A microlocal pathway to spectral asymmetry: curl and the eta invariant

The notion of eta invariant is traditionally defined by means of analytic continuation. We prove, by examining the particular case of the operator curl, that the eta invariant can equivalently be obtained as the trace of the difference of positive and negative spectral projections, appropriately regularised. Our construction is direct, in the sense that it does not involve analytic continuation, and is based on the use of pseudodifferential techniques. This provides a novel approach to the study of spectral asymmetry of non-semibounded (pseudo)differential systems on manifolds which encompasses and extends previous results.