全纯共形Cartan几何：共形引力、扭量与双扭量
Holomorphic Conformal Cartan Geometry: Conformal Gravity, Twistors, and Ambitwistors
- Department of Mathematics and Statistics, Masaryk University（马萨里克大学数学与统计系）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文在全纯共形Cartan几何框架下统一了四维共形引力、扭量与双扭量理论，构造了Cartan联络与tractor丛，证明了二次曲率作用量重现Weyl–Bach引力，并揭示了扭量空间与双扭量空间的几何意义。
AI中文摘要：
在统一的全纯共形Cartan几何框架下，综述了四维共形引力、扭量理论和双扭量理论。从几何的自旋表示出发，系统地构造了取值为$\nathfrak{sl}(4,\mathbb{C})$的规范Cartan联络、其曲率$2$-形式以及相关的tractor丛。在实Lorentzian切片上求值，二次曲率作用量在固定Weyl规范后被证明可重现四维Weyl–Bach共形引力。局部扭量和双扭量丛被构造为相伴的自旋tractor丛，说明了平行移动如何在平坦空间中产生经典关联关系，并在弯曲背景中产生手性可积性约束：对$\alpha$-曲面为反自对偶，对$\beta$-曲面为自对偶。此外，通过不变典范配对将局部扭量和双扭量配对，双扭量空间被证明可参数化复零测地线，并对称地容纳左手和右手Weyl曲率。该框架建立了连接共形Cartan几何与扭量及双扭量理论的直接规范理论桥梁。
英文摘要：
Four-dimensional conformal gravity, twistor theory, and ambitwistor theory are reviewed within the unified framework of holomorphic conformal Cartan geometry. Starting from the spinorial representation of the geometry, the normal $\mathfrak{sl}(4,\mathbb{C})$-valued Cartan connection, its curvature $2$-form, and the associated tractor bundles are systematically formulated. Evaluated on the real Lorentzian slice, the quadratic curvature action is shown to reproduce four-dimensional Weyl--Bach conformal gravity upon fixing a Weyl gauge. Local twistor and dual twistor bundles are constructed as associated spin tractor bundles, illustrating how parallel transport yields the classical incidence relations in flat space and chiral integrability constraints in curved backgrounds: anti-self-duality for $α$-surfaces and self-duality for $β$-surfaces. Furthermore, by pairing local twistors and dual twistors via an invariant canonical pairing, ambitwistor space is shown to parameterize complex null geodesics and symmetrically accommodate both left- and right-handed Weyl curvatures. This framework establishes a direct gauge-theoretic bridge linking conformal Cartan geometry to twistor and ambitwistor theories.
