Super Sum rules for Long-Range Models

TL;DR

This paper develops sum rules linking Regge limit behavior of 1D CFT correlators to dual bulk scattering in AdS2, identifying special solutions for long-range models and testing them across various theories.

Contribution

It introduces new sum rules for 1D CFTs related to bulk scattering, and demonstrates their effectiveness in constraining theories and predicting CFT data.

Findings

Sum rules accurately predict CFT data for long-range models.

Quadruple-twist operators significantly influence the sum rules.

Imposing sum rules reduces the allowed parameter space in numerical bootstrap.

Abstract

We study sum rules that control the Regge limit of one-dimensional conformal field theory (CFT) correlators and relate them to dual bulk scattering processes at high energies in $\mathrm{AdS}_2$. By imposing the condition that no scattering takes place in the bulk, these sum rules single out special solutions to crossing symmetry that describe long-range models, which can be understood as free fields in AdS with boundary interactions tuned to criticality. We test these sum rules perturbatively in several distinct theories, namely the 1d long-range versions of the Ising, $O(N)$ and Lee--Yang models, and find that they correctly predict the CFT data characterising these theories. Along the way we compute for the first time the leading contributions of quadruple-twist operators to the long range Ising correlator and analyse their role in the new sum rules. Finally, we explore the consequences of imposing these sum rules in a numerical bootstrap framework and find that they lead to substantial reductions in the allowed parameter space.