GKZ Hypergeometric Series for the Hesse Pencil, Chain Integrals and Orbifold Singularities
The GKZ system for the Hesse pencil of elliptic curves has more solutions than the period integrals. In this work we give different realizations and interpretations of the extra solution, in terms of oscillating integral, Eichler integral, chain integral on the elliptic curve, limit of a period of a...
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Інститут математики НАН України
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| Цитувати: | GKZ Hypergeometric Series for the Hesse Pencil, Chain Integrals and Orbifold Singularities / J. Zhou // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 46 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859802640685203456 |
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| author | Zhou, J. |
| author_facet | Zhou, J. |
| citation_txt | GKZ Hypergeometric Series for the Hesse Pencil, Chain Integrals and Orbifold Singularities / J. Zhou // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 46 назв. — англ. |
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| description | The GKZ system for the Hesse pencil of elliptic curves has more solutions than the period integrals. In this work we give different realizations and interpretations of the extra solution, in terms of oscillating integral, Eichler integral, chain integral on the elliptic curve, limit of a period of a certain compact Calabi-Yau threefold geometry, etc. We also highlight the role played by the orbifold singularity on the moduli space and its relation to the GKZ system.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 030, 32 pages
GKZ Hypergeometric Series for the Hesse Pencil,
Chain Integrals and Orbifold Singularities
Jie ZHOU
Perimeter Institute for Theoretical Physics, Waterloo, Ontario N2L 2Y5, Canada
E-mail: jzhou@perimeterinstitute.ca
Received October 01, 2016, in final form May 14, 2017; Published online May 20, 2017
https://doi.org/10.3842/SIGMA.2017.030
Abstract. The GKZ system for the Hesse pencil of elliptic curves has more solutions than
the period integrals. In this work we give different realizations and interpretations of the
extra solution, in terms of oscillating integral, Eichler integral, chain integral on the elliptic
curve, limit of a period of a certain compact Calabi–Yau threefold geometry, etc. We also
highlight the role played by the orbifold singularity on the moduli space and its relation to
the GKZ system.
Key words: GKZ system; chain integral; orbifold singularity; Hesse pencil
2010 Mathematics Subject Classification: 14J33; 14Q05; 30F30; 34M35
1 Introduction
The GKZ system [19, 20, 21] provides, among many things, a useful tool in computing the
Picard–Fuchs system for families of projective varieties. In the literature, the differential equa-
tions obtained by the GKZ system usually factor and the Picard–Fuchs system is given by the
subsystem formed by a subset of these factors. It is then natural to ask what is the reason for
the factorization, and what are the geometric objects that underly the extra solutions besides
the period integrals, which are integrals over the cycles in the fibers of the family.
1.1 GKZ system for the Hesse pencil
A large part of the discussions below can be extended to slightly more general families of Calabi–
Yau varieties, among which the Calabi–Yau hypersurfaces in toric varieties will be of particular
interest due to their appearances in mirror symmetry. For concreteness, in the present work we
shall focus on the Hesse pencil of elliptic curves as an example.
The equation of the Hesse pencil χ : E → B is given by
E :
{
F (x, ψ) := x3 + y3 + z3 − 3ψxyz = 0
}
⊆ P2 × B,
here the base B is a copy of P1 parametrized by ψ.
To define period integrals, one needs to specify a local, holomorphic section of the Hodge
line bundle
L = R0χ∗Ω
1
X |B → B.
Then one can integrate the corresponding family of holomorphic top forms over the locally
constant sections of a rank 2 local system, which is dual to R1χ∗Z→ B, to get period integrals.
This paper is a contribution to the Special Issue on Modular Forms and String Theory in honor of Noriko
Yui. The full collection is available at http://www.emis.de/journals/SIGMA/modular-forms.html
mailto:jzhou@perimeterinstitute.ca
https://doi.org/10.3842/SIGMA.2017.030
http://www.emis.de/journals/SIGMA/modular-forms.html
2 J. Zhou
A canonical choice for the local section is given by
Ω(ψ) = Res
ψµ0
F (x, ψ)
, µ0 := zdx ∧ dy + xdy ∧ dz + ydz ∧ dx. (1.1)
On each fiber Eψ of the family χ, the 2-form µ0/F gives a meromorphic 2-form on the ambient
space P2 with a pole of order one along Eψ and its residue gives a holomorphic top form on
this elliptic curve fiber Eψ. The integrals of this choice of holomorphic section, over a further
choice of the locally constant sections A,B of the above-mentioned rank 2 local system, gives
the period integrals
πA(ψ) =
∫
A
Ω(ψ), πB(ψ) =
∫
B
Ω(ψ).
The Picard–Fuchs equation can be derived, for example, by computing the Gauss–Manin
connection or by using the Griffiths–Dwork method. With respect to the choice Ω given above,
the differential operator annihilating the period integrals is given by
LPF = (θψ − 2)(θψ − 1)− ψ3θ2ψ, θψ := ψ
∂
∂ψ
. (1.2)
Henceforward we shall frequently use the θ-operator defined as above.
The details of the derivation for the GKZ system will be useful later in this work, so we recall
them here following [19].
We first extend the family a little by rewriting the equation for (the total space) of the family
as
F (x,a) := a1x
3 + a2y
3 + a3z
3 + a0xyz = 0.
Then, one considers the actions on the polynomial F (x,a) which belong to the diagonal
scalings inside the group GLx ×GLa and hence preserve the fibration structure. Here by GLx
we mean the affine transformations on C3 parametrized by x = {x, y, z} space and similarly
for GLa. Those which fixes F up to an overall scaling forms a subgroup G. By construction,
for any element in G, the scaling on a is determined by that on x. We can then choose the
generators of G to be
(x1, x2, x3; a1, a2, a3, a0) 7→
(
λx1, x2, x3;λ
−3a1, a2, a3, λ
−1a0
)
, λ ∈ C∗,
(x1, x2, x3; a1, a2, a3, a0) 7→
(
x1, λx2, x3; a1, λ
−3a2, a3, λ
−1a0
)
, λ ∈ C∗,
(x1, x2, x3; a1, a2, a3, a0) 7→
(
x1, x2, λx3; a1, a2, λ
−3a3, λ
−1a0
)
, λ ∈ C∗,
(x1, x2, x3; a1, a2, a3, a0) 7→
(
x1, x2, x3;λa1, λa2, λa3, λa0
)
, λ ∈ C∗. (1.3)
The former three also scale the meromorphic 2-form µ0 by λ, while the latter acts trivially.
Hence when acting on the 2-form
ω(a) :=
µ0
F (x,a)
, (1.4)
the infinitesimal versions of these group transformations give rise to the following annihilating
differential operators called Euler homogeneity operators,
Zi = θxi − 3θai − θa0 − degxi µ0, i = 1, 2, 3,
Z0 =
3∑
i=1
θai + θa0 − (−1).
GKZ Hypergeometric Series for the Hesse Pencil 3
Here degxi µ0 stands for the weight of µ0 under the action xi 7→ λxi, which is one in the current
case.
Now the monomials in the pencil parametrized by ai, i = 0, 1, 2, 3 satisfy the relation
x31 · x32 · x33 = (x1x2x3)
3.
This then gives the following differential operator that annihilates ω(a)
DGKZ =
3∏
i=1
∂ai − ∂3a0 .
The projection1, which we denoted by χ∗, of the above differential operators to be base
direction yields
(χ∗Zi)ω = 0, i = 0, 1, 2, 3, (χ∗DGKZ)ω = 0.
One then translates these differential equations to the ones satisfied by the period integrals
πγ(a) =
∫
γ Ω with respect to the holomorphic top forms Ω = a0ω(
3θai + θa0 + degxi −1
)
Ω = 0, i = 1, 2, 3,(
3∑
i=1
θai + θa0
)
Ω,(
1∏
ai
∏
θai −
1
a30
(θa0 − 3)(θa0 − 2)(θa0 − 1)
)
Ω = 0. (1.5)
In the present case, degxi µ0 = 1, i = 1, 2, 3. The former two equations allow one to make the
ansatz
πγ(a) = πγ(z), z = −a1a2a3
a30
. (1.6)
Now when acting on a function of z, the last differential equation gives
DGKZ πγ =
(
θ3z + z(−3θz − 3)(−3θz − 2)(−3θz − 1)
)
πγ = 0. (1.7)
Specializing to a1 = a2 = a3 = 1, a0 = −3ψ, by disregarding the overall constants which are
irrelevant throughout the discussions, we can see that (recall (1.2))
DGKZ =
(
θ3ψ − ψ−3(θψ − 3)(θψ − 2)(θψ − 1)
)
= θψ ◦
(
θ2ψ − ψ−3(θψ − 2)(θψ − 1)
)
= θψ ◦ ψ−3 ◦ LPF. (1.8)
When writing the Picard–Fuchs operator in terms of the α = ψ−3 coordinate, we use the
following normalization of the leading coefficient
L̃PF =
(
θ2α − α
(
θα +
1
3
)(
θα +
2
3
))
, (1.9)
so that up to a constant multiple we have D̃GKZ = θα ◦ L̃PF.
We remark that for families of hypersurface Calabi–Yau varieties in toric varieties in any
dimension, similar discussions apply. In particular, one always obtains LPF from the factor in
the rightmost as in (1.8).
1A more intrinsic description can be given by the D-module language.
4 J. Zhou
1.2 Calabi–Yau condition and factorization of differential operator
By examining the derivation, a few observations are in order. First, the first equation in (1.5)
is consistent with the second if and only if the following condition holds
3∑
i=1
degxi µ0 = degF. (1.10)
We call this the Calabi–Yau condition since the meromorphic form µ0 is a section ofOP2(−(2+1))
and the degree of the polynomial F matches with the degree of µ0 exactly when F = 0 defines
a Calabi–Yau hypersurface in the projective space P2.
Now instead of making the ansatz mentioned before in (1.6), one can eliminate the differential
operators θai , i = 1, 2, 3 by solving them from the Zi-operators in (1.5). Then the D-operator
in (1.5) becomes(
1∏
ai
∏
i
θa0 + degxi µ0 − 1
3
− 1
a30
(θa0 − 3)(θa0 − 2)(θa0 − 1)
)
.
This differential operator factors in the desired way when the set {degxi µ0 − 1, i = 1, 2, 3}
has a non-empty intersection with the set {0, 1, 2}. Again this is trivially true in the Calabi–Yau
case. For accuracy, we shall call it the right factorization to indicate that the Picard–Fuchs
operator is factored out from the right. This factorization is the reason that the GKZ system
gives an in-homogeneous Picard–Fuchs system.
There are natural situations where the integrand is replaced by other differential forms with
different scaling behaviors under the action of G. For example, the polynomial F could be
replaced by a Laurent polynomial, or the integrand by the multi-Mellin transform or the Mähler
measure. These situations occur in local Calabi–Yau mirror symmetry [12, 23, 38, 41] and in
scattering amplitudes [7]. For these cases, the above procedure of deriving differential equations
from GKZ symmetries still applies.
Also for other integrands, the factorization, if exists, might be different. Of direct relevance
to the GKZ system of the Hesse pencil is the GKZ system for the mirror geometry of KP2 ,
see [12, 24]. The integrand is given by
1
X1X2
(
a0 + a1X1 + a2X2 + a3X
−1
1 X−12
)
+ uv
dX1dX2
X1X2
dudv, (1.11)
where (X1, X2) are coordinates on the space (C∗)2 and u, v are valued in C. It is annihilated
by
LCY3 = LPF ◦ θψ, ψ−3 = −27
a1a2a2
a30
. (1.12)
The combinatorial data (which is conveniently encoded in the Newton polytope or toric
geometry) for its Picard–Fuchs system is identical to that of the Hesse pencil, only the scaling
behavior under the symmetries in (1.3) of the integrand is different.
1.3 Motivation of the work
From the factorization in (1.8), one can see that besides the period integrals, the GKZ sys-
tem DGKZ in (1.7) has one more extra solution. One of the goals of the present work is to
understand this extra solution.
We also aim to understand the difference and relation between the factorizations (1.8), (1.12)
of the operators involved in the Hesse pencil and in the mirror geometry of KP2 , respectively.
GKZ Hypergeometric Series for the Hesse Pencil 5
That there should be such a connection is predicted by the Landau–Ginzburg/Calabi–Yau cor-
respondence [43]. To be a little more precise, the solutions to the GKZ system were studied
in [3] (see also [9]) and were identified with oscillating integrals. Hence one would expect them
to appear in certain form from the perspective of the elliptic curve geometry by the Landau–
Ginzburg/Calabi–Yau correspondence.
In fact, the studies in [15, 16] imply that the extra solution to the GKZ system for the
Weierstrass family can be identified with certain chain integral on the elliptic curve. As will
be discussed in the present work, the chain therein is closely related to the symmetries of the
Weierstrass polynomial and to certain oscillating integral. Further evidences also include some
recent works [29, 40] which suggest that part of the information encoded in the Landau–Ginzburg
model should be visible in the Calabi–Yau model through the symmetries of the latter.
Finding a direct relation between the oscillating integrals in the singularity theory and (inte-
grals of) chains living on the elliptic curves will provide a first step towards a more conceptual
understanding of the LG/CY correspondence.
Relation to previous works
The explicit chain integral solutions to the GKZ system for hypersurface families were studied
in [3]. More general discussions in terms of chain integrals and D-modules were provided in
the beautiful works [6, 26, 27]. Similar examples were discussed in [7] in terms of mixed Hodge
structures. These works treat the extra solution to the GKZ system as a two dimensional integral
living in the ambient projective space or its blow-up. One of the main differences between the
current work and the above-mentioned ones is that we give a direct realization of the extra
solution in terms of chain integral living on the elliptic curve instead of in the ambient space.
The present paper also contains several observations offering connections between the extra
solution to the GKZ system and some geometric objects that are of interest in mirror symmetry.
A large part of the results obtained in this work have scattered in the literature but mainly
at the level of sketchy justifications, our new addition on this part is then to make them more
clear.
Outline of the paper
In Section 2 we review the known results on the realizations of the solutions to the GKZ system in
terms of 3-dimensional oscillating integrals and 2-dimensional chain integrals. We also interpret
these integrals in terms of ones living in a non-compact Calabi–Yau variety, to incorporate the
GKZ symmetries.
Section 3 discusses the realization of the solutions to the GKZ system of the Hesse pencil
in terms of objects living on the elliptic curves. First we use the Wronskian method to obtain
the Eichler integral formula for the solutions. Then we express them in terms of the Beltrami
differential and cycles with vanishing period integrals. We also construct chains on the elliptic
curves which give rise to the extra solution besides the period integrals.
In Section 4 we embed both the mirror of KP2 and the elliptic curves in the Hesse pencil into
some compact Calabi–Yau threefold and offer a connection between the Picard–Fuchs system
of the former and the GKZ system of the latter.
We conclude in Section 5 with some discussions and speculations.
2 Invariant 3d and 2d chain integrals under GKZ symmetries
The GKZ symmetries are symmetries of the polynomials F (x,a), not just the varieties they
define. Also the symmetries are for the forms instead of cohomology classes, as opposed to
6 J. Zhou
the case of the Picard–Fuchs operator derived from the Gauss–Manin connection. Hence any
invariant under these symmetries will provide a solution to the resulting differential equations.
Recall that in the above when discussing the invariance of the integrals πγ in (1.6) under
the GKZ symmetries, we used the fact that the (classes of) the cycles γ are invariant under the
scalings in (1.3). In general, chain integrals would not satisfy the differential equations, except
when they are indeed invariant under the scalings. This will be the case when they are chains
cut out by coordinate planes. This again opens the possibility that certain chain integrals could
solve the GKZ system and provide extra solutions other than the cycle integrals, namely the
period integrals.
2.1 Invariant chain integrals as solutions to GKZ system
Now we consider the so-called V-chain, see [3] and references therein, given by
D3 =
{
(x, y, z) ∈ C3 |x, y, z ≥ 0
} ∼= R3
≥0. (2.1)
It is indeed invariant under the transformations in (1.3). Here we have used the coordinates x,
y, z in place of x1, x2, x3, as we shall occasionally do throughout the work.
By applying a coordinate change, we can arrange such that ai = 1, i = 1, 2, 3, a0 = −3ψ.
Now we assume that the following condition
<ψ ≤ 0, such that <F (x, y, z;ψ) ≥ 0 on D3. (2.2)
The meaning of this condition will be discussed later in Remark 2.1. Hence the convergence of
the integral on D3 is ensured. We can then apply the absolute convergence theorem and write
I(ψ) :=
∫
D3
e−Fψdxdydz
= ψ
∞∑
k=0
∫
D3
e−x
3
e−y
3
e−z
3 (3ψ)n
n!
xnynzndxdydz = ψ
∞∑
n=0
(3ψ)n
n!
1
33
Γ
(
n+ 1
3
)3
. (2.3)
According to the residue of n modulo 3, the integral is the sum of three series∫
D3
e−Fψdxdydz =
2∑
i=0
ψ
∞∑
k=0
(3ψ)3k+i
(3k + i)!
1
33
Γ
(
3k + i+ 1
3
)3
. (2.4)
It breaks into the following three pieces
ψ
∞∑
k=0
(3ψ)3k
(3k)!
1
33
Γ
(
3k + 1
3
)3
=
1
33
·
(
2π · 3− 1
2
Γ(13)2
Γ(23)
)
ψ 2F1
(
1
3
,
1
3
;
2
3
, ψ3
)
,
ψ
∞∑
k=0
(3ψ)3k+1
(3k + 1)!
1
33
Γ
(
3k + 2
3
)3
=
1
33
·
(
2π · 3− 1
2
Γ(23)2
Γ(43)
)
ψ2
2F1
(
2
3
,
2
3
;
4
3
, ψ3
)
,
ψ
∞∑
k=0
(3ψ)3k+2
(3k + 2)!
1
33
Γ
(
3k + 3
3
)3
=
1
33
·
(
2π · 3− 1
2
Γ(1)3
Γ(43)Γ(53)
)
ψ3
3F2
(
1, 1, 1;
4
3
,
5
3
;ψ3
)
.(2.5)
There are other choices for the V -chain. In order for the condition <F > 0 to hold and the
coefficients of x3i , i = 1, 2, 3 to remain, one is led to the following three chains,
D3 := Cx × Cy × Cz = (0,∞)× (0,∞)× (0,∞),
ρD3 := Cx × ρCy × Cz = (0,∞)× (0, ρ∞)× (0,∞),
GKZ Hypergeometric Series for the Hesse Pencil 7
ρ2D3 := Cx × ρ2Cy × Cz = (0,∞)×
(
0, ρ2∞
)
× (0,∞).
Here ρ = exp(2πi3 ). Then the condition in (2.2) becomes
<(ρkψ) ≤ 0, such that <F (x, y, z;ψ) ≥ 0 on ρkD3, k = 0, 1, 2. (2.6)
Remark 2.1 (steepest descent contours). We now make a pause and explain the condition
in (2.6). Note that the condition <ψ ≤ 0 is not necessary in order for the chain integral∫
D3
e−Fdxdydz to be the convergent, nor for the elliptic curve Eψ = {F (x, y, z;ψ) = 0} to have
empty intersection with D3. For the former, due to the coefficients of x3, y3, z3, the integral is
always convergent. For the latter, suppose Eψ ∩D3 6= ∅, then one has ψ ∈ R. It is then easy to
see that this is true if only and if ψ ≥ 1 which is different from the condition in (2.6) as well.
Hence the above condition in (2.6) implies the convergence condition but is stronger. In fact,
any of the integrals obtained by ρkD3 are well-defined for any phase of ψ when ψ is close to 0,
as can be seen from the explicit hypergeometric series expressions above.
For a qualitative analysis it is enough to focus on the y-integral part since the ranges for Cx, Cz
are fixed and are given by the positive real axis. Hence we set x = z = 1. Then it amounts to
study the following type of Airy integral which occur in the study of the A2-singularity theory,∫
Cθ
e−y
3+3ψydy, Cθ = eiθR≥0.
As already mentioned before, the process of deriving equations from symmetries can be
applied to this case. In particular, the chain integrals over ρkCy, k = 0, 1, 2 are called the
so-called Scorer functions, see [39], satisfying certain 3rd order ODE. Now for the integral to be
convergent, the ray needs to sit inside one of the wedges
Wk : − π
6
+ k
2π
3
≤ arg y ≤ π
6
+ k
2π
3
, k = 0, 1, 2.
Small deformations within the wedges do not affect the integral, since the difference would
be the integral over an arc with radius R which tends to zero as R→∞. It is in general not easy
to compute the resulting integral once the chain moves out of the wedges. We hence restrict
ourselves to rays inside the wedges. For the purpose of analyzing the asymptotic behavior of the
integral as ψ → ∞, one deforms the ray into a steepest descent contour. Among the steepest
descent contours of particular importance are the ones passing through the critical points. The
asymptotic expansion of such a contour integral is then completely determined from a small
neighborhood of the critical point.2 The steepest decent contours passing through the critical
points that Cθ can deform to depends on the phase of ψ, resulting in the Stokes phenomenon.
The picture of moving integral contours to determine the asymptotics in Airy integrals also
holds for the Hesse pencil case. The condition (2.6) then indicates the steepest descent contours
that the integral contour in consideration can deform to for the given range of ψ. There are
subtleties however. For example, the singularities of the GKZ system for the Hesse pencil are
all regular and there are additional singularities at ψ3 = 1.
2.1.1 Monodromy action and functional relations
One can also rotate the x, z directions by powers of ρ, but the resulting chains are essentially
equivalent to the aforementioned three by using the actions in (1.3). For example, the chain
2The asymptotic expansion derived from steepest descent method naturally leads to the so-called canonical
coordinate (critical value) in singularity theory. While performing a change of variable w = w(y;ψ), y = y(w;φ)
such that −y(w)2 + 3ψw = −w3 leads to the flat coordinate φ.
8 J. Zhou
ρiCx× ρjCy ×Cz is equivalent to Cx× ρi+jCy ×Cz as far as the integrals are concerned. These
relations are nothing but a manifestation of the invariance of the integral under the action
(x, y, z;ψ) 7→
(
x, λ−1y, z;λψ
)
. (2.7)
But now due to the “gauge fixing condition” ai = 1, i = 1, 2, 3, the values that λ can take
reduce from C∗ to the multiplicative cyclic group µ3. It is easy to see that in order to fix ψ, the
transformation must be of the form3
(x, y, z) 7→
(
ρix, ρjy, ρkz
)
, i+ j + k ≡ 0 mod 3. (2.8)
According to the invariance under (2.7), the integrals over ρkD3 then satisfy the functional
relations
Iρ(ψ) :=
∫
ρD3
e−Fψdxdydz = I(ρψ) = ρJ1(ψ) + ρ2J2(ψ) + J3(ψ),
Iρ2(ψ) :=
∫
ρ2D3
e−Fψdxdydz = I
(
ρ2ψ
)
= ρ2J1(ψ) + ρJ2(ψ) + J3(ψ). (2.9)
On the other hand, the solutions annihilated by LGKZ in (1.8) are easily seen to be
ψ 3F2
(
1
3
,
1
3
,
1
3
;
1
3
,
2
3
;ψ3
)
= ψ 2F1
(
1
3
,
1
3
;
2
3
;ψ3
)
,
ψ2
3F2
(
2
3
,
2
3
,
2
3
;
2
3
,
4
3
;ψ3
)
= ψ2
2F1
(
2
3
,
2
3
;
4
3
;ψ3
)
,
ψ3
3F2
(
1, 1, 1;
4
3
,
5
3
;ψ3
)
. (2.10)
By comparing these with the above three chain integrals I(ψ), Iρ(ψ), Iρ2(ψ), we can see indeed
the 3d chain integrals give the full set of solutions to the GKZ system.
2.1.2 Period integrals as differences of chain integrals
Recall from (1.8) that the period integrals are solutions annihilated by LPF and hence are given
by
π1(ψ) = ψ 2F1
(
1
3
,
1
3
;
2
3
;ψ3
)
, π2(ψ) = ψ2
2F1
(
2
3
,
2
3
;
4
3
;ψ3
)
. (2.11)
These are proportional to the solutions to the GKZ system which are given in the first two
in (2.5). One can also check directly that for the extra solution to LGKZ in (2.10) one has
ψ−3LPF
(
ψ3
3F2
(
1, 1, 1;
4
3
,
5
3
;ψ3
))
=
2
9
.
Remark 2.2. A more convenient choice of basis (for the integrality of the connection matrices)
near this point is given by [17]
π̃1 = −ρ Γ(13)
Γ(23)2
ψ 2F1
(
1
3
,
1
3
;
2
3
;ψ3
)
, π̃2 = ρ2
Γ(−1
3)
Γ(13)2
ψ2
2F1
(
2
3
,
2
3
;
4
3
;ψ3
)
.
3From the perspective of the LG/CY correspondence, these symmetries should be thought of as the symmetries
of the underlying Landau–Ginzburg model defined at the orbifold singularity in the family [43].
GKZ Hypergeometric Series for the Hesse Pencil 9
In terms of the parameter α = ψ−3, the singularities of the Hesse pencil include the cusp
singularities α = 0, 1 and the orbifold singularity α = ∞. The periods corresponding to the
vanishing cycles at ψ−3 = 0, ψ−3 = 1 are given by,
ω0 = 2F1
(
1
3
,
2
3
; 1;ψ−3
)
= π̃1 + π̃2,
ω1 =
i√
3
2F1
(
1
3
,
2
3
; 1; 1− ψ−3
)
=
i√
3
(
−ρπ̃1 + ρ2π̃2
)
. (2.12)
See [40] for a collection of results.
The solutions to the GKZ system are naturally expanded around the orbifold point ψ = 0
in the base B. If we look at the monodromy around this orbifold point, the period integrals
(i.e., solutions to the Picard–Fuchs system) correspond to the first two in (2.5) which have
non-trivial monodromies under the action ψ 7→ e2πiψ. The extra solution J3 is the monodromy
invariant one which is therefore invisible from the Picard–Fuchs equation. Recall that the local
monodromy action near the orbifold point is rooted in the “gauged symmetry” in (2.8) and is
what leads to the functional relations in (2.9). All these suggest that the orbifold singularity
plays a special role and can detect more information about the family other than the vanishing
cycles which are topological. We shall say more about this later in Section 3.
The differences between the 3d chain integrals give rise to cycle integrals
Iρ(ψ)− I(ψ) =
∫
ρD3−D3
e−Fdxdydz = (ρ− 1)J1 +
(
ρ2 − 1
)
J2,
Iρ2(ψ)− Iρ(ψ) =
∫
ρ2D3−ρD3
e−Fdxdydz =
(
ρ2 − ρ
)
J1 +
(
ρ− ρ2
)
J2.
One can also check directly that these cycles correspond to cycles on the elliptic curves without
using the relations to the periods in (2.11). To do this we note that in the common region of ψ
such that for both Iρk1 (ψ) and Iρk2 (ψ), k1 6= k2 mod 3 the condition (2.6) holds, the chains
Cx × ρk1Cy, Cx × ρk2Cy have no intersection with the elliptic curve defined by F = 0. The
difference gives a tubular neighborhood of a certain branch Ck1,k2 of {(x, y, z) ∈ {F = 0} |x ∈
Cx}. Then using the residue calculus, one finds a chain integral on the elliptic curve
Iρk1 (ψ)− Iρk2 (ψ) =
∫
Cx×(ρk1Cy−ρk2Cy)
ψµ0
F
=
∫
Cx
Res
ψµ0
F
∣∣∣
Ck1,k2
, k = 0, 1, 2. (2.13)
See Fig. 1 for an illustration.
It is a classical result, see [2, 14], that the Hesse pencil arise as the equivariant embedding
of elliptic curves with 3-torsion structure to the projective plane via the theta functions. Here
equivariance means that the action of translations by the group E[3] of 3-torsion points (which
are lattice points in 1
3(Z⊕Zτ) if the elliptic curve is realized as E = C/(Z⊕Zτ)) on the curve E
gets mapped to projective transformations on P2. Using the particular choices for the theta
functions in [14], these projective transformations are
σ1 =
1
3
: (x, y, z) 7→
(
x, ρy, ρ2z
)
,
σ2 =
τ
3
: (x, y, z) 7→ (y, z, x). (2.14)
The zeros of the coordinate functions x, y, z correspond to those of the theta functions
used to define the embedding. Hence the difference between any two of them (in particular
the endpoints of Ck1,k2) are nothing but the 3-torsion points on the elliptic curve. Using the
translations in (2.14) which fixes ψ, the chain Ck1,k2 in (2.13) can then produce full cycles.
This shows that the difference of the chain integrals given in (2.13) essentially give the period
integrals.
10 J. Zhou
Cy
ρCy
ρ2Cy
Cx
x
y
F (x, y, 1) = 0
Figure 1. Differences of 2-chains give tubular neighorhoods.
2.2 Integrals on a local Calabi–Yau
In the above we explained that the 3d chain integrals in (2.4) give the full set of solutions to the
GKZ system. These integrals are the so-called oscillating integrals on C3∫
a0e
−Fdxdydz. (2.15)
Here we have omitted the chains in the integration since by construction they are invariant
under the GKZ symmetries in (1.3) and are not important in the discussions below.
The integrand is invariant under the symmetries in (1.3) which fix the polynomial F (x,a).
However, they are not invariant under the symmetries which do not fix F but result in scalings
on F . This set of symmetries is generated by
(x,a) 7→ (x, λa), (2.16)
and
(x,a) 7→ (λx,a). (2.17)
For example, the former gives the equation
3∑
i=0
θaiF = F,
3∑
i=0
θai(a0dxdydz) = (a0dxdydz).
The corresponding operators act as the Euler operators on homogeneous functions. Unlike the
µ0/F case in (1.4), one does not have a symmetry on the integrand since
3∑
i=0
θai
(
e−Fa0dxdydz
)
= −
(
e−FFa0dxdydz
)
6= −
(
e−Fa0dxdydz
)
.
GKZ Hypergeometric Series for the Hesse Pencil 11
That is, the transformations in (2.16), (2.17), which are redundant for µ0/F when the Calabi–
Yau condition (1.10) holds, do not seem to yield symmetries for e−Fdxdydz. However, one knows
from the explicit computations that the chain integrals in (2.15) do generate the full space of
solutions to the GKZ system and hence should be invariant under these transformations.
To resolve this conflict, we are led to the following more correct interpretation of the oscil-
lating integral in (2.15). First we note that the above two transformations (2.16), (2.17) are
related by a transformation which does preserve F , hence we only need to consider one of them.
For simplicity, we focus on the latter. Motivated by [43], we think of the above integral as one
on the total space of KP2 . We choose s(a) to be coordinate on the fiber with respect to the
trivialization a0µ0. Note that a0µ0 fails to represent a nonzero section precisely at the orbifold
point a0 = 0 on the base of the elliptic curve family. This is why the coordinate s(a) is moduli
dependent in order to render s(a)a0µ0 well-defined. Then the holomorphic top form on KP2 is
a0µ0 ∧ ds(a). Now we consider the differential form on the CY threefold KP2
e−sF (x,a)a0µ0 ∧ ds(a).
Since s(a)a0µ0 gives a meromorphic section of KP2 → P2, under the C∗-actions in (2.16), (2.17)
the quantity s(a)F is invariant.
We regard W = sF as a function in the coordinate ring of the variety KP2 . It follows that
the Calabi–Yau condition (1.10) simply means that W is homogeneous of degree one in the fiber
coordinate s
ν := degs(W ) = 1.
This way of looking at the scaling behavior of the form µ0 is convenient, especially when there
is no term a0
∏
xi involved in the polynomial F which was used to absorb the shift in (1.5) that
comes from the action on the µ0 part.
One can then compute the resulting integral as follows∫ ∞
0
∫
e−sFa0µ0 ∧ ds =
∫
1
F
a0µ0. (2.18)
In particular, in the patch z = 1, one has∫ ∞
0
∫
e−(sz
3) F
z3 a0d
(x
z
)
∧ d
(y
z
)
∧ ds.
Now one can formally make the change of variable sz3 7→ z3, as a computational shortcut, then
the above integral becomes∫ ∞
0
∫
e−F 3a0 dx ∧ dy ∧ dz.
This gives the oscillating integrals discussed earlier in (2.15).
In sum, in order to respect all of the GKZ symmetries, the integral in (2.15) should be
interpreted as one on KP2 . In doing actual computations, we shall however think of the integral
as if it is on C3 for convenience.
For the 3d chain integral in (2.4), according to (2.18), one gets the 2d real integral which in
the affine coordinate z = 1 becomes∫ ∞
0
∫ ∞
0
a0dxdy
F (x, y, 1;ψ)
. (2.19)
The resulting 2d chain is interpreted in [26] as an element in the relative homology H2(P2 −E,
∆−E ∩∆), where ∆ = {xyz = 0}. Similar examples are discussed in [7] in which the pencil of
cubic curves have base points lying on the integral domain and a blow-up is needed.
12 J. Zhou
3 Chains on the elliptic curves and orbifold singularities
It will be more satisfactory if one can find chains in the elliptic curve fibers that give rise to
the extra solutions to the GKZ system. As mentioned above, this will then establish a link
between the oscillating integrals (2.15) in the singularity theory and objects in the elliptic curve
geometry. Since the integral contour in (2.19) is not a tubular neighborhood of a chain on the
elliptic curve, a direct dimension reduction is not available.
Instead, we shall first derive an integral formula for the extra solution basing on the in-
homogeneous Picard–Fuchs equation and the Wronskian method. The relation to modular
forms, which is special in the current example, gives an Eichler integral. Also the integral
formula offers a nice interpretation of the extra solution in terms of the Beltrami differential
which captures the deformation of complex structures.
Independently, we obtain a chain integral on the elliptic curve for the extra solution, motivated
by the special role played by the orbifold singularity in the moduli space.
3.1 Wronskian method: Eichler integral
We use the Wronskian method to obtain an Eichler integral formula for the solution I(ψ) fol-
lowing [7, 15, 16]. Recall that the extra solution I(ψ) to DGKZ in (1.8) must solve the in-
homogeneous Picard–Fuchs equation(
θ2ψ − ψ−3(θψ − 2)(θψ − 1)
)
= C
for some constant C. Taking any basis of the periods u1, u2 annihilated by the Picard–Fuchs
operator LPF in (1.2), then according to the standard Wronskian method one has
Theorem 3.1. The solutions I(ψ) to the GKZ system for the Hesse pencil are given by
I(ψ) = au1(ψ) + bu2(ψ) + c
∫ ψ 1
(1− v−3)v2
1
W (v)
(u1(ψ)u2(v)− u2(ψ)u1(v))dv, (3.1)
for some constants a, b, c.
The lower bound in the integral does not matter: two different choices for the lower bound
result in a change on a, b.
The Wronskian
W (ψ) = (u′1(ψ)u2(ψ)− u1(ψ)u′2(ψ))
can be easily computed by using the Schwarzian of the Picard–Fuchs equation.
It is known that the Hesse pencil is parametrized by the modular curve Γ0(3)\H∗ whose
Hauptmodul α(τ) can be found, e.g., in [36]. We take the basis u1, u2 to be the periods
ω0(α), ω1(α) near the infinity cusp given in (2.12), with [5] τ = ω1/ω0. See [40] for a collection
of the formulas. Now the last term in (3.1) is∫ α 1
v2(1− v)
1
W (v)
(ω0(v)ω1(α)− ω0(α)ω1(v))dv
= ω0(α)
∫ α 1
v2(1− v)
1
W (v)
ω0(v)(τ(α)− τ(v))dv,
up to a constant multiple. Then we get
GKZ Hypergeometric Series for the Hesse Pencil 13
Corollary 3.2. Denote the normalized period I/ω0 by tGKZ, then one has the following Eichler
integral expression of tGKZ near the infinity cusp
tGKZ = a+ bτ + c
∫ α 1
v2(1− v)
1
W (v)
ω0(v)(τ(α)− τ(v))dv
= a+ bτ + c
∫ τ
(1− α(v))ω3
0(v)(τ − v)dv, (3.2)
for some constants a, b, c.
The above formula in (3.2) is consistent with the result that
L̃GKZ(ω0tGKZ) = θα ◦ L̃PF(ω0tGKZ) = θα ◦
1
(1− α)ω3
0
◦ ∂2τ tGKZ = 0.
Different choices for the reference point in the Eicher integral will affect the last term by
a quantity whose second derivative in τ vanishes and hence is a period integral.
We now relate the solutions to modular forms. From (3.2) it follows that
∂2τ tGKZ = c(1− α)ω3
0, (3.3)
for some constant c. Moreover, the quantity (1 − α)ω3
0 is equal to the following modular form
of weight 3 for the modular group Γ0(3)
B(τ) =
η(τ)3
η(3τ)
,
see [36] for details. See also [44] for detailed discussions on the computations on periods. In (3.3),
when c = 0 one gets the period integrals, otherwise one gets the extra solution to the GKZ
system. For simplicity, we set c = 1 below. The modular form B3 has a nice Eisenstein series
and hence Lambert series formula given by
B3(τ) = 1− 9
∑
n≥1
χ−3(n)
n2qn
1− qn , q = exp(2πiτ).
Here χ−3 is the Legendre symbol which takes the values 0, 1, −1 on integers of the form 3k,
3k + 1, 3k + 2, respectively. Hence we obtain
tGKZ =
1
2
τ2 + bτ + a+ 9
∑
n≥1
χ−3(n)Li2(q
n). (3.4)
Remark 3.3. The normalized period tGKZ should be contrasted to the flat coordinate t for
the mirror of the A-model geometry of KP2 , which is a normalized period solved from the 3rd
order Picard–Fuchs equation in (1.12) and arises as the integral of the Mahler measure [41]. It
satisfies θαt = ω0. By using the Schwarzian this becomes
∂τ t = cB(τ)3, (3.5)
for some constant c. By using its expected boundary behavior, one gets
et = −q
∏
n≥1
(1− qn)9nχ−3(n), q = e2πiτ .
The inversion of this quantity carries interesting enumerative meaning in Gromov–Witten theory.
See [38, 41, 45] for detailed discussions.
14 J. Zhou
By comparing (3.3) with (3.5), and using the properties of the special geometry [42] on the
moduli space, one can see that tGKZ is actually related to the quantum volumes of cycles [24]
in the A-model Calabi–Yau geometry under mirror symmetry. To be a little more detailed,
denoting the prepotential by F (t), then the quantum volumes are given by the normalized
periods 1, t, ∂tF (t), 2F (t)− t∂tF (t). The normalized solutions to the GKZ system are then, up
to unimportant terms,
1 = ∂t(t), τ = ∂t(∂tF (t)),
tGKZ(τ) = −∂t (2F (t)− t∂tF (t)) = t∂t(∂tF (t))− ∂tF (t).
An amusing observation is that tGKZ(τ) is the Legendre dual of ∂tF (t) and vice versa.
Since in the current example the Yukawa coupling, which is defined to be ∂3t F (t) = ∂tτ ,
is non-vanishing, we can write a derivatives in τ in terms of that in t. Ignoring the overall
multiplicative factors, and focusing on the normalized periods, we get the simplifications
DGKZ = ∂τ ◦
∂τ
∂t
◦ ∂2τ = ∂τ∂t∂τ ∼ ∂t ◦ ∂t∂τ , (3.6)
LCY3 = ∂t ◦
∂t
∂τ
◦ ∂t ◦ ∂t = ∂t∂τ ◦ ∂t. (3.7)
We shall say more about the relation between them in Section 4.
Remark 3.4. Since the oscillating integral ω0tGKZ appears naturally in the Landau–Ginzburg
B-model, in particular, through the Frobenius manifold structure, it is natural to ask whether the
normalized period tGKZ is also related to the enumerative geometry of the mirror LG A-model,
similar to the flat coordinate t for the mirror of the A-model geometry of KP2 .
The solution displayed in (3.4) agrees with the fact that the solutions of the GKZ system
must contain a solution with log2 α behavior. The latter reflects that the indicial equation
has three roots 0, 0, 0 at the point α = 0 (around which a basis of solutions can be obtained
via Frobenius method). The indeterminacy a, b indicates that the extra solution is subject to
addition by the other two solution which are periods and do not affect the log2 α behavior.
For a given solution, say J3, the constants a, b, c in (3.2) can be fixed following the standard
method. We shall not do this here. Instead, we discuss the representation of the extra solution
near the orbifold point ψ = 0 around which qualitatively analyzing the solutions is convenient
since the local monodromy action can be diagonalized.
We compute the Wronskian and get,
W (ψ) = ψ2
(
1− ψ3
)−1
.
We take the basis of solutions u1, u2 to be the ones π1, π2 in (2.11). Then we obtain
I(ψ) = aπ1(ψ) + bπ2(ψ) + c
∫ ψ
v−1(π1(ψ)π2(v)− π2(ψ)π1(v))dv,
The local uniformizing variable near the orbifold point ψ = 0 on the base B can be taken to be
s = π2/π1. Then in terms of s one has
Corollary 3.5. The local expansion of the solutions to the GKZ system for the Hesse pencil
near the orbifold point is given by
I(s) = aπ1(s) + bπ2(s) + c
∫ s
v−1(π1(s)vπ2(v)− sπ1(s)π1(v))
dψ
ds
(v)dv
= aπ1(s) + bsπ1(s) + cπ1(s)
∫ s
v−1π1(v)(v − s)dψ
ds
(v)dv.
GKZ Hypergeometric Series for the Hesse Pencil 15
Now it suffices to discuss the integral J3 in terms of the above form since the other two
solutions are period integrals which are solutions to the homogeneous Picard–Fuchs equation.
Hence we want to determine the constants a, b, c in the equality
ψ3
3F2
(
1, 1, 1;
4
3
,
5
3
;ψ3
)
= aψ 2F1
(
1
3
,
1
3
;
2
3
;ψ3
)
+ bψ2
2F1
(
1
3
,
1
3
;
2
3
;ψ3
)
+ c
∫ ψ
0
v−1
(
ψv2 2F1
(
1
3
,
1
3
;
2
3
;ψ3
)
2F1
(
2
3
,
2
3
;
4
3
; v3
)
− ψ2v 2F1
(
1
3
,
1
3
;
2
3
; v3
)
2F1
(
2
3
,
2
3
;
4
3
;ψ3
))
dv.
We then use the series formula for hypergeometric functions. Without doing any calculations,
we can see that due to the monodromy behavior near ψ = 0, we must have a = b = 0. Then by
comparing the coefficients of ψ3, we are led to
c = −2.
Therefore, the in-homogeneous contribution in the solution in terms of the Wronskian gives the
monodromy invariant chain integral J3.
Again this approach singles out the special role of the orbifold point where the gauged sym-
metry in (2.8) results in the monodromy (under which the solutions have different behaviors).
3.2 Wronskian method: vanishing periods and Beltrami differential
We now give a geometric interpretation of the last term in (3.1) obtained by the Wronskian
method
J(α) :=
∫ α 1
v
(u1(v)u2(α)− u1(α)u2(v))dv.
This naturally lives in the homology of the total space of the elliptic curve fibration, similar to
the integral over the Lefschetz thimble.
To see this, we choose γ1, γ2 to be any locally constant basis of H1(E,Z) which can be
thought of as coming from the marking m : H1(E,Z) ∼= Z2 for a generic reference fiber. We
again take Ω(v) to be the section of the Hodge line bundle specified in (1.1). The period integrals
over the two cycles γ1, γ2, with respect to Ω(v), gives a basis u1(v), u2(v) of solutions to the
Picard–Fuchs equation. Then we rewrite J(α) as
J(α) =
∫ α 1
v
dv
∫
γ(v;α)
Ω(v), γ(v;α) = u2(α)γ1 − u1(α)γ2.
Fixing α, the cycle γ(v;α) is locally constant in v due to parallel transport. It is singled out,
up to a constant multiple, by the condition∫
γ(α;α)
Ω(α) = 0. (3.8)
That is, away from the orbifold point in the moduli space, it is exactly the unique cycle
in H1(Eα,C) which is the Poincaré dual of Ω(α) and hence gives the vanishing period in the
fiber Eα.
It is easy to check that the cycle γ(v;α) is independent of the marking and in particular is
invariant under monodromy. It also varies holomorphically in α. We call it the singular cycle.
Note the difference between the singular cycles and vanishing cycles (defined with respect to
the cusps).
16 J. Zhou
It follows that the quantity J(α) measures the area of the 2-dimensional region “swept out”
by the singular cycle at the point α through parallel transport, with respect to the holomorphic
volume form dv
v ∧ Ω(v) on the total space of the fibration.
Again the orbifold point ψ = 0 plays a special role, it is the only point v in the moduli
space where
∫
γ(v;α) Ω(v) = 0 for any α, the vanishing of the integral is resulted from that of
the holomorphic top form Ω. Hence if we take this point as the reference point, then the
quantity J(α) is the area of the holomorphic form dv
v ∧ Ω(v) of the cylinder swept out by these
singular cycles. One can move the ψ factor in Ω to the cycle part. Then the singular cycle
vanishes at the orbifold point and the cylinder becomes a disk. See Fig. 2 for an illustration.
α
γ(v;α)
γ(α;α)
Figure 2. Region “swept out” by singular cycles.
In this way, the extra solution captures the global information of the family, as opposed to
the normalized period integrals which can be defined locally in the family and which does not
rely on the global structure. Note that the reference point can be taken to be any point in the
base of the family, the resulting chain integral carry the same amount of information through the
singular cycles (and also the Beltrami differential below), due to the algebraicity of the family.
Alternatively, the quantity∫
γ(v;α)
Ω(v) = (u1(v)u2(α)− u1(α)u2(v) =
∫
Eα
Ω(v) ∧ Ω(α)
measures the deviation of the two complex structures corresponding to Ω(v), Ω(α) determined
through the Torelli theorem. More precisely, one can parametrize Ω(v) in terms of the Beltrami
differential (in a suitable trivialization Ω(α), Ω∗(α) of H1(Eα,C) such that
∫
Eα Ω(α)∧Ω∗(α) = 1)
by
Ω(v) = h(v;α)(Ω(α)− µ(v;α)Ω∗(α)),
where h(v;α), µ(v;α) are holomorphic in v but not in α. Then one has
J(α) =
∫ α 1
v
dv
∫
γ(v;α)
Ω(v) =
∫ α 1
v
h(v;α)µ(v;α)dv.
Remark 3.6. We can do a local calculation as follows. Fixing a choice of the section Ω, we
can write Ω(v) = ω0(v)dzv, where dzv is the complex coordinate on the universal cover of the
elliptic curve Ev ∼= C/(Z ⊕ Zτ(v)). By choosing a marking on the (generic) reference fiber Eα,
the Beltrami differential is given by the Cayley transform through
dzv = h(v;α)(dzα − µ(v;α)(dz̄α)), h(v;α) =
τ(v)− τ(α)
τ(α)− τ(α)
, µ(v;α) =
τ(v)− τ(α)
τ(v)− τ(α)
.
GKZ Hypergeometric Series for the Hesse Pencil 17
It follows that, as already computed from the Wronskian method,∫
γ(v;α)
Ω(v) = −(τ(v)− τ(α))ω0(v)ω0(α).
As pointed out above, the orbifold singularity point has the special property that there are
two linearly independent vanishing periods corresponding to π1(α), π2(α) in (2.11), while for
a generic point α one has only one cycle such that (3.8) is satisfied.
The limit of the singular cycles at the orbifold point can be computed directly through
the period calculation as follows. Since the singular cycle is independent of the marking, for
computations we take A, B to be the monodromy invariant cycles at the infinity cusp and zero
cusp respectively. There is no ambiguity in A, B at the two cusps respectively, but they of course
suffer non-trivial monodromies elsewhere. Their period integrals are as displayed in (2.12)
ω0(α) = π̃1(α) + π̃2(α), ω1(α) = −ρκπ̃1(α) + ρ2κπ̃2(α).
It follows that near the orbifold point α =∞ or equivalently ψ = 0 (here κ = i/
√
3)
γ(α;α) = A
∫
B
Ω(α)−B
∫
A
Ω(α) = π̃1(α)(−ρκA−B) + π̃2(α)
(
ρ2κA−B
)
.
It is this particular linear combination of cycles that the nearby singular cycles converge to at
the orbifold singularity.
3.3 Chains on the elliptic curves and orbifold singularities
on the moduli space
In this section, we shall find a chain C(ψ) on the elliptic curve Eψ so that the resulting integral∫
C(ψ) Ω(ψ) gives the extra solution to the GKZ system other than the period integrals.
3.3.1 Weierstrass model
We first motivate the discussion by reviewing the well-studied example of Weierstrass family of
elliptic curves.
As mentioned above in Section 1, the derivation of differential operators from GKZ symme-
tries can be applied to any family of algebraic varieties. In particular, we can apply the same
discussion to the Weierstrass family
Y 2 = 4X3 − g2X − g3.
The GKZ operator is computed to be
θw
(
θw −
1
4
)(
θw −
1
2
)
− w
(
θw +
3
4
)(
θw +
1
12
)(
θw +
5
12
)
,
w = 1− 1728
j
= 27
g23
g32
.
The discussion by [15, 16] implies that the extra solution is provided by the following chain
integral∫ ∞
0
dX
Y
=
∫ [P(0),P ′(0),1]
[P(z0),P ′(z0),1]
dX
Y
, (3.9)
18 J. Zhou
where z0 is such that ±z0 are zeros of the Weierstrass P-function. When pulled back to the
complex z-plane (as the universal cover of the elliptic curve) via the Weierstrass embedding, the
extra solution above is half of the chain integral∫ z0
−z0
dz, (3.10)
where dz is the standard holomorphic top form on the complex z-plane.
This singles out the special role of the point determined by g3 = 0 corresponding to the
orbifold point w = 0 in the moduli space, at which the chain z0− (−z0) on the complex z-plane
vanishes. Intuitively, what is happening is that if one thinks of the elliptic curve as a 2 : 1
cover over the x-plane, the chain integral mentioned above measures the “distance” of the two
covering sheets. Its vanishing does not create a change in the topology as it does not lead to
a singular curve. However, since the value of g3 is non-vanishing nearby but is vanishing at the
orbifold point, the vanishing of the chain integral does reflect a change in the complex structure.
Remark 3.7. This chain integral is actually the Abel–Jacobi map attached to the divisor q− p
given by the above two points. It appears in the study of the mixed Hodge structure of the
singular curve
Y 2 =
(
4X2 − g2X − g3
)
X2,
and is the obstruction to the isomorphism between the mixed Hodge structure of this singular
curve and the Hodge structure of its normalization, i.e., the Weierstrass curve. Furthermore, it
is the limit of a period for the genus two curve obtained by deforming the above singular curve.
See [11] for a nice account of discussions on this.
A more natural way to look at the extra solution is to expand the corresponding oscillating
integral around the orbifold point w = ∞ or equivalently g2 = 0. The reason is that when
performing the oscillating integral one is led to the following procedure (by applying change of
variables)∫
e−Y
2Z+4X3−g2XZ2−g3Z3
dXdY dZ
=⇒
∫
Z−
1
2 e−Y
2
e4X
3
e−g2XZ
2
e−g3Z
3
dXdY dZ
=⇒
∫
Z−
1
2 g
− 1
3
3 e−Y
2
e4X
3
e−Z
3
∞∑
k=0
(
−g2g
− 2
3
3 XZ2
)k
k!
dXdY dZ.
Here the integral contour needs to be chosen appropriately to deal with the convergence issue.
As one can see, evaluating the integral not only provides series solutions to the GKZ system, but
also picks out the natural coordinate for the expansion. From the discussion about the gauged
symmetries in (2.8), it is easy to see that the orbifold point is always singled out according to
where the polynomial F becomes a Fermat type under a suitable coordinate change. Also the
degree of the DGKZ-operator can be read off easily from the action of the gauged symmetries
which in particular induces action on the integral contours. This is in agreement with the result
obtained by examining the linear relations in the toric data for hypersurfaces in toric varieties
for example.
Therefore, both to see the gauged symmetries and to match the expansion parameter, we
think of the Weierstrass elliptic curve as a 3 : 1 cover over the Y -plane. For a generic member,
there are four simply-branched points determined by, setting f(X,Y ) = Y 2− (4X3− g2X− g3),
f = 0, fX = 0,
GKZ Hypergeometric Series for the Hesse Pencil 19
as well as one 2-branched point at Y =∞. The Deck group action (or the Galois action) on the
covering sheets gets enhanced exactly when the simply-branched points collide. That is, when
the system
f = 0, fX = 0, fXX = 0.
has non-trivial solutions. This is possible exactly at the orbifold point where g2 = 0. The four
simply-branched points now become two 2-branched points Yo determined by
Y 2
o = −g3.
Now we can reinterpret the chain integral in (3.9) or (3.10) as the following one, which is
naturally defined on the 3 : 1 covering,
−
∫ Yo
−Yo
dY
fX
,
where the integral contour above means any sheet covering a path connecting the two points
−Yo, Yo which may or may not pass through the branch points.
Note that carrying out the same consideration to the 2 : 1 cover over the X-plane can only
see the cusp singularities other than the orbifold singularity. As a result one can not get the full
action of the gauged symmetries from the 2 : 1 cover picture.
3.3.2 Hesse pencil
The above discussion suggests that the oscillating integral sees the finest possible information
of the gauged symmetries by exhibiting the most possible solutions with different monodromy
behaviors. They are reflected via the Galois symmetries of the covering with the highest possible
degree.
By analogy, for the Hesse pencil, we look at the orbifold singularities in the moduli space
where the configuration of branch points changes. A natural candidate for the chain whose
integral gives rise to the extra solution to the GKZ system would then be the path connecting
points which are not branch points for generic values of the modulus but become so at the
orbifold point.
We regard a generic member of the Hesse pencil as a 3 : 1 cover over the x-plane, in the
affine patch z = 1. There are 6 simply-branched points xb determined by the equation(
x3b + 1
)3
= 4ψ3x3b . (3.11)
We denote the 6 solutions by
xb,1, xb,2 = ρx1, xb,3 = ρ2x1, xb,4 =
1
x1
, xb,5 =
1
x2
, xb,6 =
1
x3
.
The symmetry of the elliptic curve for a generic value of ψ given in (2.14) is closely related
to Galois symmetry of (3.11). To be more precise, the action σ1 in (2.14) induces
γ3 : (x, y) 7→ (ρx, ρ2y).
The Z2 action on the complex plane as the universal cover of the elliptic curve induces (x, y) 7→
(y, x). Combing this with the σ2 action in (2.14), one gets a symmetry of the covering
γ2 : (x, y) 7→
(
1
x
,
y
x
)
.
20 J. Zhou
These actions are the Galois symmetries µ3 × µ2 (3.11) defining the branch variety (not the
Deck group transform of the 3 : 1 covering).
Above a branch point xb, the covering has three sheets determined through
y3 − 3ψxby +
(
x3b + 1
)
= (y − yb)2(y + 2yb), y2b = ψxb.
The Deck group of the covering is µ2. This group gets enhanced at the point ψ = 0, with yb = 0.
It contains the further symmetry
(x, y, z) 7→ (x, ρy, z), (3.12)
which only exists on the fiber corresponding to orbifold singularity ψ = 0 (where the elliptic
curve has the extra symmetry). Note that there are other points ψ3 = 1,∞ such that the branch
variety (3.11) degenerates, but at these points the elliptic curve fibers become singular and the
3 : 1 coverings are only rational maps.
It is easy to see that the branch points given by xb,k, xb,k+3 collide and gives xo,k = −ρk,
l = 1, 2, 3 at the orbifold point ψ = 0. Now on a generic fiber, above the point xo,k, the
corresponding y-values of the points on the elliptic curve satisfy
y3o,k − 3ψxo,kyo,k +
(
x3o + 1
)
= y3o,k − 3ψxo,kyo,k = 0.
The solution yo,k = 0 gives a 3-torsion point on the elliptic curve. The other two solutions satisfy
y2o,k = 3ψxo,k. See Fig. 3 for an illustration of the degeneration of the branch variety.
x
Cx
ρCx
ρ2Cx
xb,1(ψ)
xb,4(ψ)
xb,2(ψ)
xb,3(ψ)
xb,6(ψ)
xb,5(ψ)
C(ψ)
xo,3
xo,2
xo,1
ψ → 0
Figure 3. Branch configuration of the 3 : 1 cover. As ψ → 0, the branch points xb,k, xb,k+3 collide to
xo,k = −ρk, k = 1, 2, 3.
Now for a generic value of ψ, we take a path C(ψ) on the elliptic curve with endpoints
[xo,k, (3ψxo,k)
1
2 , 1], [xo,l, 0, 1], as depicted in Fig. 3. Here k could be the same as l.
GKZ Hypergeometric Series for the Hesse Pencil 21
Note that a path like this must pass through a branch point. The difference between two
such paths are cycles and hence their integrals are differed by period integrals. See Fig. 4 for
an illustration.
x
y
C(ψ)
Figure 4. Chain on the elliptic curve passing through a branch point.
Note also that if instead one takes a path with endpoints [xo,k, 0, 1], [xo,l, 0, 1], k 6= l, then
one gets a chain connecting two 3-torsion points and the resulting integral is a period integral
as mentioned before in Section 2.1.2.
We consider the chain integral
K(ψ) :=
∫
C(ψ)
ψdx
3y2 − 3ψx
. (3.13)
We then have the following result.
Theorem 3.8. The integral (3.13) gives a solution to the GKZ system for the Hesse pencil and
is not a period integral.
Proof. From the Griffiths–Dwork method, we can find the exact term in the Picard–Fuchs
operator acting on the holomorphic top form to be
(
θ2ψ − ψ−3(θψ − 2)(θψ − 1)
)( ψdx
3y2 − 3ψx
)
= ψ−3 ◦ LPF
(
ψdx
3y2 − 3ψx
)
= d
(
ψx
fy
)
.
Note that by construction the endpoints of the path C(ψ), when parametrized by x, are locally
constant and hence annihilated by the derivatives. Then by Stokes theorem, one can immediately
check the in-homogeneous Picard–Fuchs equation
(
ψ−3 ◦ LPF
)
K(ψ) =
∫
C(ψ)
d
(
ψx
y
)
=
(
ψx
fy
) ∣∣∣
∂C(ψ)
=
1
6
−
(
−1
9
)
6= 0.
Therefore, these chain integrals do give solutions to the GKZ operator DGKZ = θ ◦ (ψ−3 ◦ LPF)
which are not periods. �
Remark 3.9. The endpoints of the chain belong to Eψ∩{x3+z3 = 0}. Under the uniformization
by he θ-functions, see, e.g., [14], these end points are zeros of certain theta functions and
22 J. Zhou
carry interesting arithmetic information, as is the case in the Weierstrass family discussed in
Section 3.3.1. In terms of the toric coordinates corresponding to the characters, this is similar
to the situation that appears in open mirror symmetry [30, 31, 32, 33, 37] which again seems
to indicate that the chain integral is related to the enumerative geometry in the A-model under
mirror symmetry.
Remark 3.10. One can also consider the higher Frobenius functions appearing as the coeffi-
cients in the ε-expansion of the function
ω0(α, ε) =
∞∑
n=0
Q(ε)
Γ(3n+ 3ε+ 1)
Γ(n+ ε+ 1)3
( α
33
)n+ε
:=
∞∑
k=0
fk(α)εk, Q(ε) =
Γ(ε+ 1)3
Γ(3ε+ 1)
.
This is the deformation of the period in (2.12) when applying the Frobenius method to solve for
the solutions to the Picard–Fuchs equation. It satisfies, recall (1.9),
L̃PFω0(α, ε) =
( α
33
)ε
ε2, D̃GKZω0(α, ε) =
( α
33
)ε
ε3.
Therefore, one has
L̃PFfk(α) =
(ln α
33
)k−2
(k − 2)!
, D̃GKZfk(α) =
(ln α
33
)k−3
(k − 3)!
.
Here we have used the convention that negative powers of (ln α
33
) give zero. Besides {f0, f1} which
are period integrals and {f0, f1, f2} which are chain integrals, the higher Frobenius functions
{fk, k ≥ 3}, which can be solved by the Wronskian method as in Section 3.1, are also interesting
on their own. For example, they carry interesting arithmetic meanings, corresponding to the
counting of rational points of the Hesse elliptic curves [9, 10]. Furthermore, when one regards
the variable ε as the hyperplane class of P2, then ω0(α, ε) gives Givental’s (twisted) I-function
valued in the cohomology ring and the factorQ(ε) is the Γ-class which appears naturally in mirror
symmetry, see [18, 22, 25, 35]. After passing to the equivariant cohomology corresponding to
the diagonal torus action, the higher Frobenius functions then appear as the coefficients of the
equivariant version of the I-function expanded in the equivariant parameter. We wish to discuss
their geometric meanings in a future work.
3.3.3 Legendre family
We conclude this section with some discussions on the Legendre family whose affine equation is
y2 = x(x− 1)(x− λ), j(λ) = 28
(λ2 − λ+ 1)3
λ2(λ− 1)2
. (3.14)
From the derivation of GKZ system using the GKZ symmetries, we can see that the operator
DGKZ is a 2nd order operator and hence coincides with the Picard–Fuchs operator. This also
agrees with the earlier discussion on the relation between the extra solution and orbifold sin-
gularities. Namely, in this case one can check that the Picard–Fuchs equation has no orbifold
singularity. One can also see this by using the standard fact that the base of the family is
parametrized by the modular curve Γ(2)\H∗ which has no elliptic fixed point.
However, from the evaluation of the oscillating integral, one can see that∫
e−(y
2z−(x3−(λ+1)x2z+λxz2))dxdydz
∼ λ− 1
2
∫
e−y
2
ex
3
ez
2
∞∑
k=0
(
λ+ 1
λ
1
2
)k x 3
2
k− 1
2 zk−
1
2
k!
dxdydz.
GKZ Hypergeometric Series for the Hesse Pencil 23
Therefore, the oscillating integral naturally singles out the coordinate α = (λ+ 1)/λ
1
2 for the
expansion parameter.4 Moreover, the gauged symmetries would give rise to at least 6 solutions
with different monodromy behaviors. This is however not a contradiction to the statement
that the DGKZ is of second order. The reason is that in deriving the DGKZ using the GKZ
symmetries, only scalings on the parameter λ are allowed and hence those act by scalings on
the new parameter α is not included.
Furthermore, the locus at which α = 0 corresponds to the point λ = −1 or j = 1728 according
to the formula for the j-invariant in (3.14). Hence indeed when parametrized by α the above
expansion of the oscillating integral occurs near an orbifold point.
One can again obtain chain integrals by studying the branch configuration of the 3 : 1 cover
realization for the elliptic curve. Now the enhancement of the Galois symmetry takes place at
λ = −ρ,−ρ2 where j = 0. This corresponds to a different way of performing the oscillating the
integral above by first applying the following change of variables and then evaluating
−y2z +
(
x3 − (λ+ 1)xz2 + λz3
)
= −y2z +
(
x− λ+ 1
3
)3
+
(
λ+ 1
3
)3
z3
−
(
λ2 − λ+ 1
3
)(
x− λ+ 1
3
)
z2 − (λ+ 1)(λ2 − λ+ 1)
32
z2.
In summary, the oscillating integrals offer more than the solutions to the GKZ systems. It has
the finest information about the gauged symmetries which includes the GKZ scaling symmetries
as a subset.
4 Period integrals in a compact Calabi–Yau threefold
In this section, we shall explain the relation between the two differential operators, namely DGKZ
in (1.8) and LCY3 in (1.12). We shall see that they are different pieces of the same Picard–Fuchs
system of a compact Calabi–Yau threefold.
Recall that in Section 2.2 we explained that the 3d oscillating integrals and 2d real integrals
in (2.18) should be interpreted as ones on a non-compact Calabi–Yau. We now push this idea
further.
We first note that the members in the Hesse pencil correspond to the sections of the anti-
canonical divisor of the toric variety P whose polytope is generated by (1, 0), (0, 1), (−1,−1).
This polytope is a reflective polytope and defines the following toric variety
P = P2/G, G =
{
(ρn1 , ρn2 , ρn3) |n1 + n2 + n3 = 0 mod 3
}
. (4.1)
The invariants of G are the monomials x31, x
3
2, x
3
3, x1x2x3 among all cubic monomials. The
induced action on a generic member of the Hesse pencil is the one generated by σ1 in (2.14).
The quotient therefore gives the 3-isogeny of the Hesse pencil and is the mirror of the Hesse
pencil according to [4]. It can be checked by using the GKZ symmetries or by evaluating the
oscillating integrals that these two elliptic curve families share the same GKZ operators. Hence
for the purpose of studying the solutions to the GKZ system, there is no difference between
these two families. See [46] for discussions on these facts and some arithmetic aspects of the
mirror symmetry.
Now the quotient of the Hesse pencil is naturally interpreted as sections of the canonical
bundle KP of P . There is a natural compactification [8, 12] X of KP . A certain limit of X gives
rise to the variety KP , as the mirror of KP2 , whose Picard–Fuchs operator is displayed in (1.12).
4The transformation from the λ-parameter to this α-parameter is induced by a 2-isogeny, as can be seen
through the elliptic κ-modulus.
24 J. Zhou
This idea is used frequently in the literature to study mirror symmetry for non-compact Calabi–
Yau manifolds. As we shall review in Section 4.1, this compactification also encodes the full
information of the quotient of the Hesse pencil by G, as the mirror of the Hesse pencil, including
the GKZ operator DGKZ in (1.8).
It is then natural to expect a relation between the two geometries–mirror of Hesse pencil and
mirror of KP2 – by embedding them in the same ambient space X. The properties about the
GKZ/Picard–Fuchs system should be independent of the choice for the compactification though.
4.1 Review of the compactification
We now recall the construction of the compactification following [8, 12]. The A-model is an
elliptic fibration over P2. The total space X̌ is a Calabi–Yau hypersurface in a toric variety. For
the mirror geometry X, the toric data gives the family of varieties X whose Zariski open sets
are described by the equation
Ξ = b0 + Z2
3Z
3
4
(
a1Z1 + a2Z2 + a3Z
−1
1 Z−12 + a0
)
+ a4Z
−1
3 + a5Z
−1
4 .
Switching to the homogeneous coordinates, this is
Ξ =
(∏
x−1i
) (
b0x1x2x3x4x5 + a1x
18
1 + a2x
18
2 + a3x
18
3 + a0x
6
1x
6
2x
6
3 + a4x
3
4 + a5x
2
5
)
:=
(∏
x−1i
)
ξ. (4.2)
It is an elliptic fibration over the base P which is parametrized by x1, x2, x3. We ignore the
subtitles about the group actions involved which do not affect the Picard–Fuchs systems we are
interested in. Thinking of X as a Weierstrass fibration over P , we then get the identification
for the divisor classes(
x3i = 0
)
, (x1x2x3 = 0) = OWP (1), i = 1, 2, 3,
(x4 = 0) = OWP (2)⊗K−2P , (x5 = 0) = OWP (3)⊗K−3P , (4.3)
where WP denotes the weighted projective space WP[1, 2, 3] in which the elliptic curve fibers
sit. This implies that the coefficients transform as sections of certain tensor powers of KP :
the variables a4, a5, b0 transform as sections of K−1P , while a1, a2, a3, a0 sections of K−6P .
Strictly speaking, they are sections of the corresponding relative line bundles over the base of
the fibration.
Note that setting a4 = a5 = b0 = 0 in ξ gives the equation for the Hesse pencil. The limit
b0 = 0 in ξ gives [12] the mirror of KP2 . We shall say more about this below.
By using the GKZ symmetries, one can simplify ξ into(
bx1x2x3x4x5 + x181 + x182 + x183 + ax61x
6
2x
6
3 + x34 + x25
)
.
Here5
a = (a1a2a3)
− 1
3a0, b = b0(a1a2a3)
− 1
18a
− 1
3
4 a
− 1
2
5 . (4.4)
There are interesting loci in the base of the family Ξ parametrized by the coordinates (a, b). In
particular, the point a = b = ∞ corresponds to the large complex structure limit. See [8, 12]
and also [1, 28] for details.
5We have used different notations for the parameters from those in [8].
GKZ Hypergeometric Series for the Hesse Pencil 25
4.2 Picard–Fuchs system and fundamental period
of the compactified geometry
The period integrals are the integrals of the following form over the tubular neighborhood of
cycles in X∫
b0
Ξ
dZ1dZ2dZ3dZ4
Z1Z2Z3Z4
=
∫
b0µ0
ξ
. (4.5)
where µ0 denotes the standard meromorphic 4-form in the ambient space which has a pole of
order one at infinity. The Picard–Fuchs system can be derive from the GKZ symmetries and
are given as follows6
D1 =
1
(−3)2(−2)3
θa0θb0 − ab−6(θb0 − 1)(θb0 − 5),
D2 =
1
(−18)3
(θb0 + 6θa0)3 − a−3(θa0 − 1)(θa0 − 2)θa0 .
In terms of the coordinates a, b, we get
D1 =
1
(−3)2(−2)3
θaθb − ab−6(θb − 1)(θb − 5),
D2 =
1
(−18)3
(θb + 6θa)
3 − a−3(θa − 1)(θa − 2)θa. (4.6)
The fundamental period7 can be obtained directly by manipulating the series expansion
in (4.5) with a suitable choice for the integral contour, as done in [8, 12]. It is given by
ω0(a, b) =
∞∑
n,m=0
Γ(18n+ 6m+ 1)
Γ(9n+ 3m+ 1)Γ(6n+ 2m+ 1)Γ(n+ 1)3Γ(m+ 1)
am(b−6)3n+m
=
∞∑
k=0
Γ(6k + 1)
Γ(3k + 1)Γ(2k + 1)Γ(k + 1)
b−6kUk(a) :=
∞∑
k=0
ckb
−6kUk(a), (4.7)
where
Uk(a) = ak
[ k
3
]∑
l=0
Γ(k + 1)
Γ(l + 1)3Γ(k − 3l + 1)
a−3l.
The above expansion (4.7) amounts to solving the Picard–Fuchs system (4.6) in the following
way. The degree k-piece ckb
−6kUk(a) in the sum satisfies θb = −6k. Hence the second equation
in (4.6) gives the equation for Uk(a)(
1
(−18)3
(−6k + 6θa)
3 − a−3(θa − 1)(θa − 2)θa
)
Uk(a) = 0.
This can be simplified into(
(θa − 1)(θa − 2)θa − a3
63
(−18)3
(θa − k)3
)
Uk(a) = 0.
6These Picard–Fuchs operators are derived by factoring out some differential operators from the left in the
GKZ Z-operators. One can also study the extra solutions to the GKZ system of the current Calabi–Yau threefold
by embedding it into a variety of higher dimension, similar to what will be discussed below. But we shall not
discuss them in this work.
7The unique (up to scaling) regular period near the large complex structure limit given by a = b =∞.
26 J. Zhou
The first equation in (4.6) then gives recursive relations among {ckb−6kUk(a)}k through∑
k
(−6k)b−6kckθaUk =
∑
k
(−3)2(−2)3ab−6k−6(6k + 1)(6k + 5)ckUk.
This is simplified into
θaUk+1 = (k + 1)aUk.
4.3 Embedding of the GKZ system for the Hesse pencil
and the Picard–Fuchs system for the mirror geometry of KP2
For the purpose of getting the other solutions via the Frobenius method and doing analytic
continuation, one needs to extend [8] the definition of Uk to Uν for complex values of ν
Uν(a) = aν
∞∑
l=0
Γ(ν + 1)
Γ(l + 1)3Γ(ν − 3l + 1)
a−3l = aν 3F2
(−ν
3
,
1− ν
3
,
2− ν
3
; 1, 1; a−3
)
.
It can be analytically continued to the orbifold a = 0 via the Barnes integral formula [8]
Uν(a) =
3−1−νρ
ν
2
Γ(−ν)
∞∑
n=0
Γ(n−ν3 )
Γ2(1− n−ν
3 )
(−3ρa)n
n!
. (4.8)
The recursive relation is given by
θaUν+1 = (ν + 1)aUν .
It is annihilated by the operator
Lν =
(
(θa − 1)(θa − 2)θa − a3
63
(−18)3
(θa − ν)3
)
.
Setting a = −3ψ (which makes contact with the Hesse pencil), one gets
Lν =
(
(θψ − 1)(θψ − 2)θψ − ψ3(θψ − ν)3
)
.
When ν = 0, this is the Picard–Fuchs operator LCY3 in (1.12).
When ν = −1, this is equivalent to the operator DGKZ in (1.8) and it annihilates the form µ0
F
in (1.4). The solution given in (4.8) is exactly the one in (2.3) up to a constant multiple.
In general, Lν annihilates
aν+1
0
µ0
F
. (4.9)
As explained in Section 2.2, one should think of the parameter a0 (previously denoted by sa0 in
the trivialization µ0) as the coordinate of the fiber of KP , and hence ν + 1 as the degree of the
form in (4.9) along the fiber direction.
Consider the analytic continuation of the expansion (4.7) to the orbifold point b = 0, then
one has [8]
ω0(a, b) =
1
2π
∞∑
k=0
2−
1
2
−2k3−
1
2
−3k6
1
2
+6kΓ(k + 1
6)Γ(k + 5
6)
Γ(k + 1)2
b−6kUk(a)
=
∞∑
k=0
Γ(k + 1
6)Γ(k + 5
6)
Γ(k + 1)2
(432)kb−6kUk(a) :=
∞∑
n=0
dnb
nU−n
6
(a).
GKZ Hypergeometric Series for the Hesse Pencil 27
Here {dn}n are some Gamma-values whose precise values are not important in the discussion
here. Then we can see that both U0, U−1 appear in the fundamental period as pieces in b-
expansion of different degrees. They appear naturally in the expansion around the orbifold
point b = 0 as opposed to the expansion near the point b =∞ in (4.7).
More geometrically, one can expand the differential form in (4.5) as follows
b0µ0
ξ
= b0µ0
∞∑
k=0
1(
a1x181 + a2x182 + a3x183 + a0x61x
6
2x
6
3 + a4x34 + a5x25
)k+1
× (−b0)k(x1x2x3x4x5)k.
The degree zero term in b0 in the summation gives the holomorphic volume form in (1.11). Here
we treat the prefactor b0 of µ0 as an overall normalization. Taking b0 = 0 is equivalent to the
limit when the compact Calabi–Yau threefold X degenerates to the mirror KP of KP2 . It is
actually more convenient to see the degenerating limit in the toric coordinates Zi on the torus
in the toric variety. One writes b0µ0/ξ as the following, from which one recognizes (1.11) easily,
b0(−b0)k(
Z2
3Z
3
4 (a1Z1 + a2Z2 + a3Z
−1
1 Z−12 + a0) + a4Z
−1
3 + a5Z
−1
4
)k+1
dZ1dZ2dZ3dZ4
Z1Z2Z3Z4
. (4.10)
Setting b0 to zero means that one is effectively looking at the vanishing of a section of
OWP[1,2,3](1) in the fiber weight projective space, that is, the divisor (x1x2x3) = 0 according
to (4.3). This defines the (unique) section of the Weierstrass fibration.
Alternatively, one can write
b0µ0
ξ
= b0µ0
∞∑
k=0
(−1)k
(a1x181 + a2x182 + a3x183 + a0x61x
6
2x
6
3)
k+1
×
(
b0x1x2x3x4x5 + a4x
3
4 + a5x
2
5
)k
. (4.11)
Now in the limit a4 = a5 = b0 = 0, one recovers the meromorphic 2-form µ0/F in (1.4) that
appears in the GKZ system for the Hesse pencil. Again here we have regarded the prefactor b0
of µ0 as an overall normalization. In fact, since in the integration, the contour that gives rise to
the fundamental period is parametrized in such a way that the coordinates x4, x5 take values
in S1, the above limit on the period can be induced by the limit a4 = a5 = 0.
In terms of the geometry, this amounts to setting x4 = x5 = 0, which cuts out the Hesse
pencil in X. One can also see this by examining (4.10) in the coordinates Zi, i = 1, 2, 3, 4 on
the torus. Intuitively, the Calabi–Yau threefold X admits a rational map to WP[1, 2, 3] as an
elliptic fibration, the fibers are the Hesse elliptic curves. The equations x4 = x5 = 0 defines the
fiber at the singular point with stabilizer µ6 in WP[1, 2, 3].
In either case, the degree in b0 indicates the degree ν along the fibration direction of KP .
4.4 Interpretation in the A-model
While it is straightforward to see the above degeneration limits by examining the defining
equation of the Calabi–Yau variety X, it is perhaps also helpful to study these limits in the
A-model geometry.
The family in the A-model is parametrized by the space of Kähler structures of the variety X̌.
The latter is the resolution of singularities of a degree 18 hypersurface X̌0 in the weight projective
space WP[1, 1, 1, 6, 9] parametrized by x1, x2, x3, x4, x5. The singularity occurs at x1 = x2 =
x3 = 0. The details are worked out in [8]. We now give a brief review on the intersection theory
of the geometry. One denotes the strict transform of (x1 = 0) by L, and the total transform of
28 J. Zhou
the divisor (x1x2x3 = 0) by H = 3L + E, where E is the class of the exceptional divisor. The
intersections are
H3 = 9, H2L = 3, HL2 = 1, L3 = 0.
Thinking of X̌ as the blow-up, the Kähler classes are linear combinations of L (the strictly
transform of the Kähler class on the singular variety) and the exceptional divisor class E. The
fibration structure also tells that the class of the base P2 is E, the pull back of OP2(1) gives the
class L. One has E ·E · L = −3. This is the degree of the line bundle KP2 over P2. It confirms
the statement that E is the base P2 of the elliptic curve family. The effective curve classes are
h = L · L, ` = L · E,
which represent the elliptic curve fiber and the hypersurface class in the base E, respectively.
See the illustration in Fig. 5. The dual nef cone is worked out to be the one generated by H, L.
WP[1, 1, 1, 6, 9][18]
E E
L
l
h
Figure 5. Elliptic fibration in the A-model as a blow-up.
These intersections are nicely encoded into the linear relations among the rays in the toric
fan that defines the ambient space
Q1 = (1, 1, 1,−3, 0, 0; 0), Q2 = (0, 0, 0, 1, 2, 3;−6).
They represent the curve classes `, h respectively. The toric invariant divisors correspond to the
columns of the above matrix of linear relations. More precisely, one has
L ∼
(
1
0
)
, H ∼
(
0
1
)
, E ∼
(
−3
1
)
.
The last vector (0,−6) represents the first Chern class of KWP[1,2,3], which is canonical sheaf
of the fiber weighted projective space (in which the elliptic curve fiber sits). We denote the
corresponding class by
J ∼
(
0
−6
)
.
Now a Kähler class is represented by a linear combination of these classes, with certain
positivity conditions satisfied,
K = log a1L+ log a2L+ log a3L+ log a0E + log a4(2H) + log a5(3H) + log b0J.
The parameters ai, bi are mirror to the coordinates with the same names in the B-model, up
to terms which do not affect the qualitative analysis. In the following we shall use the same
GKZ Hypergeometric Series for the Hesse Pencil 29
coordinates a, b as in (4.4). An element in the nef cone (one of the chambers in the second fan
of the toric variety8) must satisfy the condition
K = log
(
a1a2a3
a30
)
L+ log
(
a24a
3
5a0
b60
)
H
= log
(
a−3
)
L+ log
(
ab−6
)
H ∈ R>0L⊕ R>0H. (4.12)
Therefore the large volume limit corresponds to the point
a−3 = 0 = ab−6, (4.13)
which is mirror to the large complex structure limit mentioned before. Hence one can see that
in the defining equation ξ in (4.2) for the B-model, one needs to send both a, b to ∞.
Now it is easy to see that the limit b0 = 0 corresponds to the degeneration9 of Kähler structure
K → L, as when considering for example instanton expansions cycles with infinity volumes are
suppressed. In particular, the class of the elliptic curve fiber h = L · L does not survive the
limit. On the level of toric data or Mori cone of curve classes, the vector Q2 representing the
class h is invisible in the limit and hence what is left is the one Q1 which is exactly the toric data
that defines the geometry KP2 . On the level of geometry, that h has infinite volume tells that
the elliptic fibration over E gets decompactified to KP2 . This is consistent with the B-model
picture discussed below (4.10).
Similarly, for the limit a4 = a5 = 0, one has the degeneration K → L. Now the curve class 3`,
which is the one underlying the cubic in E ∼= P2 survives this limit. This is also consistent with
the B-model picture discussed below (4.11).
5 Discussions and speculations
For a general Laurent polynomial F , the relations among the monomials are conveniently de-
scribed combinatorially by the Newton polytope. This in particular gives a shortcut in deriving
the differential equations.
As mentioned in Section 2.1.2, the difference between the universal family of cubics and
the Hesse pencil is that the latter carries an extra level structure. This is what picks out
the 4 monomials appearing in the Hesse pencil among the all the 10 cubic monomials. These 4
monomials is what determines the symmetries of the family and hence the GKZ symmetries.
The linear space spanned by them encode the full data of the family.
Instead of using the Veronese embedding of P2 into PȞ0(P2,O(3)), we use only the 4 mono-
mials xyz, x3, y3, z3, i = 1, 2, 3. Consider the partial Veronese map
Φ: P2 → P3, (x, y, z) 7→ (X0, X1, X2, X3) =
(
xyz, x3, y3, z3
)
.
The image of P2 is given by the vanishing of the ideal generated by
Φ = X1X2X3 −X3
0 ,
which defines a singular cubic surface S0 of degree 3 in P3. Here we used the same notation Φ
to denote the polynomial by abuse of notation. The singular points on S0 are given by
(X0, X1, X2, X3) = (0, 1, 0, 0), (0, 0, 1, 0), (0, 0, 0, 1).
8See, e.g., [13] for a detailed review.
9However, in this limit log(ab−6) has a negative sign from the previous limit in (4.13) and the description
in (4.12) for the Kähler class fails. This means that this limit does not sit inside the nef cone of X̌ and it has
moved deeply into some other chamber in the secondary fan. This situation is typical in the so-called phase
transition process, see [43].
30 J. Zhou
Locally a neighborhood of each singular point is of the form of an A2-singularity described
by C2/Z3. This surface is nothing but the mirror of P2 given by P2/G as described in (4.1). The
elliptic curve is then mapped to the intersection of this cubic surface S0 with the hyperplane
H :=
∑
aiXi = 0.
One can alternatively make a moduli dependent embedding by using the monomials aiXi,
i = 0, 1, 2, 3, then the moduli dependence of the intersection is full encoded in the moduli de-
pendence of the singular cubic surface S0.
The above procedure linearizes the monomials in the defining equation F for the Hesse pencil.
The integrand in (2.15) then becomes (up to the a0 factor)
e−Fdxdydz =
1
33
e
−
3∑
i=0
aiXi
δS0
∏ dXi
X
2/3
i
:= e−HδS0µ,
where δ means the Dirac delta distribution. As before, the integrand e−sFµ0 ∧ ds on needs to
be suitably interpreted in order to incorporate all the Z-symmetries.
Now the original Z-symmetries become diagonal symmetries of the quadratic form H =∑
aiXi. These symmetries also leave the variety S0 invariant (the polynomial Φ is not invariant).
The original D-symmetries are manifest through the Veronese map Φ. Schematically one has
Z(H) = 0, Z(S0) = 0, Z
(
e−HδS0µ
)
= 0, D
(
e−HδS0µ
)
= e−HδS0Φµ = 0. (5.1)
Note that the characteristic variety (singular support) of D is now defined by{
p1p2p3 − p30 = 0
}
⊆ T ∗C4,
where pi, i = 0, 1, 2, 3 are the fiber coordinates of the cotangent bundle of C4 parametrized by
a0, a1, a2, a3. This takes the same form as the polynomial Φ. In fact, by restricting the field C
to the field R which does not affect the discussion on GKZ symmetries, one can regard
F(•) =
∫
e−H(•)µ
as a formal Fourier transform. Then one has
D = F(Φ).
Note that D has constant coefficients, this then relates Φ to the characteristic variety.
We can also formally write the original oscillating integral over the invariant chain D3 =
(0,∞)× (0,∞)× (0,∞) in (2.1) (which produces all solutions under the monodromy action) as
I =
∫
D3
e−Fµ0 =
∫
D3×(0,∞)
e−HδS0µ = F(δS0).
The configuration (Φ, H) fully encodes the information of the Hesse pencil. Since by a moduli
dependent embedding one can make H independent of a1, a2, a3, a0, it is therefore very natural
to expect that the GKZ system can be approached via studying the mixed Hodge structure of
S0 together with the moduli-independent hyperplane. The presentation of the GKZ symmetries
in the form displayed in (5.1) also begs for an explanation in terms of D-module in addressing
this problem. A good understanding of this matter is potentially useful in the studies of open
mirror symmetry where similar situation occurs, see, e.g., [34, 37].
GKZ Hypergeometric Series for the Hesse Pencil 31
Acknowledgements
The author dedicates this article to Professor Noriko Yui on the occasion of her birthday. The
author is grateful for her constant encouragement and support, and in particular for many
inspiring discussions on geometry and number theory. The author would like to thank Murad
Alim, An Huang, Bong Lian and Shing-Tung Yau for discussions on open string mirror symmetry
which to a large extent inspired this project. He thanks further Kevin Costello, Shinobu Hosono,
Si Li and Zhengyu Zong for their interest and helpful conversations on Landau–Ginzburg models
and chain integrals, and Don Zagier for some useful discussions on modular forms back in year
2013. He also thanks the anonymous referees whose suggestions have helped improving the
article.
This research was supported in part by Perimeter Institute for Theoretical Physics. Research
at Perimeter Institute is supported by the Government of Canada through Innovation, Science
and Economic Development Canada and by the Province of Ontario through the Ministry of
Research, Innovation and Science.
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http://arxiv.org/abs/1605.07877
1 Introduction
1.1 GKZ system for the Hesse pencil
1.2 Calabi–Yau condition and factorization of differential operator
1.3 Motivation of the work
2 Invariant 3d and 2d chain integrals under GKZ symmetries
2.1 Invariant chain integrals as solutions to GKZ system
2.1.1 Monodromy action and functional relations
2.1.2 Period integrals as differences of chain integrals
2.2 Integrals on a local Calabi–Yau
3 Chains on the elliptic curves and orbifold singularities
3.1 Wronskian method: Eichler integral
3.2 Wronskian method: vanishing periods and Beltrami differential
3.3 Chains on the elliptic curves and orbifold singularities on the moduli space
3.3.1 Weierstrass model
3.3.2 Hesse pencil
3.3.3 Legendre family
4 Period integrals in a compact Calabi–Yau threefold
4.1 Review of the compactification
4.2 Picard–Fuchs system and fundamental period of the compactified geometry
4.3 Embedding of the GKZ system for the Hesse pencil and the Picard–Fuchs system for the mirror geometry of KP2
4.4 Interpretation in the A-model
5 Discussions and speculations
References
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| id | nasplib_isofts_kiev_ua-123456789-148596 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T15:14:12Z |
| publishDate | 2017 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Zhou, J. 2019-02-18T16:26:31Z 2019-02-18T16:26:31Z 2017 GKZ Hypergeometric Series for the Hesse Pencil, Chain Integrals and Orbifold Singularities / J. Zhou // Symmetry, Integrability and Geometry: Methods and Applications. — 2017. — Т. 13. — Бібліогр.: 46 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 14J33; 14Q05; 30F30; 34M35 DOI:10.3842/SIGMA.2017.030 https://nasplib.isofts.kiev.ua/handle/123456789/148596 The GKZ system for the Hesse pencil of elliptic curves has more solutions than the period integrals. In this work we give different realizations and interpretations of the extra solution, in terms of oscillating integral, Eichler integral, chain integral on the elliptic curve, limit of a period of a certain compact Calabi-Yau threefold geometry, etc. We also highlight the role played by the orbifold singularity on the moduli space and its relation to the GKZ system. The author dedicates this article to Professor Noriko Yui on the occasion of her birthday. The author is grateful for her constant encouragement and support, and in particular for many inspiring discussions on geometry and number theory. The author would like to thank Murad Alim, An Huang, Bong Lian and Shing-Tung Yau for discussions on open string mirror symmetry which to a large extent inspired this project. He thanks further Kevin Costello, Shinobu Hosono, Si Li and Zhengyu Zong for their interest and helpful conversations on Landau–Ginzburg models and chain integrals, and Don Zagier for some useful discussions on modular forms back in year 2013. He also thanks the anonymous referees whose suggestions have helped improving the article. This research was supported in part by Perimeter Institute for Theoretical Physics. Research at Perimeter Institute is supported by the Government of Canada through Innovation, Science and Economic Development Canada and by the Province of Ontario through the Ministry of Research, Innovation and Science. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications GKZ Hypergeometric Series for the Hesse Pencil, Chain Integrals and Orbifold Singularities Article published earlier |
| spellingShingle | GKZ Hypergeometric Series for the Hesse Pencil, Chain Integrals and Orbifold Singularities Zhou, J. |
| title | GKZ Hypergeometric Series for the Hesse Pencil, Chain Integrals and Orbifold Singularities |
| title_full | GKZ Hypergeometric Series for the Hesse Pencil, Chain Integrals and Orbifold Singularities |
| title_fullStr | GKZ Hypergeometric Series for the Hesse Pencil, Chain Integrals and Orbifold Singularities |
| title_full_unstemmed | GKZ Hypergeometric Series for the Hesse Pencil, Chain Integrals and Orbifold Singularities |
| title_short | GKZ Hypergeometric Series for the Hesse Pencil, Chain Integrals and Orbifold Singularities |
| title_sort | gkz hypergeometric series for the hesse pencil, chain integrals and orbifold singularities |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/148596 |
| work_keys_str_mv | AT zhouj gkzhypergeometricseriesforthehessepencilchainintegralsandorbifoldsingularities |