Majorana and Majorana-Weyl fermions in lattice gauge theory

TL;DR

This paper investigates the challenges of formulating Majorana and Majorana-Weyl fermions in lattice gauge theories across various dimensions, highlighting fundamental obstructions related to gauge anomalies that complicate lattice implementations.

Contribution

It identifies dimension-dependent difficulties in decomposing lattice Dirac and Weyl actions into Majorana forms, linking these issues to global gauge anomalies.

Findings

Decomposition problems occur in 8n and 1+8n dimensions for Dirac fermions.

Similar issues arise in 2+8n dimensions for Weyl fermions.

Overlap formalism does not resolve these decomposition difficulties.

Abstract

In various dimensional Euclidean lattice gauge theories, we examine a compatibility of the Majorana decomposition and the charge conjugation property of lattice Dirac operators. In $8n$ and $1+8n$ dimensions, we find a difficulty to decompose a classical lattice action of the Dirac fermion into a system of the Majorana fermion and thus to obtain a factorized form of the Dirac determinant. Similarly, in $2+8n$ dimensions, there is a difficulty to decompose a classical lattice action of the Weyl fermion into a system of the Majorana--Weyl fermion and thus to obtain a factrized form of the Weyl determinant. Prescriptions based on the overlap formalism do not remove these difficulties. We argue that these difficulties are reflections of the global gauge anomaly associated to the real Weyl fermion in $8n$ dimensions. For this reason (besides other well-known reasons), a lattice formulation of the N=1 super Yang--Mills theory in these dimensions is expected to be extremely difficult to find.