Abstract
We study the zeta-function regularization of functional determinants of Laplace and Dirac-type operators in two-dimensional Euclidean AdS2 space. More specifically, we consider the ratio of determinants between an operator in the presence of background fields with circular symmetry and the free operator in which the background fields are absent. By Fourier-transforming the angular dependence, one obtains an infinite number of one-dimensional radial operators, the determinants of which are easy to compute. The summation over modes is then treated with care so as to guarantee that the result coincides with the two-dimensional zeta-function formalism. The method relies on some well-known techniques to compute functional determinants using contour integrals and the construction of the Jost function from scattering theory. Our work generalizes some known results in flat space. The extension to conformal AdS2 geometries is also considered. We provide two examples, one bosonic and one fermionic, borrowed from the spectrum of fluctuations of the holographic frac{1}{4} -BPS latitude Wilson loop.
Highlights
For general observables, semiclassical physics is our only systematic approach to probe the AdS/CFT correspondence beyond the leading classical limit
We study the zeta-function regularization of functional determinants of Laplace and Dirac-type operators in two-dimensional Euclidean AdS2 space
We apply the methods developed here to two examples borrowed from the literature on holographic Wilson loops [24, 25, 27]
Summary
It should be clear from the outset that, even though we use the same notation, m, q, V and Aμ need not be the same for bosons and fermions In the latter case we have included an extra connection, dΩ (notice the absence of i, implying it cannot be gauged away), whose origin is motivated by thinking of these operators as coming from some other geometry that is conformal to AdS2. The background fields must behave in such a way that all the integrals appearing below are finite These fall-off conditions imply that the operators become effectively free for large ρ,. Upon Fourier-transforming the τ dependence the relevant radial operators become The trouble with this expression, is that, even though the ratio det Ol det Olfree is well defined, the sum over Fourier modes typically diverges.
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