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1. Linear growth of the two-point function for the Unruh state in 1 + 1 dimensional black holes

   https://doi.org/10.1142/9789811269776_0100  

   Anderson, Paul R.; Scofield, Zachary P.; Traschen, Jennie (January 2023, Proceedings of the Sixteenth Marcel Grossmann Meeting on General Relativity)

   Vereshchagin, G.; Ruffini, R. (Ed.)

   The symmetric two-point function for a massless, minimally coupled scalar field in the Unruh state is examined for Schwarzschild-de Sitter spacetime in two dimensions. This function grows linearly in terms of a time coordinate that is well-defined on the future black hole and cosmological horizons, when the points are split in the space direction. This type of behavior also occurs in two dimensions for other static black hole spacetimes when the field is in the Unruh state, and at late times it occurs in spacetimes where a black hole forms from the collapse of a null shell. The generalization to the case of the symmetric two-point function in two dimensions for a massive scalar field in Schwarzschild-de Sitter spacetime is discussed. 

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2. Horizons and correlation functions in 2D Schwarzschild-de Sitter spacetime

   https://doi.org/10.1007/JHEP01(2022)192  

   Anderson, Paul R.; Traschen, Jennie (January 2022, Journal of High Energy Physics)

   A bstract Two-dimensional Schwarzschild-de Sitter is a convenient spacetime in which to study the effects of horizons on quantum fields since the spacetime contains two horizons, and the wave equation for a massless minimally coupled scalar field can be solved exactly. The two-point correlation function of a massless scalar is computed in the Unruh state. It is found that the field correlations grow linearly in terms of a particular time coordinate that is good in the future development of the past horizons, and that the rate of growth is equal to the sum of the black hole plus cosmological surface gravities. This time dependence results from additive contributions of each horizon component of the past Cauchy surface that is used to define the state. The state becomes the Bunch-Davies vacuum in the cosmological far field limit. The two point function for the field velocities is also analyzed and a peak is found when one point is between the black hole and cosmological horizons and one point is outside the future cosmological horizon. 

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3. Infrared effects and the Unruh state

   https://doi.org/10.1088/1361-6382/acd0fd  

   Anderson, Paul R; Gholizadeh Siahmazgi, Shohreh; Scofield, Zachary P (May 2023, Classical and Quantum Gravity)

   Abstract Detailed behaviors of the modes of quantized scalar fields in the Unruh state for various eternal black holes in two dimensions are investigated. It is shown that the late-time behaviors of some of the modes of the quantum fields and of the symmetric two-point function are determined by infrared effects. The nature of these effects depends upon whether there is an effective potential in the mode equation and what form this potential takes. Here, three cases are considered, one with no potential and two with potentials that are nonnegative everywhere and are zero on the event horizon of the black hole and zero at either infinity or the cosmological horizon. Specifically, the potentials are a delta function potential and the potential that occurs for a massive scalar field in Schwarzschild–de Sitter spacetime. In both cases, scattering effects remove infrared divergences in the mode functions that would otherwise arise from the normalization process. When such infrared divergences are removed, it is found that the modes that are positive frequency with respect to the Kruskal time on the past black hole horizon approach zero in the limit that the radial coordinate is fixed and the time coordinate goes to infinity. In contrast, when there is no potential and thus infrared divergences occur, the same modes approach nonzero constant values in the late-time limit when the radial coordinate is held fixed. The behavior of the symmetric two-point function when the field is in the Unruh state is investigated for the case of a delta function potential in certain asymptotically flat black hole spacetimes in two dimensions. The removal of the infrared divergences in the mode functions results in the elimination of terms that grow linearly in time. 

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4. Black hole perturbation theory and multiple polylogarithms

   https://doi.org/10.1007/JHEP11(2023)059  

   Aminov, Gleb; Arnaudo, Paolo; Bonelli, Giulio; Grassi, Alba; Tanzini, Alessandro (November 2023, Journal of High Energy Physics)

   A<sc>bstract</sc> We study black hole linear perturbation theory in a four-dimensional Schwarzschild (anti) de Sitter background. When dealing with apositivecosmological constant, the corresponding spectral problem is solved systematically via the Nekrasov-Shatashvili functions or, equivalently, classical Virasoro conformal blocks. However, this approach can be more complicated to implement for certain perturbations if the cosmological constant isnegative. For these cases, we propose an alternative method to set up perturbation theory for both small and large black holes in an analytical manner. Our analysis reveals a new underlying recursive structure that involves multiple polylogarithms. We focus on gravitational, electromagnetic, and conformally coupled scalar perturbations subject to Dirichlet and Robin boundary conditions. The low-lying modes of the scalar sector of gravitational perturbations and its hydrodynamic limit are studied in detail. 

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5. Scale separation on ${\mathrm{AdS}}_3\times S^3$ with and without supersymmetry

   https://doi.org/10.1103/mp1n-9vgy  

   Proust, Aymeric; Samtleben, Henning; Sezgin, Ergin (June 2025, Physical Review D)

   Six-dimensional chiral gauged Einstein-Maxwell supergravity admits a two-parameter rotating dyonic string solution whose near horizon limit is the direct product of three-dimensional Anti-De Sitter space and a squashed three-sphere $S^3$. For a particular relation between the two parameters, the solution preserves $1/2$supersymmetry. We determine the complete Kaluza-Klein spectrum of the theory around these ${\mathrm{AdS}}_3$backgrounds. For the supersymmetric backgrounds, the states organize into infinite towers of long and short multiplets of $\mathrm{OSp}\left(2\right|2)$. In a certain limit of parameters, both the supersymmetric and the nonsupersymmetric spectra exhibit scale separation. In the latter case there are five topologically massive vectors and five scalars retaining finite masses with integer conformal dimensions, and in the supersymmetric case there are supersymmetric partners with half integer conformal dimensions, while all other masses diverge. Published by the American Physical Society2025 

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