Beyond the Hodge Theorem: curl and asymmetric pseudodifferential projections

TL;DR

This paper introduces a microlocal analysis-based approach to spectral asymmetry on 3-manifolds, constructing a scalar pseudodifferential operator that generalizes the eta invariant and encodes spectral asymmetry information.

Contribution

It develops an explicit, direct method to construct an asymmetry operator for the curl operator on 3-manifolds, extending classical spectral invariants like the eta invariant.

Findings

Constructed a scalar pseudodifferential asymmetry operator of order -3.

The operator encodes spectral asymmetry and generalizes the eta invariant.

The construction is explicit and determined by the Riemannian structure.

Abstract

We develop a new approach to the study of spectral asymmetry. Working with the operator $\operatorname{curl}:=*\mathrm{d}$ on a connected oriented closed Riemannian 3-manifold, we construct, by means of microlocal analysis, the asymmetry operator -- a scalar pseudodifferential operator of order $-3$. 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.