Spectral asymmetry via pseudodifferential projections: the massless Dirac operator
Matteo Capoferri, Beatrice Costeri, Claudio Dappiaggi
TL;DR
This work develops a framework to study spectral asymmetry for the massless Dirac operator on a closed 3-manifold by constructing an asymmetry operator from spectral projections. It computes the leading principal symbol of the asymmetry operator in a Levi-Civita framing, showing a structure tied to the gradient of the Ricci tensor and a Levi-Civita tensor, analogous to known results for the curl operator. The authors define regularised local and global traces, ψ_Dir^loc and ψ_Dir, and argue that they should coincide with the local/global Dirac eta invariants, thus linking spectral asymmetry to eta invariants through a geometrically intrinsic, gauge-covariant regularisation. They provide a concrete local symbol computation, establish well-definedness of the regularised trace, and discuss alternative regularisation via sphere averages, with potential implications for η-invariants in Chern–Simons theory and topological phases of matter.
Abstract
A new approach to the study of spectral asymmetry for systems of partial differential equations (PDEs) on closed manifolds was proposed in a recent series of papers by the first author and collaborator. They showed that information on spectral asymmetry can be encoded within and recovered from a negative order pseudodifferential operator -- the asymmetry operator -- constructed from appropriately defined pseudodifferential (spectral) projections. In this manuscript we apply these techniques to the study of the massless Dirac operator; in particular, we compute the principal symbol of the asymmetry operator, accounting for the underlying gauge invariance.