On pure lattice Chern–Simons gauge theories

Abstract

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.

Generalized parity transformations in the regularized Chern-Simons theory

“…This changes the classical (odd) transformation properties of the pure CS action. This effect, already discussed for the case of a lattice regularization [1], is also present when the theory is defined in the continuum and, indeed, it is a manifestation of a more general 'anomalous' effect, since it happens for every regularization scheme. We explore the physical consequences of this anomaly.…”

“…There is, however, a remnant of the classical behaviour which is manifested by the existence of a generalized parity transformation under which the MCS action is still odd. This is the content of section 3, which presents the symmetry transformations associated with the Ginsparg-Wilson like relation suggested in [1]. In section 4 we show that essentially the same symmetry holds true for the lattice Chern-Simons theory.…”

“…It was also suggested in [1] that the lattice theory may, perhaps, verify a GinspargWilson [5] like relation. Indeed, the situation is, in more than one aspect, similar to the breaking of chiral symmetry on the lattice.…”

On the doubling phenomenon in lattice Chern-Simons theories

“…Here we shall evidence that as a result of the theorem proved in [10] the non-integrability of the CS kernel is a general feature of any gaugeinvariant, local, parity odd and cubic symmetric gauge theory on an infinite Euclidean lattice provided that under parity…”

“…In [10] we pointed out a no-go theorem in the lattice regularization of the pure CS theory, if one requires locality, gauge invariance and parity oddness on the lattice. As we already pointed out in the introduction, a doubling phenomenon has been already advocated for bosonic theories on the lattice [6] [7].…”