The Atiyah-Singer index formula for Dirac operators acting on the space of spinors put across a kind of topological invariant (\(\hat{{A}}\) genus) of a closed spin manifold \({{\mathcal {M}}}\), hence offering a bridge between geometric and analytical aspects of the original spin manifold. In this study, we prove the index theorem for a family of fractional Dirac operators in particular for complex analytic coordinates.

中文翻译：

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自旋几何上分数阶狄拉克算子族的 Atiyah-Singer 指数定理

作用于旋量空间的狄拉克算子的 Atiyah-Singer 指数公式跨越了一种封闭自旋流形\ ({{\mathcal {M} }}\)，因此提供了原始自旋流形的几何和分析方面之间的桥梁。在本研究中，我们证明了分数阶狄拉克算子族的指数定理，特别是对于复解析坐标。