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从引力路径积分出发的黑洞：超对称索引与精确反德西特几何

Black holes from the gravitational path integral: supersymmetric indices and precision holography

UNIVERSITÀ DEGLI STUDI DI PADOVA（帕多瓦大学）

Enrico Turetta

备注：
PhD thesis. Based on the author's contributions: 2208.01007, 2304.06101, 2403.02410, 2409.01332, 2507.12650. 280 pages + appendices and references. 8 figures
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AI总结：

通过引力路径积分计算超对称索引，揭示反德西特空间中黑洞热力学修正与共形场论的关联。

摘要：

通过计算弱耦合弦论或对偶超导电场理论中的超对称索引，得到具有反德西特或平坦渐近条件的超对称黑洞微状态计数。这些索引是受保护的观测量，其值可以从弱耦合可靠地外推到强耦合，其中引力描述适用。在本论文中，经过广泛的引言回顾后，我们讨论了如何在引力理论中直接制定这些受保护的观测量，通过欧几里得路径积分。在半经典极限下，索引退化为复欧几里得极点的求和。这些极点是超对称的但“非极值”的，出现在反德西特和平坦空间中。在反德西特设定中，我们研究了四导数修正对AdS_5黑洞热力学的影响。利用非壳方法，我们构建了五维加权超引力的修正作用。然后我们评估了超对称AdS_5黑洞的修正壳作用，并发现其与对偶共形场论的超对称索引的Cardy-like极限有精确的一致性。通过作用的Legendre变换，我们得到了修正的黑洞熵，并通过将Wald公式应用于修正的近视界几何来确认这一结果。然后我们转向具有渐近平坦边界条件的引力索引。我们揭示了一种广泛的新型极点家族，并基于其杆结构对其拓扑进行一般分类。这些解可能具有多个视界或三维气泡，具有棱镜拓扑，并允许锥形奇点产生进一步的几何结构，包括纺锤和分支球面。对于更简单的几何结构，使用奇数维版本的等变局部化计算壳作用。

英文摘要：

The counting of microstates of supersymmetric black holes with anti-de Sitter or flat asymptotics is obtained by computing a supersymmetric index in a weakly coupled string theory or a dual superconformal field theory. These indices are protected observables, whose value can be reliably extrapolated from weak to strong coupling, where the gravitational description applies. In this Thesis, after a broad introductory review, we discuss recent progress in formulating such protected observables directly within the gravitational theory, via the Euclidean path integral. In the semiclassical limit the index reduces to a sum over complex Euclidean saddles. These saddles are supersymmetric but ''non-extremal'', and arise in both anti-de Sitter and flat spaces. In the holographic setting, we investigate four-derivative corrections to the thermodynamics of AdS$_5$ black holes. Using off-shell methods, we construct the corrected action of five-dimensional gauged supergravity. We then evaluate the corrected on-shell action of supersymmetric AdS$_5$ black holes and find exact agreement with a Cardy-like limit of the superconformal index of the dual conformal field theory. By a Legendre transform of the action, we obtain the corrected black hole entropy, and we confirm this result by applying Wald's formula to the corrected near-horizon geometry. We then turn to the gravitational index with asymptotically flat boundary conditions. We uncover a broad family of novel saddles and present a general classification based on their rod structure, which characterizes their topology. These solutions may feature multiple horizons or three-dimensional bubbles with lens space topology, and allowing for conical singularities yields further geometries, involving spindles and branched spheres. For the simpler geometries, the on-shell action is computed using an odd-dimensional version of equivariant localization.

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