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The eta invariant on two-step nilmanifolds

Ruth Gornet, Ken Richardson

Abstract

The eta invariant appears regularly in index theorems but is known to be directly computable from the spectrum only in certain examples of locally symmetric spaces of compact type. In this work, we derive some general formulas useful for calculating the eta invariant on closed manifolds. Specifically, we study the eta invariant on nilmanifolds by decomposing the spin Dirac operator using Kirillov theory. In particular, for general Heisenberg three-manifolds, the spectrum of the Dirac operator and the eta invariant are computed in terms of the metric, lattice, and spin structure data. There are continuous families of geometrically, spectrally different Heisenberg three-manifolds whose Dirac operators have constant eta invariant. In the appendix, some needed results of L. Richardson and C. C. Moore are extended from spaces of functions to spaces of spinors.

The eta invariant on two-step nilmanifolds

Abstract

The eta invariant appears regularly in index theorems but is known to be directly computable from the spectrum only in certain examples of locally symmetric spaces of compact type. In this work, we derive some general formulas useful for calculating the eta invariant on closed manifolds. Specifically, we study the eta invariant on nilmanifolds by decomposing the spin Dirac operator using Kirillov theory. In particular, for general Heisenberg three-manifolds, the spectrum of the Dirac operator and the eta invariant are computed in terms of the metric, lattice, and spin structure data. There are continuous families of geometrically, spectrally different Heisenberg three-manifolds whose Dirac operators have constant eta invariant. In the appendix, some needed results of L. Richardson and C. C. Moore are extended from spaces of functions to spaces of spinors.

Paper Structure

This paper contains 29 sections, 24 theorems, 287 equations.

Table of Contents

  1. Introduction
  2. The eta invariant
  3. Eta and zeta functions of perturbed operators
  4. Heat Kernel Asymptotics
  5. The eta invariant for arbitrary manifolds with spectral symmetry
  6. The zeta function and the eta invariant for three-manifolds
  7. Two-step Nilmanifolds and Dirac operators
  8. Two-step Nilmanifolds and the Laplace-Beltrami operator
  9. The Dirac operator on two-step nilmanifolds
  10. Analogue of Pesce's theorem for spinors
  11. Decomposition of the Dirac operator on two-step nilmanifolds
  12. Finite-dimensional $\overline{\mathcal{H}_{\alpha }}$-irreducible subspaces: $k_{\alpha }=n$, i.e. $\alpha \left( \left[ \mathfrak{g},\mathfrak{g}\right] \right) =0$.
  13. Infinite-dimensional $\overline{\mathcal{H}_{ \alpha }}$-irreducible subspaces: $k_{\alpha }<n$, so that $\alpha \left( \left[ \mathfrak{g},\mathfrak{g}\right] \right)$ is not identically zero.
  14. Matrix choices
  15. Heisenberg Examples
  16. ...and 14 more sections

Key Result

Proposition \oldthetheorem

Let $D$ be any self-adjoint operator for which $\eta \left( s\right)$ is defined and analytic at $s=0$. Suppose in addition that there exists an interval $I\subset \mathbb{R}$ and a constant $B>0$ such that for all $c\in I$, Then the eta function $\eta _{c}\left( s\right)$ corresponding to the operator $D+c$ satisfies, on its domain, where $\zeta _{\left( D+c\right) ^{2}}$ is the zeta function co

Theorems & Definitions (44)

  • Proposition \oldthetheorem
  • proof
  • Corollary \oldthetheorem
  • Proposition \oldthetheorem
  • proof
  • Proposition \oldthetheorem
  • proof
  • Theorem \oldthetheorem
  • Proposition \oldthetheorem
  • Remark \oldthetheorem
  • ...and 34 more