Supersymmetric lattice theories on curved space

Abstract: We show how to construct Hamiltonian lattice theories with one exact supersymmetry on arbitrary triangulations of curved space in any number of dimensions. Both bosons and fermions satisfy discrete K\"{a}hler-Dirac equations. The quantization of the fermions proceeds by imposing conventional anti-commutation relations while the bosons require a modification of the usual canonical commutator. On regular lattices we construct parity, time reversal and translation-by-one (shift) symmetries. We argue that the latter are generically non-invertible symmetries. We also show how to couple these degrees of freedom to background gauge fields which leads to a theory with enhanced supersymmetry.