Infinitely many new families of complete cohomogeneity one G_2-manifolds: G_2 analogues of the Taub-NUT and Eguchi-Hanson spaces

Abstract

We construct infinitely many new 1-parameter families of simply connected complete noncompact G2-manifolds with controlled geometry at infinity. The generic member of each family has so-called 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 Calabi-Yau 3-fold. Our infinitely many new diffeomorphism types of AC G2-manifolds are particularly noteworthy: previously the three examples constructed by Bryant and Salamon in 1989 furnished the only known simply connected AC G2-manifolds.We also construct a closely related conically singular G2-holonomy space: away from a single isolated conical singularity, where the geometry becomes asymptotic to the G2-cone over the standard nearly Kähler structure on the product of a pair of 3-spheres, the metric is smooth and it has ALC geometry at infinity. We argue that this conically singular ALC G2-space is the natural G2 analogue of the Taub-NUT metric in 4-dimensional hyperKähler geometry and that our new AC G2-metrics are all analogues of the Eguchi-Hanson metric, the simplest ALE hyperKähler manifold. Like the Taub-NUT and Eguchi-Hanson metrics, all our examples are cohomogeneity one, i.e. they admit an isometric Lie group action whose generic orbit has codimension one.

Complete non-compact Spin(7) manifolds from self-dual Einstein 4-orbifolds

“…A complete geometric explanation of the physical duality would involve constructing AC Spin(7)-metrics arising as limits of the two 1-parameter families of ALC Spin(7)-metrics as well as an ALC Spin(7)-metric on R + × SU(3)/U(1) with an isolated conical singularity. This is completely analogous to the analysis of [32] in the G 2 setting. In fact, since the metrics in Theorem 4.3 admit a cohomogeneity one action of SU(3), it is likely that the methods of [32] can address these conjectures.…”

“…This is completely analogous to the analysis of [32] in the G 2 setting. In fact, since the metrics in Theorem 4.3 admit a cohomogeneity one action of SU(3), it is likely that the methods of [32] can address these conjectures. 4.2.2.…”

“…The large symmetry group affords a reduction of the PDE for the holonomy reduction to G 2 to a system of nonlinear ODEs. Studying solutions to these ODE systems is still non-trivial: in 1989 Bryant-Salamon [17] constructed three explicit AC G 2 -metrics, but only very recently [32,Theorem C] have further examples of AC G 2 -manifolds been found; these examples are not explicit and their existence is based on the qualitative analysis of the relevant ODE system.…”

“…Amongst the known examples of smooth AC G 2 -manifolds, only Λ − T * CP 2 endowed with Bryant-Salamon's AC G 2 -metric can be used in Theorem 3.35. Indeed, the other two Bryant-Salamon examples of AC G 2 -manifolds[17] have vanishing second cohomology, while all the infinitely many examples constructed in[32] only have compactly supported second cohomology. Thus the topological constraint (3.23) can be satisfied in a non-trivial way only in the case of Λ − T * CP 2 .…”

Deformations of Asymptotically Conical $G_2$-Instantons

“…3. Foscolo, Haskins and Nordström find in [13] infinitely many new diffeomorphism types of AC G 2 manifold with asymptotic link (a quotient of) S 3 × S 3 .…”