Associative Submanifolds of Squashed 3-Sasakian Manifolds

TL;DR

This paper explores associative 3-folds in 3-Sasakian manifolds with G2-structures, establishing a correspondence with pseudo-holomorphic curves and constructing numerous examples with diverse topologies.

Contribution

It introduces a new correspondence between associative 3-folds and pseudo-holomorphic curves in an 8-manifold, and constructs infinitely many topologically distinct associative 3-folds.

Findings

Associative 3-folds correspond to pseudo-holomorphic curves in an 8-manifold.

Constructed infinitely many non-trivial, compact associative 3-folds.

Examples include circle bundles over surfaces of any genus.

Abstract

Every compact 3-Sasakian 7-manifold $M$ admits a canonical 2-parameter family of co-closed $\text{G}_2$-structures $\varphi_{a,b}$ for $a,b > 0$, as well as a foliation by $\varphi_{a,b}$-associative 3-folds whose leaf space $X$ is a positive quaternion-K\"{a}hler 4-orbifold. We prove that associative 3-folds in $(M,\varphi_{a,b})$ 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-K\"{a}hler structure. As an application, we construct infinitely many topological types of non-trivial, compact associative 3-folds in the squashed 7-spheres $(S^7, \varphi_{a,b})$ and squashed exceptional Aloff-Wallach spaces $(N_{1,1}, \varphi_{a,b})$. Topologically, our examples are circle bundles over a genus $g$ surface, for any $g \geq 0$.