M Theory Lesson 336

Recall that the two factor mixing products corresponded to a new kind of magic matrix. For example, the tribimaximal values may be rearranged as where the number $6$ replaces the zero entry of the matrix. Such magic squares have both sum and product rules. For example, the tribimaximal values give us
$\frac{2}{3} \times \frac{1}{2} = \frac{1}{3} $
$\frac{1}{2} \times \frac{1}{3} = \frac{1}{6} $.
Similarly, the CKM values give us the product rules
$(ud) \times (tb) = (cs) $
$(us) \times (cb) = (td)$.
The probability matrices are obtained from these magic squares by rearranging only a few values. One way that we might draw such a magic square is:

Quote of the Decade

From the current session of U.S. Congress:

Quote of the Week

Eileen Byrne points out that if a plant doesn't succeed in a garden, we ask what it is about the soil, water, sun or fertilizer that is causing the problem. We don't blame the plant first.

M Theory Lesson 335

Carl Brannen's spin path integral paper is now available online at Foundations of Physics.

The spin path integral is closely associated with ideas of the mathematician Kapranov about noncommutative integrals, as discussed in his 2006 paper. Recall that Kapranov uses noncommuting directions to build paths, just as in Carl's paper. At the Streetfest in 2005, Kapranov discussed categorical aspects of noncommutative integrals. Now Carl's paper directly links the particle generations to the third categorical dimension. Long live the parity cube and broken monoidal structures.

Twistors Again

Last year at Oxford, I was fortunate enough to attend a number of interesting String Theory seminars in the mathematics department. Actually, the seminars were all about twistor theory, because the local string theorists said that twistor theory was where all the interesting stuff was happening. A typical seminar would begin with a T duality transformation into a twistor space, and after this neither string theory nor the usual local formulation of the Standard Model made any appearance whatsoever.

Recall that algebraic aspects of twistors include the relation between its geometry and Jordan algebra. As kneemo points out, the Koide mass matrices may be viewed as elements of a Jordan algebra. Similarly, the $R_2$ factors used to build mixing matrices may be associated to Jordan algebras.

Some time ago, we discussed the relevance of the three moduli spaces of twistor dimension (namely $M(0,6)$, $M(1,3)$ and $M(2,0)$) with a total of $12$ degrees of freedom. Euler characteristics and the index theorem for the six point sphere then tell us that the number of generations must be $3$. This is a ternary analogue of the pair $M(0,3)$ and $M(1,1)$ of the Grothendieck tower, an idea associated to the ribbon graph papers of Mulase et al.

New Rios Paper

On the subject of information theoretic black holes, Michael Rios has a new arxiv preprint: Jordan $C^{*}$ algebras and Supergravity. From the conclusion:

Surely there are further applications for Jordan algebraic structures based on the bioctonions, and it is interesting to consider the direct physical interpretations of such structures in M Theory.

Best Maths Blogs

Lieven Le Bruyn bemoans not being selected one of the Best 50 Maths Blogs, for maths majors, at onlinedegree. But Arcadian Functor made the list!