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The index of families of projective operators

Alexandre Baldare

Vol. 8 (2023), No. 3, 285–330
Abstract

Let 1 Γ 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
index theory, cohomology, pseudodifferential operators, group actions, projective operators
Mathematical Subject Classification
Primary: 19K56, 58J20
Secondary: 19K35, 57S15
Milestones
Received: 15 September 2021
Revised: 14 September 2022
Accepted: 11 July 2023
Published: 27 August 2023
Authors
Alexandre Baldare
Institut für Analysis
Leibniz Universität
Hannover
Germany