Introduction to Non-Commutative Geometry

corfu98

PROCEEDINGS

J. Madore LPT, Batiment 211, Universit´e de Paris-Sud F-91405 Orsay (France) E-mail: [email protected]

Abstract: A review is made of some recent results in noncommutative geometry, especially those aspects which might in some way be of use in the study of strings and membranes. Efforts to add a gravitational field to noncommutative models of space-time are also reviewed. Special emphasis is placed on the case which could be considered as the noncommutative analog of a parallelizable space-time. It is argued that, at least in this case, there is a rigid relation between the noncommutative structure of the space-time on the one hand and the nature of the gravitational field which remains as a ‘shadow’ in the commutative limit on the other.

Keywords: Noncommutative geometry, matrix calculus, gravity.

1. Historical introduction tion to replace a ‘point’ is a Planck cell of dimension given by the Planck area. A noncom- To control the divergences which from the very mutative space-time looks like a solid which has beginning had plagued quantum electrodynam- a homogeneous distribution of dislocations but ics, Heisenberg already in the 1930’s proposed no disclinations. We can pursue this solid-state to replace the space-time continuum by a lattice analogy and think of the ordinary Minkowski co- structure. A lattice however breaks Lorentz in- ordinates as macroscopic order parameters ob- variance and can hardly be considered as funda- tained by coarse-graining over scales less than the mental. It was Snyder [122, 123] who first had fundamental scale. They break down and must the idea of using a noncommutative structure at be replaced by elements of some noncommuta- small length scales to introduce an effective cut- tive algebra when one considers phenomena on off in field theory similar to a lattice but at the these scales. If a coherent description could be same time maintaining Lorentz invariance. His found for the structure of space-time which were suggestion came however just at the time when pointless on small length scales, then the ultra- the renormalization program finally successfully violet divergences of quantum field theory could became an effective if rather ad hoc prescription be eliminated. In fact the elimination of these for predicting numbers from the theory of quan- divergences is equivalent to coarse-graining the tum electrodynamics and it was for the most part structure of space-time over small length scales; ignored. Some time later von Neumann intro- if an ultraviolet cut-off Λ is used then the the- duced the term ‘noncommutative geometry’ to ory does not see length scales smaller than Λ−1. referingeneraltoageometryinwhichanal- When a physicist calculates a Feynman diagram gebra of functions is replaced by a noncommu- he is forced to place a cut-off Λ on the momentum tative algebra. As in the quantization of classi- variables in the integrands. This means that he cal phase-space, coordinates are replaced by gen- renounces any interest in regions of space-time erators of the algebra [42]. Since these do not of volume less than Λ−4. As Λ becomes larger commute they cannot be simultaneously diago- and larger the forbidden region becomes smaller nalized and the space disappears. One can ar- and smaller but it can never be made to van- gue [90] that, just as Bohr cells replace classical- ish. There is a fundamental length scale, much phase-space points, the appropriate intuitive no- corfu98 J. Madore larger than the Planck length, below which the definition of this field were to be to a certain ex- notion of a point is of no practical importance. tent overcome. Soon even a formulation of the The simplest and most elegant, if certainly not standard model of the electroweak forces could the only, way of introducing such a scale in a be given [30]. A simultaneous development was Lorentz-invariant way is through the introduc- a revival [99, 34, 86] of the idea of Snyder that ge- tion of noncommuting space-time ‘coordinates’. ometry at the Planck scale would not necessarily It might be argued that since we have made be described by a differential manifold. space-time ‘noncommutative’ we ought to do the One of the advantages of noncommutative same with the Poincaré group. This logic leads geometry is that smooth, finite examples [89, 90] naturally to the notion of a q-deformed Poincaré can be constructed which are invariant under the (or Lorentz) group which acts on a very particu- action of a continuous symmetry group. Such lar noncommutative version of Minkowski space models necessarily have a minimal length associ- called q-Minkowski space. The idea of a q-defor- ated to them and quantum field theory on them mation goes back to the dawn of time. Almost is necessarily finite [58, 60, 62, 13]. In general immediately after Clifford introduced his alge- this minimal length is usually considered to be in bras they were q-deformed, with q a root of unity, some way or another associated with the gravita- by Sylvester [121]. This idea was taken up later tional field. The possibility which we shall con- by Weyl [129] and Schwinger [119] to produce sider here is that the mechanism by which this a finite version of quantum mechanics. It has works is through the introduction of noncommut- also been argued, for conceptual as well as practi- ing ‘coordinates’. This idea has been developed cal, numerical reasons, that the lattice version of by several authors [66, 90, 43, 76, 75] since the space-time or of space is quite satisfactory if one original work of Snyder. It is the left-hand arrow uses a random lattice structure or graph. From of the diagram this point of view the Lorentz group is a classical A ⇐ ∗ A invariance group and is not valid at the micro- k¯ =Ω( k¯) scopic level. A survey of this from the point of ⇓⇑ (1.1) view of noncommutative geometry is to be found Cut-off Gravity in the book by Landi [81]. Interest in Snyder’s idea was revived much The Ak¯ is a noncommutative algebra and the in- later when mathematicians, notably Connes [25] dexk ¯ indicates the area scale below which the and Woronowicz [128], succeeded in generalizing noncommutativity is relevant; this would nor- the notion of differential structure to noncom- mally be taken to be the Planck area. The top ∗ mutative geometry. Just as it is possible to give arrow is a mathematical triviality; the Ω (Ak¯ )is many differential structures to a given topologi- a second algebra which contains Ak¯ and is what cal space it is possible to define many differential gives a differential structure to it. We shall give calculi over a given algebra. We shall use the examples of this below. We shall attempt, not term ‘noncommutative geometry’ to mean ‘non- completely successfully, to argue that each grav- commutative differential geometry’ in the sense itational field is the unique ‘shadow’ in the limit of Connes. Along with the introduction of a gen- ¯k → 0 of some differential structure over some eralized integral [32] this permits one in princi- noncommutative algebra. This would define the ple to define the action of a Yang-Mills field on right-hand arrow of the diagram. A hand-waving a large class of noncommutative geometries. argument can be given [93] which allows one to One of the more obvious applications was to think of the noncommutative structure of space- the study of a modified form of Kaluza-Klein the- time as being due to quantum fluctuations of the ory in which the hidden dimensions were replaced light-cone in ordinary 4-dimensional space-time. by noncommutative structures [86, 87, 44]. In This relies on the existence of quantum gravita- simple models gravity could also be defined [87, tional fluctuations. A purely classical argument 88] although it was not until much later [105, 46, based on the formation of black-holes has been 72] that the technical problems involved in the also given [43]. In both cases the classical grav-

2 corfu98 J. Madore itational field is to be considered as regularizing field, even in some ‘quasi-commutative’ approxi- the ultraviolet divergences through the introduc- mation. If this were done then the missing arrow tion of the noncommutative structure of space- in (1.1) could be drawn. The difficulty is partly time. This can be strengthened as the conjec- due to the lack of tractable noncommutative ver- ture that the classical gravitational field and the sions of curved spaces. noncommutative nature of space-time are two as- The fundamental open problem of the non- pects of the same thing. If the gravitational field commutative theory of gravity concerns of course is quantized then presumably the light-cone will the relation it might have to a future quantum fluctuate and any two points with a space-like theory of gravity either directly or via the theory separation would have a time-like separation on of ‘strings’ and ‘membranes’. But there are more a time scale of the order of the Planck time, in immediate technical problems which have not re- which case the corresponding operators would no ceived a satisfactory answer. We shall mention longer commute. So even in flat space-time quan- the problem of the definition of the curvature. It tum fluctuations of the gravitational field could is not certain that the ordinary definition of cur- be expected to introduce a non-locality in the vature taken directly from differential geometry theory. This is one possible source of noncommu- is the quantity which is most useful in the non- tative geometry on the order of the Planck scale. commutative theory. Cyclic homology groups The composition of the three arrows in (1.1) is an have been proposed by Connes as the appropri- expression of an old idea, due to Pauli, that per- ate generalization to noncommutative geometry turbative ultraviolet divergences will somehow be of topological invariants; the definition of other, regularized by the gravitational field. We refer to non-topological, invariants in not clear. It is not Garay [56] for a recent review. in fact even obvious that one should attempt to One example from which one can seek inspi- define curvature invariants. ration in looking for examples of noncommuta- There is an interesting theory of gravity, due tive geometries is quantized phase space, which to Sakharov and popularized by Wheeler, called had been already studied from a noncommuta- induced gravity, in which the gravitational field is tive point of view by Dirac [42]. The minimal a phenomenological coarse-graining of more fun- length in this case is given by the Heisenberg damental fields. Flat Minkowski space-time is uncertainty relations or by modifications thereof to be considered as a sort of perfect crystal and [76]. In fact in order to explain the supposed curvature as a manifestation of elastic tension, or Zitterbewegung of the electron Schrödinger had possibly of defects, in this structure. A deforma- proposed to mix position space with momentum tion in the crystal produces a variation in the vac- space in order to obtain a set of center-of-mass uum energy which we perceive as gravitational coordinates which did not commute. This idea energy. ‘Gravitation is to particle physics as elas- has inspired many of the recent attempts to in- ticity is to chemical physics: merely a statistical troduce minimal lengths. We refer to [48, 75] for measure of residual energies.’ The description of examples which are in one way or another con- the gravitational field which we are attempting nected to noncommutative geometry. Another to formulate using noncommutative geometry is concept from quantum mechanics which is use- not far from this. We have noticed that the use of ful in concrete applications is that of a coherent noncommuting coordinates is a convenient way state. This was first used in a finite noncom- of making a discrete structure like a lattice in- mutative geometry by Grosse & Preˇsnajder [59] variant under the action of a continuous group. and later applied [75, 17, 23] to the calculation of In this sense what we would like to propose is a propagators on infinite noncommutative geome- Lorentz-invariant version of Sakharov’s crystal. tries, which now become regular 2-point func- Each coordinate can be separately measured and tions and yield finite vacuum fluctuations. Al- found to have a distribution of eigenvalues simi- though efforts have been made in this direction lar to the distribution of atoms in a crystal. The [23] these fluctuations have not been satisfac- gravitational field is to be considered as a mea- torily included as a source of the gravitational sure of the variation of this distribution just as

3 corfu98 J. Madore elastic energy is a measure of the variation in the divergent integral density of atoms in a crystal. Z dp1dp2 2 2 2 When referring to the version of space-time I = ,p=p1+p2. p2 which we describe here we use the adjective ‘fuzzy’ to underline the fact that points are ill-defin- If one introduces a magnetic field B normal to ed. Since the algebraic structure is described the plane then the appropriately modified gauge- by commutation relations the qualifier ‘quantum’ covariant momenta no longer commute: has also been used [122, 43, 95]. This latter ex- [p1,p2]=i~eB. pression is unfortunate since the structure has no immediate relation to quantum mechanics and The points of momentum space have been re- also it leads to confusion with ‘spaces’ on which placed by ‘Landau cells’ of area ~eB. This serves ‘quantum groups’ act. To add to the confusion in general as an infrared cut-off: the word ‘quantum’ has also been used [57] to 2 & ~ designate equivalence classes of ordinary differ- p eB. ential geometries which yield isomorphic string If one were to replace the magnetic field by a theories and the word ‘lattice’ has been used [122, gaussian curvature K, ~eB 7→ ~2K then one 48, 124] to designate what we here qualify as would have the same effect; curvature in general ‘fuzzy’. acts as a mass. In this example ‘quantizing’ position-space 2. The basic idea coordinates consists in replacing them by two operators which satisfy a commutation relation of The basic idea is to ‘quantize’ the coordinates µ the form x of Minkowski space-time, to replace them by 1 2 12 µ [q ,q ]=i¯kq . generators q of a noncommutative algebra Ak¯ : Ipso facto the points of position space are re- µ ν µν ' −2 ~ [q ,q ]=i¯kq , ¯k µP = G . placed by ‘Planck cells’ of area 2π¯k and the in- We have written the area scale as an index but tegral I is completely regularized: of course this in no way characterizes the alge- I ∼ log(¯kK). (2.2) bra. The latter would be restricted but not in general uniquely defined by the structure of qµν . This vague idea can actually be implemented by The equivalent of the Heisenberg uncertainty re- explicit calculations [123, 43, 49, 75, 17, 104, 23, 2 lations would be Λ ¯k . 1. Very roughly speaking 78]. There is now however a new complication. then a fuzzy space-time would be composed of Although ~ is a constant and one usually sup- 2 cells of volume V ' (2π¯k) . In the limit µP →∞ poses the magnetic field B to be independent of we shall suppose that qµ → xµ but one could the momenta, the operator q12 in general could expect perhaps a singular ‘renormalization con- be expected a priori to be an arbitrary element of µ → µ stant’ Z: q Zx . To define the theory it the algebra Ak¯. Our working assumption is that would be important to introduce a state and to there is a one-to-one correspondence between the identify the qµ as hermitian operators on some q12 and the classical ‘gravitational’ field of the Hilbert space. commutative limit. In Wheeler’s language the As a simple example it is interesting to con- q12 determines the ‘lattice’ spacings away from sider the phase space of a particle in a plane: (flat space) equilibrium. For example, with the 1 2 (q ,q ,p1,p2). In classical mechanics one has definition of metric which we shall advocate one 12 12 four commuting operators; in quantum mechan- finds thatP if q = 1 the surface is flat, if rq = ics one has the commutation relations iq3 with (qi)2 = r2 the surface is a sphere and 12 2 2 1 2 ifkq ¯ = h(q ) the surface is a pseudosphere. If [q ,p1]=i~, [q ,p2]=i~. (2.1) q12 does not belong to the center of the algebra The points of classical phase space have been re- then Jacobi identities will imply that the canoni- placed by ‘Bohr cells’ of area 2π~. Consider the cal commutation relations (2.1) will be modified.

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As a simple illustration of how a ‘space’ can est in other, closely related, domain of physics. be ‘discrete’ in some sense and still covariant un- We have seen, for example, that without the dif- der the action of a continuous symmetry group ferential calculus, to be introduced below, the one can consider the ordinary round 2-sphere, fuzzy sphere is basically just an approximation which has acting on it the rotational group SO3. to a classical spin r by a quantum spin r with As a simple example of a lattice structure one ~ in lieu of ¯k. It has been extended in various can consider two points on the sphere, for exam- directions under various names and for various ple the north and south poles. One immediately reasons [9, 38, 68, 12, 11]. In order to explain notices of course that by choosing the two points the finite entropy of a black hole it has been con- one has broken the rotational invariance. It can jectured, for example by ’t Hooft [124], that the be restored at the expense of commutativity. The horizon has a structure of a fuzzy 2-sphere since set of functions on the two points can be iden- the latter has a finite number of ‘points’ and yet tified with the algebra C × C of diagonal 2 × 2 has an SO3-invariant geometry. The horizon of a matrices, each of the two entries on the diagonal black hole might be a unique situation in which corresponding to a possible value of a function at one can actually ‘see’ the cellular structure of one of the two points. Now an action of a group space. on the lattice is equivalent to an action of the group on the matrices and there can obviously 3. Differential calculi be no (non-trivial) action of the group SO3 on C × C. However, if one extends the algebra to In the following sections we shall discuss the alge- ∗ A the noncommutative algebra M2(C)ofall2×2 bra Ω ( ) of (1.1) and the associated differential 2 matrices one recovers the invariance. The two d, which satisfies the relation d = 0. The couple ∗ points, so to speak, have been smeared out over (Ω (A),d) is known as a differential calculus over the surface of a sphere; they are replaced by two the algebra A. The algebra A is what in ordinary cells. An ‘observable’ is an hermitian 2 × 2ma- geometry would determine the set of points one trix and has therefore two real eigenvalues, which is considering, with possibly an additional topo- are its values on the two cells. Although what we logical or measure theoretic structure. The dif- have just done has nothing to do with Planck’s ferential calculus is what gives an additional dif- constant it is similar to the procedure of replac- ferential structure or a notion of smoothness. On ing a classical spin which can take two values by a commutative algebra of functions on a lattice, a quantum spin of total spin 1/2. Only the quan- for example, it would determine the number of tum spin is invariant under the rotation group. nearest neighbours and therefore the dimension. By replacing the spin 1/2 by arbitrary spin s We give some simple examples before stating the one can describe a ‘lattice structure’ of n =2s+1 general definition. points in an SO3-invariant manner. The algebra 3.1 The Connes-Lott model becomes then the algebra Mn of n × n complex 3 2 1 matrices and there are n cells of area 2π¯k with Write C = C ⊕C and decompose M3 = M3(C) accordingly: Vol(S 2 ) n ' . + ⊕ − + ×C 2π¯k M3 = M3 M3 ,M3=M2 . In general, a static, closed surface in a fuzzy Fix   space-time as we define it can only have a fi- 00a1   ∗ − nite number of modes and will be described by η = 00a2,η(= −η ) ∈ M3 . ∗ ∗ some finite-dimensional algebra [58, 60, 62, 63, −a1−a20 64]. Graded extensions of some of these algebras For arbitrary α ∈ M3 define dα = −[η, α]witha have also been constructed [65, 61]. Although graded bracket. Then d is a graded derivation of we are interested in a matrix version of surfaces M3. In particular one finds that primarily as a model of an eventual noncommutative theory of gravity they have a certain inter- dη = −2η2,d2α=[η2,α].

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∗ We define an algebra Define an algebra Ω (Mn) by imposing the relations ∗ 0 1 2 Ω =Ω ⊕Ω ⊕Ω a b − b a a a η η η η θ θ = θ θ ,θ=θu where and a differential d as the restriction of du.Itis easily seen that 0 + 1 − 2 C Ωη = M3 , Ωη = M3 , Ωη = nM2−1 + Ω1(M ) ' M and by C we mean the second factor of M3 ,the n n i=1 entry in the lower right-hand corner of M3. This 2 definition of Ωη is the largest one which is consis- and one can show that tent with the condition that d2 = 0 since [η2,α] a 1 a b c + dθ = − C bcθ θ . necessarily belongs to the first factor of M3 if 2 ∈ 0 α Ωη. The multiplication law which defines a Introduce θ = −λaθ . Then one sees that the algebra is ordinary matrix multiplication. It is possible to chose η such that df = − [ θ, f]

dη + η2 =1. (3.1) and that dθ + θ2 =0. The right-hand side of this equation is to be con- ∗ 2 There is an obvious similarity between Ω (M ) sidered as an element of Ω . The algebra can be n η and the algebra of de Rham differential forms on extended to a Z-graded algebra by setting Ωp =0 η the group SU . for p ≥ 3. n 3.3 The noncommutative 2-torus 3.2 A matrix model The algebra P(u, v) of polynomials in u = eix, Consider M and over it the algebra n v = eiy is dense in any algebra of functions on M∞ the torus, defined by the relations 0 ≤ x ≤ 2π, ∗ i Ωu(Mn)= Ωu(Mn) 0 ≤ y ≤ 2π,wherexand y are the ordinary =1 i cartesian coordinates of R2. If one considers a i square lattice of n2 points then un =1andvn = where Ωu(Mn) is a submodule of the tensor prod- P uct of i + 1 copies of Mn and the product is ordi- 1 and the algebra is reduced to a subalgebra n 2 nary matrix multiplication of adjoining factors. of dimension n . Introduce a basis |ji1,0≤j≤ 0 − Cn | i ≡| i Define a map du of Ω (Mn)=Mn into Mn ⊗ Mn n 1, of with n 1 01 and replace u and by v by the operators ⊗ − ⊗ ∈ duf =1 f f 1,fMn. j n u|ji1 = q |ji1,v|ji1=|j+1i1,q=1. 1 Then Ω (Mn) is by definition the Mn-bimodule Then the new elements u and v satisfy the rela- generated by the image of du. Weusethesymbol tions f to underline the role of the elements of Mn as ‘functions’. There is a natural extension of du n n ∗ 2 uv = qvu, u =1,v=1 to all of Ω (Mn) such that du = 0. The couple ∗ (Ωu(Mn),du) is known as the universal calculus and the algebra they generate is the matrix alge- over Mn. It can be constructed over an arbitrary bra Mn instead of the commutative algebra Pn. associative algebra with unit. There is also basis |ji2 in which v is diagonal and Introduce an antihermitian basis λa of SUn a ‘Fourier’ transformation between the two [119] and define Introduce the forms [97] a a b θ = λbλ duλ . u 1 n −1 θ = −i 1 − |0i2h0| u du, Then one can show that n − 1 2 n −1 a a ∈ θ = −i 1 − |n − 1i1hn − 1| v dv. θuf = fθu,fMn. n − 1

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One can verify the relations There are many other infinite-dimensional algebras with associated differential calculi which a a ab−ba θ f = fθ ,θθ=θθ. have served as basis for exploring the possible ap- It follows that plications of noncommutative geometry to physics. We mention in particular the noncommuta- M2 1 a tive torus or rotation algebra [111, 112, 32, 27] Ω (Mn) ' Mn,dθ=0. 1 which extends the noncommutative torus we defined above, the quantum plane [101, 102, 127], The differential calculus has the form one would Rn the generalizations [47, 116, 50, 52] q of the expect of a noncommutative version of the torus. quantum line, the quantum sphere [107] the jor- 1 danian deformation [106, 2, 3, 80, 83, 77, 22] of 3.4 The quantum line R q SL(2, C) as well as ‘quantum’ deformations of R1 The quantum line [47, 117, 118, 48] q is the for- Minkowski space [4, 108, 109, 14, 5]. mal algebra with involution with two generators x and Λ which satisfy the relations 3.5 Differential calculi in general

∗ ∗ Consider an associative algebra A with a unit x = x , ΛΛ =1,xΛ=qΛx and a graded algebra for some real number q. There are two natu- M Ω∗(A)= Ωi(A), Ω0(A)=A ral differential calculi (Ω∗(R1),d)and(Ω¯∗(R1),d¯) q q i≥0 which are defined by the relations which is the direct sum of a family of A-bimod- dΛ=0, ules. A differential d is a graded derivation of Ω∗(A)withd2 =0.Ifα∈Ωi(A)andβ∈Ωj(A) dxΛ=qΛdx, xdx = qdxx, then αβ ∈ Ωi+j(A)andd(αβ) ∈ Ωi+j+1(A)with ¯ ¯ ¯ −1 ¯ dxΛ=qΛdx, xdx = q dxx. d(αβ)=dαβ +(−1)iαdβ. 1 ∈ 1 R1 If one introduces the element θ Ω ( q)and One usually supposes that as a bimodule Ω1(A) ¯1∈¯1 R1 θ Ω( q) defined by is generated by the image of d. A differential al- θ1 =Λ−1x−1dx, θ¯1 = q−1Λx−1dx¯ gebra is a graded algebra with a differential. We have already noted that the couple (Ω∗(A),d)is then one finds that called a differential calculus over A.Anelement p A 1 1 1 1 1 of Ω ( ) is known as a p-form. ¯ ¯ ∈R 1 θ f = fθ , θ f = fθ ,fq. Let Ω (A) be an arbitrary A-bimodule and let d be a module morphism It follows that d 1 1 R1 ' R1 ¯ 1 R1 ' R1 A −→ Ω ( A ) Ω ( q ) q, Ω ( q ) q. 1 1 from A into Ω (A). We have mentioned already There is a simple representation of R on an q the universal calculus Ω∗ (A)overA. In particu- infinite-dimensional analogue of the basis which u lar it defines a map we introduced to represent the noncommutative torus. It is given by A −→du A ⊗ A | i k| i | i | i x k = q k , Λ k = k +1 by d f =1⊗f−f⊗1,f∈A. A partial classification exists [115] of all repre- u 1 A ⊂ sentations. The representation can be extended As in the matrix case, by definition Ωu( ) to a representation of the differential calculi by A⊗A is the smallest A-bimodule which contains the identifications [15] the image of du. We can define then a map

1 7→ ¯1 7→ 1 A −→φ1 1 A θ 1, θ 1. Ωu( ) Ω ( )

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1 A 1 A ∈ 1 A of Ωu( )ontoΩ( )by It annihilates any f(x, y) Ωu( ) of second order in x − y. One such form is fdug −dugf.In φ1(duf)=df . fact

Because d1 = 0 the map is well deﬁned. We have (fdug −dugf)(x, y)=

A −→du 1 A −(f(y)−f(x))(g(y) − g(x)). Ω u ( ) k φ 1 ↓ 1 A The left-hand side does not vanish in Ωu( ) but d 1 A A −→ Ω 1 ( A ) its image in Ω ( ) under φ1 is equal to zero. The de Rham differential calculus can be character- and we can identify ized in fact by the condition that fdg −dgf =0 for arbitrary f,g ∈C(V). It follows immediately 1 A 1 A Ω ( )=Ωu( )/Ker φ1. then that df d g + dgdf = 0 and one readily sees in this example how the module structure of the Every bimodule of 1-forms can be so written. 1-forms influences the algebraic structure of the There exists a construction [25, 27, 41, 95] which algebra of forms. defines a differential calculus as the largest differential algebra consistent with the module struc- 3.7 The free-module case ture of the 1-forms. In particular the map φ1 can In order to analyze in more detail the structure be extended to a map of a differential calculus over an algebra A we ∗ A −→φ∗ ∗ A shall make three assumptions. We shall suppose Ωu( ) Ω ( ) first that as a bimodule 1 A uniquely defined by the bimodule Ω ( ). We Md shall use mainly 1-forms but on occasion 2-forms Ω1(A) ' A. and we shall need to introduce the product: 1

1 1 π 2 This is a condition which in ordinary geometry Ω (A) ⊗A Ω (A) −→ Ω ( A ) . would mean that the manifold in question is par- 3.6 Differential calculi over algebras of func- allelizable. It states in this case that globally a tions covariant vector can be identified with d functions. An example of such a manifold is the As an example let A = C(V ) be the algebra of sphere S3. If also the manifold is topologically smooth functions over a smooth manifold V .One trivial then any covariant vector can be identi- can introduce the de Rham differential calculus fied with its d components. A more restrictive 1 A ≡ 1 ∈A Ω ( ) Ω (V ). If f then duf is the func- assumption is that there exists d 1-forms θa such tion of 2 variables that [f,θa]=0, 1≤a≤d. duf(x, y)=f(y)−f(x). We shall refer to θa as a frame or Stehbein. The As we have already noticed it can be defined over existence of the frame obviously implies the par- any algebra of functions. The de Rham 1-form df allizability assumption. As a third condition we λ can be written locally as df = ∂ λ fdx in terms suppose a ‘smoothness’ condition. We assume of the differential of the coordinates. One can that there exist d elements λa of A such that the expand the function f(y)aboutthepointx: frame is dual to the associated derivations:

λ − λ ··· a a f(y)=f(x)+(x (y) x (x))∂λf + θ (eb)=δb,ea=adλa.

The map φ1 is given then by It follows in particular that

λ λ λ a a φ1(x (y) − x (x)) = dx . df = e a fθ =[λa,f]θ =−[θ, f]

8 corfu98 J. Madore and therefore that there exists a ‘Dirac operator’ even ‘smooth’. The quantum line is ‘smooth’, as is R3 [52]. Over the quantum plane mentioned in − a q θ = λaθ . Section 3.4 one can construct an infinite number of ‘smooth’ differential calculi for each value of Let θα = θαdxλ be a moving frame on a λ d [41]. In particular, when d =2andq4 =1the6 manifold and let γα = θαγλ be the associated λ choice Dirac matrices. The one can ‘almost’ make the identification θα 7→ γα.Infacttheθαanticom- 1 −2 2 λ1 = x y , mute and the γα almost anticommute. The des- q4 − 1 ignation ‘Dirac operator’ for θ comes from this 1 −2 λ2 = x identification. It yields the identity q4 − 1

α α df = θ e α f = γ e α f = − [ iD,/ f] (3.2) yields the Wess-Zumino calculus [127] which has special covariance properties. The jordanian de- where eα are the vector fields (derivations) dual formation is smooth [22]. to the moving frame. In the Connes-Lott exam- If the algebra A is a ∗-algebra then a ‘smooth’ ple of Section 3.1 we noticed that it was necessary differential calculus is real if the λa are anti- to restrict the module of 2-forms in order to have hermitian. The involution can be extended to 2 d = 0. This restriction is related to the prob- the differential calculus such that lems arising from the identification of the frame ∗ ∗ with the Dirac matrices. From (3.2) it follows in df =(df ) . fact that d2f = −[D/ 2,f] The frame is hermitian and the Dirac operator is antihermitian if one uses a graded commutator. It is not always possible to impose the ‘smooth- (θa)∗ = θa,θ∗=−θ. ness’ condition. The existence of the set of λa When imposes the consistency condition 1 P ab = (δaδb − δbδa) cd 2 c d c d 2λ λ P cd − λ F c − K =0. (3.3) c d ab c ab ab c then the F ab are hermitian and the Kab are anti-

This gives to the set of λa a sort of twisted Lie- hermitian. algebra structure with a central extension. The If one has a representation [27] of the algebra coefficients lie all in the center Z(A)(≡C). The and the differential calculus as von Neumann al- cd ∗ P ab are defined by the product in Ω (A): gebras then one can use the modular conjugation operator J to introduce a reality condition [28] a b a ⊗ b ab c ⊗ d θ θ = π(θ θ )=P cdθ θ . under more general conditions.

c a The F ab are related to the 2-form dθ : 4. Noncommutative Yang-Mills the- a 1 a b c dθ = − C bcθ θ ory 2 with The group of unitary elements of the algebra of a a (ae) C bc = F bc − 2λeP bc. functions on a manifold is the local gauge group of electromagnetism and the covariant derivative The K are related to the curvature of the ‘Dirac ab associated to the electromagnetic potential is a operator’: map D 1 2 1 H −→ ⊗ H dθ + θ = K θaθb Ω ( V ) A 2 ab from a C(V )-module H to the tensor product The Connes-Lott model of Section 3.1 is not par- 1 Ω (V ) ⊗C(V ) H, which satisﬁes a Leibniz rule allelizable but the matrix model of Section 3.2 and the noncommutative torus of Section 3.3 are D(fψ)=df ⊗ ψ + fDψ, f ∈C(V),ψ∈H.

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We shall often omit the tensor-product symbol in We recall that in this geometry a 2-form can be the following. As far as the electromagnetic po- considered as a complex number. The unit on tential is concerned we can identify H with C(V ) thee right-hand side is the unit in the right-hand itself; electromagnetism couples equally, for ex- side of (3.1). The action is given by ample, to all four components of a Dirac spinor. 1 2 2 The covariant derivative is defined therefore by V (φ)= Tr (1 −|φ| ) . 4 the Leibniz rule and the definition This describes electromagnetism on the Connes- D 1=A⊗1=A. Lott geometry. A covariant derivative can be defined as In the following we shall study electromagnetism on some noncommutative geometries. We Dψ = φψ. emphasize the fact that it is electromagnetism; Although this is gauge covariant it is not in the the geometry has changed not the theory being strict sense of the word a covariant derivative. studied. Because of the noncommutativity how- We shall return again to this point below. ever the result often looks more like nonabelian Yang-Mills theory and so we refer to it as such. 4.2 The matrix model The essential difference lies in the fact that one The same considerations can be repeated using must now distinguish between left- and right- the geometry of Section 3.2 with similar results. module structures. We choose as left module H = Mn. The gauge ⊂ ∈ 4.1 The Connes-Lott model group is now Un Mn. Let g Un.Itiseasyto see that Choose as left module H = C3 and let ψ ∈H. 0 −1 −1 −1 −1 Consider an anti-hermitian 1-form ω,elementof θ = g θg + g dg = g θg − g [θ, g]=θ. 1 Ωη introduced in Section 3.1. The gauge group is + If we write then an arbitrary connection form ω the group U2 ×U1 of unitary elements of M3 .We as ω = θ + φ we find that φ transforms under the repeat that this is the gauge group of ‘local’ elec- adjoint representation of U . The curvature or tromagnetic gauge transformations on the ‘space’ n + field strength is given by described by the algebra M3 . Let g ∈ U2 × U1 be a gauge transformation. It is easy to see that 1 Ω= Ω θaθb, Ω =[φ ,φ ]−Cc φ 2 ab ab a b ab c η0 = g−1ηg + g−1dg = g−1ηg − g−1[η, g]=η. and the associated action is given by This is a particular property of noncommutative 1 V (φ)=− Tr (Ω Ωab). geometry and is due to the fact that, at least in 4 ab the geometries which we shall study, the ‘Dirac This describes electromagnetism on the matrix operator’ is itself a 1-form. In more sophisticated geometry. The action vanishes at φ =0andon geometries [27] this will be in general no longer the gauge orbit φ = g−1θg of θ. the case. If we write A covariant derivative can be defined by one of the expressions ω = η + φ Dψ = φψ. then it is easy to see that Dψ = ψφ. φ0 = g−1φg. Dψ =[φ, ψ].

The φ transforms under the adjoint representa- Although these are all gauge covariant they × tion of U2 U1. If we consider ω as a connection are not in the strict sense of the word covariant form then its curvature or ﬁeld strength is given derivatives. The derivative by Ω=dω + ω2 =1+φ2 =1−|φ|2. Dψ = −θψ − ψφ

10 corfu98 J. Madore is covariant from the left and gauge-covariant Introduce a bilinear flip σ: from the right and the derivative 1 1 σ 1 1 Ω (A) ⊗A Ω (A) −→ Ω ( A ) ⊗ A Ω ( A ) (5.1) Dψ = ψθ + φψ We shall say that the metric is symmetric if is covariant from the right and gauge-covariant from the left. g ◦ σ ∝ g. At least in a particular case [96] the model of Of the examples we have discussed we men- the previous section can be obtained as a singular tion that the Connes-Lott model has a unique contraction of the model presented here. Elec- metric [98]. The matrix models have (more-or- tromagnetism has been studied also on infinite- less) unique metrics given by [98] dimensional noncommutative algebras. In fact the first example [32] was on the rotation alge- g(θa ⊗ θb)=gab,gab ∈ C. bra. R1 The line q has a (unique) metric which gives 5. Metrics rise to an observable: the distance between the spectral lines of x is uniform [15]. Other defini- We shall define a metric as a bilinear map tions of a metric have been given, some of which are similar to that given above but which weaken 1 A ⊗ 1 A −→g A Ω ( ) A Ω ( ) . the locality condition [16] and one [31] which defines a metric on the associated space of states. This is a ‘conservative’ definition, a straightfor- ward generalization of one of the possible definitions of a metric in ordinary differential ge- 6. Linear Connections ometry. The usual definition of a metric in the commutative case is a bilinear map An important geometric problem is that of com- paring vectors and forms defined at two different g X⊗C(V)X−→ C ( V ) points of a manifold. The solution to this problem leads to the concepts of a connection and where X is the C(V )-bimodule of vector fields on covariant derivative. There are two approaches. V . This definition is not suitable in the noncom- The traditional approach considers the connec- mutative case since the set of derivations of the tion as a primary object and the covariant deriva- algebra, which is the generalization of X ,hasno tive is defined in terms of it. But from the point natural structure as an A-module. The linearity of view of noncommutative geometry, which pla- condition is equivalent to a locality condition for ces primary importance on the algebra of func- the metric; the length of a vector at a given point tions, it is the second approach which is the more depends only on the value of the metric and the convenient and is the one which we shall consider vector field at that point. In the noncommutative here; the covariant derivative is defined as a lin- case bilinearity is the natural (and only possible) ear map between modules which satisfies certain expression of locality. It would exclude, for ex- Leibniz rules. We shall not define a noncommu- ample, a metric in ordinary geometry defined by tative generalization of a connection as a 1-form a map of the form on a principal fibre bundle [100]. Z We shall use here the expressions connection g(α, β)(x)= gx(αx,βy)G(x, y)dy. and covariant derivative synonymously. In fact V we shall distinguish three different types of con- 1 Here α, β ∈ Ω (V )andgx is a metric on the nections. A left connection or Yang-Mills con- tangent space at the point x ∈ V . The function nection is a connection on a left A-module; it G(x, y) is an arbitrary smooth function of x and satisfies a left Leibniz rule. A bimodule connec- y and dy is the measure on V induced by the tion is a connection on a general bimodule M metric. which satisfies a left and a right Leibniz rule. In

11 corfu98 J. Madore the particular case where M is the module of it is the definition of π; a 2-form can be consid- 1-forms we shall speak of a linear connection. ered as an antisymmetric tensor. Because of this Let A be an arbitrary algebra and (Ω∗(A),d) condition the torsion is a bilinear map. a differential calculus over A. One defines [79, 27] Using σ one can also construct an extension a Yang-Mills connection on a left A-module H as D2 1 amap M⊗A M−→ Ω ( A ) ⊗ A M⊗A M D 1 H−→ Ω ( A ) ⊗ A H by from a left A-module H to the tensor product 1 Ω (A) ⊗A H, which satisfies a left Leibniz rule D2(ξ ⊗ η)=Dξ ⊗ η + σ12 ◦ (ξ ⊗ Dη).

D(fψ)=df ⊗ ψ + fDψ, f ∈A,ψ∈H. We have here used the notation

It has a natural extension σ12 = σ ⊗ 1

∗ D ∗ Ω (A) ⊗A H −→ Ω ( A ) ⊗ A H (6.1) which we shall use again below. The operator D2 ◦ D is not in general left-linear. However, given by from the condition on σ follows the relation D(α ⊗ ψ)=dα ⊗ ψ +(−1)nα ⊗ Dψ D(ξ ⊗ η)=π12 ◦ D2(ξ ⊗ η)+Θ(ξ)⊗η ∈ n A if α Ω ( ). between the D given in the extension (6.1) and One defines the field strength by D2. It follows that 2 Ωψ = D ψ. 2 D = π12 ◦ D2 ◦ D +Θ. It satisfies the locality condition The left-hand side of this equation is defined for A Ω(fψ)=fΩψ. aleft -connection whereas the right-hand side is defined only in the case of a linear connection. A linear connection is a covariant derivative In particular one can conclude that π12 ◦ D2 ◦ D is left-linear. 1 A −→D 1 A ⊗ 1 A Ω ( ) Ω ( ) A Ω ( ) We introduce the notion of metric-compa- tibility exactly as in the commutative case. Let g on the A-bimodule Ω1(A) with an extra right be a metric and (D, σ) a linear connection. Both Leibniz rule g23 ◦ D2(ξ ⊗ η)anddg(ξ ⊗ η) are elements of 1 D(ξf)=σ(ξ⊗df )+(Dξ)f Ω (A). The linear connection is metric-compatible if defined using the flip σ introduced in (5.1). g23 ◦ D2 = d ◦ g. We define the torsion map We mentioned above the universal calculus Θ:Ω1(A)→Ω2(A) over an algebra A. In this case the product π is the identity map and it follows that σ = −1. − ◦ by Θ = d π D. It is left-linear. A short The ordinary differential du is clearly a covariant calculation yields derivative. The torsion vanishes and so does the curvature. Conversely let D define an arbitrary Θ(ξ)f − Θ(ξf)=π◦(1 + σ)(ξ ⊗ df ) . linear connection on the 1-forms of the univer- We shall impose the condition sal calculus. If we require the torsion to vanish then D = du. The only torsion-free linear con- π ◦ (σ + 1) = 0 (6.2) nection is the trivial one. Return to the example of Section 3.1. A covariant derivative is given by on σ. It could also be considered as a condition on the product π. In fact in ordinary geometry Dξ = −η ⊗ ξ + σ(ξ ⊗ η),ξ∈Ω1(A).

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One can show [98] that it is unique. The torsion the effective action is given by vanishes. The matrix models have a (more-or- less) unique torsion-free metric compatible con- Γ[g] ∝ Tr log ∆[g] ' R1 4 2 nections. The line q has two connections [15]. Λ Vol(V )[g]+Λ S1[g]+(logΛ)S2[g]+···. Using σ a reality condition on the metric and the linear connection can be introduced [51]. A If one identifies Λ = µP then one finds that S1[g] reality condition on the curvature can be formu- is the Einstein-Hilbert action. A problem with lated if σ satisfies the braid relation this is that it can be only properly defined on a compact manifold with a metric of euclidean σ12σ23σ12 = σ23σ12σ23. signature and Wick rotation on a curved space- time is a rather delicate if not dubious procedure. We do not insist upon this for reasons explained Another problem with this theory, as indeed with in the following section. the gravitational field in general, is that it pre- dicts an extremely large cosmological constant. 7. Gravity The expression Tr log ∆[g] has a natural generalization to the noncommutative case [70, 1, 19]. The classical gravitational field is normally sup- See also Example 7.3.5 of Madore [91]. posed to be described by a torsion-free, metric- We have defined gravity using a linear con- compatible linear connection on a smooth man- nection, which required the full bimodule struc- ifold. One might suppose that it is possible to ture of the A-module of 1-forms. We argued that formulate a noncommutative theory of (classi- this was necessary to obtain a satisfactory defi- cal/quantum) gravity by replacing the algebra nition of locality as well as a reality condition. of functions by a more general algebra and by It is possible to relax these requirements and de- choosing an appropriate differential calculus. We fine gravity as a Yang-Mills field [18, 120, 82, 20, however mentioned in the first section the prob- 24, 54] or as a couple of left and right connec- lem in defining curvature invariants. One way of tions [36, 37]. If the algebra is commutative (but circumventing this problem is to consider classi- not an algebra of smooth functions) then to a cer- cal gravity as an effective theory and the Einstein- tain extent all definitions coincide [39, 40, 81, 6]. Hilbert action as an induced action. We recall that the classical gravitational action is given by Z 8. Kaluza-Klein theory 4 2 S[g]=µPΛc+µP R. We mentioned in the Introduction that one of the first, obvious applications of noncommuta- In the noncommutative case there is a natu- tive geometry is as an alternative hidden struc- ral definition of the integral [32, 26, 27] but there ture of Kaluza-Klein theory. This means that one does not seem to be a natural generalization of leaves space-time as it is and one modifies only the Ricci scalar. One of the problems is the one the extra dimensions; one replaces their algebra we touched upon in the preceding section: the of functions by a noncommutative algebra, usu- natural generalization of the curvature form is in ally of finite dimension to avoid the infinite tower general not right-linear in the noncommutative of massive states of traditional Kaluza-Klein the- case. The Ricci scalar then will not be local. One ory. Because of this restriction and because the way of circumventing these problems is to return extra dimensions are purely algebraic in nature to an old version of classical gravity known as the length scale associated with them can be ar- induced gravity [113, 114]. The idea is to iden- bitrary [92], indeed as large as the Compton wave tify the gravitational action with the quantum length of a typical massive particle. corrections to a classical field in a curved back- The algebra of Kaluza-Klein theory is there- ground. If ∆[g] is the operator which describes fore, for example, a product algebra of the form the propagation of a given mode in presence of ametricgthen one finds that, with a cut-off Λ, A = C(V ) ⊗ Mn.

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Z Normally V would be chosen to be a manifold 1 αβ = Tr Fαβ F of dimension four, but since much of the formal- 4 Z Z ism which we shall outline is identical to that of 1 + Tr D φ D α φa − V (φ) the M(atrix)-theory of D-branes [8, 55, 33] we 2 α a shall leave the dimension unspecified. We men- where [53, 103, 44, 88] tion first electromagnetism and then gravity. Let θi =(θα,θa)beamovingframeonV. 1 ab V (φ)=− Tr (ΩabΩ ). This means that we suppose that V is paralleliz- 4 able. The matrix factor is also parallelizable. We As an example choose d =3and discussed in some detail in Section 3.2 the case C = r−1 . where the differential calculus is based on a set of abc abc derivations which form the Lie algebra of SUn. The hidden ‘space’ is a fuzzy sphere of radius According to the discussion of Section 3.7 this r. The potential V (φ) vanishes when φ lies on can be generalized. One can choose a differential a gauge orbit of a representation of SU2.There calculus such that are √ eπ 2n/3 Md ' √ 1 p(n) Ω (Mn) ' Mn 4n 3 1 such orbits. for arbitrary integer d, provided the consistency If one replaces the matrix algebra by the al- condition (3.3) is satisfied. In fact the interesting gebra of the Connes-Lott model then one obtains case is when a family of theories which includes the standard n d. model of the electroweak interactions. It, and its extensions to include the strong interactions, 1 We write Ω (A) as a direct sum have been extensively studied [30, 7, 71, 126, 85, 110]. Ω1(A)=Ω1 ⊕Ω1, h v For the simple models with a matrix exten- with sion one can use as gravitational action the Ein- stein-Hilbert action in ‘dimension’ 4 + d, includ- 1 1 ⊗ Ωh =Ω (V) Mn, ing possibly Gauss-Bonnet terms [88, 92, 91, 93, 73]. For a more detailed review we refer to a Ω1 =C(V)⊗Ω1(M ). v n lecture [94] at the 5th Hellenic school in Corfu. The differential df of an element f of A is given by 9. String Theory

df = d h f + d v f. Last, but not least, is the possible relation of non- i In terms of the frame θ we have as usual commutative geometry to string theory. We have mentioned that since noncommutative geometry d f = e fθα,df=efθa = −[θ, f]. h α v a is pointless a field theory on it will be divergence- We introduce a torsion-free linear connec- free. In particular monopole configurations will tion, compatible with the frame. have finite energy, provided of course that the geometry in which they are constructed can be i i j dθ + ω j θ =0 approximated by a noncommutative geometry, since the point on which they are localized has α α β a −1 a c with θ +ω β θ =0,andω b = 2C bc θ .Re- been replaced by an volume of fuzz, This is one ferring back to Section 4.2 we see that the electro- characteristic that it shares with string theory. magnetic action for the potential ω = A+(θ+φ) Certain monopole solutions in string theory have is given by finite energy [125] since the point in space (a D- Z Z 1 brane) on which they are localized has been re- S[A, φ]= L(A, φ)= Tr Ω Ωij 4 ij placed by a throat to another ‘adjacent’ D-brane.

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