The operator curl acting on a closed 3-manifold: spectral analysis

We study the spectrum of the operator curl acting on a connected oriented closed Riemannian 3-manifold. The spectrum is asymmetric about zero and this spectral asymmetry is the focus of our analysis.

Spectral asymmetry is a major subject in pure mathematics and theoretical physics. The traditional measure of spectral asymmetry is the so-called eta invariant, a real number. The classical definition of the eta invariant is by means of analytic continuation of the eta function, an analogue of the Riemann zeta function for non-semi-bounded operators. We prove that the eta invariant for the operator curl 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 is joint work with Giovanni Bracchi and Matteo Capoferri.