Beyond the Hodge Theorem: curl and asymmetric pseudodifferential projections
Matteo Capoferri, Dmitri Vassiliev
TL;DR
The paper introduces a microlocal approach to spectral asymmetry for the curl operator on oriented 3-manifolds by constructing explicit pseudodifferential projections P0, P± and forming the asymmetry operator A as a scalar, negative-order (−3) operator derived from a matrix trace. It provides explicit formulas for the principal symbol of A in terms of the Ricci tensor and orientation, proves a precise decomposition of A’s kernel, and develops a robust regularisation giving local and global traces that capture spectral asymmetry and link to eta invariants. This framework avoids heat-kernel methods, offering an explicit, coordinate-invariant route to quantify asymmetry via a geometric invariant. The work also outlines an extension to higher dimensions, highlighting significant analytic and algebraic challenges, and sets the stage for connections to physical notions such as Maxwell chirality and electromagnetic polarization. Altogether, the paper furnishes a concrete, computable bridge between curvature data and spectral asymmetry, with potential applications to Dirac-type operators and global geometric analysis.
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.