Motivated by Wigner's theorem, a canonical construction is described that produces an Atiyah-Singer Dirac operator with both unitary and anti-unitary symmetries. This Dirac operator includes the Dirac operator for KR-theory as a special case, filling a long-standing gap in the literature. In order to make the construction, orientifold $\mathrm{Spi}{\mathrm{n}}^{\mathrm{c}}$-structures are defined and classified using semi-equivariant Dixmier-Douady theory, and analogues of several standard theorems on the existence of $\mathrm{Spi}{\mathrm{n}}^{\mathrm{c}}$-structures are proved.

中文翻译：

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Dirac算子在方折上的构造

受 Wigner 定理的启发，描述了一种规范构造，该构造产生具有酉和反酉对称性的 Atiyah-Singer Dirac 算子。这个狄拉克算子包括KR理论的狄拉克算子作为一个特例，填补了文献中长期存在的空白。为了使建筑，东方折$\mathrm{螺线管}{\mathrm{n}}^{\mathrm{C}}$-结构使用半等变 Dixmier-Douady 理论和几个标准定理的类似物来定义和分类 $\mathrm{螺线管}{\mathrm{n}}^{\mathrm{C}}$-结构被证明。