Almost special holonomy in type IIA and M-theory Original Research Article

We consider spaces M7 and M8 of G2 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×M7 or (Minkowski)3×M8, to give supersymmetric solutions (Minkowski)4×Y6 or (Minkowski)3×Y7 in type IIA string theory with a non-singular dilaton. We study the associated six- and seven-dimensional spaces Y6 and Y7 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 Y6 means almost Kähler, together with a further condition. For Y7 we are led to introduce the notion of an “almost G2 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 G2 and Spin(7) metrics.