Superpotentials of D-branes in Calabi-Yau Manifolds with Several Moduli by Mirror Symmetry and Blown-up

A Calabi-Yau manifold can be defined as a hypersurface or complete intersection of several hyperserfaces in ambient toric variety. We refer to [7] for the background of toric geometry,construction of Calabi-Yau manifolds \parencitesBatyrev1993Batyrev1994, and GKZ system \parencitesHosono1995Hosono1993Hosono1994Stienstra2005Cox1999.

For a mirror pair of compact hypersurfaces $(X^{*},X)$, one may associate a pair of integral polyhedra $(\Delta^{*},\Delta)$ in a four-dimensional integral lattice and its dual. The n integral points of the polyhedron correspond to homogeneous coordinates $x_{i}$ on the toric ambient space and satisfy linear relations

$$\sum_{i}l^{j}_{i}v_{i}=0,\quad a=1,...,h^{2,1}$$

where $l^{j}_{i}$ is the ith component of the charge vector $l^{j}$. The integral entries of the vectors $l^{j}$ define the weights $l^{j}_{i}$ of the coordinates $x_{i}$ under the $\mathbb{C}^{*}$ action

$$x_{i}\rightarrow(\lambda_{j})^{l^{j}_{i}}x_{i},\quad\lambda_{j}\in\mathbb{C}^{*}$$

and $l^{j}_{i}$ ’s are the $U(1)$ charges of the fields in the gauged linear sigma model (GLSM)[25]. In above description, the mirror Calabi-Yau threefold is determined as a hypersurface in the dual toric ambient space with constraints

$$P=\sum_{i}a_{i}y_{i},\quad\prod_{i}y_{i}^{l^{j}_{i}}=z_{j}$$

(2.1)

Here $z_{j}$ denotes the complex structure moduli of $X$. In terms of vertices $v_{j}^{*}\in\Delta^{*},v_{i}\in\Delta$

$$P(x;a)=\sum_{v_{j}^{*}\in\Delta^{*}}a_{j}\prod_{i}x_{i}^{\langle v_{j}^{*},v_{i}\rangle+1}$$

(2.2)

The complete intersection Calabi-Yau threefolds can be constructed similarly and we omit the details. Equivalently, Calabi-Yau hypersurfaces is also given by the zero loci of certain sections of the anticanonical bundle. The toric variety contains a canonical Zariski open torus $\mathbb{C}^{4}$ with coordinates $X=(X_{1},X_{2},X_{3},X_{4})$. The sections are

$$P_{\Delta^{*}}(X,a)=\sum_{v^{*}_{i}\in\Delta^{*}\cap\mathbb{Z}^{5}}a_{i}X^{v_{i}^{*}}$$

(2.3)

After homogenization, above equation is the same as 2.2. On these manifolds, a mirror pairs of branes, defined in [1] by another N charge vectors $\hat{l}^{j}$. The special Lagrangian submanifold wrapped by the A-brane is described in terms of the vectors $\hat{l}^{j}$ satisfying

$$\sum_{i}\hat{l}^{j}_{i}|x_{i}|^{2}=c_{j},\quad\sum_{i}\hat{l}^{j}_{i}=0$$

where $c_{j}$ parametrize the brane position. And the holomorphic submanifold wrapped by mirror B-brane is defined by the following equation

$$\prod_{i}y_{i}-\hat{z}_{a}=0,\quad\hat{z}_{a}=\epsilon_{a}e^{-c_{a}}$$

The $N=2$ case, a toric curve, is more interesting to us and it is the geometry setting we are studying in this notes. To handle the toric curve case, we consider the enhanced polyhedron method proposed in [2]. It is possible in a simple manner to construct the enhanced polyhedron from the original polyhedra and the toric curve specified by two charge vectors. We denote the vertices of $\Delta$ by $v_{i}$, $i=1,...,n$, with $v_{0}$ the origin, its charge vectors by $l^{i}$, and two brane vectors by $\hat{l}^{1}$ and $\hat{l}^{2}$. We add 4 points to $\Delta^{*}$ to define a new polyhedron $\Delta^{e}$ with vertices

		$\displaystyle X:v_{i}=(1,0,v_{i},0),\quad i=0,...,n$		(2.4)
		$\displaystyle\hat{l}^{1}:v_{n+1}=(0,1,v_{1}^{-},-1),\quad v_{n+2}=(0,1,v_{1}^{+},-1)$
		$\displaystyle\hat{l}^{2}:v_{n+3}=(0,1,v_{2}^{-},-1),\quad v_{n+4}=(0,1,v_{2}^{+},-1)$

where we use the abbreviation

$$\begin{gathered}v_{1}^{+}=\sum_{\hat{l}_{i}^{1}>0}\hat{l}_{i}^{1}v_{i},\quad v_{1}^{-}=-\sum_{\hat{l}_{i}^{1}<0}\hat{l}_{i}^{1}v_{i}\\ v_{2}^{+}=\sum_{\hat{l}_{i}^{2}>0}\hat{l}_{i}^{2}\ v_{i},\quad v_{2}^{-}=-\sum_{\hat{l}_{i}^{2}<0}\hat{l}_{i}^{2}v_{i}\end{gathered}$$

(2.5)

The first line of 2.4 simply embeds the original toric data associated to $\Delta$ into $\Delta^{e}$, whereas the second and third line translate the brane data into geometric data of $\Delta^{e}$.

Given the toric data , the GKZ-system on the complex structure moduli space of $X$ is given by the standard formula

$$\begin{gathered}\mathcal{L}_{i}=\prod_{l^{i}_{j}>0}(\frac{\partial}{\partial a_{j}})^{l^{i}_{j}}-\prod_{l^{i}_{j}<0}(\frac{\partial}{\partial a_{j}})^{-l^{i}_{j}},\quad i=1,...,h^{2,1}+2\\ \mathcal{Z}_{i}=\sum_{j}(\bar{v}_{j})^{i}\vartheta_{j}-\beta_{i},\quad j=0,...,6\end{gathered}$$

(2.6)

Here $\beta=(-1,0,0,0,0)$ is the so-called exponent of GKZ system, $\vartheta_{j}=a_{j}\frac{\partial}{\partial a_{j}}$ are the logarithmic derivative and $\bar{v}_{j}=(1,v_{j})$. The operators $\mathcal{L}_{i}$’s express the trivial algebraic relations among monomials entering hypersurface constraints, $\mathcal{Z}_{0}$ expresses the infinitesimal generators of overall rescaling,$\mathcal{Z}_{i},i\neq 0$’s eare the infinitesimal generators of rescalings of coordinates $x_{j}$. All GKZ operators can annihilate the period matrix, thus determine the mirror maps and superpotentials.

This immediately yields a natural choice of complex coordinates given by

$$z^{j}=(-)^{l^{j}_{0}}\prod_{i}a_{i}^{l^{j}_{i}}$$

(2.7)

And from the operators $\mathcal{L}_{i}$ ,it is easy to obtain a complete set of Picard-Fuchs operators $\mathcal{D}_{i}$. Using monodromy information and knowledge of the classical terms, their solution can be associated to integrals over an integral basis of cycles in $H^{3}(X,\mathbb{Z})$ and given the flux quanta explicit superpotentials can be written down.

For appropriate choice of basis vector $l^{j}$, solutions to the GKZ system can be written interm of the generating functions in these variables

$$\varpi(z;\rho)=\sum\frac{\Gamma(1-\sum_{j}l^{j}_{0}(n_{j}+\rho_{j}))}{\prod_{i>0}\Gamma(1+\sum_{j}l^{j}_{i}(n_{j}+\rho_{j}))}\prod_{k}z_{k}^{n_{j}+\rho_{j}}$$

then we have a natural basis for the period vector

	$\displaystyle\omega_{0}(z)$	$\displaystyle=\varpi(z;\rho)|_{\rho\rightarrow 0},$
	$\displaystyle\omega_{1,i}(z)$	$\displaystyle=\partial_{\rho_{i}}\varpi(z;\rho)|_{\rho\rightarrow 0},$
	$\displaystyle\omega_{2,i}(z)$	$\displaystyle=\sum_{j,k}K_{ijk}\partial_{\rho_{j}}\partial_{\rho_{k}}\varpi(z;\rho)|_{\rho\rightarrow 0}$
	$\displaystyle...$

For a maximal triangulation corresponding to a large complex structure point centered at $z=0$,$\forall a$, $\omega_{0}(z)=1+\mathcal{O}(z)$ and $\omega_{1,i}(z)\sim\log(z_{i})$ that define the open-closed mirror maps

$$t_{i}(z)=\frac{\omega_{1,i}(z)}{\omega_{0}(z)}=\frac{1}{2\pi i}\log(z_{i})+S(z),\quad q_{i}=e^{2\pi it_{i}}$$

(2.8)

where $S(z)$ is a series in the coordinates $z$.In addition, the special solution $\Pi=\mathcal{W}_{open}(z)$ has further property that its instanton expansion near a large volume/large complex structure point encodes the Ooguri-Vafa invariants of the brane geometry.

$$\mathcal{W}_{inst}(q)=\sum_{\beta}G_{\beta}q^{\beta}=\sum_{\beta}\sum_{k=1}^{\infty}N_{\beta}\frac{q^{k\cdot\beta}}{k^{2}}$$

(2.9)

Blowing up in algebraic geometry is an important tool in this work. Now, we review the construction and properties of blowing up a manifold along its submanifold. Given $S$ be a curve in a Calabi-Yau threefold $X\in Z$ with $Z$ an ambient toric variety, we can blow up along $S$ to obtain a new manifold.

According to Section 2.2 (d) in [22], for this case that $X\subset Z$ is a closed irreducible non-singular subvariety of $Z$ and $X$ is transversal to $S$ at every point $S\cap X$, $\pi:Z^{\prime}\rightarrow Z$ be the blowup of $S$. Then the subvariety $\pi^{-1}(X)$ consist of two irreducible components,

$$\pi^{-1}(X)=\pi^{-1}(S\cap X)\cup X^{\prime}$$

and $\pi:X^{\prime}\rightarrow X$ defines the blow-up of $X$ with center in $S\cap X=S$,i.e.$X^{\prime}$ is the manifold obtained from blowing up $X$ along $S$. The subvariety $X^{\prime}\subset Z^{\prime}$ is called the birational transform of $X\subset Z$ under the blowup.

First, by the local construction, we consider an three dimensional multidisk in $X$ $\Delta$ with holomorphic coordinates $x_{i},i=1,2,3$, and $V$ is specified by $x_{1}=x_{2}=0$ on each $\Delta$. Then we define the smooth variety

$$\tilde{\Delta}\subset\Delta\times\mathbb{P}^{1}$$

as follows

$$\tilde{\Delta}=\{(x_{1},x_{2},x_{3},(y_{1}:y_{2}))\subset\Delta\times\mathbb{P}^{1}:x_{2}y_{1}-x_{1}y_{2}=0\}$$

Here $y_{1},y_{2}$ are the homogeneous coordinates on $\mathbb{P}^{1}$. The projection map $\pi:\tilde{\Delta}\rightarrow\Delta$ on the first factor is clearly an isomorphism away from $V$, while the inverse image of a point $z\in V$ is a projective space $\mathbb{P}^{1}$. The manifold $\tilde{\Delta}$, together with the projection map $\pi$ is the blow-up of $\delta$ along $V$; The inverse image $E=\pi^{-1}(V)$ is an exceptional divisor of the blow-up. For two coordinates patches $U_{i}=(y_{i}\neq 0),i=1,2$, they have holomorphic coordinates respectively

$$\begin{gathered}z^{(1)}_{1}=x_{1},\quad z_{2}^{(1)}=\frac{y_{2}}{y_{1}}=\frac{x_{2}}{x_{1}},\quad y_{3}=x_{3}\\ z^{(2)}_{1}=\frac{y_{1}}{y_{2}}=\frac{x_{1}}{x_{2}},\quad z_{2}^{(1)}=x_{2},\quad y_{3}=x_{3}\\ \end{gathered}$$

with transition function on $U_{1}\cap U_{2}$given by $g_{ij}=z^{(j)}_{i}=\frac{y_{i}}{y_{j}}=\frac{x_{i}}{x_{j}}$. Next we consider the global construction of the blow-up manifold. Let $X$ be a complex manifold of dimension three and $S\subset X$ be a curve. Let $\{U_{\alpha}\}$ be a collection of disks in $X$ covering $S$ such that in each disk $\Delta_{\alpha}$ the subvariety $S\cap\Delta_{\alpha}$ may be given as the locus $(x_{1}=x_{2}=0)$, and let $\pi_{\alpha}:\tilde{\Delta}_{\alpha}\rightarrow\Delta_{\alpha}$ be the blow-up of $\Delta_{\alpha}$ along $S\cap\Delta_{\alpha}$. We then have

$$\pi_{\alpha\beta}:\pi_{\alpha\beta}^{-1}(U_{\alpha}\cap U_{\beta})\rightarrow\pi_{\beta}^{-1}(U_{\alpha}\cap U_{\beta})$$

and using them, we can patch together the local blow-ups $\tilde{\Delta}_{\alpha}$ to form a manifold

$$\tilde{\Delta}=\cup_{\pi_{\alpha\beta}}\tilde{\Delta}_{\alpha}$$

Finally, sinve $\pi$ is an isomorphism away from $X\cap(\cup\Delta_{\alpha})$, we can take

$$X^{\prime}=\tilde{\Delta}\cup_{\pi}X-S$$

$X^{\prime}$, together with the projection map $\pi:X^{\prime}\rightarrow X$ extending $\pi$ on $\tilde{\Delta}$ and the identity on $X-S$, is called the blow-up of $X$ along $S$, and the inverse image $\pi^{-1}(S)$ is an exceptional divisor.

From the excision theorem of cohomology in algebraic topology[10],

$$H^{3}(X,S)\cong H^{3}(X-S)\cong H^{3}(X^{\prime}-E)\cong H^{3}(X^{\prime},E)$$

(2.10)

which means that the variation of the mixed Hodge structures of $H^{3}(X,S)$ and $H^{3}(X^{\prime},E)$ over the corresponding moduli space are equivalent. The mixed Hodge structure as follow,

$$\phi:H^{3}(X^{\prime}-E)\tilde{\longrightarrow}\oplus_{p+q=3}H^{q}(X^{\prime},\Omega^{\prime})$$

(2.11)

where $\Omega^{\prime}$ denotes the holomorphic p-forms on $X^{\prime}$. The filtrations have the form

$$F^{m}H^{3}=\oplus_{p\leq m}H^{3-p}(X^{\prime},\Omega^{\prime})$$

and

$$W_{-1}H^{3}=0,\quad W_{0}H^{3}=H^{3}(X^{\prime}),\quad W_{1}H^{3}=H^{3}(X^{\prime}-E)$$

Additionally, the mixed Hodge structure has graded weights

$$Gr^{W}_{k}H^{3}=Gr^{W}_{-k+3}H^{3}/Gr^{W}_{-(k+1)+3}H^{3}$$

that take the following form for the divisor E

$$\begin{gathered}Gr^{W}_{3}H^{3}=Gr^{W}_{0}H^{3}/Gr^{W}_{-1}H^{3}\cong H^{3}(X^{\prime}),\\ Gr^{W}_{2}H^{3}=Gr^{W}_{1}H^{3}/Gr^{W}_{0}H^{3}\cong H^{2}(E)\end{gathered}$$

The reason to consider these (graded) weights is the following: The mixed Hodge structure is defined such that the Hodge filtration $F^{m}H^{3}$ induces a pure Hodge structure on each graded weight, i.e. on $Gr^{W}_{2}H^{3}$ and $Gr^{W}_{3}H^{3}$. Thus, the following two induced filtrations on $Gr^{W}_{3}H^{3}$

$$H^{3}(X^{\prime})\cap F^{3}H^{3}\subset H^{3}(X^{\prime})\cap F^{2}H^{3}\subset H^{3}(X^{\prime})\cap F^{1}H^{3}\subset H^{3}(X^{\prime})\cap F^{0}H^{3}=H^{3}(X^{\prime})$$

(2.12)

and on $Gr^{W}_{2}H^{3}$

$$H^{2}(E)\cap F^{2}H^{3}\subset H^{2}(E)\cap F^{1}H^{3}\subset H^{2}(E)\cap F^{0}H^{3}=H^{2}(E)$$

(2.13)

lead to pure Hodge structures on $H^{3}(X^{\prime})$ and $H^{2}(E)$. $H^{3}(X^{\prime}-E)$ forms a bundle $\mathcal{H}^{3}$ over the open-closed moduli space $\mathcal{M}$ with the Gauss-Manin connection $\nabla$ satisfying the Griffish transversality condition

$$\nabla\mathcal{F}^{p}\in\mathcal{F}^{p-1}\otimes\Omega^{1}_{\mathcal{M}}$$

(2.14)

The flatness of the Gauss-Manin connection leads to N=1 special geometry and a Picard-Fuchs system of differential equations that govern the mirror maps and superpotentials.

The geometric setting we are interested in is a hypersurface $X:P=0$ with a curve $S$ on it, $S:P=0,h_{1}=h_{2}=0$. After blowing up along $S$, the blow-up manifold $X^{\prime}$ is given globally as the complete intersection in the total space of the projective bundle $\mathcal{W}=\mathbb{P}(\mathcal{O}(D_{1})\oplus\mathcal{O}(D_{2}))$,

$$P=0,\quad Q\equiv y_{1}h_{2}-y_{2}h_{1}=0$$

(2.15)

where $(y_{1},y_{2})\sim\lambda(y_{1},y_{2})$ is the projective coordinates on the $\mathbb{P}^{1}$ -fiber of the blow-up $X^{b}$. We have to emphasize that $X^{\prime}$ is not Calabi-Yau since the first Chern class is nonzero. In addition, the blow-up procedure do not introduce new degrees of freedom associated to deformations of $E$. Under blowing up map, the the open-closed moduli space of $(X,S)$ is mapped into the complex structure deformation of $X^{\prime}$. This enable us to calculate the superpotential $W_{\text{brane}}$ for B- branes wrapping rational curves via the periods on the complex structure moduli space of $X^{\prime}$ determined by Picard-Fuchs equations.