Proof of the Labastida-Mariño-Ooguri-Vafa Conjecture

Transcription

1 arxiv: v1 [math.qa] 12 Apr 2007 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 [37, 21, 19], 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 invariants. 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 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

2 2.2 Quantum group invariants of links Chern-Simons partition function Main results Hecke algebra and cabling Centralizer algebra and Hecke algebra representation Explicit formula of quantum dimension Cabling technique Proof of Theorem Pole structure of quantum group invariants Symmetry of quantum group invariants Cut-and-join analysis Proof of Theorem A ring that characterizes the algebraic structure of the partition function Multi-cover contribution and p-adic argument Integrality Concluding remarks and future research Duality between Chern-Simons gauge theory and open topological string theory from a mathematical point of view Other related problems A Appendix 44 A A

3 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, which triggered the development of geometry and topology. 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 symmetry 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 originated in physics by an idea that gauge theory should have a string theory explanation. In 1992, Witten [45] 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 [10] conjectured that, at large N, open topological A-model of N D-branes on T S 3 is dual to topological closed string theory on resolved conifold O( 1) O( 1) P 1. Later, Ooguri and Vafa [37] showed a picture on how to describe Chern-Simons invariants of a knot to 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 formular [32, 25] gives a beautiful closed formula for Hodge integral up to three Chern classes of Hodge bundle. Furthermore, using topological vertex 3

4 theory [1, 26, 27], 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 invariants, there had been a series of polynomial invariants discovered (for example, [14, 8, 18]), the generalization of which was provided by quantum group theory [42] in mathematics and by Chern-Simons path integral with the gauge group SU(N) [44] in physics. Based on the development of quantum group invariants of links and the Chern-Simons/topological string duality, Ooguri and Vafa [37] studied a reformulation of knot invariants in terms of new integral invariants capturing the spectrum of M2 branes ending on M5 branes embedded in the resolved conifold. Later, Labastida, Mariño and Vafa [21, 19] refined the analysis of [37] and conjectured the precise integrality structure for open Gromov-Witten invariants. We will call this conjecture as Labastida-Mariño-Ooguri-Vafa conjecture. This conjecture predicts a remarkable new algebraic 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 [29] 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). Firstly, let us describe the main ideas of the proof. The details can be found in Sections 4 and 5. 4

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 Z CS (L; q, t) = A P L W A (L; q, t) L s A α(x α ), where W A (L) is the quantum group invariants labeled by a sequence of partitions A = (A 1,...,A L ) which correspond to the irreducible representations of quantized universal enveloping algebra U q (sl(n, C)), s A α(x α ) is Schur function. Free energy is defined to be F = log Z CS. Use plethystic exponential, one can obtain α=1 F = 1 d f A (q d, t d ) d=1 A 0 L s A α( (x α ) d), (0.1) α=1 where (x α ) d = ( (x α 1 )d, (x α 2 )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 M AB (q) = µ χ A (C µ )χ B (C µ ) z µ l(µ) (q µ j/2 q µ j/2 ). (0.2) j=1 Conjecture (LMOV). For any A P L, (i). there exist P B (q, t) such that f A (q, t) = B = A 5 P B (q, t) L α=1 M A α Bα(q). (0.3)

6 Moreover, P B (q, t) has the following expansion P B (q, t) = N B; g,q (q 1/2 q 1/2 ) 2g 2 t Q. (0.4) g=0 Q Z/2 (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 (0.4). The integrality of the new invariants N B; g,q. If one looks at the right hand side of (0.4), one will find it very interesting that the pole of f A in (q 1/2 q 1/2 ) is actually at most of order 1 for any link and any labeling irreducible representations. However, by the calculation of quantum group invariants of links, the pole order of f A might be going to when the degrees of the labeling irreducible representations go higher and higher. This miracle cancellation implies a very special algebraic structure of quantum group invariants of links and thus the Chern-Simons partition function. in Section 2.4.3, we include an example in the simplest setting showing that this cancellation 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. In order to deal with the existence of the conjectured algebraic structure, one will encounter the problem of how to control the pole order of (q 1/2 q 1/2 ). We consider the framing generating series Z(L; q, t, τ) of Chern-Simons invariants of links which implies that the Chern-Simons 6

7 partition function satisfies the following cut-and-join equation Z(L; q, t, τ) τ = u L ( 2 α=1 i,j 1 ijp α i+j 2 p α i pα j + (i + j)p α i pα j p α i+j ) Z(L; q, t, τ). Here p α n = p n(x α ) are regarded as variables. However, a deep understanding this conjecture relies on the following log version of cut-andjoin equation F(L; q, t, τ) τ = u L ( 2 α=1 i,j 1 ijp α i+j 2 F p α i pα j + (i + j)p α i p α j F p α i+j + ijp α F i+j p α i F p α 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 nonlinear ODE systems and the non-linear part deeply reflects the recursion structure of Chern-Simons partition function and miracle cancelation of pole occurs in the free energy which can be explained as a generating function of open Gromov-Witten invariants. A powerful tool to control the pole order of (q 1/2 q 1/2 ) through log cut-and-join equation is developed in this paper as we called cut-andjoin analysis. An important feature of cut-and-join equation shows that the differential equation at partition 1 (d) can only have terms obtained from joining 2 while at (1 d ), 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 q 1/2 ). We found a rational function ring which characterizes the algebraic structure of Chern-Simons partition function and hence (open) topological string partition function by duality. 1 Here, we only take a knot as an example. For the case of links, it is then a natural extension. 2 Joining means that a partition is obtained from combining two rows of a Young diagram 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. 7

8 A similar ring characterizing closed topological string partition function firstly appears in the second author s work on Gopakumar-Vafa conjecture [38, 39]. 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 theory is more subtle than the integrality of Gopakumar-Vafa invariants 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 as power of curve classes, while in the open case, the generating function is weighted by the labeling irreducible representation which results in a Schur function. This subtlety had already been explained in [21]. To overcome this subtlety, one observation is that quantum group invariants look Schur-function-like. This had already been demonstrated in the topological vertex theory (also see [30]). We refine the ring in the closed case and get the new ring R(y; q, t) (cf. section 5.1). Correspondingly, we consider a new generating series of N 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 g=0 Q Z/2 N B; g,q (q 1/2 q 1/2 ) 2g t Q Z[(q 1 2 q 1 2 ) 2, t ±1 2 ]. The paper is organized as follows. In Section 1, we introduce some basic notations about partition and generalize this concept to simplify our calculation 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. 8

9 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 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 (λ 1, λ 2, ) such that λ 1 λ 2. The total number of parts in λ is called the length of λ and denoted by l(λ). We use m i (λ) to denote the number of times that i occurs in λ. The degree of λ is defined to be λ = λ i. i 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 = m i (λ)!. i There is another way to rewrite a partition λ in the following format (1 m 1(λ) 2 m 2(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 λ j 9

10 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) = ( ) =. For a given partition λ, denote by λ t the conjugate partition of λ. The Young diagram of λ t is transpose to the Young diagram of λ: the number of boxes on j-th column of λ t 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 (12 2 3) (15) = ( ). Using Young diagram, it looks like =. The following number associated with a partition λ is used throughout this paper, z λ = j j m j(λ) m j (λ)!, κ λ = j λ j (λ j 2j + 1). It s easy to see that κ λ = κ λ t. (1.1) A power symmetric function of a sequence of variables x = (x 1, x 2,...) is defined as follows p n (x) = x n i. i For a partition λ, l(λ) p λ (x) = p λj (x). j=1 10

11 It is well-known that every irreducible representation of symmetric group can be labeled by a partition. Let χ λ be the character of the irreducible representation corresponding to λ. Each conjugate class of symmetric group can also be represented by a partition µ such that the permutation in the conjugate class has cycles of length µ 1,..., µ l(µ). Schur function s λ is determined by s λ (x) = χ λ (C µ ) p µ (x) z µ µ = λ where C µ is the conjugate class of symmetric group corresponding to partition µ. 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 l(λ) l(µ) and λ i µ i, i. The summation is taken to be and the zero-element is (0). 11

12 (3). P n. The order of P n is defined as follows: Define A, B P n, A B iff A i B i, i = 1,..., n. A B = (A 1 B 1,..., A n B n ). ((0), (0),..., (0)) is the zero-element. It s easy to check that P n 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 m β (Λ) the number of times that β occurs in the parts of Λ, we then have Aut Λ = β S m β (Λ)!. Introduce the following quantities associated with Λ, u Λ = l(λ)! AutΛ, θ Λ = ( 1)l(Λ) 1 u Λ. (1.2) l(λ) 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 a nt n, then ( ) f A β p β (x) = a l(λ) A Λ p Λ (x) u Λ, where Proof. Note that β 0, β S Λ P(S) l(λ) l(λ) p Λ = p Λj, A Λ = A Λj. ( j=1 β S, β 0 η β ) n = Direct calculation completes the proof. j=1 Λ P(S), l(λ)=n 12 η Λ u Λ.

13 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 (C ij ). Quantized universal enveloping algebra U q (g) is generated by {H i, X +i, X i } together with the following defining relations: q H i/2 i q H i/2 i [H i, H j ] = 0, [H i, X ±j ] = ±C ij X ±j, [X +i, X j ] = δ ij, q 1/2 i q 1/2 i 1 C ij { 1 ( 1) k Cij } X 1 C ij k ±i X k ±j X±i k = 0, for all i j, q i k=0 where and {k} q = q k 2 q k 2 q 1 2 q 1 2 { a b } q, {k} q! = k {i} q, i=1 = {a} q {a 1} q {a b + 1} q {b} q! The ribbon category structure associated with U q (g) is given by the following datum: 1. For any given two U q (g)-modules V and W, there is an isomorphism satisfying for U q (g)-modules U, V, W. Ř V, W : V W W V Ř U V, W = (ŘU, W id V )(id U ŘV, W) Ř U, V W = (id V ŘU, W)(ŘU, V id W ) Given f Hom Uq(g)(U, Ũ), g Hom U q(g)(v, Ṽ ), one has the following naturality condition:. (g f) ŘU, V = Ř eu, e V (f g). 13

14 2. There exists an element K 2ρ U q (g), called the enhancement of Ř, such that K 2ρ (v w) = K 2ρ (v) K 2ρ (w) for any v V, w W. 3. For any U q (g)-module V, the ribbon structure Θ V : V V associated to V satisfies Θ ±1 V = tr V ر1 V,V. The ribbon structure also holds the following naturality condition for any x Hom Uq(g)(V, Ṽ ). x Θ V = Θ e V x. Definition 2.1. Given z = i x i y i End Uq(g)(U V ), the quantum trace of z is defined as follows tr V (z) = i tr(y i K 2ρ ) End Uq(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 S 1 embedded in S 3. A theorem of J. Alexander asserts that any oriented link is isotopic to the closure of some braid. A braid group B n is defined by generators σ 1,, σ n 1 and defining relation: { σi σ j = σ j σ i, if i j 2 ; σ i σ j σ i = σ j σ i σ j, if i j = 1. Let L be a link with L components K α, α = 1,..., L, represented by the closure of an element of braid group B m. We associate each component of 3 In the following context, we will briefly write sl N. 14

15 L, K α, an irreducible representation R α of quantized universal enveloping algebra U q (sl N ), labeled by its highest weight Λ α. Denote the corresponding module by V Λα. The j-th strand in the braid will be associated with the irreducible module V j = V Λα, if this strand belongs to the component K α. The braiding is defined through the following universal R-matrix of U q (sl N ) R = q 1 2 Pi,j C 1 ij H i H j positive root β where (C ij ) is the Cartan matrix and exp q [(1 q 1 )E β F β ], exp q (x) = k=0 q 1 4 k(k+1) x k {k} q!. Define braiding by Ř = P 12R, where P 12 (v w) = w v. Now for a given link L of L components, one chooses a closed braid representation of L in braid group B m. In the case of no confusion, we also use L to denote the braid representation in B m. We will associate each crossing by the braid defined above. Let U, V be the U q (sl N )-modules labeling the two outgoing strands of the crossing, the braiding ŘU,V (resp. Ř 1 V,U ) is assigned as following U V U V Ř U,V Ř 1 V,U This provides an isomorphism For example, the following link L h(l) End Uq(sl N )(V 1 V m ). 15

16 U V U L will give a homomorphism h(l) = (ŘV, U id U )(id V Ř 1 U, U )(ŘU, V id U ). Let K 2ρ be the enhancement of Ř in the sense of [40], where ρ is one half of the sum of positive roots of sl N. Associate Young diagram A α (also see it as a partition) with the irreducible representations R α, α = 1,..., L. Definition 2.2. The quantum group invariant is defined as follows: where W (A 1,...,A L )(L) = q d(l) tr V1 V m (h(l)), d(l) = 1 L ω(k α )(Λ α, Λ α + 2ρ) N α=1 lk(k α, K β ) A α A β α<β and lk(k α, K β ) is the linking number of components K α and K β. A substitution of t = q N 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 q 1/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). 16

17 A direct computation 4 shows that after removing the terms of q 1 N, q d(l) can be simplified as follows q P L α=1 κ A αw(kα)/2 t P L α=1 Aα w(k α)/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, W A ( ; q, t) = tr VA (id VA ) is equal to the quantum dimension of V A which will be denoted by dim q V A. (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 invariants of links is related to the HOMFLY polynomial of the link, P L (q, t), in the following way: ( ) W (,, ) (L; q, t) = t lk(l) t 1 2 t 1 2 P q 1 2 q 1 L (q, t). 2 (3). If L is a disjoint union of L knots, i.e., L = K 1 K 2 K L, the quantum group invariants of L is simply the multiplication of quantum group invariants of K α W (A 1,...,A L )(L; q, t) = L W A α(k α ; q, t). α=1 4 It also can be obtained from the ribbon structure. 17

18 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 = (A 1, A 2,...,A L ), µ = (µ 1, µ 2,...,µ L ) P L. Let x = (x 1,..., x L ) be L arbitrarily chosen variables. throughout the paper. The following notations will be used l(λ) [n] q = q n n 2 q 2, [λ]q = [λ j ] q, z µ = A = ( A 1,..., A L ), A = j=1 A t = ( (A 1 ) t,...,(a L ) t), χ A ( µ) = L z µ α, α=1 L A α, l( µ) = α=1 L χ A α(c µ α), s A ( x) = α=1 L l(µ α ), α=1 L s A α(x α ). Chern-Simons partition function can be defined to be the following generating function of quantum group invariants of L, Z CS (L) = 1 + A 0 W A (L; q, t)s A ( x). α=1 Define free energy F = log Z = µ 0 F µ p µ ( x). (2.2) We rewrite Chern-Simons partition function as Z CS (L) = 1 + µ 0 Z µ p µ ( x) where Z µ = A χ A ( µ) z µ W A. (2.3) By Lemma 1.3, we have F µ = θ Λ Z Λ. (2.4) Λ P(P L ), Λ = µ P L 18

19 2.4 Main results Two theorems that answer LMOV conjecture Let P B (q, t) be the function defined by (0.3), which can be uniquely obtained by the following formula, P B (q, t) = A = B f A (q, t) L α=1 µ χ A α(c µ )χ B α(c µ ) z µ l(µ α ) j=1 1 q µ j/2 q µ j/2. Theorem 1. There exist topological invariants N B; g,q Q such that expansion (0.4) holds. Theorem 2. Given any B P(P L ), the generating function of N B; g,q, P B (q, t), satisfies (q 1/2 q 1/2 ) 2 P B (q, t) Z [( q 1/2 q 1/2)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 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 [41] 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 [5, 6] 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 (0.4) 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 P L. Any term that has power of q 1/2 q 1/2 lower than 2 will be canceled! 19

20 Without the motivation of Chern-Simons/topological string duality, this mysterious cancellation is hardly able to be realized from knot-theory point of view. We will see later through an example that even in the simplest case, the cancellation has already given beautiful consequences Geometric interpretation of the new integer invariants The following interpretation is taken in physics literature from string theoretic point of view [21]. Quantum group invariants of links can be expressed as vacuum expectation 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 geometric description of the new integer invariants N B; q,t in terms of open Gromov-Witten invariants (also see [19] for more detailed description). The geometric picture of f A is proposed in [21]. One can rewrite the free energy as F = l( µ) 1 λ 2g 2+l( µ) F g, µ (t)p µ. (2.5) µ g=0 The quantities F g, µ (t) can be interpreted in terms of the Gromov-Witten invariants of Riemann surface with boundaries. It was conjectured in [37] that for every link L in S 3, one can canonically associate a lagrangian submanifold C L in the resolved conifold O( 1) O( 1) P 1. The first Betti number b 1 (C L ) = L, the number of components of L. Let γ α, α = 1,...,L, be one-cycles representing a basis for H 1 (C L, Z). Denote by M g,h,q the moduli space of Riemann surfaces of genus g and h holes embedded in the resolved conifold. There are h α holes ending on the nontrivial cycles γ α for α = 1,..., L. The product of symmetric groups Σ h1 Σ h2 Σ hl acts on the Riemann surfaces by exchanging the h α holes that end on γ α. The integer N B; q,t is then interpreted as N B; q,t = χ(s B (H (M g,h,q ))) (2.6) 20

21 where S B = S B 1 S B L, and S B α is the Schur functor. The recent progress in the mathematical definitions of open Gromov- Witten invariants [17, 22, 23] may be used to put the above definition on a rigorous setting An application to knot theory This subsection is roughly a copy from [21, 19]. Consider associating the fundamental representation to each component of the given link L. As discussed above, quantum group invariants of L will reduce to the classical HOMFLY polynomial P L (q, t) of L except for a universal factor. HOMFLY has the following expansion P L (q, t) = g 0 p 2g+1 L (t)(q 1 2 q 1 2 ) 2g+1 L. (2.7) The lowest power of q 1/2 q 1/2 is 1 L, which was proved in [28] (or one may directly derive it from Lemma 4.1). After a simple algebra calculation, one will have ( ) t 1/2 t 1/2 F (,..., ) = q 1/2 q 1/2 Lemma 4.3 states that g 0) p 2g+1 L (t)(q 1 2 q 1 2 ) 2g+1 L. p 1 L (t) = p 3 L (t) = = p L 3 (t) = 0, which implies that the p k (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 p 1 L (t) = 0. A direct comparison of the coefficients of F = log Z CS immediately leads to the following theorem proved by Lickorish and Millett [28]. This connection was first obtained in [21]. Theorem 2.5 (Lickorish-Millett). Let L be a link with L components. Its HOMFLY polynomial P L (q, t) = g 0 ( p L 2g+1 L (t) q q 2 ) 2g+1 L 21

22 satisfies ( ) p L 1 L(t) = t lk t 1 1 L 1 L 2 t 2 α=1 p Kα 0 (t) where p Kα 0 (t) is HOMFLY polynomial of the α-th component of the link L with q = 1. In [28], Lickorish and Millett obtained the above theorem by skein analysis. Here as the consequence of higher order cancellation phenomenon, one sees how easily it can be achieved. Note that we only utilize the vanishing of p 1 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 cancellation of higher order poles. 3 Hecke algebra and cabling 3.1 Centralizer algebra and Hecke algebra representation We review some facts about centralizer algebra and Hecke algebra representation and their relation to the representation of braid group. Let V be the fundamental representation of U q (sl N ) 5. The centralizer algebra of V n is defines as follows Let C n = End Uq(sl N )(V n ) = { x End(V n ) : xy = yx, y U q (sl N ) }. {K ±1 i, E i, F i : 1 i N 1} denote the standard generators of the quantized universal enveloping algebra U q (sl N ). Under a suitable basis {X 1,..., X N } of V, the fundamental repre- 5 We will reserve V to denote the fundamental representation of U q (sl N ) from now on. 22

23 sentation is given by the following matrices E i E i,i+1 F i E i+1,i K i q 1/2 E i,i + q 1/2 E i+1,i+1 + i j E jj where E i,j denotes the N N matrix with 1 at the (i, j)-position and 0 elsewhere. Direct calculation shows K 2ρ (X i ) = q N+1 2i 2 X i (3.1) and q 1 2N Ř(Xi X j ) = q 1/2 X i X j, i = j, X j X i, i < j, X j X i + (q 1/2 q 1/2 )X i X j, i > j. Hecke algebra H n (q) of type A n 1 is the complex algebra with n 1 generators g 1,..., g n 1, together with the following defining relations g i g j = g j g i, i j 2 g i g i+1 g i = g i+1 g i g i+1, i = 1, 2,..., n 2, (g i q 1/2 )(g i + q 1/2 ) = 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 H n (q) is just the group algebra CΣ n of symmetric group Σ n. A very important feature of the homomorphism is that h factors through H n (q) via h : CB n C n q 1 2N σi g i q 1 2N h(σi ). (3.2) It is well-known that the irreducible modules S λ (Specht module) of H n (q) are in one-to-one correspondence to the partitions of n. 23

24 Any permutation π in symmetric group Σ n can express as a product of transpositions π = s i1 s i2 s il. If l is minimal in possible, we say π has length l(π) = l and g π = g i1 g i2 g il. It is not difficult to see that g π is well-defined. All of such {g π } form a basis of H n (q). Minimal projection p is an element in C n such that pv n is some irreducible representation V λ. We denote it by p λ. When N is large enough, one has C n = H n (q). The minimal projections of Hecke algebras are well studied (for example [13]), which is a Z(q ±1 )-linear combination {q 1 2g i }. 3.2 Explicit formula of quantum dimension An explicit formula for quantum dimension of any irreducible representation of U q (sl N ) can be computed via decomposing V n into permutation modules. A composition of n is a sequence of non-negative integer such that b = (b 1, b 2,...) b i = n. i 1 We will write it as b n. The largest j such that b j 0 is called the end of b and denoted by l(b). Let b be a composition such that l(b) N. Define M b to be the subspace of V n spanned by the vectors X j1 X jn (3.3) such that X i occurs precisely b i times. It is clear that M b is an H n (q)-module and is called permutation module. Moreover, by explicit matrix formula of {E i, F i, K i } acting on V under the basis {X i }, we have M b = {X V n : K i (X) = q b i b i+1 X}. 24

25 The following decomposition is very useful V n = M b. b n, l(b) N Let A be a partition and V A the irreducible representation labeled by A. The Kostka number K Ab is defined to be the weight of V A in M b, i.e., K Ab = dim(v A M b ). Schur function has the following formulation through Kostka numbers s A (x 1,...,x N ) = b A, l(b) N K Ab N j=1 x b j j. By (3.1), K 2ρ is acting as a scalar N j=1 q 1 2 (N+1 2j)b j. Thus dim q V A = tr VA id VA = b A dim(v A M b ) N j=1 q 1 2 (N+1 2j)b j ( ) = s A q N 1 2,...,q N 1 2 = χ A (C µ ) ( ) p µ q N 1 2,...,q N 1 2 z µ µ = A = µ = A χ A (C µ ) z µ l(µ) j=1 Here in the last step, we use the substitution t = q N. t µ j/2 t µ j/2 q µ j/2 q µ j/2. (3.4) 3.3 Cabling technique Given irreducible representations V A 1,..., V A L to each component of link L. Let A α = d α, d = (d 1,..., d L ). The cabling braid of L, L d, is obtained by substituting d α parallel strands for each strand of K α, 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 [29]. 25

26 Lemma 3.2 ([29], Lemma 3.3). Let V i = p i V d i for some minimal projections, p i = p j if the i-th and j-th strands belong to the same knot. Then tr V1 V m (h(l)) = tr V n(h(l (d1,...,d L )) p 1 p n ). where m is the number of strands belonging to L, n = L α=1 d αr α, r α is the number of strands belong to K α, the α-th component of L. 4 Proof of Theorem Pole structure of quantum group invariants By an observation from the action of Ř on V V, we define X (i1,...,i n) = q #{(j,k) j<k,i j >i k } 2 X i1 X in. { X (i1,...,i n)} form a basis of V n. By (3.2), g j H n (q), we have { q 1 2 gj X(...,ij,i j+1,...) = X(...,ij+1,ij,...), i j i j+1, q X (...,ij+1,i j,...) + (1 q) X (...,ij, i j+,...), i j i j+1. Lemma 4.1. Let be the unknot. Given any A = (A 1,...,A L ) P L, lim q 1 (4.1) L W A (L; q, t) W A ( L ; q, t) = ξ Kα (t) dα, (4.2) where A α = d α, K α is the α-th component of L, and ξ Kα (t), α = 1,...,L, are independent of A. Proof. Choose β B m 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 components of L. Let r α be the number of the strands which belong to K α. n = α d αr α is equal to the number of components in the cabling link L d. Let α=1 Y = tr V P L α=1 dα(rα 1) 26 ( L d ). (4.3)

27 Y is both a central element of End Uq(sl N ) ( V (d d L ) ) and a Z[q ±1 ]-matrix under the basis { X (i1,...,i n)}. On the other hand, p A = p A 1... p A L is a Z (q ±1 )-matrix under the basis { X (i1,...,i n)}. 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 transcendental q, ǫ must stay in Z[q ±1 ]. By Lemma 3.2, tr V n(p r 1 A 1 p r L A L L d ) = tr V (d 1 + d L )(p A Y) = ǫ tr V (d 1 + d L )(p A ) The definition of quantum group invariants of L gives = ǫ W A ( L ). (4.4) W A (L; q, t) = q P α κ A αw(kα)/2 t P α dαw(kα)/2 ǫ W A ( ; q, t) When q 1, L d reduces to an element in symmetric group Σ n of d cycles. Moreover, when q 1, the calculation is actually taken in individual knot component while the linkings of different components have no effect. By Example 2.4 (1) and (3), we have This implies L W A ( L ; q, t) = W A α( ; q, t). α=1 lim q 1 L W A (L; q, t) W A ( L ; q, t) = lim q 1 α=1 W A α(k α ; q, t) W A α( ; q, t). (4.5) Let s consider the case when K is a knot. A is the partition of d associated with K and K d is the cabling of K. Each component of K d is a copy of K. q 1, K d reduces to an element in Σ dr. To calculate the Y, it is then equivalent to discussing d disjoint union of K. Say K has r strands. Consider Y 0 = tr V (r 1) K. 27

28 The eigenvalue of Y 0 is then P K (1, t), where P K (q, t) is the HOMFLY polynomial for K. We will rewrite ξ K (t) = P K(1, t) t w(k) (4.6) Therefore, the eigenvalue of tr V d(r 1) K d will be ξ K (t) d. Combined with (4.5), the proof is then completed. 4.2 Symmetry of quantum group invariants Define φ µ (q) = L α=1 Comparing (2.2) and (0.1), we have (q µα j /2 q µα j /2 ). l(µ α ) j=1 F µ = 1 χ A ( µ/d) ( f A q d, t d) d z µ/d d µ A = 1 χ A ( µ/d) L P B (q d, t d ) M A d z α B α(qd ) µ/d d µ A B α=1 = 1 d φ µ/d(q d ) χ B ( µ/d)p B (q d, t d ). (4.7) z µ/d d µ Apply Möbius inversion formula, P B (q, t) = µ B χ B ( µ) φ µ (q) d µ µ(d) d F µ/d(q d, t d ) (4.8) where µ(d) is the Möbius function defined as follows { ( 1) µ(d) = r, if d is a product of r distinct prime numbers; 0, otherwise. To prove the existence of formula (0.4), we need to prove Symmetry of P B (q, t) = P B (q 1, t). 28

29 The lowest degree ( q 1/2 q 1/2) in P B is no less than 2. Combine (4.7), (2.4) and (2.3), 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 A t(q, t) = ( 1) A W A (q 1, t). Proof. The following irreducible decomposition of U q (sl N ) modules is wellknown: V n = B n d B V B, where d B = χ B (C (1 d )). Let d A = L α=1 d Aα. Combined with Lemma 3.2 and eigenvalue of Y in Lemma 4.1 and (2.1), we have W (,..., ) (L d ) = W A (L)d A A = d = d A q P α κ A αw(kα)/2 t P α Aα w(k α)/2 ǫ A A = d L dim q V A α, (4.9) Where ǫ A is the eigenvalue of Y on L α=1 V Aα as defined in the proof of Lemma 4.1. Here if we change A α to (A α ) t, we have κ (A α ) t = κ Aα, which is equivalent to keep A α 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 (2.7) and Example 2.4 (2), we have W (,..., ) (L d ; q 1, t) = ( 1) P α Aα W (,..., ) (L d ; q, t). (4.10) 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, K 2ρ, acts on X i (see (3.1)) as a scalar q 1 2 (N+1 2i). We can actually generalize this scalar to any zi α where α corresponds the strands belonging to the α-th component (cf. [42]). It s not difficult to see that (4.9) still holds. The quantum dimension of V A α thus α=1 29

30 becomes s A α(z1 α,...,zα N ) which has been shown in the computation of the explicit formula of quantum dimension in Section 3.2. We rewrite the above generalized version of quantum group invariants of links as W A (L; q, t; z), where z = {z α }. (4.10) becomes W (,..., ) (L d ; q 1, t; z) = ( 1) P α Aα W (,..., ) (L d ; q, t; z) (4.11) Now, combine (4.11), (4.9) and (4.10), we obtain A t d A t q P κ(a α ) tw(k α)/2 t P (A α ) t w(k α)/2 ǫ A t(q 1 ; z) L s (A α ) t( zα ) α=1 = ( 1) P α Aα d A q P κ A αw(k α)/2 t P A α w(k α)/2 ǫ A (q; z) Note the following facts: where the second formula follows from A L s A α(z α ) α=1 (4.12) s A t( z) = ( 1) A s A (z), (4.13) d A t = d A. (4.14) χ A t(c µ ) = ( 1) µ l(µ) χ A (C µ ). (4.15) Apply (4.13) and (4.14) to (4.12). Let zi α = q 1 2 (N+1 2i), then using z instead of z is equivalent of substituting q by q 1 while keeping t in the definition of quantum group invariants of links. This can be seen by comparing l(µ) ( ) p µ z α i = q N 2i+1 t µ j/2 t µ j/2 2 = q µ j/2 q µ j/2 with Therefore, we have j=1 p µ ( z α ) = ( 1) l(µ) p µ (z α ). ǫ A t(q 1, t) = ǫ A (q, t). (4.16) 30

31 By the formula of quantum dimension, it s then easy to verify that W A t( L ; q, t) = ( 1) A W A ( L ; q 1, t) (4.17) Combining (4.4), (4.16) and (4.17), the proof of the Lemma is then completed. By Lemma 4.2, we have the following expansion about P B P B (q, t) = N B,g,Q (q 1/2 q 1/2 ) 2g 2N 0 t Q g 0 Q Z/2 for some N 0. We will show N 0 1. Let q = e u. The pole order of (q 1/2 q 1/2 ) in P B is the same as pole order of u. Let f(u) be a Laurent series in u. Denote deg u f to be the lowest degree of u in the expansion of u in f. Combined with (4.7), N 0 1 follows from the following lemma. Lemma 4.3. deg u F µ l( µ) 2. Lemma 4.3 can be proved through the following cut-and-join analysis. 4.3 Cut-and-join analysis Cut-and-join operators Substitute W A (L; q, t, τ) = W A (L; q, t) q P L α=1 κ A ατ/2 in the Chern-Simons partition function, we have the following framing series Z(L; q, t, τ) = 1 + A 0 W A (L; q, t, τ) s A ( x). Similarly, we can have framing version of Z µ and F µ Z(L; q, t, τ) = 1 + µ 0 Z µ (q, t, τ) p µ ( x), F(L; q, t, τ) = µ 0 F µ (q, t, τ)p µ ( x). 31

32 Define exponential cut-and-join operator E E = (ijp 2 i+j + (i + j)p i p j p i,j 1 i p j p i+j and log cut-and-join operator L LF = ( 2 F F F F ijp i+j + (i + j)p i p j + ijp i+j p i,j 1 i p j p i+j p i p j ), (4.18) ). (4.19) Here {p i } are regarded as independent variables. Schur function s A (x) is then a function of {p i }. Schur function s A is an eigenfunction of exponential cut-and-join with eigenvalue κ A [7, 11, 48]. Therefore, Z (L; q, t, τ) satisfies the following exponential cut-and-join equation Z(L; q, t, τ) τ = u 2 L E α Z(L; q, t, τ), α=1 or equivalently, we also have the log cut-and-join equation F(L; q, t, τ) = u L L α F(L; q, t, τ). (4.20) τ 2 In the above notation, E α and L α correspond to variables {p i } which take value of {p i (x α )}. α=1 (4.20) restricts to µ will be of the following form F µ τ = u ( ) α µ ν F ν + nonlinear terms, (4.21) 2 ν = µ, l( ν)=l( µ)±1 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 join = cut =. For µ = (µ 1,...,µ L ) and ν = (ν 1,..., ν L ), cutting and joining happens only for one partition. 32

33 4.3.2 Degree of u A direct algebraic calculation of expansion of log(z CS (L; q, t)) shows that the coefficient of the lowest degree of u in F µ is a polynomial g(t, τ) in τ and t ±1 2. Moreover, the degree of g(t, τ) in τ is equal to deg u F µ + µ. If µ = 1, only one component of L is taken into consideration. Note that F µ is now a HOMFLY polynomial. By (2.7) and Example 2.4 (2), we have deg u F µ = 1. Consider µ > 1. If we prove that we are able to claim deg u F µ > µ, (4.22) ( F µ ) deg u = deg τ u F µ. (4.23) Denote by D u W A the term of u µ in W A. Let s first look at Z µ. By Lemma 4.1 and (2.3), we know the possible term of u µ in Z µ can be computed from the following formula A χ A ( µ) z µ D u W A = u µ L ξ Kα (t) dα α=1 where ξ Kα (t) is defined as in (4.2). A χ A ( µ) z µ α ( χ A α (1 d α ) ) d α! (4.24) The right hand side of (4.24) can survive only if µ = ( (1 d 1 ),...,(1 d L ) ) by the orthogonality relation of characters of symmetric group. Combine with (2.4), we have deg u F µ > µ, if l( µ) µ. Now, we are one step away from the claim (4.23): to verify it holds when l( µ) = µ, i.e., µ = ( (1 d 1 ),..., (1 d L ) ). By (2.4), the claim will be reduced to the situation of a knot. Note that for partition (1 d ), (2.4) can be rewritten as follows F (1 d ) = µ d l(µ) θ µ Z (1 µ j ) Combined with (4.24), it s equivalent to the following lemma. j=1 33

34 Lemma 4.4. For any d > 1, we have Proof. Note that µ d θ µ l(µ) j=1 µ j! = 0. ( t = log(exp t) = log 1 + n=1 t n ). n! ( Expand log 1 + ) t n n=1 by the method of Lemma 1.3, one will have n! t = d 1 t d µ d θ µ l(µ) j=1 µ j!. The proof of the lemma is completed by comparing the coefficients Induction procedure If µ = 1, it has just been verified through degree of u in HOMFLY polynomial. If µ > 1, by the argument in the above subsection, we have ( F µ ) deg u = deg τ u F µ. Suppose Lemma 4.3 holds for any ν < d. Now, consider S = { ν : ν = d}, where d = (d 1,...,d L ) and L α=1 dα = d. If µ S, one important fact is that degree of u in the nonlinear terms in (4.21) is no less than l( µ) 3. We will prove Lemma 4.3 holds for µ S by contradiction. and Denote by S r = { ν : ν S and l( ν) = r} S D r = min{deg u F µ : µ S r }. 34

35 Assume k is the smallest integer such that there exists a ν S k satisfying deg u F ν < l( ν) 2 = k 2. Choose µ S k such that deg u F µ = D k. Rewrite (4.21) into three parts F µ τ = u 2 ν = µ, l( ν)=l( µ) 1 α µ ν F ν + u 2 ν = µ, l( ν)=l( µ)+1 α µ ν F ν + (4.25) where represents the nonlinear part and deg u = k 2. Here in the equation, ν runs in S. However, F µ deg u τ = deg u F µ = D k < k 2. D k 1 (k 1) 2, so the first part of the sums in the r.h.s of (4.25) is of the lowest order of u at least k 2 and the lowest order of u in F µ must come τ from the part of length l( µ) + 1. Then we have and 1 + D k+1 D k, (4.26) D k+1 D k 1 < k 2 1 < (k + 1) 2. Choose ξ S k+1 such that deg u F ξ = D K+1, run the similar analysis as above. Since (4.26), one will get Thus we obtain 1 + D k+2 D k+1. D k > D k+1 > D k+2 >...D d 1 > D d. Remember that there is still one last equation we have not used yet. Consider the cut-and-join equation for the following partition η = ( (1 d 1 ),...,(1 d L ) ) S d. Note that this partition has the longest length in S d, there will be only the first summand in (4.25) left. In particular, no non-linear terms will appear in the equation. Therefore, F η τ = u 2 l( ν)=d 1 35 α η ν F ν.

36 This implies D d D d 1 + 1, which contradicts with D d 1 > D d. The proof of Lemma 4.3 is completed. 5 Proof of Theorem A ring that characterizes the algebraic structure of the partition function Let d = (d 1,..., d L ), y = (y 1,...,y L ). 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 d k d µ(k) k A P(P n ), A = d/k θ A W A (q k, t k )s A (z k ) (5.2) However, T d = s B (y)p B (q, t) B = d = s B (y) N B; g,q (q 1/2 q 1/2 ) 2g 2 t Q = B = d g=0 Q Z/2 ( g=0 Q Z/2 B = d ) N B; g,q s B (y) (q 1/2 q 1/2 ) 2g 2 t Q. 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 N B; g,q s B (y) Ω(y). B = d 36

37 Let v = [1] 2 q. It s easy to see that [n]2 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; q,t. { a(y; v, t) R(y; v, t) = b(v) : a(y; v, t) Ω(y)[v, t ±1/2 ], b(v) = n k [n k ] 2 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. { f(y; q, t) M(y; q, t) = : f Ω(y)[q ±1/2, t ±1/2 ], b(v) = } [n k ] 2 q Z[v]. b(v) n k 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 define A = ( A 1, A 2,...) P(P L ), ( ) A (d) = A 1,..., A }{{} 1, A 2,..., A }{{} 2,.... d d Lemma 5.2. If θ A = c, d > 1 and gcd(c, d) = 1, then for any r d, we can d find B P ( P L) such that A = B (r). 37

38 Proof. Let l = l(a), we have Let θ A = ( 1)l(A) 1 u A = ( 1)l(A) 1 (l(a) 1)!. l(a) AutA ( ) A = A 1,..., A }{{} 1,..., A n,..., A n, }{{} m 1 m n so l(a) = m m n. Note that ( ) l u A =. m 1, m 2,, m n Let η = gcd(m 1, m 2,, m n ). We have A = Ã(η), where ( ) Ã = A 1,..., A }{{} 1,..., A n,..., A n. }{{} m 1 /η m n/η By Corollary A.4, l η u A and gcd(c, d) = 1, one has d η. We can take B = Ã ( η r ). This completes the proof. Remark 5.3. By the choice of η in the above proof, we know A α 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 gcd(r, s) = 1, f(y; q, t) is a primitive polynomial in s b(v) q ±1/2 and t ±1/2. f(y; q, t) is also a symmetric functions in y. Let p be any prime number, define Ord p ( r s f (y; q, t)) ) ( r ) = Ord p. (5.4) b(v) s Lemma 5.4. Given A P ( P L), p any prime number and f A (y; q, t) M(y; q, t), we have Ord p (θ A (p)f A (p)(y; q, t) 1 ) p θ Af A (y p ; q p, t p ) 0. 38

39 Proof. Assume θ A = r a, b p 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(P L ) such that A = B (pr). Therefore, f A = f pr B. Let s = l(b) and ( ) B = B 1,..., B }{{ } 1,..., B k,..., B n. }{{} s 1 s k Note that l(a (p) ) = p l(a). By Theorem A.2, Ord p (θ A (p) 1 ) p θ A [( ) ( )] 1 p r+1 s p r s = Ord p p l(a) p r+1 s 1,, p r+1 s k p r s 1,, p r s k 2(r + 1) (1 + r) (5.5) > 0. (5.6) Combine (5.6) with Theorem A.6, we have Ord p (θ A (p)f A (p)(y; q, t) 1 ) p θ Af A (y p ; q p, t p ) = Ord p [(θ A (p) 1 ) p θ A f A (p)(y; q, t) + 1 )] p θ A (f A (p)(y; q, t) f A (y p ; q p, t p. However, Ord p [ 1 p θ A (f A (p)(y; q, t) f A (y p ; q p, t p )] [ b = Ord p (f p r+1 B (y; q, t) pr+1 f B (y p ; q p, t p ) pr)] a 0. The proof is completed. Apply the above Lemma, we have Ord p (θ A (p)w A (p)(q, t)s A (p)(z) 1 ) p θ AW A (q p, t p )s A (z p ) 0. (5.7) 39