ODE/IM correspondence and supersymmetric affine Toda field equations

Abstract

We study the linear differential system associated with the supersymmetric affine Toda field equations for affine Lie superalgebras, which has a purely odd simple root system. For an affine Lie algebra, the linear problem modified by conformal transformation leads to an ordinary differential equation (ODE) that provides the functional relations in the integrable models. This is known as the ODE/IM correspondence. For the affine Lie superalgebras, the linear equations modified by a superconformal transformation are shown to reduce to a couple of ODEs for each bosonic subalgebra. In particular, for $osp(2,2{)}^{(2)}$, the corresponding ODE becomes the second-order ODE with squared potential, which is related to the $\mathcal{N}=1$ supersymmetric minimal model via the ODE/IM correspondence. We also find ODEs for classical affine Lie superalgebras with purely odd simple root systems.

arXiv comment28 pages; published version

References in corpus (22)

• Hidden Symmetries of the AdS_5 x S^5 Superstring
• Integrable Structure of Conformal Field Theory, Quantum KdV Theory and Thermodynamic Bethe Ansatz
• The ODE/IM Correspondence
• Anharmonic oscillators, the thermodynamic Bethe ansatz, and nonlinear integral equations
• Y-system for Scattering Amplitudes
• Spectral determinants for Schroedinger equation and Q-operators of Conformal Field Theory
• Spinons in Magnetic Chains of Arbitrary Spins at Finite Temperatures
• Pseudo-differential equations, and the Bethe Ansatz for the classical Lie algebras
• Representation theory of sl(2|1)
• Bethe Ansatz and the Spectral Theory of affine Lie algebra-valued connections I. The simply-laced case
• Analytic Bethe ansatz and functional equations for Lie superalgebra sl(r+1|s+1)
• Superconformal Field Theory and SUSY N=1 KdV Hierarchy II: The Q-operator
• Notes on parafermionic QFT's with boundary interaction
• Finite size effects and the supersymmetric sine-Gordon models
• ODE/IM correspondence and modified affine Toda field equations
• Superconformal Field Theory and SUSY N=1 KdV Hierarchy I: Vertex Operators and Yang-Baxter Equation
• Bethe Ansatz equations for the classical $A_n^{(1)}$ affine Toda field theories
• Analytic Bethe ansatz and functional relations related to tensor-like representations of type II Lie superalgebras B(r|s) and D(r|s)
• Wall-crossing of TBA equations and WKB periods for the third order ODE
• ODE/IM correspondence for affine Lie algebras: A numerical approach
• Massive ODE/IM Correspondence and Non-linear Integral Equations for $A_r^{(1)}$-type modified Affine Toda Field Equations
• Suzuki equations and integrals of motion for supersymmetric CFT

Cited by in corpus (5)

• Folding QQ-relations and transfer matrix eigenvalues: towards a unified approach to Bethe ansatz for super spin chains
• WKB analysis of the linear problem for modified affine Toda field equations
• Integrals of motion in conformal field theory with W-symmetry and the ODE/IM correspondence
• Complex D($2,1;\zeta$) and spin chain solutions from Chern-Simons theory
• The ODE/IM Correspondence between $C(2{)}^{(2)}$-type Linear Problems and 2d $\mathcal{N}=1$ SCFT