Beyond the Hodge theorem: Curl and asymmetric pseudodifferential projections

Source abstract

Abstract We develop a new approach to the study of spectral asymmetry. Working with the operator on a connected oriented closed Riemannian 3‐manifold, we construct, by means of microlocal analysis, the asymmetry operator — a scalar pseudodifferential operator of order . The latter is completely determined by the Riemannian manifold and its orientation, and encodes information about spectral asymmetry. The asymmetry operator generalises and contains the classical eta invariant traditionally associated with the asymmetry of the spectrum, which can be recovered by computing its regularised operator trace. Remarkably, the whole construction is direct and explicit.