  We describe a novel way to think about bosonic lattice theories in
Hamiltonian form where each lattice site has only a half boson degree of
freedom. The construction requires a non-trivial Poisson bracket between
neighboring sites and leads to gapless theories with non-invertible symmetries.
We also describe a bosonic version of Kahler-Dirac fermions, dubbed
Kahler-Dirac bosons that can be performed on any triangulation of a manifold.
This also leads to a straightforward implementation of supersymmetry on the
lattice and one immediately deduces the Dirac equation of the corresponding
Kahler-Dirac fermions.
