[Paper Review] Lectures on Special Lagrangian Submanifolds

Nigel Hitchin|ArXiv.org|Jul 6, 1999

Geometry and complex manifolds10 references156 citations

TL;DR

This paper develops a gerbe-theoretic framework for understanding special Lagrangian submanifolds in 3-dimensional Calabi-Yau manifolds, providing a geometric foundation for the Strominger-Yau-Zaslow (SYZ) mirror symmetry conjecture. It constructs the SYZ mirror as a moduli space of special Lagrangian tori equipped with flat trivializations of a B-field gerbe, and shows that the resulting metric structure is Kähler via a Legendre transform duality between the original and mirror spaces.

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.

Motivation & Objective

• To provide a geometric framework using gerbes to describe special Lagrangian submanifolds in 3-dimensional Calabi-Yau manifolds.
• To clarify the role of the B-field in the Strominger-Yau-Zaslow (SYZ) mirror symmetry program.
• To define the SYZ mirror as a moduli space of special Lagrangian tori with flat trivializations of a gerbe.
• To establish a duality between the metric structures on the original and mirror Calabi-Yau manifolds via Legendre transforms.

Proposed method

• Using Čech cocycles with values in the circle group to define gerbes, capturing cohomology classes in $H^3(X,\mathbf{Z})$.
• Modeling the B-field as a flat gerbe with trivial holonomy on special Lagrangian torus fibers.
• Constructing the SYZ mirror $\check{Z}$ as the moduli space of pairs $(M,T)$, where $M$ is a special Lagrangian torus and $T$ is a flat trivialization of the gerbe on $M$.
• Defining a metric on $\check{Z}$ by combining the base metric from the moduli space $\mathcal{B}$ and the dual torus metric on fibers.
• Applying the Legendre transform to show that the metric on $\check{Z}$ is Kähler with potential $\check{\phi}$, dual to the original potential $\phi$.
• Using the Gauss-Manin connection to split the tangent bundle of $\check{Z}$ into horizontal and vertical components, enabling the construction of the Kähler metric.

Experimental results

Research questions

• RQ1 How can gerbes be used to describe the B-field in the context of special Lagrangian submanifolds in 3-dimensional Calabi-Yau manifolds?
• RQ2 What is the precise geometric structure of the SYZ mirror when a nontrivial B-field is present?
• RQ3 How does the flatness of the gerbe on special Lagrangian torus fibers affect the construction of the mirror manifold?
• RQ4 Can the metric on the mirror manifold be shown to be Kähler using a duality between the original and mirror potential functions?
• RQ5 What is the role of the Gauss-Manin connection in defining the splitting of the tangent bundle of the mirror space?

Key findings

• The SYZ mirror $\check{Z}$ of a Calabi-Yau manifold with a B-field is constructed as the moduli space of pairs $(M,T)$, where $M$ is a special Lagrangian torus and $T$ is a flat trivialization of the gerbe ${\bf B}$ on $M$.
• When the B-field is trivial, the construction recovers the original SYZ mirror, showing consistency with the standard framework.
• The restriction of the B-field to any special Lagrangian torus fiber has trivial holonomy, allowing for flat trivializations that define the mirror moduli space.
• The metric on $\check{Z}$ is shown to be Kähler, with the Kähler potential $\check{\phi}$ related to the original potential $\phi$ via a Legendre transform.
• The tangent space decomposition $T{\check{Z}} \cong H^1(M,\mathbf{R}) \otimes \mathbf{C}$ naturally gives rise to an almost complex structure, and the metric takes the form $\check{g} = \sum_{ij} \frac{\partial^2 \check{\phi}}{\partial \xi_i \partial \xi_j} (d\xi_i d\xi_j + d\eta_i d\eta_j)$, confirming its Kähler nature.
• The duality between the original and mirror metrics, mediated by the Legendre transform, reveals a deep symmetry between the metric structures on $Z$ and $\check{Z}$, even though neither is a true compact Calabi-Yau metric due to high symmetry.