1. Moduli of special Lagrangian submanifolds [02YU]

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1. Moduli of special Lagrangian submanifolds

The first step in understanding the SYZ conjecture is to examine the structures which arise on the base of a special Lagrangian fibration. These structures arise from McLean’s theorem on the moduli space of special Lagrangian submanifolds [60], and these structures and their relationships were explained by Hitchin in [41]. We outline some of these ideas here. McLean’s theorem says that the moduli space of deformations of a compact special Lagrangian submanifold of a compact Calabi-Yau manifold $X$ is unobstructed. Further, the tangent space at the point of moduli space corresponding to a special Lagrangian $M\subseteq X$ is canonically isomorphic to the space of harmonic $1$ -forms on $M$ . This isomorphism is seen explicitly as follows. Let $\nu\in\Gamma(M,N_{M/X})$ be a normal vector field to $M$ in $X$ . Then the restriction of the contractions $(\iota(\nu)\omega)|_{M}$ and $(\iota(\nu)\operatorname{Im}\Omega)|_{M}$ are both seen to be well-defined forms on $M$ : one needs to lift $\nu$ to a vector field but the choice is irrelevant because $\omega$ and $\operatorname{Im}\Omega$ restrict to zero on $M$ . McLean shows that if $M$ is special Lagrangian then

\iota(\nu)\operatorname{Im}\Omega=-*\iota(\nu)\omega,

where $*$ denotes the Hodge star operator on $M$ . Furthermore, $\nu$ corresponds to an infinitesimal deformation preserving the special Lagrangian condition if and only if $d(\iota(\nu)\omega)=d(\iota(\nu)\operatorname{Im}\Omega)=0$ . This gives the correspondence between harmonic $1$ -forms and infinitesimal special Lagrangian deformations.

Let $f:X\rightarrow B$ be a special Lagrangian fibration with torus fibres, and assume for now that all fibres of $f$ are non-singular. Then we obtain three structures on $B$ , namely two affine structures and a metric, as we shall now see.

Definition 1.1.

Let $B$ be an $n$ -dimensional manifold. An affine structure on $B$ is given by an atlas $\{(U_{i},\psi_{i})\}$ of coordinate charts $\psi_{i}:U_{i}\rightarrow\mathbb{R}^{n}$ , whose transition functions $\psi_{i}\circ\psi_{j}^{-1}$ lie in ${\rm Aff}(\mathbb{R}^{n})$ . We say the affine structure is tropical if the transition functions lie in $\mathbb{R}^{n}\rtimes GL(\mathbb{Z}^{n})$ , i.e., have integral linear part. We say the affine structure is integral if the transition functions lie in ${\rm Aff}(\mathbb{Z}^{n})$ .

If an affine manifold $B$ carries a Riemannian metric $g$ , then we say the metric is affine Kähler or Hessian if $g$ is locally given by $g_{ij}=\partial^{2}K/\partial y_{i}\partial y_{j}$ for some convex function $K$ and $y_{1},\ldots,y_{n}$ affine coordinates.

Hessian and Monge-Ampére metrics were first discussed by Cheng and Yau in [12].

We obtain the three structures as follows:

Affine structure 1. For a normal vector field $\nu$ to a fibre $X_{b}$ of $f$ , $(\iota(\nu)\omega)|_{X_{b}}$ is a well-defined $1$ -form on $X_{b}$ , and we can compute its periods as follows. Let $U\subseteq B$ be a small open set, and suppose we have submanifolds $\gamma_{1},\ldots,\gamma_{n}\subseteq f^{-1}(U)$ which are families of 1-cycles over $U$ and such that $\gamma_{1}\cap X_{b},\ldots,\gamma_{n}\cap X_{b}$ form a basis for $H_{1}(X_{b},\mathbb{Z})$ for each $b\in U$ . Consider the $1$ -forms $\omega_{1},\ldots,\omega_{n}$ on $U$ defined by fibrewise integration:

\omega_{i}(\nu)=\int_{X_{b}\cap\gamma_{i}}\iota(\nu)\omega,

for $\nu$ a tangent vector on $B$ at $b$ , which we can lift to a normal vector field of $X_{b}$ . We have $\omega_{i}=f_{*}(\omega|_{\gamma_{i}})$ , and since $\omega$ is closed, so is $\omega_{i}$ . Thus there are locally defined functions $y_{1},\ldots,y_{n}$ on $U$ with $dy_{i}=\omega_{i}$ . Furthermore, these functions are well-defined up to the choice of basis of $H_{1}(X_{b},\mathbb{Z})$ and constants. Finally, they give well-defined coordinates, as follows from the fact that $\nu\mapsto\iota(\nu)\omega$ yields an isomorphism of $\mathcal{T}_{B,b}$ with $H^{1}(X_{b},\mathbb{R})$ by McLean’s theorem. Thus $y_{1},\ldots,y_{n}$ define local coordinates of a tropical affine structure on $B$ .

Affine structure 2. We can play the same trick with $\operatorname{Im}\Omega$ : choose submanifolds

\Gamma_{1},\ldots,\Gamma_{n}\subseteq f^{-1}(U)

which are families of $n-1$ -cycles over $U$ and such that $\Gamma_{1}\cap X_{b},\ldots,\Gamma_{n}\cap X_{b}$ form a basis for $H_{n-1}(X_{b},\mathbb{Z})$ . We define $\lambda_{i}$ by $\lambda_{i}=-f_{*}(\operatorname{Im}\Omega|_{\Gamma_{i}})$ , or equivalently,

\lambda_{i}(\nu)=-\int_{X_{b}\cap\Gamma_{i}}\iota(\nu)\operatorname{Im}\Omega.

Again $\lambda_{1},\ldots,\lambda_{n}$ are closed $1$ -forms, with $\lambda_{i}=d\check{y}_{i}$ locally, and again $\check{y}_{1},\ldots,\check{y}_{n}$ are affine coordinates for a tropical affine structure on $B$ .

The McLean metric. The Hodge metric on $H^{1}(X_{b},\mathbb{R})$ is given by

g(\alpha,\beta)=\int_{X_{b}}\alpha\wedge*\beta

for $\alpha$ , $\beta$ harmonic $1$ -forms, and hence induces a metric on $B$ , which can be written as

g(\nu_{1},\nu_{2})=-\int_{X_{b}}\iota(\nu_{1})\omega\wedge\iota(\nu_{2})\operatorname{Im}\Omega.

A crucial observation of Hitchin [41] is that these structures are related by the Legendre transform:

Proposition 1.2.

Let $y_{1},\ldots,y_{n}$ be local affine coordinates on $B$ with respect to the affine structure induced by $\omega$ . Then locally there is a function $K$ on $B$ such that

g(\partial/\partial y_{i},\partial/\partial y_{j})=\partial^{2}K/\partial y_{i}\partial y_{j}.

Furthermore, $\check{y}_{i}=\partial K/\partial y_{i}$ form a system of affine coordinates with respect to the affine structure induced by $\operatorname{Im}\Omega$ , and if

\check{K}(\check{y}_{1},\ldots,\check{y}_{n})=\sum\check{y}_{i}y_{i}-K(y_{1},\ldots,y_{n})

is the Legendre transform of $K$ , then

y_{i}=\partial\check{K}/\partial\check{y}_{i}

and

\partial^{2}\check{K}/\partial y_{i}\partial y_{j}=g(\partial/\partial\check{y}_{i},\partial/\partial\check{y}_{j}).

Proof.

Take families $\gamma_{1},\ldots,\gamma_{n},\Gamma_{1},\ldots,\Gamma_{n}$ as above over an open neighbourhood $U$ with the two bases being Poincaré dual, i.e., $(\gamma_{i}\cap X_{b})\cdot(\Gamma_{j}\cap X_{b})=\delta_{ij}$ for $b\in U$ . Let $\gamma_{1}^{*},\ldots,\gamma_{n}^{*}$ and $\Gamma_{1}^{*},\ldots,\Gamma_{n}^{*}$ be the dual bases for $\Gamma(U,R^{1}f_{*}\mathbb{Z})$ and $\Gamma(U,R^{n-1}f_{*}\mathbb{Z})$ respectively. From the choice of $\gamma_{i}$ ’s, we get local coordinates $y_{1},\ldots,y_{n}$ with $dy_{i}=\omega_{i}$ , so in particular

\delta_{ij}=\omega_{i}(\partial/\partial y_{j})=\int_{\gamma_{i}\cap X_{b}}\iota(\partial/\partial y_{j})\omega,

hence $\iota(\partial/\partial y_{j})\omega$ defines the cohomology class $\gamma_{j}^{*}$ in $H^{1}(X_{b},\mathbb{R})$ . Similarly, let

g_{ij}=-\int_{\Gamma_{i}\cap X_{b}}\iota(\partial/\partial y_{j})\operatorname{Im}\Omega;

then $-\iota(\partial/\partial y_{j})\operatorname{Im}\Omega$ defines the cohomology class $\sum_{i}g_{ij}\Gamma_{i}^{*}$ in $H^{n-1}(X_{b},\mathbb{R})$ , and $\lambda_{i}=\sum_{j}g_{ij}dy_{j}$ . Thus

	$\displaystyle g(\partial/\partial y_{j},\partial/\partial y_{k})$	$\displaystyle=$	$\displaystyle-\int_{X_{b}}\iota(\partial/\partial y_{j})\omega\wedge\iota(\partial/\partial y_{k})\operatorname{Im}\Omega$
		$\displaystyle=$	$\displaystyle g_{jk}.$

On the other hand, let $\check{y}_{1},\ldots,\check{y}_{n}$ be coordinates with $d\check{y}_{i}=\lambda_{i}$ . Then

{\partial\check{y}_{i}/\partial y_{j}}=g_{ij}=g_{ji}={\partial\check{y}_{j}/\partial y_{i}},

so $\sum\check{y}_{i}dy_{i}$ is a closed 1-form. Thus there exists locally a function $K$ such that $\partial K/\partial y_{i}=\check{y}_{i}$ and $\partial^{2}K/\partial y_{i}\partial y_{j}=g(\partial/\partial y_{i},\partial/\partial y_{j})$ . A simple calculation then confirms that $\partial\check{K}/\partial\check{y}_{i}=y_{i}$ . On the other hand,

$\displaystyle g(\partial/\partial\check{y}_{i},\partial/\partial\check{y}_{j})$	$\displaystyle=$	$\displaystyle g\left(\sum_{k}{\partial y_{k}\over\partial\check{y}_{i}}{\partial\over\partial y_{k}},\sum_{\ell}{\partial y_{\ell}\over\partial\check{y}_{j}}{\partial\over\partial y_{\ell}}\right)$
	$\displaystyle=$	$\displaystyle\sum_{k,\ell}{\partial y_{k}\over\partial\check{y}_{i}}{\partial y_{\ell}\over\partial\check{y}_{j}}g(\partial/\partial y_{k},\partial/\partial y_{\ell})$
	$\displaystyle=$	$\displaystyle\sum_{k,\ell}{\partial y_{k}\over\partial\check{y}_{i}}{\partial y_{\ell}\over\partial\check{y}_{j}}{\partial\check{y}_{k}\over\partial y_{\ell}}$
	$\displaystyle=$	$\displaystyle{\partial y_{j}\over\partial\check{y}_{i}}={\partial^{2}\check{K}\over\partial\check{y}_{i}\partial\check{y}_{j}}.$

∎

Thus we introduce the notion of the Legendre transform of an affine manifold with a multi-valued convex function.

Definition 1.3.

Let $B$ be an affine manifold. A multi-valued function $K$ on $B$ is a collection of functions on an open cover $\{(U_{i},K_{i})\}$ such that on $U_{i}\cap U_{j}$ , $K_{i}-K_{j}$ is affine linear. We say $K$ is convex if the Hessian $(\partial^{2}K_{i}/\partial y_{j}\partial y_{k})$ is positive definite for all $i$ , in any, or equivalently all, affine coordinate systems $y_{1},\ldots,y_{n}$ .

Given a pair $(B,K)$ of affine manifold and convex multi-valued function, the Legendre transform of $(B,K)$ is a pair $(\check{B},\check{K})$ where $\check{B}$ is an affine structure on the underlying manifold of $B$ with coordinates given locally by $\check{y}_{i}=\partial K/\partial y_{i}$ , and $\check{K}$ is defined by

\check{K}_{i}(\check{y}_{1},\ldots,\check{y}_{n})=\sum\check{y}_{j}y_{j}-K_{i}(y_{1},\ldots,y_{n}).

Exercise 1.4.

Check that $\check{K}$ is also convex, and that the Legendre transform of $(\check{B},\check{K})$ is $(B,K)$ .

Curiously, this Legendre transform between affine manifolds with Hessian metric seems to have first appeared in a work in statistics predating mirror symmetry, see [2].