Action integrals for quantum BTZ black holes

Black holes exactly incorporating quantum matter backreaction effects, namely, quantum black holes, are notoriously difficult to construct, let alone study their horizon thermodynamics. Here, we derive the thermodynamics of three-dimensional charged and rotating quantum black holes via the tree-level gravitational partition function. Specifically, we primarily focus on holographic quantum BTZ black holes, dual to $(3+1)$-dimensional accelerating black holes in anti-de Sitter space that localize on Karch-Randall end-of-the-world (ETW) branes. To derive their horizon thermodynamics, we regulate the bulk Euclidean geometry by adding a second ETW brane at asymptotic spatial infinity. We compute the on-shell action of the complexified accelerating black hole in the grand canonical ensemble and derive the quantum BTZ black hole thermodynamics, where the thermal entropy is equal to the generalized entropy. This provides a first principles derivation of the generalized entropy of three-dimensional quantum black holes. Further, we construct charged and rotating quantum black holes in three-dimensional de Sitter and Minkowski space using Randall-Sundrum ETW branes, and compute their horizon thermodynamics.

💡 Research Summary

This paper presents a first‑principles derivation of the thermodynamics of three‑dimensional quantum BTZ black holes by exploiting the holographic braneworld construction. The authors consider a four‑dimensional Einstein–Maxwell–AdS bulk containing a charged, rotating C‑metric, which describes a black hole uniformly accelerating in AdS₄ due to a cosmic string. Two codimension‑one end‑of‑the‑world (ETW) branes are introduced: one sits on the totally umbilic surface x = 0, automatically satisfying the Israel junction conditions and fixing the brane tension τ = 1/(2πG₄ℓ); the second is placed at a large radial coordinate to serve as an infrared regulator for the Euclidean geometry. By Wick‑rotating the bulk solution and imposing regularity (removing conical singularities by fixing the azimuthal period), the authors compute the full Euclidean on‑shell action, including bulk Einstein–Hilbert, Maxwell, Gibbons‑Hawking‑York terms, and contributions from both branes.

The on‑shell action takes the form I_E = β M − S_gen, where M is the ADM mass of the four‑dimensional C‑metric and S_gen is identified as the generalized entropy of the three‑dimensional quantum BTZ black hole. Expanding S_gen in the small dimensionless parameter ν ≡ G₃ℏc/ℓ₃³ < 1, they obtain
S_gen = S_BH^{(3)} + ν S_CFT + ν² S_IW + …,
where the first term is the usual Bekenstein–Hawking area entropy, the second term is the fine‑grained entropy of the conformal field theory living on the brane, and the third term is the Iyer‑Wald entropy arising from higher‑derivative corrections. This decomposition demonstrates that the generalized entropy naturally incorporates quantum matter loops and higher‑curvature effects, extending earlier semiclassical treatments that only captured the area term.

The methodology is then applied to construct charged, rotating quantum black holes in three‑dimensional de Sitter (dS₃) and Minkowski (Mink₃) spacetimes using Randall–Sundrum‑type ETW branes. By appropriately tuning the C‑metric parameters (mass μ, rotation a, charge q) and ensuring a “slowly accelerating” regime (ℓ ≫ ℓ₄), the authors obtain regular brane‑localized solutions without acceleration horizons. The same Euclidean action computation yields the generalized entropy and thermodynamic relations (first law, temperature, chemical potentials) for these dS₃ and Mink₃ quantum black holes, explicitly showing how the cosmological constant and brane tension modify the entropy.

Technical challenges addressed include (i) simultaneous removal of conical defects via the dual‑brane setup, (ii) handling the complexified rotation parameter during Wick rotation, and (iii) achieving a finite on‑shell action without ambiguous background subtraction thanks to the second regulator brane. The paper also discusses the consistency of the semiclassical limit, the role of the parameter ν as an expansion control, and the exact equivalence between the bulk canonical partition function and the induced brane theory’s partition function (“double holography”).

In the discussion, the authors outline future directions such as extending to multiple or asymmetric brane configurations, computing higher‑order ν corrections to capture full matter‑gravity interaction effects, and applying the framework to recent quantum information questions (Page curve, entanglement islands, generalized entropy inequalities). Overall, the work provides a robust and exact holographic derivation of generalized entropy for quantum BTZ black holes and its extensions to dS₃ and Mink₃, offering a powerful new tool for studying quantum aspects of black hole thermodynamics.