## Journal of Differential Geometry

### Fractional analytic index

#### Abstract

For a finite rank projective bundle over a compact manifold, so associated to a torsion, Dixmier-Douady, 3-class, $w$, on the manifold, we define the ring of differential operators ‘acting on sections of the projective bundle’ in a formal sense. In particular, any oriented even-dimensional manifold carries a projective spin Dirac operator in this sense. More generally the corresponding space of pseudodifferential operators is defined, with supports sufficiently close to the diagonal, i.e., the identity relation. For such elliptic operators we define the numerical index in an essentially analytic way, as the trace of the commutator of the operator and a parametrix and show that this is homotopy invariant. Using the heat kernel method for the twisted, projective spin Dirac operator, we show that this index is given by the usual formula, now in terms of the twisted Chern character of the symbol, which in this case defines an element of K-theory twisted by $w$; hence the index is a rational number but in general it is not an integer.

#### Article information

Source
J. Differential Geom. Volume 74, Number 2 (2006), 265-292.

Dates
First available in Project Euclid: 30 March 2007

https://projecteuclid.org/euclid.jdg/1175266205

Digital Object Identifier
doi:10.4310/jdg/1175266205

Mathematical Reviews number (MathSciNet)
MR2258800

Zentralblatt MATH identifier
1115.58021

#### Citation

Mathai, V.; Melrose, R. B.; Singer, I. M. Fractional analytic index. J. Differential Geom. 74 (2006), no. 2, 265--292. doi:10.4310/jdg/1175266205. https://projecteuclid.org/euclid.jdg/1175266205