Analytic bounds and emergence of AdS2 physics from the conformal bootstrap

Abstract

We study analytically the constraints of the conformal bootstrap on the lowlying spectrum of operators in field theories with global conformal symmetry in one and two spacetime dimensions. We introduce a new class of linear functionals acting on the conformal bootstrap equation. In 1D, we use the new basis to construct extremal functionals leading to the optimal upper bound on the gap above identity in the OPE of two identical primary operators of integer or half-integer scaling dimension. We also prove an upper bound on the twist gap in 2D theories with global conformal symmetry. When the external scaling dimensions are large, our functionals provide a direct point of contact between crossing in a 1D CFT and scattering of massive particles in large AdS 2 . In particular, CFT crossing can be shown to imply that appropriate OPE coefficients exhibit an exponential suppression characteristic of massive bound states, and that the 2D flat-space S-matrix should be analytic away from the real axis.

Cut-touching linear functionals in the conformal bootstrap

“…6 In this respect the notation of eqs. (2.8) and (2.22) of [12] is confusing, while that in [13] is OK. the conformal blocks in 1d are known explicitly. Property P2 is more tricky.…”

“…3 Recently Mazáč [12] introduced a new class of linear functionals. Unlike the functionals of [5], his functionals involve integrals of f over regions of the z space approaching the analyticity cuts where the conformal block decomposition seizes to converge.…”

“…The functionals like in [12], or even more complicated ones, may well become widespread in the conformal bootstrap. Anticipating these developments, we will show here a minimal standard of rigor in dealing with such functionals.…”

“…We derive some general criteria guaranteeing that such functionals obey finiteness and swapping. In section 5 we use our criteria to prove swapping for the functionals used in [12], at least for…”

“…the particular cases of low-lying ∆ φ where [12] provides sufficient details. The general case remains incomplete.…”