Renormalized Index Formulas for Elliptic Differential Operators on Boundary Groupoids

Abstract

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 $q\ge 3$ is odd, $K_0(C^{\ast }(\mathcal{G}))\cong \mathbb{Z}$, 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 $q\ge 3$, 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 $M$.

20 pages. arXiv admin note: substantial text overlap with arXiv:1804.10426