Lectures on Integrable Structures in Quantum Field Theory and Massive ODE/IM Correspondence

Abstract

This review was born as notes for a lecture given at the YRIS school on integrability in Durham, in the summer of 2015. It deals with a beautiful method, developed in the mid-nineties by V.V. Bazhanov, S.L. Lukyanov and A.B. Zamolodchikov and, as such, called BLZ. This method can be interpreted as a field theory version of the quantum inverse scattering (QIS), also known as algebraic Bethe ansatz (ABA). Starting with the case of conformal field theories (CFT) we show how to build the field theory analogues of commuting transfer T matrices and Baxter Q-operators of integrable lattice models. These objects contain the complete information of the integrable structure of the theory, viz. the integrals of motion, and can be used, as we will show, to derive the thermodynamic Bethe ansatz (TBA) and non-linear integral (NLIE) equations. This same method can be easily extended to the description of integrable structures of certain particular massive deformations of CFTs; these, in turn, can be described as quantum group reductions of the quantum sine-Gordon model and it is an easy step to include this last theory in the framework of BLZ approach. Finally we show an interesting and surprising connection of the BLZ structures with classical objects emerging from the study of classical integrable models via the inverse scattering transform method. This connection goes under the name of ODE/IM correspondence and we will present it for the specific case of quantum sine-Gordon model only.

51 pages, 4 figures. Minor revision, updated references

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Cited by in corpus (14)

• Lectures on Yangian Symmetry
• An Integrability Primer for the Gauge-Gravity Correspondence: an Introduction
• Introduction to the thermodynamic Bethe ansatz
• Flow equations for generalised $T\bar{T}$ deformations
• Quasilocal charges and progress towards the complete GGE for field theories with non-diagonal scattering
• On KdV characters in large c CFTs
• Massive ODE/IM Correspondence and Non-linear Integral Equations for $A_r^{(1)}$-type modified Affine Toda Field Equations
• A study of integrable form factors in massless relativistic AdS_3
• On $Ad{S}_2/CF{T}_1$ transfer matrices, Bethe ansatz and scale invariance
• Kondo line defects and affine Gaudin models
• Demystifying integrable QFTs in AdS: No-go theorems for higher-spin charges
• Deterministic Chaos vs Integrable Models
• Notes on index of quantum integrability
• Complete Minimal Form Factors for Irrelevant Deformations of Integrable Quantum Field Theory