## Tuesday, July 5, 2011

### Theory Update 89

As Mitchell points out, on page $91$ of the Witten and Gaiotto paper we see a pair of M5 branes. Recall that such a tetractys stands for the commutative form of length three qutrit paths, and is thus associated to the $3 \times 3$ $27$ dimensional Jordan algebra over the octonions. A necessary complexification leads to the $54$ dimensional bioctonion algebra.

There are no $11$ dimensional spacetimes in sight when discussing qutrits: just a few categorical diagrams and some very nice algebra. Why make M Theory harder than it needs to be?

1. You know, if they would just rename Strings 2012 to Braids 2012, I think that would be a pleasant conference to attend.

2. There's a lot of your kind of mathematics being applied in this paper to theories descended from M5-branes - theories in 2+1 dimensions, as it turns out; the M5-branes are compactified on the product of a circle and a Riemann surface. Some highlights:

Associahedra classify triangulations of the Riemann surface, each of which specifies a set of "WKB curves" which minimizes the tension of a string on an M5-brane (the string in question is actually one end of a topologically cylindrical M2-brane stretching between two M5-branes), and corresponds to a "BPS state" whose mass is exactly calculable.

Moduli spaces of these theories are described using "Fock-Goncharov coordinates".

All of this is used to rederive a "wall-crossing formula" of motivic origin. In sction 12 they discuss how their methods might be described categorically.

In reference 11, one of the formulae potentially derived from a simple associahedron is used to calculate gluon scattering amplitudes.

3. Yes, they really should have cited my thesis and my blog, shouldn't they? And it is a highly cited paper. You have not been paying attention, Mitchell.