AI 中文总结
本文针对带弯曲边界的空间，利用Roe代数与KK-理论变体，证明了体边对应关系与广义久保公式，明确了相关指标配对的等价性及重数的拓扑度来源。
AI 中文摘要
强拓扑绝缘体由整数不变量分类，这些不变量具有不同的实空间表达式。受最近研究带弯曲边界空间的此类不变量的工作启发，我们重新探讨了各种表达式的等价性问题，包括Fredholm指标配对和涉及半空间投影的久保公式。利用Roe代数和适用于不可分C*-代数的KK-理论变体，我们证明了在任意边界存在时，具有一般位置空间Dirac算子的指标配对的一般形式体边对应关系。对于粗等价于ℝᵈ的空间，我们证明这些配对在重数下等于用标准对偶Dirac算子得到的配对，重数由符号函数的拓扑度明确给出。此外，我们还证明了一个广义久保公式，该公式通过实空间公式计算一般指标配对。
英文摘要
Strong topological insulators are classified by integer invariants which admit different real-space expressions. Inspired by recent work studying these invariants for spaces with curved boundaries, we revisit the problem of equivalence of the various expressions, including Fredholm index pairings and Kubo formulas involving half-space projections. Using Roe algebras and a variant of KK-theory suitable for non-separable $C^*$-algebras, we prove a general form of bulk-edge correspondence for the index pairing with a general position-space Dirac operator in the presence of arbitrary boundaries. For spaces coarsely equivalent to $\mathbb{R}^d$, we show that up to multiplicity, these pairings are equal to those obtained with the standard dual Dirac operator, with the multiplicity explicitly given as the topological degree of the symbol function. In addition, we prove a generalized Kubo formula which computes the general index pairing by a real-space formula.
