June 16, 2004
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.
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