2. Semi-flat mirror symmetry [02Z0]

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2. Semi-flat mirror symmetry

Let’s forget about special Lagrangian fibrations for the moment. Instead, we will look at how the structures found on $B$ in the previous section give a toy version of mirror symmetry.

Definition 2.1.

Let $B$ be a tropical affine manifold.

(1)

Denote by $\Lambda\subseteq\mathcal{T}_{B}$ the local system of lattices generated locally by $\partial/\partial y_{1},\ldots,\partial/\partial y_{n}$ , where $y_{1},\ldots,y_{n}$ are local affine coordinates. This is well-defined because transition maps are in $\mathbb{R}^{n}\rtimes GL_{n}(\mathbb{Z})$ . Set

$X(B):=\mathcal{T}_{B}/\Lambda.$

This is a torus bundle over $B$ . In addition, $X(B)$ carries a complex structure defined locally as follows. Let $U\subseteq B$ be an open set with affine coordinates $y_{1},\ldots,y_{n}$ , so $\mathcal{T}_{U}$ has coordinate functions $y_{1},\ldots,y_{n}$ , $x_{1}=dy_{1},\ldots,x_{n}=dy_{n}$ . Then

$q_{j}=e^{2\pi i(x_{j}+iy_{j})}$

gives a system of holomorphic coordinates on $T_{U}/\Lambda|_{U}$ , and the induced complex structure is independent of the choice of affine coordinates. This is called the semi-flat complex structure on $X(B)$ .

Later we will need a variant of this: for $\epsilon>0$ , set

$X_{\epsilon}(B):=\mathcal{T}_{B}/\epsilon\Lambda.$

This has a complex structure with coordinates given by

$q_{j}=e^{2\pi i(x_{j}+iy_{j})/\epsilon}.$

(As we shall see later, the limit $\epsilon\rightarrow 0$ corresponds to a “large complex structure limit.”)
(2)

Define $\check{\Lambda}\subseteq\mathcal{T}^{*}_{B}$ to be the local system of lattices generated locally by $dy_{1},\ldots,dy_{n}$ , with $y_{1},\ldots,y_{n}$ local affine coordinates. Set

$\check{X}(B):=\mathcal{T}^{*}_{B}/\check{\Lambda}.$

Of course $\mathcal{T}^{*}_{B}$ carries a canonical symplectic structure, and this symplectic structure descends to $\check{X}(B)$ .

∎

We write $f:X(B)\rightarrow B$ and $\check{f}:\check{X}(B)\rightarrow B$ for these torus fibrations; these are clearly dual.

Now suppose in addition we have a Hessian metric $g$ on $B$ , with local potential function $K$ . Then the following propositions show that in fact both $X(B)$ and $\check{X}(B)$ become Kähler manifolds.

Proposition 2.2.

$K\circ f$ is a (local) Kähler potential on $X(B)$ , defining a Kähler form $\omega=2i\partial\bar{\partial}(K\circ f)$ . This metric is Ricci-flat if and only if $K$ satisfies the real Monge-Ampère equation

\det{\partial^{2}K\over\partial y_{i}\partial y_{j}}=constant.

Proof.

Working locally with affine coordinates $(y_{j})$ and complex coordinates

z_{j}={1\over 2\pi i}\log q_{j}=x_{j}+iy_{j},

we compute $\omega=2i\partial\bar{\partial}(K\circ f)={i\over 2}\sum{\partial^{2}K\over\partial y_{j}\partial y_{k}}dz_{j}\wedge d\bar{z}_{k}$ which is clearly positive. Furthermore, if $\Omega=dz_{1}\wedge\cdots\wedge dz_{n}$ , then $\omega^{n}$ is proportional to $\Omega\wedge\bar{\Omega}$ if and only if $\det(\partial^{2}K/\partial y_{j}\partial y_{k})$ is constant. ∎

We write this Kähler manifold as $X(B,K)$ .

Dually we have

Proposition 2.3.

In local canonical coordinates $y_{i},\check{x}_{i}$ on $\mathcal{T}^{*}_{B}$ , the complex coordinate functions $z_{j}=\check{x}_{j}+i\partial K/\partial y_{j}$ on $\mathcal{T}^{*}_{B}$ induce a well-defined complex structure on $\check{X}(B)$ , with respect to which the canonical symplectic form $\omega$ is the Kähler form of a metric. Furthermore this metric is Ricci-flat if and only if $K$ satisfies the real Monge-Ampère equation

\det{\partial^{2}K\over\partial y_{j}\partial y_{k}}=constant.

Proof.

It is easy to see that an affine linear change in the coordinates $y_{j}$ (and hence an appropriate change in the coordinates $\check{x}_{j}$ ) results in a linear change of the coordinates $z_{j}$ , so they induce a well-defined complex structure invariant under $\check{x}_{j}\mapsto\check{x}_{j}+1$ , and hence a complex structure on $\check{X}(B)$ . Then one computes that

\omega=\sum d\check{x}_{j}\wedge dy_{j}={i\over 2}\sum g^{jk}dz_{j}\wedge d\bar{z}_{k}

where $g_{ij}=\partial^{2}K/\partial y_{j}\partial y_{k}$ . Then the metric is Ricci-flat if and only if $\det(g^{jk})=constant$ , if and only if $\det(g_{jk})=constant$ . ∎

As before, we call this Kähler manifold $\check{X}(B,K)$ .

This motivates the definition

Definition 2.4.

An affine manifold with metric of Hessian form is a Monge-Ampère manifold if the local potential function $K$ satisfies the Monge-Ampère equation $\det(\partial^{2}K/\partial y_{i}\partial y_{j})=constant$ .

Monge-Ampère manifolds were first studied by Cheng and Yau in [12].

Exercise 2.5.

Show that the identification of $\mathcal{T}_{B}$ and $\mathcal{T}^{*}_{B}$ given by a Hessian metric induces a canonical isomorphism $X(B,K)\cong\check{X}(\check{B},\check{K})$ of Kähler manifolds, where $(\check{B},\check{K})$ is the Legendre transform of $(B,K)$ .

There is a key extra parameter which appears in mirror symmetry known as the $B$ -field. This appears as a field in the non-linear sigma model with Calabi-Yau target space, and is required mathematically to make sense of mirror symmetry. Mirror symmetry roughly posits an isomorphism between the complex moduli space of a Calabi-Yau manifold $X$ and the Kähler moduli space of $\check{X}$ . If one interprets the Kähler moduli space to mean the space of all Ricci-flat Kähler forms on $\check{X}$ , then one obtains only a real manifold as moduli space, and one needs a complex manifold to match up with the complex moduli space of $X$ . The $B$ -field is interpreted as an element ${\bf B}\in H^{2}(\check{X},\mathbb{R}/\mathbb{Z})$ , and one views ${\bf B}+i\omega$ as a complexified Kähler class on $\check{X}$ for $\omega$ a Kähler class on $\check{X}$ .

In the context of our toy version of mirror symmetry, we view the $B$ -field as an element ${\bf B}\in H^{1}(B,\Lambda_{\mathbb{R}}/\Lambda)$ , where $\Lambda_{\mathbb{R}}=\Lambda\otimes_{\mathbb{Z}}\mathbb{R}$ . This does not quite agree with the above definition of the $B$ -field, as this group does not necessarily coincide with $H^{2}(\check{X},\mathbb{R}/\mathbb{Z})$ . However, in many important cases, such as for simply connected Calabi-Yau threefolds with torsion-free integral cohomology, these two groups do coincide. More generally, including the case of K3 surfaces and abelian varieties, one would need to pass to generalized complex structures [42], [38], [7], [3], [43], which we do not wish to do here.

Noting that a section of $\Lambda_{\mathbb{R}}/\Lambda$ over an open set $U$ can be viewed as a section of $\mathcal{T}_{U}/\Lambda|_{U}$ , such a section acts on $\mathcal{T}_{U}/\Lambda|_{U}$ via translation, and this action is in fact holomorphic with respect to the semi-flat complex structure. Thus a Čech 1-cocycle $(U_{ij},\beta_{ij})$ representing ${\bf B}$ allows us to reglue $X(B)$ via translations over the intersections $U_{ij}$ . This is done by identifying the open subsets $f^{-1}(U_{ij})\subseteq f^{-1}(U_{i})$ and $f^{-1}(U_{ij})\subseteq f^{-1}(U_{j})$ via the automorphism of $f^{-1}(U_{ij})$ given by translation by the section $\beta_{ij}$ . This gives a new complex manifold $X(B,{\bf B})$ . If in addition there is a multi-valued potential function $K$ defining a metric, these translations preserve the metric and yield a Kähler manifold $X(B,{\bf B},K)$ .

Thus the full toy version of mirror symmetry is as follows:

Construction 2.6 (The toy mirror symmetry construction).

Suppose given an affine manifold $B$ with potential $K$ and $B$ -fields ${\bf B}\in H^{1}(B,\Lambda_{\mathbb{R}}/\Lambda)$ , $\check{\bf B}\in H^{1}(B,\check{\Lambda}_{\mathbb{R}}/\check{\Lambda})$ . It is not difficult to see, and you will have seen this already if you’ve done Exercise 2.5, that the local system $\check{\Lambda}$ defined using the affine structure on $B$ is the same as the local system $\Lambda$ defined using the affine stucture on $\check{B}$ . So we say the pair

(X(B,{\bf B},K),\check{\bf B})

is mirror to

(X(\check{B},\check{\bf B},\check{K}),\bf B).

This provides a reasonably fulfilling picture of mirror symmetry in a simple context. Many more aspects of mirror symmetry can be worked out in this semi-flat context, see [54] and [3], Chapter 6. This semi-flat case is an ideal testing ground for concepts in mirror symmetry. However, ultimately this only sheds limited insight into the general case. The only compact Calabi-Yau manifolds with semi-flat Ricci-flat metric which arise in this way are complex tori (shown by Cheng and Yau in [12]). To deal with more interesting cases, we need to allow singular fibres, and hence, singularities in the affine structure of $B$ . The existence of singular fibres are fundamental for the most interesting aspects of mirror symmetry.