Yang-Yang functions, Monodromy and knot polynomials

We derive a structure of $\mathbb{Z}[t,t^{-1}]$-module bundle from a family of Yang-Yang functions. For the fundamental representation of the complex simple Lie algebra of classical type, we give explicit wall-crossing formula and prove that the monodromy representation of the $\mathbb{Z}[t,t^{-1}]$-module bundle is equivalent to the braid group representation induced by the universal R-matrices of $U_{h}(g)$. We show that two transformations induced on the fiber by the symmetry breaking deformation and respectively the rotation of two complex parameters commute with each other.


Introduction
Yang-Yang function was named by N. Nekrasov, A. Rosly and S. Shatashvili in [1]. Originated from C. N. Yang and C. P. Yang's paper [2] [3], it was used for the analysis of the non-linear Schrödinger model. Behind this function hides a quantum integrable system [1] [4], thus it aroused much interest. Yang-Yang function can also be realized as the power in the correlation function of the free field realization of Virasoro vertex operators [5]. D. Gaitto and E. Witten used this realization to derive Jones polynomial for knots from four dimensional Chern-Simons gauge theory. Our interest is to figure out the structure underlying the derivation of knot invariants from the Yang-Yang function.
Similar to the work of V. G. Drinfeld and T. Kohno [6] [7] [8], where the monodromy of Knizhnik-Zamolodchikov system was proved to be equivalent to the braid group representation, induced by the universal R-matrices of the quantum enveloping algebra U h (g) of semisimple Lie algebra g, we prove that the monodromy representation of the Z[t, t −1 ]-module bundle constructed from a family of Yang-Yang functions associated with the fundamental representation of the classical complex simple Lie algebra is equivalent to the braid group representation, induced by the universal R-matrices of U h (g). Furthermore, transformations induced on the fiber by two parameter deformations, the symmetry breaking parameter c from c = 0 to c → ∞ and respectively the rotation of two singular complex parameters z 1 and z 2 , commute with each other. By studying the monodromy, creation and annihilation matrices from the parameter deformations, one can derive the HOMFLY-PT polynomial and Kauffman polynomial for knots. We expect the existence of new knot invariants, different from HOMFLY-PT and Kauffman polynomials, from general representations of Lie algebras. It will be investigated elsewhere.
This paper is organized as follows. Firstly we give the definition of Yang-Yang function in general case and derive from it the Z[t, t −1 ]-module bundle structure, then state two main theorems in section 2. In section 3, the parameter rotation for Yang-Yang function is considered. Four types of variations of critical point of Yang-Yang function are studied in detail. For the fundamental representation of classical complex simple Lie algebra, general wall-crossing formula and monodromy representation are derived. The first main theorem follows. In section 4, we discuss the relation between two monodromy representations of c = 0 and c → +∞, which leads to a proof of the second main theorem.
2 Yang-Yang function, monodromy of its associated bundle and main theorems

Yang-Yang function is defined by
It is a function of complex variables w = (w 1 , ..., w l ), distinct complex parameters z = (z 1 , ..., z m ), weights λ and n partition l of l. To each z a and w j , a dominant integral weight λ a ∈ P + and a primary root α ij are associated respectively, where i j ∈ {1, ..., n}. Let l k = #{j | 1 ≤ j ≤ l, i j = k} be the number of α k in {α ij } j=1,...,l . Denote by α(l) the sum of all these primary roots α(l) = l 1 α 1 + ...l n α n . Then W (w, z, λ, 0) is a constant function of w.
The critical points of W (w, z, λ, l) satisfy ∂W (w, z, λ, l) ∂w j = 0, j = 1, ..., l equivalently, By definition, if w is a solution of ∂W ∂wj = 0 and (α ij , α is ) = 0, then w j = w s . Also if (α ij , λ a ) = 0, then w j = z a . Under the permutation of all the coordinates w j associated with the same primary root, the critical point equation above is invariant, thus we do not distinguish these critical points.
The critical points of Yang-Yang function have a close relation with the singular vectors of the tensor product space V λ = V λ1 ⊗ V λ2 ⊗ ... ⊗ V λm . Let SingV = {v ∈ V | E i v = 0, ∀i} be the subspace of singular vectors in V . The following fact is known.
If λ 1 +...+λ m −α(l) ∈ P + , for each critical point of Yang-Yang function, Bethe vector can be constructed [9]. Shapovarov form of Bethe vector is proved to be equal to Hessian of Yang-Yang function at the critical point [4]. Thus nondegeneration of critical point guarantees Bethe vector is nonzero. By theorem 11.1 in [10], each nonzero Bethe vector belongs to SingV λ . For details of proof, we refer to [4][9] [10]. The master function in the reference is just the exponential of Yang-Yang function W (w, z, λ, l).
Formally denote by w 0 the critical point of constant functionW (w, z, λ, 0). Theorem 2 Let g be a classical complex simple Lie algebra. If z = (z 1 , z 2 ), λ = (ω 1 , ω 1 ), the nondegenerate critical points of the family of Yang-Yang functions together with the formal degenerate critical point w 0 of W (w, z, λ, 0) one to one correspond to the weight vectors in SingV λ .
Proof By Littelmann-Littlewood-Richardson rule [11], the following direct sum decompositions can be derived for each case of classical Lie algebras.
For B n Lie algebra, the singular vectors of By theorem 1, there is an injective map between the set of the nondegenerate critical points and SingV ω1 ⊗ V ω1 . Corresponding to each singular vector, the explicit solution of the critical point equation can be derived by lemma in section 3.2. They are nondegenerate except when l = 0. Thus the map is bijective on SingV ω1 ⊗ V ω1 \v 2ω1 . When l = 0, the formal degenerate critical point w 0 corresponds to the singular vector v 2ω1 in SingV ω1 ⊗ V ω1 . Therefore, one to one correspondence is proved. For A n , C n , D n , the proofs are similar.
With an additional deformation parameter c ∈ R, symmetry breaking Yang-Yang function is defined by It is clear that W 0 (w, z, λ, l) = W (w, z, λ, l). In fact, when c ∈ Z ≥0 , W c (w, z, λ, l) is just a limitation of the parameter deformation of η with an additional parameter z ∞ = 1 and the dominant integral weight λ ∞ = cρ ∈ P + associated with it.
The critical point equation of W c (w, z, λ, l) is Lemma 1 If w j is the coordinate of the critical point of W c (w, z, λ, l), then lim c→+∞ w j ∈ {z a } a=1,...,m .
Proof Because (ρ, α ij ) > 0, it is clear for any j, Divide the set {w 1 , ..., w l } into a disjoint union of Z 1 , ..., Z m and M , where Z a , a = 1, ..., m contains the coordinates w j which tend to z a and M the rest of them. Assume M = {w m1 , ..., w mp } = ∅, then p > 1. Sum the equations of When c → +∞, the left hand side of (8) is bounded.
where the summation in M is canceled, thus also bounded. (8) leads to contradiction. Therefore, M = ∅. The lemma follows.
Denote also by w 0 the critical point of the constant functionW c (w, z, λ, 0). For z = z 1 and λ = ω 1 , we have the following theorem Theorem 3 Let g be a classical complex simple Lie algebra. If c ∈ Z ≥0 and c ≥ 2, the nondegenerate critical points of the family of Yang-Yang functions together with the formal degenerate critical point w 0 one to one correspond to the weight vectors in V ω1 .
Proof By Littelmann-Littlewood-Richardson rule [11], if c ∈ Z ≥0 and c ≥ 2, the following direct sum decompositions can be derived.
For A n , it is clear that the singular vectors of in V ω1 . By theorem 1, there is an injective map between the set of the nondegenerate critical points and SingV ω1 ⊗ V cρ . Corresponding to each singular vectors, l satisfies the admissible condition and the explicit solutions of critical points with two parameters z 1 = 0, z 2 = 1 are given in [12]. They are nondegenerate except when l = 0. Thus the map is bijective on SingV ω1 ⊗V cρ \v ω1+cρ . When l = 0, the formal degenerate critical point w 0 corresponds to the singular vector v ω1+cρ in SingV ω1 ⊗V cρ . By (6), the critical point of W c (w, z 1 , ω 1 , l) are the limitation of the parameter deformation of those solutions. We give the explicit expressions of them in the lemma 4 in section 3.2.2. Therefore, one to one correspondence is proved. For B n , C n , D n , the proofs are similar.
Remark 1 Except for B n , the direct sum decompositions above are also correct if c = 1 .
For any dominant integral weight λ ∈ P + , there exists a linear automorphism of V λ ⊗ V λ called R-matrix: To study R-matrix, m = 2 is sufficient. In the following sections, we consider the fundamental representation λ = ω 1 of complex simple Lie algebra of classical types. Assume that z 1 and z 2 have the same real part and Imz 1 > Imz 2 .
2.2.1 V λ ⊗ V λ and the thimble space of Yang-Yang functions By lemma 1, when c → +∞, equations of (7) splits into two separated sets involving only z 1 or z 2 and its nondegenerate solutions space is just the tensor product of two nodegenerate solutions spaces with only one parameter z 1 or z 2 . Therefore, by theorem3, when c → +∞, the set of nondegenerate critical points of (7) together with formal critical point w 0 one to one correspond to the weight vectors in V ω1 ⊗ V ω1 .
The thimble of W c (w, z, λ, l), defined [14] as the cycle formulated by the gradient flow of the real part of Yang-Yang function starting from the critical point is a l dimensional manifold. Therefore, for all vectors v ω1−α(k) ⊗ v ω1−α(l) ∈ V ω1 ⊗ V ω1 , there exists a unique thimble of W c (w, z, λ, k + l) denoted by J k,l and J k,l = J k × J l , where J k and J l are thimbles of W c (w, z 1 , ω 1 , k) and W c (w, z 2 , ω 1 , l) respectively. Therefore, we have Theorem 4 When c → +∞, the basis of V ω1 ⊗ V ω1 one to one correspond to thimbles generated from a family of Yang-Yang functions Note that the solution of critical point equation (7) is not unique. It means that there exist different thimbles corresponding to different critical solutions of the same Yang-Yang function. We will see it in the example of section 3.1.

2.2.2
The thimble space as a fiber bundle of Z[t, t −1 ]-module As in [15], [16] and [17], Yang-Yang function appears naturally as an exponent in the correlation function of Wakimoto realization of Kac-Moody algebra at arbitrary level κ : where h ∨ is the dual Coxeter number. The crucial problem is to figure out the transformation induced on the thimble space by parameter deformation , which leads us to study the extra structure of the thimble space.
Thimbles of the real part of Yang-Yang function W c (w, z, λ, l) are the same as thimbles of | e − W c (w,z,λ,l) κ+h ∨ |, except that there are infinite number of pre-images with difference 2πi in their imaginary parts. Along the thimble of its real part, conservation law [14] of the imaginary part of Yang-Yang function implies global invariance of the phase factor e − ImW c (w,z,λ,l) κ+h ∨ i . By theorem 4, when c → +∞, the basis of V ω1 ⊗ V ω1 one to one correspond to the family of thimbles {J k,l } ω1−α(k),ω1−α(l)∈Ωω 1 of {| e − W c (w,z,λ,k+l) κ+h ∨ | } ω1−α(k),ω1−α(l)∈Ωω 1 . Each of them has infinite pre-images with different global phases. Choose a branch from infinite pre-images for each thimble. Define q = e 2πi κ+h ∨ . Continuous deformation of the parameters in e − W c (w,z,λ,l) κ+h ∨ will induce a global phase factor variation along the thimble of | e − W c (w,z,λ,l) are elements of a certain relative homology group and have a natural integral structure [5]. Let Z[t, t −1 ] be the ring of Laurrent polynomials of t over Z, then these variations naturally make thimble space a module over ring Z[t, t −1 ], where t is some minimal power of q during the variation. From now on, we consider generated by the corresponding thimbles, then we have a natural decomposition of -submodule generated by all the q dimensional thimbles of J.
Because J depends on the parameters (z 1 , z 2 ), there exists a bundle J π → X 2 , where is the configuration space of the two complex parameters z 1 and z 2 . The fiber π −1 (z 1 , z 2 ) = J(z) is defined to be a Z[t, t −1 ]-module generated by all the thimbles of {| e − W c (w,z,λ,k+l) κ+h ∨ |} ω1−α(k),ω1−α(l)∈Ωω 1 . Let E q π q → X 2 be the Because of (11), we have the following decomposition of the Z[t, t −1 ]-module bundle.
In sum, the thimble space of a geometric realization. We will use it to derive R matrix.

Monodromy representation of the Z[t, t −1 ]-module fiber bundle
Fixing a point P = (z 1 , z 2 ) ∈ X 2 , we consider a continuous parameter transformation T (s) : X 2 → X 2 defined by It is a clockwise rotation around the middle point of z 1 and z 2 : Thus, T : S 1 → X 2 generates a fundamental group π 1 (X 2 , P ) of X 2 with base point P and also induces a Z[t, t −1 ]-module transformation σ is called monodromy of the bundle and it generates a monodromy group. Its representation on the fiber space is called monodromy representation. Let B : V ω1 ⊗ V ω1 → V ω1 ⊗ V ω1 be the monodromy representation of the bundle J π → X 2 induced by T : S 1 → X 2 and B Uh(g) the braid group representation induced by the universal R-matrices of the quantum enveloping algebra U h (g). Our first main theorem is as following: Theorem 5 For the fundamental representation V ω1 of classical complex simple Lie algebra g, the monodromy representation B Y Y of the Z[t, t −1 ]-module fiber bundle J π → X 2 generated by the family of functions

Remark 2
The advantage of considering thimbles as the basis for Z[t, t −1 ]module V ω1 ⊗ V ω1 is that the imaginary part of holomorphic function is conserved along the thimble defined by the gradient flow of the real part of the function [14]. When the parameter varies, there is a global variation for the thimble, which can be extracted from the critical point associated with it. Therefore, it is convenient to calculate the monodromy of J π → X 2 valued in Z[t, t −1 ] only by considering the variation of the critical point of Yang-Yang function.
Denote by J 0 µ → X 2 the thimble space generated by the family By theorem 2, it is a fiber bundle of Z[t, t −1 ]-module with its fiber J 0 (P ) isomorphic to SingV ω1 ⊗ V ω1 . Denote also by B Y Y the monodromy representation on it induced by clockwise rotation T (1) in equation (13). In section 4, we will define transformation S : J 0 (P ) → J(P ) induced by the parameter deformations c → +∞ from 0. Our second main theorem is as following:

A simple example of wall-crossing phenomena
The key for the derivation of the monodromy representation is to figure out the wall-crossing formula. To illustrate it, we start from an example of A 1 Lie algebra g = sl(2, C), dim V ω1 = 2 and Ω ω1 = {ω 1 , ω 1 − α}. The inner product on the weight space is Its critical point equation is which has two solutions w 1 (c) and w 2 (c) for c ≥ 2. Assume With large c, let J 1,0 and J 0,1 be two thimbles associated respectively to w 1 (c) and w 2 (c). The continuous clockwise transformation T (1) induces a continuous deformation on the thimble J 1,0 : is from the phase factor difference of the critical values and it is equal to the phase factor difference of(z 1 − z 2 ) κ+h ∨ under the 1 2 clockwise rotation. The transformation of J 0,1 is more interesting. In the process of clockwise rotation T (s), z 1 will pass through J 0,1 from the right hand side of z 2 , when s = 1 2 , the imaginary parts of two critical values equals: There is a gradient flow connecting w 2 to w 1 as shown in figure 14 of [5]. The homotopic class of J 0,1 after deformation is equivalent to a zig-zag Z[t, t −1 ]linear combination of J 1,0 and J 0,1 Because in general situation thimble is of high dimension, it is convenient to just draw the variation of its critical point along the homotopic class of the thimble after rotation, rather than to draw the thimble itself. Here, the zig-zag thimble has three parts and the relations of their critical points are shown in figure 1. In the following, whenever we draw the figure of the variation of the critical point, we are showing the relations between the critical points of the different thimbles in the homotopic class of the thimble after the deformation.
. As shown in figure 1, b differs from a by an additional movement of the critical point from the right of z 2 to the right of z 1 and the negative sign before b is from the orientation reverse of the thimble. c differs from b by an anti-clockwise rotation around z 1 . Thus, The minimal phase factor under T is t = q 1 4 and therefore the monodromy is valued in C[q 1 4 , q − 1 4 ]. In sum, The phenomena is called wall-crossing phenomena and the formula (17) describing it is called wall-crossing formula. Terms with coefficients b and c are wall-crossing terms. The following properties are clear. Firstly, because critical points of the thimbles in the wall-crossing phenomena are different solutions from the same Yang-Yang function, thus the types and total number of primary roots will not be created or annihilated in the process of wall-crossing, but only be transfered from one point to another point. We call this property conservation law of wall-crossing. Secondly, clockwise transformation T make a constraint on the direction of the primary roots transfer: primary roots only move in the direction of positive real axis from z 2 to z 1 . Based on the above two properties, E q is a σ invariant sub-module. The monodromy representation B is naturally decomposed into a direct sum of the sub-representations on E q and all matrices of the sub-representation are triangular and valued in Z[t, t −1 ].

Variations of critical points, wall-crossing formula and monodromy representation
To derive wall-crossing formula, it is necessary to analyze the variation of critical points under the transformation T . As in the previous example, we focus on the homotopic class of the thimble after deformation T and see how the primary roots move from z 2 to z 1 . For the fundamental representation V ω1 of g ∈ A n , B n , C n , D n , we conclude as following four types of variations of critical points during wall-crossing and the details of derivation can be found in each case in section 3. of the finite dimensional irreducible representation of A 1 or in the case of B n Lie algebra. Type II Coordinates of the critical point near z 2 have the same imaginary parts formulated as a straight line paralleled to the real axis. The variation is just translation as shown in figure 3. The homotopic class of the thimble after deformation is equivalent to three parts. There is only one dimension in the sub-thimble reversing its orientation, thus the sign before the coefficient b is always minus in this case. Type III Coordinates of the critical point near z 2 have the same imaginary parts formulated as a straight line paralleled to the real axis. The variation from z 2 to z 1 as shown in figure 4 has a clockwise self rotation of π, then followed by a translation. In this case, the orientation of each dimension of the sub-thimble connecting z 2 to z 1 is reversed. Thus, the sign before b is (−1) j , where j is the dimension of the sub-thimble or the number of primary roots moving from z 2 to z 1 . Opposite to the sign before b, it is (−1) j+1 before c . Type IV Coordinates of the critical point near z 2 have the same imaginary parts formulated as a straight line paralleled to the real axis. The variation leaves a trace along the homotopic class of the integration cycle like a "snake" keeping the relative order of the coordinates invariant during the moving. The coefficient b is from the variation of primary roots moving from z 2 to z 1 . c 1 , c 2 and c 3 differs from b respectively by 1, j − 1 and j primary roots rotating around z 1 and other primary roots near z 1 in a specific manner, as is shown in the pictures of figure 5. Thus, the sign before b, c 1 , c 2 and c 3 are Fig. 5 The origin of c 1 , c 2 and c 3 of type IV.
In the following, we derive the monodromy representations for A n , B n , C n and D n respectively.
Its critical point equation is as following: It is obvious that z and the solution , the coordinates of the critical point are distributed respectively along two horizontal straight lines started from z 1 and z 2 on the W plane, as shown in the first picture of figure 6.
When i ≥ j, there is no wall-crossing under the transformation T .
When i < j, the variation is moving primary roots {α i+1 , α i+2 , ..., α j } from z 2 to z 1 . Let w a k , a = 1, 2 be the coordinates of α k . The type of variation can be seen from the deformation of parameter c → +∞ from c = 0. The critical point equation with two singularities w 1 i and w 2 i is as following: For c = 0, , Note that the process above is independent of the position of w 1 i and w 2 i . As shown in figure 7, when c → +∞, the coordinates of the critical point at c = 0 are continuously moving to w 1 i or w 2 i within the same horizontal line on W plane. In the process of continuous transformation T (s), when s = 1/2, the imaginary parts of two critical values equals. Thus there will be two thimbles connecting w 2 i to w 1 i from above and below z 1 and they are homotopically equivalent to the thimbles at c = 0 with Imw 1 i < Imw 2 i and Imw 1 i > Imw 2 i respectively. The variation of {w i+1 , ..., w j } is just a translation and thus a type II variation. As shown in figure 8, the homotopic class of BJ i,j is equivalent to Z[t, t −1 ]-linear combination of three parts i.e. that of (z 1 −z 2 ) (λ i ,λ j ) . Comparing with a, b has an additional phase factor q 1 2 (λ i ,λ i −λ j ) of translating {α i+1 , α i+2 , ..., α j } from z 2 to z 1 . c differs from b by translation of {α i+1 , α i+2 , ..., α j } around z 1 and {α 1 , α 2 , ..., α i }. Thus Therefore, we get the following wall-crossing formula for V ω1 :

B n
Denote the weights of the fundamental representation of B n Lie algebra by Their inner products are: where s, t = 0, 1, .., 2n. For any weight vector v λ l ∈ V ω1 , the corresponding Yang-Yang function is as following: where i j = j, j ≤ n; 2n − j + 1, n < j ≤ 2n.
By this lemma, the following property is clear.
Lemma 5 ∆ k > 0 for k = 2n + 1 − l, ..., n − 1 and ∆ n < 0. The coordinates of critical solutions satisfy the following order: When l ≤ n, 0 < w 1 < w 2 < ... < w l ; When l ≥ n + 1, assume w k < w 2n+1−k for k = 2n + 1 − l, ..., n − 1, then  If the critical points of W c (w, 0, ω 1 , l) are w, then the critical points of W c (w, z, ω 1 , l) are z + w. By the previous lemma, it is obvious that z and {w k } k =n,n+1 are on the same horizontal line of W c plane except w n and w n+1 vertically symmetrical about the center pointw n 2 . In sum, we draw the distribution of the critical point near z corresponding to v λ i ∈ V ω1 in different cases in figure 9.
In the following ,we derive monodromy representation for v λ i ⊗v λ j ∈ V ω1 ⊗ V ω1 . it is convenient to consider the case i + j = 2n firstly.
i)i + j = 2n&i ≥ j There is no wall-crossing, ii)i + j = 2n&i < j&i = n By the same method used in the case of A n , the variation is of type II.
Proof The equations imply . By this lemma, the following critical point equation with c = 0 can be solved: where w 1 n , w 2 n are coordinates of α n and w 1 n−1 , w 2 n−1 coordinates of α n−1 . The solution isw n =w n−1 , w 1 n w 2 n = (wn−1) 2 −w 1 n−1 w 2 n−1

3
. If Rew 1 n−1 = Rew 2 n−1 , then ∆ n > 0. The thimble and its critical point are shown in figure 10. When c → +∞, the condition i = n&j = n + 1 corresponds to the case of w 1 n , w 2 n tending to w 2 n−1 . After rotation of w 1 n−1 and w 2 n−1 , the homotopic class of the thimble connecting w 2 n−1 with ∞ is shown in figure 11. Therefore, as is shown in figure 12, the variation here is of type I with only one primary root α n moving. αn) ) is the sum of phase factors of moving one of two α n to the right of z 1 and c 1 + c 2 = (b 1 + b 2 ) · q −(λ n−1 ,αn)+ 1 2 (αn,αn) from anti-clockwise rotation of α n , 2π around {z 1 , α 1 , ..., α n−1 } and π around α n . Minus before b 1 and b 2 is from the reverse of the direction of the dimension one sub-thimble connecting z 2 to z 1 . Thus, iv)i + j = 2n&i = n&j > n + 1 The variation is combination of type I and II. As shown in figure 13, similar to the previous case, there are two possible way of moving α n from z 2 to z 1 , but α n is now accompanied by {α n−1 , ..., α 2n+1−j } horizontally.
Minus is always from reverse of orientation of the dimension one sub-thimble. In sum, when i + j = 2n, When i + j = 2n, the situation is more interesting. The critical point equation of two singularities z 1 and z 2 with c = 0 is as following: Summing up all the equations except the first pair gives By using lemma 6 inductively,w 1 = ... =w n = z 1 + z 2 . Then substitutingw l into (37), we have the following solution: It is clear that ∆ k < 0, k = 1, ..., n − 1 and ∆ n > 0. The distribution of the coordinates of critical point is shown in figure 19. Similar to the case of one singularity in figure 9, the coordinates are symmetric on the same line connecting z 1 and z 2 except w 1 n and w 2 n vertically in the middle. Assume Imw 1 i > Imw 2 i , i = 1, ..., n − 1. The different imaginary parts of {w 1 i , w 2 i } and z 1 , z 2 give the coordinates a partial order: Because of the existence of the thimble above, when c → +∞, there is a wallcrossing for every i = 0 during the transformation T and its homotopic class will keep this order.
To be precise, we redefine the index i and j. Let i be the number of primary roots near z 2 and j be the number of primary roots crossing wall. Assume that then from the properties of wall-crossing, Phase factor from the rotation gives The coefficients B i−j,2n−i+j 2n−i,i (j > 0) of wall-crossing formula can be computed as in the following cases.
i) 0 < j ≤ i ≤ n − 1 Keeping the order of the coordinates, moving of {α i−j+1 , ..., α i } from z 2 to z 1 is accompanied by the self clockwise rotation of π, thus the variation is of type III.
is from moving of {α n−j+1 , ..., α n } from z 2 to the position indicated by the green dotted circles as shown in figure 14. Additional anti-clockwise rotation to the position indicated by the red dotted circles gives This variation is different from type III by the position of two α n . The variation of α n is of type I. In this sense, we call the variation a combination of type I and III. Thus, iii) i > n&i − j = n As shown in figure 15, because of two α n , there are two possible cases of moving. They give coefficients b 1 and b 2 . αn) ). Additional anti-clockwise rotation around z 1 and other 2n − i primary roots gives c 1 + c 2 = (b 1 + b 2 ) · q −(λ 2n−i ,λ i−j −λ i ) . The variation is also a combination of type I and III. Thus, is exactly the same as (41).
The variation is of type IV . i − j = 2n − i means that the moving primary roots are of double multiplicity and symmetrically distributed. Same as type III, moving of them from z 2 to z 1 is accompanied by the self clockwise rotation of π. But because of double multiplicity and symmetric distribution, the order of these primary roots will give three additional terms in the wall-crossing formula, as is shown in figure  5.
The variation is of type I.
In summarization, Monodromy representation is as following: We have given the description for four types of variations in detail. In the following cases of C n and D n , the methods for proofs are similar, so we omit them except for some extraordinary cases.

C n
Denote the weights of the fundamental representation of C n Lie algebra by The inner products of them are: where s, t = 0, 1, .., 2n − 1.
For any weight vector v λ l ∈ V ω1 , Yang-Yang function W c (w, z, ω 1 , l) corresponding to it is as following: where i j = j, j ≤ n; 2n − j, n < j ≤ 2n − 1.
By this notation, when l ≥ n + 1, {w k , w 2n−k } 2n−l≤k≤n−1 are pairs of symmetric coordinates of α k in the function.
Lemma 7 For the fundamental representation V ω1 of C n Lie algebra, the solutions of the critical point equation (7) of the corresponding Yang-Yang functions W c (w, 0, ω 1 , l) are as following: When l < n, When l = n, .
By this lemma, z and the critical coordinates {w j } j=1,...,l of W c (w, z, ω 1 , l) are on the same horizontal line of W plane as shown in figure 16.
i) When i + j = 2n − 1&i ≥ j, there is no wall-crossing, ii) When i+j = 2n−1&i < j, the variation of critical point in wall-crossing is of type II.
In sum, When i + j = 2n − 1, the critical point equation of two singularities z 1 and z 2 with c = 0 is as following: ...
where 2 ≤ k ≤ n − 2. By lemma 6, its solution is as following: The distribution of the coordinates of critical point is shown in figure 20. Similar to the case of one singularity in figure 16, the coordinates are symmetric on the same line connecting z 1 and z 2 , which gives coordinates of the thimble and singularities z 1 and z 2 an order: Because of the existence of the thimble above, there is a wall-crossing whenever i = 0 and its homotopic class will keep this order. For can be computed as in the following cases.
In sum, Monodromy representation is as following:
By this notation, when l ≥ n + 1, {w k , w 2n−1−k } 2n−1−l≤k≤n−2 are pairs of symmetric coordinates of α k in the function. Definew k = w k + w 2n−1−k and Lemma 9 For the fundamental representation V ω1 of D n Lie algebra, the solutions of the critical point equation (7) of the corresponding Yang-Yang functions W c (w, 0, ω 1 , l) are as following: When l ≤ n − 1, When l = n − 1 , When l = n, When l ≥ n + 1, and for 2n − 1 − l ≤ k ≤ n − 2.
By this lemma, z and the critical coordinates {w j } j=1,...,l of W c (w, z, ω 1 , l) are on the same horizontal line of W plane as shown in figure 17.
When l = 2n − 2, the critical point equation of two singularities z 1 and z 2 with c = 0 is as following: where 2 ≤ k ≤ n − 3. By lemma 6, the solution is as following: The distribution of the coordinates of critical point is shown in figure 21. Similar to the case of one singularity in figure 17, the coordinates are symmetric on the same line connecting z 1 and z 2 , which gives coordinates of the thimble and singularities z 1 and z 2 a total order: Because of the existence of the thimble above, there is a wall-crossing whenever o(i) = 0 and its homotopic class will keep this order. Firstly, we consider the case v o(l) > n + 1 Fig. 17 Coordinates distribution of the Dn critical point on W c(w, z, λ, l) plane near z .
There is no wall-crossing, The variation is of type II.
The variation is of type I. As is shown in figure 12, is the sum of phase factors of moving one of two α n to the right of z 1 .
The variation is a combination of type I and II. As shown in figure 13, there are two possible ways of moving α n from z 2 to z 1 . But at this time, α n is followed by {α n−1 , ..., α 2n+1−j }.
In sum, For v λ i ⊗v λ j ∈ V ω1 ⊗V ω1 , o(i)+o(j) = 2n−1, the coefficients of equation Let j > 0 be the difference of the order, then B can be computed as in the following cases.
The variation is of type III and j is the number of moving primary roots.
The variation is of type III. At this time, the number of primary roots moving from z 2 to z 1 is not j, but j − 1. The sign before b is (−1) j−1 and the phase factor from −π self rotation of j − 1 primary roots equals to q j−2 The variation is of type IV . The number of primary roots moving from z 2 to z 1 is j − 1. Thus, The variation is of type III. The number of primary roots moving from z 2 to z 1 is j − 1. Thus, The variation is of type III. The number of primary roots moving from z 2 to z 1 is j − 1. Thus, Monodromy representation is as following:

Similarity transformation Q
By comparing the monodromy representation with the braid group representation induced by the universal R-matrices (see 7.3B and 7.3C of [6]) of quantum For C n and D n , Q = This proves the first main theorem.

Commutation of two parameter deformations
In the previous section, we have studied the transformation T (1) of parameters z 1 and z 2 . In this section, we consider the continuous deformation of the real parameter c and its relation with T (1). As shown in theorem 2, for the fundamental representation V ω1 of each classical complex simple Lie algebra, the lowest weight vectors in SingV ω1 ⊗ V ω1 are For A n , the Yang-Yang function with c = 0 corresponding to the singular vector v 2ω1−α1 is W (w, z, λ, 1) = 2 a=1 (α 1 , ω 1 ) log(w 1 − z a ) − (ω 1 , ω 1 ) log(z 1 − z 2 ).
Its unique critical point is w 1 = z1+z2 2 . For B n , C n and D n , the corresponding critical point equations (37) (57) (73) are already solved. Thus, corresponding to the lowest weight vector in SingV ω1 ⊗ V ω1 of each classical Lie algebra, there exists an unique thimble J ⊂ J 0 (P ) for e − W k+h v connecting z 1 and z 2 . The distributions of the coordinates of the critical point on W plane are shown in figures 18, 19, 20 and 21 respectively. As shown in section 2.3, J 0 (P ) is a Z[t, t −1 ]-module. Thus, J naturally generates a one dimensional Z[t, t −1 ]submodule of J 0 (P ). We denote also by B Y Y the monodromy representation induced by T (1) on the one dimensional Z[t, t −1 ]-submodule generated by J.
Although theorem 3 demands that c ∈ Z ≥0 , the thimble structure can be defined continuously on c ≥ 0, thus we can consider the continuous deformation of c → +∞ from c = 0. For any J ⊂ J 0 (P ), when c → +∞, by lemma 1, the coordinates of its critical point tend either to z 1 or to z 2 , thus it gives several possible thimbles in J a,b ⊂ J(P ). Multiply to each thimble J a,b the phase factor generated from the deformation and sum them up. Define the symmetry breaking transformation S by where coefficients e a,b are the phase factor difference of e − W (w,z,λ,l) κ+h ∨ in the process of c → +∞ from 0. For J ⊂ J 0 (P ) corresponding to the lowest weight vector in SingV ω1 ⊗ V ω1 , by a direct calculation, we have By the monodromy representation B c,d a,b in section 3.2 and the lemma above, it is straight forward to prove that e a,b is the eigenvector of the monodromy representation.

Lemma 13
where d is defined in (90).
By this lemma, The second main theorem 6 follows. From the equation (91), it is natural to define a Z[t, t −1 ] linear operator We can also define a creation matrix associated with M as M a,b = e a,b .

Annihilation matrix M a,b is its inverse satisfying
With these two matrices, quantum trace of any (m, m) tensor T j1j2...jm i1i2...im is just as following: T r q T = i1,i2,...im,j1,j2,...,jm T j1j2...jm i1i2...im η i1 j1 η i2 j2 ...η im jm , where η i k j k = l M i k ,l M l,j k . By Alexander's theorem (p91 I.7 of [18]), the ambient knots invariants defined by contraction of B c,d a,b , M a,b and M a,b in the decomposition of some knot projection diagram coincide with the quantum trace of the (m, m) tensor associated with the braid. In [14] and [19], knots invariants associated with the fundamental representation of A n Lie algebra and B n , C n , D n Lie algebra are proved to be HOMFLY-PT polynomial and Kauffman polynomial respectively.
Remark 3 Knots invariant depends on representations of Lie algebra. Different representations may give different knots invariants. Generally, the corresponding knots invariant is not necessary to be HOMFLY-PT polynomial or Kauffman polynomial. We will focus on this point elsewhere.