Majorana and Majorana-Weyl fermions in lattice gauge theory

In various dimensional Euclidean lattice gauge theories, we examine acompatibility of the Majorana decomposition and the charge conjugation propertyof lattice Dirac operators. In $8n$ and $1+8n$ dimensions, we find a difficultyto decompose a classical lattice action of the Dirac fermion into a system ofthe Majorana fermion and thus to obtain a factorized form of the Diracdeterminant. Similarly, in $2+8n$ dimensions, there is a difficulty todecompose a classical lattice action of the Weyl fermion into a system of theMajorana--Weyl fermion and thus to obtain a factrized form of the Weyldeterminant. Prescriptions based on the overlap formalism do not remove thesedifficulties. We argue that these difficulties are reflections of the globalgauge anomaly associated to the real Weyl fermion in $8n$ dimensions. For thisreason (besides other well-known reasons), a lattice formulation of the N=1super Yang--Mills theory in these dimensions is expected to be extremelydifficult to find.Comment: 29 pages, uses JHEP3.cls, the final version to appear in JHE