Difference Equation for Quintic 3-Fold
In this paper, we use the Mellin-Barnes-Watson method to relate solutions of a certain type of -difference equations at = 0 and = ∞. We consider two special cases; the first is the -difference equation of the -theoretic -function of the quintic, which is degree 25; we use Adams' method...
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| description | In this paper, we use the Mellin-Barnes-Watson method to relate solutions of a certain type of -difference equations at = 0 and = ∞. We consider two special cases; the first is the -difference equation of the -theoretic -function of the quintic, which is degree 25; we use Adams' method to find the extra 20 solutions at = 0. The second special case is a Fuchsian case, which is confluent to the differential equation of the cohomological -function of the quintic. We compute the connection matrix and study the confluence of the -difference structure.
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 043, 25 pages
Difference Equation for Quintic 3-Fold
Yaoxinog WEN
Korea Institute for Advanced Study, Seoul, 02455, Republic of Korea
E-mail: y.x.wen.math@gmail.com
Received September 28, 2021, in final form June 04, 2022; Published online June 14, 2022
https://doi.org/10.3842/SIGMA.2022.043
Abstract. In this paper, we use the Mellin–Barnes–Watson method to relate solutions of
a certain type of q-difference equations at Q = 0 and Q = ∞. We consider two special
cases; the first is the q-difference equation of K-theoretic I-function of the quintic, which is
degree 25; we use Adams’ method to find the extra 20 solutions at Q = 0. The second special
case is a fuchsian case, which is confluent to the differential equation of the cohomological
I-function of the quintic. We compute the connection matrix and study the confluence of
the q-difference structure.
Key words: q-difference equation; quantum K-theory; Fermat quintic
2020 Mathematics Subject Classification: 14N35; 33D90; 39A13
1 Introduction
Since the 1990s, the development of mirror symmetry has changed how people work on enumer-
ative geometry and has made some surprising predictions in algebraic geometry. Calabi–Yau
manifolds are essential in mirror symmetry. Among them, the quintic threefold was the first
example for which mirror symmetry was used to make enumerative predictions [3].
There are several ways to state mirror symmetry. In Givental’s approach to mirror symmetry,
two cohomology valued formal functions play a crucial role, i.e., the so-called J-function and
I-function. The J-function, by definition, encodes all the genus zero Gromov–Witten invariants,
so it is essential. However, it is pretty hard to obtain an explicit formula. On the other hand, the
I-function given by the oscillatory integral is computable. In [7] Givental proved that I-function
lies on the range of big J-function, and up to a change of coordinate, we can obtain J-function
from I-function.
Let X be the Fermat quintic, considered as a degree 5 hypersurface in P4, the cohomological
I-function of X is as follows
IcohX (ℏ, et) =
∞∑
d=0
∏5d
k=1(5H + kℏ)∏d
k=1(H + kℏ)5
et(
H
ℏ +d),
where H is the hyperplane class of P4, and ℏ is the equivariant parameter, since H4 = 0 in the
cohomology of X, the I-function of quintic satisfies the following degree 4 differential equation
which is called the Picard–Fuchs equation[(
ℏ
d
dt
)4
− 55et
(
ℏ
d
dt
+
1
5
ℏ
)(
ℏ
d
dt
+
2
5
ℏ
)(
ℏ
d
dt
+
3
5
ℏ
)(
ℏ
d
dt
+
4
5
ℏ
)]
IcohX
(
ℏ, et
)
= 0.
Let Q = 55et, the above differential equation becomes the following form[(
Q
d
dQ
)4
−Q
(
Q
d
dQ
+
1
5
)(
Q
d
dQ
+
2
5
)(
Q
d
dQ
+
3
5
)(
Q
d
dQ
+
4
5
)]
IcohX (ℏ, Q) = 0. (1.1)
mailto:y.x.wen.math@gmail.com
https://doi.org/10.3842/SIGMA.2022.043
2 Y. Wen
The fundamental solutions at Q = 0 are given by the expansion of IcohX (ℏ, Q), i.e.,
IcohX (ℏ, Q) = I0 · 1 + I1 ·H + I2 ·H2 + I3 ·H3 mod
(
H4
)
.
More preciously, the coefficients of the I-function relative to the cohomology basis give the
fundamental solutions of (1.1). Moreover, the fundamental solutions at Q = ∞ are related to
FJRW theory [4] and could be constructed explicitly via the Frobenius method.
Around 2000, Givental [8] and Lee [11] introduced the K-theoretic Gromov–Witten (GW)
invariants, these invariants are defined by replacing cohomological definitions by their K-theore-
tical analogs. The K-theoretic J-function and I-function are also defined and studied; unlike
cohomological GW theory, the K-theoretic I-function satisfies a difference equation instead of
a differential equation. For example, let us still consider quintic X, denote by qQ∂Q the difference
operator shifting Qk by qkQk, the K-theoretic I-function of X is as follows
IKX (q,Q) = P lq(Q)
∞∑
d=0
∏5d
k=1
(
1− P 5qk
)∏d
k=1
(
1− Pqk
)5Qd, (1.2)
where P = O(−1) on P4, and
lq(Q) = −Q
θ′q(Q)
θq(Q)
is the q-logarithm function. Here θq(Q) is the Jacobi’s theta function and the q-logarithm
function satisfies
qQ∂Q(lq(Q)) = lq(Q) + 1.
Since (1 − P )5 = 0 in K
(
P4
)
, the K-theoretic I-function of X satisfies the following difference
equation[(
1− qQ∂Q
)5 −Q
5∏
k=1
(
1− q5Q∂Q+k
)]
IKX (q,Q) = 0. (1.3)
It is a degree 25 difference equation but not fuchsian (Definition 2.15).
The characteristic equation (see (3.1) for definition) at Q = ∞ is(
1− q−1x5
)(
1− q−2x5
)(
1− q−3x5
)(
1− q−4x5
)(
1− q−5x5
)
= 0,
with 25 distinct roots. Using Frobenius method, we obtain 25 solutions (at Q = ∞) of (1.3)
given by
Wl,m(1/Q) = e
q,q
l
5 ξm
(1/Q)
∑
d≥0
∏d−1
k=0
(
1− ξ−mq−k− l
5
)5∏5d−1
k=0
(
1− q−k−l
) Q−d,
where
ξ5 = 1, l = 1, . . . , 5, m = 0, . . . , 4,
and
eq,λq(Q) =
θq(Q)
θq(λqQ)
∈ M(C∗).
The function eq,λq satisfies the q-difference equation qQ∂Qeq,λq(Q) = λqeq,λq(Q).
Difference Equation for Quintic 3-Fold 3
The characteristic equation at Q = 0 is
(1− x)5 = 0,
with 5 multiple roots. One may obtain explicit formulas for solutions via the Frobenius method.
However, we could obtain 5 solutions at Q = 0 from IKX (q,Q) as mentioned above. We use
Adams’ method to obtain the rest solutions.
Proposition 1.1. The exceptional 20 solutions of (1.3) are given as follows: for each 20th root
of unity ξ, we have a solution of form
ep,ξp−1/2(z)F (z) = ep,ξp−1/2(z)
∑
n≥0
fnz
n,
where z = Q
1
20 , p = q
1
20 . Let σp = pz∂z , then F (z) satisfies[(
z − ξp−
9
2σp
)(
z − ξp−
7
2σp
)(
z − ξp−
5
2σp
)(
z − ξp−
3
2σp
)(
z − ξp−
1
2σp
)
−
(
z5−ξ5p−
25
2 σ5
p
)(
z5−ξ5p−
15
2 σ5
p
)(
z5−ξ5p−
5
2σ5
p
)(
z5−ξ5p
5
2σ5
p
)(
z5−ξ5p
15
2 σ5
p
)]
F (z)=0.
To the best of the author’s knowledge, the connection matrix is critical in classifying fuchsian
difference equations, and very little is known about the non-fuchsian case. Even in the fuchsian
case, the connection matrix is hard to obtain if the characteristic equation has multiple roots.
We construct the following K
(
Pn−1
)
valued q-series motived by [4] to obtain connection
matrix
Fm,n(Q) = P lq(Q)
∞∑
d=0
∏m
i=1(Pαi; q)d
(Pq; q)nd
Qd. (1.4)
Here we use q-Pochhammer symbol notation:
(a; q)d := (1− a)(1− qa) · · ·
(
1− qd−1a
)
for d > 0.
Since (1− P )n = 0 in K
(
Pn−1
)
, then (1.4) satisfies the following difference equation[(
1− qQ∂Q
)n −Q
m∏
i=1
(
1− αiq
Q∂Q
)]
Fm,n(Q) = 0 mod
(
(1− P )n
)
. (1.5)
Suppose αi /∈ αjq
Z\{0} then we could find the explicit formula for m solutions at Q = ∞ denoted
by {Wk(1/Q)}mk=1 and we use Mellin–Barnes–Watson method to related solutions at Q = 0
and Q = ∞.
Theorem 1.2. For m ≥ n, the K
(
Pn−1
)
valued q-series has the following analytic continuation:
P lq(Q)
∞∑
d=0
∏m
i=1(Pαi; q)d
(Pq; q)nd
Qd = P lq(Q)
∏m
i=1(Pαi; q)∞
(Pq; q)n∞
m∑
j=1
(q, q, PαjQ, q/(PαjQ); q)∞
(Pαj , q/(Pαj), Q, q/Q; q)∞
×
(
α−1
j q; q
)n
∞e−1
q,αj
(1/Q)∏m
i=1,i ̸=j(αi/αj ; q)∞(q; q)∞
Wj(1/Q).
As for applications, if we take n = 5, m = 25 and {αi}25i=1 =
{
ξlq
k
5 | k, l = 1, 2, 3, 4, 5
}
,
then (1.4) becomes (1.2). And if we take n = 4, m = 4 and {αi}4i=1 =
{
q
i
5
}4
i=1
, then (1.5)
becomes[(
1− qQ∂Q
)4 −Q
4∏
i=1
(
1− q
i
5 qQ∂Q
)]
F (Q) = 0. (1.6)
4 Y. Wen
This difference equation is a lift of the differential equation (1.1), i.e., if we let q → 1, then (1.6)
becomes (1.1). This phenomenon is called confluence which was studied first by J. Sauloy
in 2000 [15]. Under the above specific choice, the formula in Theorem 1.2 becomes
P lq(Q)
∞∑
d=0
∏4
i=1
(
Pq
i
5 ; q
)
d
(Pq; q)4d
Qd = P lq(Q)
∏4
i=1
(
Pq
i
5 ; q
)
∞
(Pq; q)4∞
4∑
j=1
(
q, q, Pq
j
5Q, q/
(
Pq
j
5Q
)
; q
)
∞(
Pq
j
5 , q/
(
Pq
j
5
)
, Q, q/Q; q
)
∞
×
(
q−
j
5 q; q
)4
∞e−1
q,qj/5
(1/Q)∏4
i=1,i ̸=j
(
q
i−j
5 ; q
)
∞(q; q)∞
Wj(1/Q),
where {Wj(1/Q)}4j=1 are the fundamental solutions at Q = ∞. If we expand two sides with
respect to K-group basis (1 − P )k, k = 0, 1, 2, 3, we obtain the connection matrix, for more
details, see Section 5.1. Besides, the fundamental solutions of (1.6) at Q = 0 and Q = ∞ are
confluent to the solutions of (1.1), finally, we compute the confluence of the connection matrix.
The paper is arranged as follows. Section 2 reviews some basic definitions and concepts of
difference equations and introduces some special functions. In Section 3, we use the difference
equation of quintic as an example, and we use Adams’ method and Frobenius method to solve
the degree 25 difference equation at Q = 0 and Q = ∞ respectively. In Section 4, we generalize
the difference equation for the quintic and construct a K-group valued series, and then we use
the Mellin–Barnes–Watson method to relate solutions at Q = 0 and Q = ∞. In Section 5, we
apply the results in Section 4 to a particular fuchsian case, and we expand the formula with
respect to theK-group basis (1−P )k to find the connection matrix. Since the particular fuchsian
case is confluent to the differential equation of quintic. In Section 6, we study the confluence of
the connection matrix.
2 Preliminaries
In this section, we define some basic notions in the theory of q-difference equations. The main
references are [13, 15, 16].
Notations 2.1. Here are some standard notations of general use:
– Q and q are complex variables and |q| < 1, q ̸= 0,
– C({Q}) is the field of meromorphic germs at 0, is the quotient field of C{Q},
– M(C∗) is the field of meromorphic functions on C∗,
– M(C∗, 0) is the ring of germs at punctured neighborhood of Q = 0,
– M (Eq) is the field of meromorphic functions on elliptic curve Eq = C∗/qZ, i.e, the field of
elliptic functions.
– (a; q)d = (1− a)(1− qa) · · ·
(
1− qd−1a
)
for d ∈ N ∪ {+∞} is the q-Pochhammer symbol.
Definition 2.2. A difference field is a pair (K,σ), where K is a field, and σ is a field automor-
phism of K.
Example 2.3. We will focus on the fields in the above notations,
M(C∗) ⊂ M(C∗, 0),
they are all endowed with the q-shift operator σq := qQ∂Q : f(Q) 7→ f(qQ). Let K = M(C∗)
or M(C∗, 0). Usually, we denote the field of constants of the difference field (K,σq) as K
σq . For
example, M(C∗)σq = M(C∗)σq = M(Eq). This is the main reason that the modular form such
as elliptic function appears naturally in the theory of q-difference equation.
Difference Equation for Quintic 3-Fold 5
2.1 Regular singular q-difference equations
Definition 2.4. Let (Eq) : q
Q∂QXq(Q) = Aq(Q)Xq(Q) be a q-difference system, with Aq ∈
GLn(K). We define the solution space of this q-difference equation by
Sol(Eq) =
{
Xq ∈ Kn | qQ∂QXq(Q) = Aq(Q)Xq(Q)
}
.
Remark 2.5. From now on, we will focus on the local solutions at Q = 0, and the results will
also hold for Q = ∞. The reason why we don’t consider solutions at other singular points is
that: if a function f(Q) is a solution of a q-difference equation qQ∂Qf(Q) = a(Q)f(Q) and has
a singularity at some Q0 ̸= 0,∞, then f(Q) has a singularity at any complex number Q0q
k.
Proposition 2.6 ([16, Theorem 2.3.1, p. 118]). Let (Eq) : q
Q∂QXq(Q) = Aq(Q)Xq(Q) be a q-
difference system. Then, we have
dimM(Eq)
(
Sol(Eq)
)
≤ rank(Aq).
Definition 2.7. Let qQ∂QXq(Q) = Aq(Q)Xq(Q) be a q-difference system. A fundamental solu-
tion of this system is an invertible matricial solution Xq ∈ GLn(K) such that qQ∂QXq(Q) =
Aq(Q)Xq(Q).
Definition 2.8. Let qQ∂QXq(Q) = Aq(Q)Xq(Q) be a q-difference system. Consider a matrix
Pq ∈ GLn(K). The gauge transform of the matrix Aq by the gauge transformation Pq is the
matrix
Pq · [Aq] :=
(
qQ∂QPq
)
AqP
−1
q .
A second q-difference system qQ∂QXq(Q) = Bq(Q)Xq(Q) is said to be equivalent (over K) by
gauge transform to the first one if there exists a matrix Pq ∈ GLn(K) such that
Bq = Pq · [Aq].
Let us define the regular singular q-difference equation. We shall start from the local analytic
study, i.e., taking field C({Q}), and then look for solutions in the field K = M(C∗) or M(C∗, 0).
Definition 2.9. Let Aq ∈ GLn(C({Q})), a system qQ∂QXq(Q) = Aq(Q)Xq(Q) is said to be
regular singular at Q = 0 if there exists a q-gauge transform Pq ∈ GLn(C({Q})) such that the
matrix (Pq · [Aq])(0) is well-defined and invertible: Pq · [Aq](0) ∈ GLn(C).
Definition 2.10. Consider a regular singular q-difference system qQ∂QXq(Q) = Aq(Q)Xq(Q).
Suppose Aq(0) ∈ GLn(C) and denote by (λi) the eigenvalues of the matrix Aq(0). This q-
difference system is said to be non q-resonant if for every i ̸= j, we have λi
λj
/∈ qZ\{0}, where
qZ\{0} :=
{
qk | k ∈ Z\{0}
}
⊂ C.
Let’s introduce some special functions which are needed to solve regular singular q-difference
equations.
We define Jacobi’s theta function by
θq(Q) =
∑
d∈Z
q
d(d−1)
2 Qd.
This function satisfies the q-difference equation qQ∂Qθq(Q) = 1
Qθq(Q). And it has a famous
Jacobi’s triple identity
θq(Q) = (q; q)∞(−Q; q)∞(−q/Q; q)∞.
In the following, we define two special functions which are essential in solving regular singular
(irregular) q-difference equations.
6 Y. Wen
Definition 2.11. Let λq ∈ C∗. The q-character associated to λ is the function eq,λq ∈ M (C∗)
defined by
eq,λq(Q) =
θq(Q)
θq(λqQ)
∈ M(C∗).
The function eq,λq satisfies the q-difference equation qQ∂Qeq,λq(Q) = λqeq,λq(Q).
Definition 2.12. The q-logarithm is the function ℓq ∈ M (C∗) defined by
ℓq(Q) = −Q
θ′q(Q)
θq(Q)
.
By a little computation, one could know that the function ℓq satisfies the following q-difference
equation
qQ∂Qℓq(Q) = ℓq(Q) + 1.
Now we can state the existence of a fundamental solution for regular singular q-difference
equations under certain conditions.
For a q-difference system qQ∂QXq(Q) = Aq(Q)Xq(Q), without loss of generality, we assume
Aq(0) ∈ GLn(C) and moreover that it is non-resonant. We can recursively build a gauge trans-
form Fq ∈ GLn(C({Q})) which sends the matrix Aq(0) to the constant matrix Aq(Q), for details,
see [16, Corollary 3.2.4]. Then we take the Jordan–Chevalley decomposition of Aq(0) = AsAu,
where As is semi-simple, Au is unipotent and As, Au commute.
Since N = Au − In is nilpotent, we can define
A
ℓq
u := (In +N)ℓq :=
∑
k≥0
(
ℓq
k
)
Nk, (2.1)
where(
ℓq
k
)
:=
ℓq(ℓq − 1) · · · (ℓq − (k − 1))
k!
.
Note that (2.1) is actually a finite sum and A
ℓq
u is unipotent, and we have
qQ∂QA
ℓq
u = AuA
ℓq
u = A
ℓq
u Au.
Thus we set
eq,Au := A
ℓq
u .
Take a basis change P to diagonalise As = P−1 diag(λi)P . We define
eq,As := P−1 diag(eq,λi
(Q))P, (2.2)
which satisfies
qQ∂Qeq,As = Aseq,As = eq,AsAs.
Then one can check that the product Fqeq,Aseq,Au =: Xq(Q) is a fundamental solution of the
q-difference system qQ∂QXq(Q) = Aq(Q)Xq(Q). We arrive at the following theorem.
Proposition 2.13 ([16, Theorem 3.3.1]). The q-difference system σqXq(Q) = Aq(Q)Xq(Q),
regular singular at Q = 0, admits a fundamental matricial solution X := Meq,C ∈
GLn(M(C∗, 0)), where C ∈ GLn(C) and where M ∈ GLn(C({Q})). The eq,C is defined by
Jordan–Chevalley decomposition of C as above.
Remark 2.14. Let A,P ∈ GLn(C), one can check that eq,PAP−1 = Peq,AP
−1. Thus, (2.2) is
independent of the choice of P .
Difference Equation for Quintic 3-Fold 7
2.2 Monodromy of regular singular q-difference equations
Definition 2.15. A q-difference system qQ∂QXq(Q) = Aq(Q)Xq(Q) is called fuchsian if it is
regular singular both at Q = 0 and Q = ∞.
It is easy to see the difference equation (1.3) is not fuchsian since it is not regular singular at
Q = 0. But we will see it is regular singular at Q = ∞ (see (3.5)).
Definition 2.16. Let qQ∂QXq(Q) = Aq(Q)Xq(Q) be a fuchsian q-difference system. This q-
difference system admits a fundamental solution X0(Q) at Q = 0 and a second one X∞(1/Q) at
Q = ∞. Birkhoff’s connection matrix (or q-monodromy) Pq is the ratio
Mq(Q) = (X∞(1/Q))−1X0(Q).
Since the connection matrix relates two fundamental matrix solutions. It is invariant by
difference operator qQ∂Q , i.e.,
Mq(Q) ∈ GLn(M(Eq)).
However, it is not well defined: it depends on the choice of fundamental matrix solutions. To get
rid of this dependence, we need to consider the following triple.
Definition 2.17. A Birkhoff connection triple is a triple(
A(0),Mq, A
(∞)
)
∈ GLn(C)×GLn(Eq)×GLn(C)
up to certain equivalent. Where A(0) and A(∞) are related to the fundamental solutions at Q = 0
and Q = ∞ respectively, for more details, see [16, p. 133].
The data of Birkhoff’s connection triples classifies fuchsian q-difference systems up to gauge
transformations.
Proposition 2.18 ([16, Theorem 3.4.9]). Rational classes (under rational equivalence, i.e.,
over field C(Q)) of fuchsian rational systems are in bijection with equivalence classes of Birkhoff
connection triples.
2.3 Confluence of regular singular q-difference equations
First, let us introduce some interesting formulas we needed when considering the confluence of
difference equations. We fix τ0 such that Im τ0 > 0 and q0 := e−2iπτ0 and |q0| < 1. This defines
a discrete logarithmic spiral qZ0 :=
{
qk0 | k ∈ Z
}
⊂ C and a continuous spiral qR0 :=
{
qk0 | k ∈ R
}
⊂ C. Let Ω = C∗−qR0 . Denote by log(Q) the logarithm on Ω such that 1 7→ 0. LetQµ := eµ log(Q).
Lemma 2.19 ([15, Section 3.1.7, Corollaire 1]). Let q(t) = qt0, t ∈ (0, 1]. Assume there exist
complex numbers α0, α1 ∈ C so that Qi(q(t)) = Q0q
αit+o(t)
0 , Q0 ∈ Ω. Then, on Ω, we have the
uniform convergence when t → 0
lim
q→1
θq(t)(Q1(q(t)))
θq(t)(Q2(q(t)))
= Qα2−α1
0 .
Proposition 2.20 ([15, Sections 3.1.3 and 3.1.4]). As the above notation, consider λq(t), µ ∈ C∗
such that
λq(t)−1
q−1 → µ. Then we have the asymptotics:
1. We have the uniform convergence on any compact of Ω
lim
t→0
(q(t)− 1)ℓq(t)(−Q) = log(Q).
8 Y. Wen
2. We have the uniform convergence on any compact of Ω
lim
t→0
eq(t),λq(t)
(−Q) = Qµ.
Now, let’s introduce the definition of confluence.
Definition 2.21 ([15, Section 3.2]). Let q(t) = qt0, for t ∈ (0, 1]. A regular singular, non q-
resonant difference system qQ∂QXq(Q) = Aq(Q)Xq(Q) is said to be confluent if it satisfies four
conditions below. Set Bq(Q) =
Aq(Q)−Id
q−1 , whose coefficients have poles Q1(q), . . . , Qk(q) in the
input Q. We require that
1. The q-spirals satisfy
⋂k
i=1Qi(q0)q
R
0 = ∅.
2. There exists a matrix B̃ ∈ GLn(C(Q)) such that
lim
t→0
Bq(t) = B̃,
uniformly in Q on any compact of C∗ −
⋃k
i=0Qiq
R
0 , set Q0 = 1.
3. This limit defines a regular singular, non resonant differential system
Q
d
dQ
X̃ = B̃X̃.
4. There exists, for each t, a Jordan decompositions Bq(t)(0) = P−1
q(t)Jq(t)Pq(t) as well as
B̃(0) = P̃−1J̃ P̃ . We ask that
lim
t→0
Pq(t) = P̃ .
If the difference system is confluent, then there is a confluence of the solutions.
Proposition 2.22 ([13, Theorem V.2.4.7]). Let q(t) = qt0, for t ∈ [0, 1]. Consider a regu-
lar singular confluent q-difference system qQ∂QXq(Q) = Aq(Q)Xq(Q), whose limit system is
Q∂QX̃(Q) = B̃(Q)X̃(Q).
Assume that there exists a vector X0 ∈ Cn\{0}, independent of q, such that Aq(t)X0 = X0 for
all t ∈ (0, 1]. We also assume that we have a solution Xq(Q) of the q-difference system satisfying
the initial condition Xq(0) = X0.
Let X̃ (Q) be the unique solution of Q∂QX(Q) = B̃(Q)X(Q) satisfying the initial condition
X̃ (0) = X0. We have
lim
t→0
Xq(t)(Q) = X̃ (Q)
uniformly in Q on any compact of C∗ −
⋃k
i=0Qiq
R
0 .
3 The difference equation for quintic
3.1 General technique: Newton polygon
Let’s consider the equation
n∑
i=0
ai(Q)(σq)
if(Q) = 0,
Difference Equation for Quintic 3-Fold 9
with
ai(Q) = ai,0 + ai,1Q+ ai,2Q
2 + · · · .
We call the following equation the characteristic equation
an,0x
n + an−1,0x
n−1 + · · ·+ a1,0x+ a0,0 = 0, (3.1)
which plays an important role in constructing solutions.
Denote by ai,ji the first nonzero coefficient in ai(Q), and choosing i- and j-axes as horizontal
and vertical axes respectively, plot the points (n− i, ji). Construct a broken line, convex down-
ward, such that both ends of each segment of the line are points of the set (n− i, ji). Then we
obtain a Newton polygon as follows
Note that the horizontal segment corresponds to the characteristic equation
ak,0x
k + ak−1,0x
k−1 + · · ·+ ad,0x
d = 0.
The degree of the above characteristic equation is 1 less than the number of points on or above
that segment.
Example 3.1. Consider the following equation:[(
Q4 + 2Q7
)
σ6
q +
(
Q+ 3Q5
)
σ5
q +
(
3 + 2Q3
)
σ4
q + 2σ3
q + 3Qσ2
q +Q2σq
]
f(Q) = 0.
Then the associated Newton polygon is
(0,4)
(1,1)
(2,0) (3,0) (4,0)
(5,2)
The general technique to construct solutions is as follows:
10 Y. Wen
� Horizontal segment: As mentioned above, it corresponds to characteristic equation. Using
the non-zero roots, we could construct the associated solutions as regular singular cases.
� Non-horizontal segment: For each non-horizontal segment of slope µ, a rational number.
– If µ = r is an integer, we consider a formal series solution of the form
θrq(Q)
∞∑
n=0
fn(q)Q
n.
– If µ = t/s is a rational number with s positive, then we consider a formal series
solution of the form
θqt/s
(
Qt/s
) ∞∑
n=0
fn(q)Q
n/s.
For more details, see [1, 2, 14].
Remark 3.2. In Adams’ works, he used q
µ
2
(t2−t), where t = lnQ
ln q .
3.2 Solutions at Q = 0
The K-theoretic I-function of quintic is as follows [9]
IK = P lq(Q)
∑
d≥0
∏5d
k=1
(
1− P 5qk
)∏d
k=1
(
1− Pqk
)5Qd,
it satisfies the following degree 25 difference equation[(
1− qQ∂Q
)5 −Q
5∏
k=1
(
1− qkq5Q∂Q
)]
IK = 0 mod
(
(1− P )5
)
. (3.2)
Remark 3.3. See [5] for additional discussion on q-deformed Picard–Fuchs equation.
The characteristic equation at Q = 0 is
(1− x)5 = 0.
We can construct only 5 solutions by expanding IKX (q,Q) with respect to the K-group basis
(1 − P )k, k = 0, 1, 2, 3, 4. Next we use Adams’ method to find other solutions at Q = 0. The
Newton’s polygon of the above difference equation (3.2) is as follows
(0,1)
(20,0)
Then we have solutions at Q = 0 of the form
θ
q
1
20
(
Q
1
20
)
G
(
Q
1
20
)
= θ
q
1
20
(
Q
1
20
) ∞∑
d=0
gdQ
d
20 ,
Difference Equation for Quintic 3-Fold 11
where θ
q
1
20
(
Q
1
20
)
satisfies
qQ∂Qθ
q
1
20
(
Q
1
20
)
=
(
1
Q
) 1
20
θ
q
1
20
(
Q
1
20
)
,
then
q5Q∂Qθ
q
1
20
(
Q
1
20
)
= q−
1
2
(
1
Q
) 1
4
θ
q
1
20
(
Q
1
20
)
.
Substituting into (3.2), we find that G
(
Q
1
20
)
satisfies the following difference equation:[(
Q
1
20 − q−
4
20 qQ∂Q
)(
Q
1
20 − q−
3
20 qQ∂Q
)(
Q
1
20 − q−
2
20 qQ∂Q
)(
Q
1
20 − q−
1
20 qQ∂Q
)(
Q
1
20 − qQ∂Q
)
−
(
Q
1
4 − q−
2
4 q5Q∂Q
)(
Q
1
4 − q−
1
4 q5Q∂Q
)(
Q
1
4 − q5Q∂Q
)(
Q
1
4 − q
1
4 q5Q∂Q
)(
Q
1
4 − q
1
2 q5Q∂Q
)]
×G
(
Q
1
20
)
= 0.
Let z = Q
1
20 , p = q
1
20 and σp = pz∂z , then the above difference equation takes the following form[(
z − p−4σp
)(
z − p−3σp
)(
z − p−2σp
)(
z − p−1σp
)
(z − σp)
−
(
z5 − p−10σ5
p
)(
z5 − p−5σ5
p
)(
z5 − σ5
p
)(
z5 − p5σ5
p
)(
z5 − p10σ5
p
)]
G(z) = 0. (3.3)
For z = 0, we obtain the characteristic equation
p−10x5 − x25 = 0,
i.e.,
x = ξp−
1
2 , for ξ20 = 1.
Consider a solution of the form
ep,ξp−1/2(z)F (z) = ep,ξp−1/2(z)
∑
n≥0
fnz
n,
then F (z) satisfies[(
z − ξp−
9
2σp
)(
z − ξp−
7
2σp
)(
z − ξp−
5
2σp
)(
z − ξp−
3
2σp
)(
z − ξp−
1
2σp
)
−
(
z5 − ξ5p−
25
2 σ5
p
)(
z5 − ξ5p−
15
2 σ5
p
)(
z5 − ξ5p−
5
2σ5
p
)(
z5 − ξ5p
5
2σ5
p
)(
z5 − ξ5p
15
2 σ5
p
)]
× F (z) = 0.
After a short computation, we expand the above difference equation as follows[
ξ5p−
25
2
(
σ25
p − σ5
p
)
+ 5ξ4p−8zσ4
p − 10ξ3p−
9
2 z2σ3
p + 10ξ2p−2z3σ2
p − 5ξp−
1
2 z4σp
+ z5
(
1−
(
1 + p
15
2 + p
55
2 + p
95
2 + p
135
2
)
σ20
p
)
+ ξ15
(
p
15
2 + p
55
2 + 2p
95
2 + 2p
135
2 + 2p
175
2 + p
215
2 + p
255
2
)
z10σ15
p
− ξ10
(
p10 + p30 + 2p50 + 2p70 + 2p90 + p110 + p130
)
z15σ10
p
+ ξ5
(
p
15
2 + p
55
2 + p
95
2 + p
135
2 + p
175
2
)
z20σ5
p − z25
]
F (z) = 0. (3.4)
Thus we arrive at the following proposition.
12 Y. Wen
Proposition 3.4. The exceptional 20 solutions of (3.3) are given as follows: for each 20th root
of unity ξ, we have a solution of form
ep,ξp−1/2(z)F (z) = ep,ξp−1/2(z)
∑
n≥0
fnz
n,
where F (z) satisfies[(
z − ξp−
9
2σp
)(
z − ξp−
7
2σp
)(
z − ξp−
5
2σp
)(
z − ξp−
3
2σp
)(
z − ξp−
1
2σp
)
−
(
z5 − ξ5p−
25
2 σ5
p
)(
z5 − ξ5p−
15
2 σ5
p
)(
z5 − ξ5p−
5
2σ5
p
)(
z5 − ξ5p
5
2σ5
p
)(
z5 − ξ5p
15
2 σ5
p
)]
× F (z) = 0.
Proof. From (3.4), only f0 is a free variable, and fn is determined by {fk}k≤n. Then we
obtain 20 solutions. ■
Remark 3.5. The exceptional solutions are linearly independent over M(Eq), since their Wron-
skian matrix (see [16, Lemma 2.3.3]) does not equal to 0. Actually, it is sufficient to see the
Wronskian matrix for {ep,ξkp−1/2}19k=0 and it turns out to be the Vandermonde matrix.
3.3 Solutions at Q = ∞
Let w = 1/Q, (3.2) becomes[
5∏
k=1
(
1− q−kq5w∂w
)
− q10wq20w∂w
(
1− qw∂w
)5]
F (w) = 0, (3.5)
which is regular singular. The characteristic equation is as follows
5∏
k=1
(
1− q−kx5
)
= 0,
with 25 distinct roots
q
l
5 ξm, l = 1, . . . , 5, m = 0, . . . , 4.
Here ξ is the fifth root of unity. For each root q
l
5 ξm, we construct the following solution
Wl,m(w) = e
q,q
l
5 ξm
(w)
∑
d≥0
fdw
d.
Substituting the above formula into (3.5), since
qw∂we
q,q
l
5 ξm
(w) = q
l
5 ξme
q,q
l
5 ξm
(w),
then
∑
d≥0 fdw
d satisfies the following difference equation[
5∏
k=1
(
1− ql−kq5w∂w
)
− q10+4lwq20w∂w
(
1− q
l
5 ξmqw∂w
)5]
G(w) = 0.
Difference Equation for Quintic 3-Fold 13
Then one obtain 25 solutions at Q = ∞ as follows
Wl,m(w) = e
q,q
l
5 ξm
(w)
∑
d≥0
∏d−1
k=0
(
1− ξ−mq−k− l
5
)5∏5d−1
k=0
(
1− q−k−l
) wd
= e
q,q
l
5 ξm
(1/Q)
∑
d≥0
∏d−1
k=0
(
1− ξ−mq−k− l
5
)5∏5d−1
k=0
(
1− q−k−l
) Q−d.
Since we require |q| < 1 and
lim
n→∞
∏(n+1)−1
k=0
(
1− ξ−mq−k− l
5
)5∏5(n+1)−1
k=0
(
1− q−k−l
) ∏5n−1
k=0
(
1− q−k−l
)∏n−1
k=0
(
1− ξ−mq−k− l
5
)5 = lim
n→∞
(
1− ξ−mq−n− l
5
)5∏5n+4
k=5n
(
1− q−k−l
) = 0.
The 25 solutions are convergent.
Remark 3.6. These 25 solutions may relate to K-theoretic FJRW theory, for hints, see [10].
4 Auxiliary q-series and analytic continuation
In this section, we construct a K-group valued q-series, which is a generalization of the se-
ries (1.2), and it satisfies a difference equation like (3.2). Besides, we use Mellin–Barnes–Watson
method to relate the solutions at Q = 0 and Q = ∞.
4.1 Auxiliary q-series
We construct the following K
(
Pn−1
)
valued q-series motivated by [4]:
Fm,n(α⃗, Q) = P lq(Q)
∞∑
d=0
∏m
i=1(Pαi; q)d
(Pq; q)nd
Qd, m ≥ n. (4.1)
Since (1− P )n = 0 in K
(
Pn−1
)
, then it satisfies the following difference equation:[(
1− qQ∂Q
)n −Q
m∏
i=1
(
1− αiq
Q∂Q
)]
Fm,n(Q) = 0 mod
(
(1− P )n
)
. (4.2)
Suppose αi /∈ αjq
Z\{0}, i.e., the difference equation is (q-)non-resonant, the characteristic equa-
tion at Q = ∞ is as follows
m∏
i=1
(
1− α−1
i x
)
= 0,
with m-distinct roots {αi}mi=1, the same as the discussion in previous section, for each root αi,
we construct a solution as follows
Wi(1/Q) = eq,αi(1/Q)
∑
d≥0
fdQ
−d.
Recall
qQ∂Qeq,αi(1/Q) = α−1
i eq,αi(1/Q),
14 Y. Wen
then
∑
d≥0 fdQ
−d satisfies the following difference equation[(
1− α−1
i qQ∂Q
)n −Q
m∏
j=1
(
1− αj/αiq
Q∂Q
)]
G
(
Q−1
)
= 0.
Thus, we obtain
Wj(1/Q) = eq,αj (1/Q)
∑
d≥0
∏d−1
k=0(1− αjq
k)nq(m−n)d(d−1)/2∏m
i=1
∏d
k=1(1− αj/αiqk)
×
(
(−1)m−n
(
m∏
i=1
αj/αi
)
α−n
j qm/Q
)d
= eq,αj (1/Q)
∑
d≥0
∏d−1
k=0
(
1− α−1
j q−k
)n∏m
i=1
∏d
k=1
(
1− αi/αjq−k
)Q−d.
Remark 4.1. The above solutions are linearly independent over M(Eq). The same reason as
Remark 3.2.
Remark 4.2. If we take n = 5, m = 25 and {αi}25i=1 =
{
ξlq
k
5 | k, l = 1, 2, 3, 4, 5
}
, then
F25,5
({
ξlq
k
5
}
, Q
)
= P lq(Q)
∑
d≥0
∏5d
k=1
(
1− P 5qk
)∏d
k=1
(
1− Pqk
)5Qd,
and for αj = q
l
5
ξm , we have
Wj(1/Q) = e
q,q
l
5 ξm
(1/Q)
∏d−1
k=0
(
1− q−
l
5 ξ−mq−k
)n∏d
k=1
(
1− q1−lq−5k
)
· · ·
(
1− q5−lq−5k
)Q−d
= e
q,q
l
5 ξm
(1/Q)
∑
d≥0
∏d−1
k=0
(
1− ξ−mq−k− l
5
)5∏5d−1
k=0
(
1− q−k−l
) Q−d
= Wl,m(1/Q).
4.2 Analytic continuation
For the sake of simplicity, we shall assume in this section that 0 < q < 1 and write
q = e−w, w > 0.
The results can be extended to complex q in the unit disc using analytic continuation.
Consider the following contour integral. We follow the argument of [6, pp. 115–118] to show
that this integral is well defined. For |Q| < 1, we can close the contour to the right, it equals
to (4.1),
P lq(Q)
∏m
i=1(Pαi; q)∞
(Pq; q)n∞
∫
C
(
Pqs+1; q
)n
∞∏m
i=1(Pαiqs; q)∞
π(−Q)s
sinπs
ds
−2πi
. (4.3)
Here we view P = e−H . Although H is the hyperplane class, we consider it as a formal variable
valued in C. C is a curve from −i∞ to +i∞ such that only the non-negative zeros of sinπs lie
on the right side of C.
Difference Equation for Quintic 3-Fold 15
By the triangle inequality,∣∣1− ∣∣a∣∣e−ωRe(s)
∣∣ ≤ |1− aqs| ≤ 1 + |a|e−ωRe(s),
we have∣∣∣∣
(
Pqs+1; q
)n
∞∏m
i=1(Pαiqs; q)∞
∣∣∣∣ ≤ ∞∏
k=0
(
1 + |P |e−(k+1+Re(s)w)
)n∏m
i=1
(
1− |Pαi|e−(k+Re(s)w)
) ,
which is bounded on the contour C. Hence the integral (4.3) converges if | arg(−Q)| < π.
Let CR+ be a large clockwise-oriented semicircle of radius R with a center at the origin that
lies to the right of C. The semicircle is terminated by C and bounded away from the poles.
Now consider the contour integral over CR+ instead of C.
Setting s = Reiθ, we have for |s| < 1 that
Re
[
log
(−Q)s
sinπs
]
= R[cos θ log |Q| − sin θ arg(−Q)− π| sin θ|] +O(1)
≤ −R[sin θ arg(−Q) + π| sin θ|] +O(1).
Hence, when |Q| < 1 and | arg(−Q)| < π − δ, 0 < δ < π, we have
(−Q)s
sinπs
= O[exp(−δR| sin θ|)],
as R → ∞, then the integral on CR+ tends to zero as R → ∞. Therefore, by applying Cauchy’s
theorem, we can prove (4.3) equals to (4.1) through tedious computation.
Similarly, if we replace the contour C by a contour CR− consisting of a large counterclockwise-
oriented semicircle of radius R with center at the origin that lies to the left of C. From an
asymptotic formula
Re[log(qs; q)∞] =
ω
2
(Re(s))2 +
ω
2
Re(s) +O(1),
as R → −∞. Without loss of generality, we assume P = qh, αi = qai and let h, ai be real
numbers.
Then
n(1 + h+Re(s))2 + n(1 + h+Re(s))−
m∑
i=1
[
(ai + h+Re(s))2 + (ai + h+Re(s))
]
= 2
[
n(1 + h)−
m∑
i=1
(ai + h)
]
Re(s) + (n−m)
(
Re(s) + Re2(s)
)
+ const.
Note that q = e−w, w > 0, then the asymptotic formula for (qs; q)∞ implies that(
Pqs+1; q
)n
∞∏m
i=1(Pαiqs; q)∞
= O
(∣∣∣∣qm−n
2
∏m
i=1 Pαi
(Pq)n
∣∣∣∣Re(s)
)
,
when Re(s) → −∞ with s bounded away from the zeros and poles.
Similarly, it can be shown that if |Q| is big enough, we can close the contour the left, i.e., the
integral (4.3) on CR− tends to zero as R → −∞. Thus, (4.3) equals to the sum of residues at
s = w−1(−H + logαj + l · 2πi)− k and s = −1− h,
16 Y. Wen
where j = 1, . . . ,m, l ∈ Z, n, h ∈ N. The residue at s = −1− h contains a term
(1− P )n
from (
Pqs+1; q
)n
∞.
And the residue at s = w−1(−H + logαj + l · 2πi)− k is
Ress=w−1(−H+logαj+l·2πi)−k
(
Pqs+1; q
)n
∞∏m
i=1(Pαiqs; q)∞
π(−Q)s
sinπs
=
(
α−1
j q1−k; q
)n
∞∏m
i=1,i ̸=j
(
αi/αjq−k; q
)
∞
(qk+1; q)∞
(q, q; q)∞
(−1)kq
k(k+1)
2
× π(−Q)w
−1(−H+logαj)−kw−1
sinπ
(
w−1(−H + logαj)− k + w−12lπi
) exp{2lπiw−1 log(−Q)
}
.
If we sum over k, we obtain
∞∑
k=0
(
α−1
j q1−k; q
)n
∞∏m
i=1,i ̸=j
(
αi/αjq−k; q
)
∞
(
qk+1; q
)
∞
(q, q; q)∞
q
k(k+1)
2 Q−k
=
(
α−1
j q; q
)n
∞∏m
i=1,i ̸=j(αi/αj ; q)∞(q; q)∞
∞∑
k=0
(
α−1
j ; q−1
)n
k∏m
i=1
(
αi/αjq−1; q−1
)
k
Q−k
=
(
α−1
j q; q
)n
∞∏m
i=1,i ̸=j(αi/αj ; q)∞(q; q)∞
e−1
q,αj
(1/Q)Wj(1/Q).
If we sum over l, we have
∞∑
l=−∞
exp
{
2lπiw−1 log(−Q)
}
sin(w−1
(
−H + logαj)π + 2lπ2iw−1
)(−Q)w
−1(−H+logαj)
= −w(q, q, PαjQ, q/(PαjQ); q)∞
π(Pαj , q/(Pαj), Q, q/Q; q)∞
,
which comes from the following lemma.
Lemma 4.3 ([6, equation (4.3.9), p. 119]).
∞∑
m=−∞
csc
(
απ − 2mπ2iw−1
)
exp
{
2mπiw−1 log(−Q)
}
(−Q)−α =
w(q, q, aQ, q/(aQ); q)∞
π(a, q/a,Q, q/Q; q)∞
,
where a = qα = e−wα.
Summing up the above discussion, we arrive at the following theorem.
Theorem 4.4. Suppose αi /∈ αjq
Z\{0}. For m ≥ n, the K
(
Pn−1
)
valued q-series has the
following analytic continuation:
P lq(Q)
∞∑
d=0
∏m
i=1(Pαi; q)d
(Pq; q)nd
Qd = P lq(Q)
∏m
i=1(Pαi; q)∞
(Pq; q)n∞
m∑
j=1
(q, q, PαjQ, q/(PαjQ); q)∞
(Pαj , q/(Pαj), Q, q/Q; q)∞
×
(
α−1
j q; q
)n
∞ · e−1
q,αj
(1/Q)∏m
i=1,i ̸=j(αi/αj ; q)∞(q; q)∞
Wj(1/Q).
for |Q| < 1 and | arg(−Q)| < π − δ, 0 < δ < π.
Remark 4.5. In general, the above formula only contains a part of solutions at Q = 0.
Difference Equation for Quintic 3-Fold 17
5 A special fuchsian case
Consider the following difference equation:[(
1− qQ∂Q
)4 −Q
(
1− q
1
5 qQ∂Q
)(
1− q
2
5 qQ∂Q
)(
1− q
3
5 qQ∂Q
)(
1− q
4
5 qQ∂Q
)]
F (Q) = 0. (5.1)
By definition, it is fuchsian. One could easily construct the solutions at Q = ∞, indeed, let
w = 1/Q, then (5.1) becomes[(
1− q−
1
5 qw∂w
)(
1− q−
2
5 qw∂w
)(
1− q−
3
5 qw∂w
)(
1− q−
4
5 qw∂w
)
− q2w
(
1− qw∂w
)4]
G(w) = 0.
The characteristic equation of the above difference equation at w = 0 is(
1− q−
1
5x
)(
1− q−
2
5x
)(
1− q−
3
5x
)(
1− q−
4
5x
)
= 0,
with 4 different roots
x = q
i
5 , i = 1, 2, 3, 4.
So the difference equation is non-resonant. By using Frobenius method, we could construct
solutions of the form
Wi(w) = e
q,q
i
5
(w)W̃i(w) = e
q,q
i
5
(w)
∞∑
n=0
ginw
n.
After a short computation one obtain four solutions as follows
W1(1/Q) = e
q,q
1
5
(1/Q)4ϕ3
(
q
1
5 , q
1
5 , q
1
5 , q
1
5 ; q
4
5 , q
3
5 , q
2
5 ; q; q2/Q
)
, (5.2)
W2(1/Q) = e
q,q
2
5
(1/Q)4ϕ3
(
q
2
5 , q
2
5 , q
2
5 , q
2
5 ; q
6
5 , q
4
5 , q
3
5 ; q; q2/Q
)
, (5.3)
W3(1/Q) = e
q,q
3
5
(1/Q)4ϕ3
(
q
3
5 , q
3
5 , q
3
5 , q
3
5 ; q
7
5 , q
6
5 , q
4
5 ; q; q2/Q
)
, (5.4)
W4(1/Q) = e
q,q
4
5
(1/Q)4ϕ3
(
q
4
5 , q
4
5 , q
4
5 , q
4
5 ; q
8
5 , q
7
5 , q
6
5 ; q; q2/Q
)
. (5.5)
Remark 5.1. By a small trick, we could find the formula for W̃i(w) easily. Note that, for α /∈ N,
we have(
1− qw∂w
)k
eq,qα(w)W̃ (w) = eq,qα(w)
(
1− qα · qw∂w
)k
W̃ (w).
For example, W̃1(w) satisfies the following difference equation[(
1− qw∂w
)(
1− q−
1
5 qw∂w
)(
1− q−
2
5 qw∂w
)(
1− q−
3
5 qw∂w
)
− q2w
(
1− q
1
5 qw∂w
)4]
W̃1(w) = 0.
From the above explicit form, it’s quite easy to find a q-hypergeometric series representation
for W̃1(w).
Let’s consider the solutions of (5.1) at Q = 0, the characteristic equation is as follows
(1− x)4 = 0.
From the general theorem (see [2, 14] for details), we have solutions of the form
Gi(Q) = lq(Q)Gi−1(Q) + gi(Q), i = 2, 3, 4, (5.6)
18 Y. Wen
where
G1(Q) =
∞∑
d=0
∏4
i=1
(
q
i
5 ; q
)
d
(q; q)4d
Qd = 4ϕ3
(
q
1
5 , q
2
5 , q
3
5 , q
4
5 ; q, q, q; q;Q
)
(5.7)
is a solution of (5.1) and gi(Q) are power series.
For general q-hypergeometric function 4ϕ3(a1, a2, a3, a4; b1, b2, b3; q; z), we have the following
famous transformation formula.
Proposition 5.2 ([6, equation (4.5.2), p. 120]).
4ϕ3(a1, a2, a3, a4; b1, b2, b3; q; z)
=
(a2, a3, a4, b1/a1, b2/a1, b3/a1, a1z, q/a1z; q)∞
(b1, b2, b3, a2/a1, a3/a1, a4/a1, z, q/z; q)∞
× 4ϕ3
(
a1, a1q/b1, a1q/b2, a1q/b3; a1q/a2, a1q/a3, a1q/a4; q;
qb1b2b3
za1a2a3a4
)
+ idem(a1, a2, a3, a4). (5.8)
The symbol “ idem(a1, a2, a3, a4)” after an expression stands for the sum of the 3 expressions
obtained from the preceding expression by interchanging a1 with ak, k = 2, 3, 4.
So in our special case, (5.7) can be written as a combination of (5.2)–(5.5). As mentioned
before, the other solutions has the form of (5.6) which is hard to compute. Thus, it’s very hard
to find the connection matrix.
5.1 Connection matrix
Notice that (5.1) is a special case of (4.2) with n = m = 4 and
αi = q
i
5 , i = 1, 2, 3, 4,
then
F4,4
({
q
i
5
}4
i=1
, Q
)
= P lq(Q)
∞∑
d=0
∏4
i=1
(
Pq
i
5 ; q
)
d
(Pq; q)4d
Qd. (5.9)
The solutions of difference equation (5.1) atQ = 0 are given by the expansion of F4,4
({
q
i
5
}4
i=1
, Q
)
with respect to (1− P )i, i = 0, 1, 2, 3. From Theorem 4.4, then we have
Corollary 5.3.
P lq(Q)
∞∑
d=0
∏4
i=1
(
Pq
i
5 ; q
)
d
(Pq; q)4d
Qd = P lq(Q)
∏4
i=1
(
Pq
i
5 ; q
)
∞
(Pq; q)4∞
4∑
j=1
(
q, q, Pq
j
5Q, q/
(
Pq
j
5Q
)
; q
)
∞
×
(
Pq
j
5 , q/
(
Pq
j
5
)
, Q, q/Q; q
)
∞
(
q−
j
5 q; q
)4
∞e−1
q,qj/5
(1/Q)∏4
i=1,i ̸=j
(
q
i−j
5 ; q
)
∞(q; q)∞
×Wj(1/Q). (5.10)
Remark 5.4. Taking P = 1 in the above formula, we obtain
∞∑
d=0
∏4
i=1
(
q
i
5 ; q
)
d
(q; q)4d
Qd =
4∑
k=1
(
q
1
5 · · · k̂ · · · q
4
5 , q1−
k
5 , q1−
k
5 , q1−
k
5 ; q
)
∞(
q
1−k
5 , . . . k̂ . . . , q
4−k
5 , q, q, q; q
)
∞
(
q
k
5Q; q
5−k
5 /Q
)
∞
(Q, q/Q)∞
× 4ϕ3
(
q
k
5 , q
k
5 , q
k
5 , q
k−1
5
+1; q
k−2
5
+1, . . . k̂ . . . , q
k−4
5
+1; q; q2/Q
)
.
It agrees with (5.8).
Difference Equation for Quintic 3-Fold 19
In order to obtain the connection matrix, we need to expand (5.10) with respect to
{(1− P )k}3k=1. Notice that
P lq(Q) = (1− (1− P ))lq(Q) =
∑
k≥0
(−1)k
(
ℓq(Q)
k
)
(1− P )k,
where(
ℓq(Q)
k
)
=
1
k!
k−1∏
r=0
(ℓq(Q)− r).
Then
P lq(Q)
∞∑
d=0
∏4
i=1
(
Pq
i
5 ; q
)
d
(Pq; q)4d
Qd =
3∑
m=0
∑
a+b=m
(−1)a
(
ℓq(Q)
a
)
Xb(q,Q)(1− P )m,
where Xb(q,Q) is the coefficient of
∞∑
d=0
∏4
i=1
(
Pq
i
5 ; q
)
d
(Pq; q)4d
Qd =
3∑
b=0
Xb(q,Q)(1− P )b. (5.11)
Let’s consider the expansion of (Pqα; q)∞ and
(
P−1qα; q
)
∞. By definition
(Pqα; q)∞ =
∞∏
k=0
(
1− Pqα+k
)
.
For (1− Pqα+k), we have
(
1− Pqα+k
)
=
(
1− qα+k + qα+k(1− P )
)
=
(
1− qα+k
)(
1 +
qα+k
1− qα+k
(1− P )
)
.
Then
∞∏
k=0
(
1− qα+k
)(
1 +
qα+k
1− qα+k
(1− P )
)
= (qα; q)∞
∞∏
k=0
[
1 +
∞∑
k=0
qα+k
1− qα+k
(1− P ) +
∞∑
i<j
q2α+i+j
(1− qα+i)(1− qα+j)
(1− P )2
+
∞∑
i<j<l
q3α+i+j+l
(1− qα+i)(1− qα+j)(1− qα+l)
(1− P )3 +O
(
(1− P )4
)]
.
Similarly,
(
1− P−1qα+k
)
= 1− qα+k
1− (1− P )
=
(
1− qα+k
)
− qα+k(1− P )− qα+k(1− P )2
− qα+k(1− P )3 +O
(
(1− P )4
)
.
Then
∞∏
k=0
((
1− qα+k
)
− qα+k(1− P )− qα+k(1− P )2 − qα+k(1− P )3 +O
(
(1− P )4
))
20 Y. Wen
= (qα; q)∞
∞∏
k=0
[
1−
∞∑
k=0
qα+k
1− qα+k
(1− P )
+
( ∞∑
i<j
q2α+i+j
(1− qα+i)(1− qα+j)
−
∞∑
k=0
qα+k
1− qα+k
)
(1− P )2
+
(
−
∞∑
i<j<l
q3α+i+j+l
(1− qα+i)(1− qα+j)(1− qα+l)
+ 2
∞∑
i<j
q2α+i+j
(1− qα+i)(1− qα+j)
−
∞∑
k=0
qα+k
1− qα+k
)
(1− P )3
]
+O
(
(1− P )4
)
.
In order to simplify the computation, we introduce the following notations
f1(x) =
∞∑
k=0
xqk
1− xqk
,
f2(x) =
∞∑
i<j
x2qi+j
(1− xqi)(1− xqj)
,
f3(x) =
∞∑
i<j<l
x3qi+j+l
(1− xqi)(1− xqj)(1− xql)
and
F1(x1, x2, x3, x4) = −
4∑
k=1
f1(xk),
F2(x1, x2, x3, x4) = −
4∑
k=1
f2(xk) +
4∑
i<j
f1(xi)f1(xj),
F3(x1, x2, x3, x4) = −
4∑
k=1
f3(xk) +
4∑
i<j
(f1(xi)f2(xj) + f2(xi)f1(xj))
−
4∑
i<j<l
f1(xi)f1(xj)f1(xl).
With a little computation, we obtain∏4
i=1
(
Pq
i
5 ; q
)
∞
(Pq; q)4∞
=
∏4
k=1
(
q
k
5 ; q
)
∞
(q; q)4∞
[
1 +
(
F1
(
q
•
5
)
− F1(q)
)
(1− P )
+
(
F1(q)
2 − F1
(
q
•
5
)
F1(q)− F2(q) + F2
(
q
•
5
))
(1− P )2
+
(
F1(q)
3 + F1
(
q
•
5
)(
F1(q)
2 − F2(q)
)
+ 2F1(q)F2(q)− F1(q)F2
(
q
•
5
)
− F3(q) + F3
(
q
•
5
))
(1− P )3
]
+O
(
(1− P )4
)
. (5.12)
Here we use the notations:
Fi
(
q
•
5
)
:= Fi
(
q
1
5 , q
2
5 , q
3
5 , q
4
5
)
,
Fi(q) := Fi(q, q, q, q).
Difference Equation for Quintic 3-Fold 21
For simplicity, we write the above formula as∏4
i=1
(
Pq
i
5 ; q
)
∞
(Pq; q)4∞
=
∏4
k=1
(
q
k
5 ; q
)
∞
(q; q)4∞
[
1 +
3∑
k=1
Fk · (1− p)k +O
(
(1− P )4
)]
, (5.13)
where Fk stands for the coefficient of (1− P )k in (5.12).
Similarly, we consider
(Px; q)∞
(
P−1x−1q; q
)
∞.
Then (
1− Pxqd
)(
1− qP−1x−1qd
)
=
(
1− xqd + xqd(1− P )
)((
1− qx−1qd
)
− qx−1qd(1− P )− qx−1qd(1− P )2
− qx−1qd(1− P )3 +O
(
(1− P )4
))
=
(
1− xqd
)(
1− qx−1qd
)[
1 +
xqd
(
1− qx−2
)(
1− xqd
)(
1− qx−1qd
)(1− P )
− qd+1x−1
(1− xqd)(1− qx−1qd)
(1− P )2 − qd+1x−1
(1− xqd)(1− qx−1qd)
(1− P )3+O
(
(1− P )4
)]
.
Set
g1(x) =
∞∑
d=0
xqd
(
1− qx−2
)
(1− xqd)
(
1− qx−1qd
) ,
g2(x) =
∞∑
i<j
∏
k=i,j
(
xqk
(
1− qx−2
)
(1− xqk)
(
1− qx−1qk
))−
∞∑
d=0
qd+1x−1
(1− xqd)
(
1− qx−1qd
) ,
g3(x) =
∞∑
i<j<l
∏
k=i,j,l
(
xqk
(
1− qx−2
)
(1− xqk)
(
1− qx−1qk
))−
∑
i ̸=j
qi+j+1
(
1− qx−2
)∏
k=i,j(1− xqk)
(
1− qx−1qk
)
−
∞∑
d=0
qd+1x−1
(1− xqd)
(
1− qx−1qd
) .
Then
(Px; q)∞
(
P−1x−1q; q
)
∞
=
θq(−x)
(q; q)∞
(
1 + g1(x)(1− P ) + g2(x)(1− P )2 + g3(x)(1− P )3 +O
(
(1− P )4
))
.
So we obtain(
Pq
k
5Q,P−1Q−1q1−
k
5 ; q
)
∞(
Pq
k
5 , P−1q1−
k
5 ; q
)
∞
=
θq
(
−q
k
5Q
)
θq
(
−q
k
5
) [1 + (g1(q k
5Q
)
− g1
(
q
k
5
))
(1− P )
+
(
−g1
(
q
k
5Q
)
g1
(
q
k
5
)
+g21
(
q
k
5
)
+g2
(
q
k
5Q
)
−g2
(
q
k
5
))
(1−P )2
+
(
−g31
(
q
k
5
)
−g1
(
q
k
5
)
g2
(
q
k
5Q
)
+g1
(
q
k
5Q
)(
g21
(
q
k
5
)
−g2
(
q
k
5
))
+ 2g1
(
q
k
5
)
g2
(
q
k
5
)
+ g3
(
q
k
5Q
)
− g3
(
q
k
5
))
(1− P )3
]
+O
(
(1− P )4
)
. (5.14)
22 Y. Wen
For simplicity, we write the above formula as follows(
Pq
k
5Q,P−1Q−1q1−
k
5 ; q
)
∞(
Pq
k
5 , P−1q1−
k
5 ; q
)
∞
=
θq
(
−q
k
5Q
)
θq
(
−q
k
5
) [1 + 3∑
k=1
Gk · (1− P )k +O
(
(1− P )4
)]
, (5.15)
where Gk stands for the coefficient of (1− P )k in (5.14).
In conclusion, we arrive at the following corollary.
Corollary 5.5. The 4 solutions of (5.1) at Q = 0 are given as the expansion of (5.9), i.e.∑
a+b=m
(−1)a
(
ℓq(Q)
a
)
Xb(q,Q), m = 0, 1, 2, 3,
where Xb(q,Q) is defined in (5.11). The 4 solutions of (5.1) at Q = ∞ are given explicitly as
(5.2)–(5.5). The connection matrix is as follows
• X0(q,Q) =
∞∑
d=0
∏4
i=1
(
q
i
5 ; q
)
d
(q; q)4d
Qd
=
4∑
k=1
(
q
1
5 · · · k̂ · · · q
4
5 , q1−
k
5 , q1−
k
5 , q1−
k
5 ; q
)
∞(
q
1−k
5 , . . . k̂ . . . , q
4−k
5 , q, q, q; q
)
∞
(
q
k
5Q; q
5−k
5 /Q
)
∞
(Q, q/Q)∞
× e−1
q,qk/5
(1/Q)Wk(1/Q).
• X1(q,Q) =
4∑
k=1
(
q
1
5 · · · k̂ · · · q
4
5 , q1−
k
5 , q1−
k
5 , q1−
k
5 ; q
)
∞(
q
1−k
5 , . . . k̂ . . . , q
4−k
5 , q, q, q; q
)
∞
(
q
k
5Q; q
5−k
5 /Q
)
∞
(Q, q/Q)∞
× (G1 + F1)e
−1
q,qk/5
(1/Q)Wk(1/Q).
• X2(q,Q) =
4∑
k=1
(
q
1
5 · · · k̂ · · · q
4
5 , q1−
k
5 , q1−
k
5 , q1−
k
5 ; q
)
∞(
q
1−k
5 , . . . k̂ . . . , q
4−k
5 , q, q, q; q
)
∞
(
q
k
5Q; q
5−k
5 /Q
)
∞
(Q, q/Q)∞
× (G2 + F2 +G1F1)e
−1
q,qk/5
(1/Q)Wk(1/Q).
• X3(q,Q) =
4∑
k=1
(
q
1
5 · · · k̂ · · · q
4
5 , q1−
k
5 , q1−
k
5 , q1−
k
5 ; q
)
∞(
q
1−k
5 , . . . k̂ . . . , q
4−k
5 , q, q, q; q
)
∞
(
q
k
5Q; q
5−k
5 /Q
)
∞
(Q, q/Q)∞
× (G3 + F3 +G2F1 +G1F2)e
−1
q,qk/5
(1/Q)Wk(1/Q).
Here, for simplicity, we use the notations Gk and Fk defined in (5.13) and (5.15).
6 Confluence of the q-difference structure
Notice that
lim
q→1
1− qQ∂Q
1− q
= Q
d
dQ
,
then one could easily see that the following difference equation is confluent to (1.1), i.e.,
lim
q→1
[(
1− qQ∂Q
)4 −Q
(
1− q
1
5 qQ∂Q
)(
1− q
2
5 qQ∂Q
)(
1− q
3
5 qQ∂Q
)(
1− q
4
5 qQ∂Q
)]
/(1− q)4
=
[(
Q
d
dQ
)4
−Q
(
Q
d
dQ
+
1
5
)(
Q
d
dQ
+
2
5
)(
Q
d
dQ
+
3
5
)(
Q
d
dQ
+
4
5
)]
.
In the following, we set q(t) = e−t and P = qH = e−tH .
Difference Equation for Quintic 3-Fold 23
Lemma 6.1.
lim
t→0
P lq(Q)
∞∑
d=0
∏4
i=1
(
Pq
i
5 ; q
)
d
(Pq; q)4d
Qd = QH
∞∑
d=0
∏4
i=1
(
H + i
5
)
d
(H + 1)4d
Qd mod
(
H4
)
.
Proof. Since
lim
t→0
∏d
k=1
(
1− Pqk
)∏d
k=1
(
1− qk
) =
∏d
k=1(H + k)
d!
, lim
t→0
(1− P )k
(1− q)k
= Hk,
and
P lq(Q) =
∑
k≥0
(−1)k
(
ℓq(Q)
k
)
(1− P )k =
∑
k≥0
(q − 1)k
(
ℓq(Q)
k
)
(1− P )k
(1− q)k
.
Then from Proposition 2.20, we arrive at the conclusion. ■
The q-Gamma function is defined as follows
Γq(x) =
(q; q)∞
(qx; q)∞
(1− q)1−x.
It has a nice property
lim
q→1
Γq(x) = Γ(x).
Using q-Gamma function, we rewrite (5.10) in the following form
P lq(Q)
∞∑
d=0
∏4
i=1
(
Pq
i
5 ; q
)
d
(Pq; q)4d
Qd
= P lq(Q) Γq(H + 1)4
Γq
(
H + 1
5
)
Γq
(
H + 2
5
)
Γq
(
H + 3
5
)
Γq
(
H + 4
5
) 4∑
k=1
Γq
(
1−k
5
)
· · · k̂ · · ·Γq
(
4−k
5
)
Γq
(
1− k
5
)4
× Γq
(
H +
k
5
)
Γq
(
−H + 1− k
5
)
θq
(
−qH+k/5Q
)
θq(−Q)
e−1
q,qk/5
(1/Q)Wk(1/Q). (6.1)
After taking limit, we arrive at the following proposition.
Proposition 6.2.
QH
∑
d≥0
∏5d
k=1(5H + k)∏d
k=1(H + k)5
(
Q/55
)d
=
55HΓ(H + 1)5
Γ(5H + 1)
4∑
k=1
5k−1Γ(5− k)∏4
i=1,i ̸=k(i− k)Γ
(
1− k
5
)5 πe−πi(H+ k
5 )
sin
(
π
(
H + k
5
))W̃k(1/Q), (6.2)
where
W̃k =
∑
d≥0
∏d−1
l=0
(
k
5 + l
)5∏5d−1
l=0 (k + l)
(
55/Q
)d
.
24 Y. Wen
Proof. After taking limit, the left hand side of (6.1) becomes
QH
∞∑
d=0
∏4
i=1
(
H + i
5
)
d
(H + 1)4d
Qd = QH
∞∑
d=0
4∏
i=1
(
Γ
(
H + i
5 + d
)
Γ
(
H + i
5
) )(
Γ(H + 1)
Γ(H + d+ 1)
)4
Qd.
Recall some formulas of Gamma function:
Γ(x)Γ(1− x) =
π
sin(πx)
,
Γ(nx)(2π)(n−1)/2 = nnx− 1
2Γ(x)Γ
(
x+
1
n
)
· · ·Γ
(
x+
n− 1
n
)
.
Then
4∏
i=1
(
Γ
(
H + i
5 + d
)
Γ
(
H + i
5
) )
Γ(H + d+ 1)
Γ(H + 1)
=
1
55d
Γ(5H + 5d+ 1)
Γ(5H + 1)
.
Thus, we arrive at the left-hand side of (6.2).
After taking limit, the right-hand side of (6.1) becomes
QH Γ(H + 1)4
Γ
(
H + 1
5
)
· · ·Γ
(
H + 4
5
) 4∑
k=1
Γ
(
1−k
5
)
· · · k̂ · · ·Γ
(
4−k
5
)
Γ
(
1− k
5
)4 Γ
(
H +
k
5
)
Γ
(
−H + 1− k
5
)
× (−Q)−H− k
5 W̃k(1/Q) =
Γ(H + 1)4
Γ
(
H + 1
5
)
· · ·Γ
(
H + 4
5
) 4∑
k=1
Γ
(
1−k
5
)
· · · k̂ · · ·Γ
(
4−k
5
)
Γ
(
1− k
5
)4
× Γ
(
H +
k
5
)
Γ
(
−H + 1− k
5
)
eπi(−H− k
5 )W̃k(1/Q),
similarly,
Γ(H + 1)4
Γ
(
H + 1
5
)
· · ·Γ
(
H + 4
5
) =
Γ(H + 1)5
Γ(5H + 1)
55H+1/2
(2π)2
,
and
Γ
(
1−k
5
)
· · · k̂ · · ·Γ
(
4−k
5
)
Γ
(
1− k
5
)4 =
53
(1− k) · · · k̂ · · · (4− k)
∏4
i=1 Γ
(
1− k
5 + i
5
)
Γ
(
1− k
5
)
Γ
(
1− k
5
)5
=
5k−1−1/2(2π)2
(1− k) · · · k̂ · · · (4− k)
Γ(5− k)
Γ
(
1− k
5
)5 .
Thus we arrive at the right-hand side of (6.2). ■
Remark 6.3. Recall that in the introduction, we have the change of variables
Q = 55et.
Under the above change of variables, (6.2) becomes
etH
∑
d≥0
∏5d
k=1(5H + k)∏d
k=1(H + k)5
etd =
Γ(H + 1)5
Γ(5H + 1)
4∑
k=1
5k−1Γ(5− k)∏4
i=1,i ̸=k(i− k)Γ
(
1− k
5
)5 πe−πi(H+ k
5 )
sin
(
π
(
H + k
5
))W̃k.
Difference Equation for Quintic 3-Fold 25
From [4], we know
Γ(H + 1)5
Γ(5H + 1)
= 1 + C(2πi)2H2 − E(2πi)3H3 +O
(
H4
)
,
where C = 5/12 and E = −ξ(3)40/(2πi)3 with ξ(3) equals to Apéry’s constant, i.e., it is related
to the intersection theory of the quintic three-fold. We hope the expansion of the above equation
on both sides with respect to the basis {H i}3i=0 will match the result in [4, formula (53)] up
to the monodromy at 0 and ∞. For additional discussion on confluence, see [13] for projective
spaces, and [12] for weak Fano manifolds.
Acknowledgements
The author would like to thank Professor Yongbin Ruan for suggesting this problem and for
valuable discussions. Thanks are also due to Professor Shuai Guo and Dr. Yizhen Zhao for
their helpful discussion. This work was initiated during the author’s stay at the Institute For
Advanced Study In Mathematics (IASM) at Zhejiang University. The author would like to
express his thanks to IASM, Professor Bohan Fan, Professor Huijun Fan, and Peking University
for their helpful support during this visit. The author wants to thank the anonymous referees who
help improve the paper a lot. The author is supported by a KIAS Individual Grant (MG083901)
at Korea Institute for Advanced Study.
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1 Introduction
2 Preliminaries
2.1 Regular singular q-difference equations
2.2 Monodromy of regular singular q-difference equations
2.3 Confluence of regular singular q-difference equations
3 The difference equation for quintic
3.1 General technique: Newton polygon
3.2 Solutions at Q=0
3.3 Solutions at Q=infty
4 Auxiliary q-series and analytic continuation
4.1 Auxiliary q-series
4.2 Analytic continuation
5 A special fuchsian case
5.1 Connection matrix
6 Confluence of the q-difference structure
References
|
| id | nasplib_isofts_kiev_ua-123456789-211625 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2026-03-17T20:38:36Z |
| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Wen, Yaoxinog 2026-01-07T13:40:24Z 2022 Difference Equation for Quintic 3-Fold. Yaoxinog Wen. SIGMA 18 (2022), 043, 25 pages 1815-0659 2020 Mathematics Subject Classification: 14N35; 33D90; 39A13 arXiv:2011.07527 https://nasplib.isofts.kiev.ua/handle/123456789/211625 https://doi.org/10.3842/SIGMA.2022.043 In this paper, we use the Mellin-Barnes-Watson method to relate solutions of a certain type of -difference equations at = 0 and = ∞. We consider two special cases; the first is the -difference equation of the -theoretic -function of the quintic, which is degree 25; we use Adams' method to find the extra 20 solutions at = 0. The second special case is a Fuchsian case, which is confluent to the differential equation of the cohomological -function of the quintic. We compute the connection matrix and study the confluence of the -difference structure. The author would like to thank Professor Yongbin Ruan for suggesting this problem and for valuable discussions. Thanks are also due to Professor Shuai Guo and Dr. Yizhen Zhao for their helpful discussion. This work was initiated during the author’s stay at the Institute for Advanced Study in Mathematics (IASM) at Zhejiang University. The author would like to express his thanks to IASM, Professor Bohan Fan, Professor Huijun Fan, and Peking University for their helpful support during this visit. The author wants to thank the anonymous referees who helped improve the paper a lot. The author is supported by a KIAS Individual Grant (MG083901) at the Korea Institute for Advanced Study. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications Difference Equation for Quintic 3-Fold Article published earlier |
| spellingShingle | Difference Equation for Quintic 3-Fold Wen, Yaoxinog |
| title | Difference Equation for Quintic 3-Fold |
| title_full | Difference Equation for Quintic 3-Fold |
| title_fullStr | Difference Equation for Quintic 3-Fold |
| title_full_unstemmed | Difference Equation for Quintic 3-Fold |
| title_short | Difference Equation for Quintic 3-Fold |
| title_sort | difference equation for quintic 3-fold |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211625 |
| work_keys_str_mv | AT wenyaoxinog differenceequationforquintic3fold |