Real Representation in Chiral Gauge Theories on the Lattice

TL;DR

This paper demonstrates a lattice formulation for chiral gauge theories using Weyl fermions in real representations, showing explicit measure construction and equivalence to Majorana fermion formulations, avoiding Witten anomaly issues.

Contribution

It provides an explicit construction of the fermion measure for real representation Weyl fermions on the lattice and establishes their equivalence to Majorana fermion formulations.

Findings

Explicit measure construction over gauge configurations

No global obstruction from Witten anomaly

Equivalence to Majorana fermion formulation with Pfaffian

Abstract

The Weyl fermion belonging to the real representation of the gauge group provides a simple illustrative example for L\"uscher's gauge-invariant lattice formulation of chiral gauge theories. We can explicitly construct the fermion integration measure globally over the gauge-field configuration space in the arbitrary topological sector; there is no global obstruction corresponding to the Witten anomaly. It is shown that this Weyl formulation is equivalent to a lattice formulation based on the Majorana (left--right-symmetric) fermion, in which the fermion partition function is given by the Pfaffian with a definite sign, up to physically irrelevant contact terms. This observation suggests a natural relative normalization of the fermion measure in different topological sectors for the Weyl fermion belonging to the complex representation.