Strict Total Positivity of a Gauss Hypergeometric Kernel, and the Sharp Threshold for the One-Dimensional $SL(2,\mathbb{R})$ Conformal Block

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 $\mathcal{F}(Δ,z) = {}_2F_1(Δ,Δ;2Δ;z)$ is proved to be strictly totally positive of order infinity for $Δ> 0$ and $z \in (0,1)$. Second, the sharp lower $Δ$-parameter threshold for the associated one-dimensional $SL(2,\mathbb{R})$ conformal block $G_Δ(z) = z^Δ\mathcal{F}(Δ,z)$ is shown to be $1/2$: the conformal-block kernel $G_Δ(z)$ is strictly totally positive of order infinity for $Δ\geq 1/2$. For every $τ\in (0,1/2)$, 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 $Δ> 0$.