Conserved Currents in the Six-Vertex and Trigonometric Solid-On-Solid Models

We construct quasi-local conserved currents in the six-vertex model with anisotropy parameter $\eta$ by making use of the quantum-group approach of Bernard and Felder. From these currents, we construct parafermionic operators with spin $1+i\eta/\pi$ that obey a discrete-integral condition around lattice plaquettes embedded into the complex plane. These operators are identified with primary fields in a $c=1$ compactified free Boson conformal field theory. We then consider a vertex-face correspondence that takes the six-vertex model to a trigonometric SOS model, and construct SOS operators that are the image of the six-vertex currents under this correspondence. We define corresponding SOS parafermionic operators with spins $s=1$ and $s=1+2i\eta/\pi$ that obey discrete integral conditions around SOS plaquettes embedded into the complex plane. We consider in detail the cyclic-SOS case corresponding to the choice $\eta=i\pi (p-p')/p$, with $p'<p$ coprime. We identify our SOS parafermionic operators in terms of the screening operators and primary fields of the associated $c=1-6(p-p')^2/pp'$ conformal field theory.