Orientifolds and slumps in G2 and Spin(7) metrics

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 S7, described as a triaxially squashed S3 bundle over S4. The behaviour in the S3 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 S4. We then consider new G2 metrics which we denote by C 7 , which are complete on an R 2 bundle over T1,1, with principal orbits that are S3×S3. 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 S2 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(∞) Toda equation, which can provide a way of studying their interior structure.