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arXiv · 1603.02263

Spectral $\zeta$-invariants lifted to coverings

Sara Azzali·Sylvie Paycha
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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.

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Sara Azzali, Sylvie Paycha. 2016-03-07. Spectral $\zeta$-invariants lifted to coverings. https://doi.org/10.1090/tran%2F8067

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