Geometric aspects of the ODE/IM correspondence

TL;DR

This review explores the geometric connections between classical integrable models and quantum field theories through the ODE/IM correspondence, highlighting recent off-critical generalizations and their implications in AdS/CFT.

Contribution

It provides a comprehensive overview of the off-critical extension of the ODE/IM correspondence and its geometric interpretation involving surfaces and Bethe Ansatz equations.

Findings

Link between Lax operators and embedded surfaces

Extension of ODE/IM correspondence off-critical

Implications for AdS/CFT and minimal surfaces

Abstract

This review describes a link between Lax operators, embedded surfaces and Thermodynamic Bethe Ansatz equations for integrable quantum field theories. This surprising connection between classical and quantum models is undoubtedly one of the most striking discoveries that emerged from the off-critical generalisation of the ODE/IM correspondence, which initially involved only conformal invariant quantum field theories. We will mainly focus of the KdV and sinh-Gordon models. However, various aspects of other interesting systems, such as affine Toda field theories and non-linear sigma models, will be mentioned. We also discuss the implications of these ideas in the AdS/CFT context, involving minimal surfaces and Wilson loops. This work is a follow-up of the ODE/IM review published more than ten years ago by JPA, before the discovery of its off-critical generalisation and the corresponding geometrical interpretation. (Partially based on lectures given at the Young Researchers Integrability School 2017, in Dublin.)