We propose a method of computing one-loop determinants in black hole spacetimes (with emphasis on asymptotically anti-de Sitter black holes) that may be used for numerics when completely-analytic results are unattainable. The method utilizes the expression for one-loop determinants in terms of quasinormal frequencies determined by Denef, Hartnoll and Sachdev in \cite{Denef:2009kn}. A numerical evaluation must face the fact that the sum over the quasinormal modes, indexed by momentum and overtone numbers, is divergent. A necessary ingredient is then a regularization scheme to handle the divergent contributions of individual fixed-momentum sectors to the partition function. To this end, we formulate an effective two-dimensional problem in which a natural refinement of standard heat kernel techniques can be used to account for contributions to the partition function at fixed momentum. We test our method in a concrete case by reproducing the scalar one-loop determinant in the BTZ black hole background. We then discuss the application of such techniques to more complicated spacetimes.
In this note we compute the 1-loop partition function of spin-$s$ fields on Euclidean de Sitter space $S^{2n+1}$ using the quasinormal mode method. Instead of computing the quasinormal mode frequencies from scratch, we use the analytic continuation prescription $L_{\text{AdS}}\to iL_{\text{dS}}$, appearing in the dS/CFT correspondence, and Wick rotate the normal mode frequencies of fields on thermal $\text{AdS}_{2n+1}$ into the quasinormal mode frequencies of fields on de Sitter space. We compare the quasinormal mode and heat kernel methods of calculating 1-loop determinants, finding exact agr…
We extend the results of arXiv:1401.7016, computing one loop partition functions for massive fields with spin half in AdS_2 using the quasinormal mode method proposed by Denef, Hartnoll, and Sachdev in arXiv:0908.2657. We find the finite representations of SO(2,1) for spin zero and spin half, consisting of a highest weight state |h\rangle and descendants with non-unitary values of h. These finite representations capture the poles and zeroes of the one loop determinants. Together with the asymptotic behavior of the partition functions (which can be easily computed using a large mass heat kernel…
In this note, we calculate the one-loop determinant for a massive scalar (with conformal dimension $Δ$) in even-dimensional AdS$_{d+1}$ space, using the quasinormal mode method developed in arXiv:0908.2657 by Denef, Hartnoll, and Sachdev. Working first in two dimensions on the related Euclidean hyperbolic plane $H_2$, we find a series of zero modes for negative real values of $Δ$ whose presence indicates a series of poles in the one-loop partition function $Z(Δ)$ in the $Δ$ complex plane; these poles contribute temperature-independent terms to the thermal AdS partition function computed in arX…
We compute the path integral for a particle on the covering group of SL(2,R) using a decomposition of the Lie algebra into adjoint orbits. We thus intuitively derive the Hilbert space of the particle on the group including discrete and continuous representations. Next, we perform a Lorentzian hyperbolic orbifold of the partition function and relate it to the Euclidean BTZ partition function. We use the particle model to inform further discussion of the spectral content of the one loop vacuum amplitude for strings on BTZ black hole backgrounds. We argue that the poles in the loop integrand code…
An integral representation of the 1-loop partition function for charged scalars and spinors, minimally coupled to a uniform $U(1)$ field on $S^2$, is given in terms of $SO(1,2)$ Harish-Chandra group characters and evaluated exactly in terms of Hurwitz $ζ$-functions. Analytically continuing the $U(1)$ field, we interpret the path integrals as quasicanonical partition functions in $dS_2$ with an electric field. The character itself is obtained as a trace over states living at the future boundary of de Sitter and has a quasinormal mode expansion. The imaginary part of the partition function captu…
We study the stability under scalar perturbations, and we compute the quasinormal modes of the Einstein-Born-Infeld dilaton spacetime in 1+3 dimensions. Solving the full radial equation in terms of hypergeometric functions, we provide an exact analytical expression for the spectrum. We find that the frequencies are purely imaginary, and we confirm our results by computing them numerically. Although the scalar field that perturbs the black hole is electrically neutral, an instability similar to that seen in charged scalar perturbations of the Reissner-Nordström black hole is observed.
We connect two different approaches for calculating functional determinants on quotients of hyperbolic spacetime: the heat kernel method and the quasinormal mode method. For the example of a rotating BTZ background, we show how the image sum in the heat kernel method builds up the logarithms in the quasinormal mode method, while the thermal sum in the quasinormal mode method builds up the integrand of the heat kernel. More formally, we demonstrate how the heat kernel and quasinormal mode methods are linked via the Selberg zeta function. We show that a 1-loop partition function computed using t…
The quasinormal modes (QNMs) of a black hole spacetime are the free, decaying oscillations of the spacetime, and are well understood in the case of Kerr black holes. We discuss a method for computing the QNMs of spacetimes which are slightly deformed from Kerr. We mention two example applications: the parametric, turbulent instability of scalar fields on a background which includes a gravitational QNM, and the shifts to the QNM frequencies of Kerr when the black hole is weakly charged. This method may be of use in studies of black holes which are deformed by external fields or are solutions to…
We present new analytic results on black hole perturbation theory. Our results are based on a novel relation to four-dimensional $\mathcal{N}=2$ supersymmetric gauge theories. We propose an exact version of Bohr-Sommerfeld quantization conditions on quasinormal mode frequencies in terms of the Nekrasov partition function in a particular phase of the $Ω$-background. Our quantization conditions also enable us to find exact expressions of eigenvalues of spin-weighted spheroidal harmonics. We test the validity of our conjecture by comparing against known numerical results for Kerr black holes as w…
We compute numerically the quasinormal modes of Kerr-Newman black holes in the scalar case, for which the perturbation equations are separable. Then we study different approximations to decouple electromagnetic and gravitational perturbations of the Kerr-Newman metric, computing the corresponding quasinormal modes. Our results suggest that the Teukolsky-like equation derived by Dudley and Finley gives a good approximation to the dynamics of a rotating charged black hole for Q<M/2. Though insufficient to deal with Kerr-Newman based models of elementary particles, the Dudley-Finley equation shou…
We perform the stability analysis on scalarized charged black holes in the Einstein-Maxwell-Scalar (EMS) theory by computing quasinormal mode spectrum. It is noted that the appearance of these black holes with scalar hair is closely related to the instability of Reissner-Nordström black holes without scalar hair in the EMS theory. The scalarized black hole solutions are classified by the node number of $n=0,1,2,\cdots$, where $n=0$ is called the fundamental branch and $n=1,2,\cdots$ denote the $n$ excited branches. Here, we show that the $n=1,2$ excited black holes are unstable against against…
We give explicit expressions for the finite frequency greybody factor, quasinormal modes and Love numbers of Kerr black holes by computing the exact connection coefficients of the radial and angular parts of the Teukolsky equation. This is obtained by solving the connection problem of the confluent Heun equation in terms of the explicit expression of irregular Virasoro conformal blocks as sums over partitions via the AGT correspondence. In the relevant approximation limits our results are in agreement with existing literature. The method we use can be extended to solve the linearized Einstein …
A large portion of the studies concerning the quasinormal mode frequencies of a Kerr black hole have focused only on achieving higher numerical accuracy with limited emphasis on providing their physical interpretation. In this article, we partially address this issue by computing the quasinormal mode frequency spectrum of a Kerr black hole using the theory of Regge poles. By considering the retarded Green's function of the Teukolsky equation, we establish for scalar, electromagnetic and gravitational perturbations an equation linking the Regge poles to the quasinormal frequencies and we solve …
We investigate the decay of a scalar field outside a Schwarzschild anti de Sitter black hole. This is determined by computing the complex frequencies associated with quasinormal modes. There are qualitative differences from the asymptotically flat case, even in the limit of small black holes. In particular, for a given angular dependence, the decay is always exponential - there are no power law tails at late times. In terms of the AdS/CFT correspondence, a large black hole corresponds to an approximately thermal state in the field theory, and the decay of the scalar field corresponds to the de…
We obtain relaxation times for field theories with Lifshitz scaling and with holographic duals Einstein-Maxwell-Dilaton gravity theories. This is done by computing quasinormal modes of a bulk scalar field in the presence of Lifshitz black branes. We determine the relation between relaxation time and dynamical exponent z, for various values of boundary dimension d and operator scaling dimension. It is found that for d>z+1, at zero momenta, the modes are non-overdamped, whereas for d<=z+1 the system is always overdamped. For d=z+1 and zero momenta, we present analytical results.
We derive a quantization formula of Bohr-Sommerfeld type for computing quasinormal frequencies for scalar perturbations in an AdS black hole in the limit of large scalar mass or spatial momentum. We then apply the formula to find poles in retarded Green functions of boundary CFTs on $R^{1,d-1}$ and $RxS^{d-1}$. We find that when the boundary theory is perturbed by an operator of dimension $Δ>> 1$, the relaxation time back to equilibrium is given at zero momentum by ${1 \over ΔπT} << {1 \over πT}$. Turning on a large spatial momentum can significantly increase it. For a generic scalar operator …
We investigate the linearized equations of motion in three dimensional Born-Infeld gravity theory. Motivated by this model, we calculate the quasinormal modes of BTZ black hole solutions of parity-even gravity theories in three dimensions by using numerical methods. The results are classified in three families and they are so accurate such that it allows us to propose analytical form for the quasinormal frequencies. We find new quasinormal modes which have been missing in the literature of the analytical studies of three dimensional massive gravitons. These new modes do not have the known towe…
Quasinormal modes of rotating black holes beyond general relativity in the WKB approximation
Observatoire de Paris, Universite PSL（巴黎天文台，巴黎文理研究大学）；CENTRA, Departamento de Física, Instituto Superior Técnico – IST, Universidade de Lisboa – UL（里斯本大学，技术研究所物理系与理论研究中心）；Max Planck Institute for Gravitational Physics (Albert Einstein Institute)（马克斯·普朗克引力物理学研究所（爱因斯坦研究所））；Johns Hopkins University（约翰斯·霍普金斯大学）
With the recently discovered formula, which relates the off shell Euclidean one loop determinant to the on shell quantities such as normal and quasinormal modes in real spacetime, we work out the one loop partition function for $\mathcal{N}=1$ supergravity in AdS$_3$ from scratch. In passing, we also provide the explicit expression for one loop determinant of a field with arbitrary spin in AdS$_3$. To achieve this, we firstly derive the determinant expression for the one loop partition function in question using the powerful decomposition technique, and then we construct the normal modes in a …
Geodesic motion determines important features of spacetimes. Null unstable geodesics are closely related to the appearance of compact objects to external observers and have been associated with the characteristic modes of black holes. By computing the Lyapunov exponent, which is the inverse of the instability timescale associated with this geodesic motion, we show that, in the eikonal limit, quasinormal modes of black holes in any dimensions are determined by the parameters of the circular null geodesics. This result is independent of the field equations and only assumes a stationary, spherica…
The heat kernel and quasinormal mode methods of computing 1-loop partition functions of spin $s$ fields on hyperbolic quotient spacetimes $\mathbb{H}^{3}/\mathbb{Z}$ are related via the Selberg zeta function. We extend that analysis to thermal $\text{AdS}_{2n+1}$ backgrounds, with quotient structure $\mathbb{H}^{2n+1}/\mathbb{Z}$. Specifically, we demonstrate the zeros of the Selberg function encode the normal mode frequencies of spin fields upon removal of non-square-integrable modes. With this information we construct the 1-loop partition functions for symmetric transverse traceless tensors …
We study the instability of near extremal and extremal four-dimensional AdS charged hairy black hole to radial neutral massive and charged massless scalar field perturbations. We solve the scalar field equation by using the improved asymptotic iteration method and the time domain analysis and we find the quasinormal frequencies. For the charged scalar perturbations, we find the superradiance condition by computing the reflection coefficient in the low-frequency limit and we show that in the superradiance regime, which depends on the scalar hair charge, all modes of radial charged massless pert…
We establish the relation of partition functions of conformal higher spin fields on Weyl equivalent spaces in $d=4$ dimension. We express the partition function of Weyl graviton and conformal higher spin fields as an integral over characters on $S^1\times AdS_3$, $S^4$, and $AdS_4$. We observe that the partition function of conformal higher spins on hyperbolic cylinders differs from the partition function on $S^4$ by the `edge' contribution. The logarithmic coefficient obtained from the character integral of the partition function of conformal higher spins on $AdS_4$ is the half of that obtain…
Motivated by the prospect of constraining microscopic models, we calculate the exact one-loop corrected de Sitter entropy (the logarithm of the sphere partition function) for every effective field theory of quantum gravity, with particles in arbitrary spin representations. In doing so, we universally relate the sphere partition function to the quotient of a quasi-canonical bulk and a Euclidean edge partition function, given by integrals of characters encoding the bulk and edge spectrum of the observable universe. Expanding the bulk character splits the bulk (entanglement) entropy into quasinor…
We relate the heat kernel and quasinormal mode methods of computing the 1-loop partition function of arbitrary spin fields on a rotating (Euclidean) BTZ background using the Selberg zeta function associated with $\mathbb{H}^{3}/\mathbb{Z}$, extending (1811.08433). Previously, Perry and Williams showed for a scalar field that the zeros of the Selberg zeta function coincide with the poles of the associated scattering operator upon a relabeling of integers. We extend the integer relabeling to the case of general spin, and discuss its relationship to the removal of non-square-integrable Euclidean …
When computing the ideal gas thermal canonical partition function for a scalar outside a black hole horizon, one encounters the divergent single-particle density of states (DOS) due to the continuous nature of the normal mode spectrum. Recasting the Lorentzian field equation into an effective 1D scattering problem, we argue that the scattering phases encode non-trivial information about the DOS and can be extracted by "renormalizing" the DOS with respect to a reference. This defines a renormalized free energy up to an arbitrary additive constant. Interestingly, the 1-loop Euclidean path integr…
We compute dispersion relations of non-hydrodynamic and hydrodynamic modes in a non-relativistic strongly coupled two-dimensional quantum field theory. This is achieved by numerically computing quasinormal modes (QNMs) of a particular analytically known black brane solution to 3+1-dimensional Hořava Gravity. Hořava Gravity is distinguished from Einstein Gravity by the presence of a scalar field, termed the khronon, defining a preferred time-foliation. Surprisingly, for this black brane solution, the khronon fluctuation numerically decouples from all others, having its own set of purely imagina…
The most recent studies on the supersymmetric localization reveal many non-trivial features of supersymmetric field theories in diverse dimensions, and 3d gauge theory provides a typical example. It was conjectured that the index and the partition function of a 3d N=2 theory are constructed from a single component: the holomorphic block. We prove this conjecture for non-abelian gauge theories by computing exactly the 3d partition functions and holomorphic blocks.
In a recent paper, the partition function (character) of ten-dimensional pure spinor worldsheet variables was calculated explicitly up to the fifth mass-level. In this letter, we propose a novel application of Pade approximants as a tool for computing the character of pure spinors. We get results up to the twelfth mass-level. This work is a first step towards an explicit construction of the complete pure spinor partition function.
We develop an efficient method for computing thermal partition functions of weakly coupled scalar fields in AdS. We consider quartic contact interactions and show how to evaluate the relevant two-loop vacuum diagrams without performing any explicit AdS integration, the key step being the use of Kallen-Lehmann type identities. This leads to a simple method for extracting double-trace anomalous dimensions in any spacetime dimension, recovering known first-order results in a streamlined fashion.
We show that the elliptic genus of the higher rank E-strings can be computed based solely on the genus 0 Gromov-Witten invariants of the corresponding elliptic geometry. To set up our computation, we study the structure of the topological string free energy on elliptically fibered Calabi-Yau manifolds both in the unrefined and the refined case, determining the maximal amount of the modular structure of the partition function that can be salvaged. In the case of fibrations exhibiting only isolated fibral curves, we show that the principal parts of the topological string partition function at gi…
The Selberg zeta function and trace formula are powerful tools used to calculate kinetic operator spectra and quasinormal modes on hyperbolic quotient spacetimes. In this article, we extend this formalism to non-hyperbolic quotients by constructing a Selberg zeta function for warped AdS$_3$ black holes. We also consider the so-called self-dual solutions, which are of interest in connection to near-horizon extremal Kerr. We establish a map between the zeta function zeroes and the quasinormal modes on warped AdS$_3$ black hole backgrounds. In the process, we use a method involving conformal coor…
For strongly interacting systems holographic duality is a powerful framework for computing e.g. dispersion relations to all orders in perturbation theory. Using the standard Reissner-Nordstöm black hole as a holographic model for a (strange) metal, we obtain exotic dispersion relations for both plasmon modes and quasinormal modes for certain intermediate values of the charge of the black hole. The obtained dispersion relations show dissipative behavior which we compare to the generic expectations from the Caldeira-Leggett model for quantum dissipation. Based on these considerations, we investi…
