Details
| Original language | English |
|---|---|
| Pages (from-to) | 285-330 |
| Number of pages | 46 |
| Journal | Annals of K-Theory |
| Volume | 8 |
| Issue number | 3 |
| Publication status | Published - 27 Aug 2023 |
Abstract
Let 1 → Ɣ → ˜G → G → 1 be a central extension by an abelian finite group. We compute the index of families of ˜G-transversally elliptic operators on a G-principal bundle P. We then introduce families of projective operators on fibrations equipped with an Azumaya bundle A. We define and compute the index of such families using the cohomological index formula for families of SU(N)-transversally elliptic operators. More precisely, a family A of projective operators can be pulled back in a family à of SU(N)-transversally elliptic operators on the PU(N)-principal bundle of trivialisations of A. Through the distributional index of Ã, we can define an index for A and using the index formula in equivariant cohomology for families of SU(N)-transversally elliptic operators, we derive an explicit cohomological index formula in de Rham cohomology. Once this is done, we define and compute the index of families of projective Dirac operators. As a second application of our computation of the index of families of ˜G-transversally elliptic operators on a G-principal bundle P, we consider the special case of a family of Spin(2n)-transversally elliptic Dirac operators over the bundle of oriented orthonormal frames of an oriented fibration and we relate its distributional index with the index of the corresponding family of projective Dirac operators.
Keywords
- cohomology, group actions, index theory, projective operators, pseudodifferential operators
ASJC Scopus subject areas
- Mathematics(all)
- Analysis
- Mathematics(all)
- Geometry and Topology
Cite this
- Standard
- Harvard
- Apa
- Vancouver
- BibTeX
- RIS
In: Annals of K-Theory, Vol. 8, No. 3, 27.08.2023, p. 285-330.
Research output: Contribution to journal › Article › Research › peer review
}
TY - JOUR
T1 - The index of families of projective operators
AU - Baldare, Alexandre
PY - 2023/8/27
Y1 - 2023/8/27
N2 - Let 1 → Ɣ → ˜G → G → 1 be a central extension by an abelian finite group. We compute the index of families of ˜G-transversally elliptic operators on a G-principal bundle P. We then introduce families of projective operators on fibrations equipped with an Azumaya bundle A. We define and compute the index of such families using the cohomological index formula for families of SU(N)-transversally elliptic operators. More precisely, a family A of projective operators can be pulled back in a family à of SU(N)-transversally elliptic operators on the PU(N)-principal bundle of trivialisations of A. Through the distributional index of Ã, we can define an index for A and using the index formula in equivariant cohomology for families of SU(N)-transversally elliptic operators, we derive an explicit cohomological index formula in de Rham cohomology. Once this is done, we define and compute the index of families of projective Dirac operators. As a second application of our computation of the index of families of ˜G-transversally elliptic operators on a G-principal bundle P, we consider the special case of a family of Spin(2n)-transversally elliptic Dirac operators over the bundle of oriented orthonormal frames of an oriented fibration and we relate its distributional index with the index of the corresponding family of projective Dirac operators.
AB - Let 1 → Ɣ → ˜G → G → 1 be a central extension by an abelian finite group. We compute the index of families of ˜G-transversally elliptic operators on a G-principal bundle P. We then introduce families of projective operators on fibrations equipped with an Azumaya bundle A. We define and compute the index of such families using the cohomological index formula for families of SU(N)-transversally elliptic operators. More precisely, a family A of projective operators can be pulled back in a family à of SU(N)-transversally elliptic operators on the PU(N)-principal bundle of trivialisations of A. Through the distributional index of Ã, we can define an index for A and using the index formula in equivariant cohomology for families of SU(N)-transversally elliptic operators, we derive an explicit cohomological index formula in de Rham cohomology. Once this is done, we define and compute the index of families of projective Dirac operators. As a second application of our computation of the index of families of ˜G-transversally elliptic operators on a G-principal bundle P, we consider the special case of a family of Spin(2n)-transversally elliptic Dirac operators over the bundle of oriented orthonormal frames of an oriented fibration and we relate its distributional index with the index of the corresponding family of projective Dirac operators.
KW - cohomology
KW - group actions
KW - index theory
KW - projective operators
KW - pseudodifferential operators
UR - http://www.scopus.com/inward/record.url?scp=85169698356&partnerID=8YFLogxK
U2 - 10.48550/arXiv.2109.06782
DO - 10.48550/arXiv.2109.06782
M3 - Article
AN - SCOPUS:85169698356
VL - 8
SP - 285
EP - 330
JO - Annals of K-Theory
JF - Annals of K-Theory
SN - 2379-1683
IS - 3
ER -