Harmonic forms on ALF gravitational instantons

TL;DR

This paper investigates the space of square-integrable harmonic forms on ALF gravitational instantons of types A_{K-1} and D_K, providing dimension calculations, explicit bases, and geometric interpretations.

Contribution

It computes the dimension of harmonic forms on ALF gravitational instantons and constructs explicit bases, linking cohomology of the instantons to their compactifications.

Findings

Calculated the dimension of harmonic forms on ALF instantons.

Constructed explicit bases for these harmonic forms.

Provided geometric interpretations of the cohomological relations.

Abstract

We study the space of square-integrable harmonic forms over ALF gravitational instantons of type $A _{ K -1 } $ and of type $D _K $. We first calculate its dimension making use of a result by Hausel, Hunsicker and Mazzeo which relates the Hodge cohomology of a gravitational instanton $M$ to the singular cohomology of a particular compactification $X _M $ of $M$. We then exhibit an explicit basis, exact for $A _{ K -1 } $ and approximate for $D _K $, and interpret geometrically the relations between $M$, $X _M $ and their cohomologies.