Abstract
In this note, we calculate the one-loop determinant for a massive scalar (with conformal dimension $\Delta$) 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 $\Delta$ whose presence indicates a series of poles in the one-loop partition function $Z(\Delta)$ in the $\Delta$ complex plane; these poles contribute temperature-independent terms to the thermal AdS partition function computed in arXiv:0908.2657. Our results match those in a series of papers by Camporesi and Higuchi, as well as Gopakumar et al. in arXiv:1103.3627 and Banerjee et al. in arXiv:1005.3044. We additionally examine the meaning of these zero modes, finding that they Wick-rotate to quasinormal modes of the AdS$_2$ black hole. They are also interpretable as matrix elements of the discrete series representations of $SO(2,1)$ in the space of smooth functions on $S^1$. We generalize our results to general even dimensional AdS$_{2n}$, again finding a series of zero modes which are related to discrete series representations of $SO(2n,1)$, the motion group of $H_{2n}$.
Highlights
Several known ways to calculate the heat kernel
Working first in two dimensions on the related Euclidean hyperbolic plane H2, we find a series of zero modes for negative real values of ∆ whose presence indicates a series of poles in the one-loop partition function Z(∆) in the ∆ complex plane; these poles contribute temperature-independent terms to the thermal AdS partition function computed in [1]
In the case of AdS2, these zero modes are the Wick rotation of quasinormal modes for the AdS2 black hole
Summary
We use the zero-mode method to calculate the 1-loop partition function for a complex scalar φ with mass m in Euclidean AdS2 with AdS length L. We find the poles in Z(∆) by searching for values of ∆ where det −∇2 + m2 becomes zero These ∆⋆ occur whenever there exists a φ⋆ solving. Where φ⋆ is smooth in the interior of the (Euclidean) AdS, and satisfies the boundary condition (2.3). That is, at these (generally complex) ∆⋆, φ⋆ is a zero mode. As we will show for AdS2, even a noncompact space may have a discrete set of poles, when we choose the appropriate boundary condition. Find the polynomial Pol(∆) by matching the zeta-function regularization at large ∆ to the local heat kernel curvature expansion expression for log Z(∆)
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