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The ODE/IM Correspondence

Patrick Dorey, Clare Dunning, Roberto Tateo

Abstract

This article reviews a recently-discovered link between integrable quantum field theories and certain ordinary differential equations in the complex domain. Along the way, aspects of PT-symmetric quantum mechanics are discussed, and some elementary features of the six-vertex model and the Bethe ansatz are explained.

The ODE/IM Correspondence

Abstract

This article reviews a recently-discovered link between integrable quantum field theories and certain ordinary differential equations in the complex domain. Along the way, aspects of PT-symmetric quantum mechanics are discussed, and some elementary features of the six-vertex model and the Bethe ansatz are explained.

Paper Structure

This paper contains 43 sections, 353 equations, 27 figures.

Table of Contents

  1. Introduction
  2. Prelude: three reality conjectures in ${\cal P}{\cal T}$-symmetric quantum mechanics
  3. Integrable models and functional relations
  4. Generalities
  5. The six-vertex model and Bethe ansatz equations
  6. Adding a twist
  7. The $XXZ$ model
  8. Baxter's TQ relation
  9. The quantum Wronskian
  10. The fusion hierarchy and its truncation
  11. Continuum limit of lattice models
  12. TQ equations in continuum CFT: the BLZ approach
  13. Summary
  14. Ordinary differential equations and functional relations
  15. General eigenvalue problems in the complex plane
  16. ...and 28 more sections

Figures (27)

  • Figure 1: A wavefunction decaying at $x=\pm\infty$.
  • Figure 2: ${\cal H}_M=p^2-(ix)^{2M}$ : real eigenvalues as a function of $M$.
  • Figure 3: ${\cal H}_{M,l}=p^2-(ix)^{2M}+l(l{+}1)\,x^{-2}$ : real eigenvalues as a function of $M$ for $l=-0.025$, $l(l{+}1)=-0.024735$.
  • Figure 4: Real eigenvalues of $p^2-(ix)^{2M}+l(l{+}1)/x^2$ as functions of $M$, for various values of $l$.
  • Figure 5: ${\cal H}_{3,\alpha,l}$ : region of unreality in the $(\alpha,\rho)$ plane, where $\rho=\sqrt{3}(2l{+}1)$.
  • ...and 22 more figures