arXiv:0910.4215v1 [math.AG] 22 Oct 2009
Picard-Fuchs Equations for Relative Periods and Abel-Jacobi Map for Calabi-Yau Hypersurfaces
Si Li, Bong H. Lian, Shing-Tung Yau
Abstract
We study the variation of relative cohomology for a pair consisting of a smooth projective hypersurface and an algebraic subvariety in it. We construct an inhomoge- neous Picard-Fuchs equation by applying a Picard-Fuchs operator to the holomorphic top form on a toric Calabi-Yau hypersurface, and deriving a general formula for the d-exact form on one side of the equation. We also derive a double residue formula, giving a purely algebraic way to compute the inhomogeneous Picard-Fuchs equations for Abel-Jacobi map, which has played an important role in recent study of D-branes [25]. Using the variation formalism, we prove that the relative periods of toric B-branes on a toric Calabi-Yau hypersurface satisfy the enhanced GKZ-hypergeometric system proposed in physics literature [6], and discuss the relations between the works [25] [21]
[6] in recent study of open string mirror symmetry. We also give the general solutions to the enhanced hypergeometric system.
1 Introduction
Mirror symmetry connects symplectic geometry of Calabi-Yau manifold to complex geometry of its mirror manifold. In closed string theory, this has led to predictions on counting curves on projective Calabi-Yau threefolds [9][8]. In open string theory, mirror symmetry has led to predictions on counting holomorphic discs, first in the non-compact case studied in [4][3], and more recently in the compact quintic example, where the instanton sum of disc amplitude with non-trivial boundary on the real locus of the real quintic is shown to be identical to the normalized Abel-Jacobi map on the mirror quintic via mirror map [31][25][27].
In physics, the Abel-Jacobi map serves as the domain-wall tension of D-branes on the B-model, and is obtained via reduction of the holomorphic Chern-Simons action on curves [4]. It is conjectured to have remarkable integrality structure [26]. A key for calculating the Abel-Jacobi map is through inhomogeneous Picard-Fuchs equations [25]. Let Xz be a family of Calabi-Yau threefolds parameterized by variablez, and Ωzbe a family of nonzero holomorphic 3-forms on Xz. Assume that there is a family of pairs of holomorphic curves Cz+, Cz− in Xz. LetD(∂z) be a Picard-Fuchs operator. Then there exists a 2-formβz such that
D(∂z)Ωz=−dβz (1.1)
The exact term dβzdoes not contribute when it is integrated over a closed 3-cycle Γ inXz. The so-called closed-string periodR
ΓΩzthen satisfies a homogeneous Picard-Fuchs equation.
In open string theory, it is necessary to consider the integral of Ωz over a 3-chain Γ inXz
which is not closed, but whose boundary is C+−C−. Because of contributions from the boundary, this so-called open-string period R
ΓΩz satisfies an inhomogeneous Picard-Fuchs equation. Solving the equation gives a precise description of the Abel-Jacobi map up to closed-string periods. To study this map,βz plays an essential role since it is this form that
gives rise to one side of the inhomogeneous Picard-Fuchs equation:
Z
ΓD(∂z)Ωz=− Z
∂Γ
βz. (1.2)
The inhomogeneous term on the right side turns out also to encodes important information for predicting the number of holomorphic disks on a mirror Calabi-Yau manifold.
There have been several proposals for constructing the inhomogeneous Picard-Fuchs equation and its solutions. In the case of 1-moduli family [25], it was done by first computing βzusing the Griffith-Dwork reduction procedure, and then by doing an explicit (but delicate) local analytic calculation of appropriate boundary integrals. Based on the notion of off- shell mirror symmetry, two other proposals [21][6] have been put forth. Roughly speaking, their setup begins with a family of divisors Yz,u which deforms inXz under an additional parameteru. For each relative homology class Γ∈H3(Xz, Yz,u), one considers the integral
Z
Γ
Ωz (1.3)
which is called a relative period for B-brane. It is proposed that the open-string periods above be recovered as a certain critical value of the relative period, regarded as a function of u. To calculate the relative periods, [21] proposed a procedure similar to the Griffith-Dwork reduction. In [6], an enlarged polytope is proposed to encode both the geometry of the Calabi-YauXzand the B-brane geometry. This gives rise to a GKZ hypergeometric system for the relative periods, and a special solution at a critical point inuthen leads to a solution to the original inhomogeneous Picard-Fuchs equation.
Our goal in this paper is to further develop the mathematical structures underlying inho- mogeneous Picard-Fuchs equations and the Abel-Jacobi map, and to clarify the relationships between the three approaches mentioned above. Here is an outline. We begin, in section 2, with a description of a residue formalism for relative cohomology of a family of pairs (Xz, Yz), including a number of variational formulas on the local systemHn(Xz, Yz). In section 3, we derive a general formula for the exact form (the β-term) appearing in the inhomogeneous Picard-Fuchs equation for toric Calabi-Yau hypersurfaces, generalizing GKZ-type differen- tial equation to the level of differential forms instead of cohomology classes. This gives a much more uniform approach to computing theβ-term than the Griffith-Dwork reduction.
In section 4, we prove a purely algebraic a double residue formula for the inhomogeneous term of the Picard-Fuchs equation that governs the Abel-Jacobi map. This uniform approach also allows us to bypass the delicate local analytical calculation of boundary integrals in a previous approach [25][23]. In section 5, using the residue formalism in section 2, we give a simple interpretation of the relative version of the Griffith-Dwork reduction used in [21].
In particular, this gives a mathematical justification for the appearance of log divisor, and elucidates the relationship between relative periods and the Abel-Jacobi map. We also give a uniform description for the enhanced polytope method for describing toric B-brane geom- etry in a general toric Calabi-Yau hypersurface, and show that relative periods satisfy the corresponding enhanced GKZ system. Finally, we give a general formula, modeled on the closed string case [16][17], for solution to the enhanced GKZ system.
Acknowledgement. S.L. would like to thank J. Walcher for many stimulating discussions, and thank M.Soroush for answering many questions on his paper. After the completion of a preliminary draft of our paper, three other papers [2][5][15] with some overlap with ours have since been posted on the arXiv.
2 Variation of Relative Cohomology
Local System of Relative Cohomology and Gauss-Manin Connection
Let π:X →S be a smooth family ofn-dimensional projective varieties, and Y →S be a family of smooth subvarietyY ⊂ X. Lets∈S be a closed point, and denote byXs, Ysthe corresponding fiber overs. Consider the family of relative cohomology class
Hn(Xs, Ys) given by the cohomology of the complex of pairs:
Γ(Ωn(Xs))⊕Γ(Ωn−1(Ys)) with the differential
d(α, β) = (dα, α|Ys−dβ) (2.1)
Here Ωn(Xs) and Ωn−1(Ys) are sheaves of De Rham differentialn-forms onXsand (n−1)- form on Ys, and Γ is the smooth global section. Therefore an element of Hn(Xs, Ys) is represented by a differential n-form onXs whose restriction to Ys is specified by an exact form.
Lemma 2.1. Hn(Xs, Ys)forms a local system onS.
Proof. The proof is similar to the case withoutY by choosing a local trivialization ofX →S which also trivializesY →S. See e.g.[30].
We denote this local system byHn(X,Y), and let∇GM be the Gauss-Manin connection.
There’s a well-defined natural pairing
Hn(Xs, Ys) ⊗ Hn(Xs, Ys) → C Γ ⊗ (α, β) 7→ <Γ,(α, β)>≡R
Γα−R
∂Γβ. (2.2)
Given a family (αs, βs) ∈ Hn(Xs, Ys) varying smoothly, which gives a smooth section of Hn(X,Y) denoted by [(αs, βs)], and Γs ∈ Hn(Xs, Ys) a smooth family of relative cycles, we get a function onS given by the pairing
<Γs,(αs, βs)>=
Z
Γs
αs− Z
∂Γs
βs
Letv be a vector field onS. We consider the variation Lv <Γs,(αs, βs)>
where Lv is the Lie derivative with respect tov. Suppose we have a lifting ˜α,β, which are˜ differential forms onX,Y respectively, such that
˜
α|Xs =αs, β˜|Ys =βs
and that Γs moves smoothly to form a cycle ˜Γ onX: Γs= ˜Γ∩Xs, ∂Γ⊂ Y. Let ˜vX be a lifting of v onX, ˜vY be a lifting of vonY.
Proposition 2.2 (Variation Formula).
Lv<Γs,(αs, βs)>=<Γs,(ιv˜Xyd˜α, ιv˜Yy(dβ˜−α))˜ > (2.3) where ιv˜X is the contraction with ˜vX, and similarly forι˜vY.
Proof. We fix a points0∈S, and letσ(t) be a local integral curve ofv such thatσ(0) =s0. Let ˜Γσ(t) denote the one-dimensional family of cycles overσ(s), 0≤s≤t, and we denote by Γtthe cycle over the pointσ(t). Also let∂Γfσ(t)be the family of boundary cycle∂Γsover σ(s),0≤s≤t. Then we have
∂ Γ˜σ(t)
= Γt−Γ0−∂Γfσ(t) (2.4)
therefore
Z
Γ˜σ(t)
d˜α= Z
Γt
˜ α−
Z
Γ0
˜ α−
Z
∂Γfσ(t)
˜ α Similarly
Z
∂Γf
dβ˜= Z
∂Γt
β˜− Z
∂Γ0
β˜ Taking the derivative with respect to t, we get
∂
∂t Z
Γt
˜ α−
Z
∂Γt
β˜
= Z
Γt
ιv˜Xyd˜α+ Z
∂Γt
ιv˜Yy
˜ α−dβ˜ The proposition follows.
Note that (ιv˜Xyd˜α, ιv˜Yy(dβ˜−α)) is nothing but the Gauss-Manin connection˜
∇GMv [(αs, βs)] =h
(ιv˜Xyd˜α)|Xs,(ιv˜Yy(dβ˜−α))˜ |Ys
i
(2.5) and it is straightforward to check using the variation formula that the right side of (2.5) is independent of the choice of ˜α,β,˜ v˜X,v˜Y, and that the connection is flat. The following corollary also follows from (2.5).
Corollary 2.3.
Lv <Γs,(αs, βs)>=<Γs,∇GMv [(αs, βs)]> (2.6)
Residue Formalism for Relative Cohomology
In this section, we assume that Xz moves as a family of hypersurfaces in a fixedn+ 1-dim ambient projective space M with defining equationPz = 0. Here Pz ∈ H0(M,[D]) for a fixed divisor class [D], and zis holomorphic coordinate onS parametrizing the family. Let
ωz∈H0(M, KM(Xz))
be a rational (n+1,0)-form on M with pole of order one alongXz, then we get a famliy of holomorphic (n,0)-form on Xz given by
ResXzωz∈H0(Xz, KXz)
and also a family of relative cohomology classes
(ResXzωz,0)∈Hn(Xz, Yz)
Here our convention for Res is that if Xz is locally given by w = 0, and ωz = dww ∧φ, where φ is locally a smooth form, then ResXzωz = φ|Xz. Note that the map ResXz : H0(M, KM(Xz))→Hn,0(Xz) is at the level of forms, not only as cohomology classes since the order of pole is one. While in the residue formalism of ordinary cohomology, we can ignore exact forms to reduce the order of the pole [14], it is important to keep track of the order of the pole in considering periods of relative cohomology because of the boundary term.
We choose a fixed open cover {Uα} ofM and a partition of unity{ρα} subordinate to it. LetPz,α= 0 be the defining equation ofXs onUα. Then we can write
ωz=X
α
dMPz,α
Pz,α ∧φz,α (2.7)
where φz,α is a smooth (n,0)-form with supp(φz,α)⊂Uα. On the trivial familyM×S, we will use dM to denote the differential alongM only and usedto denote the differential on the total space. Let
φz=X
α
φz,α (2.8)
Then φz is a smooth form onM×S such that φz|Xz =ResXzωz
Consider the variation
∂
∂zωz = dM
X
α
∂zlog(Pz,α)φz,α
!
+X
α
dMPz,α
Pz,α ∧∂zφz,α−∂zPz,α
Pz,α
dMφz,α
Let
∂˜zX = ∂
∂z+nz,X
be a lifting of ∂z∂ toX, wherenz,X ∈Γ(TM|Xz) is along the fiber, which is a normal vector field corresponding to the deformation ofXz inM with respect toz. Then in eachUα, we have
ιnXzydMPz,α
|Xz =−∂zPz,α|Xz (2.9) It follows easily that
ι∂˜zX
ydφz|Xz =ResXz(∂zωz−dM(∂zlog(Pz)φz)) (2.10) Note that since the transition function of [D] is independent of z, ∂zlog(Pz) is globally well-defined. In general, ∂zωz will have a pole of order two along Xz, but the substraction ofdM(∂zlog(Pz)φz) makes it logarithmic alongXz, hence the residue above is well-defined.
Next we choose arbitrary lifting of ∂z∂ toY, and write it as
∂˜z,Y = ∂
∂z +nz,Y
where nz,X ∈Γ(TM|Yz) is along the fiber, which is a normal vector field corresponding to the deformation of Yz in M with respect toz. Then the variation formula implies that
Proposition 2.4 (Residue Variation Formula).
∇GM∂z (ResXzωz,0) = ResXz(∂zωz−dM(∂zlog(Pz)φz)),−ιnz,Yyφz
(2.11) To see the effect of the second component on the right side, let us assume that Yz = Xz∩H, whereH is a fixed hypersurface in M with defining equationQ= 0. Then we can choose aǫ-tubeTǫ(Γ) [14] of Γ∈Hn(Xz, Yz) with∂Tǫ(Γ)⊂H. Hence
1 2πi
Z
Tǫ(Γ)−dM(∂zlog(Pz)φz) = 1 2πi
Z
Tǫ(∂Γ)
(∂zlog(Pz)φz)|H
= Z
∂Γ
ResYz(∂zlog(Pz)φz)|H
= −
Z
∂Γ
ιnz,Yyφz
which cancels exactly the second component on the right side of (2.11). Therefore,
∂z<Γ,(ResXzωz,0)>= 1 2πi
Z
Tǫ(Γ)
∂zωz
We can localize the above observation and consider the following situation: on each Uα, suppose we can choose Qα independent of z such that Pz,α, Qα are transversal and Yz∩Uα⊂ {Qα= 0, Pz,α = 0}. Suppose we have a relative cycle Γ∈Hn(Xz, Yz) where we can choose a ǫ-tubeTǫ(Γ) such that∂Tǫ(Γ)∩Uαlies in{Qα= 0}. We have the pairing
<Γ,(ResXzωz,0)>= lim
ǫ→0
1 2πi
Z
Tǫ(Γ)
ωz
However, the right hand side doesn’t depend onǫ. In fact, letTδǫ(Γ) be a solid annulus over Γ. By Stokes’s theorem, we have
Z
Tǫ(Γ)
ωz− Z
Tσ(Γ)
ωz= Z
∂Tδǫ(Γ)
ωz
since∂Tδǫ(Γ)∩Uα⊂ {Qα= 0}for eachα, the above integral vanishes. Therefore the integral R
Tǫ(Γ)ωz doesn’t depend on the position of theǫ-tube if we impose the boundary condition as above. It follows immediately that
(∂z)k <Γ,(ResXzωz,0)>= 1 2πi
Z
Tǫ(Γ)
(∂z)kωz
where ∂Tǫ(Γ)∩Uαlies in{Qα= 0}. Applying this to a Picard-Fuchs operator, we get Proposition 2.5. [Inhomogeneous Picard-Fuchs Equation] Let D=D(∇GM∂z )be a Picard- Fuchs operator, i.e.
D(∂z)ωz=−dβz (2.12)
for some rational(n−1,0)-formβz with poles alongXz. Then under the above local choices, we have
D(∂z)<Γ,(ResXzωz,0)>= 1 2πi
Z
Tǫ(∂Γ)
βz. (2.13)
In the next section, we will derive a general formula forβz using toric method.
More generally, suppose that{Qα}depends onz which is denoted byQz,α, and we put ourǫ-tube Tǫ(Γz) inside{Qz,α}= 0 onUα. Then
Z
Γz
ResXzωz= 1 2πi
Z
Tǫ(Γz)
ωz=X
α
1 2πi
Z
Tǫ(Γz)
ραωz
and the integration doesn’t depend onǫassuming the boundary condition as above. Apply- ing the variation formula (2.3) we get
∂
∂z X
α
1 2πi
Z
Tǫ(Γz)
ραωz=X
α
1 2πi
Z
Tǫ(Γz)
ρα∂zωz−X
α
1 2πi
Z
Tǫ(∂Γz)
ρα ιnz,Qαyωz
|Qz,α=0
wherenz,Qαis the normal vector field corresponding to the deformation of{Qz,α = 0}inside M. In particular, we haveιnz,QαydQz,α|{Qz,α=0} =−∂zQz,α|{Qz,α=0}. Hence
ιnz,Qαyωz
|Qz,α=0=−ResQz,α=0(∂zlog(Qz,α)ωz) Putting together the last three equations, we arrive at
Proposition 2.6 (cf. [32]).
∂k
∂zk Z
Γz
ResXzωz
= 1 2πi
Z
Tǫ(Γz)
∂zkωz+ Xk l=1
∂k−l
∂zk−l X
α
1 2πi
Z
Tǫ(∂Γz)
ραResQz,α=0
∂zlog(Qz,α)∂l−1
∂zl−1ωz
3 Exact GKZ Differential Equation and Toric geometry
In this section, we study the Picard-Fuchs differential operators arising from a generalized GKZ hypergeometric systems [16][17] for toric Calabi-Yau hypersurfaces and derive a gen- eral formula for theβ-term of an inhomogeneous Picard-Fuchs equation, from toric data.
We first consider the special case of a weighted projective space, where β-term will be much simpler than in the general case, which will be considered at the end of this section.
Let P4(w) =P4(w1, w2, w3, w4, w5). We assume that w5 = 1 and it’s of Fermat-type, i.e., wi|dfor each i, whered=w1+w2+w3+w4+w5. There’s associated 4-dimensional integral convex polyhedron given by the convex hull of the integral vectors
∆ = (
(x1,· · · , x5)∈R5| X5 i=1
wixi= 0, xi≥ −1 )
If we choose the basis{ei= (1,0,0,0,−wi), i= 1..4}, then the vertices is given by
∆ : v1= ( d
w1 −1,−1,−1,−1) v2= (−1, d
w2−1,−1,−1) v3= (−1,−1, d
w3 −1,−1) v4= (−1,−1,−1, d
w4 −1) v5= (−1,−1,−1,−1)
and the vertices of its dual polytope is given by
∆∗: v∗1 = (1,0,0,0) v∗2 = (0,1,0,0) v∗3 = (0,0,1,0) v∗4 = (0,0,0,1)
v∗5 = (−w1,−w2,−w3,−w4)
Letv∗k, k= 0,1,2, ..,be integral points of ∆∗, wherev0∗= (0,0,0,0). We write f∆∗(x) = X
v∗k∈∆∗
akXv∗k (3.1)
which is the defining equation for our Calabi-Yau hypersurfaces in the anti-canonical divisor class. Here X = {X1, X2, X3, X4} is the toric coordinate, Xv∗k =
Q4 j=1
Xv
∗ k,j
j . If we use homogeneous coordinate [10] {zρ,1 ≤ρ≤ 5} corresponding to the one-dim cone {vρ,1 ≤ ρ≤5}, then the toric coordinate can be written by homogeneous coordinate
Xj= zjd/wj Q5 ρ=1
zρ
, j= 1,2,3,4 (3.2)
The relevant rational form with pole of order one along the hypersurface is given by
Π(a) = Q5 i=1
wi
d3 P 1
v∗k∈∆∗akXv∗k Y4 j=1
dXj
Xj
= Ω0
P
v∗k∈∆∗
ak
Q5 ρ=1
zρ<v∗k,vρ>+1
= Ω0
a0
Q5 ρ=1
zρ+ P5 ρ=1
aρzρd/wρ+ P
v∗k∈∆∗,k>5
ak
Q5 ρ=1
zρ<v∗k,vρ>+1
(3.3)
where Ω0=P5
ρ=1(−1)ρ−1wρzρdz1∧ · · ·dzˆρ∧ · · ·dz5. Define the relation lattice by L={l= (l0, l1, ...)∈Z|∆∗|+1|X
i
liv¯∗i = 0}, where ¯v∗i = (1, vi∗), vi∗∈∆∗
The moduli variable associated with the choice of a basis{l(k)}forLis given by [16]
xk = (−1)l(k)0 al(k)
The key idea here is to consider the following 1-parameter family of automorphisms φt:zρ→(a0
aρ
)wρdtzρ, 1≤ρ≤5.
Put
Πt(a) =φ∗tΠ(a).
It satisfies the differential equation
∂t(φ∗tΠ(a)) =LVφ∗tΠ(a) whereV = P5
ρ=1 wρ
d (logaa0
ρ)zρ ∂
∂zρ is the generating vector field forφt,LV is the Lie derivative.
This is solved by
φ∗tΠ(a) =etLVΠ(a) (3.4)
Define
Π(x) =˜ a0Π1(a) (3.5)
Π(x) is a function of˜ {xk} only. Indeed,
Π(x) =˜ Ω0
Q5 ρ=1
zρ+ Q5 ρ=1
(aaρ0)wρ/d
! P5 ρ=1
zd/wρ ρ + P
vk∗∈∆∗,k>5 ak
a0
Q5 ρ=1
a0
aρ
wρd<v∗k,vρ> Q5 ρ=1
zρ<vk∗,vρ>+1
Since we have
X5 ρ=1
wρvρ= 0, vk∗= X5 ρ=1
wρ
d < vk∗, vρ> v∗ρ
we see that both Q5
ρ=1
(aaρ
0)wρ/d
! and aak
0
Q5 ρ=1
a0
aρ
wρd<v∗k,vρ>
can be written in terms ofxk’s as an algebraic function.
Given an integral pointl ∈L, consider the GKZ operator (it differs from the standard GKZ operator by a factor ofa0Q
li>0alii) Dl = a0
(Y
li>0
alii ∂
∂ai
li
−alY
li<0
a−li i ∂
∂ai
−li)
=
l0
Y
j=1
(a0 ∂
∂a0 −j) Y
i6=0,li>0 lYi−1
j=0
(ai ∂
∂ai −j)−al
−l0
Y
j=1
(a0 ∂
∂a0 −j) Y
i6=0,li<0
−lYi−1 j=0
(ai ∂
∂ai −j)
a0
= D˜la0
where we use the convention that Qm i=1
(· · ·) = 1 ifm≤0. From DlΠ(a) = 0
We get
eLVD˜le−LVΠ(x) = 0˜ (3.6)
Lemma 3.1.
eLV(ai
∂
∂ai
)e−LV =ai
∂
∂ai
+δ1≤i≤5Lwidzi ∂
∂zi −δi,0LP5
ρ=1 wρ
dzρ ∂
∂zρ
here δ1≤i≤5= 1 if1≤i≤5 and otherwise0.
Proof. Since
V, ai
∂ ai
= δ1≤i≤5
wi
d zi
∂
∂zi −δi,0
X5 ρ=1
wρ
d zρ
∂
∂zρ
V,
V, ai∂
ai
= 0 The lemma follows from the formula
eLV(ai
∂
∂ai
)e−LV =ai
∂
∂ai
+ X∞ k=1
1
k!L(adV)k“ ai ∂
∂ai
”
It follows from the lemma that
l0
Y
j=1
(a0
∂
∂a0 −j) Q
i6=0,li>0 lYi−1
j=0
(ai
∂
∂ai −j+δ1≤i≤5Lwidzi ∂
∂zi)
−al
−l0
Y
j=1
(a0 ∂
∂a0 −j) Q
i6=0,li<0
−lYi−1 j=0
(ai ∂
∂ai −j+δ1≤i≤5Lwidzi ∂
∂zi)
Π(x) = 0 (3.7)˜ where we have usedLP5
ρ=1 wρ
dzρ ∂
∂zρ
Π(x) = 0 to eliminate the terms with˜ LP5
ρ=1 wρ
d zρ ∂
∂zρ
. Observe that each Lie derivativeLwidzi ∂
∂zi commutes with all other operators appearing on the left side of (3.7). So we can move every term involvingLwidzi ∂
∂zi to the right side, so that (3.7) can now be explicitly written as
D˜lΠ(x) =˜ − X5 i=1
Lwidzi ∂
∂ziαi
where the αi are (easily computable)d-closed 4-forms depending on l. By the Cartan-Lie formulaLX=dιX+ιXd, we obtain the formula
Proposition 3.2. [β-term Formula]
D˜lΠ(x) =˜ −dβl
where βl=P
iιwi
dzi ∂
∂ziαi.
Next we consider the differential operators of the extended GKZ system induced by the automorphism of the ambient toric variety. The corresponds to the root of ∆∗ [17]. Let
v∗i ∈R(∆∗), < vi∗, vρi>=−1, < v∗i, vρ>≥0 for ρ6=ρi
then we obtain an equation
X
v∗k∈∆∗
(< vk∗, vρi>+1)av∗k ∂
∂av∗k+v∗i
1
a0e−LVΠ(x) =˜ L(Q5 ρ=1
z<v
∗ i,vρ>
ρ
) zρi ∂
∂zρi
1
a0e−LVΠ(x)˜
where in the formula we have identified ak ≡ av∗k for convenience, and avk∗+vi∗ just corre- sponds to the pointvk∗+vi∗. Note that
avi∗
P
v∗k∈∆∗(< vk∗, vρi >+1)avk∗ ∂
∂av∗ k+v∗
i
1 a0
= P
vk∗∈∆∗(< vk∗, vρi >+1)aavk∗avi∗
0av∗ i+v∗
k
avk∗+v∗i ∂
∂av∗ k+v∗
i
−2av∗iaa2−v∗i 0
where the last term is zero if −v∗i 6= ∆∗. On the other hand, it’s easy to compute eLVL(Q5
ρ=1
z<v
∗ i,vρ>
ρ
) zρi∂zρi∂
e−LV = Y5 ρ=1
a0
aρ
wρd <v∗i,vρ>
L(Q5
ρ=1
z<v
∗ i,vρ>
ρ
) zρi∂zρi∂
therefore we get the following
X
v∗k∈∆∗
(< v∗k, vρi >+1) av∗kav∗i
a0av∗i+vk∗
av∗k+v∗i
∂
∂av∗k+vi∗
+δ1≤v∗i+vk∗≤5Lwidzv∗ i+v∗
k
∂
∂zv∗ i+v∗
k
−2av∗ia−vi∗
a20
Π(x)˜
= av∗i
a0
Y
ρ
a0
aρ
wρd <v∗i,vρ>
LQ
ρz<v
∗ i,vρ>
ρ
ff zρi∂zρi∂
Π(x)˜
Again, by the Cartan-Lie formula we can easily write the right side as a d-exact form.
Example: P (1 , 1 , 1 , 1 , 1)
We will compute the β-term for mirror quintic [31], wherew= (1,1,1,1,1), and f∆∗ =a0
Y5 i=1
zi+ X5 i=1
aizi5.
The relation lattice is generated by
l= (−5,1,1,1,1,1) and the moduli variable is
.x= (−1)l0al=− Q5 i=1
ai
a50 (3.8)
Put
Π(x) =˜ ω
Q5 i=1
zi−x15 P5
i=1
zi5
(3.9)
Then ourβ-term formula Proposition 3.2 for the Picard-Fuchs equation yields ( 5
Y
i=1
Θx+1
5Lzi∂zi
−x Y5 i=1
(5Θx+i) )
Π(x) = 0˜ (3.10)
or
Θ5x−x Y5 i=1
(5Θx+i)
!
Π(x) =˜ −dβx, dβx= X
I⊂{1,...,5},|I|≥1
Θ5−|I|x
5|I|
Y
k∈I
(Lzk∂k) ˜Π(x).
Note that the sum of terms for |I| = 1 in the expression of dβx is zero. A choice of βx
relevant in the next section will be βx = Θ3x
52 {z1ι∂1Lz2∂2+z3ι∂3Lz4∂4+ (z3ι∂3+z4ι∂4)(Lz1∂1+Lz2∂2)}Π(x)˜ +Θ2x
53 {(z3ι∂3+z4ι∂4)Lz1∂1Lz2∂2+z3ι∂3(Lz1∂1+Lz2∂z2)Lz4∂4}Π(x)˜ +Θx
54z3ι∂3Lz4∂4Lz1∂1Lz2∂2Π(x) +˜ X
I⊂{1,...,4},|I|≥1
Θ4−|I|x
5|I|+1z5ι∂5
Y
k∈I
(Lzk∂k) ˜Π(x)
where ι∂i is the contraction with the vector ∂z∂
i.
Example: P(2, 2, 2, 1, 1)
The mirror of degree 8 hypersurface inP(2,2,2,1,1) has two complex moduli. The integral points of its dual polytope ∆∗ is given by
∆∗: v∗0= (0,0,0,0) v∗1= (1,0,0,0) v∗2= (0,1,0,0) v∗3= (0,0,1,0) v∗4= (0,0,0,1) v∗5= (−2,−2,−2,−1) v∗6= (−1,−1,−1,0) and in homogeneous coordinate
f∆∗ =a0z1z2z3z4z5+a1z14+a2z24+a3z34+a4z48+a5z85+a6z44z54 A basis of relation lattice is given by
l(1)= (−4,1,1,1,0,0,1), l(2)= (0,0,0,0,1,1,−2) We get the moduli coordinates
x1= a1a2a3a6
a40 , x2= a4a5
a26 The rational 4-form is given by
Π(x) =˜ Ω0
z1z2z3z4z5+x114x218(z14+z42+z34+z48+z58) +x114x−238z44z54 The exact GKZ equation froml(1) andl(2) can be read
(Θx1−2Θx2) Y3 j=1
(Θx1+1
4Lzi∂zi)−x1
Y4 j=1
(4Θx1+j)
Π(x)˜ = 0
Θx2+1
8Lz4∂z4 Θx2+1 8Lz5∂z5
−x2(Θx1−2Θx2) (Θx1−2Θx2−1)
Π(x)˜ = 0
Example: P(7 , 2 , 2 , 2 , 1)
The mirror of degree 14 hypersurface in P(7,2,2,2,1) has also two complex moduli. The integral points of its dual polytope ∆∗ is given by
∆∗: v∗0= (0,0,0,0) v∗1= (1,0,0,0) v∗2= (0,1,0,0) v∗3= (0,0,1,0) v∗4= (0,0,0,1) v∗5= (−7,−2,−2,−2) v∗6= (−3,−1,−1,−1) v∗7= (−4,−1,−1,−1) v∗8= (−1,0,0,0) and in homogeneous coordinate
f∆∗ =a0z1z2z3z4z5+a1z12+a2z72+a3z73+a4z47+a5z514+a6z1z75+a7z2z3z4z58+a8z22z23z24z52 A basis of relation lattice is given by [17]
l(1)= (−1,0,0,0,0,−1,1,1,0), l(2)= (0,1,0,0,0,1,−2,0,0) l(3)= (0,0,1,1,1,0,0,1,−4), l(4)= (0,0,0,0,0,1,0,−2,1)
We get the ”moduli” coordinates before eliminating the automorphism generated by the roots
x1=−a6a7
a0a5
, x2=a1a5
a26 , x3=a2a3a4a7
a48 , x4= a5a8
a27 The rational 4-form is given by
Π(x) =˜ Ω0
z1z2z3z4z5−x1x212x317x447(z21+z27+z73+z47+z514)−x1x317x447z1z57−x1x212z2z3z4z85−x1x212x−317x437z22z32z42z52 We consider the GKZ operator corresponding to the following two relation vectors ([17])
l{1,5}= (0,1,0,0,0,1,−2,0,0), l{2,3,4,6}= (−1,0,1,1,1,0,1,0,−3) which gives two exact GKZ equations
Θx2+1
2Lz1∂z1 −Θx1+ Θx2+ Θx4+ 1 14Lz5∂z5
−x2(Θx1−2Θx2) (Θx1−2Θx2−1)
Π(x)˜ = 0 (
(Θx1−2Θx2) Y4 i=2
Θx3+1 7Lzi∂zi
−x1x3x4(Θx1+ 1) Y2 i=0
(Θx4−4Θx3−i) )
Π(x)˜ = 0 and the rootsv7∗, v∗8 give two exact equations
(Θx1+ Θx3−2Θx4)−2x1x2(Θx1−2Θx2)−x1
−Θx1+ Θx2+ Θx4+ 1 14Lz5∂z5
+x1x212Lz57∂z1
Π(x)˜ = 0 n
(Θx4−4Θx3)−2x21x2x4(Θx1+ 1)−x1x4(Θx1+ Θx3−2Θx4) +x1x212x−317x437Lz2z3z4z5∂z1
oΠ(x)˜ = 0
Calabi-Yau hypersurfaces in a general toric variety
For Calabi-Yau hypersurfaces of non-Fermat type, we can repeat the derivation of the β- term above by replacing the homogeneous coordinates by the toric coordinates Xi. Write the defining equation of a general Calabi-Yau hypersurface is in toric coordinates:
f∆∗(X) = X
v∗k∈∆∗
akXv∗k.
Here we will assume thatv∗0 = (0,0,0,0), v∗1= (1,0,0,0), v2∗(0,1,0,0), v∗3 = (0,0,1,0), v∗4= (0,0,0,1). Put
Π(a) = 1 f∆∗(X)
Y4 j=1
dXj
Xj
As in the Fermat case above, we use the automorphism φ:Xi→ a0
aiXi, 1≤i≤4
to transform Π(a) into a form parameterized only by moduli variables xk = (−1)l(k)0 al(k). We can now repeat the derivation of the β-term for each GKZ operatorDl, using the form Π(x) =˜ a0φ∗Π(a). To summarize, we have the following result.
Proposition 3.3. LetΠ(x) =˜ a0φ∗Π(a), and for eachl∈L, let D˜l=
l0
Y
j=1
(a0
∂
∂a0 −j) Y
i6=0,li>0 lYi−1
j=0
(ai
∂
∂ai−j)−al
−l0
Y
j=1
(a0
∂
∂a0 −j) Y
i6=0,li<0
−lYi−1 j=0
(ai
∂
∂ai −j).
Then Π(x)˜ is killed by the differential operator obtained fromD˜l by the substitutions:
a0 ∂
∂a0 7→ a0 ∂
∂a0− LP4
j=1
Xj ∂
∂Xj
aj
∂
∂aj 7→ aj
∂
∂aj +LXj ∂
∂Xj, 1≤j≤4 aj ∂
∂aj 7→ aj ∂
∂aj, j >4.
From this, the Cartan-Lie formula then explicitly yields aβ-term for each GKZ operator:
D˜lΠ(x) =˜ −dβl.
4 Application to Abel-Jacobi Map And Inhomogeneous Picard-Fuchs Equation
In this section, we apply the inhomogeneous Picard-Fuchs equation Proposition 2.5 to study the Abel-Jacobi map for toric Calabi-Yau hypersurfaces that arise in open string mirror symmetry [25]. We will derive a purely algebraic double residue formula for computing the boundary integral of theβ-term.
We will keep the same notation as in section 1: Xz will be a family of hypersurfaces moving in fixed 4-dimensional ambient space M parametrized byz, and we consider two
fixed hypersurfaces Y1 : Q1 = 0, Y2 : Q2 = 0, where Q1, Q2 are sections of some fixed divisors. We consider a pair of family of curves Cz+, Cz− such that they are homologically equivalent as cycles
Cz±֒→Xz∩Y1∩Y2, [Cz+] = [Cz−]∈H2(Xz,Z) We denote byC± he corresponding families
C± BB BB BB BB
//X
S
Let (H3Z, F∗H3C) be the integral Hodge structure for the family X → S, and J3 be the intermediate Jacobian fibration
J3= H3C
F2H3C⊕ H3Z
The fiber of J3overz can be identified with
Jz3= F2H3(Xz,C)∗
H3(Xz,Z) .
The Abel-Jacobi map associated toC± is a normal function ofJ3given by Z Cz+
C−z
:S→ J3.
Suppose we are given a family of rational 4-form with pole of order one alongXz
ωz∈H0(M, KM(Xz)).
Then we obtain a sectionResXzωz ofF3H3C. The integral that we will study is Z Cz+
Cz−
ResXzωz,
which is well-defined up to periods of closed cycles onXz.
We can assume thatCz+∪Cz−moves equisingularly aboveS, for otherwise we shrinkS to some smaller open subset to achieve this. We can always choose local trivialization ofX over a small disk around any point inS which also trivializesC+∪ C−. Therefore it’s easy to see that H3(Xz, Yz) still forms a local system in this case byYz=Cz+∪Cz− and the discussion in section one is still valid. Let{pz,A}be the singular points ofY1∩Y2∩Xz corresponding to non-transversal intersections. It containsCz+∩Cz−and possibly some other points onCz+ or Cz− when there are components ofY1∩Y2other thanCz±. We assume that at each pz,A, one of Y1, Y2 is intersecting transversely with Xz. Therefore we can always put anǫ-tube around each ofCz± such that theǫ-tube lies onY1∩Y2outside a small disk centered at each pz,A∈Cz+∪Cz−, where theǫ-tube lies in one ofY1or Y1 around that small disk.
SupposeD=D(∂z) is a Picard-Fuchs operator, and D(∂z)ωz=−dβz
