Orientifolds and Slumps in G_2 and Spin(7) Metrics

TL;DR

This work constructs and analyzes new complete metrics of special holonomy, notably Spin(7) and G_2, denoted ${\mathbb C}_8$ and ${\mathbb C}_7$, respectively. The Spin(7) ${\mathbb C}_8$ metrics live on a complex line bundle over ${\mathbb{CP}}^3$ with $S^7$ principal orbits and exhibit Atiyah–Hitchin–like slumping, enabling an orientifold interpretation with D6-branes wrapped on $S^4$; the G_2 ${\mathbb C}_7$ metrics live on an ${\mathbb R}^2$-bundle over $T^{1,1}$ and admit a nonvanishing, finite-length $U(1)$ Killing vector, permitting a completely non-singular M-theory to IIA reduction with RR flux through two $S^2$’s. The paper also connects four-dimensional hyper-Kähler metrics to su(\infty) Toda structures and surveys ALG gravitational instantons, monodromy, and negative-mass phenomena, illustrating how Todalike equations encode interior geometry. Together, these results provide new geometric backgrounds for string/M-theory, reveal novel non-singular dimensional reductions, and establish links between explicit cohomogeneity-one constructions and integrable Toda systems. The work thus advances the understanding of how special-holonomy metrics can encode physically meaningful brane configurations and dualities.

Abstract

We discuss some new metrics of special holonomy, and their roles in string theory and M-theory. First we consider Spin(7) metrics denoted by C_8, which are complete on a complex line bundle over CP^3. The principal orbits are S^7, described as a triaxially squashed S^3 bundle over S^4. The behaviour in the S^3 directions is similar to that in the Atiyah-Hitchin metric, and we show how this leads to an M-theory interpretation with orientifold D6-branes wrapped over S^4. We then consider new G_2 metrics which we denote by C_7, which are complete on an R^2 bundle over T^{1,1}, with principal orbits that are S^3\times S^3. We study the C_7 metrics using numerical methods, and we find that they have the remarkable property of admitting a U(1) Killing vector whose length is nowhere zero or infinite. This allows one to make an everywhere non-singular reduction of an M-theory solution to give a solution of the type IIA theory. The solution has two non-trivial S^2 cycles, and both carry magnetic charge with respect to the R-R vector field. We also discuss some four-dimensional hyper-Kahler metrics described recently by Cherkis and Kapustin, following earlier work by Kronheimer. We show that in certain cases these metrics, whose explicit form is known only asymptotically, can be related to metrics characterised by solutions of the su(\infty) Toda equation, which can provide a way of studying their interior structure.