Calabi-Yau monopoles, special Lagrangians, and Fueter sections

This talk investigates two types of proposed invariants of Calabi-Yau 3-folds:

• from gauge theory: Calabi-Yau monopoles,
• from calibrated geometry: count of special Lagrangians weighted with their Fueter sections.

Here, we focus on three conjectures central to the definition of these invariants and their relations:

1. The Donaldson-Segal conjecture on gauge theory/calibrated geometry duality: Calabi-Yau monopoles = weighted count of special Lagrangians,
2. The Donaldson-Scaduto conjecture: the existence of the pair of pants special Lagrangians, related to the formation of special Lagrangian singularities,
3. A hyperkähler variation of the Atiyah-Floer conjecture for Fueter sections: monopole Fueter Floer homology = Lagrangian Fueter Floer homology.

The discussion explores recent progress on these conjectures. This is joint work with Yang Li.