Free massive fermions inside the quantum discrete sine-Gordon model

We extend the notion of space shifts introduced in [FV3] for certain quantum light cone lattice equations of sine-Gordon type at root of unity (e.g. [FV1, FV2, BKP, BBR]). As a result, we obtain a compatibility equation for the roots of central elements: within the algebra of observables (also called current algebra). The equation, which is obtained by exponentiating these roots, is exactly the evolution equation for the "classical background" as described in [BBR]. As an application for the introduced constructions, we derive a,one to one correspondence between a special case of the quantum light cone lattice equations of sine-Gordon type and free massive fermions on a lattice, as a special case of the lattice Thirring model constructed in [DV].