Website Noncommutative Geometry and Particle Physics

A new website on noncommutative geometry has been created, connected to the workshop Noncommutative Geometry and Particle Physics organized at the Lorentz Centre in Leiden in October 2013. As this type of workshop only allows for a limited number of participants, this website will form the virtual portal for a wider audience.


It will contain updates during the workshop, documents with background material, discussions that take place during the workshop, a glossary, and much more.

In the future, we plan to extend this website as a repository for noncommutative geometry and its applications to particle physics.

A-expansions

As I have written about before, the integers Z play a dual role in arithmetic. On the one hand, they are obviously scalars in terms of the fields of definitions of varieties etc.; yet, on the other hand, they are also operators, as in the associated Z-action on multiplicative groups (or the groups of rational points of abelian varieties etc.). This is absolutely so basic that we do not notice it in day-to-day mathematics.

Yet these dual notions are there and are highlighted by the curious cases of similar phenomena in the arithmetic of function fields. This is what I want to discuss here. So, as usual in the characteristic $p$ game, let $A$:=F_q$[\theta]$, $K$:=F_q$(\theta)$ and $K$_$\infty$:=F_q$((1/\theta))$. Recall also the Carlitz module $C$ given by $C$_$\theta$(z):=$\theta z+z$^q; one always views $C$ as the analog of the multiplicative group G_m (indeed its division values generate abelian extensions etc.).

Of course every algebra lies over Z and thus one can always study the corresponding "Z-invariants'' such as class groups or class numbers etc. But the analogy between Z and $A$ really calls out for "$A$-invariants'' also. When I mentioned the possibility of such $A$-objects way back in 1980 at a conference, the participants looked at me like I had lost my mind. Be that as it may, within the past few years such $A$-invariants have indeed been produced in the seminal work of Lenny Taelman; so we now have "class $A$-modules'' and "$A$ class numbers'' (really generators of the Fitting ideals of these finite class $A$-modules). In fact, these notions fit beautifully into the special values of $L$-series in direct analogy with algebraic number theory.

In a similar way, the very notion of  "analytic function'' is clearly Z-based; i.e., based on the notion of power series $\sum$ a_i xⁱ; built directly into the definition of power series is the standard Z-action arising from multiplication (i.e., the mapping $(i,x)$ $\mapsto x$ⁱ).

So the idea of this blog is that there should analogously be "$A$-expansions'' where we now sum over the monic elements of $A$ (and not elements of  Z) in the theory and, remarkably, such things do exist. We are, by no means, close to a full theory of such expansions but rather we have a number of highly intriguing results.

Here is a very cool example of what I am talking about (essentially due to Greg Anderson in his famous "log-algebraicity" paper: Journal Number Theory 60, 165-209 (1996)): We begin by recalling, from Calculus 1, the basic expansion
$\log (1+x)=x$-x²/2+x³/3+^...
Now let $C$ be the Carlitz module and $\log$_C(z) it's logarithm. For i $\geq$ 1, we put $[i]:=\theta$^qi-$\theta\in A$ and also L_i:=[i][i-1]^...[1]. One can easily see that L_i is the least common multiple of the monic elements of $A$ of degree $i$. One then has the Z-expansion
$\log$_C(z)=$\sum$_i z^qi/L_i.
For the $A$-expansion we have Greg's formula
$\log$_C (z)=$\hat\sum$_a C_a(z)/a,
where $a$ runs over the monics of $A$ and $\hat \sum$ means that we compute the sum as the limit of  {S_d(z)} where S_d(z) is the above sum truncated over the (finite number of) monics of degree $\leq$ d (so, alas, we have not fully removed Z here after all!). Note also that without such a truncation, the convergence of the sum is extremely tricky and rare! The analogies between the $A$-expansion of $\log$_C(z) and the usual expansion for $\log (1+x)$ are very clear....

(Greg calls the power series x-x²/2+x³/3+^... "log-algebraic" since it is clearly the log of an algebraic
function. Once one views power series this way, many examples spring to mind; indeed Dwork's famous result on points over finite fields can be viewed in this optic. Greg's log-alg ideas are currently having an extremely large impact on research; for more, see Rudy's blog
https://rudyperkins.wordpress.com/   .)

For a monic $a$, the additive polynomial $C$_a(z) has derivative identically equal to $a$. As such one can find a formal composition inverse denoted $C$_a^-1 (z) as an F_q-linear power series. To obtain an $A$-expansion for the Carlitz exponential we then have the beautiful, unpublished, formula of Federico Pellarin:
$\exp$_C(z)=$\hat\sum$_a$C$_a^-1 (az),
where one must now "renormalize'' the sum in two steps: First of all, we truncate the sum over the monics of degree $d$, as before, and then we also truncate the resulting expression (which is an additive power series) to only include the terms of degree $\leq$ q^d. Again, as before, without these operations there is no hope of convergence.

Next let's move on to $L$-series in finite characteristic.  Again we find that there is a mix between $A$-expansions and Z-expansions. For purposes of illustration we only treat the simplest case; thus given a monic $a$ in $A$ of degree $d$, we set
$\langle a\rangle$:= $a/\theta$^d.
Notice that $\langle a \rangle$ is a $1$-unit in K_$\infty$, and, as such, the expression $\langle a \rangle$^y makes sense for $y \in$ Z_pvia the Binomial Theorem (and with the usual exponential properties). We put S_$\infty$:=K_$\infty$^*$\times$ Z_p with its obvious abelian group structure, and for s=(x,y)$\in$ S_$\infty$, a^s:=x^d$\langle a\rangle$^y . One then has the zeta function of $A$ defined by the $A$-expansion
$\zeta$_A(s):=$\sum$_a a^-s.
For x not in F_q[[$1/\theta$]], this expansion converges without further manipulation. For the rest of S_$\infty$ we rewrite $\zeta$_A(s)=$\hat \sum$_a a^-s where, to guarantee convergence, we again truncate by the degree $d$ and take the limit....

There is yet a third place where $A$-expansions are now playing a very interesting role  and which is presumably somehow related to the above cases. Let $f(\tau)$ be a classical elliptic modular form on the upper half plane associated to SL₂(Z). As everybody knows, the form $f$ has an expansion $f=\sum$a_nqⁿ where q:=e^{$2\pi i \tau}. Now let $g(z)$ be a modular form on the Drinfeld upper half-plane. In particular, $g(z)$ is, by definition, invariant under transformations of the form $z\mapsto z+h$ for $h\in A$; as such I showed long ago that $g(z)$ has a Z-expansion $\sum$c_n uⁿ where $u(z):=\exp$_C$(\pi z)$^-1. Noting that
$\sum$ a_nqⁿ=$\sum$ a_ne^{$n 2\pi i \tau$}
leads one to suspect that $g(z)$ might also have an expression of the form $\sum$_ad_au_a where $a$ runs over the monics and u_a:=u(az). In fact, this is almost the correct idea: Let $G(X)$ be a fixed function (so far only polynomials have been considered). Then we call an expansion of the form c₀+$\sum$_ac_aG(u_a) an "$A$-expansion''. While it turns out that not all forms in finite characteristic have such expansions (at least for the class of functions G considered up till now), it has recently become very clear that a great many important ones do!

For instance, all Eisenstein series have such expansions. More importantly, the two basic cusp forms $\Delta$ and $h$ also have them: $\Delta$=$\sum$_a a^q(q-1) u_a^q-1 and $h$ (which is a $q-1$-st root of $\Delta$) has the expansion $h=\sum$_a a^q u_a. These expansions are due to B. L'opez, Arch. Math. 95 (2010), 143–150. Very recently, in Journal of Number Theory 133 (2013) 2247–2266, A. Petrov has shown how to construct families of cusp forms
$\sum$_a a^t G_n(u _a)
for certain positive integers $t$ and polynomials {G_n(X)}. Moreover he proves that these forms are, in fact, all Hecke eigenforms with easily computed eigenvalues. In arXiv:1306.4344 I showed how these forms give rise to non-trivial interpolations at the finite primes $\mathfrak v$ of $A$ in the sense of Serre's construction of p-adic modular forms (something I have long wanted to do). This also fits perfectly in to the theory of such forms created by C. Vincent in her 2012 Wisconsin thesis.

I would like to finish by explaining how Petrov's sums, just above, have elliptic modular analogs. Put G(X):=X/(1-X) and q_n:=qⁿ. Then, indeed, the normalized Eisenstein series of weight 2k has the Lambert expansion
1+2/$\zeta(1-2k)$ $\sum$_n n^2k-1 G(q_n) .

It is my pleasure to thank Rudy Perkins and Federico Pellarin for their invaluable input.

Wronski, Vandermonde, and Moore!

This post is based on a recent letter by Matt Papanikolas outlining some results he has discovered whilst writing a (highly anticipated!) monograph on $L$-values in finite characteristic. In staring at Matt's letter, I realized that he allowed one to relate the big 3 matrices (Wronski, Vandermonde and Moore) in one simple formula which I will present below and then pose a related question.

I apologize if this is challenging to read; I am having a devil of a time getting my tex to compile correctly.

Matt's perspicacious insight was based on a paper by Felipe Voloch which was published in the Journal of Number Theory  in 1998. (Full disclosure: I had previously played with Voloch's paper and totally missed these ideas...)

Let me begin with some standard notation. As is now becoming standard, following Greg Anderson, let us put $A:= F$_q$[\theta]$. For each nonnegative integer $i$, let's put $A(i):= \{a\in A$ $\mid $ $ \deg (a)\leq  i\}$. Moreover, for such $i$, following Carlitz, we set $[i]:=\theta$^{q^i}-$\theta$. These are basic building blocks of $A$; for instance, one easily sees that $[i]$ is the product of all monic primes of degree dividing $i$. Various types of factorials are products of these elements....

Now recall that the Wronskian matrix is made up out of the derivatives of a function. Obviously in finite characteristic one can only differentiate a little bit before ending up with identically zero functions. As such we modify the Wronskian slightly here and use the "divided derivatives" $\partial$_i (i.e., those elements actually giving the coefficients of Taylor series etc.) in order to end up with nontrivial matrices.

Let $x$_0,..., $x$_i be i indeterminates. We define the Vandermonde matrix $V(i)(x$_0,..,$x$_i) as usual by having the $i$-th row consist of $(1$,$a$ ,..., $a$^i$)$, where $a=x$_{i-1}, and so on.

Finally of course the Moore matrix is made up out of the $q$^i powers of elements in a very similar
fashion to both the Wronski and Vandermonde matrices.

Let $g\in F[[\theta]]$. Notice that $g$^{q^i} is simply $g(\theta$^{q^i}) as the elements of $F$ are
fixed.  Thus, as Felipe observed, a bit of calculus (e.g., Taylor expansions) immediately gives $g$^{q^i}=$\sum$_j $\partial$_j(g) [i]^j.

Let $a\in A(i)$. Let M(i)=(a, a^q,..., a^{q^i})^t be the column vector consisting of a and its q^j-th powers and let W(i):=(a, $\partial a$, ..., $\partial$_ia)^t be the column vector consisting of the divided derivatives of a. Matt realized that Felipe's result implies
$$M(i)=V(i)([0],[1],...,[i])W(i).$$
Now simply choose a bunch more elements in A(i) and sling the above equations together. We immediately deduce an equality of matrices of the form M=VW where M is a Moore matrix, V is the Vandermonde and W the Wronskian. Thus taking determinants allows us, in this case, to calculate the Wronskian determinant in terms of the Moore and Vandermonde determinants.

It is very wild that one can relate the q-th power mapping, which is a field embedding, in this fashion with the divided derivatives which satisfy the Leibniz identity and I wonder if there is much more here.

In any case, the Moore determinant is intimately related to having elements linearly independent over F_q.  In classical theory, the Wronskian of analytic functions determines whether they are linearly independent (a result evidently due to Peano); see arXiv:1301.6598 for a modern approach. In characteristic p, the p-th power mapping clearly wreaks some havoc as usual (think of the Wronskian of x^p and x^{p^2}). But I wonder if there might not be some clear statement about linear dependence involving both the Wronskian and the p-th power mapping...?

(Updated 4-22-13: If you choose as your elements $a\in A(i)$ the set $\{1,\theta,...,\theta$^i}$, you
obtain a Wronskian with determinant 1 thus giving an equality between the determinant of Matt's Vandermonde matrix and the determinant of the Moore matrix. As in Carlitz's original paper back in the 1930's, this Moore determinant is then easily computed to be a Carlitz factorial... It should also be possible to compute the determinant of Matt's matrix directly using the well-known formula for the Vandermonde determinant.)

(Updated 4-23-13: I thank Sangtae Jeong for pointing out the work of F.K. Schmidt, in 1939, and the work of Felipe Voloch and Arnaldo Garcia, in 1987, that goes very far with Wronskians in finite characteristic. See "Wronskians and linear independence in fields of prime characteristic", Manuscripta Math., 59, 1987, 457-469.)

(:1301.6598arXiv:1301.6598

Informal video series on the Carlitz module

Dear All: My student, Rudy Perkins, and his fellow graduate student, Tim All, are creating an informal video lecture series on the Carlitz module. If you are interested, please check out http://rudyperkins.wordpress.com/ . David

CYCLIC HOMOLOGY AND ARITHMETIC

Cyclic homology has recently revealed its potential in relation to the description of Serre's Archimedean local factors in the Hasse-Weil L-function of an arithmetic variety as shown in the paper by A. Connes and C. Consani :  Cyclic homology, Serre's local factors and the lambda-operations. The elaboration of this topic constitutes one of the two leading themes of the course that AC is developing at the Collège de France this year. Cyclic cohomology was introduced and widely publicized in 1981  as an essential tool in noncommutative differential geometry. The talk  Spectral sequence and homology of currents for operator algebras  given by AC at the 1981 Oberwolfach meeting introduced for instance the SBI long exact sequence and described the cyclic cohomology of the NC torus. In the context of algebraic geometry instead, the application of cyclic homology to schemes has a more recent evolution  and it has been mainly promoted in the work of C. Weibel.
 In number-theory, there are three fundamental sources of  L-functions 1) arithmetic varieties 2) geometric  Galois representations 3) automorphic representations.
To a smooth, projective variety X  defined over a number field K, corresponds its Hasse--Weil zeta function which is the alternate product of the factors $L(H^m,s)$ attached to the Galois representation on the $\ell$-adic \'etale cohomology $H^m(X_{\bar K},\Q_{\ell})$. The function $L(H^m,s)$ is defined as an infinite Euler product whose non-archimedean factors have an immediate geometric meaning  at places $\nu$ of K of good reduction for X (we assume here that $\ell$ does not divide $\nu$), by implementing the action of the geometric Frobenius on the étale cohomology  of the reduction of $X$ at $\nu$.
At an archimedean place v instead, the local L-factor is roughly a product of powers of (shifted) Gamma functions whose definition depends upon the Hodge structure on the m-th Betti cohomology of the complex  variety $X_{v}=X\times_K\C$.
The main result of the above paper states that the alternate product (as $m$ ranges from $0$ to twice the dimension of X) of Serre's Archimedean factors is the inverse graded determinant of the action on cyclic homology of $X$, with coefficients in infinite adeles $\prod_{v|\infty} K_v$, of the operator $(2\pi)^{-1}(s-\Theta)$, where $\Theta$ generates the lambda operations which are the analogue in cyclic homology of the Adams operations in algebraic K-theory.
 Several subtle features ought to be mentioned at this point. One of them is the nuance between cyclic homology and  archimedean cyclic homology (this is the one taken up in the above result) which parallels and reflects, in cyclic homology, the difference between reduced and unreduced real Deligne cohomology (we refer to the paper for more details).
This result promotes the development of the archimedean cyclic homology as a theory playing a natural role in the theory of  motives in algebraic geometry, in view of its connection to algebraic K-theory by the regulator maps. Moreover, it also suggests the study of a generalization of the above result at the non-archimedean places and the existence of a global Lefschetz formula in cyclic homology.
The second part of the course  will focus on the description of the Archimedean counterpart of the rings of periods in p-adic Hodge theory (especially $B_{cris}$ and $B_{dR}$), by elaborating on the results contained in the recent collaborative paper The universal thickening of the field of real numbers.   Fontaine's rings of p-adic periods play a fundamental role in arithmetic in view of the comparison theorems relating étale cohomology with coefficients in p-adic numbers, with the other fundamental cohomological theories of arithmetic varieties and in particular with the de-Rham cohomology thus realizing Grothendieck's  idea of ``foncteur mysterieux''.
At a real place, the transposition of the construction of the rings of p-adic periods yields non-trivial relevant rings endowed with a canonical one parameter group of automorphisms which replaces the Frobenius in Fontaine's construction. At a complex place, this construction produces fundamental algebraic structures whose applications  transcend the realm of arithmetic by producing a natural framework in which Feynman integrals in quantum field theory should be understood.
The class will be entirely given as blackboard-chalk talks.
AC and K. Consani.

THE MUSIC OF SPHERES

The title of this post, the music of spheres, refers to a talk The music of shapes which I gave in Lille, on the 26th of September, on the occasion of a joint meeting with the Fields Institute. The talk is an introduction to the spectral aspect of noncommutative geometry and its implications in physics.
The starting point is the naive question "Where are we?", or how is it possible to communicate to aliens our position in the Universe. This question leads, in the Riemannian framework of geometry, to that of determining a complete set of geometric invariants, both for a space and for a point in a space. The theme of Mark Kac, "Can one hear the shape of a drum" associates to a shape its musical scale which is the spectrum of the square root of the Laplacian, or better of the Dirac operator. After illustrating this familiar theme by many concrete examples we give a hint of the additional invariant which allows one to recover the geometric picture, namely the CKM invariant, and illustrate it, in a simplified form, in the simplest possible example of isospectral but non congruent shapes.
What about the relation with music? One finds quickly that music is best based on the scale (spectrum) which consists of all positive integer powers $q^n$ for the real number $q=2^{\frac{1}{12}}\sim 3^{\frac{1}{19}}$. Due to the exponential growth of this spectrum, it cannot correspond to a familiar shape but to an object of dimension less than any strictly positive number. As explained in the talk, there is a beautiful space which has the correct spectrum: the quantum sphere of Poddles, Dabrowski, Sitarz, Brain, Landi et all. Its spectrum consists of a slight variant of the $q^j$ where each appears with multiplicity $O(j)$. (See the original  paper of  Dabrowski and Sitarz arXiv:math/0209048 (Banach Center Publications, 61, 49-58, 2003) for the precise formula, and the paper of Brain and Landi arXiv:math/1003.2150 for a variant and the many references to the mathematicians involved, my apologies to each of them for not puting the list here.)
 We experiment in the talk with this spectrum and show how well suited it is for playing music.
The new geometry  which encodes such new spaces, is then introduced in its spectral form, it is noncommutative geometry, which is then confronted with physics.  There the core is  the spectral Standard Model of A. Chamseddine and the author which goes back to 1996. We tell the tale of the resilience of this model in its successive confrontations with experiments.
Both the start and the end part of the talk are unusual. The previous talk was a talk by Alain Aspect on his recent experiments, with his collaborators, confirming experimentally the "delayed choice" Gedankenexperiment of John Wheeler. So the very beginning of my talk refers to Aspect's point about the subtelty of  the concept of  "reality" implied by the quantum. The thesis which I defend briefly is that the total lack of control that we have on the outcome of a quantum experiment (we control only the probabilities), is a "variability" which is more primordial than the classical variability governed by the passing of time (on which we have no control either). I also explain briefly why time will emerge from the quantum variability.
The end part, in the question session, is also unusual, it is a long answer to a question which was posed by Alain Aspect.

The three speakers, Lille 9/26: E. Ghys, A. Aspect, A. Connes

 

Update : The talk of Alain Aspect is now also available at the  conference website

Carlitz's formalism and Euler's $\Gamma$-function

It was always my fondest hope that the arithmetic of function fields in finite characteristic would finally become sophisticated enough so that it could be developed somewhat in tandem with classical arithmetic. In the recent past, this hope appears to becoming real. In particular, I would like to draw your attention to the new preprint by Federico Pellarin arXiv:1210.2490 "On the generalized Carlitz-module" where the formalism due to Carlitz is applied to deduce new functional equations for Euler's $\Gamma$-function (as well as applications to other classical special functions).

Let ${\mathbf F}_q$ be the finite field with $q$ elements and let $\theta$ an indeterminate; as usual we put $A:={\mathbf F}_q[\theta]$. When Carlitz was originally developing his module in the 1930's he chose the mapping generated by $\theta \mapsto \theta -\tau$ where $\tau$ is the $q$-th power mapping. I changed this to $\theta + \tau$ simply because this made computing Frobenius elements a bit easier.

As the theory of the Carlitz module, and general Drinfeld modules, began to develop, it was realized that one could abstract from the original settings of function fields over finite fields and this was developed e.g., by Mumford (1977) and Hellgouarch (1992 and 1997). In the paper by Federico this formalism is applied to complex analytic functions in a very concrete fashion.

More specifically, Let $F$ be the field of (complex valued) meromorphic functions in $s$ which are periodic of period $1$. Obviously, the function $s$ is not in $F$ and so one forms the polynomial ring (of functions) ${\mathbf A}:=F[s]$; notice that the huge field $F$ is now playing the role of the finite field $\mathbf F_q$....

The question is what plays the role of the $q$-th power mapping; here Federico makes the crucial choice $\tau (f):=f(s+1)$ where $f$ is a meromorphic function (so that obviously $F$ is the fixed field of this action). One then has the (generalized) Carlitz module associated to $(F[s], \tau)$ simply by following Carlitz's idea of composing actions and his original formalism mentioned above! In particular, one is now able to talk about the torsion elements associated to an element $a\in F[s]$. 

So what does this have to do with the $\Gamma$-function? Well, virtually everything! Indeed, of course, everyone knows the (almost!) definitional functional equation
$$\Gamma(s+1)=s\Gamma(s)\,.$$
Federico now simply rewrites this as
$$s\Gamma-\tau(\Gamma)=0\,$$
In other words, the $\Gamma$-function IS an $s$-torsion element for this Carlitz module. In fact, one sees that Carlitz-torsion is intimately connected with the $\Gamma$-function in one form or another. (Indeed, I would not be surprised if there is yet another characterization of the $\Gamma$-function which involves Carlitz-Pellarin torsion.)

But much more is true. In a previous blog I have discussed the applications of the Anderson-Thakur function $\omega(t)$ to $L$-series. The function $\omega(t)$ can abstractly be obtained as the Akheizer-Baker function associated to certain data. In the $F[s]$-case, Federico shows that the Akheizer-Baker function is precisely $\Gamma(s-t)$ and this leads to other functional relations in line with the analogy to $\omega(t)$.