标题： 三维共形场理论中的旋量和扭量形式主义讲座 显示英文标题

标题： Lectures on the Spinor and Twistor Formalism in 3D Conformal Field Theory

摘要： 这些笔记基于我在2025年在IISER Bhopal举行的$\text{ST}^4$讲座的内容。 在这些笔记中，我们研究了旋量和扭量方法在三维共形场理论中的应用。 它们分为三部分，分别涉及旋量动量、扭量和超扭量。 在第一部分中，我们介绍了非定域旋量动量形式，并将其应用于多个上下文，包括双复制关系、与四维散射振幅的联系、Chern-Simons物质理论中的关联函数以及手性高自旋理论的全息对应。 笔记的第二部分介绍了扭量空间形式。 在讨论了扭量空间的几何之后，我们推导了守恒流、任意标度维度的标量以及一般非守恒算符的Penrose变换。 我们还明确展示了旋量和扭量方法之间的关系。 我们讨论了这些算符和守恒流的关联函数在扭量空间中显著简化，揭示了它们隐藏的简洁性。 我们还将我们的构造扩展到超共形场理论，并发展了一个显式的超扭量空间形式，并推导了超对称的Penrose变换。 我们发现超对称的关联函数是其非超对称对应物的简单且自然的推广。 这些笔记旨在自成体系，并包含超过$50$个练习来说明该形式。 显示英文摘要

摘要： These notes are based on my lectures given at $\text{ST}^4$ 2025 held at IISER Bhopal. We study the application of spinor and twistor methods to three dimensional conformal field theories in these notes. They are divided into three parts dealing with spinor helicity, twistors and super-twistors respectively. In the first part, we introduce the off-shell spinor helicity formalism and apply it in several contexts including double copy relations, connection to four dimensional scattering amplitudes, correlators in Chern-Simons matter theories and the holography of chiral higher spin theory. The second part of the notes introduces the twistor space formalism. After discussing the geometry of twistor space, we derive the Penrose transform for conserved currents, scalars with arbitrary scaling dimension as well as generic non-conserved operators. We also explicitly show how the spinor and twistor approaches are related. We discuss how correlators of these operators and conserved currents in particular drastically simplify in twistor space unveiling their hidden simplicity. We also extend our construction to super-conformal field theories and develop a manifest super-twistor space formalism and derive the supersymmetric Penrose transform. We find that the supersymmetric correlators are simple and natural generalizations of their non-supersymmetric counterparts. The notes are made to be self-contained and also include over $50$ exercises that illustrate the formalism.