Parallel spinors and holonomy groups on pseudo-Riemannian spin manifolds
The authors describe the possible holonomy groups \(H\) of simply connected irreducible non locally-symmetric pseudo-Riemannian spin manifolds which admit parallel spinors and calculate the dimension \(N\) of the space of parallel spinors. More precisely, they prove that if \(N > 1\), then the holonomy \(H\) belongs to one of the series: \[ (SU(p,q),\;N=2) \qquad (Sp(p,q),\;N= p+q+1) \] or is the exceptional group \(G_2^{\mathbb C} \subset SO(7,7)\), \(N=2\). If \(N=1\), then \(H\) is one of the following linear groups: \( G_2 \subset SO(7)\), \(G^*_{2 (2)} \subset SO(4,3)\), \(\text{Spin}(7) \subset SO(8)\), \(\text{Spin}(4,3) \subset SO(4,4)\), \(\text{Spin}(7)^{\mathbb C}\subset SO(8,8).\)
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