Thursday, June 3, 2010

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.


  1. "...and after this neither string theory nor the usual local formulation of the Standard Model made any appearance whatsoever."

    I don't quite understand what you're getting at here. The recent developments, in which twistor theory has played a big role, are to do with scattering amplitudes in field theories with 16 or 32 supercharges -- nothing directly to do with string theory or the standard model, or an attempt to replace either.

  2. On the contrary, Rhys, they are very much about replacing both.

  3. Do you mind elaborating? Perhaps I have misunderstood what is going on.

  4. Rhys, for decades now the twistor theorists have tried to understand scattering amplitudes using twistor geometry, as opposed to the usual messy Feynman picture. In the last year or two, there have been a number of interesting breakthroughs. The Hodges' diagrams are now better understood, and amplitudes that were previously intractable may now be computed. I recommend the introductory lectures by Arkani-Hamed at Pirsa.

    For further motivation from gravity, I also recommend reading my thesis some time. The combinatorics of BCFW recursion relations are closely related to the associahedra polytopes whose motivic cohomology gives rise to, for instance, the Veneziano n-point functions. This brings us back to an S matrix picture, with roots in the particel physics programs that existed before quarks made the SM king.

  5. I really do think that these twistorial developments are continuous with string theory, rather than an overthrow of it. N=4 SYM and N=8 SUGRA are limits of string theory, or, to put it another way, string theory is a possible UV completion of these field theories. So it must be possible to ask what happens to the change of variables and the new perspective when we add the string degrees of freedom, and I rather doubt that the answer is "inconsistent, you can't do it".

    The aspect of this that I've read about is an object that Arkani-Hamed calls L(n,k), which is a product of a twistor-valued delta function with an integral over a "Grassmannian manifold" G(n,k), which itself has an intriguing history. If you view the talks from Strings 2010 of a few months ago, you'll see that Juan Maldacena of AdS/CFT fame is working on complementary issues to do with Wilson loops in AdS space.

    Speaking more speculatively (i.e. about issues where my personal ignorance is even greater), it's rather interesting that 12-dimensional F-theory formally has a two-time signature, just like twistor space. So I wonder if it also offers the proper framework for extrapolating the twistorial perspective into the full superstring theory.

  6. Mitchell, many people would agree with you, but I hope you can see that these developments do alter the position of string theory in physics, in that it is no longer absolutely fundamental in and of itself. I agree that it is 'still there', as is the SM, which it must be in order to convince physicists that it is the right way to go.


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