Wednesday, October 31, 2007


Katia's last post ended with a provocative question motivated by Grothendieck's description in Récoltes et Semailles of the "heart of the heart" of arithmetic geometry, namely the theory of motives. Her question was formulated like this:
--------What is the "heart of the heart" of noncommutative geometry?-------
I'll try to explain here that there is a definite "supplément d'âme" obtained in the transition from classical (commutative) spaces to the noncommutative ones. The main new feature is that "noncommutative spaces generate their own time" and moreover can undergo thermodynamical operations such as cooling, distillation etc...
This opens up completely new ways of handling geometric spaces and our work with Matilde Marcolli and Katia Consani is just one example of potential applications to number theory. It is closely related to the Riemann zeta function and is very close in spirit to Grothendieck's ideas on motives so that it is not out of place in the present discussion of Katia's question.
The story starts by a qualitative distinction between spaces which comes from the classification (by von Neumann) of noncommutative algebras in types I, II and III. The commutative spaces are all of type I. When encoding a space X by an algebra A of (complex valued) functions on X one uses some structure on X to restrict the class of functions (e.g. to smooth functions on a smooth space) and the above distinction between types uses the coarsest possible structure which is the measure theory. The corresponding algebras (called von Neumann algebras) are quite simple to characterize abstractly: they are commutants in Hilbert space of some unitary representation.
Since one can take the direct sum of algebras A and B, one can mix algebras of different types. More precisely any von Neumann algebra decomposes uniquely as an integral of algebras which cannot be decomposed further and are called factors. A factor is a von Neumann algebra whose center is as small as it can be, namely is reduced to the complex numbers. The factors of type I are Morita equivalent to the complex numbers, and thus a type I factor really corresponds to the classical notion of "point" in a space X.
To understand geometrically what factors of type II and III look like, it is useful to describe the (von Neumann) algebra A associated to the leaf space of a foliated manifold: (V,F). An element T of A assigns to each leaf an operator in the Hilbert space of square integrable functions on the leaf, and it makes sense to say that T is bounded, measurable, or zero almost everywhere. The algebraic operations are done leaf per leaf, and the algebra of bounded measurable elements modulo the negligible ones is a von Neumann algebra. The simplest example corresponds to the foliation whose leaf space is the noncommutative torus. It is the foliation of the two torus by the equation "dy= a dx" in flat coordinates. The corresponding von Neumann algebra is a factor when "a" is irrational and this factor is not of type I but of type II. To obtain type III examples one can take any codimension one foliation whose Godbillon-Vey invariant does not vanish. The integrable subbundle F defining a codimension one foliation is the orthogonal of a one form v and integrability gives dv as the wedge product of v by a one form w. The Godbillon-Vey invariant is the integral over V of the wedge product of w by dw when V is compact oriented of dimension three. In essence the form w is the logarithmic derivative of a transverse volume element and the GV invariant is an obstruction to finding a holonomy invariant tranverse volume element ie one which does not change when one moves along a leaf keeping track of the way the nearby leaves are developing.
More generally the factors of type II are those which possess a trace and those of type III are those which are neither of type I nor of type II. In the foliation context, a holonomy invariant tranverse volume element allows one to integrate the ordinary trace of operators and this yields a trace on the von Neumann algebra of the foliation.
Until the work of the Japanese mathematician Minoru Tomita, very few positive results existed on type III factors. The key result of Tomita is that a cyclic and separating vector v for a factor A in a Hilbert space H generates a one parameter group of automorphisms of A by the following recipee: one considers the modulus square S*S of the closable operator S which sends xv to S(xv)=x*v for any x in A, and then raises it to the purely imaginary power "it". Tomita showed that the resulting unitary operator normalizes A and hence defines an automorphism of A. One obtains in this way a one parameter group of automorphisms of A associated to the choice of a cyclic and separating vector v. He also showed that the phase J of the above closable operator S yields an antiisomorphism of A with its commutant A' which coincides with JAJ. In his account of Tomita's work, Takesaki characterized the relation between the state defined by the cyclic and separating vector v and the one parameter group of automorphisms of Tomita as the Kubo-Martin-Schwinger (KMS) condition, which had been formulated in C*-algebraic terms by the physicists Haag, Hugenholtz and Winnink.
The key result of my thesis (in 1972) is that the class modulo inner automorphisms of the Tomita automorphism group is in fact independent of the choice of the (faithful normal) state that is used in its construction. Needless to say it is this uniqueness that allows to define invariants of factors. The simplest is the subgroup T(A) of R which is formed of the periods, namely the set of times t for which the corresponding automorphism is inner. This, together with the spectral invariant S(A), led me to the classification of type III factors into subtypes III_s for s in [0,1] and the reduction from type III to type II and automorphisms done in my thesis except for the case III_1 which was later completed by Takesaki. All of this goes back to the beginning of the seventies and will suffice for this first heart beat. It is only the beginning of a long saga which is far from over hopefully, and whose main theme is this mysterious generation of an intrinsic "time" that emerges from the noncommutativity of a von Neumann algebra. Exactly as manifolds come with a natural "smooth" measure class, a noncommutative space X generally gives rise to a von Neumann algebra A which encodes the natural measure class on X. It is thus a totally new feature of the noncommutative world that the corresponding time evolution is well defined and gives a canonical homomorphism:

where the second line gives the definition of the group of outer automorphisms Out(A) of A as the quotient of the group Aut(A) of automorphisms by the normal subgroup Int(A) of inner automorphisms (which are obtained by conjugating by a unitary element of the algebra A).

Report on the AMS Special Session on Noncommutative Geometry and Arithmetic Geometry

Let us start this report on this meeting in a light way with a picture, featuring the subject of this blog and David Goss...

Connes opened the meeting with a talk on some analogies between two grand challenges in mathematics and physics: On the one hand, the search for a geometric setting in which the methods of Weil's proof of the Riemann hypothesis (RH) for curves over finite fields could be applied to prove the original RH; and on the other hand, the search for a quantum theory of gravity starting from the NCG approach to the standard model of particle physics based on the spectral action principle of Connes and Chamseddine. If my memory from an earlier talk of Connes is correct, these analogies were discovered while Connes and Marcolli were finishing an early draft of their tome "Noncommutative geometry, quantum fields, and motives" (draft here), giving rise to the final part of the book that ties (conjecturally) the two major mathematics and physics strands mentioned above. Let me try to give a (somewhat disjointed) indication of the breadth of the analogies, while leaving the bigger picture completely in the fog.

First, Connes gave an overview of the Tomita-Takesaki theory (the "secret weapon" of operator algebraicists, in the words of Jack Morava), emphasizing its novelty and stark contrast with the commutative case: noncommutative operator algebras -- but not commutative ones! -- come endowed with a non-trivial, canonical (that is, up to alteration by an inner automorphism) time evolution. Surprisingly, the theory of Tomita-Takesaki also provided the correct framework for operator-algebraic quantum statistical mechanics. Now, these are old results from the 70's (and late 60's), but around 1992 in collaboration with C. Rovelli the two points of view were considered together in a novel way: Is there a thermodynamic basis for the origin of time? In particular, what should be the (noncommutative) algebra of observables of a quantum theory of gravity?

Connes also gave a rapid summary of his on-going project with Consani and Marcolli to build a geometric world in characterstic 0 hospitable to the methods of Weil's proof of the Riemann hypothesis for curves over finite fields. Apparently, there are fruitful analogies between the necessary ingredients for quantum gravity (QG) and aspects of the space of Q-lattices, the geometric space underlying the GL(2)-system of Connes-Marcolli and the Bost-Connes system (see the summary of Laca's talk below). For example, the moduli space of Dirac operators on the QG side, being described by a double quotient space of complex algebraic groups, is mirrored on the Q-lattice side by Shimura varieties (certain double quotient spaces of adelic algebraic groups). One will find a condensed dictionary of many more analogies in the last part of the book by Connes and Marcolli. It would be desirable if some knowledgeable reader of this blog could elaborate on this (perhaps even the authors themselves, the huge job of having written 700 or so pages notwithstanding).

Continuing in the bridge-building spirit of the meeting, van Suijlekom gave a talk on his recent work with S. Mahanta on their study of the noncommutative torus from the point of view of noncommutative algebraic geometry. This is a very natural undertaking: for while noncommutative tori have long been studied from a topological and differential perspective, classical tori can also be realized as 1-dimensional complex abelian varieties (a.k.a. elliptic curves) which have rich algebraic and arithmetic structures, so it is natural to try to examine noncommutative tori as noncommutative algebraic varieties of sorts. But whereas in the differential-topological approach pioneered by Connes and Rieffel a noncommutative space is a certain kind of noncommutative topological algebra, in the current algebraic-geometry approach, a noncommutative variety is regarded as a certain type of category. Indeed, from the work of A. Rosenberg, Bondal, and Orlov it is known that smooth (irreducible) projective varieties are characterized up to isomorphism by their bounded derived categories of quasi-coherent sheaves. The work of Mahanta-van Suijlekom is an attempt to connect these two worlds for NC tori. What they have done is to define a category that, roughly speaking, interpolates between the categories reflecting the differential and algebraic nature of the NC tori. Additionally, they've shown that this interpolating category is a Tannakian category equivalent to the category of representations of Z^2.

It would be interesting to see whether the categorical approach to noncommutative tori sheds any light on the conjectured relevance of noncommutative tori to an explicit class field theory for real quadratic fields (in analogy with the theory of complex multiplication, as suggested by Manin), or clarifies what it should mean for a noncommutative torus to be defined over Q or a number field (cf. the recent thesis of J. Plazas).

Laca gave a report on his recent work with N. Larsen and S. Neshveyev. This was an especially pleasing talk to attend as this work finally wraps up an analytic problem that has remained open for more than 10 years, namely the classification of KMS states for the Bost-Connes C*-dynamical system for number fields. Avoiding all details of what the Bost-Connes system is exactly -- an excellent summary is given in the book of Connes and Marcolli -- let me mention only that its most "fabulous" feature is that it admits an action of the abelianized absolute Galois group of Q on its so-called KMS infinity states, and upon evaluation of theses KMS states on a natural rational subalgebra, this Galois action coincides with the usual Galois action on the maximal cyclotomic extension of Q. (KMS-beta states were discussed by Connes in his talk and are surely discussed elsewhere on this blog as well. To describe them quickly, albeit in a rather cryptic manner: KMS-beta states are analogues of infinite-volume limits of Gibbs states in quantum statistical mechanics; beta, in the physical context, is inverse temperature.) A natural problem is to construct C*-dynamical systems with analogous Fabulous Features for general number fields. For the case of imaginary quadratic fields, this was accomplished about three years ago by Connes-Marcolli-Ramachandran. Paugam and one of the blog posters have defined a candidate analogue of Bost-Connes for general number fields, without, however, being able to show that it is fabulous. What Laca and his collaborators have done is overcome a key analytic obstacle towards establishing "fabulousness" of the Bost-Connes system for general number fields: namely, for all beta they have classified the KMS-beta states. The result is essentially the same as for the original Bost-Connes system, though the proof follows the ergodic-theoretic techniques developed by Neshveyev, later enhanced by Laca, Larsen, and Neshveyev to clean up the KMS states classification for the Connes-Marcolli GL(2)-system. To get truly "fabulous" systems in the general number field case capable of manifesting the Galois action, it remains to find an appropriate rational structure for such C*-systems. This is a problem of a different nature, which is not likely to fall without deep arithmetic insight, given that it has implications for Hilbert's 12th problem.

The second day started with two talks by Kreimer and Yeats discussing results obtained in (perturbative) quantum field theories, in particular on quantum electrodynamics. The recursive structures that appear are by now well-known to be captured by the structure of a Hopf algebra. On the analytical side, one can expand the probability amplitudes of interest in physics (such as the vacuum self-energy of the photon) as a series in certain functions gamma_k of the coupling constant. One then writes a recursive relation for the $\gamma_k$ and tries to (numerically) solve a differential equation for the $\gamma_1$. This involved only the computation of the amplitudes of primitive graphs, which was carried out up to fourth order in the loop number. Several vector flow diagrams were presented in the second talk, corresponding to the differential equation. Striking was the difference when moving to 4th loop order, where a separatrix
appeared. Although not yet completely understood, it was observed that the fine structure constant $\alpha = 1/137... lies on this separatrix!

In addition to the talks mentioned above there were also talks on the meeting by Ramachandran on computing Beilinson's ring of correspondences at the generic point of a smooth projective variety over a finite field; by Marcolli on her joint work with Manin on the pseudomeasure formalism for modular symbols (a manifestation of a "modular shadow" in their terminology); by Moscovici on twisted spectral triples (though, unfortunately, there wasn't enough time for him to go deeper into applications to the GL(2)-system of Connes-Marcolli-Moscovici); by Goss on Hecke operators and distributions (in the sense of probability theory) in characteristic p, and some work of Boeckle; and by Zhao on improving the Deligne-Goncharov upper bounds for the dimension of spaces of multiple zeta values (of a given weight). On the second day there were additional talks by Retakh on a construction of Lie algebras and Lie groups over noncommutative rings; by Gangl on Polygons and mixed Tate motives (with Brown and Levin); and finally Zhang on differential renormalization for multiple zeta values (joint with Guo).

Eugene Ha
Walter van Suijlekom