Exact Schwarzian Metric Factor and Holographic Wilson-Loop Screening
Abstract
We evaluate exactly the radial metric factor h(ζ) generated by Schwarzian averaging in the AdS2 throat of an extremal Reissner--Nordström AdS5 black brane. The result is a Gaussian integral against coth(πy), valid at all radial depths, which Mordell's identity turns into an exact Appell--Lerch q/q∗-series representation. The dual series identifies the nonperturbative scale e−π2 C/ζ missed by any finite near-boundary truncation. The third parametric derivative required by the evaluation generates the quasimodular Eisenstein series E2, absent from the classical Mordell identity. From the integral representation we prove that 𝒢0(ζ):=h(ζ)/ζ2 is completely monotone and hence has no interior minimum, so any confining minimum produced by a finite near-boundary truncation is an artifact. We also compute the exact relative variance of the Schwarzian kernel, which makes the averaged-metric approximation error quantitative and shows that the absence of a confining minimum is robust across moment-based effective geometries. Applied to the temporal rectangular Wilson loop, the exact throat gives algebraic screening, E(L)∼ −κIR/L2, the Wilson-loop diagnostic of the semi-local quantum-liquid IR of the extremal RN brane. A numerical check in a simple matched geometry confirms that the screened saddle is the dominant string configuration, and an exact-versus-truncated force comparison shows that the apparent constant-force regime of the fourth-order truncation is not a feature of the exact geometry.
Versions
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BibTeX
@misc{tierz2026exact,
title = {{Exact Schwarzian Metric Factor and Holographic Wilson-Loop Screening}},
author = {Miguel Tierz},
howpublished = {arXiv},
year = {2026},
month = jul,
doi = {10.48550/arxiv.2607.03120},
eprint = {2607.03120},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2607.03120},
}
RIS
TY - UNPB
TI - Exact Schwarzian Metric Factor and Holographic Wilson-Loop Screening
AU - Miguel Tierz
PY - 2026
DA - 2026/07/03
PB - arXiv
DO - 10.48550/arxiv.2607.03120
UR - https://arxiv.org/abs/2607.03120
AB - We evaluate exactly the radial metric factor h(ζ) generated by Schwarzian averaging in the AdS_2 throat of an extremal Reissner--Nordström AdS_5 black brane. The result is a Gaussian integral against coth(πy), valid at all radial depths, which Mordell's identity turns into an exact Appell--Lerch q/q^∗-series representation. The dual series identifies the nonperturbative scale e^−π^2 C/ζ missed by any finite near-boundary truncation. The third parametric derivative required by the evaluation generates the quasimodular Eisenstein series E_2, absent from the classical Mordell identity. From the integral representation we prove that 𝒢_0(ζ):=h(ζ)/ζ^2 is completely monotone and hence has no interior minimum, so any confining minimum produced by a finite near-boundary truncation is an artifact. We also compute the exact relative variance of the Schwarzian kernel, which makes the averaged-metric approximation error quantitative and shows that the absence of a confining minimum is robust across moment-based effective geometries. Applied to the temporal rectangular Wilson loop, the exact throat gives algebraic screening, E(L)∼ −κ_IR/L^2, the Wilson-loop diagnostic of the semi-local quantum-liquid IR of the extremal RN brane. A numerical check in a simple matched geometry confirms that the screened saddle is the dominant string configuration, and an exact-versus-truncated force comparison shows that the apparent constant-force regime of the fourth-order truncation is not a feature of the exact geometry.
ER -