边界群胚上椭圆微分算子的重整化指标公式
Renormalized Index Formulas for Elliptic Differential Operators on Boundary Groupoids
AI总结：
本文针对特定形式的边界群胚，采用类似重整化迹的方法推导椭圆微分算子的数值指标公式，证明q≥3时eta项消失、指标仅由Atiyah-Singer项给出，q=1时结果依赖奇异集嵌入方式。
摘要：
我们研究形如$\boldsymbol{\nmathcal{G} = M _0 \times M _0 \ncup \nmathbb{R}^q \times M _1 \times M _1}$的一类边界群胚的指标问题。已有研究表明，当$q \ngeq 3$且为奇数时，$K _0 (C^* (\nmathcal{G})) \ncong \nZ$，且$K$理论指标与Fredholm指标一致，因此本文旨在推导$\nmathcal{G}$上椭圆微分算子的数值公式。我们的方法与Moroianu和Nistor的重整化迹方法（见文献[Nistor;Hom2]）类似，但发现当$q \ngeq 3$时eta项消失，因此这些群胚上椭圆（对应地，完全椭圆）拟微分算子的$K$理论指标与Fredholm指标仅由Atiyah-Singer项给出。对于$q=1$的情形，我们发现结果依赖于奇异集$M_1$在$M$中的嵌入方式。
英文摘要：
We consider the index problem of certain boundary groupoids of the form $\mathcal{G} = M _0 \times M _0 \cup \mathbb{R}^q \times M _1 \times M _1$. Since it has been shown that for the case that $q \geq 3$ is odd, $K _0 (C^* (\mathcal{G})) \cong \bbZ $, and moreover the $K$-theoretic index coincides with the Fredholm index, we attempt in this paper to derive a numerical formula for elliptic differential operators on $\mathcal{G}$. Our approach is similar to that of renormalized trace of Moroianu and Nistor \cite{Nistor;Hom2}. However, we find that when $q \geq 3$, the eta term vanishes, and hence the $K$-theoretic and Fredholm indices of elliptic (respectively fully elliptic) pseudo-differential operators on these groupoids are given only by the Atiyah-Singer term. As for the $q=1$ case we find that the result depends on how the singularity set $M_1$ lies in $M$.
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