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I am looking for examples of 2- and 3-dimensional flat spin manifolds with Euclidean and Lorentzian signatures, which admit parallel spinors and the dimension of the space of the parallel spinors is equal to 1. At the moment I have examples only for the Lorentzian case:$\mathbb{R}^{1,1}_{*}/(Z_{2})$ (for dimension 2) and $\mathbb{R}^{1,1}_{*}/(Z_{2}) \times \mathbb{R}$ (for dimension 3). Here $\mathbb{R}^{1,1}_{*}$ means $\mathbb{R}^{1,1}\setminus \{(0,0)\}$ and the generator of $Z_2$ acts as $x \mapsto (-x)$. However, I am not sure if it is correct.

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Your both examples are correct. – Vladimir Oct 3 '13 at 12:10

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