scholarly journals BTZ one-loop determinants via the Selberg zeta function for general spin

2020 ◽  
Vol 2020 (10) ◽  
Author(s):  
Cynthia Keeler ◽  
Victoria L. Martin ◽  
Andrew Svesko
Keyword(s):  
Scalar Field ◽  
Partition Function ◽  
Zeta Function ◽  
Heat Kernel ◽  
Arbitrary Spin ◽  
Quasinormal Mode ◽  
Scattering Operator ◽  
Selberg Zeta Function ◽  
Zero Modes ◽  
Spin Fields

Abstract 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 ℍ3/ℤ, extending (arXiv:1811.08433) [1]. Previously, Perry and Williams [2] 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 zero modes.

scholarly journals Normal modes in thermal AdS via the Selberg zeta function

2020 ◽  
Vol 9 (1) ◽  
Author(s):  
Victoria Martin ◽  
Andrew Svesko
Keyword(s):  
Zeta Function ◽  
Heat Kernel ◽  
Normal Mode ◽  
Kernel Method ◽  
Normal Modes ◽  
Quasinormal Mode ◽  
Partition Functions ◽  
Selberg Zeta Function ◽  
Exact Agreement ◽  
Spin Fields

The heat kernel and quasinormal mode methods of computing 1-loop partition functions of spin ss fields on hyperbolic quotient spacetimes \mathbb{H}^{3}/\mathbb{Z}ℍ3/ℤ are related via the Selberg zeta function. We extend that analysis to thermal \text{AdS}_{2n+1}AdS2n+1 backgrounds, with quotient structure \mathbb{H}^{2n+1}/\mathbb{Z}ℍ2n+1/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 in terms of the Selberg zeta function and find exact agreement with the heat kernel method.

scholarly journals Connecting quasinormal modes and heat kernels in 1-loop determinants

2020 ◽  
Vol 8 (2) ◽  
Author(s):  
Cynthia Keeler ◽  
Victoria Martin ◽  
Andrew Svesko
Keyword(s):  
Zeta Function ◽  
Heat Kernel ◽  
Kernel Method ◽  
Quasinormal Modes ◽  
Heat Kernels ◽  
Quasinormal Mode ◽  
Selberg Zeta Function ◽  
Hyperbolic Spacetime ◽  
Mode Method ◽  
Thermal Sum

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 the heat kernel method may be cast as a Selberg zeta function whose zeros encode quasinormal modes. We discuss how our work may be used to predict quasinormal modes on more complicated spacetimes.

scholarly journals Notes on massless scalar field partition functions, modular invariance and Eisenstein series

2021 ◽  
Vol 2021 (12) ◽  
Author(s):  
Francesco Alessio ◽  
Glenn Barnich ◽  
Martin Bonte
Keyword(s):  
Scalar Field ◽  
Partition Function ◽  
Low Temperature ◽  
Eisenstein Series ◽  
Chemical Potential ◽  
Proper Time ◽  
Series Representation ◽  
Massless Scalar ◽  
Massless Scalar Field ◽  
Spacetime Manifold

Abstract The partition function of a massless scalar field on a Euclidean spacetime manifold ℝd−1 × 𝕋2 and with momentum operator in the compact spatial dimension coupled through a purely imaginary chemical potential is computed. It is modular covariant and admits a simple expression in terms of a real analytic SL(2, ℤ) Eisenstein series with s = (d + 1)/2. Different techniques for computing the partition function illustrate complementary aspects of the Eisenstein series: the functional approach gives its series representation, the operator approach yields its Fourier series, while the proper time/heat kernel/world-line approach shows that it is the Mellin transform of a Riemann theta function. High/low temperature duality is generalized to the case of a non-vanishing chemical potential. By clarifying the dependence of the partition function on the geometry of the torus, we discuss how modular covariance is a consequence of full SL(2, ℤ) invariance. When the spacetime manifold is ℝp × 𝕋q+1, the partition function is given in terms of a SL(q + 1, ℤ) Eisenstein series again with s = (d + 1)/2. In this case, we obtain the high/low temperature duality through a suitably adapted dual parametrization of the lattice defining the torus. On 𝕋d+1, the computation is more subtle. An additional divergence leads to an harmonic anomaly.

scholarly journals Static black holes without spatial isometries: From AdS multipoles to electrovacuum scalarization

2020 ◽  
Vol 29 (11) ◽  
pp. 2041016
Author(s):  
Carlos Herdeiro ◽  
Eugen Radu
Keyword(s):  
Black Holes ◽  
Scalar Field ◽  
General Framework ◽  
Bound State ◽  
Back Reaction ◽  
Zero Modes ◽  
Spherical Topology ◽  
Continuous Symmetries

We review recent results on the existence of static black holes (BHs) without spatial isometries in four spacetime dimensions and propose a general framework for their study. These configurations are regular on and outside a horizon of spherical topology. Two different mechanisms allowing for their existence are identified. The first one relies on the presence of a solitonic limit of the BHs; when the solitons have no spatial isometries, the BHs, being a nonlinear bound state between the solitons and a horizon, inherit this property. The second one is related to BH scalarization, and the existence of zero modes of the scalar field without isometries around a spherical horizon. When the zero modes have no spatial isometries, the back-reaction of their nonlinear continuation makes the scalarized BHs inherit the absence of spatial continuous symmetries. A number of general features of the solutions are discussed together with possible generalizations.

scholarly journals A NOTE ON THE CASIMIR ENERGY OF A MASSIVE SCALAR FIELD IN POSITIVE CURVATURE SPACE

2004 ◽  
Vol 19 (02) ◽  
pp. 111-116 ◽  
Author(s):  
E. ELIZALDE ◽  
A. C. TORT
Keyword(s):  
High Temperature ◽  
Scalar Field ◽  
Zeta Function ◽  
Vacuum Energy ◽  
Positive Curvature ◽  
Zero Point ◽  
Casimir Energy ◽  
Massive Scalar ◽  
Zero Temperature ◽  
Massive Scalar Field

We re-evaluate the zero point Casimir energy for the case of a massive scalar field in R1×S3 space, allowing also for deviations from the standard conformal value ξ=1/6, by means of zero temperature zeta function techniques. We show that for the problem at hand this approach is equivalent to the high temperature regularization of the vacuum energy, as conjectured in a previous publication. The analytic continuation can be performed in two ways, which are seen to be equivalent.

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