Picard-Fuchs Equations for Relative Periods and Abel-Jacobi Map for Calabi-Yau Hypersurfaces

31 Pages·2009·0.3 MB·English

Picard-Fuchs Equations for Relative Periods and
Abel-Jacobi Map for Calabi-Yau Hypersurfaces
9
0
0 Si Li, Bong H. Lian, Shing-Tung Yau
2

t
c
Abstract
O
We study the variation of relative cohomology for a pair consisting of a smooth
2
projective hypersurface and an algebraic subvariety in it. We construct an inhomoge-
2
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
G d-exact form on one side of the equation. We also derive a double residue formula,
A 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
.
h [25]. Usingthevariationformalism,weprovethattherelativeperiodsoftoricB-branes
t on a toric Calabi-Yau hypersurface satisfy the enhanced GKZ-hypergeometric system
a
m proposedin physicsliterature[6],anddiscusstherelations betweentheworks[25][21]
[6] in recent study of open string mirror symmetry. We also give thegeneral solutions
[
to theenhanced hypergeometric system.

1
v
1 Introduction
5
1
2 MirrorsymmetryconnectssymplecticgeometryofCalabi-Yaumanifoldtocomplexgeometry
4
ofitsmirrormanifold. Inclosedstringtheory,this hasledtopredictionsoncountingcurves
.
0 onprojectiveCalabi-Yauthreefolds[9][8]. Inopenstringtheory,mirrorsymmetryhasledto
1 predictionsoncountingholomorphicdiscs,firstinthenon-compactcasestudiedin[4][3],and
9
more recently in the compact quintic example, where the instanton sum of disc amplitude
0
with non-trivial boundary on the real locus of the real quintic is shown to be identical to
:
v
the normalized Abel-Jacobi map on the mirror quintic via mirror map [31][25][27].
i
X 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
r
a [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 X be a
z
family of Calabi-Yau threefolds parameterizedby variablez, and Ω be a family of nonzero
z
holomorphic 3-forms on X . Assume that there is a family of pairs of holomorphic curves
z
C+,C− in X . Let (∂ ) be a Picard-Fuchs operator. Then there exists a 2-form β such
z z z D z z
that
(∂ )Ω = dβ (1.1)
z z z
D −
The exactterm dβ does not contribute when it is integratedover a closed3-cycle Γ in X .
z z
Theso-calledclosed-stringperiod Ω thensatisfiesahomogeneousPicard-Fuchsequation.
Γ z
In open string theory, it is necessary to consider the integral of Ω over a 3-chain Γ in X
z z
R
which is not closed, but whose boundary is C+ C−. Because of contributions from the
−
boundary, this so-called open-string period Ω satisfies an inhomogeneous Picard-Fuchs
Γ z
equation. Solving the equation gives a precise description of the Abel-Jacobi map up to
R
closed-stringperiods. To study this map,β plays anessentialrolesince it is this formthat
z
1
gives rise to one side of the inhomogeneous Picard-Fuchs equation:
(∂ )Ω = β . (1.2)
z z z
D −
ZΓ Z∂Γ
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
equationanditssolutions. Inthecaseof1-modulifamily[25],itwasdonebyfirstcomputing
β usingtheGriffith-Dworkreductionprocedure,andthenbydoinganexplicit(butdelicate)
z
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 Y which deforms in X under an additional
z,u z
parameter u. For each relative homology class Γ H (X ,Y ), one considers the integral
3 z z,u
∈
Ω (1.3)
z
ZΓ
which is called a relative period for B-brane. It is proposed that the open-string periods
aboveberecoveredasacertaincriticalvalueoftherelativeperiod,regardedasafunctionof
u. Tocalculatethe relativeperiods,[21]proposedaproceduresimilarto theGriffith-Dwork
reduction. In [6], an enlarged polytope is proposed to encode both the geometry of the
Calabi-YauX andthe B-branegeometry. This givesrise to a GKZ hypergeometricsystem
z
fortherelativeperiods,andaspecialsolutionatacriticalpointinuthenleadstoasolution
to the original inhomogeneous Picard-Fuchs equation.
Ourgoalinthispaperistofurtherdevelopthemathematicalstructuresunderlyinginho-
mogeneousPicard-FuchsequationsandtheAbel-Jacobimap,andtoclarifytherelationships
between the three approaches mentioned above. Here is an outline. We begin, in section 2,
withadescriptionofaresidueformalismforrelativecohomologyofafamilyofpairs(X ,Y ),
z z
including anumberofvariationalformulasonthe localsystemHn(X ,Y ). Insection3,we
z z
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 the β-term than the Griffith-Dwork reduction.
In section 4, we prove a purely algebraic a double residue formula for the inhomogeneous
termofthePicard-FuchsequationthatgovernstheAbel-Jacobimap. Thisuniformapproach
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 relationshipbetween relative periods and the Abel-Jacobi map. We also give
a uniform descriptionfor the enhancedpolytope method for describing toricB-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.wouldliketothankJ.Walcherformanystimulatingdiscussions,
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
2 Variation of Relative Cohomology
Local System of Relative Cohomology and Gauss-Manin Connection
Let π : S be a smooth family of n-dimensional projective varieties, and S be a
X → Y →
family of smooth subvariety . Let s S be a closed point, and denote by X ,Y the
s s
Y ⊂X ∈
corresponding fiber over s. Consider the family of relative cohomology class
Hn(X ,Y )
s s
given by the cohomology of the complex of pairs:
Γ(Ωn(X )) Γ(Ωn−1(Y ))
s s
⊕
with the differential
d(α,β)=(dα,α dβ) (2.1)
|Ys −
Here Ωn(X ) and Ωn−1(Y ) are sheavesofDe Rham differentialn-forms onX and (n 1)-
s s s
−
form on Y , and Γ is the smooth global section. Therefore an element of Hn(X ,Y ) is
s s s
represented by a differential n-form on X whose restriction to Y is specified by an exact
s s
form.
Lemma 2.1. Hn(X ,Y ) forms a local system on S.
s s
Proof. Theproofissimilartothecasewithout bychoosingalocaltrivializationof S
Y X →
which also trivializes S. See e.g.[30].
Y →
We denote this local system by n , and let GM be the Gauss-Manin connection.
H(X,Y) ∇
There’s a well-defined natural pairing
H (X ,Y ) Hn(X ,Y ) C
n s s ⊗ s s → (2.2)
Γ (α,β) <Γ,(α,β)> α β.
⊗ 7→ ≡ Γ − ∂Γ
R R
Given a family (α ,β ) Hn(X ,Y ) varying smoothly, which gives a smooth section of
s s s s
∈
n denoted by [(α ,β )], and Γ H (X ,Y ) a smooth family of relative cycles, we
H(X,Y) s s s ∈ n s s
get a function on S given by the pairing
<Γ ,(α ,β )>= α β
s s s s s
−
ZΓs Z∂Γs
Let v be a vector field on S. We consider the variation
<Γ ,(α ,β )>
v s s s
L
where is the Lie derivative with respect to v. Suppose we have a lifting α˜,β˜, which are
v
L
differential forms on , respectively, such that
X Y
α˜ =α , β˜ =β
|Xs s |Ys s
and that Γ moves smoothly to form a cycle Γ˜ on :
s
X
Γ =Γ˜ X , ∂Γ .
s s
∩ ⊂Y
Let v˜ be a lifting of v on , v˜ be a lifting of v on .
X Y
X Y
3
Proposition 2.2 (Variation Formula).
<Γ ,(α ,β )>=<Γ ,(ι ydα˜,ι y(dβ˜ α˜))> (2.3)
Lv s s s s v˜X v˜Y −
where ι is the contraction with v˜ , and similarly for ι .
v˜X X v˜Y
Proof. We fix a point s S, andlet σ(t) be alocalintegralcurve ofv suchthat σ(0)=s .
0 0
Let Γ˜ denote the one-∈dimensional family of cycles over σ(s), 0 s t, and we denote
σ(t)
≤ ≤
byΓ the cycleoverthe pointσ(t). Alsolet∂Γ be thefamilyofboundarycycle∂Γ over
t σ(t) s
σ(s),0 s t. Then we have
≤ ≤
f
∂ Γ˜ =Γ Γ ∂Γ (2.4)
σ(t) t 0 σ(t)
− −
(cid:16) (cid:17)
therefore f
dα˜ = α˜ α˜ α˜
ZΓ˜σ(t) ZΓt −ZΓ0 −Z∂Γσ(t)
Similarly f
dβ˜= β˜ β˜
−
Z∂Γ Z∂Γt Z∂Γ0
Taking the derivative with respefct to t, we get
∂
α˜ β˜ = ι ydα˜+ ι y α˜ dβ˜
∂t − v˜X v˜Y −
(cid:18)ZΓt Z∂Γt (cid:19) ZΓt Z∂Γt (cid:16) (cid:17)
The proposition follows.
Note that (ι ydα˜,ι y(dβ˜ α˜)) is nothing but the Gauss-Manin connection
v˜X v˜Y −
GM[(α ,β )]= (ι ydα˜) ,(ι y(dβ˜ α˜)) (2.5)
∇v s s v˜X |Xs v˜Y − |Ys
h(cid:16) (cid:17)i
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˜ ,v˜ , and that the connection is flat. The following
X Y
corollary also follows from (2.5).
Corollary 2.3.
<Γ ,(α ,β )>=<Γ , GM[(α ,β )]> (2.6)
Lv s s s s ∇v s s
Residue Formalism for Relative Cohomology
In this section, we assume that X moves as a family of hypersurfaces in a fixed n+1-dim
z
ambient projective space M with defining equation P = 0. Here P H0(M,[D]) for a
z z
∈
fixed divisor class [D], and z is holomorphic coordinate on S parametrizing the family. Let
ω H0(M,K (X ))
z M z
∈
be a rational (n+1,0)-form on M with pole of order one along X , then we get a famliy of
z
holomorphic (n,0)-form on X given by
z
Res ω H0(X ,K )
Xz z ∈ z Xz
4
and also a family of relative cohomology classes
(Res ω ,0) Hn(X ,Y )
Xz z ∈ z z
Here our convention for Res is that if X is locally given by w = 0, and ω = dw φ,
z z w ∧
where φ is locally a smooth form, then Res ω = φ . Note that the map Res :
Xz z |Xz Xz
H0(M,K (X )) Hn,0(X ) is at the level of forms, not only as cohomology classes since
M z z
→
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
orderofthepoleinconsideringperiodsofrelativecohomologybecauseoftheboundaryterm.
We choose a fixed open cover U of M and a partition of unity ρ subordinate to
α α
{ } { }
it. Let P =0 be the defining equation of X on U . Then we can write
z,α s α
d P
M z,α
ω = φ (2.7)
z z,α
P ∧
z,α
α
X
where φ is a smooth (n,0)-formwith supp(φ ) U . On the trivialfamily M S, we
z,α z,α α
⊂ ×
will use d to denote the differential along M only and use d to denote the differential on
M
the total space. Let
φ = φ (2.8)
z z,α
α
X
Then φ is a smooth form on M S such that
z
×
φ =Res ω
z|Xz Xz z
Consider the variation
∂ d P ∂ P
M z,α z z,α
ω = d ∂ log(P )φ + ∂ φ d φ
z M z z,α z,α z z,α M z,α
∂z α ! α (cid:18) Pz,α ∧ − Pz,α (cid:19)
X X
Let
∂
∂˜ = +n
zX ∂z z,X
be a lifting of ∂ to , where n Γ(T ) is along the fiber, which is a normal vector
∂z X z,X ∈ M|Xz
field corresponding to the deformation of X in M with respect to z. Then in each U , we
z α
have
ι yd P = ∂ P (2.9)
nXz M z,α |Xz − z z,α|Xz
It follows easily that (cid:0) (cid:1)
ι ydφ =Res (∂ ω d (∂ log(P )φ )) (2.10)
∂˜zX z|Xz Xz z z− M z z z
Note that since the transition function of [D] is independent of z, ∂ log(P ) is globally
z z
well-defined. In general, ∂ ω will have a pole of order two along X , but the substraction
z z z
ofd (∂ log(P )φ ) makesit logarithmicalongX , hence the residueaboveis well-defined.
M z z z z
Next we choose arbitrary lifting of ∂ to , and write it as
∂z Y
∂
∂˜ = +n
z,Y z,Y
∂z
where n Γ(T ) is along the fiber, which is a normal vector field corresponding to
z,X ∈ M|Yz
the deformation of Y in M with respect to z. Then the variation formula implies that
z
5
Proposition 2.4 (Residue Variation Formula).
GM(Res ω ,0)= Res (∂ ω d (∂ log(P )φ )), ι yφ (2.11)
∇∂z Xz z Xz z z− M z z z − nz,Y z
(cid:0) (cid:1)
To see the effect of the second component on the right side, let us assume that Y =
z
X H, where H is a fixed hypersurface in M with defining equation Q=0. Then we can
z
∩
choose a ǫ-tube T (Γ) [14] of Γ H (X ,Y ) with ∂T (Γ) H. Hence
ǫ n z z ǫ
∈ ⊂
1 1
d (∂ log(P )φ ) = (∂ log(P )φ )
M z z z z z z H
2πi − 2πi |
ZTǫ(Γ) ZTǫ(∂Γ)
= Res (∂ log(P )φ )
Yz z z z |H
Z∂Γ
= ι yφ
− nz,Y z
Z∂Γ
which cancels exactly the second component on the right side of (2.11). Therefore,
1
∂ <Γ,(Res ω ,0)>= ∂ ω
z Xz z 2πi z z
ZTǫ(Γ)
We can localize the above observation and consider the following situation: on each
U , suppose we can choose Q independent of z such that P ,Q are transversal and
α α z,α α
Y U Q = 0,P = 0 . Suppose we have a relative cycle Γ H (X ,Y ) where we
z α α z,α n z z
∩ ⊂ { } ∈
can choose a ǫ-tube T (Γ) such that ∂T (Γ) U lies in Q =0 . We have the pairing
ǫ ǫ α α
∩ { }
1
<Γ,(Res ω ,0)>= lim ω
Xz z ǫ→02πiZTǫ(Γ) z
However,the righthandside doesn’t depend onǫ. In fact, let Tǫ(Γ) be a solidannulus over
δ
Γ. By Stokes’s theorem, we have
ω ω = ω
z z z
−
ZTǫ(Γ) ZTσ(Γ) Z∂Tδǫ(Γ)
since∂Tǫ(Γ) U Q =0 foreachα,theaboveintegralvanishes. Thereforetheintegral
δ ∩ α ⊂{ α }
ω doesn’t depend on the position of the ǫ-tube if we impose the boundary condition
Tǫ(Γ) z
as above. It follows immediately that
R
1
(∂ )k <Γ,(Res ω ,0)>= (∂ )kω
z Xz z 2πi z z
ZTǫ(Γ)
where ∂T (Γ) U lies in Q =0 . Applying this to a Picard-Fuchs operator, we get
ǫ α α
∩ { }
Proposition 2.5. [Inhomogeneous Picard-Fuchs Equation] Let = ( GM) be a Picard-
D D ∇∂z
Fuchs operator, i.e.
(∂ )ω = dβ (2.12)
z z z
D −
for somerational (n 1,0)-formβ withpoles along X . Then undertheabove local choices,
z z
−
we have
1
(∂ )<Γ,(Res ω ,0)>= β . (2.13)
D z Xz z 2πi z
ZTǫ(∂Γ)
6
In the next section, we will derive a general formula for β using toric method.
z
More generally, suppose that Q depends on z which is denoted by Q , and we put
α z,α
{ }
our ǫ-tube T (Γ ) inside Q =0 on U . Then
ǫ z z,α α
{ }
1 1
Res ω = ω = ρ ω
Xz z 2πi z 2πi α z
ZΓz ZTǫ(Γz) α ZTǫ(Γz)
X
andthe integrationdoesn’tdependonǫassumingthe boundaryconditionas above. Apply-
ing the variation formula (2.3) we get
∂ 1 1 1
ρ ω = ρ ∂ ω ρ ι yω
∂z 2πi α z 2πi α z z− 2πi α nz,Qα z |Qz,α=0
α ZTǫ(Γz) α ZTǫ(Γz) α ZTǫ(∂Γz)
X X X (cid:0) (cid:1)
wheren isthenormalvectorfieldcorrespondingtothedeformationof Q =0 inside
M. In pza,Qrtαicular, we have ι ydQ = ∂ Q . Hen{cez,α }
nz,Qα z,α|{Qz,α=0} − z z,α|{Qz,α=0}
ι yω = Res (∂ log(Q )ω )
nz,Qα z |Qz,α=0 − Qz,α=0 z z,α z
(cid:0) (cid:1)
Putting together the last three equations, we arrive at
Proposition 2.6 (cf. [32]).
∂k 1 k ∂k−l 1 ∂l−1
Res ω = ∂kω + ρ Res ∂ log(Q ) ω
∂zk Xz z 2πi z z ∂zk−l 2πi α Qz,α=0 z z,α ∂zl−1 z
(cid:18)ZΓz (cid:19) ZTǫ(Γz) l=1 α ZTǫ(∂Γz) (cid:18) (cid:19)
X X
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 the β-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(w ,w ,w ,w ,w ). We assume that w = 1 and it’s of Fermat-type, i.e.,
1 2 3 4 5 5
w dforeachi,whered=w +w +w +w +w . There’sassociated4-dimensionalintegral
i 1 2 3 4 5
|
convex polyhedron given by the convex hull of the integral vectors
5
∆= (x , ,x ) R5 w x =0,x 1
1 5 i i i
( ··· ∈ | ≥− )
i=1
X
If we choose the basis e =(1,0,0,0, w ),i=1..4 , then the vertices is given by
i i
{ − }
d
∆: v =( 1, 1, 1, 1)
1
w − − − −
1
d
v =( 1, 1, 1, 1)
2
− w − − −
2
d
v =( 1, 1, 1, 1)
3
− − w − −
3
d
v =( 1, 1, 1, 1)
4
− − − w −
4
v =( 1, 1, 1, 1)
5
− − − −
7
and the vertices of its dual polytope is given by
∆∗ : v∗ =(1,0,0,0)
1
v∗ =(0,1,0,0)
2
v∗ =(0,0,1,0)
3
v∗ =(0,0,0,1)
4
v∗ =( w , w , w , w )
5 − 1 − 2 − 3 − 4
Let v∗,k =0,1,2,.., be integral points of ∆∗, where v∗ =(0,0,0,0). We write
k 0
f∆∗(x)= akXvk∗ (3.1)
v∗∈∆∗
kX
whichisthedefiningequationforourCalabi-Yauhypersurfacesintheanti-canonicaldivisor
class. Here X = {X1,X2,X3,X4} is the toric coordinate, Xvk∗ = 4 Xjvk∗,j. If we use
j=1
homogeneous coordinate [10] z ,1 ρ 5 corresponding to the onQe-dim cone v ,1
ρ ρ
{ ≤ ≤ } { ≤
ρ 5 , then the toric coordinate can be written by homogeneous coordinate
≤ }
zd/wj
X = j , j =1,2,3,4 (3.2)
j 5
z
ρ
ρ=1
Q
The relevant rational form with pole of order one along the hypersurface is given by
5
w
i 4
1 dX
Π(a) = i=1 j
Qd3 vk∗∈∆∗akXvk∗ jY=1 Xj
P Ω0
=
a 5 z<vk∗,vρ>+1
k ρ
v∗∈∆∗ ρ=1
k
P Q
Ω
0
= (3.3)
a 5 z + 5 a zd/wρ+ a 5 z<vk∗,vρ>+1
0 ρ ρ ρ k ρ
ρ=1 ρ=1 v∗∈∆∗,k>5 ρ=1
k
Q P P Q
where Ω = 5 ( 1)ρ−1w z dz dˆz dz . Define the relation lattice by
0 ρ=1 − ρ ρ 1∧··· ρ∧··· 5
L=Pl=(l ,l ,...) Z|∆∗|+1 l v¯∗ =0 , where v¯∗ =(1,v∗), v∗ ∆∗
{ 0 1 ∈ | i i } i i i ∈
i
X
The moduli variable associated with the choice of a basis l(k) for L is given by [16]
{ }
xk =( 1)l0(k)al(k)
−
The key idea here is to consider the following 1-parameter family of automorphisms
φt :zρ (a0)wdρtzρ, 1 ρ 5.
→ a ≤ ≤
ρ
Put
Π (a)=φ∗Π(a).
t t
8
It satisfies the differential equation
∂ (φ∗Π(a))= φ∗Π(a)
t t LV t
5
whereV = wρ(log a0)z ∂ isthegeneratingvectorfieldforφ , istheLiederivative.
ρ=1 d aρ ρ∂zρ t LV
This is solvePd by
φ∗Π(a)=etLVΠ(a) (3.4)
t
Define
Π˜(x)=a Π (a) (3.5)
0 1
Π˜(x) is a function of x only. Indeed,
k
{ }
Ω
Π˜(x)= 0
ρ=51zρ+ ρ=51(aaρ0)wρ/d!ρ=51zρd/wρ +v∗∈∆∗,k>5aak0 ρ=51 aaρ0 wdρ<vk∗,vρ>ρ=51zρ<vk∗,vρ>+1
k (cid:16) (cid:17)
Q Q P P Q Q
Since we have
5 5
w
w v =0, v∗ = ρ <v∗,v >v∗
ρ ρ k d k ρ ρ
ρ=1 ρ=1
X X
we see that both ρ=51(aaρ0)wρ/d! and aak0 ρ=51 aaρ0 wdρ<vk∗,vρ> can be written in terms of xk’s
(cid:16) (cid:17)
as an algebraic funcQtion. Q
Given an integral point l L, consider the GKZ operator (it differs from the standard
GKZ operator by a factor of a∈ ali)
0 li>0 i
∂ li Q ∂ −li
D = a ali al a−li
l 0(lYi>0 i (cid:18)∂ai(cid:19) − lYi<0 i (cid:18)∂ai(cid:19) )
l0 ∂ li−1 ∂ −l0 ∂ −li−1 ∂
= (a j) (a j) al (a j) (a j) a
 0∂a − i∂a − − 0∂a − i∂a −  0
0 i 0 i
jY=1 i6=0Y,li>0jY=0 jY=1 i6=0Y,li<0 jY=0 
= D˜ a
l 0 
m
where we use the convention that ( )=1 if m 0. From
··· ≤
i=1
Q
D Π(a)=0
l
We get
eLVD˜ e−LVΠ˜(x)=0 (3.6)
l
Lemma 3.1.
∂ ∂
eLV(ai∂ai)e−LV =ai∂ai +δ1≤i≤5Lwdizi∂∂zi −δi,0Lρ=51wdρzρ∂∂zρ
P
here δ =1 if 1 i 5 and otherwise 0.
1≤i≤5
≤ ≤
9
Proof. Since
5
∂ w ∂ w ∂
i ρ
V,a = δ z δ z
i 1≤i≤5 i i,0 ρ
a d ∂z − d ∂z
(cid:20) i(cid:21) i ρ=1 ρ
X
∂
V, V,a = 0
i
a
(cid:20) (cid:20) i(cid:21)(cid:21)
The lemma follows from the formula
∞
∂ ∂ 1
eLV(a )e−LV =a +
i∂ai i∂ai k=1k!L(adV)k ai∂∂ai
X “ ”
It follows from the lemma that
l0 ∂ li−1 ∂
jY=1(a0∂a0 −j) i6=0Q,li>0 jY=0(ai∂ai −j+δ1≤i≤5Lwdizi∂∂zi)
−l0 ∂ −li−1 ∂
−aljY=1(a0∂a0 −j) i6=0Q,li<0 jY=0 (ai∂ai −j+δ1≤i≤5Lwdizi∂∂zi)Π˜(x)=0 (3.7)
where we have used Π˜(x)=0 to eliminate the terms with  .
Lρ=51wdρzρ∂∂zρ Lρ=51wdρzρ∂∂zρ
Observe that each LPie derivative Lwdizi∂∂zi commutes with all other opPerators appearing
on the left side of (3.7). So we can move every term involving Lwdizi∂∂zi to the rightside, so
that (3.7) can now be explicitly written as
5
D˜lΠ˜(x)=− Lwdizi∂∂ziαi
i=1
X
where the α are (easily computable) d-closed 4-forms depending on l. By the Cartan-Lie
i
formula =dι +ι d, we obtain the formula
X X X
L
Proposition 3.2. [β-term Formula]
D˜ Π˜(x)= dβ
l l
−
where βl = iιwdizi∂∂ziαi.
P
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∗ R(∆∗), <v∗,v >= 1, <v∗,v > 0 for ρ=ρ
i ∈ i ρi − i ρ ≥ 6 i
then we obtain an equation
∂ 1 1
vk∗X∈∆∗(<vk∗,vρi >+1)avk∗∂avk∗+vi∗a0e−LVΠ˜(x)=L(ρ=51zρ<vi∗,vρ>)zρi∂z∂ρi a0e−LVΠ˜(x)
Q
 
10