Ricci-Flat Manifolds, Parallel Spinors and the Rosenberg Index

TL;DR

This paper extends the understanding of Ricci-flat spin manifolds by linking Rosenberg index to the existence of parallel spinors and special holonomy, generalizing previous results involving scalar curvature and invariants.

Contribution

It generalizes the criterion for parallel spinors to manifolds with non-vanishing Rosenberg index, broadening the class of manifolds where special holonomy can be deduced.

Findings

Manifolds with non-zero Rosenberg index admit parallel spinors on finite covers.

Every Ricci-flat spin manifold with non-zero Rosenberg index has special holonomy.

The result connects index theory with geometric structures of manifolds.

Abstract

Every closed connected Riemannian spin manifold of non-zero $\hat{A}$-genus or non-zero Hitchin invariant with non-negative scalar curvature admits a parallel spinor, in particular is Ricci-flat. In this note, we generalize this result to closed connected spin manifolds of non-vanishing Rosenberg index. This provides a criterion for the existence of a parallel spinor on a finite covering and yields that every closed connected Ricci-flat spin manifold of dimension $\geq 2$ with non-vanishing Rosenberg index has special holonomy.