We give estimates for the eigenvalues of multi-form modified Dirac operators which are constructed from a standard Dirac operator with the addition of a Clifford algebra element associated to a multi-degree form. In particular such estimates are presented for modified Dirac operators with a $k$-degree form $0\leq k\leq 4$, those modified with multi-degree $(0,k)$-form $0\leq k\leq 3$ and the horizon Dirac operators which are modified with a multi-degree $(1,2,4)$-form. In particular, we give the necessary geometric conditions for such operators to admit zero modes as well as those for the zero modes to be parallel with a respect to a suitable connection. We also demonstrate that manifolds which admit such parallel spinors are associated with twisted covariant form hierarchies which generalize the conformal Killing-Yano forms.

中文翻译：

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多形式修正狄拉克算子的特征值估计

我们给出了多形式修正狄拉克算子的特征值的估计，这些算子是从标准狄拉克算子构造的，并添加了与多度形式相关的 Clifford 代数元素。特别地，对于具有 $k$-degree 形式 $0\leq k\leq 4$ 的修改后的 Dirac 算子，以及使用多度 $(0,k)$-形式 $0\leq k\leq 3$ 修改的那些，提出了这样的估计以及使用多阶 $(1,2,4)$-形式修改的地平线狄拉克算子。特别地，我们给出了这些算子允许零模以及零模平行于适当连接的必要几何条件。我们还证明了允许这种平行旋量的流形与扭曲的协变形式层次结构相关联，这些层次结构概括了共形 Killing-Yano 形式。