Renormalized Index Formulas for Elliptic Differential Operators on Boundary Groupoids

TL;DR

This paper derives a simplified numerical index formula for elliptic operators on certain boundary groupoids, showing that the eta term vanishes for odd q ≥ 3, leaving only the Atiyah-Singer term.

Contribution

It provides a renormalized index formula for elliptic operators on boundary groupoids, extending previous K-theoretic results to explicit numerical formulas.

Findings

Eta term vanishes for odd q ≥ 3, simplifying the index formula.

Index is given solely by the Atiyah-Singer term in these cases.

The result depends on the position of the singularity set for q=1.

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 $q \geq 3$ 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 $M$.