Black holes with quantum corrections in $3d$: The case of Page curve in Lindblad, greybody factor, and Lyapunov exponent

Abstract: In this work, first we discuss black holes with quantum corrections as an open quantum system and apply Lindblad formalism to explain the ``zig-zag" behavior in the Hawking- Page curve and radiation process, specially by considering the effects of exceptional points. Then, we calculate quantum corrections for various parameters of $3d$ Cotler-Jensen theory which is $\text{AdS}_3$ with reparametrized modes. At each step of the calculations, we compare the results with the case of JT. We then calculate quantum-corrected greybody factor for various black hole solutions, and specially for the case of Cotler-Jensen theory. Finally, we study the effects of quantum corrections on Lyapunov exponent, potential, and quantum information structure of black holes.

• The paper develops a unified framework incorporating quantum corrections to black hole evaporation, greybody factors, and chaotic Lyapunov exponents in 3D.
• It applies Lindblad open quantum system methods and Cotler-Jensen reparameterization techniques to derive one-loop corrections and capture boundary graviton effects.
• Results indicate that quantum fluctuations modify the Page curve, suppress greybody factors, and adjust chaos measures, impacting black hole thermodynamics.

Black Holes with Quantum Corrections in Three Dimensions: Page Curve, Lindblad Formalism, Greybody Factors, and Lyapunov Exponents

Introduction and Framework

This work investigates quantum corrections to black hole physics in $(2+1)$-dimensional gravity, specifically focusing on $\mathrm{AdS}_3$ backgrounds and the Cotler-Jensen theory of reparameterization modes as an analog to the Schwarzian sector in JT gravity. The analysis holistically incorporates the structure of quantum radiation emission (notably, the Page curve), the precise formulation and consequences of open quantum dynamics via Lindblad equations, and explicit quantum corrections to both greybody factors and chaotic measures such as the Lyapunov exponent. Central emphasis is placed on the interplay between near-horizon quantum fluctuations, described as boundary graviton (reparameterization) modes, and thermodynamic/chaotic signatures in black hole evaporation.

Black Holes as Open Quantum Systems: Lindblad Approach and Exceptional Points

The radiation environment near the black hole horizon is modeled as an open quantum system, facilitating the application of Lindblad formalisms. This perspective, inspired by quantum statistical mechanics and the study of non-Hermitian dynamics, connects the presence of exceptional points (EPs)—degeneracies of dissipative quantum dynamics—with anomalies in the information-dynamics of black holes, such as the non-monotonic "zig-zag" structure of the quantum-corrected Page curve.

The analogy with the Lindblad SYK models is rendered precise: path integrals over zero modes of gravitational fluctuations are shown to map onto the structure of system-bath coupling in open quantum systems. The dissipative gap as a function of bath coupling exhibits non-monotonic behavior, mirroring the emergence of EP-driven transitions as noted in related studies of the Lindblad SYK and JT models. The quantization of boundary reparameterization modes produces both effective modifications to Green's functions and tracks the nature of black hole energy fluxes under both canonical and microcanonical conditions.

Quantum Corrections in AdS$_3$ Boundary Dynamics: Cotler-Jensen Theory

Key to this analysis is a detailed engagement with the Cotler-Jensen formalism, providing a $3$d analog to Schwarzian physics by capturing reparameterization modes of the AdS$_3$ boundary. The boundary action is established as a Virasoro coadjoint orbit theory, realized as a constrained WZW model or as a boundary Liouville action, encoding the full gravitational dynamics of the boundary gravitons.

Parametrization of fields as elements of $$\mathrm{Diff}(\mathbb{S}^1)/\mathrm{PSL}(2,\mathbb{R})$$ allows for explicit loop calculations leading to quantum corrections in the propagators, Hamiltonians, and effective actions. At large central charge $C$, diagrammatic expansions in $1/C$ directly capture the leading quantum gravitational corrections. The distinction between classical (tree-level) behavior and quantum-corrected (one-loop) observables is systematically constructed, with explicit measure factors, symplectic structure, and ghost field content integrated in the path integral approach.

Quantum-Corrected Greybody Factors

The evaluation of greybody factors—the probability for Hawking quanta to propagate through the nontrivial spacetime geometry to infinity—is extended to include quantum corrections from reparameterization modes. The 1-loop correction is realized as an explicit shift in the absorption cross section, with the transmission amplitude $\mathcal{T}(\omega)$ receiving perturbative corrections inherited from the graviton-reparameterization kernel. The master formula for the quantum-corrected greybody factor is given by: $$\sigma(\omega) = \sigma_{\text{cl}}(\omega) \left[1 + \delta_q(\omega)\right],$$ where $\mathrm{AdS}_3$0 is $\mathrm{AdS}_3$1 and encodes overlap integrals of classical wavefunctions and the Cotler-Jensen kernel.

Numerical and analytic results demonstrate that in several cases (notably $\mathrm{AdS}_3$2d BTZ, higher-dimensional black holes, and warped geometries), quantum corrections can suppress or enhance the greybody factor depending on the magnitude of parameters such as the central charge, black hole size, or the $\mathrm{AdS}_3$3 parameter associated with conical defect/ boundary softness. A particularly strong result is established for BTZ: quantum corrections at one loop generically suppress greybody factors, consistent with the expectation that quantum fluctuations raise effective potential barriers.

Lyapunov Exponent and Chaos in Quantum-Corrected $\mathrm{AdS}_3$4 Gravity

Quantum corrections to the classical Lyapunov exponent—characterizing the rate of growth of commutators in OTOCs and thus the signature of quantum chaos—are thoroughly analyzed in the reparameterized AdS$\mathrm{AdS}_3$5 setup. The bilocal correlators are expanded to next-to-leading order in $\mathrm{AdS}_3$6, and corrections to the Lyapunov exponent $\mathrm{AdS}_3$7 are extracted by functional differentiation. The dependence on $\mathrm{AdS}_3$8 (and, equivalently, on the heavy operator dimension $\mathrm{AdS}_3$9) is made explicit:

• Increasing $_3$0 generally increases $_3$1, indicating enhanced chaoticity as the boundary becomes softer.
• For large central charge, the Lyapunov exponent is robust against quantum corrections, while for small $_3$2, the correction becomes non-negligible.
• The leading effect of quantum corrections is to slow the decay of the bilocal correlator, consistent with the notion that quantum effects stabilize coarse-grained information against classical chaotic spreading.

Quantum Corrections in Black Hole Thermodynamics and Information Properties

The study extends to broader thermodynamic and informational characteristics, including quantum-corrected black hole potentials, the emergence of quantum width (interpreted as off-shell contributions in the path integral corresponding to fuzzy geometries), and violation or modification of entropy inequalities (holographic entropy cone constraints). The analysis connects the inclusion of complex BTZ saddles to violations of monogamy inequalities and the failure of saturation in error-correcting codes, reflecting the breakdown of geometric interpretations for certain off-shell states.

For Reissner-Nordström-AdS and other higher-dimensional black holes, explicit calculation of the one-loop corrected potential, phase structure, and transition behavior is provided. The inclusion of quantum effects is shown to shrink the first-order transition region and induce new zero-order transitions, altering the canonical picture of black hole thermodynamics.

Implications and Future Directions

The results substantiate the crucial role of quantum gravitational fluctuations in both local (greybody, Lyapunov exponent) and global (Page curve, entropy inequalities) observables. This work demonstrates the practical calculability of $_3$3 and $_3$4 corrections in tractable $_3$5-dimensional settings, and their interpretability in both statistical and quantum information theoretic language, including the identification of exceptional points as dynamical obstructions to monotonicity in open system evolution.

Potential future developments include:

• Systematic mapping of EP landscapes in near-horizon dynamics and their correlation with nontrivial features of the Page curve.
• Extension of Lindblad and coadjoint orbit techniques to higher-spin and higher-dimensional gravitational settings.
• Cross-comparison with laboratory analogs in quantum optics and condensed matter systems, where Lindblad-like open system physics is experimentally realized.

Conclusion

This paper provides a unified quantitative framework for the inclusion of quantum corrections in the dynamics of black hole evaporation, greybody spectra, and quantum chaos, within the context of three-dimensional gravity and reparameterization field theories. The explicit computational toolkit offers a robust platform to study quantum, statistical, and information-theoretic effects in low-dimensional gravity and their implications for universality in black hole physics.