Associative Submanifolds of Squashed 3-Sasakian Manifolds

Abstract

Every compact 3-Sasakian 7-manifold $M$ admits a canonical 2-parameter family of co-closed ${\text{G}}_2$-structures ${\phi }_{a,b}$ for $a,b>0$, as well as a foliation by ${\phi }_{a,b}$-associative 3-folds whose leaf space $X$ is a positive quaternion-Kähler 4-orbifold. We prove that associative 3-folds in $(M,{\phi }_{a,b})$ that are ruled by a certain type of geodesic are in correspondence with pseudo-holomorphic curves in the almost-complex 8-manifold $Z\times S^2$, where $Z$ is the twistor space of $X$ equipped with its strict nearly-Kähler structure. As an application, we construct infinitely many topological types of non-trivial, compact associative 3-folds in the squashed 7-spheres $(S^7,{\phi }_{a,b})$ and squashed exceptional Aloff-Wallach spaces $(N_{1,1},{\phi }_{a,b})$. Topologically, our examples are circle bundles over a genus $g$ surface, for any $g\ge 0$.

39 pages, 2 figures