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 AdS2. 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.
Highlights
Some of the most exciting consequences of the conformal bootstrap equations are constraints on the low-lying spectrum of operators
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
Summary
We start by explaining the basic idea of the conformal bootstrap. See [19, 20] for more complete reviews. It vanishes at ∆ = 0 and has a first-order zero and positive slope at the lowest non-identity operator with dimension ∆∗. Higher-lying scalar operators in the spectrum sit at second-order zeros since the functional must vanish there without ever becoming negative for ∆ > ∆∗. We expect both the extremal functional and the corresponding extremal solution of (2.6) to be unique up to an overall positive rescaling. There is no reason for all the zeros of the extremal functional to correspond to operators appearing in the solution to crossing with nonzero OPE coefficient. The extremal functionals for the 1D bootstrap bound will be shown to naturally lead to the physics of QFT in AdS2 of large radius when the external scaling dimensions are large
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