# The Analytic Functional Bootstrap I: 1D CFTs and 2D S-Matrices

## TL;DR

This paper develops an analytic approach to the 1D conformal bootstrap, identifying extremal functionals related to generalized free fermions and large-dimension limits, revealing connections to 2D S-matrices and integrable theories.

## Contribution

It introduces explicit extremal functionals for 1D CFTs, linking bootstrap bounds to integrable models and 2D S-matrix bootstrap results.

## Key findings

- Analytically finds extremal functionals for generalized free fermion spectrum.
- Establishes a connection between 1D conformal bootstrap and 2D S-matrix bootstrap.
- Demonstrates extremal solutions arising from integrable field theories in large AdS2 limit.

## Abstract

We study a general class of functionals providing an analytic handle on the conformal bootstrap equations in one dimension. We explicitly identify the extremal functionals, corresponding to theories saturating conformal bootstrap bounds, in two regimes. The first corresponds to functionals that annihilate the generalized free fermion spectrum. In this case, we analytically find both OPE and gap maximization functionals proving the extremality of the generalized free fermion solution to crossing. Secondly, we consider a scaling limit where all conformal dimensions become large, equivalent to the large $AdS$ radius limit of gapped theories in $AdS_2$. In this regime we demonstrate analytically that optimal bounds on OPE coefficients lead to extremal solutions to crossing arising from integrable field theories placed in large $AdS_2$. In the process, we uncover a close connection between asymptotic extremal functionals and S-matrices of integrable field theories in flat space and explain how 2D S-matrix bootstrap results can be derived from the 1D conformal bootstrap equations. These points illustrate that our formalism is capable of capturing non-trivial solutions of CFT crossing.