Almost Special Holonomy in Type IIA&M Theory

TL;DR

The paper develops a systematic framework for relating $G_2$ and Spin$(7)$ holonomy manifolds with a $U(1)$ isometry to type IIA backgrounds via Kaluza-Klein reduction, focusing on the weak-coupling regime where the six- and seven-dimensional bases remain Ricci-flat but acquire almost-holonomy structures. By analyzing linear perturbations around Gromov-Hausdorff limits, the authors construct explicit, non-singular examples of almost-$K\ähler$ and almost-$G_2$ geometries that can be lifted back to higher-dimensional Spin$(7)$ metrics, thereby connecting Calabi–Yau perturbations to non-compact $G_2$ manifolds and their Spin$(7)$ relatives. They provide concrete analytic realizations for deformed and resolved conifolds, complex line bundles over $S^2\times S^2$, and higher-dimensional line-bundle geometries, yielding regular perturbations that reproduce and extend previously known numerical results (e.g., ${\mathbb{D}}_7$, ${\widetilde{{\mathbb{C}}}}_7$, and ${\mathbb{B}}_7$ families). The work highlights how almost-holonomy geometries arise naturally in perturbative regimes and offers analytic tools for exploring string/M-theory dualities via circle reductions and near-GH limits.

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 (Minkowksi)_4\times M_7 or (Minkowksi)_3\times M_8, to give supersymmetric solutions (Minkowksi)_4\times Y_6 or (Minkowksi)_3\times Y_7 in type IIA string theory with a non-singular dilaton. We study the associated six-dimensional 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.