AI 中文总结
本文通过SL(2,R)×SL(2,R)陈-西蒙斯理论的欧几里得路径积分，构造了有限k下BTZ黑洞的量子微态，解析延拓后将永恒洛伦兹黑洞对应为模系综，并重现了半经典面积涨落。
AI 中文摘要
考虑半无限实心圆柱上的k级SL(2,R)×SL(2,R)陈-西蒙斯理论，在空间圆盘中心沿时间方向插入一条属于幺正主连续系列的威尔逊线。我们论证，在τ=0割处，该欧几里得路径积分制备了BTZ阈值之上重Virasoro基元的体对偶，其中Drinfel'd-Sokolov（DS）归约在将仿射模转换为Virasoro模的过程中发挥关键作用。半经典层面，DS约束转化为DS规范，这是渐近AdS₃引力中熟知的内容。这些基元结合边界引力子，为有限k下的BTZ黑洞微态提供了体构造。在大k的WKB极限下，控制单个基元的鞍点是具有奇异视界的BTZ黑洞。我们研究微态的（巨）正则配分函数，其态密度由模不变性决定。温度和化学势随后将完整的鞍点固定为具有平滑可收缩热循环的欧几里得BTZ黑洞。重基元的态密度按e^(2πγQP)增长，这会导致缺陷（或过剩）鞍点，除非γ=1，而γ=1是由模不变性固定的Cardy值。经过解析延拓，具有平滑视界的永恒洛伦兹黑洞因此不只是任意热系综，而是（半经典上奇异的）无视界微态的“模系综”。我们计算该系综中的热涨落，并重现半经典广义相对论预期形式的面积涨落。在我们的所有讨论中，理论系综不发挥直接作用。我们还注意到，这些微态与模糊球不同，不会破坏黑洞的等距性。
英文摘要
Consider level-$k$ $SL(2,R) \times SL(2,R)$ Chern-Simons theory on a semi-infinite solid cylinder, with a Wilson line in the unitary principal continuous series inserted along the time direction at the center of the spatial disc. We argue that at the $τ=0$ cut, this Euclidean path integral prepares the bulk dual of a heavy Virasoro primary above the BTZ threshold, with Drinfel'd-Sokolov (DS) reduction playing a crucial role in converting the affine module into a Virasoro module. Semi-classically, the DS constraint turns into the DS gauge, familiar from asymptotically AdS$_3$ gravity. Together with boundary gravitons, these primaries provide a bulk construction of BTZ black hole microstates at finite $k$. In the large-$k$ WKB limit, the saddle that controls an $individual$ primary is the BTZ black hole with a $singular$ horizon. We study the (grand-)canonical partition function of the microstates with density of states dictated by modular invariance. The temperature and chemical potential then fix the holonomy saddle to be the Euclidean BTZ black hole with a $smoothly$ contractible thermal cycle. A density of heavy primaries growing as $e^{2πγQP}$ would instead lead to a defect (or excess) saddle, except at the Cardy value $γ=1$ fixed by modular invariance. After analytic continuation, the eternal Lorentzian black hole with a smooth horizon is therefore not just any thermal ensemble, but the ``modular ensemble" of (semi-classically singular) horizonless microstates. We compute the thermal fluctuations in this ensemble, and reproduce the fluctuations in the area of the form expected from semi-classical general relativity. Ensembles of theories do not play a direct role in any of our discussions. We also note that these microstates, unlike fuzzballs, do $not$ break the isometries of the black hole.
Comments49+9 pages
