Bianchi IX Self-dual Einstein Metrics and Singular G_2 Manifolds

TL;DR

The paper constructs explicit cohomogeneity-two $G_2$ holonomy metrics built from $S^2$ twistor-space bundles over self-dual Einstein 4-metrics of Bianchi IX type, deriving first-order equations and a unifying superpotential. It provides a thorough biaxial analysis, phase-plane classifications, and reveals a broader triaxial Tod–Hitchin connection, including explicit solutions that encompass $S^4$, ${\bf CP}^2$, Taub-NUT–de Sitter, and Eguchi-Hanson–de Sitter branches. The work clarifies how singularities in these spaces influence M-theory dynamics, highlighting co-dimension-four NUT-type singularities with potential gauge enhancements and co-dimension-two bolts with subtler interpretations, while showing supersymmetry remains preserved. Overall, the results deepen our understanding of how $G_2$ holonomy spaces with controlled singularities can serve as laboratories for non-perturbative aspects of M-theory and gauge structure arising from singular geometries.

Abstract

We construct explicit cohomogeneity two metrics of G_2 holonomy, which are foliated by twistor spaces. The twistor spaces are S^2 bundles over four-dimensional Bianchi IX Einstein metrics with self-dual (or anti-self-dual) Weyl tensor. Generically the 4-metric is of triaxial Bianchi IX type, with SU(2) isometry. We derive the first-order differential equations for the metric coefficients, and obtain the corresponding superpotential governing the equations of motion, in the general triaxial Bianchi IX case. In general our metrics have singularities, which are of orbifold or cosmic-string type. For the special case of biaxial Bianchi IX metrics, we give a complete analysis their local and global properties, and the singularities. In the triaxial case we find that a system of equations written down by Tod and Hitchin satisfies our first-order equations. The converse is not always true. A discussion is given of the possible implications of the singularity structure of these spaces for M-theory dynamics.

Figures (2)

• Figure 1: The phase plane for the first-order system of equation (\ref{['uvfo']}). The heavy blue ellipse corresponds to the ${{\Bbb C}{\Bbb P}}^2$ flow, and the three heavy red ellipses to $S^4$ flows. The dashed green circle is $u^2+v^2=1$; all solutions that cross this do so horizontally. To label distinct metrics it is sufficient to consider flows lying within the positive quadrant. We label qualitatively similar flows by A, B, C and D, which indicate the regions they occupy and their initial points. Thus the regions A, B and C indicate starting-points for solutions on the $v$ axis. Region A ranges from $v=0$ to the intersection of the ${{\Bbb C}{\Bbb P}}^2$ ellipse with the $v$ axis. Region B ranges from this intersection to the intersection of the outer ellipse (the $S^4$ solution (\ref{['s4again']})) with the $v$ axis. Region C ranges from this point to $v=+\infty$. Region D denotes (singular) starting points on the $u$ axis for solutions, in the range $0<u<1$. For clarity we have plotted complete ellipses for the $S^4$ an ${{\Bbb C}{\Bbb P}}^2$ special cases, but only flows in the upper half-plane for the other representative examples.
• Figure 2: The phase plane for the first-order system of equation (\ref{['neglam']}). The heavy blue hyperbola corresponds to the Bergmann flow, and the three heavy red hyperbolae to $H^4$ flows. The discussion of other flows is analogous to that for $\Lambda>0$; some representative examples are depicted.