AI 中文总结
研究带电反德西特黑洞热力学分支结构中拓扑变化与响应奇点，通过重新分析零点分支几何结构发现，总拓扑数变化的逆温度$\tau_a$与分支响应奇异的逆温度$\tau_b$追踪不同局部结构，二者是对相同几何结构的不同诊断，起源不同。
AI 中文摘要
不同的黑洞热力学分支结构诊断方法并不总是能识别出相同的局部事件。我们在已知发生热力学拓扑相变的带电反德西特黑洞示例中研究了这个问题。在先前于[JHEP 06 (2024) 213]中报道的这些示例中，总拓扑数变化时的逆温度$\tau_a$与这里考虑的固定电荷参数分支响应变得奇异时的逆温度$\tau_b$是分开的。重新分析相同的零点分支几何结构，我们发现这两种诊断方法追踪的是不同的局部结构。$\tau_a$处的变化是正半径物理分支集的边界成员资格和指标流事件：一个具有非零缠绕的零点进入或离开计数域。而$\tau_b$处的奇点是由有限半径处的一个转折点引起的，在那里一对分支湮灭。由于这两个分支携带相反的缠绕，此事件改变分支计数但保持总拓扑数。因此，在所分析的示例中，热力学拓扑相变和分支响应奇点是对相同零点分支几何结构的不同诊断，具有不同的局部起源。
英文摘要
In this paper, we investigate how different diagnostics characterize the thermodynamic branch structure of black holes. In three charged AdS black hole examples, the thermodynamic topological transition point $τ_a$, where the total topological number changes, is separated from the point $τ_b$ where the branch response considered here, evaluated along a path with fixed charge parameters, becomes singular. Using the zero-point curves $τ(r_h)$, we identify the local mechanisms behind this separation. At $τ_a$, a zero point with nonzero winding enters or leaves the positive-radius physical branch set through the boundary of the counted domain, changing $W$ without making the corresponding positive-radius branch singular. At $τ_b$, the zero-point curve develops a finite-radius turning point, where two branches with opposite winding numbers annihilate and the branch response diverges. Thus, in the analyzed examples, a change in the topological number records a change in branch membership, whereas a response singularity records a loss of local invertibility along an admitted branch. The two diagnostics therefore probe different local events of the same zero-point branch geometry.
Comments8 pages, 1 figure, 2 tables
