4. Local mirror symmetry and Legendre transform [05CV]

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4. Local mirror symmetry and Legendre transform

4.1. Linear algebra of Legendre transform and Monge-Ampére equations

The classical fact, implicitly used in [GW00], is that solving the two-dimensional Laplace equation is equivalent to solving the real two-dimensional Monge-Ampère equation. This can be easily generalized to higher dimensions. Namely, the chain rule and elementary linear algebra for a particular coordinate transformation yields the following.

Lemma 4.1.

If $K(s,t)$ is a local solution to the split Monge-Ampère equation (17), then $\Psi(y)$ is a (local) solution of the classical (real) Monge-Ampère equation

(19)

\det\frac{\partial^{2}\Psi}{\partial y_{i}\partial y_{p}}=1,\quad 1\leq i,j\leq n,

where

(20)

y_{i}=\frac{\partial K}{\partial s_{i}},\ 1\leq i\leq k,\qquad y_{p}=t_{p},\ k+1\leq p\leq n,

and $\Psi(y)$ is the partial Legendre transform of $K(s,t)$ , defined by

(21)

\frac{\partial\Psi}{\partial y_{i}}=s_{i},\ 1\leq i\leq k,\qquad\frac{\partial\Psi}{\partial y_{p}}=-\frac{\partial K}{\partial t_{p}},\ k+1\leq p\leq n.

Proof.

First, we check the very existence of the partial Legendre transform. Consider the following Jacobian and Hessian matrices:

(22)

\frac{\partial(y_{i},y_{p})}{\partial(s,t)}=\left(\begin{array}[]{cc}\frac{\partial^{2}K}{\partial s_{j}\partial s_{i}}&\frac{\partial^{2}K}{\partial t_{q}\partial s_{i}}\\ 0&\mathbbm{1}\end{array}\right)=\left(\begin{array}[]{cc}V&B\\ 0&\mathbbm{1}\end{array}\right),\quad\operatorname{Hess}K(s,t)=\left(\begin{array}[]{cc}V&B\\ {}^{t}B&-W\end{array}\right).

Then

(23)

\frac{\partial(s,t)}{\partial(y_{i},y_{p})}=\left(\begin{array}[]{cc}V^{-1}&-V^{-1}B\\ 0&\mathbbm{1}\end{array}\right),\quad\operatorname{Hess}\Psi(y)=\left(\begin{array}[]{cc}V^{-1}&-V^{-1}B\\ {}^{t}(-V^{-1}B)&W+{{}^{t}B}V^{-1}B\end{array}\right).

This shows that if both $V$ and $W$ are symmetric and positive definite, then $\operatorname{Hess}\Psi$ is also a symmetric positive definite matrix, so there exists (locally) a convex function $\Psi$ – the partial Legendre transform.

On the other hand the inverse of a non-degenerate $2\times 2$ block matrix with $\det A\neq 0$ and $\det D\neq 0$ is:

(24)

\left(\begin{array}[]{cc}A&B\\ C&D\end{array}\right)^{-1}=\left(\begin{array}[]{cc}(A-BD^{-1}C)^{-1}&(-A+BD^{-1}C)^{-1}BD^{-1}\\ (-D+CA^{-1}B)^{-1}CA^{-1}&(D-CA^{-1}B)^{-1}\end{array}\right),

which implies that

(25)

\det\left(\begin{array}[]{cc}A&B\\ C&D\end{array}\right)=1\quad\Longleftrightarrow\quad\det A^{-1}=\det(D-CA^{-1}B).

Applied to $\operatorname{Hess}\Psi$ the last observation shows that $\Psi$ is a local Monge-Ampère solution if and only if $\det V=\det W$ . ∎

4.2. Monge-Ampère manifolds

Definition.

A Riemannian manifold $(Y,g)$ is called Monge-Ampère if it possesses an integral affine structure and the metric is, locally in affine coordinates, given by a potential $\Psi$ , that is, $g_{ij}=\frac{\partial^{2}\Psi}{\partial y_{i}\partial y_{j}}$ , that satisfies the real Monge-Ampère equation $\det g_{ij}=1$ .

Cheng and Yau [CY82] proved that every compact Monge-Ampère manifold is diffeomorphic to a torus and the Monge-Ampère structure is a deformation of the standard flat structure on $\mathbb{R}^{n}/\mathbb{Z}^{n}$ . To enrich this fairly boring class of manifolds we will allow certain singularities along a codimension 2 discriminant locus.

The Monge-Ampère condition in the affine geometry is an analog of the Ricci-flatness condition on a Kähler manifold with similar global rigidity properties. Roughly speaking, one should expect some sort of the Calabi conjecture saying that each topological class of the metric has a unique Monge-Ampère representative (cf. [HZ03] and [KT02]).

Here we are concerned with the local picture in which many Monge-Ampère structures may exist. However, for applications to the metric collapse of toric Calabi-Yau hypersurfaces we will be interested only in a certain special class of the Monge-Ampère structures, called bi-PIKAS in [HZ03]. We will show that the latter always arise from singular split Monge-Ampère solutions of the corresponding type.

The base $(\mathbb{R}^{n}\times\mathbb{R}^{l})\setminus(\Pi(\tau)\times\Pi(\sigma))$ has an open covering $\{U_{v_{i}},U_{w_{j}}\}$ where $U_{v_{i}}:=\mathbb{R}^{n}\times Q^{\sigma}_{i}$ and $U_{w_{j}}:=Q^{\tau}_{j}\times\mathbb{R}^{l}$ . The nerve of this covering is the complete bipartite graph on the vertices $v_{i}$ of $\sigma$ and $w_{j}$ of $\tau$ . We assume that the affine structure on $(\mathbb{R}^{n}\times\mathbb{R}^{l})\setminus(\Pi(\tau)\times\Pi(\sigma))$ is polyhedral of type $(\{U_{v}\},\partial\Sigma^{\vee})$ . That is, there is a homeomorphism $\mathbb{R}^{n}\times\mathbb{R}^{l}\to\partial\Sigma^{\vee}$ which provides affine coordinates on every chart $U_{v_{i}}$ by identifying it with the maximal dimensional face of $\partial\Sigma^{\vee}$ orthogonal to $v_{i}$ . Also we assume that the dual affine structure is polyhedral of type $(\{U_{w}\},\partial\mathcal{T}^{\vee})$ . Moreover, the transformation maps between the affine coordinates on $U_{v_{i}}$ and $U_{w_{j}}$ are assumed to be the natural projections from the subspaces $v_{i}^{\perp}:=\{y\in N_{\mathbb{R}}\ :\ \langle v_{i},y\rangle=0\}$ to the quotients $N_{\mathbb{R}}/w_{j}$ . This data constitutes a bi-polyhedral integral Kähler affine structure (bi-PIKAS for short) on $(\mathbb{R}^{n}\times\mathbb{R}^{l})\setminus(\Pi(\tau)\times\Pi(\sigma))$ of type $(\partial\Sigma^{\vee},\partial\mathcal{T}^{\vee})$ (cf. [HZ03]). We will abbreviate the type to be $(\sigma,\tau)$ .

It is straight forward to see that the monodromy of the (linear part of the) affine structure along a loop $(v_{i_{1}}w_{j_{1}}v_{i_{2}}w_{j_{2}})$ is given by $y\mapsto y+\langle v_{i_{2}}-v_{i_{1}},y\rangle(w_{j_{2}}-w_{j_{1}})$ (cf. [HZ02] for details). Since the radiance obstruction class vanishes locally (cf. [GS02]) we can always choose the affine coordinates such that there is no translational part of the monodromy.

Given a convex domain $R$ in $\mathbb{R}^{n}\times\mathbb{R}^{l}$ which contains the origin, a Monge-Ampère bi-PIKAS on $R$ of type $(\sigma,\tau)$ will be a restriction of a Monge-Ampère bi-PIKAS of the same type on $\mathbb{R}^{n}\times\mathbb{R}^{l}$ .

Theorem 4.2.

There is a bijection between the sets of $(\sigma,\tau)$ -type split Monge-Ampère solutions on a domain $R$ and $(\sigma,\tau)$ -type Monge-Ampère bi-PIKAS on $R$ . The Riemannian metric on $R^{\circ}:=R\setminus\Pi(sigma)\times\Pi(\tau)$ is given by $V^{ij}ds_{i}ds_{j}+W^{pq}dt_{p}dt_{q}$ , where $(V^{ij},W^{pq})$ is the corresponding split MA solution.

Proof.

Let $K_{\alpha}$ be local potentials for a given split Monge-Ampère solution $(V,W)$ . We will use the partial Legendre transform to define new coordinates $y_{i},y_{p}$ as in Lemma 4.1. We claim that these are affine coordinates, that is the transition maps are affine linear.

Indeed, by comparing the two local potentials in the overlap $U_{\alpha}\cap U_{\beta}$ we see that $K_{\alpha}-K_{\beta}$ has to be in the form $f(s)g(t)$ where both $f(s)$ and $g(t)$ are affine functions. Thus $y_{\alpha}-y_{\beta}$ are affine functions of $t_{p}$ , hence affine functions of $y_{p}$ .

Lemma 4.1 also guarantees that $\Psi_{\alpha}$ are local Monge-Ampère potentials in the affine coordinates $y$ . The polyhedral property follows from noticing that the charts $U_{v}$ are bounded by linear inequalities in $t_{p}$ , and hence by (the same) linear inequalities in the new (affine) coordinates $y_{p}$ .

Applying the partial Legendre transform to the other half of the variables $(s,t)$ gives the dual MA structure on $R^{\circ}$ . Same considerations as above show that the dual affine structure is polyhedral of type $(\{U_{w}\},\partial\mathcal{T}^{\vee})$ .

The final step is to show that the resulting Monge-Ampère bi-PIKAS has the right type, that is, to compute its monodromy around a loop $(v_{i_{1}}w_{j_{1}}v_{i_{2}}w_{j_{2}})$ . For this we consider the closed $(N^{*}_{\tau})_{\mathbb{R}}\otimes(N_{\sigma})_{\mathbb{R}}$ -valued 1-form on $R^{\circ}$ :

\beta^{iq}=\frac{\partial W^{pq}}{\partial s_{i}}dt_{p}-\frac{\partial^{2}V^{ij}}{\partial t_{q}}ds_{j}.

Making use of the distributional equation (18) and applying Stokes’ theorem we can compute the integral of $\beta$ along the loop $(v_{i_{1}}w_{j_{1}}v_{i_{2}}w_{j_{2}})=:\partial S$ :

\oint\beta=\iint_{S}d\beta=2\pi\iint_{S}\gamma_{\sigma}\gamma_{\tau}=2\pi(v_{i_{2}}-v_{i_{1}})\otimes(w_{j_{2}}-w_{j_{1}}).

Thus, $\beta$ has holonomy which depends only on the homotopy class of the path. That is, we can think of $\beta$ as given by the differential of a multi-valued function

f^{iq}=\frac{\partial^{2}K}{\partial s_{i}\partial t_{q}}.

The affine coordinates $y_{p},\ n+1\leq p\leq n+l,$ are single valued. Hence, there is no monodromy in $dy_{p}$ . On the other hand, the ambiguity in the remaining differentials

dy_{i}=\frac{\partial^{2}K}{\partial s_{i}\partial t_{q}}dt_{q}+\frac{\partial^{2}K}{\partial s_{i}\partial s_{j}}ds_{j}

is coming exactly from the multi-valuedness of $f^{iq}$ . Thus, the monodromy around the loop $(v_{i_{1}}w_{j_{1}}v_{i_{2}}w_{j_{2}})$ is given by $\mathbbm{1}+2\pi(v_{i_{2}}-v_{i_{1}})\otimes(w_{j_{2}}-w_{j_{1}})$ . The factor of $2\pi$ can be absorbed into a redefinition of the affine coordinates.

Tracing backwards through the above argument shows the converse statement is also true. Namely, given a Monge-Ampère bi-PIKAS of type $(\sigma,\tau)$ we can use it together with its dual to define single-valued coordinates $(s,t)$ on $R^{\circ}$ which will obviously extend to $R$ . Moreover, due to the polyhedral properties of these bi-PIKAS the discriminant locus in $R$ will be exactly given by $\partial\mathcal{U}_{v}\cap\partial\mathcal{U}_{w}=\Pi(\sigma)\times\Pi(\tau)$ , and the prescribed monodromy of the affine structure will guarantee the $(\sigma,\tau)$ -type asymptotics of the Monge-Ampère potential $K(s,t)$ . ∎

Remark.

The discriminant locus $D=\partial\mathcal{U}_{v}\cap\partial\mathcal{U}_{w}$ does not have to be affine linear. Even though $D$ lies in the polyhedral boundary $\partial\mathcal{U}_{v}$ , the boundary $\partial\mathcal{U}_{w}$ will be wiggled in the affine coordinates unless the partial Legendre transform is linear.

We would like to finish by mentioning an obvious application of the construction in this section to mirror symmetry. As was noted in [Hit97] the mirror duality in the semi-flat case is provided by the Legendre transform. This statement continues to hold in the neighborhood of the discriminant locus as well. Namely, in the single-valued coordinates (which are not affine) the full Legendre transform takes a singular split Monge-Ampère solution of type $(\sigma,\tau)$ into that of type $(\tau,\sigma)$ .