when A is nonunital. Notice that M (A)+ and M (A+ ) are homeomorphic. That no information is lost in passing from spaces to C  -algebras can be seen as follows. If x 2 X , the evaluation f 7! f (x) denes a character x 2 M (C (X )), and the map X : x 7! x : X ! M (C (X )) is a homeomorphism. If a 2 A, its Gelfand transform a^:  7! (a) : M (A) ! C is a continuous function on M (A), and the map G : a 7! a^ : A ! C (M (A)) is a -isomorphism of C  -algebras, that preserves identities if A is unital. These maps are functorial (or \natural") in the sense that the following diagrams commute: f  X? ! Y? A? ! B? X ? y ? yY GA ? y ? yGB M (C (X )) MCf ! M (C (Y )) C (M (A)) CM ! C (M (B )) For instance, given a unital -homomorphism : A ! B , then for any a 2 A and  2 M (B ), we get ((CM  GA )a) = ((CM)^a) = a^(M () ) = a^(  ) (a)( ) = ((GB  )a); =  ((a)) = d by unpacking the various transpositions. This \equivalence of categories" has several consequences. First of all, two commu- tative C  -algebras are isomorphic if and only if their character spaces are homeomorphic. (If : A ! B and : B ! A are inverse -isomorphisms, then M: M (B ) ! M (A) and M : M (A) ! M (B ) are inverse continuous proper maps.) Secondly, the group of automorphisms Aut(A) of a commutative C  -algebra A is isomorphic to the group of homeomorphisms of its character space. Note that, since A is commutative, there are no nontrivial inner automorphisms in Aut(A). Thirdly, the topology of X may be dissected in terms of algebraic properties of C0(X ). For instance, any ideal of C0 (X ) is of the form C0 (U ) where U  X is an open subset (the closed set X n U being the zero set of this ideal). If Y  X is a closed subset of a compact space X , with inclusion map j : Y ! X , then Cj : C (X ) ! C (Y ) is the restriction homomorphism (which is surjective, by Tietze's extension theorem). In general, f : Y ! X is injective i Cf : C (X ) ! C (Y ) is surjective. 6 AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY We may summarize several properties of the Gelfand{Namark cofunctor with the following dictionary, adapted from [119, p. 24]: TOPOLOGY ALGEBRA locally compact space C  -algebra compact space unital C  -algebra compactication unitization continuous proper map -homomorphism homeomorphism automorphism open subset ideal closed subset quotient algebra second countable separable measure positive functional The C  -algebra viewpoint also allows one to study the topology of non-Hausdor spaces, such as arise in probing a continuum where points are unresolved: see the book by Landi on noncommutative spaces [79]. A commutative C  -algebra has an abundant supply of characters, one for each point of the associated space. Looking ahead to noncommutative algebras, we can anticipate that characters will be fairly scarce, and we need not bother to search for points. There is, however, one r^ole for points that survives in the noncommutative case: that of zero- dimensional elements of a homological skeleton or cell decomposition of a topological space. For that purpose, characters are not needed; we shall require functionals that are only traces on the algebra, but are not necessarily multiplicative. The functor Continuous functions determine a space's topology, but to do geometry we need at least a dierentiable structure. Thus we shall assume from now on that our \commutative space" is in fact a dierential manifold M , of dimension n. For convenience, we shall usually assume that M is compact , even though this leaves aside important examples such as Minkowski space. (It turns out that noncommutative geometry has been developed so far almost entirely in the Euclidean signature, where compactness can be seen as a simplifying technical assumption. How to adapt the theory to deal with spaces with indenite metric is very much an open problem at this stage.) The C  -algebra A = C (M ) of continuous functions must then be replaced by the algebra A = C 1 (M ) of smooth functions on the manifold M . This is not, of course, a C  - algebra, and although it is a Frechet algebra in its natural locally convex topology, we never use the theory of locally convex algebras: our tactic is to work with the dense subalgebra A of A in a purely algebraic fashion. We think of A as the subspace of \suciently regular" elements of A. A character of A is a distribution  on M that is positive, since (aa) = j(a)j2  0, and as such is a measure [53] that extends to a character of C (M ); hence A also determines the point-space M .  1. COMMUTATIVE GEOMETRY FROM THE NONCOMMUTATIVE POINT OF VIEW 7 To study a given compact manifold M , one uses the category of (complex) vector bundles E ! M ; its morphisms are bundle maps  : E ! E 0 satisfying 0   =  and so dening brewise maps x : Ex ! Ex0 (x 2 M ) that are required to be linear. Given any vector bundle E ! M , write (E ) := C 1 (M; E ) for the space of smooth sections of M . If  : E ! E 0 is a bundle map, the composition  : s 7!   s : (E ) ! (E 0) satises, for a 2 A, x 2 M ,  (sa)(x) = x (s(x)a(x)) = x (s(x)) a(x) = (  (s)a)(x) so  (sa) =  (s)a; that is,  : (E ) ! (E 0) is a morphism of (right) A-modules. Vector bundles over M admit operations such as duality, direct sum (i.e., Whit- ney sum) and tensor product; the -functor carries these to analogous operations on A-modules; for instance, if E , E 0 are vector bundles over M , then (E E 0 ) ' (E ) A (E 0); P where the right hand side is formed by nite sums j sj s0j subject to the relations sa s0 s as0 = 0, for a 2 A. One can show that any A-linear map from (E ) to (E 0) is of the form  for a unique bundle map  : E ! E 0. It remains to identify what the image of the -functor is. First note that if E = M C r is a trivial bundle, then (E ) = Ar is a free A-module. Since M is compact, we can nd nonnegative functions 1 ; : : :; q 2 A with 12 +


+ q2 = 1 (a partition of unity) such that E is trivial over the set Uj where j > 0, for each j . If fij : Ui \ Uj ! GL(r; C ) are the transition functions for E , satisfying fik fkj = fij on P Ui \ Uj \ Uk , then the functions pij = i fij j (dened to be zero outside Ui \ Uj ) satisfy k pik pkj = pij , and so assemble into a qr  qr matrix p 2 Mqr (A) such that p2 = p. A section in (E ), given locally by smooth functions sj : Uj ! C r such that si = fij sj on Ui \ Uj , can be regarded as a column vector s = ( 1 s1; : : :; q sq )t 2 C 1 (M )qr satisfying ps = s. In this way, one identies (E ) with pAqr . The Serre{Swan theorem [111] says that this is a two-way street: any (right) A-module of the form pAm, for an idempotent p 2 Mm(A), is of the form (E ) = C 1 (M (A); E ). The bre at the point  2 M (A) is the vector space pAm A (A= ker ) whose (nite) dimension is the trace of the matrix (p) 2 Mm(C ). In general, a (right) A-module of the form pAm is called a nite projective module (more correctly, a nitely generated projective module). We summarize by saying that is a (covariant) functor from the category of vector bundles over M to the category of nite projective modules over C 1 (M ). The Serre{Swan theorem gives a recipe to construct an inverse functor going the other way, so that these categories are equivalent. (See the discussion by Brodzki [9] for more details in a modern style.) What, then, is a noncommutative vector bundle ? It is simply a nite projective right module E for a (not necessarily commutative) algebra A, which will generally be a dense subalgebra of a C  -algebra A. 8 AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY Hermitian metrics and spinc structures Any complex vector bundle can be endowed (in many ways) with a Hermitian metric . The conventional practice is to dene a positive denite sesquilinear form (
j
)x on each bre Ex of the bundle, which must \vary smoothly with x". The noncommutative point of view is to eliminate x, whereupon what remains is a pairing E  E ! A on a nite projective (right) A-module with values in the algebra A that is A-linear in the second variable, conjugate-symmetric and positive denite. In symbols: (r j s + t) = (r j s) + (r j t); (r j sa) = (r j s) a; (r j s) = (s j r); (s j s) > 0 for s 6= 0; (1:1) with r; s; t 2 E , a 2 A. Notice the consequence (rb j s) = b (r j s) if b 2 A. With this structure, E is called a pre-C  -module or \prehilbert module". More pre- cisely, a pre-C  -module over a dense subalgebra A of a C  -algebra A is a right A-module E (not necessarily nitely-generated or projective) with a sesquilinear pairing E  E ! A satisfying (1.1). If desired, one can complete it in the norm p jjjsjjj := k(s j s)k where k
k is the C  -norm of A; the resulting Banach space is then a C  -module . In the case E = C 1 (M; E ), the completion is the Banach space of continuous sections C (M; E ). Indeed, in general this completion is not a Hilbert space. For instance, one can take E = A itself, by dening (a j b) := a b; then jjjajjj equals the C  -norm kak, so the completion is the C  -algebra A. The free A-module Am is a pre-C  -module in the obvious way: (r j s) := m P  j =1 rj sj . This column-vector scalar product also works for pAm if p = p2 2 Mm(A), provided that p = p also. If q = q2 2 Mm(A), one can always nd a projector p = p2 = p in Mm(A) that is similar and homotopic to q: see, for example, [119, p. 102]. (The choice of p selects a particular Hermitian structure on the right module qAm.) Thus we shall always assume from now on that the idempotent p is also selfadjoint. One can similarly study left A-modules. In fact, if E is any right A-module, the conjugate space E is a left A-module: by writing E = f s : s 2 E g, we can dene a s := (sa) : For E = pAm, we get E = Am p where entries of Am are to be regarded as \row vectors". Morita equivalence. Finite projective A-modules with A-valued scalar products play a r^ole in noncommutative geometry as mediating structures that is partially hidden in com- mutative geometry: they allow the emergence of new algebras related, but not isomorphic, to A. Consider the \ket-bra" operators on E of the form jrihsj : t 7! r (s j t) : E ! E ;  1. COMMUTATIVE GEOMETRY FROM THE NONCOMMUTATIVE POINT OF VIEW 9 for r; s 2 E . Since r (s j ta) = r (s j t) a for a 2 A, these operators act \on the left" on E and commute with the right action of A. Composing two ket-bras yields a ket-bra: jrihsj
jtihuj = jr(s j t)ihuj = jrihu (t j s)j; so all nite sums of ket-bras form an algebra B = EndA (E ). When E = pAm, we have B = p Mm(A) p. Now E becomes a left B-module, and we say that E is a \B-A-bimodule ". One can also regard EndA (E ) as E A E , by jrihsj $ r s. On the other hand, we can form E B E , which is isomorphic to A as an A-bimodule via r s $ (r j s). This is an instance of Morita equivalence . In general, we say that two algebras A, B are Morita-equivalent if there is a B-A-bimodule E and an A-B-bimodule F such that E A F ' B; F B E ' A (1:2) as B- and A-bimodules respectively. With E = Am and F ' Am , we see that any full matrix algebra over A is Morita-equivalent to A; nontrivial projectors over A oer a host of more \twisted" examples of algebras that are equivalent to A in this sense. The importance of Morita equivalence of two algebras is that their representations match. More precisely, suppose that there is a Morita equivalence of two algebras A and B, implemented by a pair of bimodules E , F as in (1.2). Then the functors H 7! E A H and H0 7! F B H0 implement opposing correspondences between representation spaces of A and B. Moral : if we study an algebra A only through its representations, we must simul- taneously study the various algebras Morita-equivalent to A. In particular, we package together the commutative algebra C 1 (M ) and the noncommutative algebra Mn (C 1 (M )) for the purpose of doing geometry. In the category of C  -algebras, one replaces nite projective modules by arbitrary  C -modules and obtains a much richer theory; see, for instance, [78, 100]. The notion analogous to (1.2) is called \strong Morita equivalence". In particular, let us note that two C  -algebras A and B are strongly Morita equivalent if and only if A K ' B K, where K is the elementary C  -algebra of compact operators on a separable, innite-dimensional Hilbert space [11]. Spinc structures. Returning once more to ordinary manifolds, suppose that M is an n-dimensional orientable Riemannian manifold with a metric g on its tangent bundle TM . We build a Cliord algebra bundle C ` (M ) ! M whose bres are full matrix algebras (over C ) as follows. If n is even, n = 2m, then C ` x (M ) := C`(Tx M; gx) R C ' M2m (C ) is the complexied Cliord algebra over the tangent space Tx M . If n is odd, n = 2m +1, the analogous bre splits as M2m (C )  M2m (C ), so we take only the even part of the Cliord algebra: C ` x (M ) := C`even (Tx M ) R C ' M2m (C ). The price we pay for this choice is that we lose the Z2-grading of the Cliord algebra bundle in the odd-dimensional case. What we gain is that in all cases, the bundle C ` (M ) ! M is a locally trivial eld of (nite-dimensional) elementary C  -algebras. Such a eld is classied, up to equivalence, by a third-degree C ech cohomology class (C ` (M )) 2 H 3(M; Z) called the Dixmier{Douady class [38]. Locally, one nds trivial bundles with bres Sx such that C ` x (M ) ' End(Sx ); the class (C ` (M )) is precisely the obstruction to patching them together (there is no 10 AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY obstruction to the existence of the algebra bundle C ` (M )). It was shown by Plymen [95] that (C ` (M )) = W3 (TM ), the integral class that is the obstruction to the existence of a spinc structure in the conventional sense of a lifting of the structure group of TM from SO(n) to Spinc(n): see [83, Appendix D] for more information on W3 (TM ). Thus M admits spinc structures if and only if (C ` (M )) = 0. But in the Dixmier{ Douady theory, (C ` (M )) is the obstruction to constructing (within the C  -category) a B -A-bimodule S that implements a Morita equivalence between A = C0 (M ) and B = C0 (M; C ` (M )). Let us paraphrase Plymen's redenition of a spinc structure, in the spirit of noncommutative geometry: Denition. Let M be a Riemannian manifold, A = C0 (M ) and B = C0 (M; C ` (M )). We say that the tangent bundle TM admits a spinc structure if and only if it is orientable and (C ` (M )) = 0. In that case, a spinc structure on TM is a pair (; S ) where  is an orientation on TM and S is a B -A-equivalence bimodule. Following an earlier terminology introduced by Atiyah, Bott and Shapiro [2] in their seminal paper on Cliord modules, the pair (; S ) is also called a K -orientation on M . Notice that K -orientability demands more than mere orientability in the cohomological sense. What is this equivalence bimodule S ? By the Serre{Swan theorem, it is of the form (S ) for some complex vector bundle S ! M that also carries an irreducible left action of the Cliord algebra bundle C ` (M ). This is the spinor bundle whose existence displays the spinc structure in the conventional picture. We call (S ) = C 1 (M; S ) the spinor module ; it is an irreducible Cliord module in the terminology of [2], and has rank 2m over C 1 (M ) if n = 2m or 2m + 1. Another matter is how to t into this picture spin structures on M (liftings of the structure group of TM from SO(n) to Spin(n) rather than Spinc(n)). These are distin- guished by the availability of a conjugation operator J on the spinors (which is antilinear); we shall take up this matter later. To summarize : the language of bimodules and Morita equivalence gives us direct access to noncommutative (or commutative) vector bundles without ever invoking the concept of a \principal bundle". Although several proposals for dening a noncommutative principal bundle are available |see, for instance, [62]| for now we must pass them by. The Dirac operator and the distance formula As soon as a spinor module makes its appearance, one can introduce the Dirac operator. This is a selfadjoint rst-order dierential operator D= dened on the space H := L2(M; S ) of square-integrable spinors, whose domain includes the smooth spinors S = C 1 (M; S ). If M is even-dimensional, there is a Z2-grading S = S + S arising from the grading of the Cliord algebra bundle (C ` (M )), which in turn induces a grading of the Hilbert space H = H+  H ; let us call the grading operator , so that 2 = 1 and H are its (1)-eigenspaces. The Dirac operator is fabricated by composing the natural covariant derivative on the modules S  (or just on S in the odd-dimensional case) with the Cliord multiplication by 1-forms that reverses the grading. We repeat that in more detail. The Riemannian metric g = [gij ] denes isomorphisms Tx M ' Tx M and induces a metric g 1 = [gij ] on the cotangent bundle T  M . Via this  1. COMMUTATIVE GEOMETRY FROM THE NONCOMMUTATIVE POINT OF VIEW 11 isomorphism, we can redene the Cliord algebra as the bundle with bres C ` x (M ) := C`(Tx M; gx 1) R C (replacing C` by C`even when dim M is odd). Let A1(M ) := (T  M ) be the A-module of 1-forms on M . The spinor module S is then a B-A-bimodule on which the algebra B = (C ` (M )) acts irreducibly and obeys the anticommutation rule f (); ( )g = 2g 1(;  ) = 2gij i j for ;  2 A1(M ): The action of (C ` (M )) on the Hilbert-space completion H of S is called the spin representation . The metric g 1 on T  M gives rise to a canonical Levi-Civita connection rg : A1(M ) ! A1(M ) A A1(M ) that, as well as obeying the Leibniz rule rg (!a) = rg (!) a + ! da; preserves the metric and is torsion-free. The spin connection is then a linear operator rS : (S ) ! (S ) A A1(M ) satisfying two Leibniz rules, one for the right action of A and the other, involving the Levi-Civita connection, for the left action of the Cliord algebra: rS ( a) = rS ( ) a + da; rS ( (!) ) = (rg !) + (!) rS ; (1:3) for a 2 A, ! 2 A1(M ), 2 S . Once the spin connection is found, we dene the Dirac operator as the composition  rS ; more precisely, the local expression D= := (dxj ) rS@=@xj (1:4) is independent of the coordinates and denes D= on the domain S  H. One can check that this operator is symmetric; it extends to an unbounded selfadjoint operator on H, also called D= . If M is compact, the latter D= is a Fredholm operator. Since the kernel ker D= is nite-dimensional, on its orthogonal complement we may dene D= 1 , which is a compact operator. The distance formula. The Dirac operator may be characterized more simply by its Leibniz rule. Since the algebra A is represented on the spinor space H by multiplication operators, we may form D= (a ), for a 2 A and 2 H. It is an easy consequence of (1.3) and (1.4) that D= (a ) = (da) + a D= : (1:5) This is the rule that we need to keep in mind. We can equivalently write it as [D= ; a] = (da): In particular, since a is smooth and M is compact, the operator k[D= ; a]k is bounded , and its norm is simply the sup-norm kdak1 of the dierential da. This also equals the Lipschitz norm of a, dened as kakLip := sup ja(pd)(p; qa)(q)j ; p6=q 12 AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY where d(p; q) is the geodesic distance between the points p and q of the Riemannian mani- fold M . This might seem to be an unwelcome return to the use of points in geometry; but in fact this simple observation (by Connes) led to one of the great coups of noncommutative geometry [21]. One can simply stand the previous formula on its head: d(p; q) = supf ja(p) a(q)j : a 2 A; kakLip  1 g; = supf j(^p q^)(a)j : a 2 A; k[D= ; a]k  1 g; (1:6) and one discovers that the metric on the space of characters M (A) is entirely determined by the Dirac operator . This is, of course, just a tautology in commutative geometry; but it opens the way forward, since it shows that what one must carry over to the noncommutative case is precisely this operator, or a suitable analogue. One still must deal with the fact that for noncommutative algebras the characters will be scarce. The lesson that (1.6) teaches [24] is that the length element ds is in some sense inversely proportional to D= ; we shall return to this matter later. For a general overview of the many ways in which the noncommutative point of view enriches our insight at all levels: measurable, topological, dierential and metric, consult the recent review [67]. The ingredients for a reformulation of commutative geometry in algebraic terms are almost in place. We list them brie y: an algebra A; a representation space H for A; a selfadjoint operator D= on H; a conjugation operator J , still to be discussed; and, in even- dimensional cases, a Z2-grading operator  on H. This package of four or ve terms is called a real spectral triple or a real K -cycle or, more simply, a geometry. Our task will be to study, to exemplify, and if possible, to parametrize these geometries. 2. Spectral Triples on the Riemann Sphere We now undertake the construction of some spectral triples (A; H; D; J; ) for a very familiar commutative manifold, the Riemann sphere S2. This is an even-dimensional Rie- mannian spin manifold, indeed it is the simplest nontrivial representative of that class. Nevertheless, the associated spectral triples are not completely transparent, and their con- struction is very instructive. The sphere S2 can also be regarded as the complex projective line C P 1 , or as the compactied plane C 1 = C [ f1g. As such, it is described by two charts, UN and US , that omit respectively the north and south poles, with the respective local complex coordinates z = ei cot 2 ;  = e i tan 2 ; related by  = 1=z on the overlap UN \ US . We write q(z) := 1 + zz for convenience. The Riemannian metric g and the area form are given by g = d2 + sin2  d2 = 4q(z) 2 dz
dz = 4q( ) 2 d
d; = sin  d ^ d = 2i q(z) 2 dz ^ dz = 2i q( ) 2 d ^ d: Line bundles and the spinor bundle Hermitian line bundles over S2 correspond to nite projective modules over A := C (S2), of rank one ; these are of the form E = pAn where p = p2 = p 2 Mn (A) is a 1 projector of constant rank 1. (Equivalently, E is of rank one if EndA (E ) ' A.) It turns out that it is enough to consider the case p 2 M2(A). We follow the treatment of Mignaco et al [87]. Using Pauli matrices 1 , 2 , 3 , we may write any projector in M2 (A) as   1 1 + n 3 n 1 in 2 p = 2 n + in 1 n = 12 (1 + ~n
~) 1 2 3 where ~n is then a smooth function from S2 to S2. Any homotopy between two such functions yields a homotopy between the corresponding projectors p and q; and one can then construct a unitary element u 2 M4(A) such that u(p  0)u 1 = q  0. Thus inequivalent nite projective modules are classied by the homotopy group 2(S2) = Z, the corresponding integer m being the degree of the map ~n. If f (z) = (n1 + in2 )=(1 n3) is the corresponding map on C 1 after stereographic projection, then m is also the degree of f . As a representative degree-m map, one could choose f (z) = zm or f (z) = 1=zm . Let us examine the projector corresponding to f (z) = z, of degree 1. We get     1 z z  z 1 pB = 1 + zz z 1 = 1 +      ; 1  which is the well-known Bott projector that plays a key r^ole in K -theory [119]. In general, if m > 0, suitable projectors for the modules E(m) , E( m) of degrees m are  m zm   m zm  1 ( z z  ) pm = 1 + (zz)m zm 1 ; p m = 1 + (zz)m zm 1 : 1 (z z  ) 13 14 AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY One can identify E(1) with the space of sections of the tautological line bundle L ! C P 1 (the bre at the point [v] 2 C P 1 being the subspace C v of C 2 ), and E( 1) with the space of sections of its dual, the so-called hyperplane bundle H ! C P 1 . In general, the integer m is the Chern class of the corresponding line bundle [54]. Let us choose basic local sections mN (z), mS ( ) for the module E(m). We take, for m > 0, m     mN (z) := p 1 m z1 ; mS ( ) := p 1  m 1m ; 1 + (zz) 1 + (  ) normalized so that (mN j mN ) = (mS j mS ) = 1. A global section  = fN mN = fS mS is thus determined by a pair of functions fN (z; z) and fS (; ) that are related on the overlap UN \ US by the gauge transformation fN (z; z) = (z =z)m=2fS (z 1 ; z 1 ): (2:1) Denition. The spinor bundle S = S +  S over S2 has rank two and is Z2-graded; the spinor module S = S + S over A is likewise graded by S  := (S  ). With the chosen conventions, we have S + ' E(1), S ' E( 1) . Thus a spinor can be regarded as a pair of functions on each chart, N (z; z) and S (; ), related by + (z; z) = pz=z + (z 1 ; z 1 ); ( z; z  ) = p z=z S (z 1 ; z 1 ): (2:2) N S N The spin connection. This is the connection rS on the spinor module S determined by the Leibniz rule rS ( () ) = (rg ) + () rS ; where rg is the Levi-Civita connection on the cotangent bundle, determined by rgq@z dzq = z dzq ; rgq@z dqz = z dqz ;     (2:3) rgq@z dqz = z dqz ; rgq@z dzq = z dzq ;     and () is the Cliord action of the 1-form  on the spinor , given by the spin representation. Concretely, we may use the gamma-matrices     1 := 0 1 1 0 ; 2 := 0i 0i : It is convenient to introduce the complex combinations  := 21 ( 1  i 2). The grading operator for the spinor module S = S +  S is then given by   3 := i 1 2 = [ +; ] = 10 0 ; 1  2. SPECTRAL TRIPLES ON THE RIEMANN SPHERE 15 and we note that  3 =  . The Cliord action of 1-forms must satisfy f (dz); (dz)g = 2g 1 (dz; dz) = 0; f (dz); (dz)g = 2g 1(dz; dz) = q(z)2 ; so we take simply (dz) := q(z) and (dz) := q(z) +. (This choice eliminates the natural ambiguity of the matrix square root of q(z)2 1, and so is a gauge xing.) Thus  +    +   (dz) = q(z) 0 + ; (dz) = q(z) 0 : (2:4) From (2.3) and (2.4) we get rS@z = @z + 2zq 3; rS@z = @ z 2zq 3: (2:5) These operators commute with 3, and thus act on the rank-one modules S + , S by r@z = @z  2zq ; r@z = @ z  2zq : (2:6) The Dirac operator on the sphere Denition. The Dirac operator D= := (dxj )rS@j on S2 may be rewritten in complex coordinates as D= = (dz) rS@z + (dz) r@Sz = (d ) rS@ + (d) rS@ : Recalling the form (2.5) of the spin connection, we get D= = rSq@z + + rSq@z = (q @z + 12 z 3) + + (q @ z 1 3 2z ) = (q @z 12 z) + (q @ z 1 z) +: 2 The g operator. At this point, it is handy to employ a rst-order dierential operator introduced by Newman and Penrose [92]: 1 1 3=2 1=2 gz 2 z  q @z 2 z = q
@z
q := (1 + zz) @z (2:7) and its complex conjugate gz := q @ z 12 z. Then   D= = gz + gz + = 0 gz : (2:8) gz 0 This operator is selfadjoint, since gz is skewadjoint: h+ j gz i = hgz + j i; 16 AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY on the Hilbert space L2 (C ; 2i q 2 dz ^ dz), in view of gz = q3=2
@z
q 1=2 . The scalar product of spinors is then given by Z h 1 j 2i = h + + j 2 i := ( 1+ 2+ + 1 2 ) : 1 j 2 i+h 1 C D= thus extends to a selfadjoint operator on this Hilbert space of spinors, which we call H := L2(S2; S ). Moreover, 3 extends to a grading operator (also called 3) on H for which H = L2 (S2; S ), and it is immediate that D= 3 = 3D= . Denition. The conjugation operator J on the Hilbert space H of spinors is dened as follows:  +    J := + (2:9) To see that J is well-dened, it suces to recall that the gauge transformation rules for upper and lower spinors are conjugate (2.2). Clearly J 2 = 1 and J is antilinear, indeed antiunitary in the sense that hJ 1 j J 2 i = h 2 j 1i for all 1 ; 2 2 H. Moreover, J anticommutes with the grading: J 3 = 3 J . Finally, J commutes with the Dirac operator: JD= = D= J . Here it is convenient to introduce the antilinear adjoint operator J y, dened by h 1 j J y 2 i := h 2 j J 1i; of course, J y = J 1 = J since J is antiunitary. The desired identity JD= J y = D= now follows from  +  y    gz +   gz   + JD= J = JD= + = J gz  = gz + = D= : In the next chapter we shall see that the three signs that appear in the commutation relations for J , namely J 2 = 1, J 3 = 3 J and JD= = +D= J , are characteristic of dimension two . Denition. We call the data set (C 1 (S2); L2(S2; S ); D= ; 3; J ) the fundamental spec- tral triple, or fundamental K -cycle , for the commutative spin manifold S2. The Lichnerowicz formula. This formula [7] relates the square of the Dirac operator on the spinor module to the spinor Laplacian; the dierence between the two is one quarter of the scalar curvature K of the underlying spin manifold: D= 2 =
S + 41 K . For the sphere S2 with the metric already chosen, the scalar curvature (or Gaussian curvature) is K = gij Rikj k = 2, so that the Lichnerowicz formula in this case is just D= 2 =
S + 21 : (2:10) The spinor is the generalized Laplacian [7] on the spinor module:
S = gij rS@i rS@j k rS
; ij @k which in the isotropic basis f@z ; @ z g reduces to
S = q2 @z @ z + 14 zz + 21 q(z @z z @ z ) 3:  2. SPECTRAL TRIPLES ON THE RIEMANN SPHERE 17 On the other hand, from (2.7) one gets directly   D= 2 = gz gz 0 = ( q 2 @z2 @ 2z + 1 z z + 1 ) + 1 q (z @z z @ z ) 3 0 gz gz 4 2 2 =
S + 12 : Spinor harmonics and the spectrum of D= The eigenspinors of D= can now be found by turning up appropriate matrix elements of well-known representations of SU (2); but a more pedestrian approach is quicker. These eigenspinors appear already in Newman and Penrose [92] under the name spinor harmon- ics , and were further studied by Goldberg et al [55]. Their construction is based on two simple observations. The rst is an elementary calculation with the g operator:
q l zr ( z)s = (l + 21 r)q l zr ( z)s+1 + rq l zr 1 ( z)s ; gz
gz q l z r ( z)s = (l + 12 s)q l zr+1 ( z)s + sq l zr ( z)s 1 ; (2:11) where q = 1 + zz. The rst is easily checked, the second follows by complex conjugation. One sees at once that suitable combinations of the functions q l zr ( z)s, with l and (r s) held xed, will form eigenvectors for the operator D= on account of its presentation (2.8). The other matter is that compatibility with gauge transformationsP of spinors (2.2) imposes restrictions on the exponents l; r; s. Indeed, if (z; z) := r;s0 a(r; s)q lzr ( z)s, then X (z=z)1=2 (z 1 ; z 1 ) = ( 1)l+ a(r; s)q lzl r ( z)l+ s ; 1 2 1 2 1 2 r;s0 so that  2 S + i l + 12 is a positive integer, and a(r; s) 6= 0 only for r = 0; 1; : : :; l 21 and s = 0; 1; : : :; l + 12 . To have  2 S , interchange the restrictions on r and s. If we set m := r s  12 , the corresponding restriction is m = l; l + 1; : : :; l 1; l, a very familiar sight in the theory of angular momentum; but with the important dierence that here l and m are half-integers but not integers , so the corresponding matrix elements do not drop to matrix elements of representations of SO(3). We can now display the spinor harmonics. They form two families, Ylm + and Y , lm corresponding to upper and lower spinor components; they are indexed by l 2 N + 21 = f 21 ; 32 ; 25 ; : : :g; m 2 f l; l + 1; : : :; l 1; lg; and the formulae are l 12 l + 21 zr ( z)s ; X    Y + (z; z) := Clm q l lm r s r s=m 1 2 l + 21 l 12 zr ( z)s ; X    Ylm (z; z) := Clm q l r s (2:12) r s=m+ 1 2 18 AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY where the normalization constants Clm are dened as r s Clm := ( 1)l m 2l4+ 1 (l + m1 )! (l m 1 )! : (l + 2 )! (l 2 )! Eigenspinors. The coecients in (2.12) are chosen so as to satisfy gz + Ylm = (l + 12 )Ylm and z + = (l + 1 )Y : Ylm g 2 lm in view of (2.11). If we then form normalized spinors by  +  +   0 := p1 Ylm Ylm ; Y 00 := p1 Ylm ; 2 Ylm lm 2 Ylm we get an orthonormal family of eigenspinors for the Dirac operator: 0 = (l + 1 )Y 0 ; D= Ylm 00 = (l + 1 )Y 00 ; D= Ylm 2 lm 2 lm where the eigenvalues are nonzero integers and each eigenvalue (l + 21 ) has multiplicity (2l + 1). In fact, these are all the eigenvalues of D= ; for that, we need the following completeness result, established in [55]: Y lm (z; z) Ylm X  (z 0 ; z0 ) =  ( 0 )  (cos  cos 0 ): l;m 00 : l 2 N + 1 ; m 2 f l; : : :; lg g form an orthonormal 0 ; Ylm Consequently, the spinors f Ylm 2 basis for the Hilbert space H = L2 (S2; S ). The spectrum. We have thus computed the spectrum of the Dirac operator: sp(D= ) = f (l + 21 ) : l 2 N + 12 g = Z n f0g; with the aforementioned multiplicities (2l + 1). Notice that, since the zero eigenvalue is missing, the Dirac operator D= is invertible and it has index zero . The Lichnerowicz formula (2.10) gives at once the spectrum of the spinor Laplacian: sp(
S ) = f l2 + l 41 : l 2 N + 12 g: with respective multiplicities 2(2l + 1). Twisted spinor modules To dene other spectral triples over A = C 1 (M ), we may twist the spinor module S by tensoring it with some other nite projective A-module E , the Cliord action on S A E being given by ()( ) := ( () )  for 2 S;  2 E:  2. SPECTRAL TRIPLES ON THE RIEMANN SPHERE 19 We call S A E , with this action of the algebra (C ` (M )), a twisted spinor module . We now show, in the context of our example A = C 1 (S2), how one can create new K -cycles by twisting the fundamental one. However, these K -cycles will not always respect the \real structure" J , as we shall see. We examine rst the case where E = E(m) is a module of sections of a complex line bundle of rst Chern class m. Then the twisted spinor module is also Z2-graded; in fact, S A E(m) ' E(m+1)  E(m 1) . The twisted Dirac operator. The half-spinor modules S  = E(1) have connections r given by (2.6). Now E(m) ' E(1) A


A E(1) (m times) if m > 0 and E(m) ' E( 1) A


A E( 1) (jmj times) if m < 0, so we can dene a connection r(m) on E(m) by jmj X ( ) r (s1


sjmj) := s1


r (sj )


sjmj; m j =1 and from (2.6) it follows that r(@mz ) = @z + m2qz ; r@(mz ) = @ z mz 2q : On the module S A E(m) , we dene the twisted spin connection re S := rS 1 + 1 r(m) . We obtain re S@z = @z + 2zq (m + 3); re @Sz = @ z 2zq (m + 3): The corresponding Dirac operator is

D= m = re Sq@z + + re Sq@z = q(z) @z + 12 (m 1)z + q(z) @ z 12 (m + 1)z + = (gz + 12 mz) + (gz 12 mz) + or more pictorially,    0 1 mz  D= m = D=0+ D=0m = gz 1 mz gz 0 2 : m 2 This extends to a selfadjoint operator on the Z2-graded Hilbert space H(m) where H(m) = L2(S2; Lm1). Computation of the index. Notice that D= +m = q(z)(m+3)=2
@ z
q(z) (m+1)=2 ; so that a half-spinor + lies in ker D= +m if and only if N+ (z; z) = q(z)(m+1)=2 a(z) where a is an entire holomorphic function. The gauge transformation rule (2.1) shows that the function S+ (z 1 ; z 1 ) = (z=z)(m+1)=2 N+ (z; z) is regular at z = 1 only if either a = 0 or m < 0 and a(z) is a polynomial of degree < jmj. Thus dim ker D= +m = jmj if m < 0 and 20 AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY equals 0 for m  0. A similar argument shows that dim ker D= m = m if m > 0 and is 0 for m  0. We conclude that D= m is a Fredholm operator on H(m) , whose index is ind D= m := dim ker D= +m dim ker D= m = m which, up to a sign, is the rst Chern class of the twisting bundle. Incompatibility with the real structure. The twisting by E(m) loses the property of commutation with the spinor conjugation J (2.9). In fact, it is easy to check that gz + 21 mz   JD= mJ y = 0 = D= m : gz + 12 mz 0 In conventional language, we could say that the twisted spinor bundle S Lm is associated to a spinc structure on T S2, and that this is a spin structure only if m = 0. Conjugation by J exchanges the spinc structures, xing only the spin structure; this exemplies the general fact [25] that commutation (or anticommutation) of D= with J picks out a spin structure when these are available. In view of this circumstance, we shall say that J denes a real structure on (A; H). In summary, (C 1 (S2); H(m); D= m; 3) is a (complex) spectral triple, but is not a \real spectral triple" if m 6= 0. A reducible spectral triple The twisted spinor modules discussed above are irreducible for the action of the Clif- ford algebra B = (C ` (S2)). On the other hand, B acts reducibly on the algebra of dierential forms A (S2)) by ()! :=  ^ ! (] )! for  2 A1(S2); where ] is the vector eld determined by ] (f ) := g 1(; df ), f 2 A. On the algebra of forms we can use the Hodge star operator , dened as the involutive A-module isomorphism determined by ?1 = i , ?d = i sin  d (the coecient i is inserted to make ?? = 1); in complex coordinates, ?1 = 2q2 dz ^ dz; ?dz = dz; ?dz = dz: The codierential  = ?d? is the adjoint of the dierential d with respect to the scalar product of forms: Z h j  i = i( 1)k(k 1)=2  ^ ? for ;  2 Ak (S2); (2:13) S2 with which A(S2) may be completed to a Hilbert space L2;(S2) := nk=0 L2;k (S2). L The Hodge{Dirac operator. One can construct a Dirac operator on this Hilbert space by twisting, along the following lines. One can identify A (S2) with S A S 0 as B-A- bimodules, where S 0 denotes the spinor module with the opposite grading: (S 0 ) = S  .  2. SPECTRAL TRIPLES ON THE RIEMANN SPHERE 21 A detailed comparison of these bimodules and their Dirac operators is given in [114]. The spin connection on S 0 is given by (compare (2.5)): rS@z0 = @z 2zq 3; rS@z0 = @ z + 2zq 3; and re := rS 1 + 1 rS0 gives the tensor product connection on S A S 0 . The Dirac operator on this twisted module is then De= := (dz) re @z + (dz) re @z := ( 1) re q@z + ( + 1) re q@z = D= 1 + rSq@z + + rSq@z : 0 0 The Lichnerowicz formula for this operator is [114]: De= 2 =
e + 21 + 12 ( 3 3); (2:14) where the term 21 ( 3 3) is the \twisting curvature" [7]. Ugalde [114] has shown that, via an appropriate A-module isomorphism S A S 0 ' A(S2), the corresponding operator on A(S2) is precisely the operator d + , that we call the \Hodge{Dirac operator". Its square is the Hodge Laplacian
H := (d + )2 = d + d. Under the aforementioned isomorphism, (2.14) transforms to (d + )2 =
H + 21 21 . Spectrum of d + . The eigenforms for the Hodge Laplacian on spheres have been de- termined by Folland [49]. For n = 2, the eigenvalues of
H are f l(l + 1) : l 2 N g with multiplicities 4(2l + 1) for l = 1; 2; 3; : : :; for l = 0, there is a 2-dimensional kernel of har- monic forms, generated by 1 and i . The other eigenforms are interchanged by d and , and so may be combined to get a complete set of eigenvectors for d + ; this yields p sp(d + ) = f  l(l + 1) : l 2 N g; (2:15) with respective multiplicities 2(2l + 1). Grading and real structure. We have two Z2-grading operators at our disposal on the Hilbert space of forms H = L2;(S2): the even/odd form-degree grading " and the Hodge star operator ?. In dierential geometric language, these correspond to selecting the de Rham complex or the signature complex as the object of interest [54]. The Dirac operator (d + ) is odd for either grading. Thus there are two (complex) K -cycles (A; H; d + ; ") and (A; H; d + ; ?) from the A-module of forms. To distinguish them, we look for a real structure J : an antilinear isometry, satisfying J 2 = 1 and JaJ y = a if a 2 C 1 (S2), that commutes with d +  and anticommutes with the grading operator. In particular, J must preserve the two- dimensional space ker(d + ). Since the harmonic forms have even degree, any such J cannot anticommute with the grading ". However, it turns out that the eigenspaces for d+ can be split into selfdual and antiselfdual subspaces of dimension 2l +1 each. One can then nd a conjugation J that does anticommute with the Hodge star operator: J?J y = ?. This yields a real spectral triple: (C 1 (S2); L2;(S2); d + ; ?; J ): This is the \Dirac{Kahler" geometry that has been studied by Mignaco et al [87]. It also appears prominently in [51]. 3. Real Spectral Triples: the Axiomatic Foundation Having exemplied how dierential geometry may be made algebraic in the commu- tative case of Riemannian spin manifolds, we now extract the essential features of this formulation, with a view to relaxing the constraint of commutativity on the underlying algebra. We shall follow quite closely the treatment of Connes in [25, 26], wherein an axiomatic scheme for noncommutative geometries is set forth. Indeed, one could say that these lectures are essentially an extended meditation on those axioms. The data set The fundamental object of study is a K -cycle (so called because it is a building block for a K -homology theory) or a spectral triple . This consists of three pieces of data (A; H; D), sometimes accompanied by other data and J , satisfying several compatibility conditions which we formulate as axioms. If is present, we say that the K -cycle is even , otherwise it is odd . If J is present, we say the K -cycle is real ; if not, we can call it \complex". We shall, however, concentrate on the real case. Denition. An even, real, spectral triple or K -cycle consists of a set of ve objects (A; H; D; J; ), of the following types: (1) A is a pre-C  -algebra ; (2) H is a Hilbert space carrying a faithful representation  of A by bounded opera- tors; (3) D is a selfadjoint operator on H, with compact resolvent; (4) is a selfadjoint unitary operator on H, so that 2 = 1; (5) J is an antilinear isometry of H onto itself. Before introducing the further relations and properties that these objects must satisfy, we make some comments on the data themselves. Pre-C  -algebras. (1) A pre-C  -algebra A is a dense involutive subalgebra of a C  -algebra A that is stable under the holomorphic functional calculus . This means that for any a 2 A and any function f holomorphic in a neighbourhood of the spectrum sp(a) in A, the element f (a) 2 A actually belongs to A. (We may suppose that A has a unit, otherwise we adjoin one and work with the unitized algebras A+ and A+ .) Here f (a) is dened by the Dunford integral I 1 f (a) := 2i f ( )( a) 1 d; where is any circuit winding once T around sp(a). This happens whenever A = 1 k=1 Ak , where the Ak form a decreasingly nested sequence of Banach algebras with continuous inclusions Ak ,! A. For example, if A is the set of smooth elements of A under the action of a one-parameter group of automorphisms with generator L, one can take Ak := Dom(Lk ). In the case A = C 1 (M ) with M a spin manifold, one can use Ak := Dom(D= 2k ). 22  3. REAL SPECTRAL TRIPLES: THE AXIOMATIC FOUNDATION 23 The major consequence of stability under the holomorphic functional calculus is that the K -theories of A and of A are the same. That is to say, the inclusion j : A ! A induces isomorphisms j : K0(A) ! K0 (A) and j : K1(A) ! K1 (A). Thus, one may use the technology of C  -algebraic K -theory [119] with the dense subalgebra A. For more information on this point, see [22, III.C], and the appendices of [17, 19]. (2) When a 2 A and  2 H, we shall usually write a := (a) . On a few occasions, though, we shall need to refer explicitly to the representation . (3) That D has \compact resolvent" means that (D ) 1 is compact, whenever  2= R . Equivalently, D has a nite-dimensional kernel, and the inverse D 1 (dened on the orthogonal complement of this kernel) is compact. In particular, D has a discrete spectrum of eigenvalues of nite multiplicity. This generalizes the case of a Dirac operator on a compact spin manifold; thus the K -cycles discussed here represent \noncommutative compact manifolds". Noncompact manifolds can be treated in a parallel manner by sup- posing that the algebra has no unit, whereupon we require only that for each a 2 A, the operator a(D ) 1 has compact resolvent [24]. On the basis of the distance formula (1.6), we shall interpret the inverse of D as a length element . Since D need not be positive, one may prefer the inverse of its modulus jDj = (D2 )1=2; we shall write ds := jDj 1. (Actually, the point of view advocated in [25] is that one should treat ds as an abstract symbol adjoined to the algebra A and consider D 1 as its representative on H; but we shall ignore this distinction here.) (4) The grading operator , available for even K -cycles, splits the Hilbert space as H = H+  H , where H is the (1)-eigenspace of . In this case, we suppose that the representation of A on H is even and that the operator D is odd , that is, a = a for a 2 A and D = D. We display  this symbolically  as  a 0 (a) = 0 a ; D = D+ 0 ; 0 D where D+ : H+ ! H and D : H ! H+ are adjoints. (5) The real structure J must satisfy J 2 = 1 and commutation relations JD = DJ , J =  J ; for the signs, see the reality axiom below. Its adjoint is J y = J 1 = J . The recipe 0(b) := J(b)J y (3:1) denes an antirepresentation of A, that is, it reverses the product. It is convenient to think of 0 as a true representation of the opposite algebra A0, consisting of elements f a0 : a 2 A g with product a0b0 = (ba)0. Thus we shall usually abbreviate (3.1) to b0 = Jb J y. The important property that we require is that the representations  and 0 commute ; that is, [a; b0] = [a; JbJ y ] = 0 for all a; b 2 A: (3:2)  y When A is commutative, we demand also that J(b )J = (b), whereupon (3.2) is auto- matic. This requires that A act as scalar multiplication operators on the spinor space, as exemplied in x2. The stage is set. We now deal with the further conditions needed to ensure that these data underlie a geometry . 24 AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY Innitesimals and dimension Axiom 1 (Dimension). There is an integer n, the dimension of the K -cycle, such that the length element ds := jDj 1 is an innitesimal of order 1=n. By \innitesimal" we mean simply a compact operator on H. Since the days of Leibniz, an innitesimal is conceptually a nonzero quantity smaller than any positive . Since we work on the arena of an innite-dimensional Hilbert space, we may forgive the violation of the requirement T <  over a nite-dimensional subspace (that may depend on ). T must then be an operator with discrete spectrum, with any nonzero  in sp(T ) having nite multiplicity; in other words, the operator T must be compact. The singular values of T , i.e., the eigenvalues of the positive compact operator jT j := (T  T )1=2, are arranged in decreasing order: 0  1  2 


. We then say that T is an innitesimal of order  if k (T ) = O(k ) as k ! 1: Notice that innitesimals of rst order have singular values that form a logarithmically divergent series : k (T ) = O k1 =) N (T ) :=   X k (T ) = O(log N ): (3:3) k<N The dimension axiom can then be reformulated as: \there is an integer n for which the singular values of D n form a logarithmically divergent series". R R The coecient of logarithmic divergence will be denote by jDj n , where denotes the noncommutative integral ; we shall have more to say about it later. Example. Let us compute the dimension of the sphere S2 from its fundamental K -cycle. From the spectrum of D= we get the eigenvalues of the positive operator D= 2 : sp(D= 2 ) = f (l + 21 )2 : l 2 N + 21 g = f k 2 : k = 1; 2; 3; : : : g where the eigenvalue k 2 has multiplicity 4k = 2(2l + 1). For N = 2M (M + 1), we get N (D= 2) = X 4k  4 log M  2 log N as N ! 1; 2 k<M k so that D= 1 is an innitesimal of order 21 and therefore the dimension is 2 (surprise!). Also, the coecient of logarithmic divergence is Z Z ds2 := D= 2 = 2: As we shall see later on, this coecient equals 1=2 times the area in the case of any 2-dimensional surface, so the area of the sphere is hereby computed to be 4.  3. REAL SPECTRAL TRIPLES: THE AXIOMATIC FOUNDATION 25 Exercise. The Dirac operator for the circle S1 is just i d=d. Use the Fourier series expansion of functions in C 1 (S1) to check that jd=dj 1 is an innitesimal of order 1; the circle thus has dimension 1. } The order-one condition Axiom 2 (Order one). For all a; b 2 A, the following commutation relation holds: [[D; a]; JbJ y ] = 0: (3:4) This could be rewritten as [[D; a]; b0] = 0 or, more precisely, [[D; (a)]; 0(b)] = 0. Using (3.2) and the Jacobi identity, we see that this condition is symmetric in the repre- sentation  and 0, since [a; [D; b0]] = [[a; D]; b0] + [D; [a; b0]] = [[D; a]; b0] = 0: In the commutative case, the condition (3.4) expresses the fact that the Dirac operator is a rst-order dierential operator: [[D= ; a]; JbJ y] = [[D= ; a]; b] = [ (da); b] = 0: (Contrast this with a second-order operator like a Laplacian, that satises [[
; a]; b] = 2g 1(da; db), generally nonzero [7].) We can interpret (3.4) as saying that the operators 0(b) commute with the subalgebra of operators on H generated by all operators (a) and [D; (a)]. This gives rise to a linear representation of the tensor product of several copies of A: D (a a1 a2


an ) := (a) [D; (a1)] [D; (a2)] : : : [D; (an)]; or, more simply, a0 [D; a1] [D; a2] : : : [D; an]. In view of the order one condition, we could even replace the rst entry a 2 A by a b0 2 A A0, writing D ((a b0) a1 a2


an ) := (a)0(b) [D; (a1)] [D; (a2)] : : : [D; (an)]; (3:5) Now Cn (A; A A0) := (A A0) A n is a bimodule over the algebra A, and this recipe represents it by operators on H. Its elements are called Hochschild n-chains with coecients in the A-bimodule A A0. Smoothness of the algebra Axiom 3 (Regularity). For any a 2 A, [D; aT] is1 a bounded operator on H, and both a and [D; a] belong to the domain of smoothness k=1 Dom(k ) of the derivation  on L(H) given by (T ) := [jDj; T ]. In the commutative case, where [D= ; a] = (da), this condition amounts to saying that a has derivatives of all orders, i.e., that A  C 1 (M ). This can be proved with 26 AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY pseudodierential calculus, since the principal symbol of the modulus of the Dirac ope- rator T1 is jDj (x;  ) = j j 1. From there one obtains that all multiplication operators in k k=1 Dom( ) are multiplications by smooth functions. Hochschild cycles and orientation Axiom 4 (Orientability). There is a Hochschild cycle c 2 Zn (A; A A0) whose repre- sentative on H is  (c) = 1;; ifif nn isis even, odd. Here c is a Hochschild n-chain as dened above, and (c) is given by (3.5). We say c is a cycle if its boundary is zero, where the Hochschild boundary operator is b(m0 a1


an ) := m0 a1 a2


an m0 a1 a2


an +


+ ( 1)n 1 m0 a1


an 1 an + ( 1)n an m0 a1


an 1 : (Here m0 2 A A0.) This satises b2 = 0 and thus makes C (A; A A0) a chain complex, whose homology is the Hochschild homology H (A; A A0). (For the full story, see [84] or [120].) Notice that if x = (a b0 ) a1


an , then, by telescoping:
D (bx) = ( 1)n 1 ab0 [D; a1] : : : [D; an 1]an an ab0 [D; a1] : : : [D; an 1] This Hochschild cycle c is the algebraic equivalent of a volume form on our non- commutative manifold. To see that, let us look brie y at the commutative case, where we may replace A A0 simply by A. A dierential form in Ak (M ) is a sum of terms a0 da1 ^


^ dak , but in the noncommutative case the antisymmetry of the wedge product is lost, so we replace such a form with X c0 := ( 1) a0 a(1)


a(n) (3:6)  (sum over n-permutations) in A (n+1) = Cn (A; A). Then bc0 = 0 by cancellation since A is commutative; for instance: b(a a0 a00 a a00 a0 ) = (aa0 a0 a) a00 a (a0a00 a00a0 ) + (a00 a aa00) a0: In the commutative case A = C 1 (M ), chains are represented by Cliord products: D= (a0 a1


an ) = a0 (da1) : : : (dan). The Riemannian volume form on M can be written as = ib(n+1)=2c1 ^


^ n where f1; : : :; ng is an oriented orthonormal basis of 1-forms, and the corresponding cycle c is represented by D= (c) = ib(n+1)=2c (1) : : : (n). It is now an easy exercise in Cliord algebra [83] to check that D= (c) = 1 if n is odd and D= (c) =  if n is even, where  is the grading operator of the spin representation.  3. REAL SPECTRAL TRIPLES: THE AXIOMATIC FOUNDATION 27 Finiteness of the K -cycle Axiom 5 (Finiteness). The space of smooth vectors H1 := T1k=1 Dom(Dk ), is a nite projective left A-module with a Hermitian structure (
j
) given by Z ( j ) dsn := h j i: The representation : A ! L(H) and the regularity axiom already make H1 a left A-module. It is clear how to adapt the denition (1.1) of Hermitian structures for right A-modules to the case of left A-modules; for instance, one has ( j a) = a ( j ), so that the previous equation entails Z a ( j ) dsn := h j ai: (3:7) To see how (3.7) denes a Hermitian structure implicitly, notice that whenever a 2 A then a dsn = ajDj n is an innitesimal of rst order, so that the left hand side is dened provided ( j ) 2 A. As a nite projective left A-module, H1 ' Amp with p =Pp2 = p in some Mm(A), so we can write  2 H1 as a row vector (1; : : :; m) satisfying j j pjk = k . Furthermore, m X ( j ) = j j 2 A: j =1 In the commutative case, Connes's trace theorem (see below) shows that ( j ) is just the hermitian product of spinors given by the metric on the spinor bundle. A point to notice is that Z Z Z a ( j ) dsn = h j ai = ha j i = (a j ) dsn = ( j ) a dsn; R so this axiom implies that (
)jDj n denes a nite trace on the algebra A. As shown by Cipriani et al [15, Prop. 1.6], the tracial property follows from the regularity axiom (one only requires that both a and [D; a] lie in Dom()); this refutes an earlier complaint [117] that one needed an extra assumption of \tameness" on the K -cycle. The existence of a nite trace on A implies that the von Neumann algebra A00 gene- rated by A also has a nite normal trace, so it cannot have components of types I1 , II1 or III [68, x8.5]. The niteness and regularity axioms entail [25] that  1 \  A = T 2 A00 : T 2 Dom(k ) : (3:8) k=1 As such, A becomes automatically a pre-C  -algebra, so this assumption of ours is in fact redundant. 28 AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY Poincare duality and K -theory Axiom 6 (Poincare duality). The Fredholm index of the operator D yields a nonde- generate intersection form on the K -theory ring of the algebra A A0. On a compact oriented n-dimensional manifold M , Poincare duality is usually formu- lated [36] as an isomorphism of cohomology (in degree k) with homology (in degree n k), or equivalently as a nondegenerate bilinear pairing on the cohomology ring H  (M ). (For noncompact manifolds, the pairing is between the ordinary and the compactly supported cohomologies.) If  2 Ak (M ) and  2 An k (M ) are closed forms, integration over M pairs them by Z (; ) 7!  ^ ; M since the right hand side depends only on the cohomology classes of  and  (it vanishes if either  or  is exact), it gives a bilinear map H k (M )  H n k (M ) ! C . Now each Ak (M ) carries a scalar product (
j
) induced by the metric and orientation on M , given by  ^ ? =: k ( j  ) for ;  2 Ak (M ); where k = 1 or i and is the volume form on M . [Compare (2.13)]. This pairing is nondegenerate since Z Z  ^ (k 1 ?) = ( j ) > 0 for  6= 0: M M In view of the existence of isomorphisms between K  (M ) Z Q and H  (M ; Q ) given by the Chern character, one could hope to reformulate this as a canonical pairing on the K -theory ring. This can be done if M is a spinc manifold; the r^ole of the orientation [ ] in cohomology is replaced by the K -orientation, so that the corresponding pairing of K -rings is mediated by the Dirac operator: see [22, IV.1. ] or [19] for a discussion of how this pairing arises. We leave aside the translation from K -theory to cohomology (by no means a short story) and explain brie y how the intersection form may be computed in the K -context. K -theory  of algebras. There are two abelian groups, K0 (A) and K1(A), associated to a pre-C -algebra A, as follows [119]. The group K0(A) gives a rough classication of nite projective modules over A. If M1 (C ) is the algebra of compact operators of nite rank, then M1 (A) = A M1 (C ) is a pre-C  -algebra dense in A K. Two projectors in M1 (A) have a direct sum   p pq = 0 q : 0 Two such projectors p and q are equivalent if p = uqu 1 for some unitary u in some M1 (A) (this makes sense if A is unital; otherwise, we work with A+). Adding the equivalence classes by [p]+[q] := [p  q], we get a semigroup, and the group K0 (A) is the corresponding group of formal dierences [p] [q].  3. REAL SPECTRAL TRIPLES: THE AXIOMATIC FOUNDATION 29 The other group K1(A) is generated by classes of unitary matrices over A. We nest the unitary groups of various sizes by identifying u 2 Um(A) with u  S1 2 Um+k (A), and call u, v equivalent if v 1 u lies in the identity component of U1 (A) := m1 Um (A). The equivalence classes form the discrete group of components of U1 (A): this is K1(A). It turns out that [uv] = [u  v] = [v  u] = [vu], so that K1 (A) is abelian. (This is the standard denition of K1 for a C  -algebra A; to dene it thus for a pre-C  -algebra A is a slight abuse of notation on our part, that amounts to conferring on A a \topological" K -theory. In \algebraic" K -theory, the denition of the K1 group is not the same and may give dierent groups. In fact, one can equivalently dene K1 (A) with invertible rather than unitary matrices, as the quotient of GL1 (A) by its neutral component; for K1alg (A), we forget the topology of GL1 (A) and factor by the smaller subgroup generated by its commutators. See [9, 104] for the algebraic theory.) Both groups are homotopy invariant: if f pt : 0  t  1 g is a homotopy of projectors in M1 (A) and if f ut : 0  t  1 g is a homotopy in U1 (A), then [p0 ] = [p1 ] in K0(A) and [u0 ] = [u1 ] in K1(A). In the commutative case, we have Kj (C 1 (M )) = K j (M ) for j = 0; 1. When A is represented on a Z2-graded Hilbert space H = H+ H , any odd selfadjoint Fredholm operator D on H denes an index map D : K0(A) ! Z, as follows. Denote by a 7! a+  a the representation of Mm(A) on Hm+  Hm = Hm = H 


 H (m times); write Dm := D 


 D, acting on Hm . Then p Dm p+ is a Fredholm operator from Hm+ to Hm , whose index depends only on the class [p] in K0(A). We dene
D [p] := ind( (p)Dm+ (p)): On the other hand, when A is represented on an ungraded Hilbert space H, a selfadjoint Fredholm operator D on H denes an index map D : K1(A) ! Z. Let H> be the range of the spectral projector P> = P(0;1) determined by the positive part of the spectrum of D. Then if u 2 Um(A), P> u P> is a Fredholm operator on Hm> , whose index depends only on the class [u] in K1(A). We dene
D [u] := ind(P> u P> ): Finally, it is possible to work with K0 alone since there is a natural isomorphism (\sus- pension") from K1(A) to K0(A C0 (R )) for any C  -algebra A [119, 7.2.5]. The intersection form. Coming back now to the spectral triple under discussion, we dene a pairing on K(A) = K0(A)  K1(A) as follows. The commuting representations , 0 determine a representation of the algebra A A0 on H by a b0 7 ! aJbJ y = Jb J ya: If [p]; [q] 2 K0(A), then [p q0 ] 2 K0 (A A0), and the intersection form due to D is 
[p]; [q] := D [p q0 ] : Combined with the suspension isomorphism, we get three other maps Ki(A)  Kj (A0) ! Ki+j (A A0) ! Z, where the second arrow is D . 30 AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY Poincare duality is the assertion that this pairing on K(A) is nondegenerate. For the case of the nite-dimensional algebra A = C  H  M3 (C ) that acts on the space of fermions of the Standard Model [106], the intersection form has been computed in [24] (see also [86, x6.2]). For more general nite-dimensional algebras, see [77, 93]. In that context, Poincare duality is a very ecient discriminator that rules out several plausible alternatives to the Standard Model. The real structure Axiom 7 (Reality). There is an antilinear isometry J : H ! H such that the represen- tation 0(b) := J(b)J y commutes with (A), satisfying J 2 = 1; JD = DJ; J =  J; (3:9) where the signs are given by the following tables: n even: n odd: n mod 8 0 2 4 6 n mod 8 1 3 5 7 J 2 = 1 + + J 2 = 1 + + JD = DJ + + + + JD = DJ + + J = J + + These tables, with their periodicity in steps of 8, arise from the structure of real Clif- ford algebra representations that underlie real K -theory. See [2, 83] for the algebraic foun- dation of this real Bott periodicity. We claim that, in the commutative case of Riemannian spin manifolds, one can nd conjugation operators J on spinors that satisfy the foregoing sign rules. Thus, for instance, J 2 = 1 for Dirac spinors over 4-dimensional spaces with Euclidean signature. (We make no attempt to extend the theory to Minkowskian spaces at this stage.) Notice also that the signs for dimension two are those we have used in the example of the Riemann sphere. The Tomita involution. It is time to explain where the antilinear operator J comes from in the noncommutative case. The bicommutant A00 of the involutive algebra A, or more precisely of (A), is a weakly closed algebra of operators on H, i.e., a von Neumann algebra (generally much larger than the norm closure A). Let us assume that H contains a cyclic and separating vector 0 for A00, that is, a vector such that (i) A00 0 is a dense subspace of H (cyclicity) and (ii) a0 = 0 in H only if a = 0 in A00 (separation). A basic result of operator algebras, Tomita's theorem [68, 112], says that the antilinear mapping a0 7 ! a0 (3:10) extends to a closed antilinear operator S on H, whose polar decomposition S = J
1=2 determines an antiunitary operator J : H ! H with J 2 = 1 such that a 7! JaJ y is an isomorphism from A00 onto its commutant A0. (Since these commuting operator algebras  3. REAL SPECTRAL TRIPLES: THE AXIOMATIC FOUNDATION 31 are isomorphic, the space H can be neither too small nor too large; this is what the cyclic and separating vector ensures.) When the state of A00 given by a 7! h0 j a0i is a trace, the Roperator
= S S is just 1, and so the mapping (3.10) is J itself. From (3.7), the trace (
) jDj n on A gives rise to a tracial vector state on A00 . Thus the Tomita theorem already provides us with an antiunitary operator J satisfying [a; JbJ y] = 0; we shall see in the next chapter how to modify it to obtain J 2 = 1 when that is required. We sum up our discussion with the basic denition. Denition. A noncommutative geometry is a real spectral triple (A; H; D; J; ) or (A; H; D; J ), according as its dimension is even or odd, that satises the seven axioms set out above. Riemannian spin manifolds provide the commutative examples. It is not hard to manufacture noncommutative examples with nite-dimensional matrix algebras [77, 93]; these are zero-dimensional geometries in the sense of Axiom 1. In the next chapter we study a more elaborate noncommutative example which, like the Riemann sphere, has dimension two. 4. Geometries on the Noncommutative Torus We turn now to an algebra that is not commutative, in order to see how the algebraic formulation of geometries, as laid out in the previous chapter, allows us to go beyond Riemannian spin manifolds. Of course, the mere act of moving from commutative to non- commutative algebras is a very familiar one: it is the mathematical point of departure for quantum mechanics. As was already made clear by von Neumann in his 1931 study of the canonical commutation relations [91], the Schrodinger representation arises by replac- ing convolution of functions of two real variables by a noncommutative variant nowadays called \twisted convolution" [72]. This was reinterpreted by Moyal [90] as a variant of the ordinary product of functions on a two-dimensional phase space, related to Weyl's method of quantization. We begin, then, with Weyl quantization. We wish to quantize a compact phase space, since our formalism so far relies heavily on compactness, e.g., by demanding that the length element ds = D 1 be a compact operator. The Riemann sphere would seem to be a good candidate, since several studies exist of its quantization both in the Moyal framework [116] and from the noncommutative geometry point of view [60, 85]. However, all of these involve an approximating sequence of algebras rather a single algebra and so do not dene a solitary K -cycle. We turn instead to the torus T2 (with the at metric). Via the Gelfand cofunctor, this is determined by an algebra of doubly periodic functions on R 2 (or on C ). If !1 , !2 are the periods, with ratio  := !2=!1 in the upper half plane C + , so that = > 0, one identies T2 with the quotient space C =(Z!1 + Z!2). These \complex tori" are homeomorphic but are not all equivalent as complex manifolds. In fact, if one chooses to study these tori through the algebras of meromorphic functions with the required double periodicity, the resulting elliptic curves E are classied by the orbit of  under the action  7! (a + b)=(c + d) of the modular group PSL(2; Z) on C + . Algebras of Weyl operators Our starting point is the canonical commutation relations of quantum mechanics on a one-dimensional conguration space R . In Weyl form [34, 42], these are represented by a family of unitary operators on L2 (R): W (a; b) : t 7 ! e iab e2ibt (t a) for a; b 2 R : (4:1) Here  is a nonzero real parameter; the reader is invited to think of  as 1=h, the reciprocal of the Planck constant. The linear space generated by f W (a; b) : a; b 2 R g is an involutive algebra, wherein
[W (a; b); W (c; d)] = 2i sin (ad bc) W (a + c; b + d); so that W (a; b) and W (c; d) commute if and only if (ad bc) 2 Z: (4:2) 32  4. GEOMETRIES ON THE NONCOMMUTATIVE TORUS 33 Moreover, the unitary operators U := W (1; 0) and V := W (0; 1) obey the commutation relation V U = e2i U V : The full set of operators f W (a; b) : a; b 2 R g acts irreducibly on L2 (R), but by restricting to integral parameters we get reducible actions. We examine rst the von Neu- mann algebra N := f W (m; n) : m; n 2 Z g00; whose commutant can be shown [99] to be N0 := f W (r=; s=) : r; s 2 Z g00: The change of scale given by (R )(t) :=  1=2 (t=) transforms these operators to R W (r=; s=) R 1 = W1= (r; s); from which we conclude that N0 ' N1= : The centre Z (N ) = N \ N0 depends sensitively on the value of . Suppose  is rational,  = p=q where p, q are integers with gcd(p; q) = 1. Then Z (Np=q ) = f Wp=q (qm; qn) : m; n 2 Z g00: This commutative von Neumann algebra is generated by the translation (t) 7! (t q) and the multiplication (t) 7! e2ipt (t), and can be identied to the multiplication operators on periodic functions (of period q); thus Z (Np=q ) ' L1 (S1). On the other hand, if  is irrational , then N is a factor , i.e., it has trivial centre Z (N ) = C 1. If we try to apply the Tomita theory at this stage, we get an unpleasant surprise [42]: N has a cyclic vector i jj  1, whereas it has a separating vector i jj  1. This tells us that (if  6= 1) the space L2 (R) is not the right Hilbert space, and we should represent the algebra generated by the W (m; n) somewhere else. To nd the good Hilbert space, we must rst observe that there is a faithful normal trace on N determined by  0(W (m; n)) := 10;; if (m; n) = (0; 0), otherwise. The GNS Hilbert space associated to this trace is what we need. Before constructing it, notice that, for  irrational, N is a factor with a nite normal trace; and it will soon become clear that its relative dimension function [22, V.1. ] has range [0; 1], so that N is a factor of type II1 in the Murray{von Neumann classication. 34 AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY The algebra of the noncommutative torus We now leave the \measure-theoretic" level of von Neumann algebras and focus on the pre-C  -algebra A generated by the operators W (m; n). Since W (m; n) = eimn Um Vn and since we shall need to use a GNS representation, it is better to start afresh with a more abstract approach. We redene A as follows. Denition. For a xed irrational real number , let A be the unital C  -algebra generated by two elements u, v subject only to the relations uu = u u = 1, vv = vv = 1, and vu =  uv where  := e2i : (4:3) Let S (Z2) denote the double sequences a = farsg that are rapidly decreasing in the sense that sup (1 + r2 + s2)k jarsj2 < 1 for all k 2 N : r;s2Z The irrational rotation algebra A is dened as  A := a = Pr;s ars ur vs : a 2 S (Z2) : (4:4) It is a pre-C  -algebra that is dense in A . The product and involution in A are computable from (4.3): ab = r;s ar n;m mn bn;s m ur vs ; a = r;s rs a r; s ur vs: P P (4:5) The Weyl operators U , V are unitary and obey (4.3); thus they generate a faithful concrete representation of this pre-C  -algebra. On the other hand, the relation (4.3) alone generates the algebra C  [u; v] known to quantum-group theorists as the Manin q-plane [71] for q =  2 T. Here we also require unitarity of the generators, so that A is (a completion of) a quotient algebra of this q-plane. The irrational rotation algebra gets its name from another representation on L2(T): the multiplication operator U and the rotation operator V given by (U )(z) := z (z) and (V )(z) := (z) satisfy (4.3). In the C  -algebraic framework, U generates the C  -algebra C (T) and conjugation by V gives an automorphism  of C (T). Under such circumstances, the C  -algebra generated by C (T) and the unitary operator V is called the crossed product of C (T) by  (more pedantically, by the automorphism group f n : n 2 Z g). In symbols, A ' C (T) o Z: The corresponding action by the rotation angle 2 on the circle is ergodic and minimal (all orbits are dense); it is known [96] that the C  -algebra A is therefore simple . One advantage of using the abstract presentation by (4.3) and (4.4) to dene the algebras A is that certain isomorphisms become evident. First of all, A ' A+n for any n 2 Z, since  is the same for both. (Please note, however, that their representations by Weyl operators, while equivalent, are not identical: indeed, V+n = e2int V . Hence the von Neumann algebras N and N+n do not coincide.)  4. GEOMETRIES ON THE NONCOMMUTATIVE TORUS 35 Next, A ' A  via the isomorphism determined by u 7! v, v 7! u. There are no more isomorphisms among the A , however: by computing the K0-groups of these algebras, Rieel [98] has shown that the abelian group Z + Z is an isomorphism invariant of A . Some automorphisms of A are also easy to nd. The map u 7! u 1 , v 7! v 1 is one such. More generally, if a; b; c; d 2 Z, then (u) := ua vb ; (v) := uc vd (4:6) yields (v)(u) = ad bc (u)(v), so this map extends to an automorphism of A if and only if ad bc = 1. For rational , we can dene A in the same way. When  = 0, we identify u, v with multiplications P by z1 = e2i , z2 = e2i on T2 , so that A0 = C 1 (T2 ). The presentation 1 2 a = r;s ars ur vs is the expansion of a smooth function on the torus in a Fourier series : X a(1; 2 ) = ars e2ir e2is ; 1 2 r;s2Z and (4.6) gives the hyperbolic automorphisms of the torus. For other rational values  = p=q, it turns out that Ap=q is Morita-equivalent to C (T2 ); for an explicit construction of the equivalence bimodules, we remit to [101]. The 1 algebras Ap=q , for 0  p=q < 21 , are mutually non-isomorphic. The normalized trace. The linear functional  : A ! C given by 0(a) := a00 P is positive denite since 0(a a) = r;s jarsj2 > 0 for a 6= 0; it satises 0(1) = 1 and is a trace, since 0(ab) = 0 (ba) from (4.5). Also, 0 extends to a faithful continuous trace on the C  -algebra A ; and, in fact, this normalized trace on A is unique. The Weyl operators (4.1) allow us to quantize A0 in such a way that a(1 ; 2) is the R symbol of the operator a 2 A . Then it can be proved [34, Thm. 3] that 0(a) = T a(1 ; 2 ) d1 d2 , so that 0 is just the integral of the classical symbol. 2 The GNS representation space H0 = L2(A ; 0) may be described as the completion of the vector space A in the Hilbert norm p kak2 := 0 (aa): Since 0 is faithful, the obvious map A ! H0 is injective; to keep the bookkeeping straight, we shall denote by a the image in H0 of a 2 A . The GNS representation of A is just (a): b 7! ab. Notice that the vector 1 is obviously cyclic and separating, so the Tomita involution is given by J0 (a) := a : The commuting representation 0 is then given by 0(a) b := J0 (a)J0y b = J0 ab = ba: 36 AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY To build a two-dimensional geometry, we need to have a Z2-graded Hilbert space on which there is an antilinear involution J that anticommutes with the grading and satises J 2 = 1. There is a simple device that solves all of these requirements: we simply double the GNS Hilbert space by taking H := H0  H0 and dene   J := J0 J0 : 0 0 It remains to introduce the operator D. Before doing so, we make a brief excursion into two-dimensional topology. The skeleton of the noncommutative torus An ordinary 2-torus may be built up from the following ingredients: (i) a single point; (ii) two lines, adjoined at their ends to the point to form a pair of circles; and (iii) a plane sheet, attached along its borders to the two circles. If we take the torus apart again, we get its \cell decomposition" into a 0-cell, two 1-cells and a 2-cell. In more technical language, these cells form the skeleton of the torus, and are repre- sented by independent homology classes: one in H0(T2 ), two in H1 (T2 ) and one in H2 (T2 ). The Euler characteristic of the torus is then computed as 1 2 + 1 = 0. Guided by the Gelfand cofunctor, homology of spaces is replaced by cohomology of algebras; thus the skeleton of the noncommutative torus will consist of a 0-cocycle, two 1-cocycles and a 2-cocycle on the algebra A . The appropriate theory for that is cyclic cohomology [9, 19, 22]. It is a topological theory insofar as it depends only on the algebra A and not on the geometries determined by its K -cycles. Denition. A cyclic n-cochain over an algebra A is an (n + 1)-linear form : An+1 ! C that satises the cyclicity condition (a0; : : :; an 1; an) = ( 1)n (an ; a0; : : :; an 1): It is a cyclic n-cocycle if b = 0, where the coboundary operator b is dened by b(a0; : : :; an+1) := (a0 a1; a2; : : :; an+1)


+ ( 1)j (a0 ; : : :; aj aj+1; : : :; an+1) +


+ ( 1)n+1 (an+1 a0; a1; : : :; an): One checks that b2 = 0, so that the cyclic cochains form a complex whose n-th cyclic cohomology group is denoted HC n (A). The normalized trace 0 is a cyclic 0-cocycle on A , since b0(a; b) := 0(ab) 0(ba) = 0: In fact, a cyclic 0-cocycle is clearly the same thing as a trace. The uniqueness of the normalized trace shows that HC 0 (A ) = C [0 ].  4. GEOMETRIES ON THE NONCOMMUTATIVE TORUS 37 The two basic derivations. The partial derivatives @=@1, @=@2 on the algebra of Fourier series A0 = C 1 (T2 ) can be rewritten as 1 (ars ur vs) := 2ir ars ur vs ; 2 (ars ur vs) := 2is ars ur vs : (4:7) These formulae also make sense on A , and dene derivations 1 , 2 , i.e., j (ab) = (j a) b + a (j b); j = 1; 2: They are symmetric, i.e., (j a) = j (a). Each j extends to an unbounded operator on A whose smooth domain is exactly A . Notice that  (1 a) =  (2a) = 0 for all a. The two cyclic 1-cocycles we need are then given by: 1 (a; b) := 0 (a 1 b); 2 (a; b) := 0 (a 2 b): These are cocycles because 1 , 2 are derivations: b j (a; b; c) = 0(ab j c a j (bc) + a (j b)c) = 0: It turns out [19] that HC 1 (A ) = C [ 1 ]  C [ 2 ]. Next, there is a 2-cocycle obtained by promoting the trace 0 to a cyclic trilinear form: S0(a; b; c) := 0(abc): In fact, one can always promote a cyclic m-cocycle on an algebra to a cyclic (m +2)-cocycle by the periodicity operator S of Connes [22, III,1. ]. For instance, for m = 1 we get S j (a; b; c; d) := j (abc; d) j (ab; cd) + j (a; bcd): However, there is another cyclic 2-cocycle that is not in the range of S : (a; b; c) := 21i 0(a 1b 2 c a 2b 1c): (4:8) Its cyclicity (a; b; c) = (c; a; b) and the condition b = 0 are easily veried. It turns out that HC 2 (A ) = C [S0 ]  C []. For m  3, the cohomology groups are stable under repeated application of S , i.e., HC (A ) = S (HC m 2(A )) ' C  C . The inductive limit of these groups yields a Z2- m graded ring HP 0(A )  HP 1 (A ) called periodic cyclic cohomology , with HP 0 generated by [0] and [], while HP 1 is generated by [ 1] and [ 2]. (This ring is the range of the Chern character in noncommutative topology: for that, we refer to [9, 89] or to [22, III].) In this way, the four cyclic cocycles dened above yield a complete description, in algebraic terms, of the homological structure of the noncommutative torus. The cocycle (4.8) plays an important r^ole in the theory of the integer quantum Hall eect [22, IV.6. ]: for a comprehensive review, see Bellissard et al [5]. In essence, the Brillouin zone T2 of a periodic two-dimensional crystal may be replaced in the nonperiodic case by a noncommutative Brillouin zone that is none other than the algebra A (where  is a magnetic ux in units of h=e). Provided that a certain parameter  (the Fermi level) lies in a gap of the spectrum of the Hamiltonian, with corresponding spectral projector E 2 A , the Hall conductivity is given by the Kubo 2  formula: H = (e =h) (E ; E ; E ). What happens is that (E ; E; E) = []; [E] where the latter is an integer-valued pairing of HP 0 (A ) with K0(A ), so H is predicted to be an integral multiple of e2 =h. 38 AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY A family of geometries on the torus We search now for suitable operators D so that (A ; H; D; J; ) is a two-dimensional geometry. Here is the grading operator on H = H+  H where H+ , H are two copies of L2 (A ; 0). The known operators on H are       = 10 01 ; (a) = a0 a0 ; J = J0 0J0 : 0 Also, in order that D be selfadjoint and anticommute with , it must be of the form  y 0 D= @ 0 ; @ for a suitable closed operator @ on L2(A ; 0). The order-one axiom demands that [D; (a)] commute with each 0(b) = J(b)J y, so that [D; (a)] 2 (0(A ))0 = (A )00 , by Tomita's theorem. The regularity axiom and the niteness property (3.8) now show that 1 \ [D; (a)] 2 (A )00 \ Dom(k ) = (A ): k=1 We conclude that  y    [D; (a)] = [@;0a] [@ 0; a] = @a 0 @ a 0 where @ , @  are linear maps of A into itself. Let us assume also that @ (1) = @ y (1) = 0. It follows that @ (b) = [@; b](1) = @b, and that 0(a @ b) = ha j @ y bi = h@ a j bi = 0((@a) b): (4:9) Since [D; (ab)] = [D; (a)] (b) + (a) [D; (b)], the maps @ , @  are derivations of A . Then @ (1) = 0 and (4.9) shows that   @  = 0, so 0 (b (@a)) = 0((@ b) a) = 0(b @ a); and therefore (@a) = @  a . The reality condition JDJ y = D is equivalent to the condition that J0 @ J0 = @ y on H0 = L2(A ; 0). Since [J0 @ J0 ; a] = (@a) = @ a = [@ y ; a]; the operator J0 @ J0 + @ y commutes with A00 and kills the cyclic vector 1, so it vanishes identically. The derivation @ . For concreteness, we take @ to be a linear combination of the basic derivations 1 , 2 of (4.7). Apart from a scale factor, the most general such derivation is @ = @ := 1 + 2 with  2 C : (4:10)  4. GEOMETRIES ON THE NONCOMMUTATIVE TORUS 39 (We shall soon see that real values of  must be excluded.) Also, since we could replace @ by  1 @ = 2 +  1 1 , we may assume that = > 0. It follows from (4.9) that @ = 1 2 . To verify that this putative geometry is two-dimensional, we must check that ds = D 1 is an innitesimal of order 21 . Notice that D2 = @ y @   @  @ y and that the vectors um vn form an orthonormal basis of eigenvectors for both @ y @  and @  @ y . In fact, @@ (umvn ) = @ @(umvn ) = (1 + 2 )(1 + 2 )(umvn ) = 42jm + n j2 um vn : Thus D 2 has a discrete spectrum of eigenvalues (42) 1 jm + n j 2, each with multipli- city 2, and hence is a compact operator. Now the Eisenstein series X0 1 G2k ( ) := ( m + n ) 2k ; m;n with primed summation ranging over integer pairs (m; n) 6= (0; 0),Pconverges absolutely for k > 1 and only conditionally for k = 1. We shall see below that 0 m;n jm + n j 2 in fact diverges logarithmically, thereby establishing the two-dimensionality of the geometry. The orientation cycle. In terms of the generators u = e2i , v = e2i of A0, the usual 1 2 volume form on the torus T2 is d1 ^ d2 = (2i) 2u 1 v 1 du ^ dv, with the corresponding Hochschild cycle
(2i) 2 u 1v 1 u v u 1 v 1 v u : (4:11) In A , this formula must be modied since u 1 and v 1 do not commute. Consider c 2 C2 (A ; A A0 ) of the form c := m u v n v u: Then bc = (mu v m uv + vm u) (nv u n vu + un v); so that c is a 2-cycle if and only if mu = un, vm = nv and m = n in A A0 . For instance, since A ' A 0(1)  A A0 , we can take m = v 1 u 1 , n = u 1 v 1 with a suitable constant  2 C . The representative (c) on H satises  1 (c) = (v 1u 1 ) [D ; (u)] [D ; (v)] (u 1 v 1) [D ; (v)] [D ; (u)]  1 1  1 v 1 @ v @ u  v u @  u @  v u 0 = 0 v 1 u 1 @ u @v u 1 v 1 @ v @u : Since @ u = 2iu; @u = 2iu; @ v = 2iv; @v = 2iv; this reduces to      0 (c) = 4  0   = 42( ) : 2 40 AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY Thus the orientation cycle is given by c := 42(1 ) (v 1 u 1 u v u 1 v 1 v u): (4:12) This makes sense only if   6= 0, i.e.,  2= R , which explains why we chose = > 0. Thus (= ) 1 is a scale factor in the metric determined by D . There is, however, a dierence with the commutative volume form (4.11): since v 1 u 1 =  u 1 v 1, there is also a phase factor  = e2i in the orientation cycle. The area of the noncommutative torus. To determine the total area, we compute the coecient of logarithmic divergence of the series given by sp(D 2 ). Partially summing over lattice points with m2 + n2  R2 , we get, with  = s + it: Z X0 D 2 = 42 2 Rlim 1 !1 2 log R m +n R jm + n j2 1 2 2 2 Z R Z  1 = 42 Rlim 1 r dr d !1 log R 1 r Z  2  (cos  + s sin )2 + t2 sin2  = 41 2 d  (cos  + s sin )2 + t2 sin2  = 42 t = ( i ) ; 1  2  R after an unpleasant contour integration. The area is then 2 D 2 = 1== . This area is inversely proportional to the area of the period parallelogram of the elliptic curve E with periods f1;  g. A second method of computing the area relies on the existence of a Chern character homomorphism from K -homology (a classication ring for K -cycles) to cyclic cohomology: see [22, IV.1] for a brief discussion of this. The general theory suggests that the area will be given by pairing the orientation class [c] 2 H2(A ; A A0 ) with a class [] 2 HC 2 (A ), where  is the cocycle (4.8) that represents the highest level of the skeleton of the torus. (This is the image under the Gelfand cofunctor of the familiar process of integrating the volume form over the fundamental cycle of the manifold.) At the level of cocycles and cycles, the pairing is dened by h; a0 a1 a2i := (a0; a1; a2). Thus h; ci = 42(1 ) (v 1 u 1 ; u; v) (u 1 v 1 ; v; u)
(2 i ) 1
= 42( ) 0 v 1u 1 (1 u 2 v 2 u 1 v) u 1 v 1 (1 v 2 u 2 v 1 u) Z 2 i = 42( ) 0(v u uv + u v vu) = ( ) = D 2 : 1 1 1 1 i K -theory and Poincare duality. The noncommutative torus provides an example of a pre-C  -algebra, which is neither commutative nor approximately nite but has an inter- esting and computable K -theory [98]. The group K1(A ) is fairly easy to nd. There are  4. GEOMETRIES ON THE NONCOMMUTATIVE TORUS 41 two generating unitaries, u and v, and all the um vn are mutually non-homotopic in U (A ). Indeed, since 0(1) = 1 and 0(um vn ) = 0 for (m; n) 6= (0; 0), there cannot be a continuous path in U (A ) from 1 to um vn . Passing to matrices in Mk (A ) cannot remedy this, since the same argument works, with 0 replaced by 0 tr. Thus K1 (A ) = K1(A ) = Z[u]  Z[v]. To determine K0(A ), we seek projectors in Mk (A ) not equivalent to e := 1  0k 1. In fact, due to the irrationality of , such projectors may be found in A itself: the Powers{Rieel projector p 2 A has the characteristic property that 0(p) = . Given this projector, the map m[e]+ n[p] 7! m0 (e) + n0 (p) = m + n denes a map from K0(A ) to Z + Z which, by a theorem of Pimsner and Voiculescu [94], is an isomorphism of ordered groups. P The P projector p is constructed as follows. Write elements P of A  as a = s s fs v , where r fs = r ars u is a Fourier series expansion of fs (t) = r ars e 2 irt 1 in C (T). Now we look for p of the form p = gv + f + hv 1 : Since p = p, the function f is real and h(t) = g(t + ). Assuming 21 <  < 1, as we may, we choose f to be a smooth increasing function on [0; 1 ], dene f (t) := 1 if t 2 [1 ; ], and f (t) := 1 f (t ) ifpt 2 [; 1]; then let g be the smooth bump function supported in [; 1] given by g(t) := f (t) f (t)2 for   t  1. One checks that these conditions guarantee p2 = p (look at the coecients of v2 , v and 1 in the expansion of p2 ). Moreover, Z 1 Z 1  Z 1 0(p) = a00 = f (t) dt = f (t) dt + (2 1) + f (t) dt = : 0 0  The existence of these projectors (variation of f on the interval [0; 1 ] gives rise to many homotopic projectors) shows that the topology of the noncommutative torus is very disconnected, in contrast to the ordinary torus A0, whose only projectors are 0 and 1. [The commutative torus C 1 (T2 ) has the same K -groups: Kj (A0) = Z  Z for j = 1; 2. However, this algebra has no Powers{Rieel projector: the second generator of K0(A0) is obtained by pulling back the Bott projector from K0(C 1 (S2)).] Thus, K(A ) has four generators: [e], [p], [u] and [v]. The intersection form is antisymmetric (this is typical of dimension two) and the nonzero pairings of the generators are [25]:     [e]; [p] = [p]; [e] = 1; [u]; [v] = [v]; [u] = 1:   0 1 The matrix of the form is a direct sum of two 1 0 blocks, so it is nondegenerate, and Poincare duality holds. We have constructed a geometry (A ; H; D ; ; J ), which may be denoted by T2; . When  = 0,  plays the r^ole of the modular parameter of an elliptic curve. Whether or not this provides an opening to a noncommutative theory of elliptic curves is a tantalizing speculation; it remains to be seen whether T2; can yield useful arithmetic information. 5. The Noncommutative Integral One of the striking features of noncommutative geometry is how it ties together many mathematical and physical approaches in a single unifying theme. So far, our discussion of the mathematical underpinnings of the geometry has been quite \soft", emphasizing the algebraic character of the overall picture. However, when we examine more closely the interpretation of dierential and integral calculus, we see that the theory requires a considerable amount of analysis, of a fairly delicate nature. In this chapter we take up the matter of how to best dene the noncommutative integral and relate it to conventional integration on manifolds. In the course of the initial development of noncommutative geometry, integration came rst, beginning with Segal's early work with traces on operator algebras [109] and continuing with Connes' work on foliations [18]. The introduction of universal graded dierential algebras [19] shifted the emphasis to dierential calculus based on derivations, which formed the backdrop for the rst applications to particle physics [28, 39]. The pendulum has recently swung back to integral methods, due to the realization [14, 25, 65] that the Yang{Mills functionals could be obtained in this way. The primary tool for that is the noncommutative integral. The Dixmier trace on innitesimals Early attempts at noncommutative integration [109] used the ordinary trace Tr of operators on a Hilbert space as an ersatz integral, where the traceclass operators play the r^ole of integrable functions. R For example, in Moyal quantization one computes expectation values by Tr(AB ) = WA WB d, where WA , WB are Wigner functions of operators A, B and  is the normalized Liouville measure on phase space [12]. However, on the representation space of a noncommutative geometry one needs an integral that suppresses innitesimals of order higher than 1, so Tr will not do; moreover, Tr diverges for positive rst-order innitesimals, since 1 X Tr jT j = !1 n (T ) = 1 k (T ) = nlim if n (T ) = O(log n): k=0 Dixmier [37] found other (non-normal) tracial functionals on compact operators that are more suitable for our purposes: they are nite on rst order innitesimals and vanish on those of higher order. To nd them, we must look more closely at the ne structure of innitesimal operators; our treatment follows [29, Appendix A]. The algebra K of compact operators on a separable, innite-dimensional Hilbert space contains the ideal L1 of traceclass operators, on which kT k1 := Tr jT j is a norm |not to be confused with the operator norm kT k = 0 (T ). Each partial sum of singular values n is a norm on K. In fact, n (T ) = supf kTPn k1 : Pn is a projector of rank n g: 42  5. THE NONCOMMUTATIVE INTEGRAL 43 Notice that n (T )  n 0(T ) = nkT k. There is a very cute formula [29], coming from real interpolation theory of Banach spaces, that combines the last two relations: n (T ) = inf f kRk1 + nkS k : R; S 2 K; R + S = T g: (5:1) This is worth checking. It is clear that if T = R + S , then n (T )  n (R) + n (S )  kRk1 + nkS k. To show that the inmum is attained, we can assume that T is a positive operator, since both sides of (5.1) are unchanged if R, S , T are multiplied on the left by a unitary operator V such that V T = jT j. Now let Pn be the projector of rank n whose range is spanned by the eigenvectors of T corresponding to the eigenvalues P 0 ; : : :; n 1. Then R := (T n )Pn and S := n Pn + T (1 Pn ) satisfy kRk1 = k<n (k n ) = n (T ) nn and kS k = n . We can think of n (T ) as the trace of jT j with a cuto at the scale n. This scale does not have to be an integer; for any scale  > 0, we can dene  (T ) := inf f kRk1 + kS k : R; S 2 K; R + S = T g: If 0 <   1, then  (T ) = kT k. If  = n + t with 0  t < 1, one checks that  (T ) = (1 t)n (T ) + tn+1 (T ); (5:2) so  7!  (T ) is a piecewise linear, increasing, concave function on (0; 1). Each  is a norm by (5.2), and so satises the triangle inequality. For positive compact operators, there is a triangle inequality in the opposite direction:  (A) +  (B )  + (A + B ) if A; B  0: (5:3) It suces to check this for integral values  = m,  = n. If Pm , Pn are projectors of respective ranks m, n, and if P = Pm _ Pn is the projector with range Pm H + Pn H, then kAPmk1 + kBPnk1 = Tr(Pm APm) + Tr(Pn BPn )  Tr(P (A + B )P ) = k(A + B )P k1; and (5.3) follows by taking suprema over Pm , Pn . Thus we have a sandwich of norms:  (A + B )   (A) +  (B )  2 (A + B ) if A; B  0: (5:4) The Dixmier ideal. The rst-order innitesimals can now be dened precisely as the following normed ideal of compact operators: L1+ := T 2 K : kT k1+ := sup log  (T ) < 1 ; n o e  that obviously includes the traceclass operators L1 . (On the other hand, if p > 1 we have L1+  Lp , where the latter is the ideal of those T such that Tr jT jp < 1, for which  (T ) = O(1 1=p).) 44 AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY If T 2 L1+, the function  7!  (T )= log  is continuous and bounded on the interval [e; 1), i.e., it lies in the C  -algebra Cb [e; 1). We can then form the following Cesaro mean of this function: Z   (T ) := log  log 1 u (T ) du : u u e Then  7!  (T ) lies in Cb [e; 1) also, with upper bound kT k1+. From (5.4) we can derive that 0   (A) + (B )  (A + B )  kAk1+ + kB k1+ log 2 logloglog  ;
so that  is \asymptotically additive" on positive elements of L1+ . We get a true additive functional in two more steps. Firstly, let _ (A) be the class of  7!  (A) in the quotient C  -algebra B := Cb [e; 1)=C0[e; 1). Then _ is an additive, positive-homogeneous map from the positive cone of L1+ into B, and _ (UAU 1) = _ (A) for any unitary U ; therefore it extends to a linear map _ : L1+ ! B such that _ (ST ) = _ (TS ) for T 2 L1+ and any S . Secondly, we follow _ with any state (i.e., normalized positive linear form) !: B ! C . The composition is a Dixmier trace : Tr! (T ) := !(_ (T )): The noncommutative integral. Unfortunately, the C  -algebra B is not separable and there is no way to exhibit any particular state. This problem can be nessed by noticing that a function f 2 Cb [e; 1) has a limit lim!1 f () = c if and only if !(f ) = c does not depend on !. Let us say that an operator T 2 L1+ is measurable if the function  7!R (T ) converges as  ! 1, in which case any Tr! (T ) equals its limit. We denote by T the common value of the Dixmier traces: Z T := lim !1   (T ) if this limit exists: We call this value the noncommutative integral of T . Note that if T 2 K and n (T )= log n converges as n ! 1, then T lies in L1+ and is measurable. This was shown to be the case for the operators D= 2 on the Riemann sphere and D 2 on the noncommutative torus, whose integrals we have already computed. We need to do at least one integral calculation in an n-dimensional context. Suppose we try the (commutative!) torus Tn := R n =(2Z)n and consider its Laplacian
:=  @ 2


 @ 2 : 5:5 @x1 @xn Its eigenfunctions are l (x) := eil
x for l 2 Zn. We discard the zero mode 0 and regard
as an invertible operator on the orthogonal complement of the constants in L2(Tn ). The multiplicity m of the eigenvalue  = jlj2 is the number of lattice points R in Zn of length jlj. Thus the operator
s is compact for any s > 0; let us compute
s .  5. THE NONCOMMUTATIVE INTEGRAL 45 If Nr is the total number of lattice points with jlj  r, then Nr+dr Nr  n rn 1 dr and Nr  n 1 n rn , where n=2 n = 2(n=2) = vol(Sn 1) is the volume of the unit sphere. For N = NR we estimate X Z R Z R N (
s ) = jlj 2s  r 2s (Nr+dr Nr )  n rn 2s 1 dr: 1jljR 1 1 Since log NR  n log R, we see that  (
s )= log  diverges if s < n=2 and converges to 0 if s > n=2. For the borderline case s = n=2, we get  (
n=2 )= log   n =n, so  (
n=2 )  n =n also, and thus Z
n=2 = n = n=2 : (5:6) n ( n2 + 1) operator D= = (dxj ) @=@xj on Tn satises D= 2 =
, so we may rewrite (5.6) as RThenDirac R ds = jD= j n = n =n. Pseudodierential operators R In all cases considered up to now, the computation of T has required a complete determination of the spectrum of T . This is usually a fairly onerous task and is most suited to fairly R simple examples. An alternative approach is needed, which may allow us to calculate T by a general procedure. For the commutative case, such a procedure is available: it is the pseudodierential operator calculus. The extension of this calculus to the noncommutative case has already begun and is undergoing rapid development [23, 26, 29], but we cannot report on it here. We shall conne our attention to a fairly familiar case: elliptic classical pseudodierential operators ( DOs) on compact Riemannian manifolds. Denition. A pseudodierential operator A of order d on a manifold M is an operator between two Hilbert spaces of sections of Hermitian vector bundles over M , that can be written in local coordinates as ZZ Au(x) = (2) n ei(x y)
 a(x;  )u(y) dny dn ; where the symbol a(x;  ) is a matrix of smooth functions whose derivatives satisfy the growth conditions j@x @ aij (x;  )j  C (1 + j j)d jj. We simplify a little by assuming that a is scalar-valued, i.e., that the bundles are line bundles. We then say that A is a classical DO, written A 2 d (M ), if its symbol has an asymptotic expansion of the form 1 X a(x;  )  ad j (x;  ) (5:7) j =0 46 AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY where each ar (x;  ) is r-homogeneous in the variable  , that is, ar (x; t )  tr ar (x;  ). We refer to [83, 113] for the full story of these operators. Although the terms of the expansion (5.7) are generally coordinate-dependent, the principal symbol ad (x;  ) is globally dened as a function on the cotangent bundle T M , except possibly on the zero section M . We call the operator A elliptic if ad (x;  ) is invertible for  6= 0. The spaces d (M ) are decreasingly nested, the intersection being the smoothing ope- rators 1 (M ). Clearly, the symbol a determines the operator A up to a smoothing operator. The quotient algebra P := 1 (M )= 1 (M ) is called, a little improperly, the algebra of classical pseudodierential operators on M . The product AB = C of DOs corresponds to the composition of symbols given by the expansion X ( i)jj c(x;  )   ! @a @xb: 2Nn From the leading term, we see that if A 2 d (M ), B 2 r (M ), then AB 2 d+r (M ). Also, if P = [A; B ] is a commutator in 1 (M ), then X ( i)jj
p(x;  )   ! @a @xb @b @xa : (5:8) jj>0 The Wodzicki residue If M is an n-dimensional manifold, the term of order ( n) of the expansion (5.7) has a special signicance. It is coordinate-dependent, so let us x a coordinate domain U  M over which the cotangent bundle is trivial, and consider a n (x;  ) as a smooth function on T  U n U (i.e., we omit the zero section). Then  := a n (x;  ) d1 ^


^ dn ^ dx1 ^


^ dxn P is invariant under the dilations  7! t of the cotangent spaces. Thus, if R = j j @=@j is the radial vector eld on T  U n U that generates these dilations, then d R  = LR  = 0; so R  is a closed (2n 1)-form on T  U n U . (Here LR = R d + d R denotes the Lie derivative.) On abbreviating dn x := dx1 ^


^ dxn , we nd that n X R  = a n (x;  )  ^ dn x; with  := ( 1)j 1 j d1 ^


^ d cj ^


^ dn : j =1 Of course,  restricts to the volume form on the unit sphere j j = 1 in each Tx M . On integrating R  over these spheres, we get a quantity that transforms under coordinate changes x 7! y = (x),  7!  = 0 (x)t , a n (x;  ) 7! a~ n (y; ) as follows [47]: Z Z a~ n (y; )  = j det 0 (x)j a n (x;  )  : (5:9) jj=1 jj=1 The absolute value of the Jacobian det 0 (x) appears here because if 0 (x)t reverses the orientation on the unit sphere in Tx M then the integral over the sphere also changes sign.  5. THE NONCOMMUTATIVE INTEGRAL 47 The Wresidue density. As a consequence of (5.9), we get a 1-density on M , denoted wres A, whose local expression on any coordinate chart is Z  wresx A := a n (x;  )  dx1 ^


^ dxn : jj=1 This is the Wodzicki residue density . By integrating this 1-density over M , we get the Wodzicki residue [47, 70, 122]: Z Z Z Wres A := wres A = R  = a n (x;  )  dn x: (5:10) M SM SM Here SM := f (x;  ) 2 T  M : j j = 1 g is the \cosphere bundle" over M . The integral (5.10) may diverge for some A; we shall shortly identify its domain. (In the literature, this Wresidue is commonly written res A; we adjoin the W |for Wodzicki| to distinguish the density from the functional.) The tracial property of the Wresidue. It turns out that Wres is a trace on the algebra P of classical pseudodierential operators, i.e., that Wres[A; B ] = 0 always, provided that dim M > 1. The reason is that each term in the expansion is a nite sum of derivatives @p=@xj + @q=@j . For instance, the leading term of (5.8) is       @a @b i @ @b @a = @ ia @b @ ia @b : j @xj @j @xj @xj @j @j @xj We can assume that a, b are supported on a compact subset of a chart domain U of M (since we can later patch together with a partition of unity), so that all ( n)-homogeneous terms of type @p=@xj have zero integral over SM . Since we are integrating a closed (2n 1)-form over Sn 1  U , we get the same result by integration over the cylinder Sn 2  R  U : these are cohomologous cycles in (R n n f0g)  U . For any term of the form @q=@j , where q is ( n + 1)-homogeneous, we then get Z Z 1 @q =  Z @q d  0 = 0 j  jj=1 @j j0 j=1 1 @j if  0 := (1; : : :; bj ; : : :; n), since q(x;  ) ! 0 as j ! 1 because n + 1 < 0. The crucial property of Wres is that, up to scalar multiples, it is the unique trace on the algebra P . We give the gist of the beautiful elementary proof of this by Fedosov et al [47]. From the symbol calculus, derivatives are commutators, since @a ; [xj ; a] = i @ @a [j ; a] = i @x j j in view of (5.8). Hence any trace T on symbols must vanish on derivatives. For r 6= n, each r-homogeneous term ar (x;  ) is a derivative, since @=@j (j ar ) = (n + r)a by Euler's 48 AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY R theorem. Furthermore, after averaging over spheres, a n (x) := n 1 jj=1 a n (x;  )  , one can show that the centred ( n)-homogeneous term a n (x;  ) a n (x) j j n is a nite sum of derivatives. The upshot is that T (a) = T (a n (x) j j n) is a linear functional of a n (x) that kills derivatives, so it must be of the form Z T (a) = C a n (x) dnx = C Wres A: U For more general classical DOs with matrix-valued symbols, the same arguments are applicable, provided we replace a n (x;  ) by its matrix trace tr a n (x;  ) throughout. The formula Z Wres A :=  tr a n (x;  )  dx1 ^


^ dxn (5:11) S M then denes a unique (up to multiples) trace on the algebra of classical DOs whose coecients are endomorphisms of a given vector bundle over M . The trace theorem This uniqueness of the trace was exploited by Connes [20], who saw how the Dixmier traces t into this picture. The point is that pseudodierential operators of low enough order over a compact manifold are already compact operators [113], so that any Dixmier trace Tr! denes a trace on n (M ); and they all dene the same trace, since they co- incide on measurable operators. Thus all DOs of order ( n) are measurable, and the noncommutative integral is a multiple of Wres. It remains only to compute the propor- tionality constant. Moreover, since we can reduce to local calculations by patching with partitions of unity, this constant must be the same for all manifolds of a given dimension. To nd it, we can use the power
n=2 of the Laplacian on the torus Tn , whose noncommutative integral we already know (5.6). Now
is a dierential operator (5.5), with symbol b(x;  ) = j j2. Thus
n=2 is a DO with symbol j j n, which is of course ( n)-homogeneous; and better yet, j j n  1 on the cosphere bundle SM . Thus Z Z Wres
n=2 =  dn x = (2)n n : Tn jj=1 On comparing (5.6), we see that the proportionality constant is 1=n(2)n, that is, Z A = n (21)n Wres A; (5:12) for any DO of order ( n) or lower. This is Connes' trace theorem [20]. We remark that the Wodzicki residue is sometimes written with a factor n (2)n so that the noncommutative and the adjusted Wresidue coincide; that was the convention adopted in [117].  5. THE NONCOMMUTATIVE INTEGRAL 49 The commutative integral. Suppose that M is an n-dimensional spin manifold, with Dirac operator D= . The operator jD= j n is a rst-order innitesimal and is also a pseudodif- ferential operator of order ( n). Indeed, D= acts on spinors with the symbol i ( ) where denotes the spin representation, so D= 2 has symbol g 1(;  ) 1N , where g denotes the Riemannian metric; this is a scalar matrix whose size N is the rank of the spinor bundle. Recall that N = 2m if n = 2m or n = 2m + 1 (see the discussion of spinc structures in x1). The symbol of jD= j n = (D= 2) n=2 is then given locally by p det g(x) j j n 1N : More generally, when a 2 C 1 (M ) is represented as a multiplication operator on the spinor space H, thep operator a jD = j n is also pseudodierential of order ( n), with symbol  ) := a(x) det g(x) j j n 1N . Let us be mindful that the Riemannian volume form a n (x; p is = det g(x) dx1 ^


^ dxn . Invoking (5.12) and (5.11), we end up with Z a jD= j n= 1 Wres a jD= j n = 1 Z tr a (x;  )  dn x n (2)n n (2)n SM n  2 bn=2c n Z = n (2)n a(x) : M Thus the commutative integral on functions, determined by the orientation [ ], is 8 Z Z < m! (2 ) > m a D= 2m if dim M = 2m is even, a = > Z M : (2m + 1)!!  m+1 a jD= j 2m 1 if dim M = 2m + 1 is odd. In particular, since orientable 2-dimensional R manifolds always admit a spin structure [54], 2 the area of a surface (m = 1) is 2 D= , as we had previously claimed. Integrals and zeta residues We have not explained why the Wodzicki functional is called a residue. It was orig- inally discovered as the Cauchy residue of a zeta function: see Wodzicki's introductory remarks in [122]. Indeed, the following formula can be established [117] with the help of Seeley's symbol calculus for complex powers of an elliptic DO [108]: Ress=1 A (s) = n (21)n Wres A; where the zeta function of a positive compact operator A with eigenvalues k (A) may be dened as 1 X A (s) := k (A)s for <s > some s0 k=1 and extended to a meromorphic function on C by analytic continuation. 50 AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY In view of (5.12), it should be possible to prove directly that this zeta residue actually coincides with the noncommutative integral of measurable innitesimals: Z Ress=1 A(s) = A: (5:13) This can indeed be achieved by known Tauberian theorems; see, for instance, [22, IV.2. , Proposition 4] or [117, x2]. Rather than repeat these technical proofs here, we give instead a heuristic argument based on the delta-function calculus of [45], that shows why (5.13) is to be expected. Let us take A to be a compact positive operator whose eigenvalues R satisfy k (A)  L=k as k ! 1. Then n (A)  L log n, so that A is measurable with A = L. In the particular case of the operator R for which k (R) = 1=k for all k, R (s) is precisely the Riemann zeta function. P On the other hand, let us examine an interesting distribution on R , P the \Dirac comb" k2Z  (x k). It is periodic and its mean value is 1; therefore f (x) := k2Z  (x k) 1 is a periodic R1 distribution of mean zero. It can then be shown [44] that the moments m := 1 f (x) xm dx exist for all m. The same is true if we cut o the x-axis at x = 1: the distribution 1 X fR (x) := (x k) (x 1) k=1 has moments of all orders, and the function Z 1 Z 1 ZR (s) := fR (x) x s dx = X 1 x s dx = R (s) s 1 1 s R k=1 k 1 is an entire analytic function of s. (Notice how this argument shows that R is meromorphic with a single simple pole at s = 1, whose residue is 1.) Now since L=k (A)  k as k ! 1, we replace (x k) by (x L=k ) and dene 1 X fA (x) := (x L=k ) (x 1): k=1 This suggests that Z 1 Z 1 x s dx = L sA (s) s 1 1 : X ZA (s) := fA (x) x s dx = L s sk R k=1 1 From this we conclude that A (s) = LsZA (s) + Ls=(s 1) is meromorphic, analytic for <s > 1, and has a simple pole at s = 1 with residue Z Ress=1 A (s) = L = A: This surprising nexus between the Wodzicki functional, the noncommutative integral and the zeta-function residue suggests that the rst two may be protably used in quantum eld theory; see [40] for a recent example of that. 6. Quantization and the Tangent Groupoid Before embarking on the classication of geometries, let us rst explore an issue of a more topological nature, namely, the extent to which noncommutative methods allow us to achieve contact with the quantum world. To begin with, the facile but oft-repeated phrase, \noncommutative = quantum", must be disregarded. As the story of the Connes{ Lott model shows, a perfectly noncommutative geometry may be employed to produce Lagrangians for physical models at the classical level only [22, VI.5]; quantization must then proceed from this starting point. Nevertheless, the foundations of quantum mechan- ics do throw up noncommutative geometries, as we have seen with the torus. Also, the integrality features of the pairing of cyclic cohomology with K -theory give genuine exam- ples of quantizing. So, it is worthwhile to ask: what is the relation of known quantization procedures with noncommutative geometry? The rst step in the quantization of a system with nitely many degrees of freedom is to place the classical and quantum descriptions of the system on the same footing; the second step is to draw an unbroken line between these descriptions. In conventional quantum mechanics, the simplest such method is the Wigner{Weyl or Moyal quantization, which consists in \deforming" an algebra of functions on phase space to an algebra of operator kernels . In noncommutative geometry, there is a device that accomplishes this in a most economical manner, namely the tangent groupoid of a conguration space. Moyal quantizers and the Heisenberg deformation Since [4] and [6] at least, deformations of algebras have been related to the physics of quantization. We rst sketch the general scheme [56], and then illustrate it with the simplest possible example, namely, the Moyal quantization in terms of the Schrodinger representation of ordinary quantum mechanics for spinless, nonrelativistic particles. Let X be a smooth symplectic manifold,  (an appropriate multiple of) the associated Liouville measure, and H a Hilbert space. A Moyal quantizer for the triple (X; ; H) is a map
of the phase space X into the space of selfadjoint operators on H satisfying  Tr
(u) = 1; (6:1a) Tr
(u)
(v) = (u v); (6:1b) for u; v 2 X , at least in a distributional sense; and such that f
(u) : u 2 X g spans a weakly dense subspace of L(H). More precisely, the notation \(u v)" means the (distributional) reproducing kernel for the measure . An essentially equivalent denition, in the equivariant context, was introduced rst in [116]. For the proud owner of a Moyal quantizer, all quantization problems are solved in principle. Quantization of any function a on X is eected by Z a 7! Q(a) := a(u)
(u) d(u); (6:2) X and dequantization of any operator A 2 L(H) by:   A 7! WA ; with WA (u) := Tr A
(u) : 51 52 AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY This makes automatic WI = 1. Moreover, we have WQ(a) = a by (6.1b) R and therefore Q(WA) = A by irreducibility, from which it follows that Q(1) = I , i.e., X
(u) d(u) = I . We can reformulate (6.1) as Z Tr Q(a) = a(u) d(u); (6:3a) ZX   Tr Q(a)Q(b) = a(u)b(u) d(u); (6:3b) X for real functions a; b 2 L2(X; d). Moyal quantizers are essentially unique and understandably dicult to come by [56]. The standard example is given by the phase space T  (Rn ) with the canonical symplectic structure. With Planck constant h > 0 and coordinates u = (q;  ), we take d(q;  ) := (2h) n dn q dn  . We also take H = L2(R n ). Then the Moyal quantizer on T  (Rn ) is given by a family of symmetries
h (q;  ); explicitly, in the Schrodinger representation:  h 
(q;  )f (x) = 2n e2i(x q)=h f (2q x): (6:4) The twisted product a h b of two elements a; b of S (T  R n ), say, is dened by the requirement that Q(a h b) = Q(a)Q(b). We obviously have: ZZ a h b(u) = Lh (u; v; w)a(v)b(w) d(v) d(w); with Lh (u; v; w) := Tr
h (u)
h (v)
h (w) = 22n exp 2hi s(u; v) + s(v; w) + s(w; u) ;   n
o R R where s(u; u0) := q 0 q0  is the linear symplectic form on T  R n . Note that a h b = ab. By duality, then, the quantization rule can be extended to very large spaces of functions and distributions [57, 102, 115]. Moyal quantization has other several interesting uniqueness properties; for instance, it can be also uniquely obtained by demanding equivariance with respect to the linear symplectic group (upon introducing the metaplectic representation) [110]. Asymptotic morphisms. Due to its intrinsic homotopy invariance, K -theory is fairly rigid under deformations of algebras. K0 is a functor, so to any -homomorphism of C  -algebras  : A ! B there corresponds a group homomorphism  : K0(A) ! K0(B ). However, there is no need to ask for -homomorphisms; is order to have K -theory maps, it is enough to construct asymptotic morphisms of the C  -algebras. These were introduced by Connes and Higson in [27]. Denition. Let A, B be C  -algebras and h0 a positive real number. An asymptotic morphism from A to B is a family of maps T = f Th : 0 < h  h0 g, such that h 7! Th (a) is norm-continuous on (0; h0 ] for each a 2 A, and such that, for a; b 2 A and  2 C , the following norm limits apply: lim T (a) + Th (b) Th (a + b) = 0; lim h #0 h T (a) Th (a) = 0; h #0 h lim T (ab) Th (a)Th (b) = 0: h #0 h  6. QUANTIZATION AND THE TANGENT GROUPOID 53 Two asymptotic morphisms T , S are equivalent if limh #0(Th (a) Sh (a)) = 0 for all a 2 A. Thus, the equivalence classes of asymptotic morphisms from A to B corresponds to the morphisms from A to the quotient C  -algebra Be := Cb ((0; h0 ]; B )=C0((0; h0]; B ) by setting Te(a)h := Th (a). We remark that our Th is the 1=h of [22, 27]. In most cases we shall have only a preasymptotic morphisms . This is a family of maps T = f Th : 0 < h  h0 g from a pre-C  -algebra A to a C  -algebra B , such that the previous conditions hold. Such a family gives rise to a -homomorphism from A into the C  -algebra Be . Assume this homomorphism is continuous in the sup norm. Then, composing it with a section Be ! Cb ((0; h0 ]; B ) (that need not be linear nor multiplicative), we can get an asymptotic morphism. A preasymptotic morphism is real if Th (a) = Th (a) for all h; these are easier to handle. In order to dene maps of K -theory groups, it is enough to have asymptotic mor- phisms. This works as follows [63]: rst extend Th entrywise to an asymptotic morphism from Mm (A) to Mm(B ). If p is a projector in Mm(A), then by functional calculus there is a continuous family of projectors f qh : 0 < h  h0 g, whose K -theory class is well dened, such that kTh (p) qh k ! 0 as h # 0. Dene T : K0(A) ! K0 (B ) by T [p] := [qh ]. If A and B are two C  -algebras, a strong deformation from A to B is a continuous 0 eld of C  -algebras f Ah : 0  h  h0 g in the sense of [38], such that A0 = A and Ah = B for h > 0. The denition of a continuous eld involves specifying the space of norm- continuous sections h 7! s(h) 2 Ah , and guarantees that for any a 2 A0 there is such a section sa with sa (0) = a. Such a deformation gives rise to an asymptotic morphism from A to B by setting Th (a) := sa (h). The Moyal preasymptotic morphism. A very important asymptotic morphism, from C0 (T R n ) to K(L2 (R n )), is the Moyal deformation , given in terms of integral kernels by: Z Th (a)f (x) := (2h)n n a x +2 y ;  ei(x y)=h f (y) dy d: 1     (6:5) R on substituting the quantizer (6.4) in (6.2). Here a is an element of Cc1 (T  R n ) or S (T  R n ), to begin with, so the integral is well dened and the operators Th (a) are in fact trace-class; they are uniformly bounded in h. It is clear that Th (a) is the adjoint of Th (a). Connes calls it the Heisenberg deformation [22, II.B.]. In order to check the continuity of the deformation at h = 0, one can use, for instance, the following distributional identity: lim #0  n eixy= = (2)n (x) (y); (x; y 2 Rn ); from the theory of Fresnel integrals. It follows that limh #0 Lh (u; v; w) = (u v) (u w), so the twisted product reduces to the ordinary product in the limit h # 0. Now, any -homomorphism from Cc (T R n ) or S (T  R n ) into any C  -algebra B is continuous in the sup norm. When B = C , this is true since any positive distribution is a measure [53]; the general case follows from automatic continuity theorems for posi- tive mappings: see [105, V.5.6], for instance. Thus one has a real asymptotic morphism from C0 (T R n ) to K(L2(R n )). (Note that it is not claimed that the extension of Moyal quantization to elements of C0 (T  R n ) yield compact |or even bounded| operators.) 54 AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY Groupoids By denition, a groupoid G  U is a small category in which every morphism has an inverse. Its set of objects is U (often written G(0) ) and its set of morphisms is G. In practice, this means that we have a set G, a set U of \units" with an inclusion i : U ,! G, two maps r; s: G ! U , and a composition law G(2) ! G with domain G(2) := f (g; h) : s(g) = r(h) g  G  G; subject to the following rules: (i) r(gh) = r(g) and s(gh) = s(h) if (g; h) 2 G(2) ; (ii) if u 2 U then r(u) = s(u) = u; (iii) r(g)g = g = gs(g); (iv) (gh)k = g(hk) if (g; h) 2 G(2) and (gh; k) 2 G(2) ; (v) each g 2 G has an \inverse" g 1, satisfying gg 1 = r(g) and g 1g = s(g). Any group G is a groupoid, with U = feg. On the other hand, any set X is a groupoid, with G = U = X and trivial composition law x
x = x. Less trivial examples include graphs of equivalence relations, group actions, and vector bundles with brewise addition. A basic and important example is the double groupoid of a set X . Take G = X  X and U = X , included in X  X as the diagonal subset
X := f (x; x) : x 2 X g. Dene r(x; y) := x, s(x; y) := y. Then (x; y) 1 = (y; x) and the composition law is (x; y)
(y; z) = (x; z): On the strength of this example, we shall call U the diagonal of G. Notice that a disjoint union of groupoids is itself a groupoid. Denition. A smooth groupoid is a groupoid G  U where G, U and G(2) are mani- folds (possibly with boundaries), such that the inclusion i : U ,! G and the composition and inversion operations are smooth maps, and the maps r; s: G ! U are submersions . Thus, the tangent maps Tg r and Tg s are surjective at each g 2 G. In particular, this implies that rank r = rank s = dim U . Relevant examples are a Lie group, a vector bundle over a smooth manifold, and the double M  M of a smooth manifold. One can add more structure, if desired. For example, there are symplectic groupoids, where G is a symplectic manifold and U is a Lagrangian submanifold. These can be used to connect the Kostant{Kirillov{Souriau theory of geometric quantization with Moyal quantization [58, 121]. Convolution on groupoids. Functions on groupoids can be convolved in the following way [73, 97]. Suppose that on each r-bre Gu := f g 2 G : r(g) = u g there is given a measure u so that r(g) = gs(g) for all g 2 G; such a family of measures is called a \Haar system" for G. The inversion map g 7! g 1 carries each u to a measure u on the s-bre Gu := f g 2 G : s(g) = u g. The convolution of two functions a, b on G is then dened by Z Z (a  b)(g) := a(h) b(k) := a(h) b(h 1g) dr(g)(h): hk=g Gr(g)  6. QUANTIZATION AND THE TANGENT GROUPOID 55 Denition. The (reduced) C  -algebra of the smooth groupoid G  U with a given Haar system is the algebra Cr (G) obtained by completing Cc1 (G) in the norm kak := supu2U ku (a)k, where u is the representation of Cc1 (G) on the Hilbert space L2 (Gu ; u ): Z Z   u (a) (g) := a(h)  (k) := a(h) (h 1g) dr(g)(h) for g 2 Gu : hk=g Gr(g) There is a more canonical procedure to dene convolution, if one wishes to avoid hunting for suitable measures u , which is to take a, b to be not functions but half-densities on G [22, II.5]. These form a complex line bundle 1=2 ! G and one replaces Cc1 (G) by the compactly supported smooth sections Cc1 (G; 1=2) in dening Cr (G). However, for the examples considered here, the previous denition will do. When G = M  M , with M an oriented Riemannian manifold, we obtain just the convolution of kernels: Z (a  b)(x; z) := a(x; y) b(y; z) d (y) M p where d (y) = det g(y) dn y is the measure given by the volume form on M . Here Cc1 (M  M ) is the usual algebra of kernels of smoothing operators on L2 (M ), and the C  -algebra Cr (M  M ) is the completion of Cc1 (M  M ) acting as integral kernels on L2(M ), so that Cr (M  M ) ' K. When G = TM is the tangent bundle, with the operation of addition of tangent vectors, and U = M is included in TM as the zero section, r and s being of course the bering  : TM ! M , then Cr (TM ) is the completion of the convolution algebra Z p (a  b)(q; v) := a(q; u) b(q; v u) det g(q) dn u; Tq M where we may take a(q;
) and b(q;
) in Cc1 (Tq M ). The Fourier transform Z p F a(q;  ) := e iv a(q; v) det g(q) dn v Tq M replaces convolution by the ordinary product on the total space T  M of the cotangent bundle. This gives an isomorphism from Cr (TM ) to C0 (T  M ), also called F , with inverse: Z F 1 b(q; v ) = (2 ) n eiv b(q;  ) det 1=2 g(q) dn: Tq M We shall write dq (v) := det1=2 g(q) dv and dq ( ) := (2) n det 1=2 g(q) d . The tangent groupoid The asymptotic morphism involved in Moyal quantization can be given a concrete geo- metrical realization and a far-reaching generalization, by the concept of a tangent groupoid. 56 AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY To build the tangent groupoid of a manifold M , we rst form the disjoint union G0 := M  M  (0; 1] of copies of the double groupoid of M parametrized by 0 < h  1. Its diagonal is U 0 = M  (0; 1]. We also take G00 := TM , whose diagonal is U 00 = M . The tangent groupoid of M is the disjoint union GM := G0 ] G00 , with UM := U 0 ] U 00 as diagonal, whose composition law is given by (x; y; h)
(y; z; h) := (x; z; h) for h > 0 and x; y; z 2 M; (q; vq )
(q; wq ) := (q; vq + wq ) for q 2 M and vq ; wq 2 Tz M: Also, (x; y; h) 1 := (y; x; h) and (q; vq ) 1 := (q; vq ). The tangent groupoid can be given a structure of smooth groupoid in such a way that GM is a manifold with boundary, G0 contains the interior of the manifold and G00 is the nontrivial boundary. In order to see that, let us rst recall what is meant by the normal bundle over a submanifold R of a manifold M [35]. If j : R ! M is the  inclusion map, the tangent bundle of M restricts to R as the pullback j (TM ) = TM R ; this is a vector bundle over R, of which TR is a subbundle, and the normal bundle is the quotient N j := j  (TM )=TR. When M has a Riemannian metric, we may identify the bre Nqj to the orthogonal complement of Tq R in Tq M and thus regard N j as a subbundle of TM R . At each q 2 R, the exponential map expq is one-to-one from a small enough ball in Nqj into M . If the submanifold R is compact, then for some  > 0, the map (q; vq ) 7! expq (vq ) with jvq j <  is a dieomorphism from a neighbourhood of the zero section in N j to a neighbourhood of R in M (this is the tubular neighbourhood theorem). Now consider the normal bundle N
associated to the diagonal embedding of
: M ! M  M . We can identify
 T (M  M ) to TM  TM , and thereby the normal bundle over M is identied with N
= f(
(q); 21 vq ; 21 vq ) : (q; vq ) 2 TM g; which gives an obvious isomorphism between TM and N
. As in the tubular neighborhood theorem, we can (if M is compact, at any rate) nd a dieomorphism : V1 ! V2 between an open neighbourhood V1 of M in N
(considering M  N
as the zero section) and an open neighborhood V2 of
(M ) in M  M . Explicitly, we can nd r0 > 0 so that
(
(q); vq ; vq ) := expq ( 21 vq ); expq ( 12 vq ) is a dieomorphism provided vq 2 Nq
and jvq j < r0; take V1 to be the union of these open balls of radius r0 . Now we can dene the manifold structure of GM . The set G0 is given the usual product manifold structure; it has an \outer" boundary M  M  f1g. In order to attach G00 to it as an \inner" boundary, we consider U1 := f (q; vq ; h) : (q; hvq ) 2 V1 g;  6. QUANTIZATION AND THE TANGENT GROUPOID 57 which is an open subset of TM  [0; 1]; indeed, it is the union of TM  f0g and the tube of radius r0 =h around
(M )  fhg for each h 2 (0; 1]. Therefore, the map : U1 ! GM given by
(q; vq ; h) := expq ( 21 hvq ); expq ( 12 hvq ); h for h > 0; (q; vq ; 0) := (q; vq ) for h = 0; (6:6) is one-to-one and maps the boundary of U1 onto G00 . The restriction of  to U10 := f (q; vq ; h) 2 U1 : 0 < h  1 g  TM  (0; 1] is a local dieomorphism between U10 and its image U20  M  M  (0; 1]. One checks that changes of charts are smooth; thus, even if M is not compact, we can construct maps (6.6) locally and patch them together to transport the smooth structure from sets like U1 to the inner boundary of the groupoid GM . To prove that GM is a smooth groupoid, one also has to check the required properties of the inclusion UM ,! GM , the maps r and s, the inversion and the product. Actually, the present construction is a particular case of the tangent groupoid to a given groupoid given by [64] and [88]. They remark that, given a smooth groupoid G  U (in our case the double groupoid of M ), then if N is the normal bundle to U in G, the set N  f0g ] G  (0; 1] is a smooth groupoid GU with diagonal U  [0; 1], the construction (and therefore the correspondence M 7! GM ) being functorial. The smoothness properties are proven by repeated application of the following elementary result: if X , X 0 be smooth manifolds and Y , Y 0 respective closed submanifolds, and if0 f : X ! X 0 a smooth map such that f (Y )  Y 0 , then the induced map from XY to XY 0 is smooth. Moyal quantization as a continuity condition Let us now bring into play the Gelfand{Namark cofunctor C on tangent groupoids. A function on GM is rst of all a pair of functions on G0 and G00 respectively. The rst one is essentially a kernel, the second is the (inverse) Fourier transform of a function on the cotangent bundle T  M . The condition that both match to give a continuous function on GM can be seen precisely as the quantization rule. For clarity, we consider rst the case M = R n . Let a(x;  ) be a function on T  R n . Its inverse Fourier transform gives us a function on T R n : Z 1 1 F a(q; v) = (2)n n eiv a(q;  ) d: R n On R , the exponentials are given by x := expq ( 21 hv) = q + 12 hv; y := expq ( 21 hv) = q 12 hv: (6:7) Thus we can solve for (q; v): q = x +2 y ; v = x h y : (6:8) To the function a we associate the following family of kernels: Z ka (x; y; h) := h F a(q; v) = (2h)n n a x +2 y ;  ei(x y)=h d; 1   n 1 R that is, precisely the Moyal quantization formula (6.5). The factor h n is the Jacobian of the transformation (6.8). 58 AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY We get the dequantization rule by Fourier inversion: Z a(q;  ) = ka (q + 21 hv; q 12 hv) eiv dv: Rn Here ka is the kernel of the operator Q(a) of (6.2). The general Moyal asymptotic morphism. If M is a Riemannian manifold, we can now quantize any function a on T  M such that F 1a, its inverse Fourier transform in the second variable, is smooth and has compact support, say Ka . For h0 small enough, the map  of (6.6) is dened on Ka  [0; h0 ). For h < h0 , the formulae (6.7), (6.8) must be generalized to a transformation between TM and M  M  fhg whose Jacobian must be determined. We follow the treatment in [82]. Let q;v be the geodesic on M starting at q with velocity v, with an ane parameter s, i.e., q;tv (s)  q;v (ts). Locally, we may write x := q;v (s); with Jacobian matrix @@((x; y) (s): q; v) (6:9) y := q;v ( s); The Jacobian matrix can be computed from the equations of geodesic deviation [82]. In- troduce p p   n det g( q;v (s)) det g( q;v ( s))  @ (x; y)  J (q; v; s) := s det g(q)  @ (q; v ) (s): Then we have the change of variables formula: Z Z Z F (x; y) d (x) d (y) = F ( q;v ( 21 ); q;v ( 12 )) J (q; v; 12 ) dq (v) d (q): M M M Tq M The quantization/dequantization recipes are now given by ka (x; y; h) := h n J 1=2 (q; v; 21 h) F 1a(q; v);   a(q;  ) = F J 1=2(q; v; 12 h) ka (x; y; h) ; where (x; y) and (q; v) are related by (6.9) with s = 21 h. One can check that J (q; v; 12 h) = 1 + O(h2 ); a long but straightforward computation then shows that we have dened an (obviously real) preasymptotic morphism from Cc1 (T M ) to K(L2 (M )). Moreover the \tracial property" (6.3b) for the associated quantization rule is satised: Z   Tr Th (a)Th (b) = a(u)b(u) dh (u): T M The corresponding map in K -theory: T : K 0(T  M ) ! Z is an analytical index map , that in fact [88] is just the analytical index map of Atiyah{Singer theory [3].  6. QUANTIZATION AND THE TANGENT GROUPOID 59 The hexagon and the analytical index An essentially equivalent argument is done by Connes in the language of C  -algebra theory [22, II.5]. In eect, given a smooth groupoid G = G0 ] G00 which is a disjoint union of two smooth groupoids with G0 open and G00 closed in G, there is a short exact sequence of C  -algebras 0 ! C  (G0 ) ! C  (G) ! C  (G00 ) ! 0 where  is the homomorphism dened by restriction from Cc1 (G) to Cc1 (G00 ): it is enough to notice that  is continuous for the C  -norms because one takes the supremum of ku (a)k over the closed subset u 2 U 00 , and it is clear that ker  ' C  (G0 ). There is a short exact sequence of C  -algebras 0 ! C0 (0; 1] K ! C  (GM ) ! C0 (T  M ) ! 0 that yields isomorphisms in K -theory: Kj (C (GM )) !  Kj (C0(T  M )) = K j (T M ): (j = 0; 1): (6:10) This is seen as follows: since C  (M  M ) = K, the C  -algebra C  (G0 ), obtained by completing the (algebraic) tensor product Cc1 (0; 1] Cc1 (M  M ), is C0 (0; 1] K, which is contractible, via the homotopy t(f A) := f (t
) A for f 2 C0 (0; 1], 0  t  1; in particular, Kj (C0 (0; 1] K) = 0. At this point, we appeal to the six-term cyclic exact sequence in K -theory of C  -algebras [119]: K1(C0 (0x; 1] K) ! K1 (C  (GM )) ! K1(C0?(T  M )) ? ? ? y K0 (C0(T  M ))  K0 (C  (GM )) K0(C0 (0; 1] K) The two trivial groups break the circuit and leave the two isomorphisms (6.10). The restriction of elements of C  (G) to the outer boundary M  M  f1g gives a homomorphism : C (GM ) ! C  (M  M  f1g) ' K; and in K -theory this yields a homomorphism : K0(C  (GM )) ! K0(K) = Z. Finally, we have the composition ( ) 1 : K 0(T  M ) ! Z, which is just the analytical index map. Remarks on quantization and the index theorem From the point of view of quantization theory, this is not the whole story. Certainly, in the previous argument, we could have substituted any interval [0; h0] for [0; 1]. But for a given value of h0, not every reasonable function on T M can be successfully quantized. For exponential manifolds, like at phase space or the Poincare disk, everything should work ne. However, when one tries to apply a similar procedure in compact symplectic manifolds, one typically nds cohomological obstructions. This is dealt with in [46], leading back to the standard results in geometric quantization a la Kostant{Kirillov{Souriau. 60 AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY We are left with the impression that the r^ole of the apparatus of Moyal quantization in the foundations of noncommutative (topology and) geometry cannot be fortuitous. Conversely, one can ask what noncommutative geometry can do for quantization the- ory. Of course, one has to agree rst on the meaning (at least, on the mathematical side) of the word \quantization". The nearly perfect match aorded by the Moyal machinery in its particular realm is not to be expected in general. For any symplectic manifold, some kind of \quantum" deformation or star-product can always be found. However, that is mostly formal and of little use; in general a Moyal quantizer is missing. The modern temperament (see for instance [50]), that we readily adopt, is to consider that quantization is embodied in the index theorem . This dictum goes well with the original meaning of the word \quantization" in Bohr's old quantum theory. In the rare instances where it works well, though, it appears to give much more information than just a few integers (the indices of a certain Fredholm operator). The two more successful examples of quantization are Moyal quantization of nite-dimensional symplectic vector spaces and Kirillov{Kostant{Souriau geometric quantization of ag manifolds. With respect to the latter, Vergne [118] (see also [8]) has suggested to replace the concept of polarization, central to Kostant's work, by the use of Dirac-type operators |which of course takes us back to the spectral triples of noncommutative geometry. The new scheme for quantization runs more or less as follows. Let M be a smooth manifold endowed with a spinc structure (starting from a symplectic manifold, one can introduce a compatible almost complex structure in order to produce the spinc structure; the results only depend weakly of the choice made, as the set of almost complex structures is contractible). Construct prequantum line bundles L over M according to the KKS recipe, or some improved version like that of [103]. Let DL be a twisted Dirac operator for L. A quantization of M is the (virtual) Hilbert space HD;L := ker DL+ ker DL : The index theorem gives precisely the dimension of such a space. Under favourable cir- cumstances, we can do better. In the case of ag manifolds, one quantizes G-bundles and obtains G-Hilbert spaces. Then the G-index theorem gives us the character of the Kirillov representation associated to L [7], which contains all the quantum information we seek. Moyal quantization, on the other hand, is a tool of choice for the proof of the index theorem, as indicated here. The \logical" (though not the historical) way to go about the Index Theorem would be to prove the theorem in the at case rst using Moyal quantum mechanics |see [41] or [46]| and then go to analytically simpler but geometrically more involved cases. Conversely, one is left with the problem of how to recover the whole of Moyal theory from the index theorem. 7. Equivalence of Geometries We wish to classify geometries and to form some idea of how many geometries of a given type are available to us. When modelling physical systems that have an underlying geometry, we naturally wish to select the most suitable geometry from several plausible candidates. The rst question to ask, then, is: when are two geometries the same? Unitary equivalence of geometries In order to compare two geometries (A; H; D; ; J ) and (A0 ; H0; D0; 0 ; J 0), we focus rst of all on the algebras A and A0 . It is natural to ask that these be isomorphic, that is to say, that there be an involutive isomorphism : A ! A0 between their C  -algebra closures, such that (A) = A0. Since these algebras dene geometries only through their representations on the Hilbert spaces, we lose nothing by assuming that they are the same algebra A. We can also assume that the Hilbert spaces H and H0 are the same, so that A acts on H with two possibly dierent (faithful) representations. One must then match the operators D and D0 , etc., on the Hilbert space H. We are thus led to the notion of unitary equivalence of geometries. Denition. Two geometries G = (A; H; D; ; J ) and G 0 = (A; H; D0; 0 ; J 0) with the same algebra and Hilbert space are unitarily equivalent if there is a unitary operator U : H ! H such that (a) UD = D0 U , U = 0 U and UJ = J 0 U ; (b) U(a)U 1  ((a)) for an automorphism  of A. By \automorphism of A" is meant an involutive automorphism of the C  -algebra A that maps A into itself. Since UJ = JU , we also get U0(b)U 1 = UJ(b)J 1 U 1 = J((b))J 1 = J((b))J 1 = 0((b)): To be sure that this denition is consistent, let us check the following statement: given a geometry G := (A; H; D; ; J ) and a unitary operator on H such that U(A)U 1 = (A), then G 0 := (A; H; UDU 1; U U 1 ; UJU 1) is also a geometry. Firstly, (a) 7! U(a)U 1 =: ((a)) determines an automorphism  of A, since  is faithful. The operator D0 := UDU 1 has the same spectral properties as D, so the dimension is unchanged and Poincare duality remains valid for G 0 . The order-one condition is satised, since [[D0 ; (a)]; (b)0] = U [[D; a]; b0]U 1 = 0: Also, UD (c)U 1 = D0 ((c)) for c 2 Cn (A; A A0), where the action of  on A and A0 is extended to Hochschild cochains in the obvious way. In particular, if c is the orientation cycle, then D0 ((c)) = U U 1 = or D0 ((c)) = UU 1 = 1; according as the dimension is even or odd. Thus (c) is the orientation cycle for G 0 . 61 62 AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY 0 = Dom(D0 )k equals T For the niteness property, the space of smooth vectors H1 k 0 U H1 , and we may dene (U j U) := ( j ) for ;  2 H1 . This is the appropriate 0 , since (3.7) shows that hermitian structure on H1 Z Z Z (a) (U j U)0 ds0 n = Ua( j )U 1 jD0 j n = Ua( j )jDj n U 1 Z Z = a( j )jDj n= a ( j ) dsn = h j ai = hU j (a)Ui: Unitary equivalence of toral geometries. Let us now consider the eect of the \hyper- bolic" automorphism (4.6) of the algebra A on the geometry T2; of the noncommutative torus. The mapping a 7! (a) determined by (u) := ua vb , (v) := ucvd extends to a unitary operator U on L2(A ; 0), since it just permutes the orthonormal basis f umvn : m; n 2 Z g (actually, each basis vector is also multiplied by a phase factor of absolute value 1). Let U = U  U be the corresponding unitary operator on H = H+  H ; it is evident that U(a)U 1 = ((a)) for a 2 A . By construction, U = U . Also, UJ = JU on account of U J0 a = U a = (a) = J0 U a since  is involutive. The inverse-length operator transforms as ~y  y    0 @ D = @ 0 7 ! UD U = ~  1 0 @  @ 0 where @~ = U @  U 1 is given by @~ =   @   1. Since  1(u) = bd(a c 1)=2 ud v b ;  1(v) = ac(d b 1)=2 u c va ; we get at once @~ u = 2i(d b )u, @~ v = 2i(a c)v. Since @ = 1 + 2 , we arrive at @~ = (d b ) 1 + (a c) 2 : This is tantamount to replacing  by  1
 = (a c)=(d b ) in the denition of D , together with a rescaling in order to preserve the area given by the orientation cycle (4.12). [It is easy to check that h; ci = h; (c)i owing to ad bc = 1.] Harking back to elliptic curves for a moment, we see that the period parallelograms for the period pairs (1;  ) and (d b; a c) have the same area. Thus, for each geometry (A ; H; D ; ; J ), there is a family of unitarily equivalent geometries (A ; H; UD U 1 ; ; J ). If we replace the particular derivation @ of (4.10) by the most general derivation @ = 1 + 2 where =(=) > 0, we obtain a family of geometries over A , parametrized up to unitary equivalence by the fundamental domain of the modular group PSL(2; Z). Action of inner automorphisms. If u is a unitary element of the algebra A, i.e., u u = uu = 1, consider the unitary operator on H given by we write U := (u)0(u 1 ) = uJuJ y = JuJ y u:  7. EQUIVALENCE OF GEOMETRIES 63 Since J 2 = 1, we get UJ = uJu = uJ y u = J 2uJ y u = JU: The grading operator commutes with (u) and 0(u 1 ), so we also have U = U . Fur- thermore, if a 2 A, then UaU 1 = uau 1 since JuJ y commutes with a, so U implements the inner automorphism of A: u (a) := u a u 1: Such operators U provide unitary equivalences of the geometries (A; H; D; ; J ) and (A; H; uD; ; J ), where u D := UDU 1 = UDU  = JuJ y uDu Ju J y = JuJ y (D + u[D; u ])Ju J y = JuJ yDJu J y + u[D; u] = D + JuJ y[D; JuJ y] + u[D; u] = D + u[D; u]  Ju[D; u]J y: (7:1) Here we have used the order-one condition and the relation JD = DJ ; the latter gives the  sign on the right hand side, which is negative i n  1 mod 4. Notice that the operator u[D; u] = uDu D is bounded and selfadjoint in L(H). Morita equivalence and Hermitian connections The unitary equivalence of geometries helps to eliminate obvious redundancies, but it is not by any means the only way to compare geometries. For one thing, the metric (1.6) is unchanged |if we think of the right hand side of (1.6) as dening the distance between pure states p^, q^ of the algebra A. We need a looser notion of equivalence between geometries that allows to vary not just the operator data but also the algebra and the Hilbert space. Here the Morita equivalence of algebras gives us a clue as to how to proceed. We can change the algebra A to a Morita- equivalent algebra B, which also involves changing the representation space according to well-dened rules. How should we then adapt the remaining data , J and most importantly D, in order to obtain a Morita equivalence of geometries ? We start with any geometry (A; H; D; ; J ) and a nite projective right A-module E . Using the representation : A ! L(H) and the antirepresentation 0: b 7! J(b)J y, we can regard the space H as an A-bimodule. This allows us to introduce the vector space He := E A H A E : (7:2) If E = pAm , then E = Am p and He = (p)0(p)[H C m ], so that He becomes a Hilbert 2 space under the scalar product hr  q j s  ti := h j (r j s) 0(t j q)  i: If H = H+  H is Z2-graded, there is an obvious Z2-grading of He. 64 AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY The antilinear correspondence s 7! s between E and E also gives an obvious way to extend J to He: Je(s  t) := t J s: (7:3) Let B := EndA E , and recall that E is a left B-module. Then (b) : s  t 7 ! b s  t yields a representation  of B on He, satisfying 0(b) := J e (b )Jey : s  t 7 ! s  tb; where tb := bt, of course. The action , 0 of B on He obviously commute. Where connections come from. The nontrivial part of the construction of the new geometries (B; He; D; e e ; Je) is the determination of an appropriate operator De on H e . Guided by the dierential properties of Dirac operators, the most suitable procedure is to postulate a Leibniz rule : De (s  t) := (rs) t + s D t + s  (rt); (7:4) where rs, rt belong to some space whose elements can be represented on H by suitable extensions of  and 0. Consistency of (7.4) with the actions of A on E and H demands that r itself comply with a Leibniz rule. Indeed, since sa  t = s a t for all a 2 A; we get from (7.4) r(sa) t + s a D t = (rs)a t + s D a t; so we infer that r(sa) = (rs)a + [D; a]; (7:5) or more pedantically, r(sa) = (rs)(a) + [D; (a)] as operators on H. To satisfy these requirements, we introduce the space of bounded operators 1D := spanf a [D; b] : a; b 2 A g  L(H); which is evidently an A-bimodule, the right action of A being given by a [D; b]
c := a [D; bc] ab [D; c]. The notation is chosen to remind us of dierential 1-forms; indeed, for the commutative geometry (C 1 (M ); L2(M; S ); D= ; ; J ), we get 1D= = f () :  2 A1(M ) g; i.e., conventional 1-forms on M , represented on spinor space as (Cliord) multiplication operators.  7. EQUIVALENCE OF GEOMETRIES 65 Denition. We can now form the right A-module E A 1D . A connection on E is a linear mapping r: E ! E A 1D that satises the Leibniz rule (7.5). It is worth mentioning that only projective modules admit connections [32]. In the present case, if we dene linear maps j 0 !E A 1D ! E C A m! E ! 0 by j (s [D; a]) := sa 1 s a and m(s a) := sa, we get a short exact sequence of right A-modules (think of E C A as a free A-module generated by a vector-space basis of E ). Any linear map r: E ! E A 1D gives a linear section of m by f (s) := s 1 j (rs). Then f (sa) f (s)a = j (s [D; a] r(sa) + (rs)a), so f is an A-module map precisely when r satises the Leibniz rule (7.5). If that happens, f splits the exact sequence and embeds E as a direct summand of the free A-module E C A, so E is projective. Hermitian connections. The operator De must be selfadjoint on He. If ;  2 Dom(D), we get hr  q j De (s  t)i = h j D (r j rs) 0(t j q)  i + h j (r j s) 0(t j q) D i + h j (r j s) D0 (rt j q)  i; hDe (r  q) j s  ti = h j D (rr j s) 0(t j q)  i + h j D(r j s) 0(t j q)  i + h j (r j s) D0 (t j rq)  i: This reduces to the condition that (r j rs) (rr j s) = [D; (r j s)] for all r; s 2 E : (7:6) where the order one condition ensures commutation of D ( 1D ) with 0(A). We call the connection r Hermitian (with respect to D) if (7.6) holds. The minus sign is due to the presence of the selfadjoint operator D where a skewadjoint dierential operator is used in the standard denition of a metric-preserving connection [7, 83]. To sum up: two geometries (A; H; D; ; J ) and (B; He; D;e e ; Je) are Morita-equivalent if there exist a nite projective right A-module E and an 1D -valued Hermitian connection r on E , such that: B = EndA E , He and e are given by (7.2), Je by (7.3), and De by (7.4). Vector bundles over the noncommutative torus The nite projective modules over the torus C  -algebra A were dened in [16] and fully classied in [101]. (Indeed, [101] also constructs vector bundles over T2 that represent distinct classes in K0 (C 1 (T2 )) ' Z  Z; but the projective modules so obtained are unlike those of the irrational case.) To describe the latter, we return to the Weyl operators (4.1). The translation and multiplication operators W (a; 0) and W (0; b) are generated by i d=dt and t; the space of smooth vectors for these derivations is just the Schwartz space S (R ). 66 AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY Clearly S (R ) is a left A -module; but it can also be made a right A -module by making the generators act in another way. If p is any integer, one can dene
u := W (p ; 0) : t 7! (t p + );
v := W (0; 1=) : t 7! e2it (t): Therefore
vu := e i(p ) W (p ; 1=) ;
uv := e+i(p ) W (p ; 1=) ; so
vu = e2i
uv. Since the generators act compatibly with the commutation relation (4.3), this denes a right action of A on S (R ). This right module will be denoted Ep . One can dene more A -modules by a simple trick. Let q be a positive integer; the Weyl operators act on S (R q ) = S (R ) C q as W (a; b) 1q . If z 2 Mq (C ) is the cyclic shift (x1; : : :; xq ) 7! (x2; : : :; xq ; x1) and w 2 Mq (C ) is the diagonal operator (x1; : : :; xq ) 7! (x1;  2x2 ; : : :; xq ) for  := e2i=q , then zw = e2i=q wz, so that

u := W ( pq ; 0) zp ;
v := W (0; 1=) w
satisfy
vu = 
uv with  = e2i( p=q)e2ip=q = e2i . This right action of A on S (R q ) denes a right module Ep;q . It turns out that the free modules Am and these Ep;q (with p; q 2 Z, q > 0) are mutually nonisomorphic and any nite projective right A -module is isomorphic to one of them. Actually, it is perhaps not obvious that the Ep;q are nitely generated and projective. This is proved in [101], using the following Hermitian structure [30] that makes Ep;q a pre-C  -module over A : P ( j ) := r;s ur vs h
ur vs j iL (Rq) 2 for ; 2 S (R q ); where the coecients, in the case q = 1, are: Z h
ur vs j iL (R) = e 2ist (t r(p )) (t) dt: 2 (7:7) R We shall soon verify projectiveness in another way, by introducing connections. The endomorphism algebras. To reduce notational complications, let us take q = 1. The algebra B := EndA Ep is generated by Weyl operators that commute with W (p ; 0) and W (0; 1=). In view of (4.2), we can take as generators the operators U := W (1; 0); V := W (0; 1=(p )): Then V U =  UV where  = exp(2i=(p )), so that B ' A1=(p ) . For the simplest case p = 0, q = 1, we have U (t) = (t 1); V (t) = e 2it= (t); (7:8)  7. EQUIVALENCE OF GEOMETRIES 67 so that A and A 1= are Morita equivalent via E . It is known [98] that A and A are Morita equivalent if and only if either  or  lies in the orbit of  under the action of SL(2; Z), i.e., if and only if  = (a + b)=(c + d). The proof is K -theoretic: since 0(K0(A )) = Z + Z, a necessary condition is that Z + Z = r (Z + Z ) for some r > 0. Suciency is proved by exhibiting an appropriate equivalence bimodule Ep;q . Morita-equivalent toral geometries Let us now construct a Hermitian connection r (with respect to the operator D ) on the A -module E0 = S (R ). To do so, we must rst determine the bimodule 1D . Clearly    (a) [D ; (b)] = a @0 b a @0 b ;  so that 1D ' A  A as A -bimodules. Thus E0 A 1D ' E0  E0. Therefore, r = (r0 ; r00 ) where r0 , r00 are two derivations on S (R ). The corre- sponding Leibniz rules are given by (7.5): r0 (
a) = (r0 )
a +
@ a; r00 (
a) = (r00 )
a
@ a: This implies that r0 = r1 +  r2 and r00 = r1  r2, where r1 , r2 comply with Leibniz rules involving the basic derivations: rj (
a) = (rj )
a +
j a; and it is enough to check these relations for a = u; v. It will come as no surprise that r1 and r2 are just the position and momentum operators of quantum mechanics (with a scale factor of i=2 = i=2h); in fact, r1 (t) := 2it  (t); r2 (t) := 0(t): (7:9) One immediately checks that r1 (
u) (r1 )
u = [t 7! 2i (t + )] =
1 u; r1(
v) (r1 )
v = 0 =
1 v; r2 (
u) (r2 )
u = 0 =
2 u; r2(
v) (r2 )
v = [t 7! 2i e2it (t)] =
2 v: Thus r is a connection satisfying (7.5) with D = D . To see that r is Hermitian, it is enough to observe that (7.6) is equivalent to ( j r0 ) (r00  j ) = @ ( j ) for all ; 2 S (R ); 68 AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY or equivalently ( j rj ) + (rj  j ) = j ( j ) (j = 1; 2); (7:10) where the A -valued Hermitian structure on S (R ) is the special case of (7.7): Z e 2ist(t + r) (t) dt: P ( j ) := r;s ars ur vs; ars := R This can be veried by direct calculation. For instance, when j = 2, the left hand side of (7.10) equals Z e 2ist dtd (t + r) (t) dt = r;s 2is ars ur vs = 2 ( j ): P h i P r;s ur vs R The geometry on A 1= . Let us take stock of the new geometry. The algebra is A 1= , with generators U , V of (7.8). The Hilbert space is Z2-graded, with He+ = E0 A H+ A E 0, that we can identify with L2(A 1= ; 0). Under this identication, Je becomes a 7! a, as before. It remains to identify the operator De , whose general form has been determined in x4. We nd that e U ](   [D;  ) = [r; U ]
  for ;  2 S (R ),  2 H,where  00  [r; U ] = [r00; U ] [r 0; U ] : It is immediate from the denitions (7.8), (7.9) that [r1; U ] = 2i U = 1 1U; [r2 ; V ] = 2i V = 1 2 V; [r1 ; V ] = [r2; U ] = 0: Thus  y 1 De =  @ 0 :0 @  Setting aside the overall scale factor 1=, we see that the modulus  is unchanged. We conclude that the geometries T2; and T2 1=; are Morita equivalent. Gauge potentials Let us examine what Morita equivalence entails when the algebra A is unchanged, and the equivalence bimodule is A itself. The algebra A, regarded as a right A-module, carries a standard Hermitian connection with respect to D, namely AdD : A ! 1D : b 7! [D; b];  7. EQUIVALENCE OF GEOMETRIES 69 and by the Leibniz rule (7.5), any connection diers from AdD by an operator in 1D : rb =: [D; b] + A b; (7:11) where P A := j aj [D; bj ] (nite sum) lies in 1D . We call it a gauge potential if it is selfadjoint: A  = A . Hermiticity of the connection for the product (a j b) := a b demands that a rb (ra)b = [D; ab], that is, a(A A  )b = 0 for all a; b 2 A, so a Hermitian connection on A is given by a gauge potential A . On substituting the connection (7.11) in the recipe (7.4) for an extended Dirac ope- rator, one obtains De (b ) = ([D; b] + A b) + b D  bJ (r1)J y = (D + A  J A J y )(b ); (7:12) where the signs are as in (7.1). Therefore, the gauge transformation D 7! D + A  J A J y yields a geometry that is Morita-equivalent to the original. Another way of saying this is that the geometries whose other data (A; H; ; J ) are xed form an ane space modelled on the selfadjoint part of 1D . In summary, we have shown how the classication of geometries up to Morita equi- valence allows a rst-order dierential calculus to enter the picture, via the Hermitian connections. In the next section, we shall explore the various geometries on a noncommu- tative manifold from a variational point of view. 8. Action Functionals On a dierential manifold, one may use many Riemannian metrics; on a spin manifold with a given Riemannian metric, there may be many distinct (i.e., unitarily inequivalent) geometries. An important task, already in the commutative case, is to select, if possible, a particular geometry by some general criterion, such as minimization of an action func- tional, a time-honoured tradition in physics. In the noncommutative case, the minimizing geometries are often not unique, leading to the phenomenon of spontaneous symmetry breaking, an important motivation for physical applications [39]. Automorphisms of the algebra In order to classify geometries, we x the data (A; H; ; J ) and consider how the inverse distance operator D may be modied under the actions of automorphisms of the algebra A. The point at issue here is that the automorphism group of the algebra is just the noncommutative version of the group of dieomorphisms of a commutative manifold. For instance, if A = C 1 (M ) for a smooth manifold M , and if  2 Aut(A), then each character x^ of A is the image under  of a unique character y^ (that is,  1 (^x) is also a character, so it equals y^ for some y 2 M ). Write (x) := y; then  is a continuous bijection on M satisfying (f )(x) = f ( 1(x)), and the chain rule for derivatives shows that  is itself smooth and hence is a dieomorphism of M . In ne,  $  is a group isomorphism from Aut(C 1 (M )) onto Di(M ). On a noncommutative algebra, there are many inner automorphisms u (a) := u a u 1; where u lies in the unitary group U (A); these are of course trivial when A is commutative. We adopt the attitude that these inner automorphisms are hence forth to be regarded as internal dieomorphisms of our algebra A. Already in the commutative case, dieomorphisms change the metric on a manifold. To select a particular metric, some sort of variational principle may be used. In general relativity, one works with the Einstein{Hilbert action Z p Z IEH / r(x) g(x) dn x = r(x) ; M M where r is the scalar curvature of the metric g, in order to select a metric minimizing this action. In Yang{Mills R theories of particle physics, the bosonic action functional is of the form IYM / F (?F ) where F is a gauge eld, i.e., a curvature form. The question then arises as to what is the general prescription for appropriate action functionals in noncommutative geometry. Inner automorphisms and gauge potentials. Let us rst recall how inner automor- phisms act on geometries. If u 2 U (A), the operator U := uJuJ y implements a unitary equivalence (7.1) between the geometries determined by D and by u D = D + u[D; u ]  Ju[D; u ]J y : 70  8. ACTION FUNCTIONALS 71 More generally, any selfadjoint A 2 1D gives rise to a Morita equivalence (7.12) between the geometries determined by D and by D + A  J A J y . The slaying of abelian gauge elds. It is important to observe that these gauge trans- formations are trivial when the algebra A is commutative. Recall that 0(b) = (b) in the commutative case (since the action by J on spinors takes multiplication by a function to multiplication by its complex conjugate). Therefore we can write a = JaJ y when A is commutative. But then, Ja[D; b]J y = aJ [D; b]J y = J [D; b]J ya = [JDJ y; JbJ y]a = [D; b]a =  a[D; b] 
since JDJ y = D. Hence J A J y = A  for A 2 1D , and thus A  J A J y = A A  ; for a selfadjoint gauge potential, A  J A J y vanishes. As pointed out in [86], this means that, within our postulates, a commutative manifold could support gravity but not electromagnetism; in other words, even to get abelian gauge elds we need that the underlying manifold be noncommutative! Exercise. Show that the gauge potentials A = umvn [D ; v n u m ] for the toral geometry T2; satisfy A + J A J y = 0, but this is not so for A := v [D ; u ] [D ; u]v  . } In [25], Connes also raised the issue of whether A might admit further symmetries arising from Hopf algebras. We cannot go into this here, but we should mention the recent investigations [31, 33, 75] on a 27-dimensional Hopf algebra closely related to the gauge group of the Standard Model. The fermionic action In the Standard Model of particle physics, the following prescription denes the fermionic action functional: I (; A ) := h j (D + A  J A J y ) i (8:1) (with the  sign as before). Here  may be interpreted as a multiplet of spinors representing elementary particles and antiparticles [22, 86, 107]. The gauge group U (A) acts on potentials in the following way. If u 2 A is unitary and if r = AdD +A is a hermitian connection, then so is uru = u AdD u + uA u = AdD +u[D; u ] + uA u ; so that u A := uA u + u[D; u] is the gauge-transformed potential. With U = uJuJ y , we get U A U 1 = uA u 1 since JuJ y commutes with 1D , and so D + u A  J uA J y = D + u[D; u]  Ju[D; u]J y + uA u  JuA u J y = U (D + A  J A J y )U 1 : The gauge invariance of (8.1) under the group U (A) is now established by I (U; uA ) = hU j (D + u A  J uA J y )U i = hU j U (D + A  J A J y ) i = I (; A ): 72 AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY A remark on curvature. In Yang{Mills models, the fermionic action is supplemented by a bosonic action that is a quadratic functional of the gauge elds or curvatures associated to the gauge potential A . One may formulate the curvature of a connection in noncom- mutative geometry and obtain a Yang{Mills action; indeed, this is the main component of the Connes{Lott models [28]. One can formally introduce the curvature as F := dA + A 2 , where the notation means P P dA := j [D; aj ] [D; bj ] whenever A = j aj [D; bj ]: (8:2) Regrettably, this denition is awed, since the rst sum may be nonzero in cases where the second sum vanishes [22, VI.1]. For instance, in the commutative case, one may have a[D= ; a] [D= ; 21 a2] = (a da d( 21 a2)) = 0 but [D= ; a] [D= ; a] = (da)2 = (da j da) < 0. If we push ahead anyway, we can make a formal check that F transforms under the gauge group U (A) by u F = uF u . Indeed, d(u A ) = [D; u] [D; u] + j [D; uaj ] [D; bj u ]  P P j [D; uaj bj ] [D; u ] = [D; u] [D; u] + [D; u]A u uA [D; u ] + j u[D; aj ] [D; bj ]u; P whereas, using the identity u[D; u]u = [D; u], we have (u A )2 = u[D; u]u[D; u] + u[D; u]uA u + uA [D; u ] + uA 2 u = [D; u] [D; u] [D; u]A u + uA [D; u ] + uA 2 u ; and consequently u F := d(u A ) + (u A )2 = u(dA + A 2 )u = uF u : Provided that the denition (8.2) can be corrected, one can then dene a gauge-invariant action [59] as the symmetrized Yang{Mills type functional Z Z (F + J F J y )2 dsn = (F + J F J y )2 jDj n ; since Z Z Z (uF + J u F J y )2 ju Dj n = U (F + J F J y )2 jDj n U  = (F + J F J y )2 jDj n : 1 2 The ambiguity in (8.2) can be removed by introducing the P A-bimodule ( D ) =J2 , where the subbimodule J2 consists of the so-called \junk" terms j [D; aj ] [D; bj ] for which P 2 j aj [D; bj ] = 0. Then, by redening F as the orthogonal projection of dA + A on the orthogonal complement of J2 in ( 1D )2, one gets a well-dened curvature and the noncommutative integral of its square gives the desired Yang{Mills action. The spectral action principle This Yang{Mills action, evaluated on a suitable geometry, achieves the remarkable feat of reproducing the classical Lagrangian of the Standard Model. This is discussed at length in [22, VI] and in several other places [10, 13, 66, 76, 86]. However, its computation leads to fearsome algebraic manipulations and very delicate handling of the junk terms, leading one to question whether this action is really fundamental.  8. ACTION FUNCTIONALS 73 The seminal paper [25] makes an alternative proposal. The unitary equivalence D 7! D + u[D; u ]  Ju[D; u]J y is a perturbation by internal dieomorphisms, and one can regard the Morita equivalence D 7! D + A  J A J y as an internal perturbation of D. The correct bosonic action functional should not merely be dieomorphism invariant (where by dieomorphisms we mean automorphisms of A), that is to say, \of purely gravita- tional nature", but one can go further and ask that it be spectrally invariant . As stated unambiguously by Chamseddine and Connes [14]: \The physical action only depends upon sp(D)." The fruitfulness of this viewpoint has been exemplied by Landi and Rovelli [80, 81], who consider the eigenvalues of the Dirac operator as dynamical variables for general relativity. Since quantum corrections must still be provided for [1], the particular action cho- sen should incorporate a cuto scale  (roughly comparable to inverse Planck length, or Planck mass, where the commutative spacetime geometry must surely break down), and some suitable cuto function: (t)  0 for t  0 with (t) = 0 for t  1. Therefore, Chamseddine and Connes propose a bosonic action of the form B (D) = Tr (D2 =2): (8:3) This spectral action turns out to include not only the Standard Model bosonic action but also the Einstein{Hilbert action for gravity, plus some higher-order gravitational terms, thereby establishing it rmly as an action for an eective eld theory at low energies. We refer to [14, 65, 106] for the details of how all these terms emerge in the calculation. Most of these terms can also be recovered by an alternative procedure involving Quillen's superconnections [48], which seems to suggest that the Chamseddine{Connes action is in the nature of things. Here we must limit ourselves to the humble computational task of explaining the general method of extracting such terms from (8.3), by a spectral asymptotic development in the cuto parameter . Spectral densities and asymptotics We consider the general problem of providing the functional (8.3) with an asymptotic expansion as  ! 1, without prejudging the particular cuto function . In any case, as we shall see, the dependence of the nal results on  is very weak. There is, of course, a great deal of accumulated experience with the related heat kernel expansion for pseudo- dierential operators [54]. One can adapt the heat kernel expansion [14] to develop (8.3), under the tacit assumption that  is a Laplace transform. However, we take a more direct route, avoiding the detour through the heat kernel expansion. The basic idea, expounded in detail in [44], is to develop distributional asymptotics directly from the spectral density of the positive selfadjoint operator A = D2 . If the spectral projectors of A are f E () :   0 g, the spectral density is the derivative ( A) := dEd() 74 AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY that makes sense as a distribution with operatorial values in L(H1 ; H). For instance, when A has discrete spectrum fj g (in increasing order) with a corresponding orthonormal basis of eigenfunctions uj , then X 1 X E () = juj ihuj j; and so dA () = juj ihuj j ( j ): j  j =1 A functional calculus may be dened by setting Z 1 f (A) := f () ( A) d: 0 For instance, Z 1 Z 1 Ak = k ( A) d; e tA = e t ( A) d: 0 0 For further details of this calculus and the conditions for its validity, we refer to [43, 44]. Spectral densities of pseudodierential operators. The algebra of the Standard Model spectral triple is of the form C 1 (M ) AF , where M is a compactied (Euclidean) spacetime and AF is an algebra with a nite basis; in fact, AF = C  H  M3(C ), acting on a spinor multiplet space L2(M; S ) HF ' L2 (M; S HF ) through a nite-dimensional real representation [106]. Thus the operator D is of the form D= 1 + 5 DF , where DF is a matrix of Yukawa mass terms and D= is the Dirac operator on the spinor space of M . By Lichnerowicz' formula [7], D= 2 =
S + 41 r; where
S is the spinor Laplacian and r is the scalar curvature. After incorporating the terms from DF , one nds [14, 65] that D2 is a generalized Laplacian [7] with matrix-valued coecients. Thus the task is to compute an expansion for (8.3) under the assumption that A = D2 is a pseudodierential operator of order d = 2. We suppose, then, that A is a positive, elliptic, classical pseudodierential operator of order d on an n-dimensional manifold M . If A has symbol (A) = a(x;  ) in local coordinates, we ask what the symbol (( A)) might be. If A has constant coecients, the symbol of Ak is just Z a(x;  )k = k ( a(x;  )) d; so ( a(x;  )) is the symbol of ( A) in that particular case. In general, the symbol of Ak depends also on the derivatives of a(x;  ), so we arrive at the prescription [44]:
 ( A)  ( (A)) q1 0 ( (A)) + q2 00 ( (A))


+ ( 1)k qk (k) ( (A)) +


as  ! 1: (8:4)  8. ACTION FUNCTIONALS 75 R By computing k (( A)) d for k = 0; 1; 2; : : :, we get q1 = 0,

q2 (x;  ) = 21 (A2) (A)2 ; q3 (x;  ) = 16 (A3) 3(A2)(A) + 2(A)3 ; and so on. The order of the symbol q2 is  (2d 1), the order of q3 is  (3d 2), etc. Cesaro calculus. An important technical issue is how to interpret the distributional development (8.4). On subtracting the rst N terms on the right from the left hand side, one needs a distribution that falls o like N as  ! 1, with exponents N that decrease to 1. It turns out that this holds, in a Cesaro-averaged sense [44]. To be precise, a distribution f is of order  at innity, in the Cesaro sense: f () = O( ) (C ) as  ! 1; if there is, for some N , a function fN whose N th distributional derivative equalsPf1, such that fN () = p() + O(+N ) as  ! 1 with p a polynomial P1 of degree < N . If n=1 an is a Cesaro-summable series, then the distribution f () := n=1 an ( n) satises Z 1 1 X f () d  an (C ): 0 n=1 R Let us recall that the symbol of A is dened by writing Au(x) = kA (x; y)u(y) dny where the kernel is the distribution Z kA (x; y) := (2) n ei(x y)
 a(x;  ) dn; and in particular, on the diagonal: Z kA (x; x) = (2) n a(x;  ) dn: The kernel for the spectral density ( A) is then given on the diagonal by Z   dA (x; x; )  (2) n ( a(x;  )) + q2 (x;  ) 00( a(x;  ))


dn  (C ): (8:5) By the functional calculus, the action functional (8.3) may then be expressed as Z Z 1 p Tr (D2 =2 ) = (=2 )dA (x; x; ) d g(x) dn x; M 0 provided one learns the trick of integrating a Cesaro development to get a parametric development in t =  2 as t # 0. Parametric developments. Some distributions have zero Cesaro expansion, namely those f for which f () = o( 1 ) (C ) as jj ! 1. These coincide with the dual space K0 76 AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY of the space K of GLS symbols [61]: elements of K are smooth functions  such that for some , (k) () = O(jj kR) as jj ! 1. The space K includes all polynomials, so any f 2 K0 has moments k := k f () d of all orders. Indeed, K0 is precisely the space of distributions that satisfy the moment asymptotic expansion [45]: 1 X ( 1)k k (k) () f ()  k! k+1 as  ! 1: k=0 This a parametric development of f (). For a general distribution, we may have a Cesaro expansion in falling powers of : X f ()  ck k (C ) as  ! 1; k1 and the corresponding parametric development is of the form [45]: f ()  X ck ()k + X ( 1)mm (m) () as  ! 1: (8:6) k1 m0 m! m+1 (This is an oversimplication, valid only if no k is a negative integer: the general case is treated in [45] and utilized in [44].) The moral is this: if one knows the Cesaro development, the parametric development is available also, assuming that one can compute the moments that appear in (8.6). Then one can evaluate on a test function by a change of variable, obtaining an ordinary asymptotic expansion in a new parameter: Z X Z 1 X m (m) (0) tm f ()(t) d  ck t k 1 k () d + as t # 0: (8:7) k1 0 m0 m! (The integral on the right is to be regarded as a nite-part integral; also, when some k are negative integers, there are extra terms in tr log t.) The heat kernel development may be obtained by taking () := e  for   0. The spectral coecients. The coecients of the spectral density kernel (8.5), after integration over  , have an intrinsic meaning: in fact, they are all Wodzicki residues! More precisely, it has been argued in [44] that (8.5) simplies to dA (x; x; ) dnx  d (21)n wresx (A n=d )(n d)=d + wresx (A(1 n)=d )(n d 1)=d  (8:8)  +


+ wresx (A(k n)=d )(n d k)=d +


(C ) as  ! 1: It is worth indicating brie y how this comes about: we shall compute the leading term in (8.8). To integrate (8.5), we use polar coordinates  = r! with j!j = 1. Since the  8. ACTION FUNCTIONALS 77 integrand involves ( a(x; r!)) and its derivatives, we must solve a(x; r!) =  for r; denote the unique positive solution by r = r(x; !; ). To solve, we revert the expansion X  = a(x; r!)  ad j (x; !) rd j as r ! 1 (8:9) j 0 to get a development in falling powers of : X r = r(x; !; )  rk (x; !) (1 k)=d as  ! 1: (8:10) k0 Now we unpack the distribution ( a(x; r!)) = a0 (x;(rr((x;x;!!; ;)))!) : Since dn  = rn 1 dr ! , the rst term in (8.5) yields (2) n Z rn 1(x; !; )  : (8:11) ! j!j=1 a0 (x; r(x; ! ; )! ) If we retain only the rst terms in (8.9) and (8.10), this integrand becomes rn 1  r0 (x; !)n d(n d)=d  (n d)=d a (x; !) n=d ; d rd 1ad (x; !) d ad(x; !) d d since r0 (x; !) = ad (x; !) 1=d. Thus the leading term in the -development of (8.11) is (n d)=d Z n=d d (2)n j!j=1 ad (x; !) ! ; and it remains only to notice that ad (x; !) n=d is the principal symbol, of order ( n), of the operator A n=d . Spectral densities of generalized Laplacians. We can apply this general machinery to the case where A is a generalized Laplacian, with a symbol of the form a(x;  ) = gij (x) ij + bk (x)k + c(x); where bk , c are scalar functions on M . Rewriting (8.8) as d (2)n dA (x; x; )  a0(x)(n d)=d + a1(x)(n d 1)=d + a2(x)(n d 2)=d +


; we see that a0(x) is constant with value n . Also, ak = 0 for odd k since their computation involves integrating odd powers of the !j over the sphere j!j = 1. For a2(x), one can express the metric in normal coordinates [7]: gij (x)  ij + 31 Riklj (x0) (x x0)k (x x0)l +


78 AN INTRODUCTION TO NONCOMMUTATIVE GEOMETRY where Riklj is the Riemann curvature tensor. The q2 term of (8.5) extracts from this a Ricci-tensor term 31 Rkj (x) k j and integration over the unit sphere leaves the scalar curvature r(x). The upshot is that a2 (x) = (n 22) n 16 r(x) c(x) :
(8:12) On the other hand, this gives the Wresidue density of A(2 n)=2 . We thus arrive at one of the most striking results in noncommutative geometry, derived independently by Kastler [74] and Kalau and Walze [69], namely that the Einstein{Hilbert action of general relativity is a multiple of the Wodzicki residue of D= 2 on a 4-dimensional manifold: Z p Wres D= 2 / r(x) g(x) d4 x; M on combining (8.12) with the Lichnerowicz formula c = 41 r. The computation of Wres D 2 for the Standard Model is given in [14, 65]. The Chamseddine{Connes action. Pulling all the threads together, we apply the ex- pansion (8.12) to the action functional (8.3). For the Standard Model plus gravity, we take n = 4 and A = D2 , a generalized Laplacian, acting on a space of sections of a vector bundle E over M . From (8.8) we get dD (x; x; )  rank E  + 1 wres D 2 (C ) as  ! 1 2 162 324 x since the nonnegative powers of the dierential operator D2 have zero Wresidue. Applying (8.7) with t =  2 gives an expansion of the form Tr (D2 =2 )  rank E  4 + 1 Wres D 2  2 + X b (D2 ) 162 0 324 2 2m+4 2m+4  2m m0 as  ! 1, where 0 = 01 () d, 2 = 01 () d and 2m+4 = ( 1)m(m) (0) for R R m = 0; 1; 2; : : :. Thus the cuto function  plays only a minor r^ole, and these integrals and derivatives may be determined from experimental data. In [14] detailed results are given for the spectral triple associated to the Standard Model. The bosonic parts of the SM appear in the 2 and 0 terms; the Einstein{Hilbert action appears in the 2 term, as expected; and other gravity pieces and a gravity-Higgs coupling in the 0 term; the 4 term is cosmological. The 0 term is conformally invariant. Higher-order terms may be neglected. Thus the stage is set for a theory that encompasses gravity and matter elds on the same footing. However, it must, when we nd it, be a fully quantum theory; and that is for the future.  References [1] E. Alvarez, J. M. Gracia-Bonda and C. P. 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