[Paper Review] Spectral Asymmetry, Zeta Functions and the Noncommutative Residue

Raphaël Ponge|ArXiv.org|Oct 8, 2003

Advanced Operator Algebra Research39 references4 citations

TL;DR

This paper refines Wodzicki's zeta function approach to spectral asymmetry for odd-class elliptic pseudodifferential operators on compact manifolds, establishing local independence of zeta function values at integers from spectral cuttings under specific topological and order conditions. The key contribution is a geometric expression for spectral asymmetry of Dirac operators, yielding a new spectral interpretation of the Einstein-Hilbert action in gravity.

ABSTRACT

In this paper we study the spectral asymmetry of (possibly nonselfadjoint) elliptic PsiDO's in terms of the difference of zeta functions coming from different cuttings. Refining previous formulas of Wodzicki in the case of odd class elliptic PsiDO's, our main results have several consequence concerning the local independence with respect to the cutting, the regularity at integer points of eta functions and a geometric expression for the spectral asymmetry of Dirac operators which, in particular, yields a new spectral interpretation of the Einstein-Hilbert action in gravity.

Motivation & Objective

• To investigate the spectral asymmetry of non-selfadjoint elliptic pseudodifferential operators using zeta functions with different spectral cuttings.
• To refine Wodzicki's formulas for zeta function differences in the case of odd-class elliptic ΨDOs.
• To establish local independence of zeta function values at integer points from spectral cuttings under specific geometric and order conditions.
• To provide a geometric expression for the spectral asymmetry of Dirac operators, linking it to the Einstein-Hilbert action in gravity.
• To extend the regularity and independence properties of zeta functions to the local level (pointwise densities) for odd-class operators.

Proposed method

• Uses the difference of zeta functions associated with different spectral cuttings $ L_{ heta} $ and $ L_{ heta'} $ to analyze spectral asymmetry.
• Applies Wodzicki's sectorial projection $ ilde{ heta}, heta' (P) $, defined via contour integration over $ heta < \arg \lambda < \theta' $, to isolate spectral components.
• Employs the noncommutative residue $ \operatorname{Res} $ to relate the residue of the zeta function difference to the trace of the sectorial projection.
• Analyzes the symbol of the sectorial projection $ \Pi_{\theta,\theta'}(P) $ to derive local properties of the zeta functions.
• Applies results from functional analysis, including Dunford-Schwartz spectral theory and Schatten ideal properties, to establish completeness of root vectors.
• Derives local zeta functions $ \zeta_{\theta}(P;0)(x) $ as densities whose integrals yield the global zeta functions, enabling pointwise analysis.

Experimental results

Research questions

• RQ1 Under what conditions is the regular value of the zeta function $ \zeta_{\theta}(P;s) $ at integer points independent of the spectral cutting $ L_{\theta} $?
• RQ2 How does the spectral asymmetry of odd-class elliptic ΨDOs relate to the noncommutative residue and zeta function differences?
• RQ3 Can the spectral asymmetry of Dirac operators be expressed geometrically via the noncommutative residue?
• RQ4 What conditions ensure the regularity of $ \zeta_{\theta}(P;s) $ at integer points for odd-class operators?
• RQ5 Does the local zeta function density $ \zeta_{\theta}(P;0)(x) $ remain independent of the spectral cutting for odd-class operators?

Key findings

• When $ \dim M $ is odd and $ \operatorname{ord} P $ is even, $ \zeta_{\theta}(P;s) $ is regular at every integer point and its value is independent of the spectral cutting $ L_{\theta} $.
• When $ \dim M $ is even, $ \operatorname{ord} P $ is odd, and the principal symbol lies in the union of two opposite open cones $ \Lambda_{\theta,\theta'} \cup \Lambda_{\theta+\pi,\theta'+\pi} $, the difference $ \zeta_{\theta}(P;s) - \zeta_{\theta'}(P;s) $ satisfies $ \operatorname{ord} P \cdot \lim_{s \to k} (\zeta_{\theta}(P;s) - \zeta_{\theta'}(P;s)) = i\pi \operatorname{Res} P^{-k} $ for any integer $ k $.
• The regular value $ \zeta_{\theta}(P;0)(x) $ of the local zeta function at each point $ x \in M $ is independent of the spectral cutting $ L_{\theta} $ under the conditions of Theorem 5.1 or Theorem 5.2.
• For selfadjoint odd-class operators, the sectorial projection $ \Pi_{\theta,\theta'}(P) $ is the orthogonal projection onto the spectral subspace corresponding to the spectrum in the sector $ \theta < \arg \lambda < \theta' $.
• When $ P $ is normal, $ \Pi_{\theta,\theta'}(P) $ projects onto the Hilbertian direct sum of eigenspaces $ \oplus_{\lambda \in \operatorname{Sp} P \cap \Lambda_{\theta,\theta'}} \ker(P - \lambda) $.
• The spectral asymmetry of Dirac operators is geometrically expressed via the noncommutative residue, providing a new spectral interpretation of the Einstein-Hilbert action in gravity.