The moduli space of special Lagrangian submanifolds

Question

Given a special Lagrangian fibration $f:M \rightarrow B$ of a Calabi-Yau manifold $M$, one can associate to it two affine structures (symplectic and complex) on the base space $B$. A theorem of Hitchin says that there locally exists a convex function $K$ such that the coordinate system in one affine structure is mapped to the coordinate system in the other affine structure via the Legendre transform associated to $K$. Is it possible to see this theorem explicitly for elliptic curves and K3 surfaces? What is K for elliptic curves?

Answer 1

This is the classical Legendre transform, the two affine structures are symplectic affine structure and complex affine structure. The best example to see this is when the CY manifold $X=\mathbb{CP}^2\setminus D$, where $D\cong-K_X$ is the toric boundary divisor. In this case, the symplectic affine structure is the interior of an triangle, and the complex affine structure is isomorphic to $\mathbb{R}^2$. For elliptic curves you can write it down using coordinates explicitly. But for elliptic $K3$ surface, since generically the special Lagrangian fibration $f$ will have 24 singular fibers, $B\cong S^2$ is not an affine manifold, but instead a singular affine manifold. On $B-\Delta$ where $\Delta$ consists of 24 points, there is an affine structure, but the affine structure does not extend to $B$, so the Legendre transform can only be carried out on $B-\Delta$.

Answer 2

Both the elliptic curve and K3 case can be calculated explicitly, the first more than the second.

First let's recall how the structures arise. We have the 2-form $\omega$ and the $n$-form $\Im \Omega$. Let $\gamma_1,\ldots,\gamma_n$ be a basis for integral 1-cycles in the fibres of the fibration and $\Gamma_1,\ldots,\Gamma_n$ a basis for the $n-1$-cycles in the fibres. Then we can define 1-forms $\omega_1,\ldots, \omega_n$ by defining their value on a tangent vector $\nu$ on the base as $$ \omega_i(\nu):=\int_{\gamma_i} \iota(\nu)\omega, $$ where $\iota(\nu)$ denotes contraction by a lift of a tangent vector $\nu$ to a normal vector field of the corresponding slag fibre. Similarly, define 1-forms $\lambda_i$ by $$\lambda_i(\nu):=-\int_{\Gamma_i} \iota(\nu)\Im\Omega.$$ Now Hitchin points out that these forms are closed, hence locally of the form $\omega_i=dy_i$ and $\lambda_i=d\check y_i$, with $y_1,\ldots,y_n$ coordinates defining the symplectic affine structure and $\check y_1,\ldots,\check y_n$ defining the affine structure coming from the complex structure. Finally, the metric, which should be defined by a potential $K$ in either two affine structures, is defined by $$ g(\nu_1,\nu_2)=-\int \iota(\nu_1)\omega\wedge \iota(\nu_2)\Im\Omega. $$ Here the integral is over the whole fibre.

Let's do this first for an elliptic curve, say take $E={\mathbb C}/\langle 1, i\rangle$, with complex coordinate $z=x+iy$, $\Omega=dz=dx+idy$, $\omega=dx\wedge dy$. Take the fibration to be $z\mapsto y$: this is a map $E\rightarrow {\mathbb R}/{\mathbb Z}$. Taking $\nu=\partial/\partial y$, one finds easily that $\omega_1=-dy$, $\Omega_1=-dy$, so $y$ is an affine coordinate in both affine structures. Furthermore, $g$ is the constant metric with $g(\partial/\partial y, \partial/\partial y)=1$. Thus we can take $K=y^2/2$. This is viewed as a multi-valued function on the base ${\mathbb R/\mathbb Z}$. It is a good exercise to check to see what happens if we vary the complex or symplectic structures.

In the K3 case, one can't write down such an explicit global answer. However, using the hyperkaehler rotation trick, one knows that writing down a special Lagrangian fibration is the same as writing down an elliptic fibration with respect to the complex structure induced by the two-form $\Omega_K=\Im\Omega +i\omega$. Thus one can write down an elliptic fibration of your choice by specifying the holomorphic periods, and then doing the above calculation explicitly, describing affine coordinates in terms of the holomorphic periods. I won't do this here, but leave this as an exercise. For example, choose an open set $U\subseteq {\bf C}$ with holomorphic coordinate $z=y_1+iy_2$ as the base, and write down an elliptic fibration by dividing $U\times {\mathbb C}$ out by a varying family of lattices generated by $1$ and $\tau(z)$. Write coordinates on the ${\mathbb C}$ factor as $w=x_1+ix_2$. We can then choose $\Omega_K=dz\wedge dw$, and go through a similar calculation as above. One finds that the affine coordinates and the metric can be written down in terms of $\tau$. This in fact describes the general local behaviour for what happens in the K3 case. Of course, it is impossible to give an explicit global description.