富克斯共振与极点跳过的视界-边界字典
Fuchsian Resonance and a Horizon-to-Boundary Dictionary for Pole-Skipping
- POSTECH（浦项工科大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
该研究建立了富克斯共振与极点跳过的视界-边界字典，通过分析黑洞背景下的标量径向方程，揭示了共振时视界解空间的结构及边界条件与视界性质的对应关系。
中文摘要 AI 辅助
极点跳过发生在特殊的复频率和动量处，此时黑洞或黑膜背景的推迟格林函数无法被唯一确定。局部近视界机制是众所周知的：在共振时，弗罗贝尼乌斯递推矩阵失去秩，光滑视界解的空间会扩大。我们研究了一个全局问题：两个边界归一化解如何进入这个共振视界解空间。对于在视界附近解析的非极端背景上的一般二阶标量径向方程，共振阶的弗罗贝尼乌斯递推会产生一个可解性条件。我们证明，该条件的消失等价于视界递推矩阵的奇异性、对数弗罗贝尼乌斯项的缺失，以及两个独立光滑视界解的存在。在远离共振时，源零点等价于响应归一化解的视界光滑性，而响应零点等价于源归一化解的视界光滑性。随后，我们分析非共振入射解依赖于参数的简单极点在趋近共振时的行为。在移除该奇异性后，其共振极限与较大根的弗罗贝尼乌斯解成正比，比例因子由同一个可解性函数给出。延拓至边界表明，相同的条件等价于两个正则化边界连接系数同时消失。这建立了局部富克斯共振与极点跳过的边界源-响应0/0结构之间的视界-边界字典。最后，不同的趋近方向可选择不同的共振解，该选择由同一个可解性函数的一阶参数变分决定。
英文摘要
Pole-skipping occurs at special complex frequencies and momenta where the retarded Green function of a black-hole or black-brane background is not uniquely defined. The local near-horizon mechanism is well known: at resonance, a Frobenius recurrence matrix loses rank and the space of smooth horizon solutions enlarges. We address the global question of how two boundary-normalized solutions enter this resonant horizon solution space. For a general second-order scalar radial equation on a nonextremal background analytic near the horizon, the Frobenius recurrence at the resonant order yields a solvability condition. We prove that its vanishing is equivalent to singularity of the horizon recurrence matrix, absence of the logarithmic Frobenius term, and existence of two independent smooth horizon solutions. Away from resonance, the source zero is equivalent to horizon smoothness of the response-normalized solution, while the response zero is equivalent to horizon smoothness of the source-normalized solution. We then analyze the parameter-dependent simple pole of the nonresonant ingoing solution as resonance is approached. After removing this singularity, its resonant limit is proportional to the larger-root Frobenius solution, with proportionality factor given by the same solvability function. Continuing to the boundary shows that the same condition is equivalent to simultaneous vanishing of the two regularized boundary connection coefficients. This establishes a horizon-to-boundary dictionary between local Fuchsian resonance and the boundary source-response 0/0 structure of pole-skipping. Finally, different directions of approach can select different resonant solutions, with the selection governed by the first parameter variation of the same solvability function.
