$${\mathrm {SU}}(4)$$ SU ( 4 ) -holonomy via the left-invariant hypo and Hitchin flow

Abstract

The Hitchin flow constructs eight-dimensional Riemannian manifolds (M, g) with holonomy in $${\mathrm {Spin}}(7)$$ starting with a cocalibrated $$\mathrm {G}_2$$ -structure on a seven-dimensional manifold. As $${\mathrm {Sp}}(2)\subseteq {\mathrm {SU}}(4)\subseteq {\mathrm {Spin}}(7)$$ , one may also obtain Calabi–Yau fourfolds or hyperkahler 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 g with $$Hol(g)\subseteq {\mathrm {SU}}(4)$$ but $$Hol(g)\ne {\mathrm {Sp}}(2)$$ . We investigate when we actually get $$Hol(g)={\mathrm {SU}}(4)$$ and obtain many new explicit examples of Calabi–Yau fourfolds. The results rely on the connection between cocalibrated $$\mathrm {G}_2$$ -structures and hypo $${\mathrm {SU}}(3)$$ -structures and between the Hitchin and the hypo flow and on a systematic study of hypo $${\mathrm {SU}}(3)$$ -structures and the hypo flow on Lie algebras. This study gives us many other interesting results: We obtain full classifications of hypo $${\mathrm {SU}}(3)$$ -structures with particular intrinsic torsion on Lie algebras. Moreover, we can exclude reducible or $${\mathrm {Sp}}(2)$$ -holonomy for the Riemannian manifolds obtained by the hypo flow on Lie algebras with initial values lying in certain intrinsic torsion classes and show that for initial values in other torsion classes we always get Riemannian metrics with holonomy equal to $${\mathrm {SU}}(4)$$ .