We review a recipe to produce a lattice construction of fermionic phases of matter in the presence of time reversal symmetry by extending the fermionization and bosonization known in (1+1) dimensions to various setups including higher spacetime dimensions in the presence of global symmetries. As an application, we provide a state sum lattice path integral for a (1+1)-dimensional topological superconductor with time reversal symmetry generating the $\mathbb{Z}_8$ classification of the symmetry-protected topological phase. We also illustrate a state sum path integral for a (3+1)-dimensional topological superconductor with time reversal symmetry that generates the $\mathbb{Z}_{16}$ classification.

中文翻译：

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费米子拓扑相和更高维度的玻色子化

我们回顾了一个配方，通过将 (1+1) 维度中已知的费米子化和玻色子化扩展到各种设置，包括存在全局对称性的更高时空维度，在存在时间反转对称性的情况下产生物质的费米子相的晶格构造。作为一个应用，我们为具有时间反转对称性的 (1+1) 维拓扑超导体提供状态和晶格路径积分，生成对称保护拓扑相的 $\mathbb{Z}_8$ 分类。我们还说明了具有时间反转对称性的 (3+1) 维拓扑超导体的状态和路径积分，该超导体生成 $\mathbb{Z}_{16}$ 分类。