Strict Total Positivity of a Gauss Hypergeometric Kernel, and the Sharp Threshold for the One-Dimensional SL(2,ℝ) Conformal Block
Abstract
In this paper, we completely resolve the two total positivity problems raised by Li [J. High Energy Phys., 2023(7):Paper No. 167, 44 pp., 2023], thereby determining the precise scope of this positivity structure in the one-dimensional conformal bootstrap. First, the Gauss hypergeometric kernel ℱ(Δ,z) = 2F1(Δ,Δ;2Δ;z) is proved to be strictly totally positive of order infinity for Δ> 0 and z ∈ (0,1). Second, the sharp lower Δ-parameter threshold for the associated one-dimensional SL(2,ℝ) conformal block GΔ(z) = zΔℱ(Δ,z) is shown to be 1/2: the conformal-block kernel GΔ(z) is strictly totally positive of order infinity for Δ≥ 1/2. For every τ∈ (0,1/2), there exists a strictly negative odd-order minor of GΔ(z) such that all its Δ-values are in (τ,1/2) and all its z-values can be chosen arbitrarily close to one. Consequently, no restriction z > z0 with z0 < 1 can restore total positivity of order infinity over all Δ> 0.
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BibTeX
@misc{ouimet2026strict,
title = {{Strict Total Positivity of a Gauss Hypergeometric Kernel, and the Sharp Threshold for the One-Dimensional $SL(2,\mathbbR)$ Conformal Block}},
author = {Frédéric Ouimet and Donald Richards},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.21210},
eprint = {2609.21210},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.21210},
}
RIS
TY - UNPB
TI - Strict Total Positivity of a Gauss Hypergeometric Kernel, and the Sharp Threshold for the One-Dimensional SL(2,ℝ) Conformal Block
AU - Frédéric Ouimet
AU - Donald Richards
PY - 2026
DA - 2026/09/18
PB - arXiv
DO - 10.48550/arxiv.2609.21210
UR - https://arxiv.org/abs/2609.21210
AB - In this paper, we completely resolve the two total positivity problems raised by Li [J. High Energy Phys., 2023(7):Paper No. 167, 44 pp., 2023], thereby determining the precise scope of this positivity structure in the one-dimensional conformal bootstrap. First, the Gauss hypergeometric kernel ℱ(Δ,z) = _2F_1(Δ,Δ;2Δ;z) is proved to be strictly totally positive of order infinity for Δ> 0 and z ∈ (0,1). Second, the sharp lower Δ-parameter threshold for the associated one-dimensional SL(2,ℝ) conformal block G_Δ(z) = z^Δℱ(Δ,z) is shown to be 1/2: the conformal-block kernel G_Δ(z) is strictly totally positive of order infinity for Δ≥ 1/2. For every τ∈ (0,1/2), there exists a strictly negative odd-order minor of G_Δ(z) such that all its Δ-values are in (τ,1/2) and all its z-values can be chosen arbitrarily close to one. Consequently, no restriction z > z_0 with z_0 < 1 can restore total positivity of order infinity over all Δ> 0.
ER -