Immersed Lagrangian deformations of a branched covering of a special Lagrangian 3-sphere in a Calabi-Yau 3-fold and its deviation from Joyce's criteria: Potential image-support rigidity of A-branes that wrap around a sL $S^3$

TL;DR

This paper constructs a family of immersed Lagrangian deformations of a branched covering of a special Lagrangian 3-sphere in a Calabi-Yau 3-fold, analyzing their deviation from Joyce's criteria and suggesting potential rigidity of A-branes wrapping around sL 3-spheres.

Contribution

It introduces a novel construction of immersed Lagrangian deformations via hyperKähler rotation and isotropic submanifold rolling, revealing deviations from Joyce's deformation criteria.

Findings

Deformation family constructed through gluing techniques.

Deviations from Joyce's criteria indicate potential rigidity.

Resembles phenomena in Gromov-Witten theory for holomorphic curves.

Abstract

Using a hyperK\"{a}hler rotation on complex structures of a Calabi-Yau 2-fold and rolling of an isotropic 2-submanifold in a symplectic 6-manifold, we construct, by gluing, a natural family of immersed Lagrangian deformations of a branched covering of a special Lagrangian 3-sphere in a Calabi-Yau 3-fold and study how they deviate from being deformable to a family of special Lagrangian deformations by examining in detail Joyce's criteria on this family. The result suggests a potential image-support rigidity of A-branes that wrap around a special Lagrangian 3-sphere in a Calabi-Yau 3-fold, which resembles a similar phenomenon for holomorphic curves that wrap around a rigid smooth rational curve in a Calabi-Yau 3-fold in Gromov-Witten theory.