The $\bar\mu$-invariant of Seifert fibered homology spheres and the Dirac operator

TL;DR

This paper establishes a formula linking the $ar$-invariant of Seifert fibered homology spheres to the eta-invariant of their Dirac operators, leading to new vanishing results for Dirac indices on bounding 4-manifolds.

Contribution

It provides a novel explicit formula connecting the $ar$-invariant with Dirac operator eta-invariants for Seifert fibered spheres, and derives related vanishing theorems.

Findings

Derived a formula for the $ar$-invariant in terms of eta-invariants.

Established vanishing results for Dirac operator indices on bounding 4-manifolds.

Connected topological invariants with analytical spectral data.

Abstract

We derive a formula for the $\bar\mu$-invariant of a Seifert fibered homology sphere in terms of the eta-invariant of its Dirac operator. As a consequence, we obtain a vanishing result for the index of certain Dirac operators on plumbed 4-manifolds bounding such spheres.