Translational anomaly in chiral gauge theories on a torus and the overlap formalism

Abstract

We point out that a fermion determinant of a chiral gauge theory on a 2D torus has a phase ambiguity proportional to the Polyakov loops along the boundaries, which can be reproduced by the overlap formalism. We show that the requirement on the fermion determinant that a singularity in the gauge field can be absorbed by a change of the boundary condition for the fermions, is not compatible with translational invariance in general. As a consequence, the gauge anomaly for singular gauge transformations discovered by Narayanan-Neuberger actually exists in any 2D U(1) chiral gauge theory unless the theory is vector-like. We argue that the gauge anomaly is peculiar to the overlap formalism with the Wigner-Brillouin phase choice and that it is not necessarily a property required in the continuum. We also generalize our results to any even

Lattice chiral gauge theories

“…The issue of gauge invariance has been studied in detail in a two-dimensional U (1) model with four LH fermions of charge 1 and one RH fermion of charge 2 (which is anomaly free in d = 2) [21,22]. The most complete investigation in this context is that of ref.…”

Torsion, topology and CPT anomaly in two-dimensional chiral U(1) gauge theory

“…This requirement naturally leads to a decomposition of the SO(2) rotation angle ϕ(x) into topologically trivial and nontrivial parts: 6) where the real function ω(x) is taken to be strictly periodic in x 1 and x 2 , with period L. The other real function χ(x) is associated with the two generat-ing curves a and b of the homology group H 1 (T 2 , Z) = Z ⊕ Z; cf. Ref.…”

“…Since we intend to compute the Jacobians for the left-and right-moving chiral fermions separately, we can work with the following Hamiltonian: 6) which has the advantage of being Hermitian for each chirality separately, H ± = (H ± ) † . Explicitly, its components are given by…”

“…[6] and references therein). For this reason, we will consider in this paper primarily two-dimensional chiral U(1) gauge theory defined over the torus.…”

Domain wall fermion and chiral gauge theories on the lattice with exact gauge invariance

“…In the overlap formalism, reflecting chiral anomaly, the phase factor of the chiral determinant is not fixed in general and any reasonable choice of the phase factor should lead to the gauge anomaly for single Weyl fermion. The Wigner-Brillouin phase convention has been adopted for perturbative studies [19] and has also been tested numerically in a non-perturbative formulation of chiral gauge theories [20,21]. Geometrical treatment of the gauge anomaly in the overlap formalism has been discussed in detail in abelian theories [22] and non-abelian theories [23].…”