Topology of Toric Gravitational Instantons

TL;DR

This paper investigates the topology of toric gravitational instantons, expressing their signature via rod structures, deriving necessary conditions for their classification, and analyzing specific cases with three turning points.

Contribution

It introduces a method to compute the signature from rod structures and formulates necessary conditions for classifying toric ALE/ALF instantons.

Findings

Signature expressed in terms of rod structure

Necessary conditions for rod structures of instantons

Analysis of cases with three turning points

Abstract

For an asymptotically locally Euclidean (ALE) or asymptotically locally flat (ALF) gravitational instanton $(M,g)$ with toric symmetry, we express the signature of $(M,g)$ directly in terms of its rod structure. Applying Hitchin$\unicode{x2013}$Thorpe-type inequalities for Ricci-flat ALE/ALF manifolds, we formulate, as a step toward a classification of toric ALE/ALF instantons, necessary conditions that the rod structures of such spaces must satisfy. Finally, we apply these results to the study of rod structures with three turning points.