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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: $\mathbb{R}^{1,1}_* /(\mathbb{Z}_2)$case:$\mathbb{R}^{1,1}_{*}/(Z_{2})$ (for dimension 2) and $\mathbb{R}^{1,1}_{*}/(\mathbb{Z}_{2}) \mathbb{R}^{1,1}_{*}/(Z_{2}) \times \mathbb{R}$ (for dimension 3). Here $\mathbb{R}^{1,1}_{*}$ means $\mathbb{R}^{1,1}\setminus \{(0,0)\}${(0,0)}$and the generator of$\mathbb{Z}_2$Z_2$ acts as $x \mapsto (-x)$. However, I am not sure if it is correct.

 
 
 
 
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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: $\mathbb{R}^{1,1}_{*}/(\mathbb{Z}{2})$ \mathbb{R}^{1,1}_* /(\mathbb{Z}_2)$ (for dimension 2) and $\mathbb{R}^{1,1}{*}/(\mathbb{Z}{2}) \mathbb{R}^{1,1}_{*}/(\mathbb{Z}_{2}) \times \mathbb{R}$ (for dimension 3). Here $\mathbb{R}^{1,1}{*}$\mathbb{R}^{1,1}_{*}$ means $\mathbb{R}^{1,1}\setminus {(0,0)}$ \{(0,0)\}$ and the generator of$\mathbb{Z}_2$acts as$x \mapsto (-x)$. However, I am not sure if it is correct. 3 corrected typos 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:$\mathbb{R}^{1,1}_{*}/(\mathbb{Z}/2)$\mathbb{R}^{1,1}_{*}/(\mathbb{Z}{2})$ (for dimension 2) and $\mathbb{R}^{1,1}_{*}/(\mathbb{Z}/2) \mathbb{R}^{1,1}{*}/(\mathbb{Z}{2}) \times \mathbb{R}$ (for dimension 3). Here $\mathbb{R}^{1,1}_{*}$ \mathbb{R}^{1,1}{*}$means$\mathbb{R}^{1,1}\setminus {(0,0)}$and the generator of$\mathbb{Z}/2$\mathbb{Z}_2$ acts as $x \mapsto (-x)$. However, I am not sure if it is correct.

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