M Theory Lesson 282

Recall that the vertices of associahedra are described by all rooted binary trees of a certain height, such that degeneracies in the level of the nodes is permitted. For example, for trees with three vertices (including a root) there must be four leaves, and we obtain the five vertices of a pentagon. The edges of the pentagon are labelled by trees with only two nodes, which are the contractions of the trees on the boundary vertices. And the face of the pentagon itself is labelled by the single vertex four leaved tree. The associahedron for two leaves is a point and the associahedron for three leaves is a single edge. For each real dimension, there is an associahedron.

What about ternary trees? First observe that the real dimension must increase by two at each step, because ternary vertices increase the number of leaves by two at each branching. The first two ternary polytopes are described by the following trees. The second case has three points on a surface, with no marked edges, just like a Riemann sphere. The next case naturally lives in dimension four, so we only draw the seven leaved trees marking the $12$ points:

Magic Matrix

Philip Gibbs has now provided a webpage with his solution to the problem of showing that any $3 \times 3$ unitary matrix can be turned into a magic matrix by multiplication of its rows and columns by phase factors.

Mixing History

Carl just showed me these interesting papers, which should have been discussed earlier:
(1) Neutrino Mixing with Delta(27)
(2) A4 Symmetry and Neutrinos
both by Ernest Ma.

A Preprint

I think I'll leave it to Carl Brannen to put our four page preprint on mixing matrices on his website. We eagerly await referee reports.

M Theory Lesson 281

Laurent Manivel, of the CNRS, has a paper that discusses how the hyperdeterminant arises as the restriction of a quartic form for the Weyl group $W(E_7)$. Here $E_7$ means the root system of that name.

It turns out that we should be interested in a product of seven copies of $A_2$ with the automorphism group of the Fano plane. The latter group is the nice group $PSL(2, F_7)$. The hyperdeterminant is a quartic form for an eight dimensional space $A_{ijk}$ that appears in a $56$ dimensional representation of $E_7$ made of seven copies of the form $A_{ijk}$ for different $i$, $j$ and $k$. The entanglement of qudits really does have a lot of wonderful geometry associated to it.

M Theory Lesson 280

According to Oeding, the Lagrangian Grassmanian of an even dimensional symplectic space $V$ is the image of a map $f$ that takes a symmetric matrix and gives a vector of minors. There is a projection from the Grassmanian onto the variety of principal minors of all $n \times n$ matrices.

This is interesting because minors are a natural way to describe pure states in quantum mechanics. Consider a three qubit state with $8$ amplitudes. Forgetting about $a_{000}$, which we can set to $1$ projectively speaking, and letting $a_{111}$ be related somehow to the determinant of a matrix, it turns out that the other six amplitudes should be expressed as the principal minors of the matrix which has full determinant given by the entanglement measure (Cayley's hyperdeterminant)

The String Wars

Gil Kalai has written a book, entitled "Gina Says" - Adventures in the Blogosphere's String War, based on his experiences as a poster named Gina. It is full of gems, including many admirable quotes, such as:

"Gina's comments are blocked on my blog because she was posting a large number of comments there, while most of the time clearly not understanding what she was writing about", P. Woit, Backreaction 5:59, December 27, 2006

"Gina, you and quite a few others seem confused about the meaning of higher dimensions." Thomas Love, September 28th, 2006 at 2:16 pm

This could be a bestseller!

Ambitwistor Holography

One of the interesting twistor ideas that I have been hearing about lately is the Ambitwistor Lagrangian of Mason and Skinner. They give an integral for (an $N = 3$) supertwistor space, over an $8$ dimensional form, that is defined in terms of a Chern-Simons piece along with supersymmetric twistor forms.

Note that the $8$ dimensions comes from a light like component of the $10$ dimensional ambitwistor component of the $12$ dimensional twistor space for $(Z,W)$. The fermionic coordinates satisfy $(\psi \cdot \eta)^4 = 0$ (just think of the quantum Fourier transform), which is responsible for the condition $(Z \cdot W)^4 = 0$, associated to Yang-Mills solutions.

Although Lagrangians cannot possibly be fundamental in a nonlocal theory, this is pretty interesting when one thinks about three copies of it. Recall that the $24$ dimensions (and $24 = 3 \times 8$) of the CFT for the $26$ dimensional bosonic string theory is associated with the Leech lattice and the Monster group and other moonshine maths!

Emerging Holography

Last week's amazing twistor workshop ended Friday with an outstanding physics colloquium by Nima Arkani-Hamed, called Holography and the S matrix, but secretly about computing scattering amplitudes using twistor spaces.

He went to some effort to try to convince a large audience of theoretical physicists that there was a mysterious new, mind blowing holographic theory behind these magical simplifications in scattering amplitudes for both Yang Mills and gravity. However, unlike serious fans of thermodynamic gravities (for instance, Padmanabhan) he didn't seem in favour of a microscopic theory of gravity that was wildly different from string theory.

Some time was spent criticising the Standard Model emphasis on manifest locality, when locality should be an emergent property. In the fantastic results so far, twistor space is clearly doing holography for us, but there is a long way to go before emergent locality is properly understood. After all, if we can remove spacetime from particle physics, why not its boundaries too?

Quote of the Week

We believe that [this formula] encapsulates the complete n-particle tree level S-matrix of YM theory (for any gauge group) ... [we] highlight a crucial fact about the formula: namely, that it is not really an integral at all.

Roiban et al

Jordan M Theory

Baez decided to learn M Theory, and asked for some hints on how the exceptional Jordan algebra might appear in an $11$ dimensional theory, prompting some helpful advice from two people named Lubos and Kea. The whole conversation was of course quickly deleted, although I don't recall it containing any direct personal insults. Anyway, here is a fresh link to kneemo's blog, who I am quite sure knows far more about this question than anybody else.

A New Home

In good news from down under, after a successful program of pest control on Raoul Island, many of my kakariki friends have decided to make their homes there again, after 150 years.

Twistor Time

It is very difficult to keep up with arxiv preprints these days, but since kneemo hasn't mentioned it yet, in this new paper Arkani-Hamed et al study the twistor diagrams of Hodges. As the abstract states:

Our twistor transformation is inspired by Witten's, but differs in treating twistor and dual twistor variables more equally. In these variables the three and four-point amplitudes are amazingly simple.

They refer in particular to this new paper by Mason and Skinner.

A Question

Today at lunch I was asked one of the questions that nonsense theorists are often asked: so what does this have to do with the real world? Of course, one could always launch into a (now fashionable) tirade about protocols for quantum information, or two dimensional systems and topological quantum field theories. However, since the conversation was set more in the context of quantum gravity, and the asker was mostly looking for a very simple, one line answer (after having already suffered a five minute introduction to category theory), I was at a loss to find the right words.

So here is the challenge: can you summarize categorical quantum gravity in 20 catchy words or less? We assume that our readers will not be captivated by statements along the lines of Everything is made of Strings or, more pertinently, the speed of light varies (although that is, of course, true). Rather, the phrase should capture the potential of quantum gravity to describe aspects of the world completely outside the domain of established physical theory.

A Pi Groupoid

Recall that the cardinality of a groupoid involves the inverse of the cardinalities of groups. At PI, Jeff Morton told me about a very nice example involving, for instance, the cyclic groups $C_{n} \times C_{n}$, which each have cardinality $n^2$. That is, we can have a cardinality $\pi^2$, because

$\pi^{2} = 6 \sum_{k} \frac{1}{k^2}$.

Recall that this infinite sum is the number $\zeta (2)$ for the Riemann zeta function, first evaluated by Euler in 1735. Since $e$ is also a groupoid cardinality, namely for the groupoid of finite sets and bijections, it seems that transcendentals naturally appear in the context of infinite groupoids.

Cool Cats

Since the cool cats conference in Canada, I have been catching a few more Oxford seminars. Yesterday, Andrew Dancer spoke about Frenkel's loop group version of the Langlands Correspondence. He noted the category theorists in the audience, promising to discuss some category theory, but of course there was very little category theory and I only obtained the usual miniscule improvement in my understanding of this subject.

Meanwhile, I'm off to a lovely old college for lunch and it's a (relatively unusually) beautiful day here!

Back in Oxford

The PI conference is over and I have returned to Britannia. There were some excellent talks, but more importantly, there was time for discussion in the afternoons. Jeff Morton has started blogging the talks. I would like to say more, but am currently consumed by other tasks, such as fixing my computer account (which was automatically deleted on a previously valid expiry date).

CQC Monday

Day One of CQC at PI consisted of talks until lunch, a leisurely two hour break and then an afternoon question time, which quickly degenerated into a discussion on the difference between instrumental approaches (ie. the background independent point of view in the context of diagrammatic QM, roughly speaking) and foundational approaches to physics, the latter applying to proper theories (the definition of this unfortunately relying on very few previously known examples, such as GR). Anyway, hopefully the amount of discussion sets the tone for a pleasant week.

The weather, on the other hand, does not look like improving.

Rejecta Mathematica

Good News! The old short paper on Koide masses and the quantum Fourier transform has been accepted by Rejecta Mathematica, which only publishes works that have been rejected by respectable journals. Of course, given the ridiculous time frame for physics publishing, this paper is now hopelessly outdated by the enormous progress since made by Carl Brannen, who will no doubt have many papers published soon.