The geometry of wrapped M5-branes in Calabi-Yau 2-folds

TL;DR

This paper investigates the geometric structures of M5-branes wrapping special Lagrangian 2-cycles in Calabi-Yau two-folds, revealing new integrability properties and relating to known holomorphic cycle geometries.

Contribution

It introduces a characterization of wrapped M5-brane geometries using non-integrable and integrable almost complex structures in Calabi-Yau two-folds, extending previous methods from three-folds.

Findings

Identifies a non-integrable almost complex structure describing the geometry.

Shows the existence of an integrable almost complex structure for the wrapped cycle.

Establishes connections between special Lagrangian and holomorphic cycle geometries.

Abstract

We study the geometry of M5-branes wrapping a 2-cycle which is Special Lagrangian with respect to a specific complex structure in a Calabi-Yau two-fold. Using methods recently applied to the three-fold case, we are again able find a characterization of the geometry, in terms of a non-integrable almost complex structure and a (2,0) form. This time, however, due to the hyper-K{\"a}hler nature of the underlying 2-fold we also have the freedom of choosing a different almost complex structure with respect to which the wrapped 2-cycle is holomorphic. We show that this latter almost complex structure is integrable. We then relate our geometry to previously found geometries of M5-branes wrapping holomophic cycles and go further to prove some previously unknown results for M5-branes on holomorphic cycles.