M theory: life, the universe and everything

My friends were keen that I should write something about M theory. Don't worry folks: I have not abandonned category theory for a boring 11 dimensional theory. I shall explain. Of course, I could redirect you to PF, but it seems to have crashed yet again (now I wonder why that is?). Or, I could redirect you to the new n-category Cafe (see blog roll), where lots of cool stuff will be happening soon.

Short story: we know how to do M theory rigorously. This is a fairly compelling case that our ideas about quantum gravity are, er, let's say, on the right track. Personally, I do not claim to understand much of it at all. I don't see how anyone could claim to, except maybe Ed Witten and the mysterious kneemo and a few other wizards.

Physically, M theory needs to formulate a Machian duality (Witten's favourite word) that looks like supersymmetry, but not in the sense of ordinary algebras or superpartners. Years ago, when I was trying to picture this, and stumbling clumsily with varying hbar or speed of light (while Louise was working it all out), I asked myself: if horizons are like boundaries, but the holographic mirror has to turn everything inside out, then what is a boundary? How do primordial black holes and the cosmological horizon fit together? Many of the most fundamental ideas in physics are about understanding boundaries. Consider Stokes' theorem, for instance.

After playing with a little mathematics I fell wildly in love with Algebraic Topology, because Stokes' theorem could be written pretty simply! The Machian principle also suggested an interconnectedness-of-all-things idea. It slowly became clear that this was impossible without category theory, because category theory is the mathematics of relationships (and people had already tried pretty well everything else).

But I won't bore you with diagrams, which I can't draw here anyway. It turns out that to understand M theory we need several pictures at once in our mind: twistor String theory, spin foam QG and matrix models.

It is a powerful mathematical theorem that the complex moduli of Riemann surfaces with punctures is closely related to a moduli of labelled (metric) ribbon graphs. A ribbon graph is a closed diagram (graph) of flat ribbons, which are allowed to cross over and under one another. By allowing twists, one can also study matrix models for the quaternionic ensemble. The expert on this is Mulase.

Anyway, since the Bilson-Thompson preons were made of ribbons, it made sense to think about these ribbon moduli. These spaces turn out to have cell decompositions that look a lot like the special polytopes that turn up in higher category theory. This is no accident, because it is all really about operads. Now, I knew that there was a first class maverick amateur physicist named Carl Brannen, who had already calculated the neutrino and charged lepton masses based entirely on ideas from the Geometric Algebra of Hestenes. What Carl did was associate preons (not quite the same as the Bilson-Thompson ones) with idempotent eigenmatrices in Clifford algebras.

A few days ago, people started talking about the Bilson-Thompson preons, yet again, and Carl briefly outlined his vastly superior version, which could explain the number of generations. I showed up, and so did Michael Rios (kneemo), who happens to be a young wizard expert in Jordan algrebras, and just about everything else it seems. The three of us got talking, and while I was struggling to understand all the algebra it dawned on me that, geometrically, the model for the idempotents were the special points in projective space from Mulase's ribbon graph theory. As soon as we realised that the orbifold Euler characteristic of the moduli space of the 6 punctured sphere was -6 we knew that was a derivation of the number of generations of String theoretic power. The gallant kneemo got working, and I tried hard too but mostly I just went into panic!

This is all about doing not only higher categorical cohomology (boundary theory), but also Poincare duality - the mother of all Poincare dualities. Mulase had already shown how to get T duality out of the real-symplectic Penner model. This used twisty single ribbons. The triple ribbons come in because the idempotents are 3x3. Carl's preons can be built out of a basis of 2 flat ribbons (particle, antiparticle) and two twisted ribbons (1 left, 1 right), each of which effectively has 4 labels. A left handed electron must pair with a right handed electron to generate mass.

Don't worry: spin foam models are in there, too. Louis Crane's geometrization of matter proposal was about keeping the spin foam topological and relaxing the restraint on using manifolds. Matter should somehow be related to the singularities, where the nature of a point (vertex of the spin foam) is given by the surrounding 3-space. It was a lot of fun to play around with 3 dimensional hyperbolic geometry and knots, but it wasn't clear how to make the higher genus surfaces (boundaries of the 3-spaces) look exactly like black holes (or 'dark matter'). Now we can do it. The trick is to think not of an actual surface, but the whole moduli space being modelled by projective geometry (twistors).

More on this later. If anyone happens to be in Sydney, I'll be talking about ribbon graphs to some category theory guys this coming week, at 2pm on Wednesday August 23 in the Maths department at Macquarie.

M Theory Lesson 100

On page 127 of their book [1], Lambek and Scott discuss the unifying properties of topos theory, as seen from the perspective of type theory. The three traditional mathematical philosophies in question are intuitionism, Platonism and formalism (in the sense of concrete symbolism). They note that a fourth philosophy is somewhat neglected, namely neologicism: the idea that all mathematics can be formulated in terms of logic. Strict logicism would require deriving the properties of the natural numbers from more foundational principles.

But isn't that just what the physics has been telling us we must do? Lambek and Scott take the viewpoint that categorical logic treats logic pragmatically, as a part of ordinary mathematics, thus refuting logicism. Should logos theory refuse to accept this point of view? Recall that in M theory any useful logical statement has a physical meaning pertaining to a given experiment and its constraints. The numbers associated with logical statements (or rather diagrams) also take on a physical meaning (loosely speaking, a collection of measurements). Thus both logic and number theory should be derived from the physical principles of measurement, expressed as a higher dimensional topos theory.

There is a chicken-and-egg objection: that the formalism of higher toposes requires the mathematics of logic before acquiring a physical interpretation. However, since physics is undoubtedly guiding the axioms in question, this argument appears weak. Ironically, logicism in M Theory would be the ultimate in reductionism, despite the intuition of emergent schemes on different scales.

[1] J. Lambek and P.J. Scott, Introduction to higher order categorical logic, Cambridge U.P. 1986

M Theory Lesson 52

The logical necessity of weakening distributivity in logos theory forces a study of pseudomonads, not just monads. Steve Lack has shown that a good theory for pseudomonads really requires Gray categories, our favourite tricategorical toys. This is the primary reason that a quantum analogue for a topos must go higher than 2-categorical structures.

The first kind of distributivity that we learn about is that of ordinary multiplication over addition. This is fully described by monads (in particular + and x) in a (causal) square involving the categories Set, Ring, Monoid (for multiplication) and Ab (for addition). The category of rings is where the numbers actually live. Now by characterising Set as a ground 2-logos, we begin to see that a very fundamental axiomatisation of M Theory should be possible, in terms of pseudomonads for 3-logoses.

Hopefully by now it has occurred to our readers that the term M Theory does not merely refer to an 11 dimensional supergravity.

M Theory Lesson 51

The gallant kneemo has pointed out that a locale is an important concept in topos theory. A locale is a simple generalisation of the lattice of open sets for a topological space. Localic toposes are discussed in the book of Mac Lane and Moerdijk.

By restricting attention to sober topological spaces, there is an equivalence of categories between spaces and a suitable collection of locales. Alternatively, there is a duality between spaces and frames, where a frame is just a locale in the opposite category. Now the initial object in the category of frames is a two point set {0,1}, because lattices always have a top element 1 and a bottom element 0.

When considering spaces, the most basic space is really the Sierpinski space $S$ and not a single point, which is often used to describe points in spaces via maps from the one point space. This space has the property of being both a space and a frame, the initial frame. An open set in a space $M$ is considered as a continuous map $M \rightarrow S$, whereby the inverse of 1 picks out the open set.

The existence of such a self-dual object in a categorical duality turns out to be very useful. Another example is the circle $U(1)$ in Pontrjagin duality for locally compact (Hausdorff) abelian groups. This is why kneemo's remark about a ternary analogue is interesting. We have seen that generalised Fourier transforms are important in M Theory. It is not enough, however, to simply replace the two element set with the three element set {0,1,2}. The inclusion of classical locales into the 'quantum' topos theory would require a higher categorical setting, for which the three possible two point subsets are perhaps rather subcategories of the triangle category on three points. This is suggestive of the need to consider three inclusions for 2-logoses into 3-logoses.

Aside: I have inserted a button to Carl's gravity simulator on the left sidebar.

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.

M Theory Lesson 27

The planar algebras of Vaughan Jones arise from a coloured operad of empty discs in a larger disc, with string pieces (open or closed) in the surface, such that there are an even number of boundary points on each disc. The theory of Jones' subfactors is a kind of Galois theory for $II_{1}$ factors. Matti Pitkanen has considered the relevance of this to physics.

In M theory we would also like to consider higher dimensional operads. For quaternionic number theory it is appropriate to start with 3-sphere discs, plus further structure. Rather than marked boundary points, for instance, a 3-sphere can contain knots and links and also surface boundary components. If the surface pieces are punctured spheres they can be made to look like 2-disc diagrams. We could pack enormous amounts of algebraic information into such an operadic structure. Batanin's 2-level tree composition is a guide to horizontal and vertical compositions in the 2-operad case.

M Theory Lesson 238

We can think of the braid group $B_{3}$ as the general matrix group over the field with one element, associated to sets as vector spaces. It is also the fundamental group of the complement of the trefoil knot. Recall that the trefoil knot corresponds to the Pauli quandle of operators $\sigma_X$, $\sigma_Y$ and $\sigma_Z$.

This quandle can be thought of as a group ring for a field with one element. Additively, there is only one choice for the coefficients of $\sigma_X$, $\sigma_Y$ and $\sigma_Z$, and so the formal sum $\sigma_X + \sigma_Y + \sigma_Z$ represents the three element set as the union of labelled one element sets. Multiplicatively, the cyclic quandle rules hold, and these are the only rules.

What does it mean to take the fundamental group (or groupoid) not of the trefoil, but of the Pauli quandle? What is the complement of the quandle in MUB space? A truncated braid group of type $B_3$ naturally arises for the $3 \times 3$ operators. Moreover, M theory is very interested in how the Pauli operators interact with this three dimensional case. Somehow M theory doesn't mind that $B_3$ is specialised to truncated knots when considering three objects. After all, the fundamental group is really about maps of a circle into a space, but a circle is what one obtains only after considering (at least) an infinite number of objects.

M Theory Lesson 50

Fundamental to an n-logos is the concept of n-ality. Dualities belong to 2-logos structures, which are built upon the binary logic of the line element [0,1] and the parity square. In modern language, a duality expresses an equivalence of categories, which we take to be a bicategorical concept. In M Theory we see that triality is about the interplay of three parity cubes, based on ternary logic for the values 0,1 and 2.

Observe that connections between the logic values form a basic n-simplex, which may be labelled with directed faces as in Street's orientals, which describe strict n-categories. Unsurprisingly then, Kapranov's non-commutative Fourier transform is built upon simplices and cubes. This kind of Fourier transform should be a basic construction in M Theory, as it was for Heisenberg and Dirac.

Riemann Revisited II

I'm so excited by this claim of Tribikram Pati! I haven't read the paper yet, but quoting from the abstract: our analysis shows that the assumption of the truth of the Riemann Hypothesis leads to a contradiction. Maybe he's right! After all, we've seen that the zeta function really shouldn't be studied within the confines of Boolean logic. But then there is already a post by Julia Kuznetsova claiming to have found the (almost inevitably present) flaw. A reply by a K. L. Lange to this criticism, supporting the proof, states, "So the main idea of [Pati] was to show, that we need that delta to prove RH, but there is no delta, so we cannot prove RH after all ..." In other words, the paper may well show that there is no proof of the Hypothesis within standard analysis. That doesn't sound surprising at all.

If we built an L function on the surreals in the M Theory operadic landscape, would we care about the ordinary Riemann zeta function? Yes, of course we would, because it's very interesting! And the M Theory L function would contain the Riemann one anyway. I have reluctantly come to the conclusion that a decent definition for n-logoses really must sort out the meaning of Analysis in category theory. This can't be done the 1-topos way. At least surreals have infinitesimals. And now I should probably confess that I always had great difficulties with analysis. Well, I also have great difficulties with Geometry, Algebra and Logic, which is one good reason for studying category theory, because it rolls them all into one! But as Donaldson, Perelman and many other mathematicians have shown, beautiful proofs these days need analysis.

M Theory Lesson 229

Using the three dimensional Fourier operator $F$, and a generator $M_{1}$ on which it acts, one obtains the cycle of four MUB operators satisfying

$F^{\dagger} M_{1} F = M_{2}$
$F^{\dagger} M_{2} F = M_{3}$
$F^{\dagger} M_{3} F = M_{4}$
$F^{\dagger} M_{4} F = M_{1}$

where factors of 3 and $\sqrt{3}$ are ignored as usual (this is no worse than the habit of insisting that $c = 1$ all the time). Observe how the set of four matrices naturally factors into two sets of two, just like the number 4. For example, $M_{3}$ and $M_{4}$ are related by a simple two dimensional map. If we divide all entries by $\omega$, this component of $M_{i}$ is just $\sigma_{Z}$.

Origin of Species II

Todd Trimble kindly made the following comment:

I hope I'm not being too obnoxiously anticipatory by also remarking that the usual notion of permutative V-operad can be defined very concisely as a monoid in the monoidal category of V-species, where the monoidal product is species substitution.

This is far from being obnoxious, Todd! Struggling physicists appreciate perceptive comments from knowledgeable category theorists who can see where we are trying to go with M Theory. Indeed, although we tend to talk about operads loosely as one object multicategories, the species definition comes very close to what Rivasseau et al have in mind for redefining quantum field theory.

Note also that from the logos perspective (ie. M theoretic higher topos ideas) the category of finite sets (resp. vector spaces), as it sits in the topos Set (resp. Vect), plays an important role in attempting to define the generalised logic behind the simple operators that we associate to measurement algebras. The one major alteration to these structures that M theory requires, which comes up again and again in many guises, is the relaxation of the monoidal condition. This happens naturally with higher dimensional structures, as illustrated by Batanin's tower of coherence laws. For QFT, this forces a complete change of mathematical language, since too many concepts are only defined in categorical terms. Fortunately, everybody can understand simple diagrams of ribbons and trees.

Aside: Post written with wireless connection from my heavenly warm room amidst the fresh snow at the observatory. Can't seem to find any old skis lying about ...

M Theory Lesson 35

One day, many years ago, I wanted to catch a bus from Sikkim down to the plains. It was my first lesson in the theory of order-under-chaos. A major landslide had blocked the only road out of the mountains, and yet in no time at all a relay of buses was setup. Travel between buses simply required a short walk along the goat track over the landslide debris.

It looks like M Theory needs to discuss the concepts of operadification and cooperadification. Don't worry, I am not suggesting that we adopt this cumbersome terminology. Instead, let's name the 2 functors that describe these processes entropy and information respectively. Entropy describes the process of leaping up to the next quantum level, which is far more complex and intricate. Information is the dual process of dropping down whilst investigating a question. It might not be humans asking questions. The galaxy might want to ask Computer Earth a question now and again.

Baez et al have been describing a program of groupoidification. This lives in the realm of n-category theory, and we expect that such categorical structures will arise as algebras of master operads.

M Theory Lesson 129

Jacques Distler bemoans the pitiable standards of the physics blogosphere in his post on Lisi's paper. Apparently, triality for the generations is a no starter. If one sticks to ordinary classical representation theory, no doubt this is correct.

But in M Theory one does not do this. The term triality is clearly a ternary analogue of the ubiquitous stringy term duality. Now let's look at a bit more lattice theory.

Recall (see Ebeling) that the subgroup $\Gamma (3)$ of $SL(2, \mathbb{Z})$ has modular forms $\theta_0$ and $\theta_1$ of the form, for $q = e^{2 \pi i z}$,

$\theta_0 = 1 + 6(q + q^3 + q^4 + 2 q^7 + q^9 + \cdots)$
$\theta_1 = 3 q^{\frac{1}{3}} (1 + q + 2q^2 + 2q^4 + \cdots)$

The theta function for the $E8$ lattice takes the form

$\Theta = \theta_{0}^{4} + 8 \theta_{0} \theta_{1}^{3}$

and this function appears three times in the celebrated j-invariant

$j = \frac{1728 (\theta_{0}^{4} + 8 \theta_{0} \theta_{1}^{3})^{3}}{- 4^{3} (\theta_{1}^{4} - \theta_{0}^{3} \theta_{1})^{3}}$

The group $\Gamma (3)$ appears naturally when studying ternary codes. The quotient of $SL(2, \mathbb{Z})$ by this group gives the group $PSL(2, F_{3})$, which is otherwise known as the group $A_4$, studied by Ernest Ma in his derivation of the tribimaximal mixing matrix.

M Theory Lesson 188

The 1997 Broadhurst and Kreimer paper shows how knot crossing numbers correspond to the weight of the MZV. For example, the positive braid in $B_{2}$ defined by the word $\sigma_{1}^{5}$ is decorated with three chords, and this corresponds to $\zeta (5)$ at weight $w = 5$. The trefoil knot $\sigma_{1}^{3}$ is the simplest $B_{2}$ knot, which gives a three loop chord diagram (well, Feynman diagram, actually) using only two chords. The pattern of crossing and non-crossing chords gets more interesting for braids with $s$ strands where $s > 2$, via the relations of the MZV algebra, where depth corresponds to $(s - 1)$ (this is why the example of $\zeta (5)$ only has one argument). Who would have thought it was so easy to do QED with knots and number theory? Once upon a time physicists admitted that group and gauge theory was a complicated, messy business, so why bother with it? M Theory is much more fun. Observe that the number of points on the circle of the chord diagram is $2n$ (or $w + 1$) where $n$ is the number of chords, so $\zeta (5)$ is really a decorated hexagon, our favourite polygon, often used to label the vertices of the three dimensional associahedron.

M Theory Lesson 86

Smolin's slides from Loops07 are now available. Skip the stuff about The Dark Force and look in particular at slide number 38. The important thing to note here is that (a) The Loopies have enlisted the help of none other than Louis Kauffman, an absolutely brilliant knot theorist, and (b) Kauffman has invented something called the Kauffman numbers for three stranded braids, which do this: turning elementary braids into codes of the kind that appear in Carl Brannen's version of the Standard Model (eg. see Carl's comment here). Thus it appears there is a growing consensus that the three generations arise not from more complicated knots, as originally proposed, but rather from the kind of combinatorics that appear in M Theory. Category Theory is not mentioned at all in this work, despite the increasing usage of both knot theory and quantum information language.

Update: Carl Brannen points out that his scheme for the generations is far more advanced than the one outlined in Smolin's talk in later slides. I would have to agree.

M Theory Lesson 59

One of the most information packed 600+ page tomes in the library here is Sphere Packings, Lattices and Groups by J. H. Conway and N. J. A. Sloane. It looks at the 24 dimensional Leech lattice in numerous ways.

Chapter 1 covers the basics of lattice theory and sphere packings. On page 5 it is noted that for a hexagonal plane tiling, rather than basis vectors $(1,0)$ and $(0.5, 0.5 \sqrt{3})$ in two dimensions, it is useful to use the simple three dimensional coordinates $(1,-1,0)$ and $(0,1,-1)$, which lie on the plane $x + y + z = 0$. This basis gives a Dynkin diagram labelling of the $A_2$ root lattice. It corresponds to the densest sphere packing in two dimensions with a density of $\frac{\pi}{\sqrt{12}}$.

The theta function of an integral lattice is related to modular forms. For example, for the Leech lattice the series takes the form

$1 + 196560 q^4 + 16773120 q^6 + \cdots$

which follows from a term for the $E_8$ theta series minus another term which is 720 times the Ramanujan series

$q^2 - 24 q^4 + 252 q^6 -1472 q^8 + \cdots$

Defining $\theta (\tau) = \sum_{- \infty}^{\infty} e^{\pi i \tau n^{2}}$, one can use Riemann's observation that

$\theta (\frac{-1}{\tau}) = \sqrt{- i \tau} \theta (\tau)$

to prove the functional equation for the zeta function. Recall that in M theory this property of the zeta function is intimately related to the physical duality of Space and Time. In a logos style constructive approach to zeta functions it is therefore natural to view lattices and theta series as useful tools for the geometrization of operad polytopes.

Note that the currently popular j invariant may be expressed easily in terms of theta series as

$j (\tau) = 32 \frac{(\theta (\tau)^8 + \theta_{01} (\tau)^8 + \theta_{10} (\tau)^8)^{3}}{(\theta (\tau) \theta_{01} (\tau) \theta_{10} (\tau))^{8}}$

which comes from basic elliptic function theory, but smells of triality, I think.

M Theory Lesson 197

Speaking of platonic groups in neutrino physics, Lieven Le Bruyn beautifully clarifies the story in a post on Galois. As he points out, these three groups, the tetrahedral, octahedral and icosahedral,

in turn correspond to the three exceptional Lie algebras $E_6$, $E_7$, $E_8$ via the McKay correspondence (wrt. their 2-fold covers).

Yesterday we came across $\Gamma (3)$ in connection with the neutrino mixing tetrahedron. Recall that the generating function for $\Gamma (3)$ is $j^{\frac{1}{3}}$, where the dimension of $E_8$ appears in the second term of the expansion. But these connections to the exceptional Lie groups have much more to do with lattices and operads than with strings or toes, as Lieven promises to explain soon. M Theory is the theory that explains the structure of stringy geometry, not the theory that confirms so called stringy physics.

M Theory Lesson 78

The last Riemann post raised again the issue of defining numbers in terms of diagrams. In logos theory one cannot simply pull infinite sets, or real numbers, out of a hat. The rabbit prefers to live in a burrow, where any number represents an enormous variety of objects.

The Euler characteristic of a category was one natural way of assigning rational numbers to finite type diagrams. What about irrational numbers? Legend has it that Pythagoras treated Hippasus rather badly (perhaps by killing him) after Hippasus demonstrated that the square root of 2 was irrational. A proof begins by assuming that $\sqrt{2}$ is rational and hence expressible as an indecomposable ratio of two integers $a$ and $b$, and then derives a contradiction. This relies on the concept of primeness for the ordinals. From the classical assumption that, if P is false, (not P) must be true it is deduced that $\sqrt{2}$ is irrational, the only alternative to being rational. Ah, hang on a minute. In logos theory we don't want to assume that complement is an involution, so the proof doesn't quite work, but there clearly needs to be more than one kind of number. The geometric definition of $\sqrt{2}$ uses two squares of side length 1, which are both cut in half and glued together as shown. This assumes that the correct units for area are $L^{2} = L.L$ where $L$ is a unit of length, but of course $L$ could be any unit. It seems there are an awful lot of assumptions being made to define irrational numbers. In three dimensions in M Theory the complement is a triality. Thus we should probably assign a different numerical status to $\sqrt{2}$ and $\sqrt{3}$. And only when we get to weak $\omega$ categories do we get all the reals.

If we tried to measure a real edge of a square of unit area we could never show it was exactly $\sqrt{2}$ because that would require subdividing the interval further and further until the atomic, or subatomic, structure of the environment altered our sense of the edge, or altered the edge itself so that there was no longer anything to measure.

On the other hand, zeta values show up as soon as one starts thinking about knots and QFT, although the zeta values are treated as abstract basis elements for an algebra over the rational numbers. It is convenient to assign real number values to them in order to evaluate physical quantities, but this comes from the rules of ordinary complex analysis, which we would like to move beyond.

M Theory Lesson 75

So it seems the Morava paper belongs to a productive train of thought, leading to yet more incomprehensible papers by Manin et al. This is the honey that led Manin into a place hard to follow.

But the notion of extended modular operad is clearly a good one. Manin et al solve the problem of the missing 2-punctured moduli $M_{0,2}$ (which makes the operad messy) by replacing all the $M_{g,m}$ with the extended (compactified) spaces $L_{g,m,n}$. The index $n$ labels a second collection of $n$ marked points on the genus $g$ surface. There is a surjective morphism from $M_{0,m+2}$ to the $L_{0,2,m}$, which are toric varieties associated to permutahedra: phew, something we can understand!

So instead of looking for tilings of the ordinary complex moduli $M_{0,m}$ with 2-operad polytopes, we can tile the extended moduli spaces. Loday's geometric realizations of associahedra and permutohedra (from cubes) may come in handy at last. Recall that a 5-leaved labelling of the usual Stasheff associahedron in three dimensions becomes a 4-leaved labelling for permutations in a permutohedron, so Loday's shift in the number of marked points from $M$ to $L$ may clarify the abovementioned surjective mapping.

M Theory Lesson 64

Whilst on the topic of AdS/CFT, Michael Rios has an interesting post on dimension altering weak coupling phase transitions for N=4 SUSY Yang-Mills.

A continuous change in dimension from six down to five is reminiscent of Thurston's beautiful fractal 2-spheres, which are filled with a 1-dimensional curve. These arise in the study of 3-manifolds such as those with 1-punctured torus fibres over the circle. The punctures draw out a boundary for the manifold by tracing a knot. Now according to Matti, the fractional modular domains would fit into the domain for the once punctured torus moduli (the $n=2$ case) on the upper half plane. Perhaps the $n=5$ domain (or rather the theta functions) could be used to model a 5-sphere, much as the j-invariant Belyi map links the $n=2$ domain to $\mathbb{CP}^1$.

Note: For the new readers to this blog, our use of the term M Theory must not be confused with its more popular usage in string theory related papers. The letter M stands here possibly for Motive, or perhaps Monad. Although these terms do appear in the popular literature, they rarely correspond to the physical usage we would like to make of them.