Theory Update 134

In the new lectures, Arkani-Hamed reviews basic twistor geometry. As usual we work with the $N = 4$ theory, because that is the amount of supersymmetry in the complex numbers $C$. The bioctonion case of $C \cdot C$ allows us to discuss $N = 8$ supergravity. In the lectures, traditional supersymmetry is used to obscure the categorical structure of the combinatorics. Recall that a trivalent vertex may be used to represent a point associahedron, and comes in two allowed helicity triplets.
The counting of $\textrm{N}^{k}\textrm{MHV}$ terms for $n$ legs is given by the Catalan number of the appropriate associahedron. The distinct trivalent vertices may then define a directed associahedron edge, via their representation as points.

Arkani-Hamed stresses the importance of Grothendieck's mathematics and other motivic mathematics, such as the Connes-Kreimer approach to renormalisation. However, there still appears to be a belief that this will all work out without any abstract airy fairy category theory. Although he admits a previous long term allergy to all things motivic, there is obvious excitement about the new connections to the interests of mathematicians like Deligne. Unofficially, the lectures are about the positive Grassmannian (which is, of course, all about generalised associahedra).