Non-K\"ahler Special Lagrangian submanifolds and SYZ mirror symmetry

TL;DR

This paper identifies algebraic equations for special Lagrangian submanifolds in non-K"ahler Calabi-Yau manifolds, studies their properties, and explores their role in SYZ mirror symmetry, including explicit examples and mirror pairs.

Contribution

It provides a method to determine SLag distributions algebraically, analyzes their deformation theory, and constructs explicit non-K"ahler SYZ mirror pairs with detailed cohomological properties.

Findings

SLag distributions can be explicitly characterized algebraically.

Deformation theory of these SLags is unobstructed.

Existence of non-diffeomorphic semi-flat non-K"ahler mirror pairs.

Abstract

We determine purely algebraic equations to identify \textit{SLags} generated by invariant distributions in a class of non-K\"ahler Calabi-Yau manifolds. We determine SLag distributions, determine which leaves integrate to compact submanifolds and study the deformation theory, which we find to be unobstructed. We apply our results to the Iwasawa manifold, the completely solvable 6-dimensional Nakamura manifold and the complex parallelizable Nakamura manifold. Through these examples we find families of topologically distinct \textit{SLags}, including the existence of SLag torus fibrations. Following the proposal of Lau-Tseng-Yau, we compute the non-K\"ahler SYZ mirrors of Nakamura manifolds, together with their refined symplectic Bott-Chern cohomologies. As a consequence, we find the existence of semi-flat non-K\"ahler mirror pairs which are not diffeomorphic.