9 Proof of the Labastida-Mariño-Ooguri-Vafa
Conjecture
Kefeng Liu and Pan Peng
Abstract
Based on large N Chern-Simons/topological string duality, in a
series of papers [38, 22, 20], J.M.F. Labastida, M. Mariño, H. Ooguri
and C. Vafa conjectured certain remarkable new algebraic structure of
link invariants and the existence of infinite series of new integer invari-
ants. In this paper, we provide a proof of this conjecture. Moreover,
we also show these new integer invariants vanish at large genera.
Contents
0 Introduction 3
0.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
0.2 Labastida-Mariño-Ooguri-Vafa conjecture . . . . . . . . . . . . 4
0.3 Main ideas of the proof . . . . . . . . . . . . . . . . . . . . . . 6
0.4 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . 8
1 Preliminary 9
1.1 Partition and symmetric function . . . . . . . . . . . . . . . . 9
1.2 Partitionable set and infinite series . . . . . . . . . . . . . . . 11
2 Labastida-Mariño-Ooguri-Vafa conjecture 12
2.1 Quantum trace . . . . . . . . . . . . . . . . . . . . . . . . . . 12
http://arxiv.org/abs/0704.1526v3
2.2 Quantum group invariants of links . . . . . . . . . . . . . . . . 14
2.3 Chern-Simons partition function . . . . . . . . . . . . . . . . . 17
2.4 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Hecke algebra and cabling 22
3.1 Centralizer algebra and Hecke algebra representation . . . . . 22
3.2 Quantum dimension . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Cabling Technique . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Proof of Theorem 1 28
4.1 Pole structure of quantum group invariants . . . . . . . . . . . 28
4.2 Symmetry of quantum group invariants . . . . . . . . . . . . . 30
4.3 Cut-and-join analysis . . . . . . . . . . . . . . . . . . . . . . . 33
4.4 Framing and pole structures . . . . . . . . . . . . . . . . . . . 38
5 Proof of Theorem 2 41
5.1 A ring characterizes the partition function . . . . . . . . . . . 41
5.2 Multi-cover contribution and p-adic argument . . . . . . . . . 43
5.3 Integrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6 Concluding Remarks and Future Research 47
6.1 Duality from a mathematical point of view . . . . . . . . . . . 47
6.2 Other related problems . . . . . . . . . . . . . . . . . . . . . . 48
A Appendix 50
A.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
A.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
0 Introduction
0.1 Overview
For decades, we have witnessed the great development of string theory and its
powerful impact on the development of mathematics. There have been a lot
of marvelous results revealed by string theory, which deeply relate different
aspects of mathematics. All these mysterious relations are connected by a
core idea in string theory called “duality”. It was found that string theory
on Calabi-Yau manifolds provided new insight in geometry of these spaces.
The existence of a topological sector of string theory leads to a simplified
model in string theory, the topological string theory.
A major problem in topological string theory is how to compute Gromov-
Witten invariants. There are two major methods widely used: mirror symme-
try in physics and localization in mathematics. Both methods are effective
when genus is low while having trouble in dealing with higher genera due
to the rapidly growing complexity during computation. However, when the
target manifold is Calabi-Yau threefold, large N Chern-Simons/topological
string duality opens a new gate to a complete solution of computing Gromov-
Witten invariants at all genera.
The study of large N Chern-Simons/topological string duality was orig-
inated in physics by an idea that gauge theory should have a string theory
explanation. In 1992, Witten [46] related topological string theory of T ∗M
of a three dimensional manifold M to Chern-Simons gauge theory on M . In
1998, Gopakumar and Vafa [11] conjectured that, at large N , open topo-
logical A-model of N D-branes on T ∗S3 is dual to closed topological string
theory on resolved conifold O(−1) ⊕ O(−1) → P1. Later, Ooguri and Vafa
[38] showed a picture on how to describe Chern-Simons invariants of a knot
by open topological string theory on resolved conifold paired with lagrangian
associated with the knot.
Though large N Chern-Simons/topological string duality still remains
open, there have been a lot of progress in this direction demonstrating the
power of this idea. Even for the simplest knot, the unknot, Mariño-Vafa
formula [33, 26] gives a beautiful closed formula for Hodge integral up to
three Chern classes of Hodge bundle. Furthermore, using topological vertex
theory [1, 27, 28], one is able to compute Gromov-Witten invariants of any
toric Calabi-Yau threefold by reducing the computation to a gluing algorithm
of topological vertex. This thus leads to a closed formula of topological string
partition function, a generating function of Gromov-Witten invariants, in all
genera for any toric Calabi-Yau threefolds.
On the other hand, after Jones’ famous work on polynomial knot invari-
ants, there had been a series of polynomial invariants discovered (for example,
[15, 9, 19]), the generalization of which was provided by quantum group the-
ory [43] in mathematics and by Chern-Simons path integral with the gauge
group SU(N) [45] in physics.
Based on the large N Chern-Simons/topological string duality, Ooguri
and Vafa [38] reformulated knot invariants in terms of new integral invari-
ants capturing the spectrum of M2 branes ending on M5 branes embedded
in the resolved conifold. Later, Labastida, Mariño and Vafa [22, 20] refined
the analysis of [38] and conjectured the precise integrality structure for open
Gromov-Witten invariants. This conjecture predicts a remarkable new al-
gebraic structure for the generating series of general link invariants and the
integrality of infinite family of new topological invariants. In string theory,
this is a striking example that two important physical theories, topological
string theory and Chern-Simons theory, exactly agree up to all orders. In
mathematics this conjecture has interesting applications in understanding
the basic structure of link invariants and three manifold invariants, as well
as the integrality structure of open Gromov-Witten invariants. Recently,
X.S. Lin and H. Zheng [30] verified LMOV conjecture in several lower degree
cases for some torus links.
In this paper, we give a complete proof of Labastida-Mariño-Ooguri-Vafa
conjecture for any link (We will briefly call it LMOV conjecture). First, let
us describe the conjecture and the main ideas of the proof. The details can
be found in Sections 4 and 5.
0.2 Labastida-Mariño-Ooguri-Vafa conjecture
Let L be a link with L components and P be the set of all partitions. The
Chern-Simons partition function of L is given by
ZCS(L; q, t) =
~A∈PL
W ~A(L; q, t)
sAα(x
α) (0.1)
for any arbitrarily chosen sequence of variables
xα = (xα1 , x
2 , . . . , ) .
In (0.1), W ~A(L) is the quantum group invariants of L labeled by a sequence
of partitions ~A = (A1, . . . , AL) ∈ PL which correspond to the irreducible rep-
resentations of quantized universal enveloping algebra Uq(sl(N,C)), sAα(x
is the Schur function.
Free energy is defined to be
F = logZCS .
Use plethystic exponential, one can obtain
~A 6=0
f ~A(q
d, td)
(xα)d
, (0.2)
where
(xα)d =
(xα1 )
d, (xα2 )
d, . . .
Based on the duality between Chern-Simons gauge theory and topological
string theory, Labastida, Mariño, Ooguri, Vafa conjectured that f ~A have the
following highly nontrivial structures.
For any A, B ∈ P, define the following function
MAB(q) =
χA(Cµ)χB(Cµ)
ℓ(µ)∏
(q−µj/2 − qµj/2) . (0.3)
Conjecture (LMOV). For any ~A ∈ PL,
(i). there exists P ~B(q, t) for ∀ ~B ∈ PL, such that
f ~A(q, t) =
|Bα|=|Aα|
P ~B(q, t)
MAαBα(q). (0.4)
Furthermore, P ~B(q, t) has the following expansion
P ~B(q, t) =
Q∈Z/2
N ~B; g,Q(q
−1/2 − q1/2)2g−2tQ . (0.5)
(ii). N ~B; g,Q are integers.
For the meaning of notations, the definition of quantum group invariants
of links and more details, please refer to Section 2.
This conjecture contains two parts:
• The existence of the special algebraic structure (0.5).
• The integrality of the new invariants N ~B; g,Q.
If one looks at the right hand side of (0.5), one will find it very interesting
that the pole of f ~A in (q
−1/2− q1/2) is actually at most of order 1 for any link
and any labeling partitions. However, by the calculation of quantum group
invariants of links, the pole order of f ~A might be going to ∞ when the degrees
of labeling partitions go higher and higher. This miracle cancelation implies
a very special algebraic structure of quantum group invariants of links and
thus the Chern-Simons partition function. in Section 2.4.2, we include an
example in the simplest setting showing that this cancelation has shed new
light on the quantum group invariants of links.
0.3 Main ideas of the proof
In our proof of LMOV conjecture, there are three new techniques.
• When dealing with the existence of the conjectured algebraic structure,
one will encounter the problem of how to control the pole order of
(q−1/2 − q1/2). We consider the framed partition function Z(L; q, t, τ)
of Chern-Simons invariants of links which satisfies the following cut-
and-join equation
∂Z(L; q, t, τ)
i,j≥1
ijpαi+j
∂pαi ∂p
+ (i+ j)pαi p
∂pαi+j
Z(L; q, t, τ) .
Here pαn = pn(x
α) are regarded as independent variables.
However, a deeper understanding of this conjecture relies on the fol-
lowing log cut-and-join equation
∂F (L; q, t, τ)
i,j≥1
ijpαi+j
∂pαi ∂p
+ (i+ j)pαi p
∂pαi+j
+ ijpαi+j
This observation is based on the duality of Chern-Simons theory and
open Gromov-Witten theory. The log cut-and-join equation is a non-
linear ODE systems and the non-linear part reflects the essential recur-
sion structure of Chern-Simons partition function. The miracle cance-
lation of lower order terms of q−1/2 − q1/2 occurring in free energy can
be indicated in the formulation of generating series of open Gromov-
Witten invariants on the geometric side.
A powerful tool to control the pole order of (q−1/2 − q1/2) through log
cut-and-join equation is developed in this paper as we called cut-and-
join analysis. An important feature of cut-and-join equation shows that
the differential equation at partition1 (d) can only have terms obtained
from joining2 while at (1d), non-linear terms vanishes and there is no
joining terms. This special feature combined with the degree analysis
will squeeze out the desired degree of (q−1/2 − q1/2).
• We found a rational function ring which characterizes the algebraic
structure of Chern-Simons partition function and hence (open) topo-
logical string partition function by duality.
A similar ring characterizing closed topological string partition func-
tion firstly appears in the second author’s work on Gopakumar-Vafa
conjecture [39, 40]. The original observation in the closed case comes
from the structure of R-matrix in quantum group theory and gluing
algorithm in the topological vertex theory.
However, the integrality in the case of open topological string the-
ory is more subtle than the integrality of Gopakumar-Vafa invariants
1Here, we only take a knot as an example. For the case of links, it is then a natural
extension.
2Joining means that a partition is obtained from combining two rows of a Young dia-
gram and reforming it into a new partition, while cutting means that a partition is obtained
by cutting a row of a Young diagram into two rows and reforming it into a new partition.
in the closed case. This is due to the fact that the reformulation of
Gromov-Witten invariants as Gopakumar-Vafa invariants in the closed
case weighted by power of curve classes, while in the open case, the gen-
erating function is weighted by the labeling irreducible representation
of Uq(sl(N,C)). This subtlety had already been explained in [22].
To overcome this subtlety, one observation is that quantum group in-
variants look Schur-function-like. This had already been demonstrated
in the topological vertex theory (also see [31]). We refine the ring in
the closed case and get the new ring R(y; q, t) (cf. section 5.1). Corre-
spondingly, we consider a new generating series ofN ~B; g,Q, T~d, as defined
in (5.1).
• To prove T~d ∈ R(y; q, t), we combine with the multi-cover contribution
and p-adic argument therein. Once we can prove that T~d lies in R, due
to the pole structure of the ring R, the vanishing of N ~B; g,Q has to occur
at large genera, we actually proved
Q∈Z/2
N ~B; g,Q(q
−1/2 − q1/2)2gtQ ∈ Z[(q−
2 − q
2 )2, t±
2 ] .
The paper is organized as follows. In Section 1, we introduce some basic
notations about partition and generalize this concept to simplify our calcu-
lation in the following sections. Quantum group invariants of links and main
results are introduced in Section 2. In Section 3, we review some knowledge
of Hecke algebra used in this paper. In Section 4 and 5, we give the proof
of Theorem 1 and 2 which answer LMOV conjecture. In the last section, we
discuss some problems related to LMOV conjecture for our future research.
0.4 Acknowledgments
The authors would like to thank Professors S.-T. Yau, F. Li and Z.-P. Xin
for valuable discussions. We would also want to thank Professor M. Mariño
for pointing out some misleading parts. K. L. is supported by NSF grant.
P. P. is supported by NSF grant DMS-0354737 and Harvard University.
Before he passed away, Professor Xiao-Song Lin had been very interested
in the LMOV conjecture, and had been working on it. We would like to
dedicate this paper to his memory.
1 Preliminary
1.1 Partition and symmetric function
A partition λ is a finite sequence of positive integers (λ1, λ2, · · · ) such that
λ1 ≥ λ2 ≥ · · · .
The total number of parts in λ is called the length of λ and denoted by ℓ(λ).
We use mi(λ) to denote the number of times that i occurs in λ. The degree
of λ is defined to be
|λ| =
If |λ| = d, we say λ is a partition of d. We also use notation λ ⊢ d. The
automorphism group of λ, Aut λ, contains all the permutations that permute
parts of λ while still keeping it as a partition. Obviously, the order of Aut λ
is given by
|Autλ| =
mi(λ)! .
There is another way to rewrite a partition λ in the following format
(1m1(λ)2m2(λ) · · · ) .
A traditional way to visualize a partition is to identify a partition as a
Young diagram. The Young diagram of λ is a 2-dimensional graph with λj
boxes on the j-th row, j = 1, 2, ..., ℓ(λ). All the boxes are put to fit the
left-top corner of a rectangle. For example
(5, 4, 2, 2, 1) = (12245) = .
For a given partition λ, denote by λt the conjugate partition of λ. The Young
diagram of λt is transpose to the Young diagram of λ: the number of boxes
on j-th column of λt equals to the number of boxes on j-th row of λ, where
1 ≤ j ≤ ℓ(λ).
By convention, we regard a Young diagram with no box as the partition
of 0 and use notation (0). Denote by P the set of all partitions. We can
define an operation “ ∪ ” on P. Given two partitions λ and µ, λ ∪ µ is the
partition by putting all the parts of λ and µ together to form a new partition.
For example
(1223) ∪ (15) = (122235).
Using Young diagram, it looks like
∪ = .
The following number associated with a partition λ is used throughout
this paper,
jmj(λ)mj(λ)! , κλ =
λj(λj − 2j + 1) .
It’s easy to see that
κλ = −κλt . (1.1)
A power symmetric function of a sequence of variables x = (x1, x2, ...) is
defined as follows
pn(x) =
xni .
For a partition λ,
pλ(x) =
ℓ(λ)∏
pλj (x).
It is well-known that every irreducible representation of symmetric group
can be labeled by a partition. Let χλ be the character of the irreducible rep-
resentation corresponding to λ. Each conjugate class of symmetric group can
also be represented by a partition µ such that the permutation in the conju-
gate class has cycles of length µ1, . . . , µℓ(µ). Schur function sλ is determined
sλ(x) =
|µ|=|λ|
χλ(Cµ)
pµ(x) (1.2)
where Cµ is the conjugate class of symmetric group corresponding to partition
1.2 Partitionable set and infinite series
The concept of partition can be generalized to the following partitionable set.
Definition 1.1. A countable set (S,+) is called a partitionable set if
1). S is totally ordered.
2). S is an Abelian semi-group with summation “ + ”.
3). The minimum element 0 in S is the zero-element of the semi-group,
i.e., for any a ∈ S,
0+ a = a = a+ 0.
For simplicity, we may briefly write S instead of (S,+).
Example 1.2. The following sets are examples of partitionable set:
(1). The set of all nonnegative integers Z≥0;
(2). The set of all partitions P. The order of P can be defined as follows:
∀λ , µ ∈ P, λ ≥ µ iff |λ| > |µ|, or |λ| = |µ| and there exists a j such
that λi = µi for i ≤ j − 1 and λj > µj. The summation is taken to be
“ ∪ ” and the zero-element is (0).
(3). Pn. The order of Pn is defined similarly as (2): ∀ ~A, ~B ∈ Pn, ~A ≥ ~B
i=1 |Ai| >
i=1 |Bi|, or
i=1 |Ai| =
i=1 |Bi| and there is a j
such that Ai = Bi for i ≤ j − 1 and Aj > Bj. Define
~A ∪ ~B = (A1 ∪ B1, ..., An ∪ Bn).
((0), (0), ..., (0)) is the zero-element. It’s easy to check that Pn is a
partitionable set.
Let S be a partitionable set. One can define partition with respect to
S in the similar manner as that of Z≥0: a finite sequence of non-increasing
non-minimum elements in S. We will call it an S-partition, (0) the zero
S-partition. Denote by P(S) the set of all S-partitions.
For an S-partition Λ, we can define the automorphism group of Λ in a
similar way as that in the definition of traditional partition. Given β ∈ S,
denote by mβ(Λ) the number of times that β occurs in the parts of Λ, we
then have
AutΛ =
mβ(Λ)! .
Introduce the following quantities associated with Λ,
ℓ(Λ)!
|AutΛ|
, θΛ =
(−1)ℓ(Λ)−1
uΛ . (1.3)
The following Lemma is quite handy when handling the expansion of
generating functions.
Lemma 1.3. Let S be a partitionable set. If f(t) =
n≥0 ant
n, then
β 6=0, β∈S
Aβ pβ(x)
Λ∈P(S)
aℓ(Λ)AΛ pΛ(x) uΛ ,
where
ℓ(Λ)∏
pΛj , AΛ =
ℓ(Λ)∏
AΛj .
Proof. Note that
β∈S, β 6=0
Λ∈P(S), ℓ(Λ)=n
ηΛuΛ .
Direct calculation completes the proof.
2 Labastida-Mariño-Ooguri-Vafa conjecture
2.1 Quantum trace
Let g be a finite dimensional complex semi-simple Lie algebra of rank N
with Cartan matrix (Cij). Quantized universal enveloping algebra Uq(g) is
generated by {Hi, X+i, X−i} together with the following defining relations:
[Hi, Hj] = 0 , [Hi, X±j] = ±CijX±j , [X+i, X−j] = δij
−Hi/2
i − q
i − q
1−Cij∑
(−1)k
1− Cij
1−Cij−k
±i X±jX
±i = 0 , for all i 6= j ,
where
{k}q =
2 − q k2
2 − q 12
, {k}q! =
{i}q ,
and {
{a}q · {a− 1}q · · · {a− b+ 1}q
{b}q!
The ribbon category structure associated with Uq(g) is given by the fol-
lowing datum:
1. For any given two Uq(g)-modules V and W , there is an isomorphism
ŘV,W : V ⊗W →W ⊗ V
satisfying
ŘU⊗V,W = (ŘU,W ⊗ idV )(idU ⊗ŘV,W )
ŘU, V⊗W = (idV ⊗ŘU,W )(ŘU, V ⊗ idW )
for Uq(g)-modules U , V , W .
Given f ∈ HomUq(g)(U, Ũ), g ∈ HomUq(g)(V, Ṽ ), one has the following
naturality condition:
(g ⊗ f) ◦ ŘU, V = ŘeU, eV ◦ (f ⊗ g) .
2. There exists an element K2ρ ∈ Uq(g), called the enhancement of Ř,
such that
K2ρ(v ⊗ w) = K2ρ(v)⊗K2ρ(w)
for any v ∈ V , w ∈ W .
3. For any Uq(g)-module V , the ribbon structure ΘV : V → V associated
to V satisfies
Θ±1V = trV Ř
V,V .
The ribbon structure also satisfies the following naturality condition
x ·ΘV = ΘeV · x.
for any x ∈ HomUq(g)(V, Ṽ ).
Definition 2.1. Given z =
i xi⊗yi ∈ EndUq(g)(U ⊗V ), the quantum trace
of z is defined as follows
trV (z) =
tr(yiK2ρ)xi ∈ EndUq(g)(U).
2.2 Quantum group invariants of links
Quantum group invariants of links can be defined over any complex simple
Lie algebra g. However, in this paper, we only consider the quantum group
invariants of links defined over sl(N,C)3 due to the current consideration for
large N Chern-Simons/topological string duality.
Roughly speaking, a link is several disconnected S1 embedded in S3.
A theorem of J. Alexander asserts that any oriented link is isotopic to the
closure of some braid. A braid group Bn is defined by generators σ1, · · · , σn−1
and defining relation:
σiσj = σjσi , if |i− j| ≥ 2 ;
σiσjσi = σjσiσj , if |i− j| = 1 .
Let L be a link with L components Kα, α = 1, . . . , L, represented by the
closure of an element of braid group Bm. We associate each Kα an irreducible
representation Rα of quantized universal enveloping algebra Uq(slN), labeled
by its highest weight Λα. Denote the corresponding module by VΛα. The j-th
strand in the braid will be associated with the irreducible module Vj = VΛα ,
if this strand belongs to the component Kα. The braiding is defined through
the following universal R-matrix of Uq(slN )
R = q
Hi⊗Hj
positive root β
expq[(1− q−1)Eβ ⊗ Fβ] .
3In the following context, we will briefly write slN .
Here (Cij) is the Cartan matrix and
expq(x) =
k(k+1) x
{k}q!
Define braiding by Ř = P12R, where P12(v ⊗ w) = w ⊗ v.
Now for a given link L of L components, one chooses a closed braid
representative in braid group Bm whose closure is L. In the case of no
confusion, we also use L to denote the chosen braid representative in Bm.
We will associate each crossing by the braiding defined above. Let U , V
be two Uq(slN)-modules labeling two outgoing strands of the crossing, the
braiding ŘU,V (resp. Ř
V,U) is assigned as in Figure 1.
PSfrag replacements
U VŘU,V
PSfrag replacements
U VŘ−1V,U
Figure 1: Assign crossing by Ř.
The above assignment will give a representation of Bm on Uq(g)-module
V1 ⊗ · · · ⊗ Vm. Namely, for any generator, σi ∈ Bm4,
define
h(σi) = idV1 ⊗ · · · ⊗ ŘVi+1,Vi ⊗ · · · ⊗ idVN .
Therefore, any link L will provide an isomorphism
h(L) ∈ EndUq(slN )(V1 ⊗ · · · ⊗ Vm) .
For example, the link L in Figure 2 gives the following homomorphism
h(L) = (ŘV, U ⊗ idU)(idV ⊗Ř−1U, U)(ŘU, V ⊗ idU) .
PSfrag replacements
Figure 2: A braid representative for Hopf link
Let K2ρ be the enhancement of Ř in the sense of [41], where ρ is the
half-sum of all positive roots of slN . The irreducible representation Rα is
labeled by the corresponding partition Aα.
Definition 2.2. Given L labeling partitions A1, . . . , AL, the quantum group
invariant of L is defined as follows:
W(A1,...,AL)(L) = qd(L) trV1⊗···⊗Vm(h(L)) ,
where
d(L) = −
ω(Kα)(Λα,Λα + 2ρ) +
lk(Kα,Kβ)|Aα| · |Aβ| ,
and lk(Kα,Kβ) is the linking number of components Kα and Kβ. A sub-
stitution of t = qN is used to give a two-variable framing independent link
invariant.
Remark 2.3. In the above formula of d(L), the second term on the right
hand side is meant to cancel not important terms involved with q1/N in the
definition.
It will be helpful to extend the definition to allow some labeling partition
to be the empty partition (0). In this case, the corresponding invariants will
be regarded as the quantum group invariants of the link obtained by simply
removing the components labeled by (0).
4In the case of σ−1
, use Ř−1
Vi,Vi+1
instead.
A direct computation5 shows that after removing the terms of q
N , qd(L)
can be simplified as
α=1 κAαw(Kα)/2 · t
α=1 |A
α|w(Kα)/2 . (2.1)
Example 2.4. The following examples are some special cases of quantum
group invariants of links.
(1). If the link involved in the definition is the unknot ©,
WA(©; q, t) = trVA(idVA)
is equal to the quantum dimension of VA which will be denoted by
dimq VA.
(2). When all the components of L are associated with the fundamental
representation, i.e., the labeling partition is the unique partition of 1,
the quantum group invariant of L is related to the HOMFLY polynomial
of the link, PL(q, t), in the following way:
W( ,··· , )(L; q, t) = tlk(L)
2 − t 12
2 − q 12
PL(q, t) .
(3). If L is a disjoint union of L knots, i.e.,
L = K1 ⊗K2 ⊗ · · · ⊗ KL ,
the quantum group invariants of L is simply the multiplication of quan-
tum group invariants of Kα
W(A1,...,AL)(L; q, t) =
WAα(Kα; q, t) .
2.3 Chern-Simons partition function
For a given link L of L components, we will fix the following notations in
this paper. Given λ ∈ P, ~A = (A1, A2, . . . , AL), ~µ = (µ1, µ2, . . . , µL) ∈ PL.
Let x = (x1, ..., xL) where xα is a series of variables
xα = (xα1 , x
2 , · · · ) .
5It can also be obtained from the ribbon structure.
The following notations will be used throughout the paper:
[n]q = q
2 − q
2 , [λ]q =
ℓ(λ)∏
[λj ]q , z~µ =
zµα ,
| ~A| = (|A1|, ..., |AL|) , ‖ ~A ‖=
|Aα| , ℓ(~µ) =
ℓ(µα) ,
~At =
(A1)t, . . . , (AL)t
, χ ~A (~µ) =
χAα(Cµα) , s ~A(x) =
sAα(x
Denote by
1~µ = (1|µ
1|, · · · , 1|µL|) . (2.2)
Chern-Simons partition function can be defined to be the following gen-
erating function of quantum group invariants of L,
ZCS(L) = 1 +
~A 6=0
W ~A(L; q, t)s ~A(x).
Define free energy
F = logZ =
~µ 6=0
F~µp~µ (x) . (2.3)
Here in the similar usage of notation,
p~µ(x) =
pµα(x
We rewrite Chern-Simons partition function as
ZCS(L) = 1 +
~µ6=0
Z~µp~µ(x)
where
Z~µ =
χ ~A(~µ)
W ~A. (2.4)
By Lemma 1.3, we have
F~µ =
Λ∈P(PL), |Λ|=~µ∈PL
θΛZΛ . (2.5)
2.4 Main results
2.4.1 Two theorems that answer LMOV conjecture
Let P ~B(q, t) be the function defined by (0.4), which can be determined by
the following formula,
P ~B(q, t) =
| ~A|=|~B|
f ~A(q, t)
χAα(Cµ)χBα(Cµ)
ℓ(µα)∏
q−µj/2 − qµj/2
Theorem 1. There exist topological invariants N ~B; g,Q ∈ Q such that expan-
sion (0.5) holds.
Theorem 2. Given any ~B ∈ PL, the generating function of N ~B; g,Q, P ~B(q, t),
satisfies
(q−1/2 − q1/2)2P ~B(q, t) ∈ Z
q−1/2 − q1/2
, t±1/2
It is clear that Theorem 1 and 2 answered LMOV conjecture. Moreover,
Theorem 2 implies, for fixed ~B, N ~B; g,Q vanishes at large genera.
The method in this paper may apply to the general complex simple Lie
algebra g instead of only considering sl(N,C). We will put this in our future
research. If this is the case, it might require a more generalized duality picture
in physics which will definitely be very interesting to consider and reveal much
deeper relation between Chern-Simons gauge theory and geometry of moduli
space. The extension to some other gauge group has already been done in
[42] where non-orientable Riemann surfaces is involved. For the gauge group
SO(N) or Sp(N), a more complete picture had appeared in the recent work
of [6, 7] and therein a BPS structure of the colored Kauffman polynomials
was also presented. We would like to see that the techniques developed in
our paper extend to these cases.
The existence of (0.5) has its deep root in the duality between large
N Chern-Simons/topological string duality. As already mentioned in the
introduction, by the definition of quantum group invariants, P ~B(q, t) might
have very high order of pole at q = 1, especially when the degree of ~B
goes higher and higher. However, LMOV-conjecture claims that the pole at
q = 1 is at most of order 2 for any ~B ∈ PL. Any term that has power of
q−1/2−q1/2 lower than −2 will be canceled! Without the motivation of Chern-
Simons/topological string duality, this mysterious cancelation is hardly able
to be realized from knot-theory point of view.
2.4.2 An application to knot theory
We now discuss applications to knot theory, following [22, 20].
Consider associating the fundamental representation to each component
of the given link L. As discussed above, the quantum group invariant of L
will reduce to the classical HOMFLY polynomial PL(q, t) of L except for a
universal factor. HOMFLY has the following expansion
PL(q, t) =
p2g+1−L(t)(q
2 − q
2 )2g+1−L . (2.6)
The lowest power of q−1/2 − q1/2 is 1 − L, which was proved in [29] (or one
may directly derive it from Lemma 4.1). After a simple algebra calculation,
one will have
F( ,..., ) =
t−1/2 − t1/2
q−1/2 − q1/2
p̃2g+1−L(t)(q
2 − q
2 )2g+1−L . (2.7)
Lemma 4.3 states that
p̃1−L(t) = p̃3−L(t) = · · · = p̃L−3(t) = 0 ,
which implies that the pk(t) are completely determined by the HOMFLY
polynomial of its sub-links for k = 1− L, 3− L, . . . , L− 3.
Now, we only look at p̃1−L(t) = 0. A direct comparison of the coeffi-
cients of F = logZCS immediately leads to the following theorem proved by
Lickorish and Millett [29].
Theorem 2.5 (Lickorish-Millett). Let L be a link with L components. Its
HOMFLY polynomial
PL(q, t) =
pL2g+1−L(t)
2 − q
)2g+1−L
satisfies
pL1−L(t) = t
2 − t
)L−1 L∏
pKα0 (t)
where pKα0 (t) is HOMFLY polynomial of the α-th component of the link L
with q = 1.
In [29], Lickorish and Millett obtained the above theorem by skein anal-
ysis. Here as the consequence of higher order cancelation phenomenon, one
sees how easily it can be achieved. Note that we only utilize the vanishing
of p̃1−L. If one is ready to carry out the calculation of more vanishing terms,
one can definitely get much more information about algebraic structure of
HOMFLY polynomial. Similarly, a lot of deep relation of quantum group
invariants can be obtained by the cancelation of higher order poles.
2.4.3 Geometric interpretation of the new integer invariants
The following interpretation is taken in physics literature from string theo-
retic point of view [22, 38].
Quantum group invariants of links can be expressed as vacuum expec-
tation value of Wilson loops which admit a large N expansion in physics.
It can also be interpreted as a string theory expansion. This leads to a ge-
ometric description of the new integer invariants N ~B; g,Q in terms of open
Gromov-Witten invariants (also see [20] for more details).
The geometric picture of f ~A is proposed in [22]. One can rewrite the free
energy as
−1ℓ(~µ)
λ2g−2+ℓ(~µ)Fg,~µ(t)p~µ . (2.8)
The quantities Fg,~µ(t) can be interpreted in terms of the Gromov-Witten in-
variants of Riemann surface with boundaries. It was conjectured in [38] that
for every link L in S3, one can canonically associate a lagrangian submanifold
CL in the resolved conifold
O(−1)⊕O(−1) → P1 .
The first Betti number b1(CL) = L, the number of components of L. Let
γα, α = 1, . . . , L, be one-cycles representing a basis for H1(CL, Z). Denote
by Mg,h,Q the moduli space of Riemann surfaces of genus g and h holes
embedded in the resolved conifold. There are hα holes ending on the non-
trivial cycles γα for α = 1, . . . , L. The product of symmetric groups
Σh1 × Σh2 × · · · × ΣhL
acts on the Riemann surfaces by exchanging the hα holes that end on γα.
The integer N ~B; q,t is then interpreted as
N ~B; q,t = χ(S ~B(H
∗(Mg,h,Q))) (2.9)
where S ~B = SB1 ⊗ · · · ⊗ SBL , and SBα is the Schur functor.
The recent progress in the mathematical definitions of open Gromov-
Witten invariants [18, 23, 24] may be used to put the above definition on a
rigorous setting.
3 Hecke algebra and cabling
3.1 Centralizer algebra and Hecke algebra representa-
We review some facts about centralizer algebra and Hecke algebra represen-
tation and their relation to the representation of braid group.
Denote by V the fundamental representation of Uq(slN)
6. Let
{K±1i , Ei, Fi : 1 ≤ i ≤ N − 1}
be the standard generators of the quantized universal enveloping algebra
Uq(slN). Under a suitable basis {X1, ..., XN} of V , the fundamental repre-
sentation is given by the following matrices
Ei 7−→ Ei,i+1
Fi 7−→ Ei+1,i
Ki 7−→ q−1/2Ei,i + q1/2Ei+1,i+1 +
i 6=j
6We will reserve V to denote the fundamental representation of Uq(slN ) from now on.
where Ei,j denotes the N × N matrix with 1 at the (i, j)-position and 0
elsewhere. Direct calculation shows
K2ρ(Xi) = q
−N+1−2i
2 Xi (3.1)
2N Ř(Xi ⊗Xj) =
q−1/2Xi ⊗Xj, i = j ,
Xj ⊗Xi, i < j ,
Xj ⊗Xi + (q−1/2 − q1/2)Xi ⊗Xj, i > j .
The centralizer algebra of V ⊗n is defines as follows
Cn = EndUq(slN )(V ⊗n) =
x ∈ End(V ⊗n) : xy = yx, ∀y ∈ Uq(slN)
Hecke algebra Hn(q) of type An−1 is the complex algebra with n − 1
generators g1, ..., gn−1, together with the following defining relations
gigj = gjgi , |i− j| ≥ 2
gigi+1gi = gi+1gigi+1 , i = 1, 2, ..., n− 2,
(gi − q−1/2)(gi + q1/2) = 0 , i = 1, 2, ..., n− 1.
Remark 3.1. Here we use q−1/2 instead of q to adapt to our notation in the
definition of quantum group invariants of links. Note that when q = 1, the
Hecke algebra Hn(q) is just the group algebra CΣn of symmetric group Σn.
When N is large enough, Cn is isomorphic to the Hecke algebra Hn(q).
A very important feature of the homomorphism
h : CBn −→ Cn
is that h factors through Hn(q) via
2N σi 7→ gi 7→ q−
2N h(σi) . (3.2)
It is well-known that the irreducible modules Sλ (Specht module) ofHn(q)
are in one-to-one correspondence to the partitions of n.
Any permutation π in symmetric group Σn can express as a product of
transpositions
π = si1si2 · · · sil .
If l is minimal in possible, we say π has length ℓ(π) = l and
gπ = gi1gi2 · · · gil .
It is not difficult to see that gπ is well-defined. All of such {gπ} form a basis
of Hn(q).
Minimal projection S is an element in Cn such that SV ⊗n is some irre-
ducible representation Sλ. We denote it by pλ. The minimal projections of
Hecke algebras are well studied (for example [14]), which is a Z(q±1)-linear
combination {q 12gi}.
3.2 Quantum dimension
3.2.1 Explicit formula
An explicit formula for quantum dimension of any irreducible representation
of Uq(slN) can be computed via decomposing V
⊗n into permutation modules.
A composition of n is a sequence of non-negative integer
b = (b1, b2, . . .)
such that ∑
bi = n .
We will write it as b � n. The largest j such that bj 6= 0 is called the end of
b and denoted by ℓ(b).
Let b be a composition such that ℓ(b) ≤ N . DefineMb to be the subspace
of V ⊗n spanned by the vectors
Xj1 ⊗ · · · ⊗Xjn (3.3)
such that Xi occurs precisely bi times. It is clear thatM
b is anHn(q)-module
and is called permutation module. Moreover, by explicit matrix formula of
{Ei, Fi, Ki} acting on V under the basis {Xi}, we have
Mb = {X ∈ V ⊗n : Ki(X) = q−
bi−bi+1
2 X} .
The following decomposition is very useful
V ⊗n =
b�n, ℓ(b)≤N
Let A be a partition and VA the irreducible representation labeled by A.
The Kostka number KAb is defined to be the weight of VA in M
b, i.e.,
KAb = dim(VA ∩Mb) .
Schur function has the following formulation through Kostka numbers
sA(x1, . . . , xN) =
b�|A|, ℓ(b)≤N
By (3.1), K2ρ is acting on M
b as a scalar
j=1 q
(N+1−2j)bj . Thus
dimq VA = trVA idVA
b�|A|
dim(VA ∩Mb)
(N+1−2j)bj
2 , . . . , q−
(3.4)
By (1.2),
dimq VA =
|µ|=|A|
χA(Cµ)
2 , . . . , q−
|µ|=|A|
χA(Cµ)
ℓ(µ)∏
t−µj/2 − tµj/2
q−µj/2 − qµj/2
. (3.5)
Here in the last step, we use the substitution t = qN .
3.2.2 An expansion of the Mariño-Vafa formula
Here we give a quick review about Mariño-Vafa formula [33, 26] for the
convenience of Knot theorist. For details, please refer [26].
Let Mg,n denote the Deligne-Mumford moduli stack of stable curves of
genus g with n marked points. Let π :M g,n+1 →M g,n be the universal curve,
and let ωπ be the relative dualizing sheaf. The Hodge bundle E = π∗ωπ is a
rank g vector bundle over M g,n. Let si : M g,n → M g,n+1 denote the section
of π which corresponds to the i-th marked point, and let Li = s
iωπ. A Hodge
integral is an integral of the form
1 · · ·ψjnn λ
1 · · ·λkgg
where ψi = c1(Li) is the first Chern class of Li, and λj = cj(E) is the j-th
Chern class of the Hodge bundle. Let
Λ∨g (u) = u
g − λ1u+ · · ·+ (−1)gλg
be the Chern polynomial of E∨, the dual of the Hodge bundle.
Define
Cg,µ(τ) = −
−1ℓ(µ)
|Aut(µ)|
[τ(τ + 1)]ℓ(µ)−1
ℓ(µ)∏
∏µi−1
a=1 (µiτ + a)
(µi − 1)!
Mg,l(µ)
Λ∨g (1)Λ
g (−τ − 1)Λ∨g (τ)∏ℓ(µ)
i=1 (1− µiψi)
Note that
C0,µ(τ) = −
−1ℓ(µ)
|Aut(µ)|
[τ(τ + 1)]ℓ(µ)−1
ℓ(µ)∏
∏µi−1
a=1 (µiτ + a)
(µi − 1)!
M0,l(µ)
∏ℓ(µ)
i=1 (1− µiψi)
. (3.6)
The coefficient of the leading term in τ is:
−1ℓ(µ)
|Autµ|
· |µ|ℓ(µ)−3 . (3.7)
The Mariño-Vafa formula gives the following identity:
pµ(x)
u2g−2+ℓ(µ)Cg,µ(τ) = log
ρ)sA(x)
(3.8)
where qρ = (q−1/2, q−3/2, · · · , q−n+1/2, · · · ).
ρ)sA(y)q
Gµpµ.
|Λ|=µ
ℓ(Λ)∏
χAα(Λ
sAα(q
2 (3.9)
= Gµ(0) +
|Λ|=µ
Ω 6=(1Λ),A
χA(Λ)
χA(Ω)
2 − q− 12 )|µ|
|Λ|=µ
χA(Λ)
A (3.10)
The third summand of the above formula gives the non-vanishing leading
term in τ which is equal to (3.7). Therefore, we have:
|Λ|=µ
χA(Λ)
A 6= 0 (3.11)
for ∀p ≥ |µ|+ ℓ(µ)− 2.
3.3 Cabling Technique
Given irreducible representations VA1 , ..., VAL to each component of link L.
Let |Aα| = dα, ~d = (d1, ..., dL). The cabling braid of L, L~d, is obtained by
substituting dα parallel strands for each strand of Kα, α = 1, . . . , L.
Using cabling of the L gives a way to take trace in the vector space of
tensor product of fundamental representation. To get the original trace, one
has to take certain projection, which is the following lemma in [30].
Lemma 3.2 ([30], Lemma 3.3). Let Vi = SiV
⊗di for some minimal projec-
tions, Si = Sj if the i-th and j-th strands belong to the same knot. Then
trV1⊗···Vm (h(L)) = trV ⊗n(h(L(d1,...,dL)) ◦S1 ⊗ · · · ⊗Sn) .
where m is the number of strands belonging to L, n =
α=1 dαrα, rα is the
number of strands belong to Kα, the α-th component of L.
4 Proof of Theorem 1
4.1 Pole structure of quantum group invariants
By an observation from the action of Ř on V ⊗ V , we define
X̃(i1,...,in) = q
#{(j,k)|j<k,ij>ik}
2 Xi1 ⊗ · · · ⊗Xin .
{X̃(i1,...,in)} form a basis of V ⊗n. By (3.2), ∀gj ∈ Hn(q), we have
2gjX̃(...,ij,ij+1,...) =
X̃(...,ij+1,ij ,...), ij ≤ ij+1 ,
qX̃(...,ij+1,ij ,...) + (1− q)X̃(...,ij,ij+‘,...), ij ≥ ij+1.
(4.1)
Lemma 4.1. Let © be the unknot. Given any ~A = (A1, . . . , AL) ∈ PL,
W ~A(L; q, t)
W ~A(©⊗L; q, t)
ξKα(t)
dα , (4.2)
where |Aα| = dα, Kα is the α-th component of L, and ξKα(t), α = 1, . . . , L,
are independent of ~A.
Proof. Choose β ∈ Bm such that L is the closure of β, the total number
of crossings of L~d is even and the last L strands belongs to distinct L com-
ponents of L. Let rα be the number of the strands which belong to Kα.
dαrα is equal to the number of components in the cabling link L~d.
Y = tr
dα(rα−1)
. (4.3)
Y is both a central element of EndUq(slN )
V ⊗(d1+...+dL)
and a Z[q±1]-matrix
under the basis {X̃(i1,...,in)}.
On the other hand,
p ~A = pA1 ⊗ ...⊗ pAL
is a Z (q±1)-matrix under the basis {X̃(i1,...,in)}. By Schur lemma, we have
p ~A ◦ Y = ǫ · p ~A ,
where ǫ ∈ Z(q±1) is an eigenvalue of Y . Since Z[q±1] is a UFD for transcen-
dental q, ǫ must stay in Z[q±1]. By Lemma 3.2,
trV ⊗n(p
⊗ · · · ⊗ p⊗rL
◦ L~d) = trV ⊗(d1+···dL)(p ~A ◦ Y)
= ǫ · tr
⊗(d1+···dL) (p ~A)
= ǫ ·W ~A(©
⊗L) . (4.4)
The definition of quantum group invariants of L gives
W ~A(L; q, t) = q
κAαw(Kα)/2t
dαw(Kα)/2 · ǫ ·W ~A(©
⊗L; q, t) (4.5)
When q → 1, L~d reduces to an element in symmetric group Σn of ‖ ~d ‖
cycles. Moreover, when q → 1, the calculation is actually taken in individual
knot component while the linking of different components have no effect. By
Example 2.4 (1) and (3), we have
W ~A(©
⊗L; q, t) =
WAα(©; q, t) .
This implies
W ~A(L; q, t)
W ~A(©⊗L; q, t)
WAα(Kα; q, t)
WAα(©; q, t)
. (4.6)
Let’s consider the case when K is a knot. A is the partition of d associated
with K and Kd is the cabling of K. Each component of Kd is a copy of
K. q → 1, Kd reduces to an element in Σdr. To calculate the Y , it is then
equivalent to discussing d disjoint union of K. Say K has r strands. Consider
Y0 = trV ⊗(r−1) K .
The eigenvalue of Y0 is then PK(1,t)tw(K) , where PK(q, t) is the HOMFLY polyno-
mial for K. Denote by ξK(t) = PK(1, t), we have
WA(K; q, t)
WA(©; q, t)
= ξK(t)
|A| . (4.7)
Combined with (4.6), the proof is completed.
4.2 Symmetry of quantum group invariants
Define
φ~µ (q) =
ℓ(µα)∏
/2 − qµαj /2) .
Comparing (2.3) and (0.2), we have
F~µ =
χ ~A (~µ/d)
z~µ/d
qd, td
d · z~µ/d
χ ~A(~µ/d)
P ~B(q
d, td)
MAαBα(q
φ~µ/d(q
z~µ/d
χ ~B(~µ/d)P ~B(q
d, td)
= φ~µ(q)
d · z~µ/d
χ ~B(~µ/d)P ~B(q
d, td) ,
φ~µ(q)
d · z~µ/d
χ ~B(~µ/d)P ~B(q
d, td) . (4.8)
Apply Möbius inversion formula,
P ~B(q, t) =
χ ~B(~µ)
φ~µ(q)
F~µ/d(q
d, td) (4.9)
where µ(d) is the Möbius function defined as follows
µ(d) =
(−1)r, if d is a product of r distinct prime numbers;
0, otherwise.
To prove the existence of formula (0.5), we need to prove:
• Symmetry of P ~B(q, t) = P ~B(q−1, t).
• The lowest degree
q−1/2 − q1/2
in P ~B is no less than −2.
Combine (4.8), (2.5) and (2.4), it’s not difficult to find that the first
property on the symmetry of P ~B follows from the following lemma.
Lemma 4.2. W ~At(q, t) = (−1)‖
~A‖W ~A(q
−1, t).
Proof. The following irreducible decomposition of Uq(slN) modules is well-
known:
V n =
dBVB ,
where dB = χB(C(1n)).
Let d ~A =
α=1 dAα. Combined with Lemma 3.2 and eigenvalue of Y in
Lemma 4.1 and (2.1), we have
W( ,..., )(L~d) =
| ~A|=~d
W ~A(L)d ~A
| ~A|=~d
d ~A q
κAαw(Kα)/2 · t
|Aα|w(Kα)/2ǫ ~A ·
dimq VAα , (4.10)
Where ǫ ~A is the eigenvalue of Y on
α=1 VAα as defined in the proof of
Lemma 4.1. Here if we change Aα to (Aα)t, we have κ(Aα)t = −κAα , which
is equivalent to keep Aα while changing q to q−1.
Note that W( ,..., )(L~d) is essentially a HOMFLY polynomial of L~d by
Example 2.4. From the expansion of HOMFLY polynomial (2.6) and Exam-
ple 2.4 (2), we have
W( ,..., )(L~d ; q
−1, t) = (−1)
|Aα|W( ,..., )(L~d ; q, t) . (4.11)
However, one can generalize the definition of quantum group invariants
of links in the following way. Note that in the definition of quantum group
invariants, the enhancement of Ř, K2ρ, acts on Xi (see (3.1)) as a scalar
(N+1−2i). We can actually generalize this scalar to any zαi where α cor-
responds the strands belonging to the α-th component (cf. [43]). It’s not
difficult to see that (4.10) still holds. The quantum dimension of VAα thus
becomes sAα(z
1 , . . . , z
N) obtained in the same way as (3.4).
We rewrite the above generalized version of quantum group invariants of
links as W ~A(L; q, t; z), where z = {zα}. (4.11) becomes
W( ,..., )(L~d ; q
−1, t;−z) = (−1)
|Aα|W( ,..., )(L~d ; q, t; z) (4.12)
Now, combine (4.12), (4.10) and (4.11), we obtain
d ~At q
κ(Aα)tw(Kα)/2 · t
|(Aα)t|w(Kα)/2ǫ ~At(q
−1; −z)
s(Aα)t(−zα)
= (−1)
α |Aα| ·
d ~A q
κAαw(Kα)/2 · t
|Aα|w(Kα)/2ǫ ~A(q; z)
sAα(z
(4.13)
Note the following facts:
sAt(−z) = (−1)ℓ(A)sA(z) , (4.14)
d ~At = d ~A . (4.15)
where the second formula follows from
χAt(Cµ) = (−1)|µ|−ℓ(µ)χA(Cµ) . (4.16)
Apply (4.14) and (4.15) to (4.13). Let zαi = q
(N+1−2i), then using −z in-
stead of z is equivalent to substitute q by q−1 while keeping t in the definition
of quantum group invariants of links. This can be seen by comparing
zαi = q
−N−2i+1
ℓ(µ)∏
t−µj/2 − tµj/2
q−µj/2 − qµj/2
pµ(−zα) = (−1)ℓ(µ)pµ(zα) .
Therefore, we have
ǫ ~At(q
−1, t) = ǫ ~A(q, t) . (4.17)
By the formula of quantum dimension, it is easy to verify that
W ~At(©
L; q, t) = (−1)‖ ~A‖W ~A(©
L; q−1, t) (4.18)
Combining (4.4), (4.17) and (4.18), the proof of the Lemma is then com-
pleted.
By Lemma 4.2, we have the following expansion about P ~B
P ~B(q, t) =
Q∈Z/2
N ~B,g,Q(q
−1/2 − q1/2)2g−2N0tQ
for some N0. We will show N0 ≤ 1.
Let q = eu. The pole order of (q−1/2 − q1/2) in P ~B is the same as pole
order of u.
Let f(u) be a Laurent series in u. Denote degu f to be the lowest degree
of u in the expansion of u in f .
Combined with (4.8), N0 ≤ 1 follows from the following lemma.
Lemma 4.3. degu F~µ ≥ ℓ(~µ)− 2.
Lemma 4.3 can be proved through the following cut-and-join analysis.
4.3 Cut-and-join analysis
4.3.1 Cut-and-join operators
Let τ = (τ1, · · · , τL), substitute
W ~A(L; q, t; τ) =W ~A(L; q, t) · q
α=1 κAατα/2
in the Chern-Simons partition function, we have the following framed parti-
tion function
Z(L; q, t, τ) = 1 +
~A 6=0
W ~A(L; q, t, τ) · s ~A(x) .
Similarly, framed free energy
F (L; q, t, τ) = logZ(L; q, t, τ) .
We also defined framed version of Z~µ and F~µ as follows
Z(L; q, t, τ) = 1 +
~µ6=0
Z~µ(q, t, τ) · p~µ(x) ,
F (L; q, t, τ) =
~µ6=0
F~µ(q, t, τ)p~µ(x) .
One important fact of these framing series is that they satisfy the cut-and-
join equation which will give a good control of F~µ.
Define exponential cut-and-join operator E
i,j≥1
ijpi+j
∂pi∂pj
+ (i+ j)pipj
∂pi+j
, (4.19)
and log cut-and-join operator L
i,j≥1
ijpi+j
∂pi∂pj
+ (i+ j)pipj
∂pi+j
+ ijpi+j
. (4.20)
Here {pi} are regarded as independent variables. Schur function sA(x) is
then a function of {pi}.
Schur function sA is an eigenfunction of exponential cut-and-join with
eigenvalue κA [8, 12, 49]. Therefore, Z(L; q, t, τ) satisfies the following ex-
ponential cut-and-join equation
∂Z(L; q, t, τ)
EαZ(L; q, t, τ) , (4.21)
or equivalently, we also have the log cut-and-join equation
∂F (L; q, t, τ)
LαF (L; q, t, τ) . (4.22)
In the above notation, Eα and Lα correspond to variables {pi} which take
value of {pi(xα)}.
(4.22) restricts to ~µ will be of the following form:
|~ν|=|~µ|, ℓ(~ν)=ℓ(~µ)±1
α~µ~νF~ν + nonlinear terms
, (4.23)
where α~µ~ν is some constant, ~ν is obtained by cutting or jointing of ~µ.
Recall that given two partitions A and B, we say A is a cutting of B if
one cuts a row of the Young diagram of B into two rows and reform it into
a new Young diagram which happens to be the Young diagram of A, and we
also say B is a joining of A. For example, (7, 3, 1) is a joining of (5, 3, 2, 1)
where we join 5 and 2 to get 7 boxes. Using Young diagram, it looks like
=⇒ cut=⇒ .
In (4.23), cutting and joining happens only for the α-th partition.
4.3.2 Degree of u
By (2.5), it is easy to see through induction that for two links L1 and L2,
F( ,··· , )(L1 ⊗ L2) = 0 . (4.24)
For simplicity of writing, we denote by
F( ,··· , )(L) = F ◦(L)
Note that when we put the labeling irreducible representation by the
fundamental ones, quantum group invariants of links reduce to HOMFLY
polynomials except for a universal factor. Therefore, if we apply skein re-
lation, the following version of skein relation can be obtained. Let positive
crossing
used later appear between two different components, K1 and K2, of the link.
Denote by lk = lk(K1,K2), then:
− F ◦
= t−lk+
2 (q−
2 − q
2 )F ◦
(4.25)
We want to claim that
degu F
◦(L) ≥ L− 2 , (4.26)
where L is the number of components of L.
Firstly, for a knot K, a simple computation shows that:
degu F (K) = −1 .
Using induction, we may assume that claim (4.26) holds for L ≤ k. When
L = k + 1, by (4.25),
− F ◦
= 1 + degu
≥ k − 1 . (4.27)
However, if
< k − 1 ,
this will imply that in the procedure of unlinking L, the lowest degree term
of u in F ◦ are always the same. However, this unlinking will lead to F ◦ equal
to 0 due to (4.24), which is a contradiction! This implies that if the number
of components of L1 is greater than the number of components of L2, we
always have
degu F
◦(L1) > degu F ◦(L2)
Therefore, we proved the claim (4.26).
4.3.3 Induction procedure of cut-and-join analysis
Let µi = (µi1, · · · , µiℓi). We use symbol
Ẑ~µ = Z~µ · z~µ; F̂ = F~ν · z~µ .
Notice the following fact from the definition of quantum group invariants:
Ẑ(µ1,··· ,µL)(L) = Ẑ(µ11),··· ,(µ1ℓ1 ),··· ,(µL1 ),··· ,(µLℓL )(L~µ) , (4.28)
F̂(µ1,··· ,µL)(L) = F̂(µ11),··· ,(µ1ℓ1 ),··· ,(µL1 ),··· ,(µLℓL)(L~µ) . (4.29)
Let τ = (τ1, 0, · · · , 0). Similar as (3.9), we give the following formula:
F~µ =
|Λ|=~µ
ℓ(Λ)∏
χAα(Λ
W ~A(q, t)q
= F~µ(0) +
|Λ|=~µ
Ω 6=(1Λ),A
χA(Λ)
χA(Ω)
ZΩ(0)κ
+ Z1Λ(0)
|Λ|=~µ
χA(Λ)
A (4.30)
Let g
~µ (τα) be the degree of τα in the coefficient of the lowest degree of
u in F~µ. By (3.11) and (4.30), if |µα| > 1, we have
degτα g
~µ = |µ
α|+ ℓ(µα)− 2 > 0 .
By induction, assume that if |µα| ≤ d, degu F~µ ≥ ℓ(~µ) − 2 and hence by
(3.11), degτα g
~µ (τα) = |µα|+ℓ(µα)−2. Combined with (4.29), if |λα| = d+1
and ℓ(λα) > 1, we have:
degu F~λ = ℓ(
~λ)− 2 .
Without loss of generality, assume µ1 = (d, 1), we will just consider the
cut-and-join equation for τ1:
∂F(µ1,··· ,µL)
(d+1),µ2··· ,µL
ℓ(~ν)=ℓ(~µ)+1
β~νF~ν
where β~ν are some constants and ∗ represents some non-linear terms in the
cut-and-join equation. The crucial observation of this non-linear terms is
that its degree in u is equal to ℓ(~µ)− 2. Comparing the degree in u on both
sides of the equation, we have:
degu F
(d+1),µ2,··· ,µL
) = 1 +
ℓ(µα)− 2 .
This completes the induction. The proof of Lemma 4.3 follows immediately.
Define
F̃~µ =
φ~µ(q)
, Z̃~µ =
φ~µ(q)
Lemma 4.3 directly implies follows:
Corollary 4.4. F̃ are of the following form:
F̃~µ(q, t) =
finitely many nα
aα(t)
[nα]2
+ polynomial.
Remark 4.5. Combine the above Corollary, (2.7) and (2.5), we have:
F̃( ,..., )(q, t) =
+ polynomial. (4.31)
4.4 Framing and pole structures
Consider δn = σ1 · · ·σn−1. LetSA be the minimal projection fromHn → HA,
χA(Cµ)SA
We will apply a lemma of Aiston-Morton [2] in the following computation:
δnnSA = q
κASA .
(d1), . . . , (dL)
, . . . ,
Due to the cabling formula to the length of partition (4.28) and (4.29), we
can simply deal with all the color of one row without loss of generality. Take
framing τα = nα+
and choose a braid group representative of L such that
the writhe number of Lα is nα. Denote by ~τ = (τ1, . . . , τL),
L; q, t;~τ
χ ~A(C~d)W ~A(L; q, t)q
α=1 κAα
α dαnαTr
χ ~A(C~d)q
dα ⊗Lα=1 SAα
dαnαTr
χ ~A(C~d)(δd1 ⊗ · · · ⊗ δdL)⊗
α=1 SAα
dαnαTr
L~d · ⊗
α=1δα ·P
⊗ · · · ⊗P(dL)
Here, P
means that in the projection, we use qdα, tdα instead of using q, t.
If we denote by
L ∗Q~d = L~d · δ~d ·P
⊗ · · · ⊗P(dL)
we have
L; q, t;
= H(L ∗Q~d) . (4.32)
Here H is the HOMFLY polynomial which is normalized as
H(unknot) =
2 − t− 12
2 − q− 12
With the above normalization, for any given link L, we have
[1]L · H(L) ∈ Q
[1]2, t±
Substituting q by qdα in the corresponding component, it leads to the follows:
[dα] · Ẑ~d(L; q, t;~τ) ∈ Q
[1]2, t±
. (4.33)
On the other hand, given any two frames ~ω = (ω1, . . . , ωL),
Ẑ~µ(L; q, t; ~ω) =
χ ~A(C~µ)W ~A(L; q, t; ~ω)
χ ~A(C~µ)
χ ~A(C~ν)
Ẑ~ν(L; q, t)q
κAαωα .
Exchange the order of summation, we have the following convolution formula:
Ẑ~µ(L; q, t; ~ω) =
Ẑ~ν(L; q, t)
χ ~A(C~µ)χ ~A(C~ν)q
κAαωα . (4.34)
Return to (4.33). This property holds for arbitrary choice of nα, α =
1, . . . , L. By convolution formula, the coefficients of possible other poles
vanish for arbitrary integer nα. q
nα is involved through certain polynomial
relation, which implies the coefficients for other possible poles are simply
zero. Therefore, (4.33) holds for any frame.
Consider Ẑ~d(L), where each component of L is labeled by a young diagram
of one row. As discussed above, this assumption does not lose any generality.
Let K be a component of L labeled by the color (c). If we multiply Z̃~d(L)
by [c]2, we call it normalizing Z̃~d(L) w.r.t K. Similarly, we call [c]2F̃~d(L)
normalizing F̃~d(L) w.r.t K.
By (4.31), after normalizing F̃( ,..., )(L) w.r.t any component of L, one
will obtain a polynomial in terms of [1]2 and t±
2 . Note that
Z̃~d =
Aut |Λ|
Therefore, if we normalizing on both side w.r.t K, the equality still holds.
Applying (4.32), one has
[c]2F̃~d(L; q, t) ∈ Q
[1]2, t±
. (4.35)
We thus proved the following proposition:
Proposition 4.6. Notation as above, we have:
[dα] · Ẑ~d(L; q, t) ∈ Q[[1]
2, t±
2 ]; (4.36)
2F̃~d(L; q, t) ∈ Q
[1]2, t±
, ∀α . (4.37)
For a given ~d, denote by D~d the gcd{d1, . . . , dL}. Proposition 4.6 implies
that the principle part of F̃ are possible summations of
, where k divides
all dα’s, or equivalently, k|D~d.
Similar as above, choose a braid group element representative for L such
that its α-th component has writhe number nα and choose frame as ~τ =
(n1 +
, . . . , nL +
Ẑ~d(L; q, t;~τ) =
χ ~A(C~d)W ~A(L; q, t)q
κAα(nα+
dαnαTr
χ ~A(C~d)q
D~d ⊗Lα=1 SAα
α dαnαTr(L~d · ⊗
⊗Lα=1 P
(D~d)
(dα/D~d)
α dαnαẐ~d/D~d
(L~d · ⊗
; qDµ, tDµ) . (4.38)
This implies that Ẑ~d is a rational function of q
D~d and t±
D~d .
Passing this to F̃~d, we have F̃~d is a rational function of [D~d]
2 and t±
D~d .
As discussed above, the principle part of F̃~d are summations of
where
k|D~d. Combining them, we have
F̃~d(L; q, t) =
H~d/D~d
(tD~d)
[D~d]
+ polynomial in [D~d]
2 and t±
D~d . (4.39)
Once again, due to arbitrary choice of nα, we know the above pole structure
of F̃~d holds for any frame.
Now we will show H~d/D~d
(t) only depends on ~d/D~d and L. If one checks
(4.38), one finds the computation also follows if 1/D~d is replaced by 1/k for
any k|D~d. By induction, we obtain that H~d/D~d(t) only depends on
~d/D~d and
Combining (4.29), we proved the following Proposition:
Proposition 4.7. Notations are as above. Assume L is labeled by the
color ~µ = (µ1, . . . , µL). Denote by D~µ is the greatest common divisor of
{µ11, . . . , µ1ℓ(µ1), . . . , µij, . . . , µLℓ(µL)}. F̃~µ has the following structure:
F̃~µ(q, t) =
H~µ/D~µ(t
[D~µ]2
+ f(q, t) ,
where H~µ/D~µ(t) only depends on ~µ/D~µ and L, f(q, t) ∈ Q
[1]2, t±
Remark 4.8. In Proposition 4.7, it is very interesting to interpret in topo-
logical string side that H~µ/D~µ(t) only depends on ~µ/D~µ and L. The principle
term is generated due to summation of counting rational curves and indepen-
dent choice of k in the labeling color k · ~µ/D~µ. This phenomenon simply tells
us that contributions of counting rational curves in the labeling color k ·~µ/D~µ
are through multiple cover contributions of ~µ/D~µ.
5 Proof of Theorem 2
5.1 A ring characterizes the partition function
Let ~d = (d1, ..., dL), y = (y
1, . . . , yL). Define
T~d =
~B=~d
s ~B(y)P ~B(q, t) . (5.1)
By the calculation in Appendix A.1, one will get
T~d = q
A∈P(Pn),‖A‖=~d/k
θAWA(q
k, tk)sA(z
k) (5.2)
However,
T~d =
| ~B|=~d
s ~B(y)P ~B(q, t)
Q∈Z/2
| ~B|=~d
N ~B; g,Qs ~B(y)
(q1/2 − q−1/2)2g−2tQ .
Denote by Ω(y) the space of all integer coefficient symmetric functions in y.
Since Schur functions forms a basis of Ω(y) over Z, N ~B; g,Q ∈ Z will follow
from ∑
| ~B|=~d
N ~B; g,Qs ~B(y) ∈ Ω(y) .
Let v = [1]2q. It’s easy to see that [n]
q is a monic polynomial of v with
integer coefficients. The following ring is very crucial in characterizing the
algebraic structure of Chern-Simons partition function, which will lead to
the integrality of N ~B; g,Q.
R(y; v, t) =
a(y; v, t)
: a(y; v, t) ∈ Ω(y)[v, t±1/2], b(v) =
q ∈ Z[v]
If we slightly relax the condition in the ring R(y; v, t), we have the following
ring which is convenient in the p-adic argument in the following subsection.
M(y; q, t) =
f(y; q, t)
: f ∈ Ω(y)[q±1/2, t±1/2], b(v) =
q ∈ Z[v]
Given
f(y; q,t)
∈ M(y; q, t), if f(y; q, t) is a primitive polynomial in terms of
q±1/2, t±1/2 and Schur functions of y, we call
f(y; q,t)
is primitive.
5.2 Multi-cover contribution and p-adic argument
Proposition 5.1. T~d(y; q, t) ∈ R(y; v, t).
Proof. Recall the definition of quantum group invariants of links. By the
formula of universal R-matrix, it’s easy to see that
W ~A(L) ∈ M(y; q, t) .
Since we have already proven the existence of the pole structure in LMOV
conjecture, Proposition 5.1 will be naturally satisfied if we can prove
T~d(y; q, t) ∈ M(y; q, t) . (5.3)
Before diving into the proof of (5.3), let’s do some preparation. For
∀A = ( ~A1, ~A2, ...) ∈ P(PL) ,
define
A(d) =
~A1, ..., ~A1︸ ︷︷ ︸
, ~A2, ..., ~A2︸ ︷︷ ︸
, . . .
Lemma 5.2. If θA =
, d > 1 and gcd (c, d) = 1, then for any r|d, we can
find B ∈ P(PL) such that A = B(r).
Proof. Let ℓ = ℓ(A), we have
(−1)ℓ(A)−1
(−1)ℓ(A)−1(ℓ(A)− 1)!
|AutA|
~A1, ..., ~A1︸ ︷︷ ︸
, . . . , ~An, ..., ~An︸ ︷︷ ︸
so ℓ(A) = m1 + ...+mn. Note that
m1, m2, · · · , mn
Let η = gcd(m1, m2, · · · , mn). We have A = Ã(η), where
~A1, . . . , ~A1︸ ︷︷ ︸
, . . . , ~An, . . . , ~An︸ ︷︷ ︸
By Corollary A.4 , ℓ
|uA and gcd(c, d) = 1, one has d|η. We can take B =
). This completes the proof.
Remark 5.3. By the choice of η in the above proof, we know | ~Aα| is divisible
by η for any α.
By Lemma 5.2 and (5.2), we know T~d is of form f(y; q, t)/k, where f ∈
M(y; q, t), k|~d. We will show k is in fact 1.
Given r
f(y; q,t)
where
f(y; q, t)
∈ M(y; q, t)
is primitive, define
f(y; q, t)
= Ordp
. (5.4)
Lemma 5.4. Given A∈ P
, p any prime number and fA(y; q, t) ∈
M(y; q, t), we have
θA(p)fA(p)(y; q, t)−
θAfA(y
p; qp, tp)
≥ 0 .
Proof. Assume θA =
pr ·a , where gcd(p
r · a, b) = 1, p ∤ a. In (5.5), minus is
taken except for one case: p = 2 and r = 0, in which the calculation is very
simple and the same result holds. Therefore, we only show the general case
which minus sign is taken. By Lemma 5.2, one can choose B ∈ P(PL) such
that A = B(p
r). Note that fA = f
Let s = ℓ(B) and
~B1, . . . , ~B1︸ ︷︷ ︸
, . . . , ~Bk, . . . , ~Bn︸ ︷︷ ︸
Since ℓ(A(p)) = p · ℓ(A), by Theorem A.2,
θA(p) −
= Ordp
p · ℓ(A)
pr+1s
pr+1s1, · · · , pr+1sk
prs1, · · · , prsk
(5.5)
≥ 2(r + 1)− (1 + r)
> 0 . (5.6)
By Theorem A.6,
fA(p)(y; q, t)− fA(yp; qp, tp
= Ordp
pr+1a
fB(y; q, t)
pr+1 − fB(yp; qp, tp)p
≥ 0 .
Apply the above two inequalities to
θA(p)fA(p)(y; q, t)−
θAfA(y
p; qp, tp)
= Ordp
θA(p) −
fA(p)(y; q, t) +
fA(p)(y; q, t)− fA(yp; qp, tp
The proof is completed.
Apply the above Lemma, we have
θA(p)WA(p)(q, t)sA(p)(z)−
θAWA(q
p, tp)sA(z
≥ 0 . (5.7)
Φ~d (y; q, t) =
A∈P(Pn), ‖A‖=~d
θAWA(q, t)sA(z) .
By Lemma 5.2, one has
B : ‖ B ‖= p ~d and Ordp(θB) < 0
A(p) : ‖ A ‖= ~d
Therefore, By (5.7),
(y; q, t)−
Φ~d (y
p; qp, tp)
‖A‖=~d
θA(p)WA(p)(q, t)sA(p)(z)−
θAWA(q
p, tp)sA(z
≥ 0 .
We have thus proven the following Lemma.
Lemma 5.5. For any prime number p and ~d,
(y; q, t)− 1
Φ~d (y
p; qp, tp)
≥ 0 .
For any p|~d, by (5.2),
T~d = q
A∈P(PL), ‖A‖=~d/k
θAWA(q
k, tk)sA(z
k|~d, p∤k
Φ~d/k(y
n; qn, tn) +
k|~d, p∤k
µ(pk)
Φ~d/(pk)(y
pk; qpk, tpk)
k|~d, p∤k
Φ~d/k(y
k; qk, tk)− 1
Φ~d/(pk)(y
pk; qpk, tpk)
By Lemma 5.5,
Ordp T~d ≥ 0 . (5.8)
By the arbitrary choice of p, we prove that T~d ∈ M(y; q, t), hence
T~d ∈ R(y; v, t) .
The proof of Proposition 5.1 is completed.
5.3 Integrality
By (4.9) and Propositions 4.7, (note that φ~µ/d(q
d, td) = φ~µ(q, t).)
P ~B(q, t) =
χ ~B(~µ)
φ~µ(q)
F~µ/d(q
d, td)
χ ~B(~µ)
F̃~µ/d(q
d, td)
χ ~B(~µ)
d|D~µ
H~µ/D~µ(t
[D~µ]2
+ polynomial
χ ~B(~µ)δ1,D~µ
H~µ/D~µ(t
[D~µ]2
+ polynomial ,
where δ1,n equals 1 if n = 1 and 0 otherwise. It implies that P ~B is a rational
function which only has pole at q = 1. In the above computation, we used a
fact of Möbius inversion,
= δ1,n .
Therefore, for each ~B,
Q∈Z/2
N ~B; g,Q(q
−1/2 − q1/2)2gtQ ∈ Q[(q−1/2 − q1/2)2, t±
2 ] .
On the other hand, by Proposition 5.1, T~d ∈ R(y; v, t) and Ordp T~d ≥ 0
for any prime number p. We have
| ~B|=~d
N ~B; g,Qs ~B(y) ∈ Ω(y) .
This implies N ~B; g,Q ∈ Z.
Combine the above discussions, we have
Q∈Z/2
N ~B; g,Q(q
−1/2 − q1/2)2gtQ ∈ Z[(q−1/2 − q1/2)2, t±1/2] .
The proof of Theorem 2 is completed.
6 Concluding Remarks and Future Research
In this section, we briefly discuss some interesting problems related to string
duality which may be approached through the techniques developed in this
paper.
6.1 Duality from a mathematical point of view
Let p = (p1, . . . ,pL), where pα = (pα1 , p
2 , . . . , ) . Defined the following gen-
erating series of open Gromov-Witten invariants
Fg,~µ(t, τ) =
where ω is the Kähler class of the resolved conifold, τ is the framing parameter
t = e
ω, and e
= tQ .
Consider the following generating function
F (p; u, t; τ) =
u2g−2+ℓ(~µ)Fg,~µ(t; τ)
pαµα .
It satisfies the log cut-and-join equation
∂F (p; u, t; τ)
LαF (p; u, t; τ) .
Therefore, duality between Chern-Simons theory and open Gromov-Witten
theory reduces to verifying the uniqueness of the solution of cut-and-join
equation.
Cut-and-join equation for Gromov-Witten side comes from the degen-
eracy and gluing procedure while uniqueness of cut-and-join system should
in principle be obtained from the verification at some initial value. How-
ever, it seems very difficult to find a suitable initial value. For example, in
the case of topological vertex theory, cut-and-join system has singularities
when the framing parameter takes value at 0, −1, ∞ while these points are
the possible ones to evaluate at. One solution might be through studying
Riemann-Hilbert problem on controlling the monodromy at three singularity
points. When the case goes beyond, the situation will be even more compli-
cated. A universal method of handling uniqueness is required for the final
proof of Chern-Simons/topological string duality conjecture.
A new hope might be found in our development of cut-and-join analysis.
In the log cut-and-join equation (4.22), the non-linear terms reveals the im-
portant recursion structure. For the uniqueness of cut-and-join equation, it
will appear as the vanishing of all non-linear terms. We will put this in our
future research.
6.2 Other related problems
There are many other problems related to our work on LMOV conjecture.
We briefly list some problems that we are working on.
Volume conjecture was proposed by Kashaev in [16] and reformulated by
[34]. It relates the volume of hyperbolic 3-manifolds to the limits of quantum
invariants. This conjecture was later generalized to complex case [35] and
to incomplete hyperbolic structures [13]. The study of this conjecture is still
staying at a rather primitive stage [36, 17, 48, 10].
LMOV conjecture has shed new light on volume conjecture. The cut-and-
join analysis we developed in this paper combined with rank-level duality in
Chern-Simons theory seems to provide a new way to prove the existence of
the limits of quantum invariants.
There are also other open problems related to LMOV conjecture. For
example, quantum group invariants satisfy skein relation which must have
some implications on topological string side as mentioned in [20]. One could
also rephrase a lot of unanswered questions in knot theory in terms of open
Gromov-Witten theory. We hope that the relation between knot theory and
open Gromov-Witten theory will be explored much more in detail in the
future. This will definitely open many new avenues for future research.
A Appendix
Here we carry out the calculation in Section 5.1.
T~d =
| ~B|=~d
s ~B(y)P ~B(q, t)
| ~B|=~d
s ~B(y)χ ~B(~µ)
φ~µ(q)
F~µ/k
qk, tk
k|~µ ,|~µ|=~d
p~µ(y)
φ~µ(q)
Λ∈P(P
), |Λ|=~µ/k
θΛZΛ(q
k, tk)
k|~µ ,|~µ|=~d
‖Λ‖=~µ/k
k)pΛ(y
k)ZΛ(q
k, tk)
k|~µ ,|~µ|=~d
‖Λ‖=~µ/k
k)pΛ(y
ℓ(Λ)∏
χ~Aβ(
W~Aβ(q
|~Aβ |=|~Λβ |=dβ/k
(−1)ℓ(Λ)−1
ℓ(Λ)∏
φ−1~Λβ
(qk)p~Λβ(y
χ~Aβ (
W~Aβ (q
k, tk)
k|~µ ,|~µ|=~d
A=(~A1,...), ‖A‖=~d/k
(−1)ℓ(A)−1
ℓ(A)∏
W~Aβ (q
k, tk)
χ~Aβ(
φ−1~Λβ
(qk)p~Λβ(y
A∈P(PL), ‖A‖=~d/k
θAWA(q
k, tk)sA(z
where
pn(z) = pn(y) · pn(xi = qi−1) .
Lemma A.1. p is prime, r ≥ 1. Then
pr−1a
pr−1b
≡ 0 mod (p2r)
Proof. Consider the ratio
pr−1a
pr−1b
(a−b)pr+k
k∏pr−1b
(a−b)pr−1+k
gcd(k,p)=1,
1≤k≤prb
(a− b)pr + k
≡ 1 + pr(a− b)
gcd(k,p)=1,
1≤k≤prb
mod (p2r) .
Ap(n) =
1≤k≤n,
gcd(k,p)=1
Therefore, the proof of the lemma can be completed by showing
rb) =
for some c, d such that gcd(d, p) = 1.
If gcd(k, p) = 1, there exist αk, βk such that
αkk + βkp
r = 1 .
Bp(n) =
1≤k≤n,gcd(k,p)=1
By the above formula,
rb) ≡ bAp(pr) mod (pr)
≡ bBp(pr) mod (pr) ,
k − p
pr−1∑
pr(pr + 1)
pr−1(pr−1 + 1)
p2r−1(p− 1)
Here, we have 2r − 1 ≥ r since r ≥ 1. The proof is then completed.
The following theorem is a simple generalization of the above Lemma.
Theorem A.2.
i=1 ai = a, p is prime, r ≥ 1, then
pra1, · · · , pran
pr−1a1, · · · , pr−1an
≡ 0 mod (p2r)
Proof. We have
pra1, · · · , pran
pr−1a1, · · · , pr−1an
pr(a−
i=1 ai)
pr−1(a−
i=1 ai)
pr−1ak
≡ 0 mod (p2r) .
In the last step, we used the Lemma A.1. The proof is completed.
Lemma A.3. We have
gcd(a, b)
Proof. Notice that (
gcd(a, b)
gcd(a, b)
However,
gcd(a, b)
gcd(a, b)
= 1 ,
gcd(a, b)
This direct leads to the following corollary.
Corollary A.4. a = a1 + a2 + · · ·+ an. Then
gcd(a1, a2, · · · , an)
a1, a2, · · · , an
Lemma A.5. a, r ∈ N, p is a prime number, then
r − apr−1 ≡ 0 mod (pr) .
Proof. If a is p, since pr−1 ≥ r, the claim is true. If gcd(a, p) = 1, by Fermat
theorem, ap−1 ≡ 1 mod (p). We have
r − apr−1 = apr−1
(kp+ 1)p
r−1(p−1) − 1
pr−1∑
(kp)i
≡ 0 mod (pr) .
Here in the last step, we used Lemma A.3.
A direct consequence of the above lemma is the following theorem.
Theorem A.6. Given f(y; q, t) ∈ M(y; q, t), we have
f(y; q, t)p
r+1 − f(yp; qp, tp)pr
≥ r + 1 .
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Center of Mathematical Sciences
Zhejiang University, Box 310027
Hangzhou, China
Department of mathematics
University of California at Los Angeles
Box 951555
Los Angeles, CA 90095-1555
Email: liu@math.ucla.edu.
Department of Mathematics
Harvard University
One Oxford Street
Cambridge, MA, 02138
Email: ppeng@math.harvard.edu.
	Introduction
	Overview
	Labastida-Mariño-Ooguri-Vafa conjecture
	Main ideas of the proof
	Acknowledgments
	Preliminary
	Partition and symmetric function
	Partitionable set and infinite series
	Labastida-Mariño-Ooguri-Vafa conjecture
	Quantum trace
	Quantum group invariants of links
	Chern-Simons partition function
	Main results
	Hecke algebra and cabling
	Centralizer algebra and Hecke algebra representation
	Quantum dimension
	Cabling Technique
	Proof of Theorem 1
	Pole structure of quantum group invariants
	Symmetry of quantum group invariants
	Cut-and-join analysis
	Framing and pole structures
	Proof of Theorem 2
	A ring characterizes the partition function
	Multi-cover contribution and p-adic argument
	Integrality
	Concluding Remarks and Future Research
	Duality from a mathematical point of view
	Other related problems
	Appendix

  Based on large N Chern-Simons/topological string duality, in a series of
papers, J.M.F. Labastida, M. Marino, H. Ooguri and C. Vafa conjectured certain
remarkable new algebraic structure of link invariants and the existence of
infinite series of new integer invariants. In this paper, we provide a proof of
this conjecture. Moreover, we also show these new integer invariants vanish at
large genera.

Introduction 3
0.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
0.2 Labastida-Mariño-Ooguri-Vafa conjecture . . . . . . . . . . . . 4
0.3 Main ideas of the proof . . . . . . . . . . . . . . . . . . . . . . 6
0.4 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . 8
1 Preliminary 9
1.1 Partition and symmetric function . . . . . . . . . . . . . . . . 9
1.2 Partitionable set and infinite series . . . . . . . . . . . . . . . 11
2 Labastida-Mariño-Ooguri-Vafa conjecture 12
2.1 Quantum trace . . . . . . . . . . . . . . . . . . . . . . . . . . 12
http://arxiv.org/abs/0704.1526v3
2.2 Quantum group invariants of links . . . . . . . . . . . . . . . . 14
2.3 Chern-Simons partition function . . . . . . . . . . . . . . . . . 17
2.4 Main results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Hecke algebra and cabling 22
3.1 Centralizer algebra and Hecke algebra representation . . . . . 22
3.2 Quantum dimension . . . . . . . . . . . . . . . . . . . . . . . 24
3.3 Cabling Technique . . . . . . . . . . . . . . . . . . . . . . . . 27
4 Proof of Theorem 1 28
4.1 Pole structure of quantum group invariants . . . . . . . . . . . 28
4.2 Symmetry of quantum group invariants . . . . . . . . . . . . . 30
4.3 Cut-and-join analysis . . . . . . . . . . . . . . . . . . . . . . . 33
4.4 Framing and pole structures . . . . . . . . . . . . . . . . . . . 38
5 Proof of Theorem 2 41
5.1 A ring characterizes the partition function . . . . . . . . . . . 41
5.2 Multi-cover contribution and p-adic argument . . . . . . . . . 43
5.3 Integrality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6 Concluding Remarks and Future Research 47
6.1 Duality from a mathematical point of view . . . . . . . . . . . 47
6.2 Other related problems . . . . . . . . . . . . . . . . . . . . . . 48
A Appendix 50
A.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
A.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
0 Introduction
0.1 Overview
For decades, we have witnessed the great development of string theory and its
powerful impact on the development of mathematics. There have been a lot
of marvelous results revealed by string theory, which deeply relate different
aspects of mathematics. All these mysterious relations are connected by a
core idea in string theory called “duality”. It was found that string theory
on Calabi-Yau manifolds provided new insight in geometry of these spaces.
The existence of a topological sector of string theory leads to a simplified
model in string theory, the topological string theory.
A major problem in topological string theory is how to compute Gromov-
Witten invariants. There are two major methods widely used: mirror symme-
try in physics and localization in mathematics. Both methods are effective
when genus is low while having trouble in dealing with higher genera due
to the rapidly growing complexity during computation. However, when the
target manifold is Calabi-Yau threefold, large N Chern-Simons/topological
string duality opens a new gate to a complete solution of computing Gromov-
Witten invariants at all genera.
The study of large N Chern-Simons/topological string duality was orig-
inated in physics by an idea that gauge theory should have a string theory
explanation. In 1992, Witten [46] related topological string theory of T ∗M
of a three dimensional manifold M to Chern-Simons gauge theory on M . In
1998, Gopakumar and Vafa [11] conjectured that, at large N , open topo-
logical A-model of N D-branes on T ∗S3 is dual to closed topological string
theory on resolved conifold O(−1) ⊕ O(−1) → P1. Later, Ooguri and Vafa
[38] showed a picture on how to describe Chern-Simons invariants of a knot
by open topological string theory on resolved conifold paired with lagrangian
associated with the knot.
Though large N Chern-Simons/topological string duality still remains
open, there have been a lot of progress in this direction demonstrating the
power of this idea. Even for the simplest knot, the unknot, Mariño-Vafa
formula [33, 26] gives a beautiful closed formula for Hodge integral up to
three Chern classes of Hodge bundle. Furthermore, using topological vertex
theory [1, 27, 28], one is able to compute Gromov-Witten invariants of any
toric Calabi-Yau threefold by reducing the computation to a gluing algorithm
of topological vertex. This thus leads to a closed formula of topological string
partition function, a generating function of Gromov-Witten invariants, in all
genera for any toric Calabi-Yau threefolds.
On the other hand, after Jones’ famous work on polynomial knot invari-
ants, there had been a series of polynomial invariants discovered (for example,
[15, 9, 19]), the generalization of which was provided by quantum group the-
ory [43] in mathematics and by Chern-Simons path integral with the gauge
group SU(N) [45] in physics.
Based on the large N Chern-Simons/topological string duality, Ooguri
and Vafa [38] reformulated knot invariants in terms of new integral invari-
ants capturing the spectrum of M2 branes ending on M5 branes embedded
in the resolved conifold. Later, Labastida, Mariño and Vafa [22, 20] refined
the analysis of [38] and conjectured the precise integrality structure for open
Gromov-Witten invariants. This conjecture predicts a remarkable new al-
gebraic structure for the generating series of general link invariants and the
integrality of infinite family of new topological invariants. In string theory,
this is a striking example that two important physical theories, topological
string theory and Chern-Simons theory, exactly agree up to all orders. In
mathematics this conjecture has interesting applications in understanding
the basic structure of link invariants and three manifold invariants, as well
as the integrality structure of open Gromov-Witten invariants. Recently,
X.S. Lin and H. Zheng [30] verified LMOV conjecture in several lower degree
cases for some torus links.
In this paper, we give a complete proof of Labastida-Mariño-Ooguri-Vafa
conjecture for any link (We will briefly call it LMOV conjecture). First, let
us describe the conjecture and the main ideas of the proof. The details can
be found in Sections 4 and 5.
0.2 Labastida-Mariño-Ooguri-Vafa conjecture
Let L be a link with L components and P be the set of all partitions. The
Chern-Simons partition function of L is given by
ZCS(L; q, t) =
~A∈PL
W ~A(L; q, t)
sAα(x
α) (0.1)
for any arbitrarily chosen sequence of variables
xα = (xα1 , x
2 , . . . , ) .
In (0.1), W ~A(L) is the quantum group invariants of L labeled by a sequence
of partitions ~A = (A1, . . . , AL) ∈ PL which correspond to the irreducible rep-
resentations of quantized universal enveloping algebra Uq(sl(N,C)), sAα(x
is the Schur function.
Free energy is defined to be
F = logZCS .
Use plethystic exponential, one can obtain
~A 6=0
f ~A(q
d, td)
(xα)d
, (0.2)
where
(xα)d =
(xα1 )
d, (xα2 )
d, . . .
Based on the duality between Chern-Simons gauge theory and topological
string theory, Labastida, Mariño, Ooguri, Vafa conjectured that f ~A have the
following highly nontrivial structures.
For any A, B ∈ P, define the following function
MAB(q) =
χA(Cµ)χB(Cµ)
ℓ(µ)∏
(q−µj/2 − qµj/2) . (0.3)
Conjecture (LMOV). For any ~A ∈ PL,
(i). there exists P ~B(q, t) for ∀ ~B ∈ PL, such that
f ~A(q, t) =
|Bα|=|Aα|
P ~B(q, t)
MAαBα(q). (0.4)
Furthermore, P ~B(q, t) has the following expansion
P ~B(q, t) =
Q∈Z/2
N ~B; g,Q(q
−1/2 − q1/2)2g−2tQ . (0.5)
(ii). N ~B; g,Q are integers.
For the meaning of notations, the definition of quantum group invariants
of links and more details, please refer to Section 2.
This conjecture contains two parts:
• The existence of the special algebraic structure (0.5).
• The integrality of the new invariants N ~B; g,Q.
If one looks at the right hand side of (0.5), one will find it very interesting
that the pole of f ~A in (q
−1/2− q1/2) is actually at most of order 1 for any link
and any labeling partitions. However, by the calculation of quantum group
invariants of links, the pole order of f ~A might be going to ∞ when the degrees
of labeling partitions go higher and higher. This miracle cancelation implies
a very special algebraic structure of quantum group invariants of links and
thus the Chern-Simons partition function. in Section 2.4.2, we include an
example in the simplest setting showing that this cancelation has shed new
light on the quantum group invariants of links.
0.3 Main ideas of the proof
In our proof of LMOV conjecture, there are three new techniques.
• When dealing with the existence of the conjectured algebraic structure,
one will encounter the problem of how to control the pole order of
(q−1/2 − q1/2). We consider the framed partition function Z(L; q, t, τ)
of Chern-Simons invariants of links which satisfies the following cut-
and-join equation
∂Z(L; q, t, τ)
i,j≥1
ijpαi+j
∂pαi ∂p
+ (i+ j)pαi p
∂pαi+j
Z(L; q, t, τ) .
Here pαn = pn(x
α) are regarded as independent variables.
However, a deeper understanding of this conjecture relies on the fol-
lowing log cut-and-join equation
∂F (L; q, t, τ)
i,j≥1
ijpαi+j
∂pαi ∂p
+ (i+ j)pαi p
∂pαi+j
+ ijpαi+j
This observation is based on the duality of Chern-Simons theory and
open Gromov-Witten theory. The log cut-and-join equation is a non-
linear ODE systems and the non-linear part reflects the essential recur-
sion structure of Chern-Simons partition function. The miracle cance-
lation of lower order terms of q−1/2 − q1/2 occurring in free energy can
be indicated in the formulation of generating series of open Gromov-
Witten invariants on the geometric side.
A powerful tool to control the pole order of (q−1/2 − q1/2) through log
cut-and-join equation is developed in this paper as we called cut-and-
join analysis. An important feature of cut-and-join equation shows that
the differential equation at partition1 (d) can only have terms obtained
from joining2 while at (1d), non-linear terms vanishes and there is no
joining terms. This special feature combined with the degree analysis
will squeeze out the desired degree of (q−1/2 − q1/2).
• We found a rational function ring which characterizes the algebraic
structure of Chern-Simons partition function and hence (open) topo-
logical string partition function by duality.
A similar ring characterizing closed topological string partition func-
tion firstly appears in the second author’s work on Gopakumar-Vafa
conjecture [39, 40]. The original observation in the closed case comes
from the structure of R-matrix in quantum group theory and gluing
algorithm in the topological vertex theory.
However, the integrality in the case of open topological string the-
ory is more subtle than the integrality of Gopakumar-Vafa invariants
1Here, we only take a knot as an example. For the case of links, it is then a natural
extension.
2Joining means that a partition is obtained from combining two rows of a Young dia-
gram and reforming it into a new partition, while cutting means that a partition is obtained
by cutting a row of a Young diagram into two rows and reforming it into a new partition.
in the closed case. This is due to the fact that the reformulation of
Gromov-Witten invariants as Gopakumar-Vafa invariants in the closed
case weighted by power of curve classes, while in the open case, the gen-
erating function is weighted by the labeling irreducible representation
of Uq(sl(N,C)). This subtlety had already been explained in [22].
To overcome this subtlety, one observation is that quantum group in-
variants look Schur-function-like. This had already been demonstrated
in the topological vertex theory (also see [31]). We refine the ring in
the closed case and get the new ring R(y; q, t) (cf. section 5.1). Corre-
spondingly, we consider a new generating series ofN ~B; g,Q, T~d, as defined
in (5.1).
• To prove T~d ∈ R(y; q, t), we combine with the multi-cover contribution
and p-adic argument therein. Once we can prove that T~d lies in R, due
to the pole structure of the ring R, the vanishing of N ~B; g,Q has to occur
at large genera, we actually proved
Q∈Z/2
N ~B; g,Q(q
−1/2 − q1/2)2gtQ ∈ Z[(q−
2 − q
2 )2, t±
2 ] .
The paper is organized as follows. In Section 1, we introduce some basic
notations about partition and generalize this concept to simplify our calcu-
lation in the following sections. Quantum group invariants of links and main
results are introduced in Section 2. In Section 3, we review some knowledge
of Hecke algebra used in this paper. In Section 4 and 5, we give the proof
of Theorem 1 and 2 which answer LMOV conjecture. In the last section, we
discuss some problems related to LMOV conjecture for our future research.
0.4 Acknowledgments
The authors would like to thank Professors S.-T. Yau, F. Li and Z.-P. Xin
for valuable discussions. We would also want to thank Professor M. Mariño
for pointing out some misleading parts. K. L. is supported by NSF grant.
P. P. is supported by NSF grant DMS-0354737 and Harvard University.
Before he passed away, Professor Xiao-Song Lin had been very interested
in the LMOV conjecture, and had been working on it. We would like to
dedicate this paper to his memory.
1 Preliminary
1.1 Partition and symmetric function
A partition λ is a finite sequence of positive integers (λ1, λ2, · · · ) such that
λ1 ≥ λ2 ≥ · · · .
The total number of parts in λ is called the length of λ and denoted by ℓ(λ).
We use mi(λ) to denote the number of times that i occurs in λ. The degree
of λ is defined to be
|λ| =
If |λ| = d, we say λ is a partition of d. We also use notation λ ⊢ d. The
automorphism group of λ, Aut λ, contains all the permutations that permute
parts of λ while still keeping it as a partition. Obviously, the order of Aut λ
is given by
|Autλ| =
mi(λ)! .
There is another way to rewrite a partition λ in the following format
(1m1(λ)2m2(λ) · · · ) .
A traditional way to visualize a partition is to identify a partition as a
Young diagram. The Young diagram of λ is a 2-dimensional graph with λj
boxes on the j-th row, j = 1, 2, ..., ℓ(λ). All the boxes are put to fit the
left-top corner of a rectangle. For example
(5, 4, 2, 2, 1) = (12245) = .
For a given partition λ, denote by λt the conjugate partition of λ. The Young
diagram of λt is transpose to the Young diagram of λ: the number of boxes
on j-th column of λt equals to the number of boxes on j-th row of λ, where
1 ≤ j ≤ ℓ(λ).
By convention, we regard a Young diagram with no box as the partition
of 0 and use notation (0). Denote by P the set of all partitions. We can
define an operation “ ∪ ” on P. Given two partitions λ and µ, λ ∪ µ is the
partition by putting all the parts of λ and µ together to form a new partition.
For example
(1223) ∪ (15) = (122235).
Using Young diagram, it looks like
∪ = .
The following number associated with a partition λ is used throughout
this paper,
jmj(λ)mj(λ)! , κλ =
λj(λj − 2j + 1) .
It’s easy to see that
κλ = −κλt . (1.1)
A power symmetric function of a sequence of variables x = (x1, x2, ...) is
defined as follows
pn(x) =
xni .
For a partition λ,
pλ(x) =
ℓ(λ)∏
pλj (x).
It is well-known that every irreducible representation of symmetric group
can be labeled by a partition. Let χλ be the character of the irreducible rep-
resentation corresponding to λ. Each conjugate class of symmetric group can
also be represented by a partition µ such that the permutation in the conju-
gate class has cycles of length µ1, . . . , µℓ(µ). Schur function sλ is determined
sλ(x) =
|µ|=|λ|
χλ(Cµ)
pµ(x) (1.2)
where Cµ is the conjugate class of symmetric group corresponding to partition
1.2 Partitionable set and infinite series
The concept of partition can be generalized to the following partitionable set.
Definition 1.1. A countable set (S,+) is called a partitionable set if
1). S is totally ordered.
2). S is an Abelian semi-group with summation “ + ”.
3). The minimum element 0 in S is the zero-element of the semi-group,
i.e., for any a ∈ S,
0+ a = a = a+ 0.
For simplicity, we may briefly write S instead of (S,+).
Example 1.2. The following sets are examples of partitionable set:
(1). The set of all nonnegative integers Z≥0;
(2). The set of all partitions P. The order of P can be defined as follows:
∀λ , µ ∈ P, λ ≥ µ iff |λ| > |µ|, or |λ| = |µ| and there exists a j such
that λi = µi for i ≤ j − 1 and λj > µj. The summation is taken to be
“ ∪ ” and the zero-element is (0).
(3). Pn. The order of Pn is defined similarly as (2): ∀ ~A, ~B ∈ Pn, ~A ≥ ~B
i=1 |Ai| >
i=1 |Bi|, or
i=1 |Ai| =
i=1 |Bi| and there is a j
such that Ai = Bi for i ≤ j − 1 and Aj > Bj. Define
~A ∪ ~B = (A1 ∪ B1, ..., An ∪ Bn).
((0), (0), ..., (0)) is the zero-element. It’s easy to check that Pn is a
partitionable set.
Let S be a partitionable set. One can define partition with respect to
S in the similar manner as that of Z≥0: a finite sequence of non-increasing
non-minimum elements in S. We will call it an S-partition, (0) the zero
S-partition. Denote by P(S) the set of all S-partitions.
For an S-partition Λ, we can define the automorphism group of Λ in a
similar way as that in the definition of traditional partition. Given β ∈ S,
denote by mβ(Λ) the number of times that β occurs in the parts of Λ, we
then have
AutΛ =
mβ(Λ)! .
Introduce the following quantities associated with Λ,
ℓ(Λ)!
|AutΛ|
, θΛ =
(−1)ℓ(Λ)−1
uΛ . (1.3)
The following Lemma is quite handy when handling the expansion of
generating functions.
Lemma 1.3. Let S be a partitionable set. If f(t) =
n≥0 ant
n, then
β 6=0, β∈S
Aβ pβ(x)
Λ∈P(S)
aℓ(Λ)AΛ pΛ(x) uΛ ,
where
ℓ(Λ)∏
pΛj , AΛ =
ℓ(Λ)∏
AΛj .
Proof. Note that
β∈S, β 6=0
Λ∈P(S), ℓ(Λ)=n
ηΛuΛ .
Direct calculation completes the proof.
2 Labastida-Mariño-Ooguri-Vafa conjecture
2.1 Quantum trace
Let g be a finite dimensional complex semi-simple Lie algebra of rank N
with Cartan matrix (Cij). Quantized universal enveloping algebra Uq(g) is
generated by {Hi, X+i, X−i} together with the following defining relations:
[Hi, Hj] = 0 , [Hi, X±j] = ±CijX±j , [X+i, X−j] = δij
−Hi/2
i − q
i − q
1−Cij∑
(−1)k
1− Cij
1−Cij−k
±i X±jX
±i = 0 , for all i 6= j ,
where
{k}q =
2 − q k2
2 − q 12
, {k}q! =
{i}q ,
and {
{a}q · {a− 1}q · · · {a− b+ 1}q
{b}q!
The ribbon category structure associated with Uq(g) is given by the fol-
lowing datum:
1. For any given two Uq(g)-modules V and W , there is an isomorphism
ŘV,W : V ⊗W →W ⊗ V
satisfying
ŘU⊗V,W = (ŘU,W ⊗ idV )(idU ⊗ŘV,W )
ŘU, V⊗W = (idV ⊗ŘU,W )(ŘU, V ⊗ idW )
for Uq(g)-modules U , V , W .
Given f ∈ HomUq(g)(U, Ũ), g ∈ HomUq(g)(V, Ṽ ), one has the following
naturality condition:
(g ⊗ f) ◦ ŘU, V = ŘeU, eV ◦ (f ⊗ g) .
2. There exists an element K2ρ ∈ Uq(g), called the enhancement of Ř,
such that
K2ρ(v ⊗ w) = K2ρ(v)⊗K2ρ(w)
for any v ∈ V , w ∈ W .
3. For any Uq(g)-module V , the ribbon structure ΘV : V → V associated
to V satisfies
Θ±1V = trV Ř
V,V .
The ribbon structure also satisfies the following naturality condition
x ·ΘV = ΘeV · x.
for any x ∈ HomUq(g)(V, Ṽ ).
Definition 2.1. Given z =
i xi⊗yi ∈ EndUq(g)(U ⊗V ), the quantum trace
of z is defined as follows
trV (z) =
tr(yiK2ρ)xi ∈ EndUq(g)(U).
2.2 Quantum group invariants of links
Quantum group invariants of links can be defined over any complex simple
Lie algebra g. However, in this paper, we only consider the quantum group
invariants of links defined over sl(N,C)3 due to the current consideration for
large N Chern-Simons/topological string duality.
Roughly speaking, a link is several disconnected S1 embedded in S3.
A theorem of J. Alexander asserts that any oriented link is isotopic to the
closure of some braid. A braid group Bn is defined by generators σ1, · · · , σn−1
and defining relation:
σiσj = σjσi , if |i− j| ≥ 2 ;
σiσjσi = σjσiσj , if |i− j| = 1 .
Let L be a link with L components Kα, α = 1, . . . , L, represented by the
closure of an element of braid group Bm. We associate each Kα an irreducible
representation Rα of quantized universal enveloping algebra Uq(slN), labeled
by its highest weight Λα. Denote the corresponding module by VΛα. The j-th
strand in the braid will be associated with the irreducible module Vj = VΛα ,
if this strand belongs to the component Kα. The braiding is defined through
the following universal R-matrix of Uq(slN )
R = q
Hi⊗Hj
positive root β
expq[(1− q−1)Eβ ⊗ Fβ] .
3In the following context, we will briefly write slN .
Here (Cij) is the Cartan matrix and
expq(x) =
k(k+1) x
{k}q!
Define braiding by Ř = P12R, where P12(v ⊗ w) = w ⊗ v.
Now for a given link L of L components, one chooses a closed braid
representative in braid group Bm whose closure is L. In the case of no
confusion, we also use L to denote the chosen braid representative in Bm.
We will associate each crossing by the braiding defined above. Let U , V
be two Uq(slN)-modules labeling two outgoing strands of the crossing, the
braiding ŘU,V (resp. Ř
V,U) is assigned as in Figure 1.
PSfrag replacements
U VŘU,V
PSfrag replacements
U VŘ−1V,U
Figure 1: Assign crossing by Ř.
The above assignment will give a representation of Bm on Uq(g)-module
V1 ⊗ · · · ⊗ Vm. Namely, for any generator, σi ∈ Bm4,
define
h(σi) = idV1 ⊗ · · · ⊗ ŘVi+1,Vi ⊗ · · · ⊗ idVN .
Therefore, any link L will provide an isomorphism
h(L) ∈ EndUq(slN )(V1 ⊗ · · · ⊗ Vm) .
For example, the link L in Figure 2 gives the following homomorphism
h(L) = (ŘV, U ⊗ idU)(idV ⊗Ř−1U, U)(ŘU, V ⊗ idU) .
PSfrag replacements
Figure 2: A braid representative for Hopf link
Let K2ρ be the enhancement of Ř in the sense of [41], where ρ is the
half-sum of all positive roots of slN . The irreducible representation Rα is
labeled by the corresponding partition Aα.
Definition 2.2. Given L labeling partitions A1, . . . , AL, the quantum group
invariant of L is defined as follows:
W(A1,...,AL)(L) = qd(L) trV1⊗···⊗Vm(h(L)) ,
where
d(L) = −
ω(Kα)(Λα,Λα + 2ρ) +
lk(Kα,Kβ)|Aα| · |Aβ| ,
and lk(Kα,Kβ) is the linking number of components Kα and Kβ. A sub-
stitution of t = qN is used to give a two-variable framing independent link
invariant.
Remark 2.3. In the above formula of d(L), the second term on the right
hand side is meant to cancel not important terms involved with q1/N in the
definition.
It will be helpful to extend the definition to allow some labeling partition
to be the empty partition (0). In this case, the corresponding invariants will
be regarded as the quantum group invariants of the link obtained by simply
removing the components labeled by (0).
4In the case of σ−1
, use Ř−1
Vi,Vi+1
instead.
A direct computation5 shows that after removing the terms of q
N , qd(L)
can be simplified as
α=1 κAαw(Kα)/2 · t
α=1 |A
α|w(Kα)/2 . (2.1)
Example 2.4. The following examples are some special cases of quantum
group invariants of links.
(1). If the link involved in the definition is the unknot ©,
WA(©; q, t) = trVA(idVA)
is equal to the quantum dimension of VA which will be denoted by
dimq VA.
(2). When all the components of L are associated with the fundamental
representation, i.e., the labeling partition is the unique partition of 1,
the quantum group invariant of L is related to the HOMFLY polynomial
of the link, PL(q, t), in the following way:
W( ,··· , )(L; q, t) = tlk(L)
2 − t 12
2 − q 12
PL(q, t) .
(3). If L is a disjoint union of L knots, i.e.,
L = K1 ⊗K2 ⊗ · · · ⊗ KL ,
the quantum group invariants of L is simply the multiplication of quan-
tum group invariants of Kα
W(A1,...,AL)(L; q, t) =
WAα(Kα; q, t) .
2.3 Chern-Simons partition function
For a given link L of L components, we will fix the following notations in
this paper. Given λ ∈ P, ~A = (A1, A2, . . . , AL), ~µ = (µ1, µ2, . . . , µL) ∈ PL.
Let x = (x1, ..., xL) where xα is a series of variables
xα = (xα1 , x
2 , · · · ) .
5It can also be obtained from the ribbon structure.
The following notations will be used throughout the paper:
[n]q = q
2 − q
2 , [λ]q =
ℓ(λ)∏
[λj ]q , z~µ =
zµα ,
| ~A| = (|A1|, ..., |AL|) , ‖ ~A ‖=
|Aα| , ℓ(~µ) =
ℓ(µα) ,
~At =
(A1)t, . . . , (AL)t
, χ ~A (~µ) =
χAα(Cµα) , s ~A(x) =
sAα(x
Denote by
1~µ = (1|µ
1|, · · · , 1|µL|) . (2.2)
Chern-Simons partition function can be defined to be the following gen-
erating function of quantum group invariants of L,
ZCS(L) = 1 +
~A 6=0
W ~A(L; q, t)s ~A(x).
Define free energy
F = logZ =
~µ 6=0
F~µp~µ (x) . (2.3)
Here in the similar usage of notation,
p~µ(x) =
pµα(x
We rewrite Chern-Simons partition function as
ZCS(L) = 1 +
~µ6=0
Z~µp~µ(x)
where
Z~µ =
χ ~A(~µ)
W ~A. (2.4)
By Lemma 1.3, we have
F~µ =
Λ∈P(PL), |Λ|=~µ∈PL
θΛZΛ . (2.5)
2.4 Main results
2.4.1 Two theorems that answer LMOV conjecture
Let P ~B(q, t) be the function defined by (0.4), which can be determined by
the following formula,
P ~B(q, t) =
| ~A|=|~B|
f ~A(q, t)
χAα(Cµ)χBα(Cµ)
ℓ(µα)∏
q−µj/2 − qµj/2
Theorem 1. There exist topological invariants N ~B; g,Q ∈ Q such that expan-
sion (0.5) holds.
Theorem 2. Given any ~B ∈ PL, the generating function of N ~B; g,Q, P ~B(q, t),
satisfies
(q−1/2 − q1/2)2P ~B(q, t) ∈ Z
q−1/2 − q1/2
, t±1/2
It is clear that Theorem 1 and 2 answered LMOV conjecture. Moreover,
Theorem 2 implies, for fixed ~B, N ~B; g,Q vanishes at large genera.
The method in this paper may apply to the general complex simple Lie
algebra g instead of only considering sl(N,C). We will put this in our future
research. If this is the case, it might require a more generalized duality picture
in physics which will definitely be very interesting to consider and reveal much
deeper relation between Chern-Simons gauge theory and geometry of moduli
space. The extension to some other gauge group has already been done in
[42] where non-orientable Riemann surfaces is involved. For the gauge group
SO(N) or Sp(N), a more complete picture had appeared in the recent work
of [6, 7] and therein a BPS structure of the colored Kauffman polynomials
was also presented. We would like to see that the techniques developed in
our paper extend to these cases.
The existence of (0.5) has its deep root in the duality between large
N Chern-Simons/topological string duality. As already mentioned in the
introduction, by the definition of quantum group invariants, P ~B(q, t) might
have very high order of pole at q = 1, especially when the degree of ~B
goes higher and higher. However, LMOV-conjecture claims that the pole at
q = 1 is at most of order 2 for any ~B ∈ PL. Any term that has power of
q−1/2−q1/2 lower than −2 will be canceled! Without the motivation of Chern-
Simons/topological string duality, this mysterious cancelation is hardly able
to be realized from knot-theory point of view.
2.4.2 An application to knot theory
We now discuss applications to knot theory, following [22, 20].
Consider associating the fundamental representation to each component
of the given link L. As discussed above, the quantum group invariant of L
will reduce to the classical HOMFLY polynomial PL(q, t) of L except for a
universal factor. HOMFLY has the following expansion
PL(q, t) =
p2g+1−L(t)(q
2 − q
2 )2g+1−L . (2.6)
The lowest power of q−1/2 − q1/2 is 1 − L, which was proved in [29] (or one
may directly derive it from Lemma 4.1). After a simple algebra calculation,
one will have
F( ,..., ) =
t−1/2 − t1/2
q−1/2 − q1/2
p̃2g+1−L(t)(q
2 − q
2 )2g+1−L . (2.7)
Lemma 4.3 states that
p̃1−L(t) = p̃3−L(t) = · · · = p̃L−3(t) = 0 ,
which implies that the pk(t) are completely determined by the HOMFLY
polynomial of its sub-links for k = 1− L, 3− L, . . . , L− 3.
Now, we only look at p̃1−L(t) = 0. A direct comparison of the coeffi-
cients of F = logZCS immediately leads to the following theorem proved by
Lickorish and Millett [29].
Theorem 2.5 (Lickorish-Millett). Let L be a link with L components. Its
HOMFLY polynomial
PL(q, t) =
pL2g+1−L(t)
2 − q
)2g+1−L
satisfies
pL1−L(t) = t
2 − t
)L−1 L∏
pKα0 (t)
where pKα0 (t) is HOMFLY polynomial of the α-th component of the link L
with q = 1.
In [29], Lickorish and Millett obtained the above theorem by skein anal-
ysis. Here as the consequence of higher order cancelation phenomenon, one
sees how easily it can be achieved. Note that we only utilize the vanishing
of p̃1−L. If one is ready to carry out the calculation of more vanishing terms,
one can definitely get much more information about algebraic structure of
HOMFLY polynomial. Similarly, a lot of deep relation of quantum group
invariants can be obtained by the cancelation of higher order poles.
2.4.3 Geometric interpretation of the new integer invariants
The following interpretation is taken in physics literature from string theo-
retic point of view [22, 38].
Quantum group invariants of links can be expressed as vacuum expec-
tation value of Wilson loops which admit a large N expansion in physics.
It can also be interpreted as a string theory expansion. This leads to a ge-
ometric description of the new integer invariants N ~B; g,Q in terms of open
Gromov-Witten invariants (also see [20] for more details).
The geometric picture of f ~A is proposed in [22]. One can rewrite the free
energy as
−1ℓ(~µ)
λ2g−2+ℓ(~µ)Fg,~µ(t)p~µ . (2.8)
The quantities Fg,~µ(t) can be interpreted in terms of the Gromov-Witten in-
variants of Riemann surface with boundaries. It was conjectured in [38] that
for every link L in S3, one can canonically associate a lagrangian submanifold
CL in the resolved conifold
O(−1)⊕O(−1) → P1 .
The first Betti number b1(CL) = L, the number of components of L. Let
γα, α = 1, . . . , L, be one-cycles representing a basis for H1(CL, Z). Denote
by Mg,h,Q the moduli space of Riemann surfaces of genus g and h holes
embedded in the resolved conifold. There are hα holes ending on the non-
trivial cycles γα for α = 1, . . . , L. The product of symmetric groups
Σh1 × Σh2 × · · · × ΣhL
acts on the Riemann surfaces by exchanging the hα holes that end on γα.
The integer N ~B; q,t is then interpreted as
N ~B; q,t = χ(S ~B(H
∗(Mg,h,Q))) (2.9)
where S ~B = SB1 ⊗ · · · ⊗ SBL , and SBα is the Schur functor.
The recent progress in the mathematical definitions of open Gromov-
Witten invariants [18, 23, 24] may be used to put the above definition on a
rigorous setting.
3 Hecke algebra and cabling
3.1 Centralizer algebra and Hecke algebra representa-
We review some facts about centralizer algebra and Hecke algebra represen-
tation and their relation to the representation of braid group.
Denote by V the fundamental representation of Uq(slN)
6. Let
{K±1i , Ei, Fi : 1 ≤ i ≤ N − 1}
be the standard generators of the quantized universal enveloping algebra
Uq(slN). Under a suitable basis {X1, ..., XN} of V , the fundamental repre-
sentation is given by the following matrices
Ei 7−→ Ei,i+1
Fi 7−→ Ei+1,i
Ki 7−→ q−1/2Ei,i + q1/2Ei+1,i+1 +
i 6=j
6We will reserve V to denote the fundamental representation of Uq(slN ) from now on.
where Ei,j denotes the N × N matrix with 1 at the (i, j)-position and 0
elsewhere. Direct calculation shows
K2ρ(Xi) = q
−N+1−2i
2 Xi (3.1)
2N Ř(Xi ⊗Xj) =
q−1/2Xi ⊗Xj, i = j ,
Xj ⊗Xi, i < j ,
Xj ⊗Xi + (q−1/2 − q1/2)Xi ⊗Xj, i > j .
The centralizer algebra of V ⊗n is defines as follows
Cn = EndUq(slN )(V ⊗n) =
x ∈ End(V ⊗n) : xy = yx, ∀y ∈ Uq(slN)
Hecke algebra Hn(q) of type An−1 is the complex algebra with n − 1
generators g1, ..., gn−1, together with the following defining relations
gigj = gjgi , |i− j| ≥ 2
gigi+1gi = gi+1gigi+1 , i = 1, 2, ..., n− 2,
(gi − q−1/2)(gi + q1/2) = 0 , i = 1, 2, ..., n− 1.
Remark 3.1. Here we use q−1/2 instead of q to adapt to our notation in the
definition of quantum group invariants of links. Note that when q = 1, the
Hecke algebra Hn(q) is just the group algebra CΣn of symmetric group Σn.
When N is large enough, Cn is isomorphic to the Hecke algebra Hn(q).
A very important feature of the homomorphism
h : CBn −→ Cn
is that h factors through Hn(q) via
2N σi 7→ gi 7→ q−
2N h(σi) . (3.2)
It is well-known that the irreducible modules Sλ (Specht module) ofHn(q)
are in one-to-one correspondence to the partitions of n.
Any permutation π in symmetric group Σn can express as a product of
transpositions
π = si1si2 · · · sil .
If l is minimal in possible, we say π has length ℓ(π) = l and
gπ = gi1gi2 · · · gil .
It is not difficult to see that gπ is well-defined. All of such {gπ} form a basis
of Hn(q).
Minimal projection S is an element in Cn such that SV ⊗n is some irre-
ducible representation Sλ. We denote it by pλ. The minimal projections of
Hecke algebras are well studied (for example [14]), which is a Z(q±1)-linear
combination {q 12gi}.
3.2 Quantum dimension
3.2.1 Explicit formula
An explicit formula for quantum dimension of any irreducible representation
of Uq(slN) can be computed via decomposing V
⊗n into permutation modules.
A composition of n is a sequence of non-negative integer
b = (b1, b2, . . .)
such that ∑
bi = n .
We will write it as b � n. The largest j such that bj 6= 0 is called the end of
b and denoted by ℓ(b).
Let b be a composition such that ℓ(b) ≤ N . DefineMb to be the subspace
of V ⊗n spanned by the vectors
Xj1 ⊗ · · · ⊗Xjn (3.3)
such that Xi occurs precisely bi times. It is clear thatM
b is anHn(q)-module
and is called permutation module. Moreover, by explicit matrix formula of
{Ei, Fi, Ki} acting on V under the basis {Xi}, we have
Mb = {X ∈ V ⊗n : Ki(X) = q−
bi−bi+1
2 X} .
The following decomposition is very useful
V ⊗n =
b�n, ℓ(b)≤N
Let A be a partition and VA the irreducible representation labeled by A.
The Kostka number KAb is defined to be the weight of VA in M
b, i.e.,
KAb = dim(VA ∩Mb) .
Schur function has the following formulation through Kostka numbers
sA(x1, . . . , xN) =
b�|A|, ℓ(b)≤N
By (3.1), K2ρ is acting on M
b as a scalar
j=1 q
(N+1−2j)bj . Thus
dimq VA = trVA idVA
b�|A|
dim(VA ∩Mb)
(N+1−2j)bj
2 , . . . , q−
(3.4)
By (1.2),
dimq VA =
|µ|=|A|
χA(Cµ)
2 , . . . , q−
|µ|=|A|
χA(Cµ)
ℓ(µ)∏
t−µj/2 − tµj/2
q−µj/2 − qµj/2
. (3.5)
Here in the last step, we use the substitution t = qN .
3.2.2 An expansion of the Mariño-Vafa formula
Here we give a quick review about Mariño-Vafa formula [33, 26] for the
convenience of Knot theorist. For details, please refer [26].
Let Mg,n denote the Deligne-Mumford moduli stack of stable curves of
genus g with n marked points. Let π :M g,n+1 →M g,n be the universal curve,
and let ωπ be the relative dualizing sheaf. The Hodge bundle E = π∗ωπ is a
rank g vector bundle over M g,n. Let si : M g,n → M g,n+1 denote the section
of π which corresponds to the i-th marked point, and let Li = s
iωπ. A Hodge
integral is an integral of the form
1 · · ·ψjnn λ
1 · · ·λkgg
where ψi = c1(Li) is the first Chern class of Li, and λj = cj(E) is the j-th
Chern class of the Hodge bundle. Let
Λ∨g (u) = u
g − λ1u+ · · ·+ (−1)gλg
be the Chern polynomial of E∨, the dual of the Hodge bundle.
Define
Cg,µ(τ) = −
−1ℓ(µ)
|Aut(µ)|
[τ(τ + 1)]ℓ(µ)−1
ℓ(µ)∏
∏µi−1
a=1 (µiτ + a)
(µi − 1)!
Mg,l(µ)
Λ∨g (1)Λ
g (−τ − 1)Λ∨g (τ)∏ℓ(µ)
i=1 (1− µiψi)
Note that
C0,µ(τ) = −
−1ℓ(µ)
|Aut(µ)|
[τ(τ + 1)]ℓ(µ)−1
ℓ(µ)∏
∏µi−1
a=1 (µiτ + a)
(µi − 1)!
M0,l(µ)
∏ℓ(µ)
i=1 (1− µiψi)
. (3.6)
The coefficient of the leading term in τ is:
−1ℓ(µ)
|Autµ|
· |µ|ℓ(µ)−3 . (3.7)
The Mariño-Vafa formula gives the following identity:
pµ(x)
u2g−2+ℓ(µ)Cg,µ(τ) = log
ρ)sA(x)
(3.8)
where qρ = (q−1/2, q−3/2, · · · , q−n+1/2, · · · ).
ρ)sA(y)q
Gµpµ.
|Λ|=µ
ℓ(Λ)∏
χAα(Λ
sAα(q
2 (3.9)
= Gµ(0) +
|Λ|=µ
Ω 6=(1Λ),A
χA(Λ)
χA(Ω)
2 − q− 12 )|µ|
|Λ|=µ
χA(Λ)
A (3.10)
The third summand of the above formula gives the non-vanishing leading
term in τ which is equal to (3.7). Therefore, we have:
|Λ|=µ
χA(Λ)
A 6= 0 (3.11)
for ∀p ≥ |µ|+ ℓ(µ)− 2.
3.3 Cabling Technique
Given irreducible representations VA1 , ..., VAL to each component of link L.
Let |Aα| = dα, ~d = (d1, ..., dL). The cabling braid of L, L~d, is obtained by
substituting dα parallel strands for each strand of Kα, α = 1, . . . , L.
Using cabling of the L gives a way to take trace in the vector space of
tensor product of fundamental representation. To get the original trace, one
has to take certain projection, which is the following lemma in [30].
Lemma 3.2 ([30], Lemma 3.3). Let Vi = SiV
⊗di for some minimal projec-
tions, Si = Sj if the i-th and j-th strands belong to the same knot. Then
trV1⊗···Vm (h(L)) = trV ⊗n(h(L(d1,...,dL)) ◦S1 ⊗ · · · ⊗Sn) .
where m is the number of strands belonging to L, n =
α=1 dαrα, rα is the
number of strands belong to Kα, the α-th component of L.
4 Proof of Theorem 1
4.1 Pole structure of quantum group invariants
By an observation from the action of Ř on V ⊗ V , we define
X̃(i1,...,in) = q
#{(j,k)|j<k,ij>ik}
2 Xi1 ⊗ · · · ⊗Xin .
{X̃(i1,...,in)} form a basis of V ⊗n. By (3.2), ∀gj ∈ Hn(q), we have
2gjX̃(...,ij,ij+1,...) =
X̃(...,ij+1,ij ,...), ij ≤ ij+1 ,
qX̃(...,ij+1,ij ,...) + (1− q)X̃(...,ij,ij+‘,...), ij ≥ ij+1.
(4.1)
Lemma 4.1. Let © be the unknot. Given any ~A = (A1, . . . , AL) ∈ PL,
W ~A(L; q, t)
W ~A(©⊗L; q, t)
ξKα(t)
dα , (4.2)
where |Aα| = dα, Kα is the α-th component of L, and ξKα(t), α = 1, . . . , L,
are independent of ~A.
Proof. Choose β ∈ Bm such that L is the closure of β, the total number
of crossings of L~d is even and the last L strands belongs to distinct L com-
ponents of L. Let rα be the number of the strands which belong to Kα.
dαrα is equal to the number of components in the cabling link L~d.
Y = tr
dα(rα−1)
. (4.3)
Y is both a central element of EndUq(slN )
V ⊗(d1+...+dL)
and a Z[q±1]-matrix
under the basis {X̃(i1,...,in)}.
On the other hand,
p ~A = pA1 ⊗ ...⊗ pAL
is a Z (q±1)-matrix under the basis {X̃(i1,...,in)}. By Schur lemma, we have
p ~A ◦ Y = ǫ · p ~A ,
where ǫ ∈ Z(q±1) is an eigenvalue of Y . Since Z[q±1] is a UFD for transcen-
dental q, ǫ must stay in Z[q±1]. By Lemma 3.2,
trV ⊗n(p
⊗ · · · ⊗ p⊗rL
◦ L~d) = trV ⊗(d1+···dL)(p ~A ◦ Y)
= ǫ · tr
⊗(d1+···dL) (p ~A)
= ǫ ·W ~A(©
⊗L) . (4.4)
The definition of quantum group invariants of L gives
W ~A(L; q, t) = q
κAαw(Kα)/2t
dαw(Kα)/2 · ǫ ·W ~A(©
⊗L; q, t) (4.5)
When q → 1, L~d reduces to an element in symmetric group Σn of ‖ ~d ‖
cycles. Moreover, when q → 1, the calculation is actually taken in individual
knot component while the linking of different components have no effect. By
Example 2.4 (1) and (3), we have
W ~A(©
⊗L; q, t) =
WAα(©; q, t) .
This implies
W ~A(L; q, t)
W ~A(©⊗L; q, t)
WAα(Kα; q, t)
WAα(©; q, t)
. (4.6)
Let’s consider the case when K is a knot. A is the partition of d associated
with K and Kd is the cabling of K. Each component of Kd is a copy of
K. q → 1, Kd reduces to an element in Σdr. To calculate the Y , it is then
equivalent to discussing d disjoint union of K. Say K has r strands. Consider
Y0 = trV ⊗(r−1) K .
The eigenvalue of Y0 is then PK(1,t)tw(K) , where PK(q, t) is the HOMFLY polyno-
mial for K. Denote by ξK(t) = PK(1, t), we have
WA(K; q, t)
WA(©; q, t)
= ξK(t)
|A| . (4.7)
Combined with (4.6), the proof is completed.
4.2 Symmetry of quantum group invariants
Define
φ~µ (q) =
ℓ(µα)∏
/2 − qµαj /2) .
Comparing (2.3) and (0.2), we have
F~µ =
χ ~A (~µ/d)
z~µ/d
qd, td
d · z~µ/d
χ ~A(~µ/d)
P ~B(q
d, td)
MAαBα(q
φ~µ/d(q
z~µ/d
χ ~B(~µ/d)P ~B(q
d, td)
= φ~µ(q)
d · z~µ/d
χ ~B(~µ/d)P ~B(q
d, td) ,
φ~µ(q)
d · z~µ/d
χ ~B(~µ/d)P ~B(q
d, td) . (4.8)
Apply Möbius inversion formula,
P ~B(q, t) =
χ ~B(~µ)
φ~µ(q)
F~µ/d(q
d, td) (4.9)
where µ(d) is the Möbius function defined as follows
µ(d) =
(−1)r, if d is a product of r distinct prime numbers;
0, otherwise.
To prove the existence of formula (0.5), we need to prove:
• Symmetry of P ~B(q, t) = P ~B(q−1, t).
• The lowest degree
q−1/2 − q1/2
in P ~B is no less than −2.
Combine (4.8), (2.5) and (2.4), it’s not difficult to find that the first
property on the symmetry of P ~B follows from the following lemma.
Lemma 4.2. W ~At(q, t) = (−1)‖
~A‖W ~A(q
−1, t).
Proof. The following irreducible decomposition of Uq(slN) modules is well-
known:
V n =
dBVB ,
where dB = χB(C(1n)).
Let d ~A =
α=1 dAα. Combined with Lemma 3.2 and eigenvalue of Y in
Lemma 4.1 and (2.1), we have
W( ,..., )(L~d) =
| ~A|=~d
W ~A(L)d ~A
| ~A|=~d
d ~A q
κAαw(Kα)/2 · t
|Aα|w(Kα)/2ǫ ~A ·
dimq VAα , (4.10)
Where ǫ ~A is the eigenvalue of Y on
α=1 VAα as defined in the proof of
Lemma 4.1. Here if we change Aα to (Aα)t, we have κ(Aα)t = −κAα , which
is equivalent to keep Aα while changing q to q−1.
Note that W( ,..., )(L~d) is essentially a HOMFLY polynomial of L~d by
Example 2.4. From the expansion of HOMFLY polynomial (2.6) and Exam-
ple 2.4 (2), we have
W( ,..., )(L~d ; q
−1, t) = (−1)
|Aα|W( ,..., )(L~d ; q, t) . (4.11)
However, one can generalize the definition of quantum group invariants
of links in the following way. Note that in the definition of quantum group
invariants, the enhancement of Ř, K2ρ, acts on Xi (see (3.1)) as a scalar
(N+1−2i). We can actually generalize this scalar to any zαi where α cor-
responds the strands belonging to the α-th component (cf. [43]). It’s not
difficult to see that (4.10) still holds. The quantum dimension of VAα thus
becomes sAα(z
1 , . . . , z
N) obtained in the same way as (3.4).
We rewrite the above generalized version of quantum group invariants of
links as W ~A(L; q, t; z), where z = {zα}. (4.11) becomes
W( ,..., )(L~d ; q
−1, t;−z) = (−1)
|Aα|W( ,..., )(L~d ; q, t; z) (4.12)
Now, combine (4.12), (4.10) and (4.11), we obtain
d ~At q
κ(Aα)tw(Kα)/2 · t
|(Aα)t|w(Kα)/2ǫ ~At(q
−1; −z)
s(Aα)t(−zα)
= (−1)
α |Aα| ·
d ~A q
κAαw(Kα)/2 · t
|Aα|w(Kα)/2ǫ ~A(q; z)
sAα(z
(4.13)
Note the following facts:
sAt(−z) = (−1)ℓ(A)sA(z) , (4.14)
d ~At = d ~A . (4.15)
where the second formula follows from
χAt(Cµ) = (−1)|µ|−ℓ(µ)χA(Cµ) . (4.16)
Apply (4.14) and (4.15) to (4.13). Let zαi = q
(N+1−2i), then using −z in-
stead of z is equivalent to substitute q by q−1 while keeping t in the definition
of quantum group invariants of links. This can be seen by comparing
zαi = q
−N−2i+1
ℓ(µ)∏
t−µj/2 − tµj/2
q−µj/2 − qµj/2
pµ(−zα) = (−1)ℓ(µ)pµ(zα) .
Therefore, we have
ǫ ~At(q
−1, t) = ǫ ~A(q, t) . (4.17)
By the formula of quantum dimension, it is easy to verify that
W ~At(©
L; q, t) = (−1)‖ ~A‖W ~A(©
L; q−1, t) (4.18)
Combining (4.4), (4.17) and (4.18), the proof of the Lemma is then com-
pleted.
By Lemma 4.2, we have the following expansion about P ~B
P ~B(q, t) =
Q∈Z/2
N ~B,g,Q(q
−1/2 − q1/2)2g−2N0tQ
for some N0. We will show N0 ≤ 1.
Let q = eu. The pole order of (q−1/2 − q1/2) in P ~B is the same as pole
order of u.
Let f(u) be a Laurent series in u. Denote degu f to be the lowest degree
of u in the expansion of u in f .
Combined with (4.8), N0 ≤ 1 follows from the following lemma.
Lemma 4.3. degu F~µ ≥ ℓ(~µ)− 2.
Lemma 4.3 can be proved through the following cut-and-join analysis.
4.3 Cut-and-join analysis
4.3.1 Cut-and-join operators
Let τ = (τ1, · · · , τL), substitute
W ~A(L; q, t; τ) =W ~A(L; q, t) · q
α=1 κAατα/2
in the Chern-Simons partition function, we have the following framed parti-
tion function
Z(L; q, t, τ) = 1 +
~A 6=0
W ~A(L; q, t, τ) · s ~A(x) .
Similarly, framed free energy
F (L; q, t, τ) = logZ(L; q, t, τ) .
We also defined framed version of Z~µ and F~µ as follows
Z(L; q, t, τ) = 1 +
~µ6=0
Z~µ(q, t, τ) · p~µ(x) ,
F (L; q, t, τ) =
~µ6=0
F~µ(q, t, τ)p~µ(x) .
One important fact of these framing series is that they satisfy the cut-and-
join equation which will give a good control of F~µ.
Define exponential cut-and-join operator E
i,j≥1
ijpi+j
∂pi∂pj
+ (i+ j)pipj
∂pi+j
, (4.19)
and log cut-and-join operator L
i,j≥1
ijpi+j
∂pi∂pj
+ (i+ j)pipj
∂pi+j
+ ijpi+j
. (4.20)
Here {pi} are regarded as independent variables. Schur function sA(x) is
then a function of {pi}.
Schur function sA is an eigenfunction of exponential cut-and-join with
eigenvalue κA [8, 12, 49]. Therefore, Z(L; q, t, τ) satisfies the following ex-
ponential cut-and-join equation
∂Z(L; q, t, τ)
EαZ(L; q, t, τ) , (4.21)
or equivalently, we also have the log cut-and-join equation
∂F (L; q, t, τ)
LαF (L; q, t, τ) . (4.22)
In the above notation, Eα and Lα correspond to variables {pi} which take
value of {pi(xα)}.
(4.22) restricts to ~µ will be of the following form:
|~ν|=|~µ|, ℓ(~ν)=ℓ(~µ)±1
α~µ~νF~ν + nonlinear terms
, (4.23)
where α~µ~ν is some constant, ~ν is obtained by cutting or jointing of ~µ.
Recall that given two partitions A and B, we say A is a cutting of B if
one cuts a row of the Young diagram of B into two rows and reform it into
a new Young diagram which happens to be the Young diagram of A, and we
also say B is a joining of A. For example, (7, 3, 1) is a joining of (5, 3, 2, 1)
where we join 5 and 2 to get 7 boxes. Using Young diagram, it looks like
=⇒ cut=⇒ .
In (4.23), cutting and joining happens only for the α-th partition.
4.3.2 Degree of u
By (2.5), it is easy to see through induction that for two links L1 and L2,
F( ,··· , )(L1 ⊗ L2) = 0 . (4.24)
For simplicity of writing, we denote by
F( ,··· , )(L) = F ◦(L)
Note that when we put the labeling irreducible representation by the
fundamental ones, quantum group invariants of links reduce to HOMFLY
polynomials except for a universal factor. Therefore, if we apply skein re-
lation, the following version of skein relation can be obtained. Let positive
crossing
used later appear between two different components, K1 and K2, of the link.
Denote by lk = lk(K1,K2), then:
− F ◦
= t−lk+
2 (q−
2 − q
2 )F ◦
(4.25)
We want to claim that
degu F
◦(L) ≥ L− 2 , (4.26)
where L is the number of components of L.
Firstly, for a knot K, a simple computation shows that:
degu F (K) = −1 .
Using induction, we may assume that claim (4.26) holds for L ≤ k. When
L = k + 1, by (4.25),
− F ◦
= 1 + degu
≥ k − 1 . (4.27)
However, if
< k − 1 ,
this will imply that in the procedure of unlinking L, the lowest degree term
of u in F ◦ are always the same. However, this unlinking will lead to F ◦ equal
to 0 due to (4.24), which is a contradiction! This implies that if the number
of components of L1 is greater than the number of components of L2, we
always have
degu F
◦(L1) > degu F ◦(L2)
Therefore, we proved the claim (4.26).
4.3.3 Induction procedure of cut-and-join analysis
Let µi = (µi1, · · · , µiℓi). We use symbol
Ẑ~µ = Z~µ · z~µ; F̂ = F~ν · z~µ .
Notice the following fact from the definition of quantum group invariants:
Ẑ(µ1,··· ,µL)(L) = Ẑ(µ11),··· ,(µ1ℓ1 ),··· ,(µL1 ),··· ,(µLℓL )(L~µ) , (4.28)
F̂(µ1,··· ,µL)(L) = F̂(µ11),··· ,(µ1ℓ1 ),··· ,(µL1 ),··· ,(µLℓL)(L~µ) . (4.29)
Let τ = (τ1, 0, · · · , 0). Similar as (3.9), we give the following formula:
F~µ =
|Λ|=~µ
ℓ(Λ)∏
χAα(Λ
W ~A(q, t)q
= F~µ(0) +
|Λ|=~µ
Ω 6=(1Λ),A
χA(Λ)
χA(Ω)
ZΩ(0)κ
+ Z1Λ(0)
|Λ|=~µ
χA(Λ)
A (4.30)
Let g
~µ (τα) be the degree of τα in the coefficient of the lowest degree of
u in F~µ. By (3.11) and (4.30), if |µα| > 1, we have
degτα g
~µ = |µ
α|+ ℓ(µα)− 2 > 0 .
By induction, assume that if |µα| ≤ d, degu F~µ ≥ ℓ(~µ) − 2 and hence by
(3.11), degτα g
~µ (τα) = |µα|+ℓ(µα)−2. Combined with (4.29), if |λα| = d+1
and ℓ(λα) > 1, we have:
degu F~λ = ℓ(
~λ)− 2 .
Without loss of generality, assume µ1 = (d, 1), we will just consider the
cut-and-join equation for τ1:
∂F(µ1,··· ,µL)
(d+1),µ2··· ,µL
ℓ(~ν)=ℓ(~µ)+1
β~νF~ν
where β~ν are some constants and ∗ represents some non-linear terms in the
cut-and-join equation. The crucial observation of this non-linear terms is
that its degree in u is equal to ℓ(~µ)− 2. Comparing the degree in u on both
sides of the equation, we have:
degu F
(d+1),µ2,··· ,µL
) = 1 +
ℓ(µα)− 2 .
This completes the induction. The proof of Lemma 4.3 follows immediately.
Define
F̃~µ =
φ~µ(q)
, Z̃~µ =
φ~µ(q)
Lemma 4.3 directly implies follows:
Corollary 4.4. F̃ are of the following form:
F̃~µ(q, t) =
finitely many nα
aα(t)
[nα]2
+ polynomial.
Remark 4.5. Combine the above Corollary, (2.7) and (2.5), we have:
F̃( ,..., )(q, t) =
+ polynomial. (4.31)
4.4 Framing and pole structures
Consider δn = σ1 · · ·σn−1. LetSA be the minimal projection fromHn → HA,
χA(Cµ)SA
We will apply a lemma of Aiston-Morton [2] in the following computation:
δnnSA = q
κASA .
(d1), . . . , (dL)
, . . . ,
Due to the cabling formula to the length of partition (4.28) and (4.29), we
can simply deal with all the color of one row without loss of generality. Take
framing τα = nα+
and choose a braid group representative of L such that
the writhe number of Lα is nα. Denote by ~τ = (τ1, . . . , τL),
L; q, t;~τ
χ ~A(C~d)W ~A(L; q, t)q
α=1 κAα
α dαnαTr
χ ~A(C~d)q
dα ⊗Lα=1 SAα
dαnαTr
χ ~A(C~d)(δd1 ⊗ · · · ⊗ δdL)⊗
α=1 SAα
dαnαTr
L~d · ⊗
α=1δα ·P
⊗ · · · ⊗P(dL)
Here, P
means that in the projection, we use qdα, tdα instead of using q, t.
If we denote by
L ∗Q~d = L~d · δ~d ·P
⊗ · · · ⊗P(dL)
we have
L; q, t;
= H(L ∗Q~d) . (4.32)
Here H is the HOMFLY polynomial which is normalized as
H(unknot) =
2 − t− 12
2 − q− 12
With the above normalization, for any given link L, we have
[1]L · H(L) ∈ Q
[1]2, t±
Substituting q by qdα in the corresponding component, it leads to the follows:
[dα] · Ẑ~d(L; q, t;~τ) ∈ Q
[1]2, t±
. (4.33)
On the other hand, given any two frames ~ω = (ω1, . . . , ωL),
Ẑ~µ(L; q, t; ~ω) =
χ ~A(C~µ)W ~A(L; q, t; ~ω)
χ ~A(C~µ)
χ ~A(C~ν)
Ẑ~ν(L; q, t)q
κAαωα .
Exchange the order of summation, we have the following convolution formula:
Ẑ~µ(L; q, t; ~ω) =
Ẑ~ν(L; q, t)
χ ~A(C~µ)χ ~A(C~ν)q
κAαωα . (4.34)
Return to (4.33). This property holds for arbitrary choice of nα, α =
1, . . . , L. By convolution formula, the coefficients of possible other poles
vanish for arbitrary integer nα. q
nα is involved through certain polynomial
relation, which implies the coefficients for other possible poles are simply
zero. Therefore, (4.33) holds for any frame.
Consider Ẑ~d(L), where each component of L is labeled by a young diagram
of one row. As discussed above, this assumption does not lose any generality.
Let K be a component of L labeled by the color (c). If we multiply Z̃~d(L)
by [c]2, we call it normalizing Z̃~d(L) w.r.t K. Similarly, we call [c]2F̃~d(L)
normalizing F̃~d(L) w.r.t K.
By (4.31), after normalizing F̃( ,..., )(L) w.r.t any component of L, one
will obtain a polynomial in terms of [1]2 and t±
2 . Note that
Z̃~d =
Aut |Λ|
Therefore, if we normalizing on both side w.r.t K, the equality still holds.
Applying (4.32), one has
[c]2F̃~d(L; q, t) ∈ Q
[1]2, t±
. (4.35)
We thus proved the following proposition:
Proposition 4.6. Notation as above, we have:
[dα] · Ẑ~d(L; q, t) ∈ Q[[1]
2, t±
2 ]; (4.36)
2F̃~d(L; q, t) ∈ Q
[1]2, t±
, ∀α . (4.37)
For a given ~d, denote by D~d the gcd{d1, . . . , dL}. Proposition 4.6 implies
that the principle part of F̃ are possible summations of
, where k divides
all dα’s, or equivalently, k|D~d.
Similar as above, choose a braid group element representative for L such
that its α-th component has writhe number nα and choose frame as ~τ =
(n1 +
, . . . , nL +
Ẑ~d(L; q, t;~τ) =
χ ~A(C~d)W ~A(L; q, t)q
κAα(nα+
dαnαTr
χ ~A(C~d)q
D~d ⊗Lα=1 SAα
α dαnαTr(L~d · ⊗
⊗Lα=1 P
(D~d)
(dα/D~d)
α dαnαẐ~d/D~d
(L~d · ⊗
; qDµ, tDµ) . (4.38)
This implies that Ẑ~d is a rational function of q
D~d and t±
D~d .
Passing this to F̃~d, we have F̃~d is a rational function of [D~d]
2 and t±
D~d .
As discussed above, the principle part of F̃~d are summations of
where
k|D~d. Combining them, we have
F̃~d(L; q, t) =
H~d/D~d
(tD~d)
[D~d]
+ polynomial in [D~d]
2 and t±
D~d . (4.39)
Once again, due to arbitrary choice of nα, we know the above pole structure
of F̃~d holds for any frame.
Now we will show H~d/D~d
(t) only depends on ~d/D~d and L. If one checks
(4.38), one finds the computation also follows if 1/D~d is replaced by 1/k for
any k|D~d. By induction, we obtain that H~d/D~d(t) only depends on
~d/D~d and
Combining (4.29), we proved the following Proposition:
Proposition 4.7. Notations are as above. Assume L is labeled by the
color ~µ = (µ1, . . . , µL). Denote by D~µ is the greatest common divisor of
{µ11, . . . , µ1ℓ(µ1), . . . , µij, . . . , µLℓ(µL)}. F̃~µ has the following structure:
F̃~µ(q, t) =
H~µ/D~µ(t
[D~µ]2
+ f(q, t) ,
where H~µ/D~µ(t) only depends on ~µ/D~µ and L, f(q, t) ∈ Q
[1]2, t±
Remark 4.8. In Proposition 4.7, it is very interesting to interpret in topo-
logical string side that H~µ/D~µ(t) only depends on ~µ/D~µ and L. The principle
term is generated due to summation of counting rational curves and indepen-
dent choice of k in the labeling color k · ~µ/D~µ. This phenomenon simply tells
us that contributions of counting rational curves in the labeling color k ·~µ/D~µ
are through multiple cover contributions of ~µ/D~µ.
5 Proof of Theorem 2
5.1 A ring characterizes the partition function
Let ~d = (d1, ..., dL), y = (y
1, . . . , yL). Define
T~d =
~B=~d
s ~B(y)P ~B(q, t) . (5.1)
By the calculation in Appendix A.1, one will get
T~d = q
A∈P(Pn),‖A‖=~d/k
θAWA(q
k, tk)sA(z
k) (5.2)
However,
T~d =
| ~B|=~d
s ~B(y)P ~B(q, t)
Q∈Z/2
| ~B|=~d
N ~B; g,Qs ~B(y)
(q1/2 − q−1/2)2g−2tQ .
Denote by Ω(y) the space of all integer coefficient symmetric functions in y.
Since Schur functions forms a basis of Ω(y) over Z, N ~B; g,Q ∈ Z will follow
from ∑
| ~B|=~d
N ~B; g,Qs ~B(y) ∈ Ω(y) .
Let v = [1]2q. It’s easy to see that [n]
q is a monic polynomial of v with
integer coefficients. The following ring is very crucial in characterizing the
algebraic structure of Chern-Simons partition function, which will lead to
the integrality of N ~B; g,Q.
R(y; v, t) =
a(y; v, t)
: a(y; v, t) ∈ Ω(y)[v, t±1/2], b(v) =
q ∈ Z[v]
If we slightly relax the condition in the ring R(y; v, t), we have the following
ring which is convenient in the p-adic argument in the following subsection.
M(y; q, t) =
f(y; q, t)
: f ∈ Ω(y)[q±1/2, t±1/2], b(v) =
q ∈ Z[v]
Given
f(y; q,t)
∈ M(y; q, t), if f(y; q, t) is a primitive polynomial in terms of
q±1/2, t±1/2 and Schur functions of y, we call
f(y; q,t)
is primitive.
5.2 Multi-cover contribution and p-adic argument
Proposition 5.1. T~d(y; q, t) ∈ R(y; v, t).
Proof. Recall the definition of quantum group invariants of links. By the
formula of universal R-matrix, it’s easy to see that
W ~A(L) ∈ M(y; q, t) .
Since we have already proven the existence of the pole structure in LMOV
conjecture, Proposition 5.1 will be naturally satisfied if we can prove
T~d(y; q, t) ∈ M(y; q, t) . (5.3)
Before diving into the proof of (5.3), let’s do some preparation. For
∀A = ( ~A1, ~A2, ...) ∈ P(PL) ,
define
A(d) =
~A1, ..., ~A1︸ ︷︷ ︸
, ~A2, ..., ~A2︸ ︷︷ ︸
, . . .
Lemma 5.2. If θA =
, d > 1 and gcd (c, d) = 1, then for any r|d, we can
find B ∈ P(PL) such that A = B(r).
Proof. Let ℓ = ℓ(A), we have
(−1)ℓ(A)−1
(−1)ℓ(A)−1(ℓ(A)− 1)!
|AutA|
~A1, ..., ~A1︸ ︷︷ ︸
, . . . , ~An, ..., ~An︸ ︷︷ ︸
so ℓ(A) = m1 + ...+mn. Note that
m1, m2, · · · , mn
Let η = gcd(m1, m2, · · · , mn). We have A = Ã(η), where
~A1, . . . , ~A1︸ ︷︷ ︸
, . . . , ~An, . . . , ~An︸ ︷︷ ︸
By Corollary A.4 , ℓ
|uA and gcd(c, d) = 1, one has d|η. We can take B =
). This completes the proof.
Remark 5.3. By the choice of η in the above proof, we know | ~Aα| is divisible
by η for any α.
By Lemma 5.2 and (5.2), we know T~d is of form f(y; q, t)/k, where f ∈
M(y; q, t), k|~d. We will show k is in fact 1.
Given r
f(y; q,t)
where
f(y; q, t)
∈ M(y; q, t)
is primitive, define
f(y; q, t)
= Ordp
. (5.4)
Lemma 5.4. Given A∈ P
, p any prime number and fA(y; q, t) ∈
M(y; q, t), we have
θA(p)fA(p)(y; q, t)−
θAfA(y
p; qp, tp)
≥ 0 .
Proof. Assume θA =
pr ·a , where gcd(p
r · a, b) = 1, p ∤ a. In (5.5), minus is
taken except for one case: p = 2 and r = 0, in which the calculation is very
simple and the same result holds. Therefore, we only show the general case
which minus sign is taken. By Lemma 5.2, one can choose B ∈ P(PL) such
that A = B(p
r). Note that fA = f
Let s = ℓ(B) and
~B1, . . . , ~B1︸ ︷︷ ︸
, . . . , ~Bk, . . . , ~Bn︸ ︷︷ ︸
Since ℓ(A(p)) = p · ℓ(A), by Theorem A.2,
θA(p) −
= Ordp
p · ℓ(A)
pr+1s
pr+1s1, · · · , pr+1sk
prs1, · · · , prsk
(5.5)
≥ 2(r + 1)− (1 + r)
> 0 . (5.6)
By Theorem A.6,
fA(p)(y; q, t)− fA(yp; qp, tp
= Ordp
pr+1a
fB(y; q, t)
pr+1 − fB(yp; qp, tp)p
≥ 0 .
Apply the above two inequalities to
θA(p)fA(p)(y; q, t)−
θAfA(y
p; qp, tp)
= Ordp
θA(p) −
fA(p)(y; q, t) +
fA(p)(y; q, t)− fA(yp; qp, tp
The proof is completed.
Apply the above Lemma, we have
θA(p)WA(p)(q, t)sA(p)(z)−
θAWA(q
p, tp)sA(z
≥ 0 . (5.7)
Φ~d (y; q, t) =
A∈P(Pn), ‖A‖=~d
θAWA(q, t)sA(z) .
By Lemma 5.2, one has
B : ‖ B ‖= p ~d and Ordp(θB) < 0
A(p) : ‖ A ‖= ~d
Therefore, By (5.7),
(y; q, t)−
Φ~d (y
p; qp, tp)
‖A‖=~d
θA(p)WA(p)(q, t)sA(p)(z)−
θAWA(q
p, tp)sA(z
≥ 0 .
We have thus proven the following Lemma.
Lemma 5.5. For any prime number p and ~d,
(y; q, t)− 1
Φ~d (y
p; qp, tp)
≥ 0 .
For any p|~d, by (5.2),
T~d = q
A∈P(PL), ‖A‖=~d/k
θAWA(q
k, tk)sA(z
k|~d, p∤k
Φ~d/k(y
n; qn, tn) +
k|~d, p∤k
µ(pk)
Φ~d/(pk)(y
pk; qpk, tpk)
k|~d, p∤k
Φ~d/k(y
k; qk, tk)− 1
Φ~d/(pk)(y
pk; qpk, tpk)
By Lemma 5.5,
Ordp T~d ≥ 0 . (5.8)
By the arbitrary choice of p, we prove that T~d ∈ M(y; q, t), hence
T~d ∈ R(y; v, t) .
The proof of Proposition 5.1 is completed.
5.3 Integrality
By (4.9) and Propositions 4.7, (note that φ~µ/d(q
d, td) = φ~µ(q, t).)
P ~B(q, t) =
χ ~B(~µ)
φ~µ(q)
F~µ/d(q
d, td)
χ ~B(~µ)
F̃~µ/d(q
d, td)
χ ~B(~µ)
d|D~µ
H~µ/D~µ(t
[D~µ]2
+ polynomial
χ ~B(~µ)δ1,D~µ
H~µ/D~µ(t
[D~µ]2
+ polynomial ,
where δ1,n equals 1 if n = 1 and 0 otherwise. It implies that P ~B is a rational
function which only has pole at q = 1. In the above computation, we used a
fact of Möbius inversion,
= δ1,n .
Therefore, for each ~B,
Q∈Z/2
N ~B; g,Q(q
−1/2 − q1/2)2gtQ ∈ Q[(q−1/2 − q1/2)2, t±
2 ] .
On the other hand, by Proposition 5.1, T~d ∈ R(y; v, t) and Ordp T~d ≥ 0
for any prime number p. We have
| ~B|=~d
N ~B; g,Qs ~B(y) ∈ Ω(y) .
This implies N ~B; g,Q ∈ Z.
Combine the above discussions, we have
Q∈Z/2
N ~B; g,Q(q
−1/2 − q1/2)2gtQ ∈ Z[(q−1/2 − q1/2)2, t±1/2] .
The proof of Theorem 2 is completed.
6 Concluding Remarks and Future Research
In this section, we briefly discuss some interesting problems related to string
duality which may be approached through the techniques developed in this
paper.
6.1 Duality from a mathematical point of view
Let p = (p1, . . . ,pL), where pα = (pα1 , p
2 , . . . , ) . Defined the following gen-
erating series of open Gromov-Witten invariants
Fg,~µ(t, τ) =
where ω is the Kähler class of the resolved conifold, τ is the framing parameter
t = e
ω, and e
= tQ .
Consider the following generating function
F (p; u, t; τ) =
u2g−2+ℓ(~µ)Fg,~µ(t; τ)
pαµα .
It satisfies the log cut-and-join equation
∂F (p; u, t; τ)
LαF (p; u, t; τ) .
Therefore, duality between Chern-Simons theory and open Gromov-Witten
theory reduces to verifying the uniqueness of the solution of cut-and-join
equation.
Cut-and-join equation for Gromov-Witten side comes from the degen-
eracy and gluing procedure while uniqueness of cut-and-join system should
in principle be obtained from the verification at some initial value. How-
ever, it seems very difficult to find a suitable initial value. For example, in
the case of topological vertex theory, cut-and-join system has singularities
when the framing parameter takes value at 0, −1, ∞ while these points are
the possible ones to evaluate at. One solution might be through studying
Riemann-Hilbert problem on controlling the monodromy at three singularity
points. When the case goes beyond, the situation will be even more compli-
cated. A universal method of handling uniqueness is required for the final
proof of Chern-Simons/topological string duality conjecture.
A new hope might be found in our development of cut-and-join analysis.
In the log cut-and-join equation (4.22), the non-linear terms reveals the im-
portant recursion structure. For the uniqueness of cut-and-join equation, it
will appear as the vanishing of all non-linear terms. We will put this in our
future research.
6.2 Other related problems
There are many other problems related to our work on LMOV conjecture.
We briefly list some problems that we are working on.
Volume conjecture was proposed by Kashaev in [16] and reformulated by
[34]. It relates the volume of hyperbolic 3-manifolds to the limits of quantum
invariants. This conjecture was later generalized to complex case [35] and
to incomplete hyperbolic structures [13]. The study of this conjecture is still
staying at a rather primitive stage [36, 17, 48, 10].
LMOV conjecture has shed new light on volume conjecture. The cut-and-
join analysis we developed in this paper combined with rank-level duality in
Chern-Simons theory seems to provide a new way to prove the existence of
the limits of quantum invariants.
There are also other open problems related to LMOV conjecture. For
example, quantum group invariants satisfy skein relation which must have
some implications on topological string side as mentioned in [20]. One could
also rephrase a lot of unanswered questions in knot theory in terms of open
Gromov-Witten theory. We hope that the relation between knot theory and
open Gromov-Witten theory will be explored much more in detail in the
future. This will definitely open many new avenues for future research.
A Appendix
Here we carry out the calculation in Section 5.1.
T~d =
| ~B|=~d
s ~B(y)P ~B(q, t)
| ~B|=~d
s ~B(y)χ ~B(~µ)
φ~µ(q)
F~µ/k
qk, tk
k|~µ ,|~µ|=~d
p~µ(y)
φ~µ(q)
Λ∈P(P
), |Λ|=~µ/k
θΛZΛ(q
k, tk)
k|~µ ,|~µ|=~d
‖Λ‖=~µ/k
k)pΛ(y
k)ZΛ(q
k, tk)
k|~µ ,|~µ|=~d
‖Λ‖=~µ/k
k)pΛ(y
ℓ(Λ)∏
χ~Aβ(
W~Aβ(q
|~Aβ |=|~Λβ |=dβ/k
(−1)ℓ(Λ)−1
ℓ(Λ)∏
φ−1~Λβ
(qk)p~Λβ(y
χ~Aβ (
W~Aβ (q
k, tk)
k|~µ ,|~µ|=~d
A=(~A1,...), ‖A‖=~d/k
(−1)ℓ(A)−1
ℓ(A)∏
W~Aβ (q
k, tk)
χ~Aβ(
φ−1~Λβ
(qk)p~Λβ(y
A∈P(PL), ‖A‖=~d/k
θAWA(q
k, tk)sA(z
where
pn(z) = pn(y) · pn(xi = qi−1) .
Lemma A.1. p is prime, r ≥ 1. Then
pr−1a
pr−1b
≡ 0 mod (p2r)
Proof. Consider the ratio
pr−1a
pr−1b
(a−b)pr+k
k∏pr−1b
(a−b)pr−1+k
gcd(k,p)=1,
1≤k≤prb
(a− b)pr + k
≡ 1 + pr(a− b)
gcd(k,p)=1,
1≤k≤prb
mod (p2r) .
Ap(n) =
1≤k≤n,
gcd(k,p)=1
Therefore, the proof of the lemma can be completed by showing
rb) =
for some c, d such that gcd(d, p) = 1.
If gcd(k, p) = 1, there exist αk, βk such that
αkk + βkp
r = 1 .
Bp(n) =
1≤k≤n,gcd(k,p)=1
By the above formula,
rb) ≡ bAp(pr) mod (pr)
≡ bBp(pr) mod (pr) ,
k − p
pr−1∑
pr(pr + 1)
pr−1(pr−1 + 1)
p2r−1(p− 1)
Here, we have 2r − 1 ≥ r since r ≥ 1. The proof is then completed.
The following theorem is a simple generalization of the above Lemma.
Theorem A.2.
i=1 ai = a, p is prime, r ≥ 1, then
pra1, · · · , pran
pr−1a1, · · · , pr−1an
≡ 0 mod (p2r)
Proof. We have
pra1, · · · , pran
pr−1a1, · · · , pr−1an
pr(a−
i=1 ai)
pr−1(a−
i=1 ai)
pr−1ak
≡ 0 mod (p2r) .
In the last step, we used the Lemma A.1. The proof is completed.
Lemma A.3. We have
gcd(a, b)
Proof. Notice that (
gcd(a, b)
gcd(a, b)
However,
gcd(a, b)
gcd(a, b)
= 1 ,
gcd(a, b)
This direct leads to the following corollary.
Corollary A.4. a = a1 + a2 + · · ·+ an. Then
gcd(a1, a2, · · · , an)
a1, a2, · · · , an
Lemma A.5. a, r ∈ N, p is a prime number, then
r − apr−1 ≡ 0 mod (pr) .
Proof. If a is p, since pr−1 ≥ r, the claim is true. If gcd(a, p) = 1, by Fermat
theorem, ap−1 ≡ 1 mod (p). We have
r − apr−1 = apr−1
(kp+ 1)p
r−1(p−1) − 1
pr−1∑
(kp)i
≡ 0 mod (pr) .
Here in the last step, we used Lemma A.3.
A direct consequence of the above lemma is the following theorem.
Theorem A.6. Given f(y; q, t) ∈ M(y; q, t), we have
f(y; q, t)p
r+1 − f(yp; qp, tp)pr
≥ r + 1 .
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Center of Mathematical Sciences
Zhejiang University, Box 310027
Hangzhou, China
Department of mathematics
University of California at Los Angeles
Box 951555
Los Angeles, CA 90095-1555
Email: liu@math.ucla.edu.
Department of Mathematics
Harvard University
One Oxford Street
Cambridge, MA, 02138
Email: ppeng@math.harvard.edu.
	Introduction
	Overview
	Labastida-Mariño-Ooguri-Vafa conjecture
	Main ideas of the proof
	Acknowledgments
	Preliminary
	Partition and symmetric function
	Partitionable set and infinite series
	Labastida-Mariño-Ooguri-Vafa conjecture
	Quantum trace
	Quantum group invariants of links
	Chern-Simons partition function
	Main results
	Hecke algebra and cabling
	Centralizer algebra and Hecke algebra representation
	Quantum dimension
	Cabling Technique
	Proof of Theorem 1
	Pole structure of quantum group invariants
	Symmetry of quantum group invariants
	Cut-and-join analysis
	Framing and pole structures
	Proof of Theorem 2
	A ring characterizes the partition function
	Multi-cover contribution and p-adic argument
	Integrality
	Concluding Remarks and Future Research
	Duality from a mathematical point of view
	Other related problems
	Appendix
