# An octahedron of complex null rays, and conformal symmetry breaking

## TL;DR

This paper explores the geometric structure of twistor space and its relation to conformal symmetry breaking, revealing an octahedral configuration of complex light rays and its implications for loop quantum gravity.

## Contribution

It demonstrates how the manifold $T^*SU(2, 2)$ can be obtained via symplectic reduction from twistor space and introduces a mechanism for conformal symmetry breaking relevant to quantum gravity.

## Key findings

- Identification of $T^*SU(2, 2)$ as a symplectic reduction of twistor space
- Description of an octahedral configuration of complex light rays
- Proposal of a symmetry breaking mechanism to $T^*SL(2, C)$

## Abstract

We show how the manifold $T^*SU(2, 2)$ arises as a symplectic reduction from eight copies of the twistor space. Some of the constraints in the twistor space correspond to an octahedral configuration of twelve complex light rays in the Minkowski space. We discuss a mechanism to break the conformal symmetry down to the twistorial parametrisation of $T^*SL(2, C)$ used in loop quantum gravity.