Spectral $\zeta$-invariants lifted to coverings

TL;DR

This paper explores how spectral $zeta$-invariants can be lifted to universal coverings of manifolds, revealing their relation to Atiyah's $L^2$-index theorem and geometric characteristic forms.

Contribution

It introduces a method to lift spectral $zeta$-invariants using defect formulas and connects these lifts to classical index theorems and geometric invariants.

Findings

Lifted spectral $zeta$-invariants relate to Atiyah's $L^2$-index theorem.

Certain lifted invariants are expressed as integrals of Pontryagin and Chern forms.

The canonical trace and Wodzicki residue are preserved under lifting to coverings.

Abstract

The canonical trace and the Wodzicki residue on classical pseudodifferential operators on a closed manifold are characterised by their locality and shown to be preserved under lifting to the universal covering as a result of their local feature. As a consequence, we lift a class of spectral $\zeta$-invariants using lifted defect formulae which express discrepancies of $\zeta$-regularised traces in terms of Wodzicki residues. We derive Atiyah's $L^2$-index theorem as an instance of the $\mathbb Z_2$-graded generalisation of the canonical lift of spectral $\zeta$-invariants and we show that certain lifted spectral $\zeta$-invariants for geometric operators are integrals of Pontryagin and Chern forms.