标题： Majorana费米子的自由相：十重方式比较 显示英文标题

标题： Free phases of Majorana fermions: Tenfold ways compared

摘要： 我们提供了对中性自由费米子的对称性保护拓扑(SPT)相的数学严格分类。 我们的方法利用了带有负平方的Karoubi三元组，视为极化。 我们证明了由对称代数$A$保护的中性自由费米子SPT相由实数$K$-理论群$K_2(A^{op})$分类，并展示了在存在电荷的情况下，我们的分类如何重现已知结果。 我们的形式也允许使用费米群及其费米群$C^*$-代数的形式来描述对称性。 我们的分类扩展到正的空间维度，并通过晶体等价原理包括弱相。 我们的方法通过对称代数的Morita等价建立了各种现有十重方式分类的等价性，从而澄清并统一了这些分类。 我们期望我们的分类成为Freed和Hopkins提出的自由到相互作用映射的自然领域。 显示英文摘要

摘要： We provide a mathematically rigorous classification of symmetry-protected topological (SPT) phases of neutral free fermions. Our approach utilizes Karoubi triples with negative squares, thought of as polarizations. We prove that neutral free fermion SPT phases protected by a symmetry algebra $A$ are classified by the real $K$-theory group $K_2(A^{op})$, and demonstrate how our classification reproduces known results in the presence of charge. Our formalism also allows for symmetries described by groups, potentially with time-reversal, using the formalism of fermionic groups and their fermionic group $C^*$-algebras. Our classification extends to positive spatial dimensions and includes weak phases using the crystalline equivalence principle. Our approach clarifies and unifies various existing tenfold way classifications by establishing their equivalence through Morita equivalences of symmetry algebras. We expect our classification to be the natural domain for the free-to-interacting map proposed by Freed and Hopkins.