Staggered Bosons, Kahler Dirac Bosons and Supersymmetry on the Lattice

I will describe constructions of lattice field theories that assign a single bosonic variable to each site, rather a conjugate pair x,p. The information to realize a non-trivial dynamics is realized by non-trivial Poisson brackets between nearest neighbors. The construction is similar to staggered fermions in 1+1 dimensions. 

I will show how this construction readily leads to critical field theories in various dimensions and realizes naturally certain non-invertible symmetries on the lattice fied theory. A more general version can be used on triangulations of manifolds, where the Poisson bracket utilizes the homology chain complex  of the triangulation. I will show that when coupled to  fermions this construction readiy gives rise to a supersymmetric Hamiltonian where the fermions are realized as Kahler-Dirac fermions.