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 -