On Classifying HyperK\"ahler Kummer 8-Orbifolds

TL;DR

This paper constructs numerous new examples of compact hyperK"ahler orbifolds of Kummer type in 8 dimensions, identifying which can be resolved into known hyperK"ahler manifolds and exploring potential new holonomy manifolds relevant to string theory.

Contribution

It provides hundreds of new hyperK"ahler orbifold examples of Kummer type and analyzes their resolutions, extending methods to potentially construct new special holonomy manifolds.

Findings

Identified orbifolds with known holomorphic symplectic resolutions

Found that only certain orbifolds resolve into known hyperK"ahler manifolds

Extended methods to construct potential new $ ext{SU}(4)$ and $ ext{Spin}(7)$ holonomy manifolds

Abstract

HyperK\"ahler spaces, including manifolds, orbifolds and conical singularities play an important role in superstring/$M$-theory and gauge theories as well as in differential and algebraic geometry. In this paper we provide hundreds of new examples of compact hyperK\"ahler orbifolds of Kummer type $T^8/G$, where $T^8$ is the maximal torus of the compact Lie group $E_8$ and $G$ a finite group of isometries whose holonomies form a subgroup of the Weyl group of $E_8$. We show that, out of all of these examples, the only orbifolds whose singularities have a known holomorphic symplectic resolution lead to manifolds diffeomorphic to the two currently known examples of compact hyperK\"ahler 8-manifolds. We also demonstrate that these methods can, when combined with theorems of Joyce, be extended to construct potentially new manifolds of $\operatorname{SU}(4)$- and $\operatorname{Spin}(7)$- holonomy. All of these examples give rise to new vacua of string/$M$-theory in two/three dimensions.