The index of families of projective 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.