Alain Connes (pdf)
Alain Connes (pdf)
Alain Connes (pdf)
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NONCOMMUTATIVE GEOMETRY<br />
<strong>Alain</strong> <strong>Connes</strong><br />
As long as we consider geometry as intimately related to our model of space-time, Einstein’s<br />
general relativity clearly vindicated the ideas of Gauss and Riemann, allowing<br />
for variable curvature. They formulated the intrinsic geometry of a curved space independently<br />
of its embedding in Euclidean space. The two key notions are those of<br />
manifold of arbitrary dimension, whose points are labelled by finitely many real numbers<br />
x µ , and that of the line element i.e. the infinitesimal unit of length which, when<br />
carried around, allows to measure distances. The infinitesimal calculus allows to encode<br />
the geometry by the formula for the line element ds in local terms<br />
ds 2 = g µν dx µ dx ν . (1)<br />
and to generalise most of the concepts which were present either in euclidean or noneuclidean<br />
geometry while considerably enhancing the number of available interesting examples.<br />
In particular the idea of straight line does continue to make sense and is given<br />
by the equation of geodesics,<br />
d 2 x µ<br />
dt 2<br />
= − 1 2 gµα (g αν,ρ + g αρ,ν − g νρ,α ) dxν<br />
dt<br />
which, when applied in the modified Minkowski Lorentzian metric,<br />
dx ρ<br />
dt<br />
(2)<br />
dx 2 + dy 2 + dz 2 − (1 + 2V (x, y, z)) dt 2 (3)<br />
gives Newton’s law for the motion in the gravitational field deriving from the potential<br />
function V .<br />
Riemann was sufficiently cautious in his lecture on the foundation of geometry to question<br />
the validity of his hypotheses in the infinitely small. He explicitly proposed to<br />
”gradually modify the foundations under the compulsion of facts which cannot be explained<br />
by it” in case physics would find new unexplained phenomena in the exploration<br />
of smaller scales.<br />
The origin of noncommutative geometry can be traced back to the discovery of such<br />
unexplained phenomena in the phase space of the microscopic mechanical system describing<br />
an atom. This system manifests itself through its interaction with radiation<br />
1
and the basic laws of spectroscopy, as found in particular by Ritz and Rydberg, are in<br />
contradiction with the ”manifold” picture of the phase space.<br />
The very bare fact which came directly from experimental findings in spectroscopy and<br />
was unveiled by Heisenberg (and then understood at a more mathematical level by Born,<br />
Jordan, Dirac and the physicists of the late 1920’s) is that whereas when you are dealing<br />
with a manifold you can parametrize (locally) its points x by real numbers x 1 , x 2 , . . .<br />
which specify completely the situation of the system, when you turn to the phase space<br />
of a microscopic mechanical system, even of the simplest kind, the coordinates, namely<br />
the real numbers x 1 , x 2 , . . . that you would like to use to parametrize points, actually<br />
do not commute.<br />
This means that the classical geometrical framework is too narrow to describe in a<br />
faithful manner many physical spaces of great interest. In noncommutative geometry<br />
one replaces the usual notion of manifold formed of points labelled by coordinates<br />
with spaces of a more general nature (as we shall see later). Usual geometry is just<br />
a particular case of this new theory, in the same way as Euclidean and non Eulidean<br />
geometry are particular cases of Riemannian geometry. Many of the familar geometrical<br />
concepts do survive in the new theory, but they carry also a new unexpected meaning.<br />
Before describing the novel notion of space it is worthwile to explain in simple terms<br />
how noncommutative geometry modifies the measurement of distances. Such a simple<br />
description is possible because the evolution between the Riemannian way of measuring<br />
distances and the new (noncommutative) way exactly parallels the improvement of<br />
the standard of length ∗ in the metric system. The original definition of the meter at<br />
the end of the 18th century was based on a small portion (one forty millionth part)<br />
of the size of the largest available macroscopic object (here the earth circonference).<br />
Moreover this “unit of length” became concretely represented in 1799 as “metre des<br />
archives” by a platinum bar localised near Paris. The international prototype was a<br />
more stable copy of the “metre des archives” which served to define the meter. The<br />
most drastic change in the definition of the meter occured in 1960 when it was redefined<br />
as a multiple of the wavelength of a certain orange spectral line in the light emitted<br />
by isotope 86 of the krypton. This definition was then replaced in 1983 by the current<br />
definition which using the speed of light as a conversion factor is expressed in terms of<br />
inverse-frequencies rather than wavelength, and is based on a hyperfine transition in<br />
the caesium atom. The advantages of the new standard are obvious. No comparison to<br />
a localised “metre des archives” is necessary, the uncertainties are estimated as 10 −15<br />
and for most applications a commercial caesium beam is sufficiently accurate. Also we<br />
could (if any communication were possible) communicate our choice of unit of length to<br />
aliens, and uniformise length units in the galaxy without having to send out material<br />
copies of the “metre des archives”!<br />
As we shall see below the concept of “metric” in noncommutative geometry is precisely<br />
based on such a spectral data.<br />
Let us now come to “spaces”. What the discovery of Heisenberg showed is that the<br />
familiar duality of algebraic geometry between a space and its algebra of coordinates<br />
(i.e. the algebra of functions on that space) is too restrictive to model the phase space<br />
of microscopic physical systems. The basic idea then, is to stretch this duality so<br />
that the algebra of coordinates on a space, is no longer required to be commutative.<br />
Gelfand’s work on C ∗ -algebra provides the right framework to define noncommutative<br />
∗ or equivalently of time using the speed of light as a conversion factor<br />
2
topological spaces. They are given by their algebra of continuous functions which can<br />
be an arbitrary, not necessarily commutative C ∗ -algebra.<br />
It turns out that there is a wealth of examples of spaces which have obvious geometric<br />
meaning but which are best described by a noncommutative algebra of coordinates.<br />
The first examples came, as we saw above, from phase space in quantum mechanics<br />
but there are many others, such as the leaf spaces of foliations, the duals of nonabelian<br />
discrete groups, the space of Penrose tilings, the noncommutative torus which plays a<br />
role in M-theory compactification, the Brillouin zone in solid state physics, the spaces<br />
underlying ”quantum groups” obtained as deformations of classical Lie groups and<br />
finally the space of Q-lattices which is a natural geometric space with an action of<br />
the scaling group providing a spectral interpretation of the zeros of the L-functions<br />
of number theory and an interpretation of the Riemann explicit formulas as a trace<br />
formula.<br />
The second vital ingredient of the theory is the extension of geometric ideas to the<br />
noncommutative framework. The stretching of geometric thinking imposed by passing<br />
to noncommutative spaces forces one to rethink about most of our familiar notions.<br />
The difficulty is not to add arbitrarily the adjective quantum to our familiar geometric<br />
language but to develop far reaching extensions of classical concepts. As we shall see<br />
below this has been achieved with variable degrees of perfection for measure theory,<br />
topology, differential geometry, and Riemannian geometry.<br />
Let us first describe the basic principle which allows to establish the general duality,<br />
Quotient spaces ↔ Noncommutative algebra . (4)<br />
Many of the spaces that we are used to consider are not really described by listing their<br />
points but are obtained by identifications, their elements being given as equivalence<br />
classes in a larger space M.<br />
There are two ways of proceeding at the algebraic level in order to identify two points<br />
a and b in a given space M. The first way is to consider only those functions f on M<br />
which take the same value at the points a and b, namely that satisfy f(a) = f(b). Thus<br />
the usual algebra of functions associated to the quotient is<br />
A = {f; f(a) = f(b)} . (5)<br />
There is however another way of performing, in an algebraic manner, the above quotient<br />
operation. It consists, instead of taking the subalgebra given by (5), to adjoin to the<br />
algebra of functions the identification of a with b. The obtained algebra is, restricting<br />
to M = {a, b}, the algebra of two by two matrices<br />
B =<br />
{<br />
f =<br />
[ ]}<br />
faa f ab<br />
. (6)<br />
f ba f bb<br />
In other words we do not require that functions have the same value at a and b, but we<br />
allow the two points a and b to relate to each other via the off-diagonal matrix elements<br />
f a b and f b a . The effect of these off-diagonal matrix elements is to identify these two<br />
points into one point in the spectrum of the algebra, i. e. in the space of its irreducible<br />
representations. Indeed the irreducible representation corresponding to a (associated<br />
to the pure state a) has now become equivalent to the representation coming from b.<br />
3
Now in this very simple example of two points, the two different algebras that one<br />
obtains are still equivalent. The equivalence between A and B is called Morita equivalence<br />
[28] and means that the corresponding categories of right modules are equivalent.<br />
There is of course an obvious difference between A and B since, unlike A, the algebra<br />
B is no longer commutative.<br />
Let me now give another simple example, in which the two ways of doing things do not<br />
give the same answer, even up to Morita equivalence. We take two intervals, [0,1], and<br />
we want to identify the insides of these two intervals but not their endpoints.<br />
Applying the first procedure above, and working in the smooth (C ∞ ) category, we<br />
obtain the algebra of smooth functions on M = [0, 1]× {0, 1} which have equal values<br />
on the points (x, 0) and (x, 1) for any x in the open interval ]0, 1[. But, by continuity,<br />
functions which are equal inside the intervals are also equal at the endpoints. Thus<br />
what we get is simply the algebra A = C ∞ ([0, 1]), whose spectrum is the interval [0, 1],<br />
and we did not succeed to keep the endpoints apart.<br />
Applying the second procedure, we get the algebra of smooth maps from the interval<br />
[0,1] to 2 × 2 matrices, but since the endpoints are not identified with each other, we<br />
only get smooth maps which have diagonal matrices as boundary values. The obtained<br />
algebra B is quite interesting, it is the smooth group ring of the dihedral group, namely<br />
of the free product of two groups with 2 elements. This algebra is quite different from<br />
the first one, and certainly not Morita equivalent to C ∞ ([0, 1]).<br />
In general, when we consider arbitrary quotient spaces, the two algebras A and B are<br />
not Morita equivalent. The first operation (5) is of a cohomological flavor while the<br />
second (6) keeps a closer contact with the quotient space.<br />
We shall now describe a basic concrete example, the noncommutative torus whose<br />
noncommutative differential geometry was unveiled in 1980 ([8]). This example is the<br />
following: consider the 2-torus<br />
M = R 2 /Z 2 . (7)<br />
The space X which we contemplate is the space of solutions of the differential equation,<br />
where θ ∈]0, 1[ is a fixed irrational number (fig 1).<br />
dx = θdy x, y ∈ R/Z (8)<br />
1<br />
θ<br />
y<br />
dx = θdy<br />
x, y ∈ R/Z<br />
0<br />
x 1<br />
Figure 1<br />
Thus the space we are interested in here is just the space of leaves of the foliation defined<br />
by the differential equation. We can label such a leaf by a point of the transversal given<br />
by y = 0 which is a circle S 1 = R/Z, but clearly two points of the transversal which<br />
differ by an integer multiple of θ give rise to the same leaf. Thus<br />
X = S 1 /θZ (9)<br />
4
i.e. X is the quotient of S 1 by the equivalence relation which identifies any two points<br />
on the orbits of the irrational rotation<br />
R θ x = x + θ mod 1 . (10)<br />
When we deal with S 1 as a space in the various categories (smooth, topological, measurable)<br />
it is perfectly described by the corresponding algebra of functions,<br />
C ∞ (S 1 ) ⊂ C(S 1 ) ⊂ L ∞ (S 1 ) . (11)<br />
When one applies the operation (5) to pass to the quotient, one finds, irrespective of<br />
which category one works with, the trivial answer<br />
A = C . (12)<br />
The operation (6) however gives very interesting algebras, by no means Morita equivalent<br />
to C. In the above situation of a quotient by a group action the operation (6) is<br />
the construction of the crossed product familiar to algebraist from the theory of central<br />
simple algebras.<br />
An element of B is given by a power series<br />
b = ∑ n∈Z<br />
b n U n (13)<br />
where each b n is an element of the algebra (11), while the multiplication rule is given<br />
by<br />
UhU −1 = h ◦ R −1<br />
θ<br />
. (14)<br />
Now the algebra (11) is generated by the function V on S 1 ,<br />
V (α) = exp(2πiα) α ∈ S 1 (15)<br />
and it follows that B admits the generating system (U, V ) with presentation given by<br />
the relation<br />
UV U −1 = λ −1 V λ = exp2πiθ . (16)<br />
Thus, if for instance we work in the smooth category a generic element b of B is given<br />
by a power series<br />
b = ∑ Z 2 b nm U n V m , b ∈ S(Z 2 ) (17)<br />
where S(Z 2 ) is the Schwartz space of sequences of rapid decay on Z 2 .<br />
This algebra has a very rich and interesting algebraic structure. It is (canonically up<br />
to Morita equivalence) associated to the foliation (8) and the interplay between the<br />
geometry of the foliation and the algebraic structure of B begins by noticing that to<br />
a closed transversal of the foliation corresponds canonically a finite projective module<br />
over B.<br />
From the transversal x = 0, one obtains the following right module over B. The<br />
underlying linear space is the usual Schwartz space,<br />
S(R) = {ξ, ξ(s) ∈ C ∀s ∈ R} (18)<br />
of smooth functions on the real line all of whose derivatives are of rapid decay.<br />
5
The right module structure is given by the action of the generators U, V<br />
(ξU)(s) = ξ(s + θ) , (ξV )(s) = e 2πis ξ(s) ∀s ∈ R . (19)<br />
One of course checks the relation (16), and it is a beautiful fact that as a right module<br />
over B the space S(R) is finitely generated and projective (i.e. complements to a free<br />
module). It follows that it has the correct algebraic atributes to deserve the name of<br />
“noncommutative vector bundle” according to the dictionary,<br />
Space<br />
Vector bundle<br />
Algebra<br />
Finite projective module.<br />
The concrete description of the general finite projective modules over A θ is obtained<br />
by combining the results of [8] [27] [29]. They are classified up to isomorphism by a<br />
pair of integers (p, q) such that p + q θ ≥ 0 and the corresponding modules H θ p,q are<br />
obtained by the above construction from the transversals given by closed geodesics of<br />
the torus M.<br />
The algebraic counterpart of a vector bundle is its space of smooth sections C ∞ (X, E)<br />
and one can in particular compute its dimension by computing the trace of the identity<br />
endomorphism of E. If one applies this method in the above noncommutative example,<br />
one finds<br />
dim B (S) = θ . (20)<br />
The appearance of non integral dimension is very exciting and displays a basic feature<br />
of von Neumann algebras of type II. The dimension of a vector bundle is the only<br />
invariant that remains when one looks from the measure theoretic point of view (i.e.<br />
when one takes the third algebra in (11)). The von Neumann algebra which describes<br />
the quotient space X from the measure theoretic point of view is the crossed product,<br />
R = L ∞ (S 1 ) >✁ Rθ Z (21)<br />
and is the well known hyperfinite factor of type II 1 . In particular the classification of<br />
finite projective modules E over R is given by a positive real number, the Murray and<br />
von Neumann dimension,<br />
dim R (E) ∈ R + . (22)<br />
The next surprise is that even though the dimension of the above module is irrational,<br />
when we compute the analogue of the first Chern class, i.e. of the integral of the<br />
curvature of the vector bundle, we obtain an integer. Indeed the two commuting vector<br />
fields which span the tangent space for an ordinary (commutative) 2-torus correspond<br />
algebraically to two commuting derivations of the algebra of smooth functions. These<br />
derivations continue to make sense when the generators U and V of C ∞ (T 2 ) no longer<br />
commute but satisfy (16) so that they generate C ∞ (T 2 θ ). They are given by the same<br />
formulas as in the commutative case,<br />
δ 1 = 2πiU<br />
∂<br />
∂U , δ 2 = 2πiV<br />
so that δ 1 ( ∑ b nm U n V m ) = 2πi ∑ nb nm U n V m and similarly for δ 2 . One still has of<br />
course<br />
δ 1 δ 2 = δ 2 δ 1 (24)<br />
∂<br />
∂V<br />
(23)<br />
6
and the δ j are still derivations of the algebra B = C ∞ (T 2 θ ),<br />
δ j (bb ′ ) = δ j (b)b ′ + bδ j (b ′ ) ∀b, b ′ ∈ B . (25)<br />
The analogues of the notions of connection and curvature of vector bundles are straightforward<br />
to obtain since a connection is just given by the associated covariant differentiation<br />
∇ on the space of smooth sections. Thus here it is given by a pair of linear<br />
operators,<br />
∇ j : S(R) → S(R) (26)<br />
such that<br />
∇ j (ξb) = (∇ j ξ)b + ξδ j (b) ∀ξ ∈ S , b ∈ B . (27)<br />
One checks that, as in the usual case, the trace of the curvature Ω = ∇ 1 ∇ 2 − ∇ 2 ∇ 1 , is<br />
independent of the choice of the connection. Now the remarkable fact here ([8]) is that<br />
(up to the correct powers of 2πi) the integral curvature of S is an integer. In fact for<br />
the following choice of connection the curvature Ω is constant, equal to 1 so that the<br />
θ<br />
irrational number θ disappears in the integral curvature, θ × 1 θ<br />
(∇ 1 ξ)(s) = − 2πis<br />
θ<br />
ξ(s) (∇ 2 ξ)(s) = ξ ′ (s) . (28)<br />
With this integrality, one could get the wrong impression that the algebra B = C ∞ (T 2 θ )<br />
looks very similar to the algebra C ∞ (T 2 ) of smooth functions on the 2-torus. A striking<br />
difference is obtained by looking at the range of Morse functions. The range of a Morse<br />
function on T 2 is of course a connected interval. For the above noncommutative torus<br />
T 2 θ the range of a Morse function is the spectrum of a real valued function such as<br />
h = U + U ∗ + µ(V + V ∗ ) (29)<br />
and it can be a Cantor set, i.e. have infinitely many disconnected pieces. This shows<br />
that the one dimensional shadows of our space T 2 θ are truly different from what they are<br />
in the commutative case. The above noncommutative torus T 2 θ is the simplest example<br />
of noncommutative manifold, it arises naturally not only from foliations but also from<br />
the Brillouin zone in the Quantum Hall effect as understood by J. Bellissard, and in M-<br />
theory as we shall see next. Indeed both the noncommutative tori and the components<br />
∇ j of the Yang-Mills connections occur naturally in the classification of the BPS states<br />
in M-theory [15]. In the matrix formulation of M-theory the basic equations to obtain<br />
periodicity of two of the basic coordinates X i turn out to be the following<br />
U i X j Ui −1 = X j + a δ j i , i = 1, 2 (30)<br />
where the U i are unitary gauge transformations.<br />
The multiplicative commutator U 1 U 2 U1 −1 U2 −1 is then central and in the irreducible case<br />
its scalar value λ = exp 2πiθ brings in the algebra of coordinates on the noncommutative<br />
torus. The X j are then the components of the Yang-Mills connections. It is quite<br />
remarkable that the same picture emerged from the other information one has about M-<br />
theory concerning its relation with 11 dimensional supergravity and that string theory<br />
dualities could be interpreted using Morita equivalence. The latter [28] relates the<br />
values of θ on an orbit of SL(2, Z) and this type of relation, which is obvious from the<br />
foliation point of view, would be invisible in a purely deformation theoretic perturbative<br />
expansion like the one given by the Moyal product.<br />
7
We shall come back later to the natural moduli space for the noncommutative tori<br />
and to the metric aspect when we have the correct general tools. But we first need to<br />
dispell the impression that the noncommutative torus, because of its ubiquity, is the<br />
only example of noncommutative space.<br />
A very large class of examples is provided first of all by the duals of discrete groups. We<br />
saw already the dual of the dihedral group which was truly very simple, but as soon as<br />
one considers more complicated discrete groups one finds that their duals are exactly<br />
like leaf spaces of foliations. It is important to understand at least on one example why<br />
the dual of a discrete non abelian group is in essence a quotient space, like the one we<br />
analysed above for foliations. Thus, let Γ be the solvable group Γ = Z 2 >✁ α Z obtained<br />
as the semi-direct product of the additive group Z 2 by the action of Z given by the<br />
matrix<br />
[ ]<br />
1 1<br />
T = ∈ SL(2, Z) . (31)<br />
1 2<br />
By Mackey’s theory of induced representations each element x of the 2-torus Ẑ2 which<br />
is the Pontrjagin dual of the group Z 2 , determines by induction an irreducible representation<br />
of Γ,<br />
π x ∈ Irrep(Γ) . (32)<br />
Moreover the two representations π x and π y associated to x, y ∈ Ẑ2 , are equivalent iff<br />
x and y are on the same orbit of the (transposed) transformation T of Ẑ2 . One does<br />
not get all irreducible representations of Γ by this procedure but the quotient space<br />
X = Ẑ2 / ∼ , x ∼ y iff y = T n x for some n (33)<br />
does correspond in the above sense to the smooth group ring of Γ.<br />
Here is a more complete list of basic examples,<br />
Space of leaves of foliations<br />
Space of irreducible representations of discrete groups<br />
Space of Penrose tilings of the plane<br />
Brillouin zone in the quantum Hall effect<br />
Phase space in quantum mechanics<br />
Space time<br />
Another rich class of examples arises from deformation theory such as deformation of<br />
Poisson manifolds, quantum groups and their homogeneous spaces.<br />
Let me finally mention the space of Adele classes, a noncommutative space whose<br />
understanding is intimately related to the location of the zeros of Hecke L-functions in<br />
the number field case.<br />
Thus there is no shortage of examples of noncommutative spaces that beg our understanding<br />
but which are very difficult to comprehend.<br />
The reason why I started working in noncommutative geometry is that I found that<br />
even at the very coarse level of measure theory, the general noncommutative case was<br />
becoming highly non trivial. When you look at an ordinary space and you do measure<br />
theory, you use the Lebesgue theory which is a beautiful theory, but all spaces are the<br />
same, there is nothing really happening as far as classification is concerned. This is<br />
not at all the case in noncommutative measure theory. What happens there is very<br />
surprising. It is an absolutely fascinating fact that when you take a non commutative<br />
algebra M from the measure theory point of view, such an algebra evolves with time !<br />
8
What I mean is that it admits a god-given time evolution given by a canonical group<br />
homomorphism, ([6])<br />
δ : R → Out(M) = Aut(M)/Int(M) (34)<br />
from the additive group R to the center of the group of automorphism classes of M<br />
modulo inner automorphisms.<br />
This homomorphism is provided by the uniqueness of the, a priori state dependent,<br />
modular automorphism group of a state. Together with the earlier work of Powers,<br />
Araki-Woods and Krieger it was the beginning of a long story which eventually led to the<br />
complete classification ([6], [31], [26], [21], [7], [22]) of approximately finite dimensional<br />
factors (also called hyperfinite).<br />
They are classified by their module,<br />
Mod(M) ⊂<br />
∼<br />
R ∗ + , (35)<br />
which is a virtual closed subgroup of R ∗ + in the sense of G. Mackey, i.e. an ergodic<br />
action of R ∗ + .<br />
One can interpret this classification as a non-trivial Brauer theory for Archimedian<br />
local fields. Local class field theory is interesting and non trivial for local fields which<br />
are non archimedian, like p-adic numbers because such fields K are very far from being<br />
algebraically closed.<br />
But there is nothing comparable when you look at class field over complex numbers,<br />
because the field of complex numbers is algebraically closed, so there is apparently<br />
nothing going on. Now it turns out that the theory of factors gives a non-trivial<br />
replacement for the missing Brauer theory at archimedian places. This can be seen<br />
as follows. To say that a field is algebraically closed can be formulated in terms of<br />
representation theory. It is equivalent to say that when we look at representations<br />
of semisimple objects (groups, algebras...) over this field any such representation can<br />
be decomposed in direct sums of multiples of representations whose commutant is the<br />
scalars. Of course this does hold when we take finite dimensional representations over<br />
C. But as soon as we take infinite dimensional representations a new phenomenon<br />
occurs and it is in perfect analogy with the unramified extensions of p-adic fields.<br />
Non trivial factors occur, they are commutants of representations in Hilbert space and<br />
though they have trivial center, they are not Morita equivalent to C. As we saw these<br />
factors have an invariant, which is called their module, very analogous to the module<br />
of central simple algebras over local fields; this module is not really a subgroup of R ∗ + ,<br />
it’s a virtual subgroup of R ∗ + in the sense that it is a flow. And for factors which are<br />
approximately finite dimensional this flow turns out to give exactly a complete invariant<br />
of factors and all flows occur. This analogy goes quite far, noncommutative manifolds<br />
give examples of construction of the hyperfinite II 1 factor as the crossed product of the<br />
field K q of elliptic functions by a subgroup of its Galois group in perfect analogy with the<br />
Brauer theory. The occurence of the type III 1 factor from simple adelic constructions<br />
yields a spectral interpretation of the zeros of L-functions in number theory ([13]).<br />
So, we see that noncommutative measure theory is already highly non trivial, and<br />
thus we have all reasons to believe that if one goes further in the natural hierarchy of<br />
features of a space, one will discover really interesting new phenomenas.<br />
Measure theory is indeed a very coarse way of looking at a space, and a finer and finer<br />
picture is obtained by going up in the following hierarchy of points of view:<br />
9
Riemannian geometry<br />
Differential geometry<br />
Topology<br />
Measure theory<br />
The development of the topological ideas was prompted by the work of Israel Gel’fand,<br />
whose C* algebras give the required framework for noncommutative topology. The<br />
two main driving forces were the Novikov conjecture on homotopy invariance of higher<br />
signatures of ordinary manifolds as well as the Atiyah-Singer Index theorem. It has<br />
led, through the work of Atiyah, Singer, Brown, Douglas, Fillmore, Miscenko and Kasparov<br />
([1] [3] [25]) to the recognition that not only the Atiyah-Hirzebruch K-theory but<br />
more importantly the dual K-homology admit Hilbert space techniques and functional<br />
analysis as their natural framework. The cycles in the K-homology group K ∗ (X) of a<br />
compact space X are indeed given by Fredholm representations of the C* algebra A of<br />
continuous functions on X. The central tool is the Kasparov bivariant K-theory. A basic<br />
example of C* algebra to which the theory applies is the group ring of a discrete group<br />
and restricting oneself to commutative algebras is an obviously undesirable assumption.<br />
For a C ∗ algebra A, let K 0 (A), K 1 (A) be its K theory groups. Thus K 0 (A) is the<br />
algebraic K 0 theory of the ring A and K 1 (A) is the algebraic K 0 theory of the ring<br />
A ⊗ C 0 (R) = C 0 (R, A). If A → B is a morphism of C ∗ algebras, then there are induced<br />
homomorphisms of abelian groups K i (A) → K i (B). Bott periodicity provides a six<br />
term K theory exact sequence for each exact sequence 0 → J → A → B → 0 of C ∗<br />
algebras and excision shows that the K groups involved in the exact sequence only<br />
depend on the respective C ∗ algebras.<br />
Discrete groups, Lie groups, group actions and foliations give rise through their convolution<br />
algebra to a canonical C ∗ algebra, and hence to K theory groups. The analytical<br />
meaning of these K theory groups is clear as a receptacle for indices of elliptic operators.<br />
However, these groups are difficult to compute. For instance, in the case of<br />
semi-simple Lie groups the free abelian group with one generator for each irreducible<br />
discrete series representation is contained in K 0 Cr ∗G where C∗ r G is the reduced C∗ algebra<br />
of G. Thus an explicit determination of the K theory in this case in particular<br />
involves an enumeration of the discrete series.<br />
We introduced with P. Baum ([2]) a geometrically defined K theory which specializes<br />
to discrete groups, Lie groups, group actions, and foliations. Its main features are<br />
its computability and the simplicity of its definition. In the case of semi-simple Lie<br />
groups it elucidates the role of the homogeneous space G/K (K the maximal compact<br />
subgroup of G) in the Atiyah-Schmid geometric construction of the discrete series.<br />
Using elliptic operators we constructed a natural map µ from our geometrically defined<br />
K theory groups to the above analytic (i.e. C ∗ algebra) K theory groups. Much<br />
progress has been made in the past years to determine the range of validity of the<br />
isomorphism between the geometrically defined K theory groups and the above analytic<br />
(i.e. C ∗ algebra) K theory groups. We refer to the three Bourbaki seminars ([30]) for<br />
an update on this topic and for a precise account of the various contributions. Among<br />
the most important contributions are those of Kasparov and Higson who showed that<br />
the conjectured isomorphism holds for amenable groups. It also holds for real semisimple<br />
Lie groups thanks in particular to the work of A. Wassermann. Moreover the<br />
recent work of V. Lafforgue crossed the barrier of property T, showing that it holds<br />
for cocompact subgroups of rank one Lie groups and also of SL(3, R) or of p-adic<br />
10
Lie groups. He also gave the first general conceptual proof of the isomorphism for<br />
real or p-adic semi-simple Lie groups. The proof of the isomorphism for all connected<br />
locally compact groups, based on Lafforgue’s work has been obtained by J. Chabert,<br />
S. Echterhoff and R. Nest ([5]). The proof by G. Yu of the analogue (due to J. Roe) of<br />
the conjecture in the context of coarse geometry for metric speces which are uniformly<br />
embeddable in hilbert space, and the work of G. Skandalis J. L. Tu, J. Roe and N.<br />
Higson on the groupoid case got very striking consequences such as the injectivity<br />
of the map µ for exact Cr ∗ (Γ) due to Kaminker, Guentner and Ozawa. Finally the<br />
independent results of Lafforgue and Mineyev-Yu show that the conjecture holds for<br />
arbitrary hyperbolic groups (most of which have property T), and P. Julg was even<br />
able to prove the conjecture with coefficients for rank one groups which is probably the<br />
strongest existing positive result. Its possible extension to arbitrary hyperbolic groups<br />
is under intense investigation. On the negative side, recent progress due to Gromov,<br />
Higson, Lafforgue and Skandalis gives counterexamples to the general conjecture for<br />
locally compact groupoids for the simple reason that the functor G → K 0 (Cr ∗ (G)) is<br />
not half exact, unlike the functor given by the geometric group. This makes the general<br />
problem of computing K(Cr ∗ (G)) really interesting. It shows that besides determining<br />
the large class of locally compact groups for which the original conjecture is valid,<br />
one should understand how to take homological algebra into account to deal with the<br />
correct general formulation.<br />
The development of differential geometric ideas, including de Rham homology, connections<br />
and curvature of vector bundles, etc... took place during the eighties thanks to<br />
cyclic cohomology ([10] [9] [32]). This led for instance to the proof of the Novikov conjecture<br />
for hyperbolic groups [17], but got many other applications. Basically, by extending<br />
the Chern-Weil characteristic classes to the general framework it allows for many<br />
concrete computations of differential geometric nature on noncommutative spaces. It<br />
also showed the depth of the relation between the above classification of factors and<br />
the geometry of foliations. For instance, using cyclic cohomology together with the<br />
following simple fact,<br />
“A connected group can only act trivially on a homotopy<br />
invariant cohomology theory”,<br />
(36)<br />
one proves (cf. [11]) that for any codimension one foliation F of a compact manifold V<br />
with non vanishing Godbillon-Vey class one has,<br />
Mod(M) has finite covolume in R ∗ + , (37)<br />
where M = L ∞ (V, F ) and a virtual subgroup of finite covolume is a flow with a finite<br />
invariant measure.<br />
In order to understand what cyclic cohomology is about, it is worthwile to start by<br />
proving , as an exercice, the following simple fact:<br />
”Let A be an algebra and ϕ a trilinear form on A such that 1) ϕ(a 0 , a 1 , a 2 ) =<br />
ϕ(a 1 , a 2 , a 0 ) ∀a j ∈ A. 2) ϕ(a 0 a 1 , a 2 , a 3 )−ϕ(a 0 , a 1 a 2 , a 3 )+ϕ(a 0 , a 1 , a 2 a 3 )−ϕ(a 2 a 0 , a 1 , a 2 ) =<br />
0 ∀a j ∈ A. Then the scalar ϕ n (E, E, E) is invariant under homotopy for projectors<br />
(idempotents) E ∈ M n (A)”<br />
(here ϕ has been uniquely extended to M n (A) using the trace on M n (C), i.e. ϕ n =<br />
ϕ ⊗ Trace).<br />
11
This fact is not difficult to prove, the point is that a deformation of idempotents is<br />
always isospectral,<br />
Ė = [X, E] for some X ∈ M n (A) . (38)<br />
When we take A = C ∞ (M) for a manifold M and let<br />
ϕ(f 0 , f 1 , f 2 ) = 〈C, f 0 df 1 ∧ df 2 〉 ∀f j ∈ A (39)<br />
where C is a 2-dimensional closed de Rham current, the invariant given by the lemma<br />
is equal to (up to normalisation)<br />
〈C, c 1 (E)〉 (40)<br />
where c 1 is the first chern class of the vector bundle E on M whose fiber at x ∈ M<br />
is the range of E(x) ∈ M n (C). In this example we see that for any permutation of<br />
{0, 1, 2} one has:<br />
ϕ(f σ(0) , f σ(1) , f σ(2) ) = ε(σ)ϕ(f 0 , f 1 , f 2 ) (41)<br />
where ε(σ) is the signature of the permutation. However when we extend ϕ to M n (A)<br />
as ϕ n = ϕ ⊗ Tr,<br />
ϕ n (f 0 ⊗ µ 0 , f 1 ⊗ µ 1 , f 2 ⊗ µ 2 ) = ϕ(f 0 , f 1 , f 2 )Tr(µ 0 µ 1 µ 2 ) (42)<br />
the property 41 only survives for cyclic permutations. This is at the origin of the name,<br />
cyclic cohomology, given to the corresponding cohomology theory.<br />
In the example of the noncommutative torus, the cyclic cocycle that was giving an<br />
integral invariant is<br />
where τ is the unique trace,<br />
ϕ(b 0 , b 1 , b 2 ) = τ(b 0 (δ 1 (b 1 )δ 2 (b 2 ) − δ 2 (b 1 )δ 1 (b 2 )) (43)<br />
τ(b) = b 00 for b = ∑ b nm U n W m . (44)<br />
The pairing given by the lemma then gives the Hall conductivity when applied to<br />
a spectral projection of the Hamiltonian (see [12] for an account of the work of J.<br />
Bellissard).<br />
We then obtain in general the beginning of a dictionary relating usual geometrical<br />
notions to their algebraic counterpart in such a way that the latter is meaningfull in<br />
the general noncommutative situation.<br />
Space<br />
Vector bundle<br />
Differential form<br />
DeRham current<br />
DeRham homology<br />
Chern Weil theory<br />
Algebra<br />
Finite projective module<br />
(Class of) Hochschild cycle<br />
(Class of) Hochschild cocycle<br />
Cyclic cohomology<br />
Pairing 〈K(A), HC(A)〉<br />
12
Of course such a dictionary that translates standard notions of geometry in terms<br />
which do not involve commutativity, only gives a superficial idea of noncommutative<br />
geometry, because as we already saw with measure theory, what matters is not only to<br />
translate, but to uncover completely new phenomenas.<br />
For instance, unlike the de Rham cohomology, its noncommutative replacement which<br />
is cyclic cohomology is not graded but filtered. Also it inherits from the Chern character<br />
map a natural integral lattice. These two features play a basic role in the description<br />
of the natural moduli space (or more precisely, its covering Teichmüller space, together<br />
with a natural action of SL(2, Z) on this space) for the noncommutative tori T 2 θ . The<br />
discussion parallels the description of the moduli space of elliptic curves, but involves<br />
the even cohomology instead of the odd one ([15]).<br />
We shall now come to the central notion of noncommutative geometry i.e. the identification<br />
of the noncommutative analogue of the two basic concepts in Riemann’s<br />
formulation of Geometry, namely those of manifold and of infinitesimal line element.<br />
Both of these noncommutative analogues are of spectral nature and combine to give rise<br />
to the notion of spectral triple and spectral manifold, which will be described below.<br />
We shall first describe an operator theoretic framework for the calculus of infinitesimals<br />
which will provide a natural home for the infinitesimal line element ds.<br />
The stage is provided by quantum mechanics. Let us look at the first two lines of the<br />
following dictionary which translates classical notions into the language of operators in<br />
the Hilbert space H:<br />
Complex variable<br />
Operator in H<br />
Real variable<br />
Infinitesimal<br />
Selfadjoint operator<br />
Compact operator<br />
Infinitesimal of order α Compact operator with characteristic values<br />
µ n ∫<br />
satisfying µ n = O(n −α ) , n → ∞<br />
Integral of an infinitesimal<br />
− T = Coefficient of logarithmic<br />
of order 1 divergence in the trace of T .<br />
A variable (in the intuitive sense) should be thought of as an operator in Hilbert space.<br />
The set of values of the variable is the spectrum of the operator, and the number of<br />
times a value is reached is the spectral multiplicity. A real variable corresponds to<br />
a self-adjoint operator. Its spectrum is real and we can act on it by any measurable<br />
function. In general one can act on complex variables only by holomorphic functions,<br />
and this is exactly what happens for non-selfadjoint operators. Now there is in this<br />
dictionary a perfect place for infinitesimals, namely for something which is smaller than<br />
ɛ for any ɛ, without being zero. Of course requiring that the operator norm is smaller<br />
than ɛ for any ɛ, is too strong, but one can be more subtle and ask that for any ɛ positive<br />
one can condition the operator by a finite number of linear conditions, so that its norm<br />
becomes less than ɛ. This is a well known characterization of compact operators in<br />
Hilbert space and they are the obvious candidates for infinitesimals. The basic rules<br />
of infinitesimals are easy to check, for instance the sum of two compact operators is<br />
compact, the product compact times bounded is compact.<br />
The size of the infinitesimal T ∈ K is governed by the rate of decay of the decreasing<br />
sequence of its characteristic values µ n = µ n (T ) as n → ∞. In particular, for all real<br />
13
positive α the following condition defines infinitesimals of order α:<br />
µ n (T ) = O(n −α ) when n → ∞ (45)<br />
(i.e. there exists C > 0 such that µ n (T ) ≤ Cn −α<br />
also form a two–sided ideal and moreover,<br />
∀ n ≥ 1). Infinitesimals of order α<br />
T j of order α j ⇒ T 1 T 2 of order α 1 + α 2 . (46)<br />
Since the size of an infinitesimal is measured by the sequence µ n → 0 it might seem<br />
that one does not need the operator formalism at all, and that it would be enough to<br />
replace the ideal K in L(H) by the ideal c 0 (N) of sequences converging to zero in the<br />
algebra l ∞ (N) of bounded sequences. A variable would just be a bounded sequence, and<br />
an infinitesimal a sequence µ n , µ n → 0. However, this commutative version does not<br />
allow for the existence of variables with range a continuum since all elements of l ∞ (N)<br />
have a point spectrum and a discrete spectral measure. Thus, this is obviously wrong,<br />
because then you exclude variables which have Lebesgue spectrum. It turns out that<br />
it is only because of non commutativity that infinitesimals can coexist with continuous<br />
variables. As we shall see shortly, it is precisely this lack of commutativity between<br />
the line element and the coordinates on a space that will provide the measurement of<br />
distances.<br />
Another key new ingredient in this dictionary is the integral − ∫ which is obtained by the<br />
following analysis, mainly due to Dixmier, of the logarithmic divergence of the partial<br />
traces<br />
Trace N (T ) =<br />
N−1<br />
∑<br />
0<br />
µ n (T ) , T ≥ 0 . (47)<br />
(In fact, it is useful to define Trace Λ (T ) for any positive real Λ > 0 by piecewise affine<br />
interpolation for noninteger Λ).<br />
Define for all order 1 operators T ≥ 0<br />
τ Λ (T ) = 1<br />
log Λ<br />
∫ Λ<br />
e<br />
Trace µ (T )<br />
log µ<br />
dµ<br />
µ<br />
(48)<br />
which is the Cesaro mean of the function Traceµ(T )<br />
log µ<br />
over the scaling group R ∗ + .<br />
For T ≥ 0, an infinitesimal of order 1, one has<br />
Trace Λ (T ) ≤ C log Λ (49)<br />
so that τ Λ (T ) is bounded. The essential property is the following asymptotic additivity<br />
of the coefficient τ Λ (T ) of the logarithmic divergence (49):<br />
|τ Λ (T 1 + T 2 ) − τ Λ (T 1 ) − τ Λ (T 2 )| ≤ 3C<br />
log(log Λ)<br />
log Λ<br />
(50)<br />
for T j ≥ 0.<br />
An easy consequence of (50) is that any limit point τ of the nonlinear functionals τ Λ<br />
for Λ → ∞ defines a positive and linear trace on the two–sided ideal of infinitesimals<br />
of order 1,<br />
14
In practice the choice of the limit point τ is irrelevant because in all important examples<br />
T is a measurable operator, i.e.:<br />
τ Λ (T ) converges when Λ → ∞ . (51)<br />
Thus the value τ(T ) is independent of the choice of the limit point τ and is denoted<br />
∫<br />
− T . (52)<br />
The first interesting example is provided by pseudodifferential operators T on a differentiable<br />
manifold M. When T is of order 1 in the above sense, it is measurable and<br />
∫<br />
−T is the non-commutative residue of T ([33]). It has a local expression in terms of<br />
the distribution kernel k(x, y), x, y ∈ M. For T of order 1 the kernel k(x, y) diverges<br />
logarithmically near the diagonal,<br />
k(x, y) = −a(x) log |x − y| + 0(1) (for y → x) (53)<br />
where a(x) is a 1–density independent of the choice of Riemannian distance |x − y|.<br />
Then one has (up to normalization),<br />
∫ ∫<br />
− T = a(x). (54)<br />
M<br />
The right hand side of this formula makes sense for all pseudodifferential operators (cf.<br />
[33]) since one can easily see that the kernel of such an operator is asymptotically of<br />
the form<br />
k(x, y) = ∑ a k (x, x − y) − a(x) log |x − y| + 0(1) (55)<br />
where a k (x, ξ) is homogeneous of degree −k in ξ, and the 1–density a(x) is defined<br />
intrinsically since the logarithm does not mix with rational terms under a change of<br />
local coordinates.<br />
The same principle of extension of ∫ − to infinitesimals of order < 1 works for hypoelliptic<br />
operators and more generally for spectral triples whose dimension spectrum is simple,<br />
as we shall see below.<br />
With this tool we shall now come back to the two basic notions introduced by Riemann<br />
in the classical framework, those of manifold and of line element. We shall see that both<br />
of these notions adapt remarkably well to the noncommutative framework and this will<br />
lead us to the notion of spectral triple which noncommutative geometry is based on.<br />
In ordinary geometry of course you can give a manifold by a cooking recipe, by charts<br />
and local diffeomorphisms, and one could be tempted to propose an analogous cooking<br />
recipe in the noncommutative case. This is pretty much what is achieved by the general<br />
construction of the algebras of foliations and it is a good test of any general idea that<br />
it should at least cover that large class of examples.<br />
But at a more conceptual level, it was recognized long ago by geometors that the main<br />
quality of the homotopy type of an oriented manifold is to satisfy Poincaré duality<br />
not only in ordinary homology but also in K-homology. Poincaré duality in ordinary<br />
homology is not sufficient to describe homotopy type of manifolds but D. Sullivan<br />
showed (in the simply connected PL case of dimension ≥ 5 ignoring 2-torsion) that<br />
it is sufficient to replace ordinary homology by KO-homology. Moreover the Chern<br />
15
character of the KO-homology fundamental class contains all the rational information<br />
on the Pontrjagin classes.<br />
The characteristic property of differentiable manifolds which is carried over to the<br />
noncommutative case is Poincaré duality in KO-homology.<br />
Moreover, as we saw above in the discussion of topology, K-homology admits a fairly<br />
simple definition in terms of Hilbert space and Fredholm representations of algebras, as<br />
gradually emerged from the work of Atiyah, Singer, Brown-Douglas-Fillmore, Miscenko,<br />
and Kasparov.<br />
For an ordinary manifold the choice of the fundamental cycle in K-homology is a<br />
refinement of the choice of orientation of the manifold and in its simplest form is a<br />
choice of spin structure. Of course the role of a spin structure is to allow for the<br />
construction of the corresponding Dirac operator which gives a corresponding Fredholm<br />
representation of the algebra of smooth functions. The origin of the construction of<br />
the Dirac operator was the extraction of a square root for a second order differential<br />
operator like the Laplacian.<br />
What is rewarding is that this will not only guide us towards the notion of noncommutative<br />
manifold but also to a formula, of operator theoretic nature, for the line element<br />
ds. In the Riemannian case one gives the Taylor expansion of the square ds 2 of the<br />
infinitesimal line element, in our framework the extraction of square root effected by<br />
the Dirac operator allows us to deal directly with ds itself.<br />
The infinitesimal unit of length“ds” should be an infinitesimal in the above operator<br />
theoretic sense and one way to get an intuitive understanding of the formula for ds is<br />
to consider Feynman diagrams which physicists use currently in the computations of<br />
quantum field theory. Let us contemplate the diagram:<br />
y<br />
<br />
x<br />
<br />
Figure 2<br />
which is involved in the computation of the self-energy of an electron in QED. The<br />
two points x and y of space-time at which the photon (the wiggly line) is emitted and<br />
reabsorbed are very close by and our ansatz for ds will be at the intuitive level,<br />
ds = ×−−−× . (56)<br />
The right hand side has good meaning in physics, it is called the Fermion propagator<br />
16
and is given by<br />
where D is the Dirac operator.<br />
We thus arrive at the following basic ansatz,<br />
×−−−× = D −1 (57)<br />
ds = D −1 . (58)<br />
In some sense it is simpler than the ansatz giving ds 2 as g µν dx µ dx ν , the point being<br />
that the spin structure allows really to extract the square root of ds 2 (as is well known<br />
Dirac found the corresponding operator as a differential square root of a Laplacian).<br />
The first thing we need to do is to check that we are still able to measure distances<br />
with our “unit of length” ds. In fact we saw in the discussion of the quantized calculus<br />
that variables with continuous range cant commute with “infinitesimals” such as ds<br />
and it is thus not very surprising that this lack of commutativity allows to compute, in<br />
the classical Riemannian case, the geodesic distance d(x, y) between two points. The<br />
precise formula is<br />
d(x, y) = Sup {|f(x) − f(y)| ; f ∈ A , ‖[D, f]‖ ≤ 1} (59)<br />
where D = ds −1 as above and A is the algebra of smooth functions. Note that ds has<br />
the dimension of a length L, D has dimension L −1 and the above expression for d(x, y)<br />
also has the dimension of a length.<br />
Thus we see in the classical geometric case that both the fundamental cycle in K-<br />
homology and the metric are encoded in the spectral triple (A, H, D) where A is the<br />
algebra of functions acting in the Hilbert space H of spinors, while D is the Dirac<br />
operator.<br />
To get familiar with this notion one should check that one recovers the volume form of<br />
the Riemannian metric by the equality (valid up to a normalization constant [12])<br />
∫ ∫<br />
−f |ds| n = f √ g d n x (60)<br />
M n<br />
but the first interesting point is that besides this coherence with the usual computations<br />
there are new simple questions we can ask now such as ”what is the two-dimensional<br />
measure of a four manifold” in other words ”what is its area ” Thus one should<br />
compute<br />
∫<br />
− ds 2 (61)<br />
It is obvious from invariant theory that this should be proportional to the Hilbert–<br />
Einstein action, the direct computation was done by D. Kastler and gives,<br />
∫<br />
− ds 2 = −1 ∫<br />
r √ g d 4 x (62)<br />
48π 2 M 4<br />
where as above dv = √ g d 4 x is the volume form, ds = D −1 the length element, i.e. the<br />
inverse of the Dirac operator and r is the scalar curvature.<br />
In the general framework of Noncommutative Geometry the confluence of the Hilbert<br />
space incarnation of the two notions of metric and fundamental class for a manifold<br />
lead very naturally to define a geometric space as given by a spectral triple:<br />
(A, H, D) (63)<br />
17
where A is a concrete algebra of coordinates represented on a Hilbert space H and the<br />
operator D is the inverse of the line element.<br />
ds = 1/D. (64)<br />
This definition is entirely spectral; the elements of the algebra are operators, the points,<br />
if they exist, come from the joint spectrum of operators and the line element is an<br />
operator.<br />
The basic properties of such spectral triples are easy to formulate and do not make any<br />
reference to the commutativity of the algebra A. They are<br />
[D, a] is bounded for any a ∈ A , (65)<br />
D = D ∗ and (D + λ) −1 is a compact operator ∀ λ ∉ C . (66)<br />
(Of course D is an unbounded operator).<br />
There is no difficulty to adapt the above formula for the distance in the general noncommutative<br />
case, one uses the same, the points x and y being replaced by arbitrary<br />
states ϕ and ψ on the algebra A. Recall that a state is a normalized positive linear<br />
form on A such that ϕ(1) = 1,<br />
ϕ : Ā → C , ϕ(a ∗ a) ≥ 0 , ∀ a ∈ Ā , ϕ(1) = 1 . (67)<br />
The distance between two states is given by,<br />
d(ϕ, ψ) = Sup {|ϕ(a) − ψ(a)| ; a ∈ A , ‖[D, a]‖ ≤ 1} . (68)<br />
The significance of D is two-fold. On the one hand it defines the metric by the above<br />
equation, on the other hand its homotopy class represents the K-homology fundamental<br />
class of the space under consideration.<br />
There is an equally simple formula for the Yang-Mills action in general The analogue of<br />
the Yang-Mills action functional and the classification of Yang-Mills connections on the<br />
noncommutative tori were developped in [20], with the primary goal of finding a ”manifold<br />
shadow” for these noncommutative spaces. These moduli spaces turned out indeed<br />
to fit this purpose perfectly, allowing for instance to find the usual Riemannian space of<br />
gauge equivalence classes of Yang-Mills connections as an invariant of the noncommutative<br />
metric. We refer to [12] for the construction of the metrics on noncommutative tori<br />
from the conceptual point of view. All natural axioms of noncommutative geometry<br />
are fulfilled in that case.<br />
These constructions were greatly generalised to isospectral deformations of Riemannian<br />
geometries of rank > 1 in joint work with G. Landi and M. Dubois-Violette .<br />
In essence the transition from the Riemannian paradigm of geometry to the above spectral<br />
one parallels the evolution of the unit of length in physics. The ”meter” for instance<br />
was defined in the eighteen century as a small fraction of the earth circonference, and<br />
concretely represented by a platinum bar localized near Paris. The actual standard of<br />
length is given nowadays in a quite different way: it is the wave-length corresponding to<br />
specific transitions in the cesium atom. The inverse frequencies are computed from the<br />
corresponding Dirac Hamiltonian in exact analogy with the above spectral definition<br />
of ”ds”. This unit of length is no longer localized, owing to its lack of commutativity<br />
with the space-time coordinates.<br />
18
There is a very important test of our general framework of geometry which is its compatibility<br />
with what we know of space-time. The best way is to start with the hard<br />
core information one has from physics and that can be summarized by a Lagrangian.<br />
This Lagrangian is the Einstein Lagrangian plus the minimally coupled standard model<br />
Lagrangian. How do we recover this action functional which contains both the Einstein-<br />
Hilbert term as well as the standard model The answer is very simple: the Fermionic<br />
part of this action is used to determine a spectral triple (A, H, D), where the algebra<br />
A encodes not only the ordinary space-time (coming from QED) but also the internal<br />
symmetries, the Hilbert space H encodes not only the ordinary spinors (coming from<br />
QED) but all quarks and leptons, and the operator D encodes not only the ordinary<br />
Dirac operator but also the Yukawa coupling matrix. Then one recovers the bosonic<br />
part as follows. Both the Hilbert–Einstein action functional for the Riemannian metric,<br />
the Yang–Mills action for the vector potentials, the self interaction and the minimal<br />
coupling for the Higgs fields all appear with the correct signs in the asymptotic expansion<br />
for large Λ of the number N(Λ) of eigenvalues of D which are ≤ Λ (cf. [4]),<br />
N(Λ) = # eigenvalues of D in [−Λ, Λ]. (69)<br />
This step function N(Λ) is the superposition of two terms,<br />
N(Λ) = 〈N(Λ)〉 + N osc (Λ).<br />
The oscillatory part N osc (Λ) is the same as for a random matrix, governed by the<br />
statistic dictated by the symmetries of the system and does not concern us here. The<br />
average part 〈N(Λ)〉 is computed by a semiclassical approximation from local expressions<br />
involving the familiar heat equation expansion and delivers the correct terms.<br />
Other nonzero terms in the asymptotic expansion are cosmological, Weyl gravity and<br />
topological terms. We showed with G. Landi and M. Dubois-Violette that if one studies<br />
natural presentations of the algebra generated by A and D one naturally gets only metrics<br />
with a fixed volume form so that the bothering cosmological term does not enter<br />
in the variational equations associated to the spectral action 〈N(Λ)〉. It is tempting<br />
to speculate that the phenomenological Lagrangian of physics, combining matter and<br />
gravity appears from the solution of an extremely simple operator theoretic equation.<br />
In the spectral noncommutative framework the notion of curvature comes from the<br />
local computation of the analogue of Pontrjagin classes: i.e. of the components of the<br />
cyclic cocycle which is the Chern character of the K-homology class of D and which<br />
make sense in general. This result allows, using the infinitesimal calculus, to go from<br />
local to global in the general framework of spectral triples (A, H, D).<br />
The Fredholm index of the operator D determines (we only look at the odd case for<br />
ϕ<br />
simplicity but there are similar formulas in the even case) an additive map K 1 (A) → Z<br />
given by the equality<br />
ϕ([u]) = Index (P uP ) , u ∈ GL 1 (A) (70)<br />
where P is the projector P = 1+F , F = Sign (D).<br />
2<br />
It is an easy fact that this map is computed by the pairing of K 1 (A) with the following<br />
cyclic cocycle<br />
τ(a 0 , . . . , a n ) = Trace (a 0 [F, a 1 ] . . . [F, a n ]) ∀ a j ∈ A (71)<br />
19
where F = Sign D and we assume that the dimension p of our space is finite, which<br />
means that (D + i) −1 is of order 1/p, also n ≥ p is an odd integer. There are similar<br />
formulas involving the grading γ in the even case, and it is quite satisfactory that both<br />
cyclic cohomology and the chern Character formula adapt to the infinite dimensional<br />
case in which the only hypothesis is that exp(−D 2 ) is a trace class operator.<br />
The cocycle τ is however nonlocal in general because the formula (71) involves the<br />
ordinary trace instead of the local trace ∫ − and it is crucial to obtain a local form of<br />
the above cocycle.<br />
This problem is solved in our joint work with Henri Moscovici ([18]). We make the<br />
following regularity hypothesis on (A, H, D)<br />
a and [D, a] ∈ ∩ Dom δ k , ∀ a ∈ A (72)<br />
where δ is the derivation δ(T ) = [|D|, T ] for any operator T .<br />
We let B denote the algebra generated by δ k (a), δ k ([D, a]). The usual notion of dimension<br />
of a space is replaced by the dimension spectrum which is the subset of<br />
{z ∈ C, Re(z) ≥ 0} of singularities of the analytic functions<br />
ζ b (z) = Trace (b|D| −z ) Re z > p , b ∈ B . (73)<br />
The dimension spectrum of an ordinary manifold M is the set {0, 1, . . . , n}, n = dim M;<br />
it is simple. Multiplicities appear for singular manifolds. Cantor sets provide examples<br />
of complex points z /∈ R in the dimension spectrum.<br />
We assume that Σ is discrete and simple. Let (A, H, D) be a spectral triple satisfying<br />
the hypothesis (72) and (73). The local index theorem is the following, [18]:<br />
1. The equality ∫<br />
−P = Res z=0 Trace (P |D| −z )<br />
defines a trace on the algebra generated by A, [D, A] and |D| z , where z ∈ C.<br />
2. There is only a finite number of non–zero terms in the following formula which<br />
defines the odd components (ϕ n ) n=1,3,... of a cocycle in the bicomplex (b, B) of A,<br />
ϕ n (a 0 , . . . , a n ) = ∑ k<br />
c n,k<br />
∫−a 0 [D, a 1 ] (k 1) . . . [D, a n ] (kn) |D| −n−2|k| ∀ a j ∈ A<br />
where the following notations are used: T (k) = ∇ k (T ) and ∇(T ) = D 2 T − T D 2 ,<br />
k is a multi-index, |k| = k 1 + . . . + k n ,<br />
c n,k = (−1) √ (<br />
|k| 2i(k 1 ! . . . k n !) −1 ((k 1 +1) . . . (k 1 +k 2 +. . .+k n +n)) −1 Γ |k| + n )<br />
.<br />
2<br />
3. The pairing of the cyclic cohomology class (ϕ n ) ∈ HC ∗ (A) with K 1 (A) gives the<br />
Fredholm index of D with coefficients in K 1 (A).<br />
The first test of this general local index formula was the computation of the local<br />
Pontrjagin classes in the case of foliations. Their transverse geometry is, as explained<br />
20
in [18], encoded by a spectral triple. The answer for the general case [19], was obtained<br />
thanks to a Hopf algebra H(n) only depending on the codimension n of the foliation. It<br />
allowed to organize the computation and to encode algebraically the noncommutative<br />
curvature. It also dictated the correct generalization of cyclic cohomology for Hopf<br />
algebras. Another case of great interest came recently from quantum groups where<br />
the local index formula works fine and yields quite remarkable formulas even in the<br />
simplest case of SU q (2).<br />
At about the same time as the Hopf algebra H(n), another Hopf algebra was discovered<br />
by Dirk Kreimer, as the organizing concept in the computations of renormalization in<br />
quantum field theory.<br />
His Hopf algebra is commutative as an algebra and we showed that it is the dual Hopf<br />
algebra of the envelopping algebra of a Lie algebra G whose basis is labelled by the one<br />
particle irreducible Feynman graphs. The Lie bracket of two such graphs is computed<br />
from insertions of one graph in the other and vice versa. The corresponding Lie group<br />
G is the group of characters of H.<br />
We showed that the group G is a semi-direct product of an easily understood abelian<br />
group by a highly non-trivial group closely tied up with groups of diffeomorphisms.<br />
Our joint work shows that the essence of the concrete computations performed by<br />
physicists in the renormalization technique is conceptually understood as a special<br />
case of a general principle of multiplicative extraction of finite values coming from the<br />
Riemann-Hilbert problem. The Birkhoff decomposition is the factorization<br />
γ (z) = γ − (z) −1 γ + (z) z ∈ C (74)<br />
where we let C ⊂ P 1 (C) be a smooth simple curve, C − the component of the complement<br />
of C containing ∞ ∉ C and C + the other component. Both γ and γ ± are loops<br />
with values in G,<br />
γ (z) ∈ G ∀ z ∈ C<br />
and γ ± are boundary values of holomorphic maps (still denoted by the same symbol)<br />
γ ± : C ± → G . (75)<br />
The normalization condition γ − (∞) = 1 ensures that, if it exists, the Birkhoff decomposition<br />
is unique (under suitable regularity conditions).<br />
When G is a simply connected nilpotent complex Lie group the existence (and uniqueness)<br />
of the Birkhoff decomposition is valid for any γ. When the loop γ : C → G<br />
extends to a holomorphic loop: C + → G, the Birkhoff decomposition is given by<br />
γ + = γ, γ − = 1. In general, for z ∈ C + the evaluation,<br />
γ → γ + (z) ∈ G (76)<br />
is a natural principle to extract a finite value from the singular expression γ(z). This<br />
extraction of finite values coincides with the removal of the pole part when G is the<br />
additive group C of complex numbers and the loop γ is meromorphic inside C + with z<br />
as its only singularity.<br />
The main result of our joint work ([16]) is that the renormalized theory is just the<br />
evaluation at z = D of the holomorphic part γ + of the Birkhoff decomposition of the<br />
loop γ with values in G provided by the dimensional regularization.<br />
21
In fact the relation that we uncovered in [16] between the Hopf algebra of Feynman<br />
graphs and the Hopf algebra of coordinates on the group of formal diffeomorphisms of<br />
the dimensionless coupling constants of the theory allows to prove the following result<br />
which for simplicity deals with the case of a single dimensionless coupling constant.<br />
Let the unrenormalized effective coupling constant g eff (ε) be viewed as a formal power<br />
series in g and let g eff (ε) = g eff+ (ε) (g eff− (ε)) −1 be its (opposite) Birkhoff decomposition<br />
in the group of formal diffeomorphisms. Then the loop g eff− (ε) is the bare coupling<br />
constant and g eff+ (0) is the renormalized effective coupling.<br />
This allows, using the relation between the Birkhoff decomposition and the classification<br />
of holomorphic bundles, to encode geometrically the operation of renormalization. It<br />
also signals a very clear analogy between the renormalization group as an ”ambiguity”<br />
group of physical theories and the missing Galois theory at Archimedian places alluded<br />
to above. We refer to [14] for a detailed account of this analogy.<br />
I<br />
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23
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24