Extensions of the Geometric Subsystem Programme:Fermionic zero mode on a kink and translational modes of a domain wall in (2+1)D

We present two extensions of the geometric subsystem quantisation programme beyond the pure scalar kink in $(1+1)$ dimensions. First, we include a fermionic zero mode in the Jackiw--Rebbi model, obtaining a super‑moduli space whose symplectic form is the direct sum of the bosonic translational form da dP and the fermionic form i\,d c d c. Second, we lift the scalar kink to a $(2+1)$-dimensional domain wall, where the normal translation yields a relativistic particle and the tangential translation yields a decoupled cyclic degree of freedom, whose conjugate momentum vanishes in the thermodynamic limit. Both extensions are rigorous and demonstrate the programme's ability to handle fermionic and higher‑dimensional topological defects.