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 non-compact G 2 -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 G 2 -manifolds are particularly noteworthy: previously the three examples constructed by Bryant and Salamon in 1989 furnished the only known simply connected AC G 2 -manifolds.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 2 -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 G 2 -space is the natural G 2 analogue of the Taub-NUT metric in 4-dimensional hyperKähler geometry and that our new AC G 2 -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.

A circle quotient of a G2 cone

“…By analogy to the families of metrics with G 2 holonomy interpolating between highlycollapsed metrics and those asymptotic to the cone over S 3 ×S 3 in [15], one might expect M to acquire a Calabi-Yau metric and the singular R 3 's to be special Lagrangian. In our situation, there is no such collapsed limit because C has no finite circles at infinity, and our work shows that the picture painted in [4] is somewhat of an oversimplification.…”

New G2-conifolds in M-theory and their field theory interpretation

“…Modulo conventions, this is exactly the condition used in [16] to construct infinitely many new families of complete cohomogeneity one G 2 -manifolds.…”

“…However, precisely because these G 2 metrics move in 1-parameter families up to scale, it is not unreasonable to assume the existence of G 2 metrics for any size of the M -theory circle: in the limit of large radius we obtain a G 2 metric (unique up to scale) asymptotic to the G 2 cone C(S 3 × S 3 )/Γ p,N,q . In the simplest case p = 2 the existence of these asymptotically conical G 2 metrics was established in [16].…”

“…By a result of Karigiannis-Lotay [72] every smooth G 2 manifold asymptotic to C(S 3 ×S 3 ) or one of its quotients inherits all the G 2 symmetries of the cone. When p = 2 the symmetries of C(S 3 ×S 3 )/Γ 2,N,q contain SU(2)×SU(2)×U(1): the classification of complete G 2 metrics with SU(2) × SU(2) × U(1) symmetry in [16] guarantees that the classical moduli space contains no other branches than the ones in figure 7. For p > 2, the symmetry group of…”

“…Advances have been made both from the physics [9][10][11][12][13][14] and the mathematics [1,[15][16][17][18][19][20][21], and involves aspects such as a string-worldsheet interpretation [22][23][24], a better understanding of exceptional gauge bundles and instantons [25][26][27][28][29][30][31], in addition to non-perturbative aspects such as M 2 instantons and their relation to exceptional enumerative geometry [32][33][34].…”