Geometric aspects of the ODE/IM correspondence

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.)

arXiv commentPartially based on lectures given at the Young Researchers Integrability School 2017,in Dublin. 62 pages, 8 figures. Minor amendments. Version published in J. Phys A "Special Issue on Recent Advances in AdS/CFT Integrability"

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Cited by in corpus (11)

• WKB periods for higher order ODE and TBA equations
• ODE/IM correspondence for affine Lie algebras: A numerical approach
• Deforming the ODE/IM correspondence with $\mathrm{T}\bar{\mathrm{T}}$
• On the origin of the correspondence between classical and quantum integrable theories
• Power-like potentials: from the Bohr-Sommerfeld energies to exact ones
• On Chebyshev Wells: Periods, Deformations, and Resurgence
• On solutions of the Bethe Ansatz for the Quantum KdV model
• Spectrum of quantum KdV hierarchy in the semiclassical limit
• KPZ fluctuations in finite volume
• Complete Minimal Form Factors for Irrelevant Deformations of Integrable Quantum Field Theory
• $\text{T}\bar{\text{T}}$-deformed Nonlinear Schrödinger