On the Quantum K-Theory of the Quintic
Quantum K-theory of a smooth projective variety at genus zero is a collection of integers that can be assembled into a generating series 𝐽(𝑄, 𝑞, 𝘵) that satisfies a system of linear differential equations with respect to 𝘵 and 𝑞-difference equations with respect to 𝑄. With some mild assumptions on t...
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| Cite this: | On the Quantum K-Theory of the Quintic. Stavros Garoufalidis and Emanuel Scheidegger. SIGMA 18 (2022), 021, 20 pages |
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| author_facet | Garoufalidis, Stavros Scheidegger, Emanuel |
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| description | Quantum K-theory of a smooth projective variety at genus zero is a collection of integers that can be assembled into a generating series 𝐽(𝑄, 𝑞, 𝘵) that satisfies a system of linear differential equations with respect to 𝘵 and 𝑞-difference equations with respect to 𝑄. With some mild assumptions on the variety, it is known that the full theory can be reconstructed from its small 𝐽-function 𝐽(𝑄, 𝑞, 0), which, in the case of Fano manifolds, is a vector-valued 𝑞-hypergeometric function. On the other hand, for the quintic 3-fold, we formulate an explicit conjecture for the small 𝐽-function and its small linear 𝑞-difference equation expressed linearly in terms of the Gopakumar-Vafa invariants. Unlike the case of quantum knot invariants and the case of Fano manifolds, the coefficients of the small linear 𝑞-difference equations are not Laurent polynomials, but rather analytic functions in two variables determined linearly by the Gopakumar-Vafa invariants of the quintic. Our conjecture for the small 𝐽-function agrees with a proposal of Jockers-Mayr.
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| first_indexed | 2026-03-13T20:04:03Z |
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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 18 (2022), 021, 20 pages
On the Quantum K-Theory of the Quintic
Stavros GAROUFALIDIS a and Emanuel SCHEIDEGGER b
a) International Center for Mathematics, Department of Mathematics,
Southern University of Science and Technology, Shenzhen, China
E-mail: stavros@mpim-bonn.mpg.de
URL: http://people.mpim-bonn.mpg.de/stavros
b) Beijing International Center for Mathematical Research, Peking University, Beijing, China
E-mail: esche@bicmr.pku.edu.cn
Received October 21, 2021, in final form March 03, 2022; Published online March 21, 2022
https://doi.org/10.3842/SIGMA.2022.021
Abstract. Quantum K-theory of a smooth projective variety at genus zero is a collection
of integers that can be assembled into a generating series J(Q, q, t) that satisfies a system
of linear differential equations with respect to t and q-difference equations with respect
to Q. With some mild assumptions on the variety, it is known that the full theory can be
reconstructed from its small J-function J(Q, q, 0) which, in the case of Fano manifolds, is
a vector-valued q-hypergeometric function. On the other hand, for the quintic 3-fold we
formulate an explicit conjecture for the small J-function and its small linear q-difference
equation expressed linearly in terms of the Gopakumar–Vafa invariants. Unlike the case
of quantum knot invariants, and the case of Fano manifolds, the coefficients of the small
linear q-difference equations are not Laurent polynomials, but rather analytic functions in
two variables determined linearly by the Gopakumar–Vafa invariants of the quintic. Our
conjecture for the small J-function agrees with a proposal of Jockers–Mayr.
Key words: quantum K-theory; quantum cohomology; quintic; Calabi–Yau manifolds; Gro-
mov–Witten invariants; Gopakumar–Vafa invariants; q-difference equations; q-Frobenius
method; J-function; reconstruction; gauged linear σ models; 3d-3d correspondence; Chern–
Simons theory; q-holonomic functions
2020 Mathematics Subject Classification: 14N35; 53D45; 39A13; 19E20
1 Introduction
1.1 Quantum K-theory, the small J-function and its q-difference equation
The K-theoretic Gromov–Witten invariants of a compact Kähler manifold X (often omitted
from the notation) is a collection of integers (see [27, p. 6])〈
E1L
k1 , . . . , EnL
kn
〉
g,n,d
(1.1)
defined for vector bundles E1, . . . , En on X and nonnegative integers k1, . . . , kn as the holomor-
phic Euler characteristic of Ovir ⊗
(
⊗n
i=1ev
∗
i (Ei)⊗ Lk1
i
)
over the moduli space MX,d
g,n of genus g
degree d stable maps to X with n marked points. Here, L1, . . . , Ln denote the line (orbi)bundles
over MX,d
g,n formed by the cotangent lines to the curves at the respective marked points. A defi-
nition of these integers was given by Givental and Lee [22, 33]. These numerical invariants can
be assembled into a generating series which at genus zero can be used to define an associative
deformation of the product of the K-theory ring K(X) of X.
There are several ways to assemble the integers (1.1) into generating series, and reconstruction
theorems relate these generating series and often determine one from the other. This is reviewed
mailto:stavros@mpim-bonn.mpg.de
http://people.mpim-bonn.mpg.de/stavros
mailto:esche@bicmr.pku.edu.cn
https://doi.org/10.3842/SIGMA.2022.021
2 S. Garoufalidis and E. Scheidegger
in Section 2.2. Our choice of generating series will be the so-called small J-function
JX(Q, q, 0) = (1− q)Φ0 +
∑
d
∑
α
〈
Φα
1− qL
〉
0,1,d
ΦαQd ∈ K(X)⊗K−(q)[[Q]] (1.2)
(with the notation of Section 2.1), which determines the genus 0 quantum K-theory X, i.e., the
integers (1.1) [28, Theorem 1.1, Lemma 3.3] with g = 0, as well as the genus 0 permutation-
equivariant quantum K-theory X [24] (when K(X) is generated by line bundles).
The small J-function is a vector-valued function (taking values in the rational vector space
K(X)) that obeys a system of linear q-difference equations [26, 27], giving rise to matrices
Ai(Q, q, 0) ∈ K(X) ⊗ K+(q)[[Q]], for i = 1, . . . , r which can also be used to reconstruct the
genus 0 quantum K-theory of X [28, Lemma 3.3]. Concretely, for X = CPN , the small J-
function is given by a q-hypergeometric formula [26, 27, 33]
JCPN (Q, q, 0) = (1− q)
∞∑
d=0
Qd
((1− x)q; q)N+1
d
∈ K
(
CPN
)
⊗K−(q)[[Q]], (1.3)
where (z; q)d =
∏d−1
j=0
(
1− qjz
)
for d ≥ 0, and
K
(
CPN
)
= Q[x]/
(
xN+1
)
is the K-theory ring with basis
{
1, x, . . . , xN
}
where 1−x is the class ofO(1).1 The corresponding
matrix A(Q, q, 0) of the vector-valued q-holonomic function J(Q, q, 0) is given by [28, Section 4.1]
A(Q, q, 0) = I −
0 0 . . . 0 Q
1 0 . . . 0 0
0 1 . . . 0 0
...
...
. . .
...
...
0 0 . . . 1 0
(1.4)
in the above basis of K(CPN ). It is remarkable that either (1.3) or (1.4) give the complete
determination of all the integers (1.1) for CPN . Observe that the small J-function of CPN is
given by a vector-valued q-hypergeometric formula, which is always q-holonomic (as follows from
Zeilberger et al. [35, 41, 43]), and as a result the entries of A(Q, q, 0) (as well as the coefficients of
the small quantum product) are polynomials in Q and q. It turns out that the small J-function
of Grassmanianns, flag varieties, homogeneous spaces and more generally Fano manifolds is q-
hypergeometric as shown by many researchers; see, e.g., [5, 37, 38] and references therein. On
the other hand, new phenomena are expected for the case of general Calabi–Yau manifolds, and
particularly for the quintic. Our motivation to study the case of the quintic was two-fold, coming
from numerical observations concerning coincidences of quantum K-theory counts and quantum
cohomology counts (given below), as well as a comparison of the linear q-difference equations in
quantum K-theory with those in Chern–Simons theory (such as the q-difference equation of the
colored Jones polynomial of a knot [19]).
Our results give a relation between quantum K-theory and quantum cohomology of the
quintic in two different limits, namely q = 1 (see Corollary 1.3) and q = 0 (see Corollary 1.5),
and propose a linear expression of the small J-function of the quintic in terms of its Gopakumar–
Vafa invariants (see Conjecture 1.1).
1The K-theory ring is also written as [28, Section 4.1] K
(
CPN
)
= Q
[
P, P−1
]
/
(
(1−P )N+1
)
as the Grothendieck
group of locally free sheaves on projective space, where P = OPN (−1) in which case the small J-function takes
the form JCPN (Q, q, 0) = (1− q)
∑∞
d=0
Qd
(Pq;q)N+1
d
.
On the Quantum K-Theory of the Quintic 3
1.2 The small J-function for the quintic
Quantum K-theory was developed by analogy with quantum cohomology (or Gromov–Witten
theory), a theory that deforms the cohomology ring H(X) of X and whose corresponding nu-
merical invariants are rational numbers (known as Gromov–Witten invariants) or integers in
the case of a Calabi–Yau threefold (known as the Gopakumar–Vafa invariants). A standard
reference is [7] and the book [9]. For the quintic 3-fold X, the first six values of the GW and
the GV invariants are given by
d 1 2 3 4 5 6
GWd
2875
1
4876875
8
8564575000
27
15517926796875
64
229305888887648
1
248249742157695375
1
GVd 2875 609250 317206375 242467530000 229305888887625 248249742118022000
with 2875 being the famous number of rational curves in the quintic. The two sets of invariants
are related by the following multi-covering formula
GVn =
∑
d|n
µ(d)
d3
GWn/d, GWn =
∑
d|n
1
d3
GVn/d .
In [38, Section 6.5], Tonita gave an algorithm to compute the quantum K-theory of the quintic
and using it, he found that
⟨1⟩0,1,1 = 2875,
where 2875 coincides with the famous number of lines in the quintic. Going further, (see Jockers–
Mayr [29, 30] and equation (1.11) below) one finds that
⟨1⟩0,1,2 = 620750 = 609250 + 4 · 2875, (1.5a)
⟨1⟩0,1,3 = 317232250 = 317206375 + 9 · 2875, (1.5b)
⟨1⟩0,1,4 = 242470013000 = 242467530000 + 4 · 609250 + 16 · 2875, (1.5c)
⟨1⟩0,1,5 = 229305888959500 = 229305888887625 + 25 · 2875, (1.5d)
⟨1⟩0,1,6 = 248249743392434250
= 248249742118022000 + 4 · 317206375 + 9 · 609250 + 36 · 2875 (1.5e)
are nearly equal to GV invariants of the quintic, and more precisely matched with linear combi-
nations of GV invariants. Surely this is not a coincidence and suggests that the GV invariants
can fully reconstruct the quantum K-theory invariants. In [27] this “coincidence” is proven in
abstractly. Givental and Tonita give a complete solution in genus-0 to the problem of expressing
K-theoretic GW-invariants of a compact complex algebraic manifold in terms of its cohomologi-
cal GW-invariants. One motivation for our work is to give an explicit formula (see Conjecture 1.1
below) of this abstract statement. To phrase our conjecture, recall that the rational K-theory
of the quintic 3-fold X is given by
K(X) = Q[x]/
(
x4
)
(1.6)
is the K-theory ring with basis {Φα} for α = 0, 1, 2, 3 where Φα = xα. Here 1 − x is the class
of O(1)|X . We define
5a(d, r, q) =
dr
1− q
+
dq
(1− q)2
, (1.7a)
5b(d, r, q) =
rd+ r2 − d
1− q
+
d
(1− q)2
− q + q2
(1− q)3
. (1.7b)
4 S. Garoufalidis and E. Scheidegger
Conjecture 1.1. The small J-function of the quintic is expressed linearly in terms of the GV-
invariants by
1
1− q
J(Q, q, 0) = 1 + x2
∑
d,r≥1
a(d, r, qr)GVdQ
dr + x3
∑
d,r≥1
b(d, r, qr)GVdQ
dr. (1.8)
It is interesting to observe that the right hand side of (1.8) is a meromorphic function of q
with poles at roots of unity of bounded order 3. In Section 3 we verify the above conjecture
modulo O
(
Q7
)
by an explicit calculation. Without doubt, Conjecture 1.1 concerns not only the
quintic 3-fold, but Calabi–Yau 3-folds with h1,1 = 1 (there are plenty of those, see, e.g., [2])
and beyond. In contrast to the case of CPN (see (1.3)) or the case of Fano manifolds, the
small J-function of the quintic is not hypergeometric. The above conjecture was formulated
independently by Jockers–Mayr [29, p. 10] and a comparison between their formulation and ours
is given in Section 3.3. Our conjecture also agrees with the results of Jockers–Mayr presented
in [30, Table 6.1]. Let us introduce the following multi-covering notation
GV(γ)
n =
∑
d|n
dγ GVd .
Then, we have the following.
Corollary 1.2. We have
5J(Q, 0, 0) = 5 + x2
∞∑
n=1
nGV(0)
n Qn + x3
∞∑
n=1
(
nGV(0)
n +n2GV(−2)
n
)
Qn
= 5 +
(
2875Q+ 1224250Q2 + 951627750Q3 + 969872568500Q4 + · · ·
)
x2
+
(
5750Q+ 1845000Q2 + 1268860000Q3 + 1212342581500Q4 + · · ·
)
x3. (1.9)
The above corollary reproduces the invariants of equations (1.5). To extract them, let
[J(Q, q, 0)]xα denote the coefficient of xα in J(Q, q, 0). The next corollary is proven in Sec-
tion 3.2.
Corollary 1.3. We have
∑
d≥1
〈
Φα
1− qL
〉
0,1,d
Qd =
−5[J(Q, q, 0)]x2 + 5[J(Q, q, 0)]x3 if α = 0,
5[J(Q, q, 0)]x2 if α = 1,
0 if α = 2, 3.
(1.10)
Setting q = 0, it follows that
∑
d≥1
⟨1⟩0,1,dQd =
∞∑
n=1
n2GV(−2)
n Qn = 2875Q+ 620750Q2+ 317232250Q3+ 242470013000Q4
+ 229305888959500Q5 + 248249743392434250Q6 + · · · (1.11)
matching with equations (1.5) (being the generating series of the K-theoretic versions of the
GV-invariants, given in the second page and in [30, Table 6.1]), as well as
∑
d≥1
⟨Φ1⟩0,1,dQd =
∞∑
n=1
nGV(0)
n Qn = 2875Q+ 1224250Q2+ 951627750Q3+ 969872568500Q4
+ 1146529444452500Q5 + 1489498454615043000Q6 + · · · .
On the Quantum K-Theory of the Quintic 5
1.3 The linear q-difference equation for the quintic
In this section we give an explicit formula for the small linear q-difference equation for the quintic,
assuming Conjecture 1.1. A key feature of this formula is that the coefficients of this equation
are analytic (as opposed to polynomial) functions of Q and q. The small J-function J(Q, q, 0),
viewed as a vector in the vector space K(X), forms the first column of the matrix T (Q, q, 0) of
fundamental solutions of the small linear q-difference equation in the basis
{
1, x, x2, x3
}
ofK(X).
The formula (1.8) for the small J-function and that fact that it is a cyclic vector of the linear
q-difference equation allows us to reconstruct the matrix A(Q, q, 0). See also [28, Theorem 1.1,
Lemma 3.3]. To do so, let us introduce some useful notation. If f = f(d, r, q) ∈ Q(q) we denote
[f ] =
∑
d,r≥1
f(d, r, qr)GVdQ
dr.
With this notation, equation (1.8) becomes
1
1− q
J(Q, q, 0) = 1 + [a]x2 + [b]x3 =
1
0
[a]
[b]
in the basis
{
1, x, x2, x3
}
ofK(X), where a, b are given by (1.7). Further, we denote (Ef)(d, r, q)
= qdf(d, r, q), and define
5c = π+((1− E)a), 5d = π+(Ea+ (1− E)b), (1.12)
with projections π± : K(q) → K±(q) given in Section 2.1. Explicitly, we have
5c(d, r, q) =
d2
1− q
,
5e(d, r, q) =
dr
1− q
− d(dq + q − d)
(1− q)2
.
Recall the T matrix from [28, Proposition 2.3] which is a fundamental solution of the linear q-
difference equation, and whose first column is J . The proof of the next theorem and its corollary
is given in Section 4.1.
Theorem 1.4. Conjecture 1.1 implies that the small T -matrix of the quintic is given by
T (Q, q, 0) =
1 0 0 0
0 1 0 0
[a] [c] 1 0
[b] [e] 0 1
(1.13)
and the small A-matrix of the linear q-difference equation is given by
A = I −DT, D(Q, q, 0) =
0 1 [a− c− Ea] [b− e+ Ea− Eb]
0 0 1 + [c− Ec] [e+ Ec− Ee]
0 0 0 1
0 0 0 0
. (1.14)
Note that the entries of 5D(Q, q, 0) are in Z[[Q]][q] and given explicitly in equations (4.2)
below. Let us denote by cttt(Q, q, t) = 5D2,3(Q, q, t), where Di,j denotes the (i, j)-entry of the
6 S. Garoufalidis and E. Scheidegger
matrix D. In other words, we have
cttt(Q, q) = 5 +
∑
d,r≥1
d2
1− qdr
1− qr
GVdQ
dr
=
∞∑
d=1
d2GVd
(
Li0
(
Qd
)
+ Li0
(
qQd
)
+ · · ·+ Li0
(
qd−1Qd
))
,
where Lis denotes the s-polylogarithm function Lis(z) =
∑
d≥1 z
d/ds. Recall the genus 0 gener-
ating series (minus its quadratic part) of the quintic [7, 9]
F(Q) =
∞∑
n=1
GWnQ
n =
5
6
(logQ)3 +
∞∑
d=1
GVd Li3
(
Qd
)
and its third derivative
cttt(Q) = (Q∂Q)
3F(Q) = 5 +
∞∑
d=1
d3GVd Li0
(
Qd
)
, (1.15)
where ∂Q = ∂/∂Q.
The next corollary gives a second relation between the q = 1 limit of quantum K-theory and
quantum cohomology.
Corollary 1.5. The function cttt(Q, q) ∈ Z[[Q]][q] is a q-deformation of the Yukawa coupling
(i.e., 3-point function) cttt(Q) in (1.15). Indeed, we have
cttt(Q, 1) = cttt(Q), 5D2,3(Q, q, 0) = cttt(Q, q).
Thus, the q-difference equation of the quantum K-theory of the quintic is a q-deformation of
the well-known Picard–Fuchs equation of the quintic.
Let us abbreviate the four nontrivial entries of D(Q, q, 0) by
α = D1,3, β = D1,4, γ = D2,3, δ = D2,4.
Lemma 1.6 ([30, equations (8.22) and (8.23)]). The linear q-difference equation
∆
y0
y1
y2
y3
=
0 1 α β
0 0 γ δ
0 0 0 1
0 0 0 0
y0
y1
y2
y3
(where ∆ = 1− E) is equivalent to the equation
Ly0 = 0, L = ∆
(
1 + ∆
δ + Eα+∆β
γ +∆α
)−1
∆(γ +∆α)−1∆2. (1.16)
We now discuss the q → 1 limit, using the realization of the q-commuting operators E = ehQ∂Q
and Q which act on a function f(z, h) by
(Ef)(z, h) = f(z + h, h), (Qf)(z, h) = ezf(z, h), EQ = ehQE,
where Q = ez and q = eh. Then, in the limit h→ 0, the operator L is given by
L(∆, Q, q) = 1
γ(Q, 1)
∆4 + ∂2z
1
γ(Q, 1)
∂2zh
4 +O
(
h5
)
, (1.17)
where 5γ(Q, 1) = cttt(Q, 1). Thus, the coefficient of h4 is the Picard–Fuchs equation of the
quintic, whereas the coefficient of h0 (the analogue of the AJ conjecture) is a line (1− E)4 = 0
with multiplicity 4, punctured at the zeros of γ(Q, 1) = 0. It is not clear if one can apply
topological recursion on such a degenerate curve.
On the Quantum K-Theory of the Quintic 7
2 A review of quantum K-theory
2.1 Notation
In this section we collect some useful notation that we use throughout the paper. For a smooth
projective variety X, let K(X) = K0(X;Q) denote the Grothendieck group of topological com-
plex vector bundles with rational coefficients.
Although we will not use it, the Chern class map induces a rational isomorphism of rings
ch: K(X)⊗Q → Hev(X,Q)
between K-theory and even cohomology. The ring K(X) has a basis {Φα} for α = 0, . . . , N such
that Φ0 = 1 = [OX ] is the identity element. There is a nondegenerate pairing on K(X) given
by (E,F ) ∈ K(X)⊗K(X) 7→ χ(E ⊗ F ), where
χ(E) =
∫
X
ch(E)td(X)
is the holomorphic Euler characteristic of E. Let {Φα} denote the dual basis of K(X) with
respect to the above pairing. Let {P1, . . . , Pr} denote a collection of vector bundles whose first
Chern class forms a nef integral basis of H2(X,Z)/torsion, and let Q = (Q1, . . . , Qr) be the
collection of Novikov variables dual to (P1, . . . , Pr).
The vector space K(q) = Q(q) admits a symplectic form
ω(f, g) = (Resq=0 +Resq=∞)
(
f(q)g
(
q−1
)dq
q
)
and a splitting
K(q) = K+(q)⊕K−(q)
(with projections π± : K(q) → K±(q)) into a direct sum of two Lagrangian susbpaces K+(q) =
Q
[
q±1
]
and K−(q), the space of reduced functions of q, i.e., rational functions of negative degree
which are regular at q = 0.
2.2 Reconstruction theorems for quantum K-theory
In our paper we will focus exclusively on the genus 0 quantum K-theory ofX (i.e., g = 0 in (1.1)).
The collection of integers (1.1) can be encoded in several generating series. Among them is the
primary potential
FX(Q, t) =
∑
d,n
⟨t, . . . , t⟩0,n,d
Qd
n!
∈ Q[[Q, t]]
(where the summation is over d ∈ Eff(X) and n ≥ 0), the J-function
JX(Q, q, t) = (1− q)Φ0 + t+
∑
d,n
∑
α
〈
t, . . . , t,
Φα
1− qL
〉
0,n+1,d
ΦαQd ∈ K(X)⊗K(q)[[Q, t]]
(where {Φα} is a basis for K(X) for α = 0, . . . , N with Φ0 = 1), and the T matrix Tα,β(Q, q, t) ∈
End(K(X)) ⊗ K(q)[[Q, t]] and its inverse, whose definition we omit but may be found in [28,
Section 2]. We may think of FX , JX(Q, q, t) and T (Q, q, t) as scalar-valued, vector-valued and
matrix-valued invariants, respectively. JX(Q, q, t) specializes to JX(Q, q, 0) when t = 0 and
8 S. Garoufalidis and E. Scheidegger
specializes to FX(Q, t) when α = 0 (as follows from the string equation). Also, the α = 0
column of T is JX .
There are several reconstruction theorems that determine all the invariants (1.1) from others.
In [28, Theorem 1.1], it was shown that the small J-function JX(Q, q, 0) uniquely determines
the J-function JX(Q, q, t), the primary potential FX(Q, t) and the integers (1.1) (with g = 0),
under the assumption that K(X) is generated by line bundles. In [24] it was shown (under
the same assumption on X) that the small J-function JX(Q, q, 0) reconstructs a permutation-
equivariant version of the quantum K-theory of X. This theory was introduced by Givental
in [24], where this theory takes into account the action of the symmetric groups Sn on the
moduli spaces MX,d
g,n that permutes the marked points. The J function of the permutation-
equivariant quantum K-theory of X takes values in the ring K(X) ⊗ K(q) ⊗ Λ[[Q]] where Λ is
the ring of symmetric functions in infinitely many variables [34]. K(X), Q[[Q]] and Λ are λ-rings
with Adams operations ψr, so is their tensor product. Moreover, the small J function of the
permutation-equivariant quantum K-theory of X agrees with the small J-function JX(Q, q, 0)
of the (ordinary) genus 0 quantum K-theory of X. According to a reconstruction theorem of
Givental [24] one can recover all genus zero permutation-equivariant K-theoretic GW invariants
of a projective manifold X (under the mild assumption that the ring K(X) is generated by
line bundles) from any point t∗ on their K-theoretic Lagrangian cone via an explicit flow. In
fortunate situations (that apply to the quintic as we shall see below), one is given a value
JX(Q, q, t∗) ∈ K(X) ⊗ K(q)[[Q]] ⊂ K(X) ⊗ K(q) ⊗ Λ[[Q]] and t∗ ∈ K(X) ⊗ K+(q)[[Q]] (e.g.,
t∗ = 0), in which case there exists a unique ε(x,Q, q) ∈ K(X)⊗QK+(q)[[Q]] such that for all t
JX(Q, q, t) = exp
∑
r≥1
ψr(ε((1− x)E,Q, q))
r(1− qr)
JX(Q, q, t∗) ∈ K(X)⊗K(q)[[Q]], (2.1)
where E is the operator that shifts Q to qQ. The key point here is that the coefficients of
ε(x,Q, q) (for each power of Q and x) are in the subspace K+(q) of K(q) whereas the correspond-
ing coefficients of JX(Q, q, t) are in the complementary subspace K−(q) of K(q). Another key
point is that although the above formula a priori is an equality in the permutation-equivariant
quantum K-theory, in fact it is an equality of the ordinary quantum K-theory when ε is inde-
pendent of Λ.
It follows that a single value JX(Q, q, t∗) ∈ K(X) ⊗ K(q)[[Q]] uniquely determines t∗ as
well as the small J-function JX(Q, q, 0), which in turn determines the permutation-equivariant
J-function JX(Q, q, t) for all t via (2.1).
2.3 A special value for the J-function of the quintic
For concreteness, we will concentrate on the caseX of the quintic. To use the above formula (2.1)
we need the value JX(Q, q, t∗) at some point t∗. Such a value was given by Givental in [23, p. 11]
and by Tonita in [38, Theorem 1.3 and Corollary 6.8] who proved that if Jd denotes the coefficient
of Qd in JCP4(Q, q, 0) given in (1.3), then
IO(5)(Q, q) =
∞∑
d=0
Jd
(
(1− x)5q; q
)
5d
Qd = (1− q)
∞∑
d=0
(
(1− x)5q; q
)
5d
((1− x)q; q)5d
Qd (2.2)
lies on the K-theoretic Lagrangian cone of the quintic X. This means that if ι : X → CP4 is
the inclusion, and ι∗ : K(CP4) = Q[x]/(x5) → K(X) = Q[x]/(x4) is the induced map (sending
x mod x5 to x mod x4), there exists a t∗ such that ι∗IO(5)(Q, q) = JX(Q, q, t∗). In other words,
we have
J(Q, q, t∗) = (1− q)
∞∑
d=0
((1− x)q; q)5d
((1− x)q; q)5d
Qd ∈ K(X)⊗K(q)[[Q]]. (2.3)
On the Quantum K-Theory of the Quintic 9
Interestingly, the above formula has been interpreted by Jockers and Mayr as an example
of the 3d-3d correspondence of gauged linear σ-models [30]. More precisely, the disk partition
function of a 3d gauged linear σ-model is a one-dimensional (so-called vortex) integral whose
integrand is a ratio of infinite Pochhammer symbols. A residue calculation then produces the
q-hypergeometric series (2.2).
3 The flow of the J-function
3.1 Implementing the flow
In this section we explain how to obtain a formula for the small J-function of the quintic (one
power of Q at a time) using formula (2.2) and the flow (2.1). Observe that the coefficients of q
in the function J(Q, q, t∗) given in (2.3) are not in K−(q). For instance,
coeff
(
1
1− q
J(Q, q, t∗), x0
)
=
∞∑
d=0
(q; q)5d
(q; q)5d
Qd
is a power series in Q whose coefficients are in K+(q) (and even in N[q]) and not in K−(q). Note
also that the function J(Q, q, t∗) satisfies a 24th order (but not a 4th order) linear q-difference
equation with polynomial coefficients. This is discussed in detail in Section 4.2 below.
To find J(Q, q, 0) from J(Q, q, t∗), we need to apply a flow operator (2.1). To state the
theorem, recall thatK(X)⊗K(q)[[Q]] is a λ-ring with Adams operations ψ(r) given by combining
the usual Adams operations in K-theory with the replacement of Q and q by Qr and qr. More
precisely, for a positive natural number r, we have
ψ(r) : K(X)⊗K(q)[[Q]] → K(X)⊗K(q)[[Q]], ψ(r)
(
(1− x)if(q)Qj
)
= (1− x)rif(qr)Qrj
for f(q) ∈ K(q) and natural numbers i, j and x as in (1.6). Recall that the plethystic exponential
of f(x,Q, q) ∈ K(X)⊗K(q)[[Q]] (with f(x, 0, q) = 0) is given by
Exp(f) = exp
( ∞∑
r=1
ψ(r)(f)
r
)
.
It is easy to see that when f is small (i.e., f(x, 0, q) = 0), then Exp(f) ∈ K(X) ⊗ K(q)[[Q]] is
well-defined. Let E denote the q-difference operator that shifts Q to qQ, as in (1.12). By slight
abuse of notation, we denote
E : K(X)⊗K(q)[[Q]] → K(X)⊗K(q)[[Q]],
E
(
(1− x)if(Q)Qj
)
= (1− x)if(qQ)Qj . (3.1)
Throughout the paper, the operators E and Q will act on a function f(Q, q) by
(Ef)(Q, q) = f(qQ, q), (Qf)(Q, q) = Qf(Q, q), EQ = qQE. (3.2)
The theorem of Givental–Tonita asserts that there exists a unique
ε(x,Q, q) ∈ K(X)⊗QK+(q)[[Q]]
such that
Exp
(
ε((1− x)E,Q, q)
1− q
)
J(Q, q, t∗) ∈ K(X)⊗K−(q)[[Q]] (3.3)
10 S. Garoufalidis and E. Scheidegger
and then, the left hand side of the above equation is J(Q, q, 0). Equation (3.3) is a non-linear
fixed-point equation for ε that has a unique solution that may be found working on one Q-degree
at a time. Indeed, we can write
ε(x,Q, q) =
∞∑
k=1
εk(x, q)Q
k, εk(x, q) =
3∑
ℓ=0
∞∑
k=1
εk,ℓ(q)x
ℓQk.
Then for each positive integer number N we have
π+
(
exp
(
N∑
r=1
3∑
ℓ=0
N∑
k=1
ψ(r)εk,ℓ(q)
r(1− qr)
Qrk((1− x)E)ℓr
)
J(Q, q, t∗)
)
= 0.
Equating the coefficient of each power of xi for i = 0, . . . , 3 to zero in the above equation,
we get a system of four inhomogeneous linear equations with unknowns (εN,0, . . . , εN,3) (with
coefficients polynomials in εN ′,ℓ′ for N
′ < N), with a unique solution in the field K(q). A further
check (according to Givental–Tonita’s theorem) is that the unique solution lies in K+(q), and
even more, in our case we check that it lies in Q[q]. Once εN ′(x, q) is known for N ′ ≤ N ,
equation (3.3) allows us to compute Jd(q), where
J(Q, q, 0) =
∞∑
d=0
Jd(q)Q
d.
For instance, when N = 1 we have
ε1,0(q) = 1724 + 572q − 625q2 − 1941q3 − 3430q4 − 4952q5 − 6223q6 − 6755q7 − 6184q8
− 4690q9 − 2747q10 − 969q11,
ε1,1(q) = −4600− 1140q + 2485q2 + 6520q3 + 11140q4 + 15890q5 + 19860q6 + 21490q7
+ 19630q8 + 14860q9 + 8690q10 + 3060q11,
ε1,2(q) = 4025 + 555q − 3115q2 − 7255q3 − 12055q4 − 17020q5 − 21175q6 − 22850q7
− 20830q8 − 15740q9 − 9190q10 − 3230q11,
ε1,3(q) = −1150 + 10q + 1250q2 + 2670q3 + 4340q4 + 6080q5 + 7540q6 + 8120q7
+ 7390q8 + 5575q9 + 3250q10 + 1140q11,
and, consequently, we find that
J0(q) = 1− q,
J1(q) = − 575x2
−1 + q
− 1150(−1 + 2q)x3
(−1 + q)2
in agreement with [30, eqation (6.38)]. Continuing our computation, we find that
J2(q) = −
25
(
9794 + 19496q + 9725q2
)
x2
(−1 + q)(1 + q)2
−
50
(
−7380− 9748q + 14760q2 + 29244q3 + 12139q4
)
x3
(−1 + q)2(1 + q)3
and
J3(q) = −
25
(
7613022 + 15225906q + 22838859q2 + 15225860q3 + 7612953q4
)
x2
(−1 + q)
(
1 + q + q2
)2
On the Quantum K-Theory of the Quintic 11
− 50
(−1 + q)2
(
1 + q + q2
)3 (−5075440− 7612953q − 7612953q2 + 10150880q3
+ 22838859q4 + 30451812q5 + 17763442q6 + 7612953q7
)
x3.
Two further values of Jd(q) for d = 4, 5 were computed but are too long to be presented
here. Based on this data, we guessed the formula for J(Q, q, 0) given in (1.8). Finally, we
computed J6(q) and found that it is in agreement with out predicted formula (1.8).
3.2 Extracting quantum K-theory counts from the small J-function
In this section we give a proof of Corollary 1.3 for the quintic X. Recall that K(X) from
equation (1.6) has basis Φα = xα for α = 0, 1, 2, 3 with x4 = 0 and inner product
(Φa,Φb) =
∫
X
ΦaΦbtd(X) =
0 5 −5 5
5 −5 5 0
−5 5 0 0
5 0 0 0
. (3.4)
The dual basis {Φa} of K(X) is given by
Φ0 = 1
5Φ3, Φ1 = 1
5(Φ2 +Φ3), Φ2 = 1
5(Φ1 +Φ2), Φ3 = 1
5(Φ0 +Φ1 − Φ3), (3.5)
and is related to the basis {Φa} by
Φ0 = 5
(
Φ1 − Φ2 +Φ3
)
, Φ1 = 5
(
Φ0 − Φ1 +Φ2
)
,
Φ2 = 5
(
−Φ0 +Φ1
)
, Φ3 = 5Φ0.
Substituting Φα as above in equation (1.2) and collecting the powers of xα, it follows that
[J(Q, q, 0)]1 = 1− q +
1
5
∑
d≥1
〈
Φ3
1− qL
〉
0,1,d
Qd,
[J(Q, q, 0)]x =
1
5
∑
d≥1
(〈
Φ2
1− qL
〉
0,1,d
+
〈
Φ3
1− qL
〉
0,1,d
)
Qd,
[J(Q, q, 0)]x2 =
1
5
∑
d≥1
(〈
Φ1
1− qL
〉
0,1,d
+
〈
Φ2
1− qL
〉
0,1,d
)
Qd,
[J(Q, q, 0)]x3 =
1
5
∑
d≥1
(〈
Φ0
1− qL
〉
0,1,d
+
〈
Φ1
1− qL
〉
0,1,d
−
〈
Φ3
1− qL
〉
0,1,d
)
Qd.
The above is a linear system of equations with unknowns
∑
d≥1
〈
Φα
1−qL
〉
0,1,d
Qd for α = 0, 1, 2, 3.
Solving the linear system combined with equation (1.9), gives (1.10). Setting q = 0 in (1.10)
and using Corollary 1.2, we obtain (1.11) and (1.3) and conclude the proof of Corollary 1.3.
3.3 A comparison with Jockers–Mayr
In this section we give the details of the comparison of our Conjecture 1.1 with a conjecture of
Jockers–Mayr [29, p. 10].
To begin with, their IQK(t) is our J(Q, q, t) and their I(0) in [29, equation (7)] is our J(Q, q, 0).
They drop the index QK later on. From [29, equation (4)] it follows that they are working in
the same basis Φα = xa, α = 0, 1, 2, 3, as we are. Furthermore, the inner product on K(X)
12 S. Garoufalidis and E. Scheidegger
[29, equation (6)] agrees with the one given in equation (3.4) with dual basis {Φa} of K(X)
given in (3.5). By [29, equation (8)], specialized to the quintic, the function I(t) becomes
I(t) = 1 − q + tΦ1 + F 2(t)Φ2 + F 3(t)Φ3. Then, they define functions FA and F̂A by writing∑
A F
AΦA =
∑
A
(
FA,cl + F̂A
)
ΦA, where F̂A(t) =
∑
d>0Q
d
〈〈
ΦA
1−qL
〉〉
d
, cf. [29, equation (9)], and
FA,cl are “constant”, i.e., independent of Q and t. Note that only F 2, F 3 are nonzero which
implies that only F0, F1 are nonzero. Their conjecture [29, p. 10] can now be stated (in the case
of the quintic) as follows [29, equation (10)]:
F̂0 = p2 +
1
(1− q)2
[(1− 3q)F + qtF1]tn>2 ,
F̂1 = p1,1 +
1
(1− q)
[F1]tn>1 ,
where p2, p1,1, F , F1 are certain explicitly given functions of t and the Gopakumar–Vafa invari-
ants GVd, [29, equations (11) and (12)]. Combining everything so far, their conjecture reads
I(t) = 1− q + tΦ1 + (F1,cl + p1,1 +
1
(1− q)
[F1]tn>1)Φ1
+ (F0,cl + p2 +
1
(1− q)2
[(1− 3q)F + qtF1]tn>2)Φ0.
We will not spell out these functions completely, but only their value at t = 0 in order to compare
it to our formulas. First, the brackets [. . . ]tn>1 , [. . . ]tn>2 vanish for t = 0. So we are left with p2
and p1,1 [29, equation (12)]. Noting that
∑
j djtj = 0 for t = 0, these read
1
1− q
p1,1|t=0 =
∑
d>0
Qd
∑
r|d
GVd/r
d(1− qr) + d
r q
r
(1− qr)2
,
1
1− q
p2|t=0 =
∑
d>0
Qd
∑
r|d
GVd/r
r2(1− qr)2 − qr(1 + qr)
(1− qr)3
.
Next, we rewrite these sums so that they run over all values of r
1
1− q
p1,1|t=0 =
∑
d,r>0
Qdr GVr
dr(1− qr) + dqr
(1− qr)2
,
1
1− q
p2|t=0 =
∑
d,r>0
Qdr GVr
r2(1− qr)2 − qr(1 + qr)
(1− qr)3
.
Hence,
1
1− q
p1,1|t=0 = 5
∑
d,r>0
Qdr GVr a(d, r, q
r),
1
1− q
p2|t=0 = 5
∑
d,r>0
Qdr GVr (b(d, r, q
r)− a(d, r, qr)) .
The appearance of the term involving a(d, r, qr) in the second equation is due to the change of
basis Φ2 = 5
(
−Φ0 +Φ1
)
. This completes the compatibility of our conjecture and theirs.
4 q-difference equations
4.1 The small q-difference equation of the quintic
In this section we explain how Theorem 1.4 follows from Conjecture 1.1. We begin with a general
discussion. Given a collection of vector functions fj(Q, q) ∈ Q(q)[[Q]]r for j = 1, . . . , r such that
On the Quantum K-Theory of the Quintic 13
det(f1|f2| . . . |fr) is not identically zero, there is always a canonical linear q-difference equation
(Ey)(Q, q) = A(Q, q)y(Q, q)
with fundamental solution set f1, . . . , fr, where E is the shift operator of equation (3.1) that
replaces Q by qQ. Indeed, the equations Eyj = Ayj for j = 1, . . . , r are equivalent to the
matrix equation ET = AT where T = (f1|f2| . . . |fr) is the fundamental matrix solution, and
inverting T , we find that A = (ET )−1T . This can be applied in particular to the case of
a collection Ejg for j = 0, . . . , r − 1 of a vector function g(Q, q) ∈ Q(q)[[Q]] that satisfies
det
(
g|Eg| . . . |Er−1g
)
is nonzero. Said differently, every vector function g(Q, q) ∈ Q(q)[[Q]]
along with its r − 1 shifts (generically) satisfies a linear q-difference equation.
We will apply the above principle to the 4-tuple ((1 − x)E)jJ(Q, q, 0)/(1 − q) ∈ K(X) ⊗
K(q)[[Q]] for j = 0, . . . , 3 where J(Q, q, 0) ∈ K(X)⊗K−(q)[[Q]] is as in Conjecture 1.1. However,
notice that although the q-coefficients of J(Q, q, 0)/(1 − q) are in K−(q), this is no longer true
for the shifted functions ((1 − x)E)jJ(Q, q, 0)/(1 − q) for j = 1, 2, 3. In that case, we need to
apply the Birkhoff factorization [25, App.A] to the matrix
1
1− q
(
J |(1− x)EJ |((1− x)E)2J |((1− x)E)3J
)
= TU, (4.1)
where the q-coefficients of the entries of T are in K−(q) and of U are in K+(q) (compare also
with Lemma 3.3 of [28, equation (4)]). The existence and uniqueness of matrices T and U in the
above equation follows from the fact that the left hand side of the above equation is unipotent,
and the proof is discussed in detail in the above reference.
In our case, the choice
T =
1
1− q
π+
(
J |(1− x)EJ |((1− x)E)2J |((1− x)E)3J
)
together with equation (4.1) implies that the q-coefficients of the entries of U are in K+(q).
Equation (1.13) for the fundamental matrix T follows from the fact that
π+
(
qd
(
r2
1− q
− q + q2
(1− q)3
))
=
−1 + 3q − 4q2
(1− q)3
+
(−1 + d)(−1− d+ 3q + dq)
(1− q)2
+
r2
1− q
,
π+
(
qd
(
r
1− q
+
q
(1− q)2
))
=
−d+ q + dq
(1− q)2
+
r
1− q
valid for all positive natural numbers d and r.
Having computed the fundamental matrix T (1.13), we use [28, equation (2)], with P−1qQ∂Q
replaced by 1− (1− x)E to deduce the small A-matrix (1.14).
Explicitly, the four nontrivial entries of the matrix D are given by
5(a− c− Ea)(d, r, q) =
d
(
−d+ q + dq − q1+d + r − qr − qdr + q1+dr
)
(1− q)2
, (4.2a)
5(b− e+ Ea− Eb)(d, r, q) = −
q2
(
1 + 2d+ d2 − r2
)
+ q
(
1− 2d− 2d2 + 2r2
)
(1− q)3
+
d2 − r2 + qd
(
−q − q2 + r2 − 2qr2 + q2r2
)
(1− q)3
, (4.2b)
5(c− Ec)(d, r, q) =
d2
(
1− qd
)
1− q
, (4.2c)
5(e+ Ec− Ee)(d, r, q) = −
d
(
−d+ q + dq − q1+d − r + qr + qdr − q1+dr
)
(1− q)2
. (4.2d)
14 S. Garoufalidis and E. Scheidegger
Note that the entries of 5D are in Z[[Q]][q]. Moreover, the values when q = 1 are given by
5(a− c− Ea)(d, r, 1) = −1
2
d2(1 + d− 2r),
5(b− e+ Ea− Eb)(d, r, 1) = −1
6
d
(
1 + 3d+ 2d2 − 6r2
)
,
5(c− Ec)(d, r, q) = d3,
5(e+ Ec− Ee)(d, r, 1) =
1
2
d2(1 + d+ 2r).
As a further consistency check, note that our matrix D given in (1.14) equals to the matrix D
of [30, equation (8.21)].
Given the formula of (1.14), an explicit calculation shows that the entries of D are given
by (4.2). This concludes the proof of Theorem 1.4.
Proof of Corollary 1.5. It follows from equations (4.2c) and (1.15). ■
Proof of Lemma 1.6. We have
∆y0 = y1 + αy2 + βy3, ∆y1 = γy2 + δy3, ∆y2 = y3, ∆y3 = 0.
The lemma follows by eliminating y1, y2, y3 (one at a time) using the fact that
E(fg) = (Ef)(Eg), ∆(fg) = (∆f)g + f(∆g)− (∆f)(∆g).
(which follows from (Ef)(Q, q) = f(qQ, q) and ∆ = 1− E). Indeed, we have
∆2y0 = ∆(∆y0) = ∆(y1 + αy2 + βy3) = (γ +∆α)y2 + (δ + Eα+∆β)y3,
and hence,
(γ +∆α)−1∆2y0 = y2 +
δ + Eα+∆β
γ +∆α
y3,
and hence,
∆(γ +∆α)−1∆2y0 =
(
1 +
δ + Eα+∆β
γ +∆α
)
y3.
Applying ∆ once again and using ∆y3 = 0 concludes the proof of equation (1.16). Note that
the notation is such that an operator ∆ is applied to everything on the right hand side.
The q = 1 limit of L(∆, Q, q) follows from equation (1.16), the fact that
(∆f)(Q, q)|q=1 = (f(qQ, q)− f(Q, q))|q=1 = 0
and Corollary 1.5. ■
4.2 The Frobenius method for linear q-difference equations
In this section we discuss in detail the linear q-difference equation satisfied by the function
J(Q, q, t∗) of (2.2). Recall the operators E and Q that act on functions of Q and q by (3.2).
Let
J(Q, q, x) =
∞∑
n=0
an(q, x)Q
n = J0(Q, q) + J1(Q, q)x+ · · · ∈ Q(q)[[Q, x]], (4.3)
On the Quantum K-Theory of the Quintic 15
where Jn(Q, q) ∈ Q(q)[[Q]] for all n and
an(q, x) =
(
e5xq; q
)
5n(
exq; q
)5
n
,
where eax is to be understood as a polynomial in x obtained as eax +O
(
x4
)
.
The functions Jn(Q, q) are given by series whose summand ia a q-hypergeometric function
times a polynomial of q-harmonic functions. For example, we have
J0(Q, q) =
∞∑
n=0
(q; q)5n
(q; q)5n
Qn,
J1(Q, q) =
∞∑
n=0
(q; q)5n
(q; q)5n
(1 + 5H5n(q)− 5Hn(q))Q
n,
where Hn(q) =
∑n
j=1 q
j/
(
1− qj
)
is the nth q-harmonic number. Consider the 25-th order linear
q-difference operator
L5(E,Q, q) = (1− E)5 −Q
5∏
j=1
(
1− qjE5
)
(4.4)
with coefficients polynomials in Q and q. Note that L5 = (1 − E)5 −
∏5
j=1
(
1 − q5−jE5
)
Q,
hence L5 factors as 1− E times a 24-th order operator.
Lemma 4.1. With J as in (4.3) and L5 as in (4.4), we have
L5
(
ex, Q, q
)
J =
(
1− ex
)5
.
Proof. It is easy to see that
an(q, x)
an−1(q, x)
=
∏5
j=1
(
1− e5xq5n−j
)(
1− exqn
)5 .
Hence,
(
1− exqn
)5
an(q, x)Q
n = Q
5∏
j=1
(
1− e5xq5n−j
)
an−1(q, x)Q
n−1
and in operator form,
(
1− exE
)5
an(q, x)Q
n = Q
5∏
j=1
(
1− qje5xE5
)
an−1(q, x)Q
n−1.
Summing from n = 1 to infinity, we obtain that
(
1− exE
)5
(J − 1) = Q
5∏
j=1
(
1− qje5xE5
)
J.
Since
(
1− exE
)5
=
(
1− ex
)5
, the result follows. ■
16 S. Garoufalidis and E. Scheidegger
Note that the proof of Lemma 4.1 implies that J(Q, q, x) satisfies a 24-th order linear q-
difference equation but this will no play a role in our paper. Of importance is the fact that
the 25-th order equation L5f = 0 has a distinguished 5-dimensional space of solutions, given
explicitly by a q-version of the Frobenius method. Since this method is well-known for linear
differential equations, but less so for linear q-difference equations, we give more details than
usual. For additional discussion on this method, see Wen [40], and for references for the q-
gamma and q-beta functions, see De Sole–Kac [10].
First, we define an n-th derivative of an operator P (E,Q, q) by
P (n)(E,Q, q) =
∑
k=0d
knck(Q, q)E
k, P (E,Q, q) =
∑
k=0d
ck(Q, q)E
k.
In other words, we may write P (n) = (E∂E)
n(P ).
Lemma 4.2. For a linear q-difference operator P (E,Q, q) we have
P (exE,Q, q) =
∞∑
n=0
xn
n!
P (n)(E,Q, q). (4.5)
Moreover, for all natural numbers n and a function f(Q, q) we have
P ((logQ)nf) =
n∑
k=0
(
n
k
)
(logQ)n−k(log q)kP (n−k)f. (4.6)
Proof. Equations (4.5) and (4.6) are additive in P , hence it suffices to prove them when P = Ea
for a natural number a, in which case (Ea)(n) = an and both identities are clear. ■
Lemma 4.3. Suppose P (E,Q, q) is a linear q-difference operators with coefficients polynomials
in E and Q, and J(Q, q, x) ∈ Q(q)[[Q, x]] is such that
P (exE,Q, q)J(Q, q, x) = O
(
xN+1
)
(4.7)
for some natural number N . Then,
n∑
k=0
(
n
k
)
P (k)Jn−k = 0 (4.8)
for n = 0, . . . , N , where Jk = coeff
(
J(Q, q, x), xk
)
, and
Pfn = 0, fn =
n∑
k=0
(
n
k
)
(logQ)n−k(log q)kJk (4.9)
for n = 0, 1, . . . , N . In other words, the equation Pf = 0 has N+1 distinguished solutions given
by
f0 = J0,
f1 = logQJ0 + log qJ1,
f2 = (logQ)2J0 + 2 logQ log qJ1 + (log q)2J2,
· · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · ·
Proof. Equation (4.8) follows easily using (4.5) and by expanding the left hand side of equa-
tion (4.7) into power series in x and equating the coefficient of xn with zero for n = 0, 1, . . . , N .
Equation (4.9) follows from equations (4.8) and (4.6), and induction on n. ■
On the Quantum K-Theory of the Quintic 17
5 Quantum K-theory versus Chern–Simons theory
There are several hints in the physics literature pointing to a deeper relation between Quantum
K-theory and Chern–Simons gauge theory (e.g., for 3-manifolds with boundary, such as knot
complements), see for instance in [6, 13, 15, 29, 30] and in references therein. In this section
we discuss and comment on the q-difference equations in Chern–Simons theory, gauged linear
σ-models and Quantum K-theory. We will discuss three aspects of this comparison:
(a) q-holonomic systems and their q = 1 semiclassical limits,
(b) ε-deformations,
(c) matrix-valued invariants.
We begin with the case of the Chern–Simons theory. The partition function of Chern–
Simons theory with compact (e.g., SU(2)) gauge group on a 3-manifold (with perhaps nonempty
boundary) is given by a finite-dimensional state-sum whose summand has as a building block
the quantum n-factorial. This follows from existence of an underlying TQFT [36, 39, 42] which
reduces the computation of the partition function into elementary pieces. For the complement of
a knot K in S3, the partition function recovers the colored Jones polynomial of a knot which, in
the case of SU(2), is a sequence JK,n(q) ∈ Z
[
q±
]
of Laurent polynomials which can be presented
as a finite-dimensional sum whose summand has as a building block the finite q-Pochhammer
symbol (q; q)n. This ultimately boils down to the entries of the R-matrix which are given for
example in [36].
On the other hand, Chern–Simons theory with complex (e.g., SL2(C)) gauge group is not
well-understood as a TQFT. However, the partition function for a 3-manifold with boundary
can be computed by a finite-dimensional state-integral whose integrand has as a building block
Faddeev’s quantum dilogarithm function [16]. The latter is a ratio of two infinite Pochham-
mer symbols which form a quasi-periodic function with two quasi periods.
(
Recall that the
Pochhammer symbol is (x; q)∞ =
∏∞
j=0
(
1 − qjx
)
.
)
These are the state-integrals studied in
quantum Teichmüller theory by Kashaev et al. [3, 4, 31] and in complex Chern–Simons theory
by Dimofte et al. [11, 12].
The appearance of q-holonomic systems in Chern–Simons theory with compact/complex
gauge group is a consequence of Zeilberger theory [35, 41, 43] applied to finite-dimensional
state-sums/integrals whose summand/integrand has as a building block the finite/infinite q-
Pochhammer symbol. This is exactly how it was deduced that the sequence of colored Jones
polynomials JK,n(q) of a knot satisfy a linear q-difference equation AK
(
L̂, M̂ , q
)
JK = 0 (see [19]),
where L̂ and M̂ are q-commuting operators that act on a sequence f : N → Q(q) by(
L̂f
)
(n) = f(n+ 1),
(
M̂f
)
(n) = qnf(n), LM = qML.
In the case of state-integrals, the existence of two quasi-periods leads to a linear q- (and also
q̃)-difference equation, where q = e2πih and q̃ = e−2πi/h.
It is conjectured that the linear q-difference equation of the colored Jones polynomial essen-
tially coincides with the one of the state-integral, and that the classical q = 1 limit (the so-called
AJ conjecture [17]) coincides with the A-polynomial AK(L,M, 1) of the knot. The latter is the
SL2(C)-character variety of the fundamental group of the knot complement, viewed from the
boundary torus [8]. Finally, the semiclassical limit (the analogue of (1.17) is given by
AK
(
L̂, M̂ , q
)
= AK(L,M, 1) +DK(z, ∂z)h
s +O
(
hs+1
)
,
where DK(z, ∂z) is a linear differential operator of degree s where s is the order of vanishing of
AK(L, 1, 1) at L = 1. This order is typically 1 (e.g., for the 41, 52, 61 and more generally all
twist knots) but it is equal to 2 for the 818 knot.
18 S. Garoufalidis and E. Scheidegger
We now come to the feature, namely an expected “factorization” of state-integrals into a finite
sum of products of q-series and q̃-series. This factorization is computed by an ε-deformation of q-
and q̃-hypergeometric series that arise by applying the residue theorem to the state-integrals.
For a detailed illustration of this, we refer the reader to [6, 18] and [21].
Our last discussed feature, namely a matrix-valued extension of the Chern–Simons invari-
ants with compact/complex gauge group was recently discovered in two papers [20, 21]. More
precisely, it was conjectured and in some cases verified that the scalar valued quantum knot
invariants such as the Kashaev invariant [32] (an evaluation of the n-th colored Jones polyno-
mial at n-th roots of unity) and the Andersen–Kashaev state-integral [3] admit an extension
into a matrix-valued invariants. The rows and columns are labeled by the set PM of SL2(C)
boundary-parabolic representations of π1(M). In the case of a knot complement, the set PM
can be thought of as the set of branches of the A-polynomial curve above a point (where the
meridian has eigenvalues 1). Although the corresponding vector space R(M) := QPM with
basis PM has no ring structure known to us, it has a distinguished element corresponding to
the trivial SL2(C)-representation that plays an important role. A ring structure QPM might
be defined as the Grothendieck group of an appropriate category associated to flat connections
on 3-manifolds with boundary, or perhaps by contructing an appropriate logarithmic conformal
field theory use fusion rules will define the sought ring as suggested by Gukov. Alternatively,
the sought ring might be described in terms of SL(2,C)-Floer homology, suggested by Witten.
Alternatively, it might be described by the quantum K-theory of the mirror of the local Calabi–
Yau manifold uv = AM (x, y), (where AM is the A-polynomial discussed above), suggested by
Aganagic–Vafa [1].
We now discuss the above features (a)–(c) that appear in the 3d-gauged linear σ-models and
their 3d-3d correspondence studied in detail in [6, 13, 14, 15, 29, 30] and references therein.
The q-holonomic aspect is still present since the (so-called vortex) partition function is a finite-
dimensional integral whose integrand has as a building block the infinite Pochhammer symbol
(note however that q̃ does not appear). The second aspect involving ε-deformations is also
present for the same reason as in Chern–Simons theory. The third aspect is absent in general.
We finally discuss the above features in genus 0 quantum K-theory of the quintic. The first
aspect is different: the linear q-differential equation has coefficients which are analytic (and not
polynomial) functions of Q and q. The classical limit q = 1 of the linear q-difference equation
of the quintic is given by γ(Q, 1)−1∆4 (1.17) and this defines a degenerate analytic curve in
C × C that consists of a finite collection of lines with coordinates (∆, Q). On the other hand,
the semi-classical limit (i.e., the coefficient of h4 in (1.17)) is the famous Picard–Fuchs equation
of the quintic. The second feature, the ε-deformation for a nilpotent variable ε is encoded in the
fact that K(X) has nilpotent elements x. The last feature is most interesting since the matrix-
valued invariants are encoded in End(K(X)), where K(X) is not just a rational vector space,
but a ring unit 1. It follows that the linear q-difference equations have not only a distinguished
solution JX(Q, q, 0) but a basis of solutions parametrized by a basis {Φα} of K(X).
Let us end our discussion with some questions on the colored Jones polynomial JK,n(q) colored
by the n-dimensional irreducible sl2(C) representation. For simplicity, we abbreviate R
(
S3 \K
)
defined above by R(K).
Question 5.1.
(a) Does the vector space R(K) have a ring structure?
(b) If so, is the series
∑∞
n=1 JK,n(q)Q
n the coefficient of 1 in the R(K)-valued small J-function
JK(Q, q, 0) of a knot K?
(c) If so, is there a t-deformation JK(Q, q, t)?
On the Quantum K-Theory of the Quintic 19
Acknowledgements
The authors wish to thank the Max-Planck-Institute for Mathematics and the Bethe Center
for Theoretical Physics in Bonn for inviting them to their workshop on Number Theoretic Me-
thods in Quantum Physics in July 2019, where the first ideas were conceived. We also wish to
thank Gaetan Borot, Alexander Givental, Todor Milanov and Di Yang for useful conversations.
E.S. wishes to thank the University of Melbourne for having him as a guest during 2020 and
Southern University of Science and Technology for hospitality in 2021.
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1 Introduction
1.1 Quantum K-theory, the small J-function and its q-difference equation
1.2 The small J-function for the quintic
1.3 The linear q-difference equation for the quintic
2 A review of quantum K-theory
2.1 Notation
2.2 Reconstruction theorems for quantum K-theory
2.3 A special value for the J-function of the quintic
3 The flow of the J-function
3.1 Implementing the flow
3.2 Extracting quantum K-theory counts from the small J-function
3.3 A comparison with Jockers–Mayr
4 q-difference equations
4.1 The small q-difference equation of the quintic
4.2 The Frobenius method for linear q-difference equations
5 Quantum K-theory versus Chern–Simons theory
References
|
| id | nasplib_isofts_kiev_ua-123456789-211524 |
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| language | English |
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| publishDate | 2022 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Garoufalidis, Stavros Scheidegger, Emanuel 2026-01-05T12:24:47Z 2022 On the Quantum K-Theory of the Quintic. Stavros Garoufalidis and Emanuel Scheidegger. SIGMA 18 (2022), 021, 20 pages 1815-0659 2020 Mathematics Subject Classification: 14N35; 53D45; 39A13; 19E20 arXiv:2101.07490 https://nasplib.isofts.kiev.ua/handle/123456789/211524 https://doi.org/10.3842/SIGMA.2022.021 Quantum K-theory of a smooth projective variety at genus zero is a collection of integers that can be assembled into a generating series 𝐽(𝑄, 𝑞, 𝘵) that satisfies a system of linear differential equations with respect to 𝘵 and 𝑞-difference equations with respect to 𝑄. With some mild assumptions on the variety, it is known that the full theory can be reconstructed from its small 𝐽-function 𝐽(𝑄, 𝑞, 0), which, in the case of Fano manifolds, is a vector-valued 𝑞-hypergeometric function. On the other hand, for the quintic 3-fold, we formulate an explicit conjecture for the small 𝐽-function and its small linear 𝑞-difference equation expressed linearly in terms of the Gopakumar-Vafa invariants. Unlike the case of quantum knot invariants and the case of Fano manifolds, the coefficients of the small linear 𝑞-difference equations are not Laurent polynomials, but rather analytic functions in two variables determined linearly by the Gopakumar-Vafa invariants of the quintic. Our conjecture for the small 𝐽-function agrees with a proposal of Jockers-Mayr. The authors wish to thank the Max-Planck-Institute for Mathematics and the Bethe Center for Theoretical Physics in Bonn for inviting them to their workshop on Number Theoretic Methods in Quantum Physics in July 2019, where the first ideas were conceived. We also wish to thank Gaetan Borot, Alexander Givental, Todor Milanov, and Di Yang for useful conversations. E.S. wishes to thank the University of Melbourne for having him as a guest during 2020 and Southern University of Science and Technology for its hospitality in 2021. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications On the Quantum K-Theory of the Quintic Article published earlier |
| spellingShingle | On the Quantum K-Theory of the Quintic Garoufalidis, Stavros Scheidegger, Emanuel |
| title | On the Quantum K-Theory of the Quintic |
| title_full | On the Quantum K-Theory of the Quintic |
| title_fullStr | On the Quantum K-Theory of the Quintic |
| title_full_unstemmed | On the Quantum K-Theory of the Quintic |
| title_short | On the Quantum K-Theory of the Quintic |
| title_sort | on the quantum k-theory of the quintic |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/211524 |
| work_keys_str_mv | AT garoufalidisstavros onthequantumktheoryofthequintic AT scheideggeremanuel onthequantumktheoryofthequintic |