Differential geometric study of conformal and CR invariant theory via the ambient metric construction due to Fefferman and Graham

通过 Fefferman 和 Graham 的环境度量构造对共形和 CR 不变理论进行微分几何研究

• 批准号: 61601504
• 负责人: Privatdozent Dr. Felipe Leitner
• 金额: --
• 依托单位: Lehrstuhl für Differentialgeometrie
• 依托单位国家: 德国
• 项目类别: Priority Programmes
• 财政年份: 2007
• 资助国家: 德国
• 起止时间: 2006-12-31 至 2007-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/61601504?language=en
• 关键词: Differential; geometric; study; conformal; CR

Conformal and CR geometries are rigid structures, which are distinguishable by curvature and integral invariants. The invariant theory can be studied via parabolic Cartan geometry/tractor calculus. An alternative approach developed by Fefferman/Graham is the ambient metric construction, which is basically equivalent to the Poincare-Einstein model. Via these constructions invariants such as the GJMS-operators and Q-curvature can be defined for the purpose of purely mathematical studies in conformal and CR geometry. Moreover, "holographic relations" of objects on the (conformal) boundary and the interior "bulk" of the Poincare model can be established. This is of interest in physics, where the holographic principle (of quantum gravity) finds a concrete manifestation in connection with the AdS/CFT-correspondence, which aims to relate string theory/sup er gravity with sup er symmetric conformal field theories. This project considers as its main goal (geometric) realisations of Fefferman-Graham ambient and Poincare-Einstein models in curved situations. For such models explicit expressions for conformal and CR invariants (which are otherwise formally defined) shall be calculated. We aim to establish "holographic" relations between objects on the boundary and on the interior of the Poincare-Einstein model. In particular, we ask for Taylor expansions of Poincare spaces and explicit formulae for GJMS-operators and Q-curvature. A further question concerns geometric Poisson transformations for harmonic solutions of boundary problems on curved Poincare models in terms of integral formulae. It is also a task to relate symmetries such as solutions of overdetermined invariant differential equations (e.g. twistor forms/spinors) on the boundary to geometric objects on the Poincare "bulk".

共形几何和CR几何都是刚性结构，它们可以通过曲率和积分不变量来区分。不变量理论可以通过抛物卡坦几何/拖拉机微积分来研究。Fefferman/Graham提出的另一种方法是环境度量结构，它基本上相当于庞加莱-爱因斯坦模型。通过这些构造，诸如gjms算子和q曲率之类的不变量可以定义为保形几何和CR几何的纯数学研究的目的。此外，还可以建立（共形）边界上的物体与庞加莱模型内部“体”之间的“全息关系”。这在物理学中很有趣，全息原理（量子引力）在AdS/ cft对应中找到了具体的表现，其目的是将弦论/重力与对称共形场论联系起来。该项目将在弯曲情况下实现费费曼-格雷厄姆环境模型和庞加莱-爱因斯坦模型作为其主要目标（几何）。对于这些模型，应计算共形不变量和CR不变量的显式表达式（否则将被正式定义）。我们的目标是在庞加莱-爱因斯坦模型的边界和内部建立物体之间的“全息”关系。特别地，我们要求得到庞加莱空间的泰勒展开式以及gjms算子和q曲率的显式公式。另一个问题是关于曲线庞加莱模型边界问题调和解的几何泊松变换的积分公式。将边界上的对称性（如超定不变微分方程的解（如扭转形式/旋量））与庞加莱“体”上的几何物体联系起来也是一个任务。