### 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!

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!

## 3 Comments:

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.

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.

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.

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