SU(4)-holonomy via the left-invariant hypo and Hitchin flow

TL;DR

This paper explores the Hitchin flow on specific Lie algebras to produce new Calabi-Yau fourfolds with SU(4) holonomy, expanding understanding of special geometric structures and their classifications.

Contribution

It systematically studies hypo SU(3)-structures on Lie algebras and identifies conditions for SU(4) holonomy, providing new explicit examples and classifications of Calabi-Yau fourfolds.

Findings

Hitchin flow on almost Abelian Lie algebras yields SU(4) holonomy.

Classification of hypo SU(3)-structures with specific torsion.

Many new explicit Calabi-Yau fourfold examples.

Abstract

The Hitchin flow constructs eight-dimensional Riemannian manifolds (M,g) with holonomy in Spin(7) starting with a cocalibrated G_2-structure on a seven-dimensional manifold. As Sp(2)\subseteq SU(4)\subseteq Spin(7), one may also obtain Calabi-Yau fourfolds or hyperK\"ahler manifolds via the Hitchin flow. In this paper, we show that the Hitchin flow on almost Abelian Lie algebras and on Lie algebras with one-dimensional commutator always yields Riemannian metrics with Hol(g)\subseteq SU(4) but Hol(g)\neq Sp(2). We investigate when we actually get Hol(g)=SU(4) and obtain so many new explicit examples of Calabi-Yau fourfolds. The results rely on the connection between cocalibrated G_2-structures and hypo SU(3)-structures and between the Hitchin and the hypo flow and on a systematic study of hypo SU(3)-structures and the hypo flow on Lie algebras. This study gives us many other interesting results: We obtain full classifications of hypo SU(3)-structures with particular intrinsic torsion on Lie algebras. Moreover, we can exclude reducible or Sp(2)-holonomy or do get Hol(g)=SU(4) for the Riemannian manifolds obtained by the hypo flow with initial values in some other intrinsic torsion classes.