On pure lattice Chern–Simons gauge theories

We revisit the lattice formulation of the Abelian Chern–Simons model defined on an infinite Euclidean lattice. We point out that any gauge invariant, local and parity odd Abelian quadratic form exhibits, in addition to the zero eigenvalue associated with the gauge invariance and to the physical zero mode at p=0 due to translational invariance, a set of extra zero eigenvalues inside the Brillouin zone. For the Abelian Chern–Simons theory, which is linear in the derivative, this proliferation of zero modes is reminiscent of the Nielsen–Ninomiya no-go theorem for fermions. A gauge invariant, local and parity even term such as the Maxwell action leads to the elimination of the extra zeros by opening a gap with a mechanism similar to that leading to Wilson fermions on the lattice.