Almost special holonomy in type IIA and M-theory

Abstract

We consider spaces M 7 and M 8 of G 2 holonomy and Spin(7) holonomy in seven and eight dimensions, with a U(1) isometry. For metrics where the length of the associated circle is everywhere finite and non-zero, one can perform a Kaluza–Klein reduction of supersymmetric M-theory solutions (Minkowski) 4× M 7 or (Minkowski) 3× M 8, to give supersymmetric solutions (Minkowski) 4× Y 6 or (Minkowski) 3× Y 7 in type IIA string theory with a non-singular dilaton. We study the associated six- and seven-dimensional spaces Y 6 and Y 7 perturbatively in the regime where the string coupling is weak but still non-zero, for which the metrics remain Ricci-flat but that they no longer have special holonomy, at the linearised level. In fact they have “almost special holonomy”, which for the case of Y 6 means almost Kähler, together with a further condition. For Y 7 we are led to introduce the notion of an “almost G 2 manifold”, for which the associative 3-form is closed but not co-closed. We obtain explicit classes of non-singular metrics of almost special holonomy, associated with the near Gromov–Hausdorff limits of families of complete non-singular G 2 and Spin(7) metrics.