Neural Networks, Dispersion Relations and the Thermal Bootstrap

TL;DR

This paper introduces a neural network-based conformal bootstrap framework that uses dispersion relations without positivity constraints, applied to thermal correlators in CFTs.

Contribution

It presents a novel bootstrap approach combining dispersion relations and neural networks, applicable to thermal two-point functions in conformal field theories.

Findings

Applied method to Generalized Free Fields and 4d holographic CFTs.

Analyzed stability of non-convex optimization scheme.

Discussed potential links to smoothness properties in CFT correlators.

Abstract

We review a framework for the conformal bootstrap that does not rely on positivity and treats the infinite tower of high-dimension OPE contributions to conformal correlators through dispersion relations and neural networks. We apply it to scalar thermal two-point functions on $S^1\times \mathbb R^{d-1}$. We discuss the stability properties of the relevant non-convex optimisation scheme and potential relations to recent discussions of smoothness properties in CFT correlators. We illustrate the numerical application of the method to Generalized Free Fields and 4d holographic CFTs. This is a proceedings contribution to the ``Athens Workshop in Theoretical Physics: 10th Anniversary", held at the National and Kapodistrian University of Athens on December 17-19 2025.