Motif of Motives II

Carl Brannen has reminded me of Cartier's classic paper, A Mad Day's Work. He discusses everything, from Grothendieck's biography to symmetry groups for a point. In particular, he points out that a sensible notion of symmetry group for a point comes from considering points as functors between toposes. Since there are natural transformations between functors, one might find a group of invertible natural transformations between a functor and itself.

The really cool thing about all this is that the group is not fundamental. Eat your heart out Gauge Theory!

Which reminds me that I meant to say something about Grothendieck's motives. As Cartier explains, motives are a part of Grothendieck's dream, a vision of unifying number theory and modern topology, and hence almost everything else as well. The theory of motives is still mysterious, although an impressive amount of progress in the related physics and mathematics has been made in the last 30 years. Consider for example the work of Kontsevich on motives and operads in deformation quantization. It's kind of funny that the mathematicians have chosen a word (motives) that starts with M. It's their version of M-theory!

An important intuition behind motives is that of projective geometry. Motives obey powerful relations, an example of which is the equation

M(projective plane) = M(plane) + M(line) + M(point)

which expresses the usual grading of a projective plane (over any field) into an affine space with a line and point at infinity. This feature of a grading in dimension is typical of motives, as it is for categorical dimension.

Monday Motives

Somehow we have returned to Motives! According to the illustrious wikipedia: in terms of category theory, [the theory of Motives] was intended to have a definition via splitting idempotents in a category of algebraic correspondences. Oh, yes, we were wondering about those, weren't we? Unfortunately, when I google twistor, motive and n-category, I find almost no hits. Hmm. Maybe that's the fault of the hyphen. But when I throw in Riemann hypothesis and MHV for good measure ...

Anyway, I must rush off and prepare to teach some eager young minds some good old Newtonian mechanics. It is a stunning day outside. At 7am there was a thick fog laying only a few centimetres over the ground, and the Waimakariri dust clouds have subsided.

Magic Motives

The most fashionable of all phrases amongst real mathematicians studying QFT, occasionally mentioned here, are the words motivic cohomology. This is supposed to be the mother of all cohomology theories. But what is cohomology anyway?

As a basic concept, cohomology relies on the logic of ordinary sets, particularly the difference between unions and disjoint unions. The latter type of union keeps track of which set each element actually came from, so it tends to double count intersections. In the category Set, disjoint union is a coproduct. Ordinary unions are a little less natural, but intersections are pullbacks of subset monics.

Universality is a basic categorical concept. A cohomology theory usually assumes a standard set of axioms for a cohomology functor $H^{*}$ from some category of (commutative) spaces (which we want invariants for) to a category of groups (where invariants live). The existence of group inverses is related to the arbitrariness of path directions in a commutative space. A universal cohomology theory is supposed be about an elusive category of motives.

But thinking of the field with one element, what happens if we have sets with funny actions, instead of just sets? Are our spaces necessarily commutative? Perhaps the underlying logic of intersection and union should be modified to properly account for the geometry of the absolute point. This might require drawing higher dimensional limit diagrams, or looking at higher dimensional target categories, not necessarily groupoids. Fortunately, some really smart people are now thinking about noncommutative motives, but it may take a while for us poor physicists to see what is going on.

Motive Madness

There are two papers by A. B. Goncharov, namely

1. Multiple polylogarithms and mixed Tate motives
2. Periods and mixed motives,

which are referred to by Brown in his paper on multiple zeta values and period integrals, as an excellent study of the punctured sphere moduli M(0,4) $\simeq \mathbb{P}^1 \backslash \{ 0,1, \infty \}$ and its finite covers based on roots of unity.

Hmmm. Maybe this post should be called M Theory Lesson 18. These topics (motives, number theory, gluon amplitudes etc.) are getting awfully mixed up! No, never mind. I'd better get back to reading, I guess. I must be crazy. It's a gorgeous day outside...

Motive Madness

Whoa! I wasn't expecting it that soon. Motives appear already, at least conjecturally, in John Baez's lecture 6! In the diagram the vertical arrows are the decategorification of either sets or projective spaces. An isomorphism class of $n$ element sets is mapped to the number $n$. Natural numbers become $q$-numbers in the case of spaces, which is to say rational functions in the parameter $q$ corresponding to the number of elements in the finite field, or secretly really polynomials with integer coefficients. But what replaces the category of finite sets? There is more structure to the projective spaces, and we also want to understand the bottom arrow, which considers a set as a space over a one element field.

This is a rather delicate mathematical question. Baez mentions a recent paper by Durov (with lots of stuff on monads) about the idea of a one element field. When we understand this properly, do we find motives? Now, that is the question.

The plane of a (finite) field $F_{q}^{2}$ is the $q$-analogue of a two element set, which plays an important role in the Boolean topos Set, namely as the subobject classifier. The vector space version of this is commonly known as a qubit. Somehow the reason that a one element field $F_1$ doesn't usually make sense is because the logical 0 and 1 are not distinct. Since $q$ is, in the first instance, just a natural number (= $p^{k}$ for some prime $p$), we can ask ourselves first what it means to collapse a finite plane to a one element field. For the topos Set, this would amount to turning the whole category into the trivial category 1, since there is no way to distinguish a subset of a set $S$ from its complement and all sets have effectively only one element. Now this one element set is like a basis for spaces over $F_1$. But the map that takes a basis to a space is just the functor 1 $\rightarrow$ Set, which picks out a set!

But this doesn't sound right. Maybe what we need here is not the trivial 1-category, or a one point 0-category (set), but rather a -1-category. This idea always lurks in the operad heirarchy, where the left hand side of the table starts with the single leaf tree, despite the fact that a point is actually a two leaf tree.

Anyway, think of a finite set $S$ lying at the endpoints of the unit vectors in a vector space. The empty set at the origin is the smallest piece of the power set of $S$, and the one element subsets are the next smallest pieces. The power set fills out a cube of dimension $|S|$. Since the field in question is $F_1$ there is no extent to the axes. Only the elements of $S$ really exist.

Have a Nice Day

I apologise for moving off topic today, but I found this Smile Test very interesting. Apparently, most people are not as good at telling fake smiles as they think they are. The theory is that people are easily fooled because it is socially convenient not to know what people are thinking. Despite my expectations of doing badly, I actually did really well on this test (17/20). Now I realise that it's easier not to care what people think when one's ability to detect fakeness makes it impractical to take such things into account.

Ars Mathematica finally reports a retract of the claimed disproof of The Hypothesis. David Ben-Zvi has kindly provided notes on the recent Chicago conference, where Goncharov was talking about Motives, path integrals and trivalent graphs. Sounds intriguing. OK, I printed out the notes. Wow. OMG. Goncharov claims to have identified the category of mixed motives (a.k.a. the holy grail for ordinary real/complex geometry) in terms of path integrals for projective varieties. For instance, when the variety corresponds to modular subgroups indexed by $\hbar = \sqrt{N}^{-1}$, as in TGD or $N$-fold covers of moduli spaces, one gets Langlands from the cohomology. He concludes with a statement that Feynman integrals (with observables) are valued in motivic cohomology. Yeah, duh, the physicists know that. We just don't know how we're ever going to learn that much mathematics.

Ah! That means the S duality we need for the Riemann Hypothesis relies on the whole range of quantised $\hbar$, and is therefore necessarily omega-categorical. That was expected, because the surreal zeta arguments extend through the ordinals. It is fantastically exciting to have some confirmation of this link between $\hbar$ values and S duality. I wonder how string theory will deal with a variable $\hbar$. Oh, I see.

Tour De Force

Warning: make sure you are sitting down before you attempt to look at this 642 page draft of the new book, Noncommutative Geometry, Quantum Fields and Motives, by Matilde Marcolli and Alain Connes.

Not content with introducing QFT, NCG, Connes' Standard Model, the Riemann hypothesis, motives and the kitchen sink, they finish up (from page 611) with a section entitled The analogy between QG and RH. The preface makes it abundantly clear that the book is primarily about this, as yet mysterious, correspondence. Unfortunately, I suspect I will find most of the book extremely mysterious as long as I live.

Magic Motives II

If sets and vector spaces are secretly the same thing, and spaces are built out of these new sets, then there should be one big category of spaces that has everything in it, including invariants!

For instance, a vector space of dimension $n$ over a finite field may be represented by all $n$-tuplets of MUB operators representing the field. Now these tuplets are just collections of arrows in the category of fancy sets. A triplet of $3 \times 3$ matrices with cubed roots of unity would act on a $27$ element set, via Cartesian product. The collection of all finite dimensional vector spaces over $F_3$ would live in the set subcategory made up of Cartesian powers of the three element set.

Similarly, spaces for the two element field $F_2$ may occupy powers of the two element set, usually denoted by $\Omega$ in the ordinary topos. This suggests that a power set monad for fancy sets might have something to do with invariant functors for $F_2$ (or even $2$-adic numbers). How cool would that be? The world of motives would then return to Grothendieck's dream.

Motives in Tokyo

Thanks to Motivic Stuff for pointing out a wonderful workshop in December in Tokyo: International Workshop on Motives.

Time Machine

Although it caused quite a stir in the press and on the blogosphere, I didn't take much notice of the Time Machine paper until today, when I realised it was written by Irina Aref'eva and Volovich, who happen to work on p-adic strings and the quantization of the Riemann zeta function. Ultimately, they are simply speculating about new kinds of objects, related to classical causality violation, that may be visible at the LHC, and the catchy title is simply a gimmick without which it is difficult these days to get papers posted on the arxiv.

In this paper, the authors discuss some pretty hairy mathematics, in the physicist's characteristic shockingly hand-wavy manner. To quote:

[this lends] additional support to the proposal that the Beilinson conjectures on the values of L-functions of motives can be interpreted as dealing with the cosmological constant problem ... in section 6 we shall discuss an approach of how to use a Galois group and quantum L-functions instead of SUSY to improve the spectrum.

By the spectrum they are referring to their analysis, inspired by the non trivial zeroes of the Riemann zeta function, which correspond to $m^{2}$ values in Klein-Gordon operators. In other words, the zeta function is defined not on numbers, but as a pseudodifferential operator. The Hypothesis says that the zeta field is given by a sum of such Klein-Gordon Lagrangians.

Note that in M Theory, we prefer to replace $\Lambda$ with the heirarchy of Planck scales, but this idea is basically present in their work.

M Theory Lesson 102

In the 2005 lecture 6 Alain Connes points out that although his framework predicts physical couplings that match $SU(5)$ unification, it achieves this without the addition of extra fields or supersymmetry arguments. Towards the end of the lecture he summarises the situation: the problem is to combine (1) the renormalisation theory (Hopf algebras, motivic Galois group) and (2) the geometric setup from NCG operator theory, in such a way that running geometries (on different scales) are possible. Connes proposes a functional integral over geometries which is spectral in nature, and in fact looks like a matrix model. The question is, what sort of constraints should be applied to geometries? It is made clear that this is a challenge to physicists: his spectral action principle is a statement about the nature of observables, but insufficient in itself to guide, as Connes puts it, the merging of motives and NCG.

These lectures are highly recommended to physicists. Don't expect to understand all the mathematical gobbledygook, but try to take in the big picture, which is fantastically conveyed. Note that more recent work removes much of the arbitrariness of the original NCG formulation of the SM, but still puts the number of generations in by hand.

Magic Motives

Speaking of orbifold Euler characteristics, let's put the magic formula

$f(n) = \prod_{m=0}^{n - 1} \frac{m!}{(m + n)!}$

in terms of Euler characteristics. First, let $m = 2g - 2$ be the Euler characteristic of a closed surface of genus $g$. This already suggests allowing non-orientable surfaces to account for odd values of $m$. Then consider moduli spaces $M_{m,n}$ for $(n + 1)$ punctured surfaces. The orbifold Euler characteristic of such a space will be denoted by $E_{m,n}$. Using Mulase's expression for $E_{m,n}$ and assuming it may be extended to the non-orientable case, one finds a natural definition for the moment coefficients of the form

$f(n) = \frac{1}{(n + 1)!} \prod_{m=0}^{n - 1} b_{m + 2} E_{m,n}^{-1}$

which is a product over surfaces of genus $g$ limited by $n$, and where $b_{i}$ is a Bernoulli number (for even $m$ these are defined in terms of zeta values for odd negative reals). One should take more care with the non-orientable factors, but this simple exercise shows that the zeta moments are naturally dependent on categorical invariants associated to complex moduli.

Operadification III

The multicategorical analogue of the natural number diagram

$1 \rightarrow N \rightarrow N$

looks like

$\Delta^{\cap} \rightarrow \textrm{Tree} \rightarrow \textrm{Tree}$

where the category $\Delta^{\cap}$ (dimension not specified) has objects $n$ represented by single level trees, the associahedra trees. That is, since we are allowed any number of input identity arrows, the simplest one object category has an arrow for each $n$. The category Tree, by definition, extends these single level trees to $k$-ordinal trees of $k$ levels. In other words, the ordinals $N$ in Set are replaced by the levels of the $k$-ordinal trees. This is how we wanted to represent $n$ in the quantum world, in association with dimension.

Now recall that the $k$-ordinal trees can represent Batanin's polytopes, which are topological spaces. The successor map simply adds a leaf to every top branch. For example, the sequence of $k$ dimensional spheres arises as a version of the ordinals in this sense.

Multicategorical arithmetic therefore compares an ordinary cosimplicial object in $C$ with a weakened kind of
cohomological object Tree $\rightarrow C$. By truncating the categories at level $k$, one obtains a multicategorical analogue of modular number objects. There are many motives for studying this kind of arithmetic.

M Theory Lesson 193

While I was busy at Neutrino 08, the NCG blog posted an update on the Vanderbilt meeting. In particular, they note that Manin's lectures on Zeta functions and Motives are available at Katia Consani's homepage! Niranjan Ramachandran spoke about this paper at Vanderbilt. This work, originating in the physical ideas of Deninger, looks at the field over one element (which is fast becoming a popular subject). Deninger writes the zeta function, completed with the infinite prime, in the form

$\zeta (s) = \frac{R}{s (s - 1)}$

where $R$ is a regularized determinant to be viewed as an infinite dimensional analogue of a determinant of an endomorphism of a finite dimensional vector space (according to Connes and Consani).

M Theory Lesson 135

James Dolan speaks of categorification and decategorification, and of information and entropy.

In logos land, these processes have a dimension raising or lowering aspect. It is often said that categorification is ill defined, in comparison to decategorification, but with dual processes it should not be so. Therefore, categorification itself must be defined in some canonical way that generalises the turning of natural numbers into sets or spaces. One way to do this would be to put the heirarchy on a loop, such as the loop labelled by the $q$ parameters at roots of unity. There would be $n$-categories for $n \in \mathbb{N}$ and $r$-categories for $r \in \mathbb{Q}$, and $n \rightarrow \infty$ would look like the limit $q \rightarrow 1$ again, where spaces begin to look like sets.

After all, projective geometry has its horizons, and the cohomology of motives would move left and right, like the mass interaction, or Stokes' theorem, or the Riemann zeta function.

Connes Kreimer Marcolli

There is an amazing series of papers by Connes, Marcolli and others on From Physics To Number Theory. See for example here or here or here. This goes back to work of Kreimer and Broadhurst, which is now very well known. Some of the older papers are here. I particularly recommend the paper: Broadhurst and Kreimer, Association of Multiple Zeta Values with Positive Knots via Feynman Diagrams up to 9 Loops, Phys. Lett. 393 B (1997) 403-412.

Its about turning knots into simple Feynman diagrams into Multiple Zeta Values. These MZVs satisfy all sorts of crazy relations, which the mathematicans have been studying like crazy. But really they're quite simple. They act on a set of k ordinals (yes, that's right, you should be thinking 1-ordinals) and are characterised by two numbers, namely the weight n, which is the sum of these, and k itself, the so called depth. Of course these naturally show up as special integrals of something called Mixed Tate Motives (don't even ask), so we know that the weight n is the same n of M(0,n+3). Goodness, me. The Yang-Mills problem and the Riemann hypothesis seem to be related. Well, well.

The real question, however, is how to go beyond scalars to other entities in QFT. Any guesses?

Motif of Motives

It is said that Grothendieck, one of the greatest mathematicians of the 20th century, is now mad. A piece of evidence often cited in support of this hypothesis is his fixation with the speed of light, a mental exercise that might be recommended to many of the critics.

The arbitrary local numerical value of this quantity depends on the arbitrary old definition of the metre from Napolean's time. After some international political wrangling, some French guys measured the meridian from Dunkerque to Barcelona in the years 1792 to 1798. If they had chosen a different geographical location the platinum metre bar would no doubt have come out slightly differently and maybe, with a little stretch of the imagination, we would not be plagued with awkward values for c today. As Einstein said in a lecture in 1921:

In order to complete the definition of time we may employ the principle of the constancy of the velocity of light in a vacuum.

With emphasis on the word may. The constancy of c was not to be taken as a fundamental consideration, but as a convenient means of defining clocks for observers in uniform motion. To assume that the constancy of c should suffice for quantum gravitational clocks is rather stupid. Fortunately people have considered alternatives. Louise Riofrio has some very pretty pictures and graphs which use a varying c to explain away the magical Dark Energy.

Coincidence

Recall that Motives are about projective geometry. Naturally then, Motivic Cohomology should be about diagrams with lots of points and lines. We want to build very complicated geometries, perhaps to probe the region close to a Black Hole, from such diagrams. Now we see that dualities abound in motivic pictures. The most basic of these is the one between Space and Time. So there are also singularities in Time, as we observe.

Maybe this is all an April Fool's Joke? Who knows?