Renormalized Index Formulas for Elliptic Differential Operators on Boundary Groupoids
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2021
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866915059341459456 |
|---|---|
| author | Qiao, Yu So, Bing Kwan |
| author_facet | Qiao, Yu So, Bing Kwan |
| contents | 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 $M$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2108_04592 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Renormalized Index Formulas for Elliptic Differential Operators on Boundary Groupoids Qiao, Yu So, Bing Kwan Operator Algebras Differential Geometry K-Theory and Homology 58J20 (primary) 46L80, 19K56 (Secondary) 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 $M$. |
| title | Renormalized Index Formulas for Elliptic Differential Operators on Boundary Groupoids |
| topic | Operator Algebras Differential Geometry K-Theory and Homology 58J20 (primary) 46L80, 19K56 (Secondary) |
| url | https://arxiv.org/abs/2108.04592 |