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Register a new userThe index of projective families of elliptic operators: the decomposable case
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An index theory for projective families of elliptic pseudodifferential operators is developed when the twisting, i.e. Dixmier-Douady, class is decomposable. One of the features of this special case is that the corresponding Azumaya bundle can be realized in terms of smoothing operators. The topological and the analytic index of a projective family of elliptic operators both take values in the twisted K-theory of the parameterizing space. The main result is the equality of these two notions of index. The twisted Chern character of the index class is then computed by a variant of Chern-Weil theory.
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Following [44], we 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 from [7]. More precisely, a family of projective operators $A$ 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 cohomological index formula from [7], we obtain an explicit cohomological index formula. Let $1 to Gamma to tilde{G} to G to 1$ be a central extension by an abelian finite group. As a preliminary result, we compute the index of families of $tilde{G}$-transversally elliptic operators on a $G$-principal bundle $P$.
We define and study the index map for families of $G$-transversally elliptic operators and introduce the multiplicity for a given irreducible representation as a virtual bundle over the base of the fibration. We then prove the usual axiomatic properties for the index map extending the Atiyah-Singer results [1]. Finally, we compute the Kasparov intersection product of our index class against the K-homology class of an elliptic operator on the base. Our approach is based on the functorial properties of the intersection product, and relies on some constructions due to Connes-Skandalis and to Hilsum-Skandalis.
We define the Chern character of the index class of a $G$-invariant family of $G$-transversally elliptic operators, see [6]. Next we study the Berline-Vergne formula for families in the elliptic and transversally elliptic case.
We introduce a notion of cobordism of Callias-type operators over complete Riemannian manifolds and prove that the index is preserved by such a cobordism. As an application we prove a gluing formula for Callias-type index. In particular, a usual index of an elliptic operator on a compact manifold can be computed as a sum of indexes of Callias-type operators on two non-compact, but topologically simpler manifolds. As another application we give a new proof of the relative index theorem for Callias-type operators, which also leads to a new proof of the Callias index theorem.
We introduce a mathematician-friendly formulation of the physicist-friendly derivation of the Atiyah-Patodi-Singer index of our previous paper. Our viewpoint sheds some new light on the interplay among the Atiyah-Patodi-Singer boundary condition, domain-wall fermions, and edge modes.
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