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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.

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.

Reviewer: Yong Seung Cho (Seoul)

### MSC:

58J20 | Index theory and related fixed-point theorems on manifolds |

53C27 | Spin and Spin\({}^c\) geometry |