Quantum Integrability of Coupled N=1 Super Sine/Sinh-Gordon Theories and the Lie Superalgebra D(2,1;\A)

TL;DR

This paper constructs and analyzes a family of integrable (1+1)-dimensional supersymmetric quantum field theories based on the affine Lie superalgebra D(2,1;A), demonstrating their classical and quantum integrability through conserved currents.

Contribution

It introduces a new class of integrable supersymmetric models derived from the affine Lie superalgebra D(2,1;A), including their classical construction and quantum integrability proof.

Findings

Models are constructed as Toda theories based on D(2,1;A)^{(1)}.

Classical conserved currents up to spin 4 are explicitly constructed.

Quantum conserved currents confirm the models' quantum integrability.

Abstract

We discuss certain integrable quantum field theories in (1+1)-dimensions consisting of coupled sine/sinh-Gordon theories with N=1 supersymmetry, positive kinetic energy, and bosonic potentials which are bounded from below. We show that theories of this type can be constructed as Toda models based on the exceptional affine Lie superalgebra $D(2,1;\A)^{(1)}$ (or on related algebras which can be obtained as various limits) provided one adopts appropriate reality conditions for the fields. In particular, there is a continuous family of such models in which the couplings and mass ratios all depend on the parameter $\A$. The structure of these models is analyzed in some detail at the classical level, including the construction of conserved currents with spins up to 4. We then show that these currents generalize to the quantum theory, thus demonstrating quantum-integrability of the models.