全息引力与纠缠熵

The AdS/CFT correspondence deeply reveals the holographic nature of gravity, providing a new approach for understanding quantum gravity. One of the most important applications is the concept of the holographic entanglement entropy: the entanglement entropy in QFT could be obtained through evaluating the area of a minimal surface in the dual AdS spacetime, in which a deep relation between spacetime geometry and quantum entanglement is hinted. In the present project, we would like to explore how the gravity emerges from the entanglement structure in the dual CFT. Three aspects of research are planned as following. First, rebuild the full non-linear equation of gravity dynamics through the “first law” of the entanglement entropy in quantum local quench process. Second, define the metric of the holographic extra dimension locally through the renormalization group flow of the entanglement density. Third, explain the origin of the thermal entropy of higher spin black holes through entanglement entropy by generalizing the concept of differential entropy to higher spin gravity. These studies would shed new lights to the research of realizing gravity through entanglement entropy and thus helpful for further understanding of quantum gravity.

引力/规范对偶深刻揭示了引力的全息性质，为认识量子引力开拓了一条新的途径。其重要应用之一是全息纠缠熵的提出：量子场论中的纠缠熵可以通过计算对偶时空中极小曲面的面积而得，这一方案隐含了时空几何与量子纠缠的深层联系。本项目探究引力如何由对偶共形场论的纠缠结构浮现而来。我们计划开展三个方面的研究：一是非线性引力动力学方程的重建，由量子局域淬火过程的纠缠熵所满足的热力学第一定律导出；二是全息额外维几何的局域定义，通过纠缠密度的重正化群流来实现；三是利用纠缠熵来解释高自旋黑洞熵的起源，方法是将微分熵的概念推广到高自旋引力中。这些研究可为通过纠缠熵认识引力本质带来新的线索，有助于更深入地理解量子引力。

本项目利用 AdS/CFT 对应探讨了三维及四维具有挠率的引力理论的量子性质。对于四维 Einstein-Cartan-费米子理论，我们论证了体纠缠楔形区域内的准局域能量与边界上球形区域内的全息相对熵的等价关系；由量子场论中的相对熵的正定性，我们获得了轴矢流-挠率耦合参数、费米子质量及物态方程之间的新限制。我们也发展了在具有挠率的引力理论的一阶形式体系下的 Wald 形式体系，并用之计算了三维 Mielke-Baekler 引力的准局域能量。对于 Einsten 引力及手征引力中的 BTZ 黑洞，我们发现在类光能量条件满足时弱宇宙监督假设总成立；而对于 Mielke-Baekler 引力，由于挠率耦合的存在，向 BTZ 黑洞丢入物质可能导致裸圆锥奇点的出现。