The index of families of projective operators

TL;DR

This paper develops a cohomological index formula for families of projective operators on fibrations with Azumaya bundles, extending the theory of transversally elliptic operators and applying it to Dirac operators.

Contribution

It introduces a new index formula for families of projective operators using equivariant cohomology and extends the framework to projective Dirac operators.

Findings

Derived an explicit cohomological index formula in de Rham cohomology.

Computed the index of families of projective Dirac operators.

Connected the index of projective Dirac operators with transversally elliptic Dirac operators.

Abstract

Let $1 \to \Gamma \to \tilde{G} \to G \to 1$ be a central extension by an abelian finite group. In this paper, we compute the index of families of $\tilde{G}$-transversally elliptic operators on a $G$-principal bundle $P$. We then introduce the notion of families of projective operators on fibrations equipped with an Azumaya bundle $\mathcal{A}$. 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 $\tilde{A}$ of $SU(N)$-transversally elliptic operators on the $PU(N)$-principal bundle of trivialisations of $\mathcal{A}$. Through the distributional index of $\tilde{A}$, we can define an index for the family $A$ of projective operators 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 $\tilde{G}$-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.