Quantum integrability of coupled N=1 super sine/sinh-Gordon theories and the Lie superalgebra D(2,1;alpha)

We discuss certain integrable quantum field theories in 1 + 1 dimensions co nsisting of coupled sine/sinh-Gordon theories with N = 1 supersymmetry, pos itive 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; alpha)((1)) (or on relat ed algebras which can be obtained as various limits) provided one adopts ap propriate reality conditions for the fields. In particular, there is a cont inuous family of such models in which the couplings and mass ratios all dep end on the parameter alpha. The structure of these models is analyzed in so me detail at the classical level, including the construction of conserved c urrents with spins up to 4. We then show that these currents generalize to the quantum theory, thus demonstrating quantum-integrability of the models.