How to tame your (black hole) saddles: Lessons from the Lorentzian Gravitational Path Integral
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914497107591168 |
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| author | Kolanowski, Maciej Marolf, Donald |
| author_facet | Kolanowski, Maciej Marolf, Donald |
| contents | We resolve a puzzle associated with the spherically-symmetric sector of the AdS$_4$ Einstein-Maxwell partition function with inverse temperature $β$. Since charge is quantized, the semiclassical limit of the partition function is expected to be given by a sum over complex black hole solutions obtained by shifting the associated chemical potential $μ$ by $\frac{2πi n}{e β}$ in terms of the relevant charge quantum $e$. However, the sum over all such saddles turns out to diverge at any finite value of $β$. We therefore consider a definition of this partition function as an integral over a space of metrics that are real and of Lorentz-signature up to the presence of certain conical singularities. A Picard-Lefshetz analysis shows that only a finite subset of the above saddles contribute to our integral at finite $β$, and thus that the sum over such saddles converges. The low temperature limit is nonetheless associated with a convergent sum over all saddles that (as $β\rightarrow \infty$) approach the usual large real Euclidean black holes. We also analyze the analogous partition function for the (uncharged) BTZ black hole in the ensemble defined by fixing an angular velocity $Ω$ up to shifts by $\frac{2πi m}{s β}$, where $s=\frac{1}{2}$ or $s=1$ depending on the presence of absence of fermionic states. In this case, at all $β$ we find that all saddles contribute and that the sum over saddles converges. We also comment briefly on the apparent lack of utility of the so-called KSW condition in our context. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_24681 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | How to tame your (black hole) saddles: Lessons from the Lorentzian Gravitational Path Integral Kolanowski, Maciej Marolf, Donald High Energy Physics - Theory General Relativity and Quantum Cosmology We resolve a puzzle associated with the spherically-symmetric sector of the AdS$_4$ Einstein-Maxwell partition function with inverse temperature $β$. Since charge is quantized, the semiclassical limit of the partition function is expected to be given by a sum over complex black hole solutions obtained by shifting the associated chemical potential $μ$ by $\frac{2πi n}{e β}$ in terms of the relevant charge quantum $e$. However, the sum over all such saddles turns out to diverge at any finite value of $β$. We therefore consider a definition of this partition function as an integral over a space of metrics that are real and of Lorentz-signature up to the presence of certain conical singularities. A Picard-Lefshetz analysis shows that only a finite subset of the above saddles contribute to our integral at finite $β$, and thus that the sum over such saddles converges. The low temperature limit is nonetheless associated with a convergent sum over all saddles that (as $β\rightarrow \infty$) approach the usual large real Euclidean black holes. We also analyze the analogous partition function for the (uncharged) BTZ black hole in the ensemble defined by fixing an angular velocity $Ω$ up to shifts by $\frac{2πi m}{s β}$, where $s=\frac{1}{2}$ or $s=1$ depending on the presence of absence of fermionic states. In this case, at all $β$ we find that all saddles contribute and that the sum over saddles converges. We also comment briefly on the apparent lack of utility of the so-called KSW condition in our context. |
| title | How to tame your (black hole) saddles: Lessons from the Lorentzian Gravitational Path Integral |
| topic | High Energy Physics - Theory General Relativity and Quantum Cosmology |
| url | https://arxiv.org/abs/2603.24681 |