标题： Karch-Randall braneworld中的复制虫洞和纠缠岛 显示英文标题

标题： Replica Wormholes and Entanglement Islands in the Karch-Randall Braneworld

摘要： Karch-Randall膜世界提供了一个自然的框架，利用全息工具研究黑洞的霍金辐射。 这样的黑洞存在于膜上，高度量子化但仍具有一个高维经典理论的全息对偶，该理论存在于周围空间。 此外，这样的黑洞与一个非引力浴耦合，该浴吸收其霍金辐射。 这使我们能够通过研究浴使用量子极值面处方来计算霍金辐射的熵。 量子极值面在周围空间几何化为Ryu-Takayanagi面。 Ryu-Takayanagi面在时间上的拓扑相变，从连接浴的不同部分到连接浴和膜，给出了与幺正性一致的霍金辐射的Page曲线。 然而，在Karch-Randall膜世界中不存在量子极值面处方及其几何化的推导。 在本文中，我们填补了这一空白。 我们主要关注周围空间为(2+1)维的情况，在这种情况下可以在每个描述中进行显式计算。 我们表明，Ryu-Takayanagi面的拓扑相变对应于Karch-Randall膜上复制虫洞的形成，作为复制路径积分的主要贡献。 对于更高维情况，我们表明膜的几何满足与共形物质耦合的爱因斯坦方程。 我们评论了这个方程可能对引力路径积分的一般规则产生的影响。 显示英文摘要

摘要： The Karch-Randall braneworld provides a natural set-up to study the Hawking radiation from a black hole using holographic tools. Such a black hole lives on a brane and is highly quantum yet has a holographic dual as a higher dimensional classical theory that lives in the ambient space. Moreover, such a black hole is coupled to a nongravitational bath which is absorbing its Hawking radiation. This allows us to compute the entropy of the Hawking radiation by studying the bath using the quantum extremal surface prescription. The quantum extremal surface geometrizes into a Ryu-Takayanagi surface in the ambient space. The topological phase transition of the Ryu-Takayanagi surface in time from connecting different portions of the bath to the one connecting the bath and the brane gives the Page curve of the Hawking radiation that is consistent with unitarity. Nevertheless, there doesn't exit a derivation of the quantum extremal surface prescription and its geometrization in the Karch-Randall braneworld. In this paper, we fill this gap. We mainly focus on the case that the ambient space is (2+1)-dimensional for which explicit computations can be done in each description of the set-up. We show that the topological phase transition of the Ryu-Takayanagi surface corresponds to the formation of the replica wormhole on the Karch-Randall brane as the dominate contribution to the replica path integral. For higher dimensional situations, we show that the geometry of the brane satisfies Einstein's equation coupled with conformal matter. We comment on possible implications to the general rule of gravitational path integral from this equation.