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Fractional analytic index. (English) Zbl 1115.58021
The index of an elliptic operator over a compact manifold is an integer. Even if a manifold has no $$\text{spin}_\mathbb C$$ structure, there is a projective bundle associated to the Clifford algebra of the cotangent bundle which is an Azumaya bundle. Such a projective bundle may not satisfy the cocycle condition. This defines the Dixmier-Douady (:= DD), 3-dimensional torsion, invariant $$W_3$$ on the manifold.
The authors define the ring of pseudodifferential operators acting on sections of the projective bundle in a formal sense. For such operators they define the analytic index, as the trace of the commutator of the operator and a parametrix, and show that this is a homotopy invariant. Using the heat kernel method for the projective spin Dirac operator twisted by $$W_3$$, they show that the analytic index is the same as the usual topological index, in terms of the twisted Chern character of the symbol, which defines an element of $$K$$-theory twisted by the DD-invariant $$W_3$$. This index is a rational number but in general it is not an integer. The authors introduce several operators whose indices are not integers, but rationals.

##### MSC:
 58J20 Index theory and related fixed-point theorems on manifolds 53C27 Spin and Spin$${}^c$$ geometry
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