# Inverse Bootstrapping Conformal Field Theories

## TL;DR

This paper introduces a new method for studying conformal field theories by interpreting crossing-symmetric functions as generating functions of conformal data, enabling the analysis of various known CFTs through a reversed bootstrap approach.

## Contribution

It proposes a novel inverse bootstrap approach that interprets crossing-symmetric functions as generating functions, simplifying the study of conformal data in diverse CFTs.

## Key findings

- Consistent with minimal fusion rule relations in multiple CFTs
- Derived approximate relations for low-lying operator conformal data
- Validated the approach with known theories like Ising and Wilson-Fisher CFTs

## Abstract

We propose a novel approach to study conformal field theories (CFTs) in general dimensions. In the conformal bootstrap program, one usually searches for consistent CFT data that satisfy crossing symmetry. In the new approach, we reverse the logic and interpret manifestly crossing-symmetric functions as generating functions of conformal data. Physical CFTs can be obtained by scanning the space of symmetric functions. By truncating the fusion rules, we are able to concentrate on the low-lying operators and derive some approximate relations for their conformal data. It turns out that the free scalar theory, the 2d minimal model CFTs, the $\phi^{4}$ Wilson-Fisher CFT, the Lee-Yang CFTs and the Ising CFTs are consistent with the universal relations from the minimal fusion rule $\phi_1\times \phi_1=I+\phi_2+T$, where $\phi_1,\,\phi_2$ are scalar operators, $I$ is the identity operator and $T$ is the stress tensor.