Orientifolds and Slumps in G_2 and Spin(7) Metrics

TL;DR

This paper introduces new special holonomy metrics in G_2 and Spin(7) geometries, explores their string theory implications, and demonstrates how they enable non-singular M-theory to IIA reductions with charged cycles.

Contribution

It presents novel Spin(7) and G_2 metrics with specific topologies, analyzes their properties, and connects certain hyper-Kahler metrics to Toda equations for interior structure insights.

Findings

Spin(7) metrics on a complex line bundle over CP^3 with orientifold interpretation

G_2 metrics on R^2 bundle over T^{1,1} with non-singular IIA reduction

Hyper-Kahler metrics related to solutions of the su(∞) Toda equation

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.