Differential equations and integrable models: the SU(3) case

Abstract

We exhibit a relationship between the massless $a_2^{(2)}$ integrable quantum field theory and a certain third-order ordinary differential equation, thereby extending a recent result connecting the massless sine-Gordon model to the Schrödinger equation. This forms part of a more general correspondence involving $A_2$-related Bethe ansatz systems and third-order differential equations. A non-linear integral equation for the generalised spectral problem is derived, and some numerical checks are performed. Duality properties are discussed, and a simple variant of the nonlinear equation is suggested as a candidate to describe the finite volume ground state energies of minimal conformal field theories perturbed by the operators ${\varphi }_{12}$, ${\varphi }_{21}$ and ${\varphi }_{15}$. This is checked against previous results obtained using the thermodynamic Bethe ansatz.

25 pages, Latex 2e, 3 figures, uses graphics, cite. v2: Typos corrected, section 6 expanded and an error fixed,references added v3: Notes and references added, to appear in Nucl. Phys. B

References in corpus (4)

• On the relation between Stokes multipliers and the T-Q systems of conformal field theory
• Functional Relations in Stokes Multipliers and Solvable Models related to U_q(A^{(1)}_n)
• Nonlinear Integral Equation and Finite Volume Spectrum of Minimal Models Perturbed by ${\Phi }_{(1,3)}$
• Airy functions in the thermodynamic Bethe ansatz

Cited by in corpus (37)

• Spectral equivalences, Bethe Ansatz equations, and reality properties in PT-symmetric quantum mechanics
• The ODE/IM Correspondence
• Integrable structure of W_3 Conformal Field Theory, Quantum Boussinesq Theory and Boundary Affine Toda Theory
• Pseudo-differential equations, and the Bethe Ansatz for the classical Lie algebras
• Differential equations for general SU(n) Bethe ansatz systems
• Functional relations in Stokes multipliers - Fun with $x^6+\alpha x^2$ potential-
• New families of flows between two-dimensional conformal field theories
• Geometric aspects of the ODE/IM correspondence
• Finite size effects and the supersymmetric sine-Gordon models
• A reality proof in PT-symmetric quantum mechanics
• ODE/IM correspondence and modified affine Toda field equations
• The Bethe Ansatz and the Tzitzéica-Bullough-Dodd equation
• TBA equations for the Schrödinger equation with a regular singularity
• Bethe Ansatz equations for the classical $A_n^{(1)}$ affine Toda field theories
• ODE/IM correspondence and Bethe ansatz for affine Toda field equations
• Finite lattice Bethe ansatz systems and the Heun equation
• WKB periods for higher order ODE and TBA equations
• Integrable aspects of the scaling q-state Potts models II: finite-size effects
• Integrable differential systems of topological type and reconstruction by the topological recursion
• Wall-crossing of TBA equations and WKB periods for the third order ODE
• Ordinary Differential Equations and Integrable Models
• Massless flows between minimal W models
• Deforming the ODE/IM correspondence with $\mathrm{T}\bar{\mathrm{T}}$
• ODE/IM correspondence for affine Lie algebras: A numerical approach
• $T$, $Q$ and periods in $SU(3)$ $\mathcal{N}=2$ SYM
• RSOS revisited
• Opers for higher states of the quantum Boussinesq model
• The Geometry of integrable systems. Tau functions and homology of Spectral curves. Perturbative definition
• Massive ODE/IM Correspondence and Non-linear Integral Equations for $A_r^{(1)}$-type modified Affine Toda Field Equations
• Stokes Multipliers, Spectral Determinants and T-Q relations
• Loop equations from differential systems
• On the ODE/IM correspondence for minimal models
• Elementary functions in Thermodynamic Bethe Ansatz
• Integrals of motion in conformal field theory with W-symmetry and the ODE/IM correspondence
• Quasi-exact solvability, resonances and trivial monodromy in ordinary differential equations
• The ODE/IM Correspondence between $C(2{)}^{(2)}$-type Linear Problems and 2d $\mathcal{N}=1$ SCFT
• The ODE/IM correspondence and PT-symmetric quantum mechanics