# Black hole microstates One of the most important questions regarding black holes in quantum gravity is the counting of black hole microstates. ## Approaches - BPS state counting: - original [[1996#Strominger, Vafa]] - review [[2019#Zaffaroni (Lectures)]] - fuzzball: - reviewed in [[BenaMartinecMathurWarner2022]][](https://arxiv.org/abs/2204.13113) - field excitations: - [[THooft1985]][](https://www.sciencedirect.com/science/article/abs/pii/0550321385904183?via%3Dihub) - Euclidean path integral and state overlap: - [[2022#Balasubramanian, Lawrence, Magan, Sasieta (a)]] - [[2024#Geng, Jiang]]: single-sided black holes with Karch-Randall branes - [[2024#Balasubramanian, Craps, Hernandez, Khramtsov, Knysh (Dec)]]: dynamical - [[2025#Barbon, Velasco-Aja]]: instability of Euclidean wormholes - [[2025#Wang]]: de Sitter entropy ## 2D black hole microstate counting - [[AhmadainFrenkelRaySoni2022]][](https://arxiv.org/pdf/2210.11493.pdf): from boundary matrix QM - [[BetziosPapadoulaki2022]]: from string theory ## Alternatives - [[LarsonLee2021]][](https://arxiv.org/pdf/2101.08497.pdf): AdS3 grand canonical - [[BahBenaHeidmannLiMayerson2021]][](https://arxiv.org/pdf/2104.10686.pdf): horizonless microstate geometries - [[BagchiGrumillerSheikh-Jabbari2022]][](https://arxiv.org/abs/2210.10794): 3d BH microstates as tensionless null string states ## Bosonic vs fermionic microstates When counting microstates, one can insert $(-1)^{\mathsf{F}}$ which assigns $+1$ to bosonic states and $-1$ to fermionic states. Instead of the sum, this computes the difference between bosonic and fermionic states. In [[2023#Chen, Turiaci]], using the fact that $(-1)^{\mathrm{F}}=e^{ \pm 2 \pi \mathrm{i} J}$, the following quantities are computed: - $Z_{\text {spin }}(\beta)=\operatorname{Tr}(-1)^{\mathsf{F}} e^{-\beta H}=e^{-\beta F_{\text {spin }}(\beta)}$, - $\operatorname{Tr}_E(-1)^{\mathsf{F}}=e^{S_{\text {spin }}(E)}$. Instead of computing $(-1)^\mathsf{F}$, one can study the grand canonical ensemble with a specific value of the angular potential, such that e.g. $\operatorname{Tr}\left(e^{-\beta H} e^{\beta \Omega J}\right)=\operatorname{Tr}\left(e^{-\beta H} e^{2\pi i J}\right)$=Z_{\text {spin }}(\beta)$. A comment on boundary conditions: with $(-1)^{\mathsf{F}}$, the fermions have antiperiodic boundary conditions; without, periodic.