# Twistor theory The twistor theory is a non-local reorganisation of physical information in terms of geometric data. It is therefore intrinsically holographic in a certain sense. For example, the conformal structure of spacetime is manifested as holomorphic structures of the twistor space. ## Refs - original [[Penrose1967]] - good for learning - [[Rsc0053 Tim Adamo Lectures on Twistor Theory 2017]] - [[book_HuggettTod]] - standard reference [[book_PenroseRindler]] - [[book_WardWells]] more mathematical - integrability oriented: [[book_MasonWoodhouse]] and [[book_Dunakski]] - review [[review_MacCallumPenrose1972]] - useful intro - recent historic overview [[AtiyahDunajskiMason2017]] - for CCFT - [[2022#Bu, Casali]] etc - [[0421 Higher-spin gravity|higher spin]] - for chiral higher-spin gravity in AdS: [[Tran202209]][](https://arxiv.org/pdf/2209.00925.pdf) - review [[Tran202211Lectures]][](https://arxiv.org/pdf/2211.10484.pdf) - in dS or AdS - [[2024#Baumann, Mathys, Pimentel, Rost]] ## Twistor space - the twistor space, $\mathbb{P}\mathbb{T}$, is an open subset of $\mathbb{C}\mathbb{P}^3$ - $Z^I\sim sZ^I$, $s$ being complex - $Z^I=\left(\mu^{\dot{\alpha}}, \lambda_\alpha\right)$ - the complex line $\lambda=0$ is removed - relation to spacetime coordinates, known as the *incidence relation* - $\mu^{\dot{\alpha}}=x^{\dot{\alpha} \alpha} \lambda_\alpha$ ## Applications - [[Witten200312]][](https://arxiv.org/abs/hep-th/0312171) - resurrection of twistors - combines string perturbation theory to calculate the entire tree-level S-matrix of YM in 4D ## Signature dependence The number of degrees of freedoms of the twistors depend on the dimension and signature of the spacetime. In [[0056 Split signature|split signature]] (2,2), the spinors are real and independent.