Almost special holonomy in type IIA and M-theory

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) x M-7 or (Minkowski)(3) x M-8, to give supersymmetric solutions (Minkowski)(4) x Y-6 or (Minkowski)(3) x 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 Kahler, 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. (C) 2002 Elsevier Science B.V. All rights reserved.