Scalar, fermionic and supersymmetric field theories with subsystem symmetries in d+1 dimensions

We study various non-relativistic field theories with exotic symmetries called subsystem symmetries, which have recently attracted much attention in the context of fractons. We start with a scalar theory called $\phi$-theory in $d+1$ dimensions and discuss its properties studied in literature for $d\leq 3$ such as self-duality, vacuum structure, 't Hooft anomaly, anomaly inflow and lattice regularization. Next we study a theory called chiral $\phi$-theory which is an analogue of a chiral boson with subsystem symmetries. Then we discuss theories including fermions with subsystem symmetries. We first construct a supersymmetric version of the $\phi$-theory and dropping its bosonic part leads us to a purely fermionic theory with subsystem symmetries called $\psi$-theory. We argue that lattice regularization of the $\psi$-theory generically suffers from an analogue of doubling problem as previously pointed out in the $d=3$ case. We propose an analogue of Wilson fermion to avoid the ``doubling" problem. We also supersymmetrize the chiral $\phi$-theory and dropping the bosonic part again gives us a purely fermionic theory. We finally discuss vacuum structures of the theories with fermions and find that they are infinitely degenerate because of spontaneous breaking of subsystem symmetries.