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On Classifying HyperKähler Kummer 8-Orbifolds

Daniel Andrew Baldwin, Bobby Samir Acharya

TL;DR

Acharya and Baldwin systematically classify eight-dimensional compact hyperKähler orbifolds of Kummer type by analyzing finite subgroups of the Weyl group $W(E_8)$ acting on the maximal torus $T^8$ that preserve a hyperKähler structure. They construct 139 model orbifolds $T^8/H'$, provide explicit matrix realizations, and determine, via a space-group analysis and symplectic-resolution criteria, that only one case (row 24) yields a crepant resolution yielding a manifold diffeomorphic to the known compact hyperKähler eight-manifolds, with all other cases containing non-resolvable singularities. The work demonstrates the limitations of this construction for producing new hyperKähler manifolds but delivers a comprehensive catalog of hyperKähler Kummer-type orbifolds and a method to assess resolvability, which can be extended to SU$(4)$ and Spin$(7)$ holonomies via Joyce’s perturbation techniques. The results have implications for the landscape of string/$M$-theory vacua and suggest directions for finding novel holonomy manifolds through controlled crepant resolutions.

Abstract

HyperKähler 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 hyperKähler 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 hyperKähler 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.

On Classifying HyperKähler Kummer 8-Orbifolds

TL;DR

Acharya and Baldwin systematically classify eight-dimensional compact hyperKähler orbifolds of Kummer type by analyzing finite subgroups of the Weyl group acting on the maximal torus that preserve a hyperKähler structure. They construct 139 model orbifolds , provide explicit matrix realizations, and determine, via a space-group analysis and symplectic-resolution criteria, that only one case (row 24) yields a crepant resolution yielding a manifold diffeomorphic to the known compact hyperKähler eight-manifolds, with all other cases containing non-resolvable singularities. The work demonstrates the limitations of this construction for producing new hyperKähler manifolds but delivers a comprehensive catalog of hyperKähler Kummer-type orbifolds and a method to assess resolvability, which can be extended to SU and Spin holonomies via Joyce’s perturbation techniques. The results have implications for the landscape of string/-theory vacua and suggest directions for finding novel holonomy manifolds through controlled crepant resolutions.

Abstract

HyperKähler spaces, including manifolds, orbifolds and conical singularities play an important role in superstring/-theory and gauge theories as well as in differential and algebraic geometry. In this paper we provide hundreds of new examples of compact hyperKähler orbifolds of Kummer type , where is the maximal torus of the compact Lie group and a finite group of isometries whose holonomies form a subgroup of the Weyl group of . 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 hyperKähler 8-manifolds. We also demonstrate that these methods can, when combined with theorems of Joyce, be extended to construct potentially new manifolds of - and - holonomy. All of these examples give rise to new vacua of string/-theory in two/three dimensions.
Paper Structure (22 sections, 1 theorem, 55 equations, 9 tables)

This paper contains 22 sections, 1 theorem, 55 equations, 9 tables.

Table of Contents

  1. Introduction
  2. HyperKähler Geometry
  3. Compact hyperKähler manifolds
  4. Symplectic resolutions
  5. Quotients of the Maximal Torus of $E_{8}$
  6. Classification of subgroups of $W(E_{8})$ preserving a hyperKähler structure
  7. Subgroups preserving special structures
  8. Explicit matrix representations
  9. List of model hyperKähler orbifolds
  10. Studying the Singular Sets
  11. Deciding non-resolvability
  12. Other Holonomy Groups
  13. The Octonions and Triality
  14. The Integral Octonions
  15. Identities, $\operatorname{Spin}(8)$ and triality
  16. ...and 7 more sections

Key Result

Theorem 1

Let $T^{8} = \mathbb{C}^{4}/\Lambda_{E_{8}}$ be the maximal torus of $E_{8}$ and $G_{i}\subset \operatorname{Aut}(\Lambda_{E_{8}})$$(i=1,2,3)$ be as described in sec: mat reps. Let $H\subset G_{i}$ be a subgroup that is neither Abelian nor acting on $\mathbb{C}^{4}$ as $(\mathbb{C}^{2}/\Gamma_{ADE}) we have that $\pi(\hat{H}) = H$ and $\hat{H}$ contains a subgroup $T_{E_{8}} = \{(\mathbb{1}, v)\mi

Theorems & Definitions (3)

  • Example 1
  • Theorem 1
  • Remark 1