occasional meanderings in physics' brave new world

Name:
Location: New Zealand

Marni D. Sheppeard

## Monday, June 22, 2009

### 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!

kneemo said...

Ok, so the eight dimensional space you refer to seems to be the CR ambitwistor space A_E of complex null geodesics that intersect E=R^4, where A_E is homeomorphic to R^4 x CP^1 x CP^1.

June 24, 2009 4:24 AM
Kea said...

Right, so I guess that with 3 copies there would be 6 lots of CP^1 giving 12 out of the 24 dimensions of the Leech lattice. Inside there I suppose there is some nice 6d (real) torus.

June 24, 2009 8:20 PM
kneemo said...

As the 12-dimensional projective ambitwistor space is A=CP^3 x CP^3, it's possible to embed A in a 34-dimensional Freudenthal triple system (FTS) over J(4,C), e.g., F=J(4,C)+J(4,C)*+R+R, where the CP^3's are the spaces of projectors for J(4,C) and it's dual J(4,C)*. The automorphism group of the FTS would then correspond to the conformal group of A and one can proceed to classify orbits and attractors by reading off the ranks of the FTS.

June 25, 2009 4:58 AM