Infinitely many new families of complete cohomogeneity one G_2manifolds: G_2 analogues of the TaubNUT and EguchiHanson spaces
Abstract
We construct infinitely many new 1parameter families of simply connected complete noncompact G_2manifolds with controlled geometry at infinity. The generic member of each family has socalled asymptotically locally conical (ALC) geometry. However, the nature of the asymptotic geometry changes at two special parameter values: at one special value we obtain a unique member of each family with asymptotically conical (AC) geometry; on approach to the other special parameter value the family of metrics collapses to an AC CalabiYau 3fold. Our infinitely many new diffeomorphism types of AC G_2manifolds are particularly noteworthy: previously the three examples constructed by Bryant and Salamon in 1989 furnished the only known simply connected AC G_2manifolds. We also construct a closely related conically singular G_2 holonomy space: away from a single isolated conical singularity, where the geometry becomes asymptotic to the G_2cone over the standard nearly Kähler structure on the product of a pair of 3spheres, the metric is smooth and it has ALC geometry at infinity. We argue that this conically singular ALC G_2space is the natural G_2 analogue of the TaubNUT metric in 4dimensional hyperKaehler geometry and that our new AC G_2metrics are all analogues of the EguchiHanson metric, the simplest ALE hyperKähler manifold. Like the TaubNUT and EguchiHanson metrics, all our examples are cohomogeneity one, i.e. they admit an isometric Lie group action whose generic orbit has codimension one.
 Publication:

arXiv eprints
 Pub Date:
 May 2018
 arXiv:
 arXiv:1805.02612
 Bibcode:
 2018arXiv180502612F
 Keywords:

 Mathematics  Differential Geometry;
 High Energy Physics  Theory;
 53C10;
 53C25;
 53C29;
 53C80
 EPrint:
 53 pages, 1 figure. v3: Corrected Theorem 5.1. To appear in JEMS