AdS bulk locality from sharp CFT bounds

It is a long-standing conjecture that any CFT with a large central charge and a large gap ∆gap in the spectrum of higher-spin single-trace operators must be dual to a local effective field theory in AdS. We prove a sharp form of this conjecture by deriving numerical bounds on bulk Wilson coefficients in terms of ∆gap using the conformal bootstrap. Our bounds exhibit the scaling in ∆gap expected from dimensional analysis in the bulk. Our main tools are dispersive sum rules that provide a dictionary between CFT dispersion relations and S-matrix dispersion relations in appropriate limits. This dictionary allows us to apply recently-developed flat-space methods to construct positive CFT functionals. We show how AdS4 naturally resolves the infrared divergences present in 4D flat-space bounds. Our results imply the validity of twice-subtracted dispersion relations for any S-matrix arising from the flat-space limit of AdS/CFT.

Smoothness of solutions of a convolution equation of restricted type on the sphere

Let $\mathbb {S}^{d-1}$ denote the unit sphere in Euclidean space $\mathbb {R}^d$ , $d\geq 2$ , equipped with surface measure $\sigma _{d-1}$ . An instance of our main result concerns the regularity of solutions of the convolution equation $$\begin{align*}a\cdot(f\sigma_{d-1})^{\ast {(q-1)}}\big\vert_{\mathbb{S}^{d-1}}=f,\text{ a.e. on }\mathbb{S}^{d-1}, \end{align*}$$ where $a\in C^\infty (\mathbb {S}^{d-1})$ , $q\geq 2(d+1)/(d-1)$ is an integer, and the only a priori assumption is $f\in L^2(\mathbb {S}^{d-1})$ . We prove that any such solution belongs to the class $C^\infty (\mathbb {S}^{d-1})$ . In particular, we show that all critical points associated with the sharp form of the corresponding adjoint Fourier restriction inequality on $\mathbb {S}^{d-1}$ are $C^\infty $ -smooth. This extends previous work of Christ and Shao [4] to arbitrary dimensions and general even exponents and plays a key role in the companion paper [24].

A conjecture on the minimal size of bound states

We conjecture that, in a renormalizable effective quantum field theory where the heaviest stable particle has mass mm, there are no bound states with radius below 1/m1/m (Bound State Conjecture). We are motivated by the (scalar) Weak Gravity Conjecture, which can be read as a statement forbidding certain bound states. As we discuss, versions for uncharged particles and their generalizations have shortcomings. This leads us to the suggestion that one should only constrain rather than exclude bound objects. In the gravitational case, the resulting conjecture takes the sharp form of forbidding the adiabatic construction of black holes smaller than 1/m1/m. But this minimal bound-state radius remains non-trivial as M_\mathrm{P}\to \inftyMP→∞, leading us to suspect a feature of QFT rather than a quantum gravity constraint. We find support in a number of examples which we analyze at a parametric level.

Valuable Conflictness, Coping – strategy and temperament

In the article results of empirical research of connections between a valuable conflictness, coping‑strategy and properties of temperament are presented. Some connections with the sharp form of a valuable conflictness which we name semantic deadlock are certain also.

Sharp form of inequality for the constant e

(i) We determine the best possible constants α and β such that the inequalities ... are valid for all integers n ≥ 1...We determine the values a, b which provide the fastest sequence (vn)n≥1 approximating the constant e. (ii) Let the sequence vn be defined by