The moduli space of special Lagrangian submanifolds

TL;DR

This paper explores the geometric structure of the moduli space of special Lagrangian submanifolds in Calabi-Yau manifolds, revealing a Lagrangian immersion and connecting it to mirror symmetry via Legendre transform.

Contribution

It demonstrates that the moduli space's metric arises from a Lagrangian immersion into cohomology groups and interprets mirror symmetry through classical Legendre transform.

Findings

The moduli space has a natural L^2 metric induced by a Lagrangian immersion.

The approach links the geometry of the moduli space to mirror symmetry.

Provides a new interpretation of mirror symmetry via Legendre transform.

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 $H^1(L)\times H^{n-1}(L)$. Using this approach, an interpretation of the mirror symmetry discussed by Strominger, Yau and Zaslow is given in terms of the classical Legendre transform.