[Paper Review] The moduli space of special Lagrangian submanifolds

Nigel Hitchin|arXiv (Cornell University)|Nov 4, 1997

Geometric Analysis and Curvature Flows2 references41 citations

TL;DR

This paper establishes that the moduli space of special Lagrangian submanifolds in a Calabi-Yau manifold naturally carries the structure of a Lagrangian submanifold inside the product of the first and second cohomology groups, $ H^1(L,bR) imes H^{n-1}(L,bR) $, with a canonical Riemannian metric induced from McLean's theory. It further shows that this moduli space admits a natural complex structure and Kähler metric, and when the embedding is special, the metric is Calabi-Yau, linking the geometry of special Lagrangians to mirror symmetry via Legendre duality.

ABSTRACT

This paper considers the natural geometric structure on the moduli space of deformations of a compact special Lagrangian submanifold $L^n$ of a Calabi-Yau manifold. From the work of McLean this is a smooth manifold with a natural $L^2$ metric. It is shown that the metric is induced from a local Lagrangian immersion into the product of cohomology groups $H^1(L) imes H^{n-1}(L)$. Using this approach, an interpretation of the mirror symmetry discussed by Strominger, Yau and Zaslow is given in terms of the classical Legendre transform.

Motivation & Objective

• To determine the natural geometric structure on the moduli space of special Lagrangian submanifolds in a Calabi-Yau manifold.
• To show that this moduli space inherits a Riemannian metric from McLean's deformation theory via the $ L^2 $-inner product on harmonic 1-forms.
• To demonstrate that the moduli space embeds naturally as a Lagrangian submanifold in the product of dual cohomology groups $ H^1(L,bR) \times H^{n-1}(L,bR) $.
• To establish that this embedding, when special, endows the moduli space with a Calabi-Yau metric, linking it to mirror symmetry.
• To show that the moduli space can be locally described as the graph of the derivative of a function, realizing the Legendre transform symmetry central to the Strominger-Yau-Zaslow mirror symmetry proposal.

Proposed method

• Utilizes McLean's deformation theory to identify the tangent space of the moduli space with the space of harmonic 1-forms on the special Lagrangian submanifold.
• Constructs a natural embedding of the local moduli space into $ H^1(L,bR) \times H^{n-1}(L,bR) $, which are dual vector spaces, endowing it with a symplectic structure.
• Shows that the induced metric on the moduli space from McLean's $ L^2 $-inner product corresponds to the natural Riemannian metric on this Lagrangian submanifold.
• Applies the Legendre transform to relate the moduli space to the dual description, interpreting the symmetry as a manifestation of mirror symmetry.
• Constructs a Kähler metric on the moduli space by pulling back a metric from a complexified version of the moduli space, using a Kähler potential derived from a real-valued function $ ho $.
• Demonstrates that the holomorphic $ m $-form on the complexified moduli space has constant length if and only if the volume of the special Lagrangian torus is constant, implying the metric is Calabi-Yau.

Experimental results

Research questions

• RQ1 What is the natural geometric structure on the moduli space of special Lagrangian submanifolds in a Calabi-Yau manifold?
• RQ2 How does McLean's $ L^2 $-metric on harmonic 1-forms relate to the intrinsic geometry of the moduli space?
• RQ3 Can the moduli space be naturally embedded as a Lagrangian submanifold in a symplectic vector space, and if so, in which space?
• RQ4 What conditions ensure that the induced metric on the moduli space is Calabi-Yau?
• RQ5 How does the Legendre duality between the moduli space and its dual description reflect the Strominger-Yau-Zaslow mirror symmetry proposal?

Key findings

• The moduli space of special Lagrangian submanifolds embeds canonically as a Lagrangian submanifold in $ H^1(L,bR) \times H^{n-1}(L,bR) $, the product of dual cohomology groups.
• McLean's $ L^2 $-metric on harmonic 1-forms is realized as the natural induced Riemannian metric on this Lagrangian submanifold.
• The moduli space admits a natural complex structure and Kähler metric, with a Kähler potential given by $ ho/2 $, where $ ho $ is a real-valued function on the moduli space.
• The holomorphic $ m $-form on the complexified moduli space has constant length if and only if the volume of the special Lagrangian torus is constant.
• When the embedding is special, the Kähler metric on the moduli space is Calabi-Yau, meaning it admits a covariant constant holomorphic $ m $-form.
• The symmetry between the moduli space and its dual description via the Legendre transform provides a geometric realization of mirror symmetry in the context of special Lagrangian fibrations.