Analytic Bounds and Emergence of $\textrm{AdS}_2$ Physics from the Conformal Bootstrap

TL;DR

This work develops analytic, basis-oriented conformal bootstrap functionals that yield explicit, verifiable bounds in 1D and 2D CFTs. By replacing derivative-based functionals with integral kernels supported on the branch cut and expanding in a Legendre basis, the authors construct extremal functionals that saturate the 1D scalar-gap bound Δ* = 2Δ_ψ + 1 for Δ_ψ ∈ {integers, half-integers} and derive a 2D twist bound τ ≤ 2Δ_φ + 2 analytically. In the large Δ_ψ limit, these functionals reveal a direct AdS_2 interpretation, predicting exponential suppression of bound-state OPE coefficients and enforcing S-matrix analyticity away from the real axis. The paper also generalizes the construction to 2D using a factorized tensor-product basis, obtaining an analytic bound in 2D and outlining implications for Virasoro-symmetric theories and defects. Overall, the results provide a concrete analytic bridge between conformal bootstrap constraints and bulk AdS_2 physics, with potential implications for higher-dimensional islands and mixed-correlator bootstrap.

Abstract

We study analytically the constraints of the conformal bootstrap on the low-lying 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 $\textrm{AdS}_2$. 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.

Figures (12)

• Figure 1: The action of a typical extremal functional for the bound on the scalar gap on $F_{\Delta,0}$. The leading non-identity operator appears at a first-order zero with a positive slope, while higher operators lie at second-order zeros.
• Figure 2: Black dots: Numerical bootstrap bound on the gap above identity following from \ref{['eq:be1D']}, using 50 derivatives. Red dashed line: $\tilde{\Delta}=2\Delta_\psi + 1$. Figure taken from DefectBootstrap.
• Figure 3: The transformation \ref{['eq:ycoordinate']} between the $z$ and $y$ coordinates. $z$-derivatives evaluated at $z=1/2$ can reconstruct the values of $F_{\Delta}(z)$ only in the dark-blue region, while $y$-derivatives at $y=0$ can reconstruct the values everywhere away from the branch cuts.
• Figure 4: The choice of integration contour leading to a well-defined action of $\omega$, equation \ref{['eq:functionalAction2']}.
• Figure 5: The integral kernel $h(x)$ for $\Delta_\psi=1/2$, given by equation \ref{['eq:hXHalf']}.