The eta invariant on two-step nilmanifolds

TL;DR

This paper develops formulas for computing the eta invariant on nilmanifolds, especially Heisenberg three-manifolds, by decomposing the Dirac operator and analyzing its spectrum in relation to geometric data.

Contribution

It introduces a method to compute the eta invariant on nilmanifolds using Kirillov theory and extends existing results to spinor spaces, providing explicit spectral formulas.

Findings

Computed eta invariant for Heisenberg three-manifolds in terms of geometric data

Identified continuous families with constant eta invariant despite spectral differences

Extended representation theory results to spinor spaces

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.