Lectures on Special Lagrangian Submanifolds
Nigel Hitchin
TL;DR
The notes develop a higher-geometric framework for mirror symmetry in Calabi–Yau 3-folds by incorporating gerbes, especially degree-one gerbes, into the moduli of special Lagrangian submanifolds. They construct a Kähler quotient description of the moduli space of special Lagrangian submanifolds equipped with gerbes, with McLean’s deformation theory giving a real $b_1$-dimensional tangent space and a Hessian-type metric on the base. The SYZ program is extended to include a B-field via flat gerbes, yielding a mirror description as the moduli of special Lagrangian tori with gerbe trivializations and a Gauss–Manin–Legendre duality between the original and mirror geometries. The approach connects Abel–Jacobi-type maps, linear equivalence of codimension-three submanifolds, and the geometry of gerbes to provide a cohesive language for mirror symmetry, including the B-field phenomenon. Its significance lies in offering a concrete, gerbe-enriched geometric framework that links deformations, holonomy, and dual torus structures within SYZ-type mirror constructions.
Abstract
These notes consist of a study of special Lagrangian submanifolds of Calabi-Yau manifolds and their moduli spaces. The particular case of three dimensions, important in string theory, allows us to introduce the notion of gerbes. These offer an appropriate language for describing many significant features of the Strominger-Yau-Zaslow approach to mirror symmetry.