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A microlocal pathway to spectral asymmetry: curl and the eta invariant

Matteo Capoferri, Dmitri Vassiliev

Abstract

The notion of eta invariant is traditionally defined by means of analytic continuation. We prove, by examining the particular case of the operator curl, that the eta invariant can equivalently be obtained as the trace of the difference of positive and negative spectral projections, appropriately regularised. Our construction is direct, in the sense that it does not involve analytic continuation, and is based on the use of pseudodifferential techniques. This provides a novel approach to the study of spectral asymmetry of non-semibounded (pseudo)differential systems on manifolds which encompasses and extends previous results.

A microlocal pathway to spectral asymmetry: curl and the eta invariant

Abstract

The notion of eta invariant is traditionally defined by means of analytic continuation. We prove, by examining the particular case of the operator curl, that the eta invariant can equivalently be obtained as the trace of the difference of positive and negative spectral projections, appropriately regularised. Our construction is direct, in the sense that it does not involve analytic continuation, and is based on the use of pseudodifferential techniques. This provides a novel approach to the study of spectral asymmetry of non-semibounded (pseudo)differential systems on manifolds which encompasses and extends previous results.

Paper Structure

This paper contains 14 sections, 24 theorems, 198 equations, 1 figure.

Table of Contents

  1. Statement of the problem
  2. Main results
  3. The matrix trace of an operator, revisited
  4. The parameter-dependent asymmetry operator $A^{(s)}$
  5. An alternative representation for $A^{(s)}$
  6. The order of $A^{(s)}$
  7. The principal symbol of $A^{(s)}$
  8. From $A^{(s)}$ to $\eta_{\operatorname{curl}}(s)$
  9. The leading singularity of the integral kernel of $A^{(s)}$
  10. Proof of Theorem \ref{['Theorem main result']}
  11. Concluding remarks
  12. Symbol of the Hodge Laplacian
  13. Proof of Lemma \ref{['lemma Apt']}
  14. Auxiliary proposition on the leading singularity

Key Result

Theorem 1.6

The integral kernel $\mathfrak{a}(x,y)$ of the asymmetry operator $A$ is a bounded function, smooth outside the diagonal. Furthermore, for any $x\in M$ the limit exists and defines a continuous scalar function $\,\psi_{\operatorname{curl}}^\mathrm{loc}:M\to\mathbb{R}\,$. Here $\mathbb{S}_r(x)=\{y\in M|\operatorname{dist}(x,y)=r\}$ is the sphere of radius $r$ centred at $x$ and $\mathrm{d} S_y$ is

Figures (1)

  • Figure 1: Contour of integration.

Theorems & Definitions (58)

  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Definition 1.4
  • Definition 1.5
  • Theorem 1.6: curl
  • Definition 1.7
  • Remark 1.8
  • Theorem 2.1
  • Definition 2.2
  • ...and 48 more