Minuscule ABCDE Lax Operators from 4D Chern-Simons Theory

Using 4D Chern-Simons (CS) theory with gauge symmetry $G$ having minuscule coweights, we develop a suitable operator basis to deal with the explicit calculation of the Lax operator of integrable spin chain satisfying the RLL equation. Using this basis, we derive the oscillator realisations of the full list of the minuscule L-operators which are classified by the gauge symmetries A$_{N}$, B$_{N}$, C$_{N}$, D$_{N}$, E$_{6}$, E$_{7}$. We also complete missing results regarding the non simply laced $SO_{2N+1}$ and $SP_{2N}$ gauge symmetries and comment on their intrinsic features. Moreover, we investigate the properties of links reported in Yangian spin chain studies between the (A-,D-) Lax operators and the (C-,B-) homologue. We show that these links are due to discrete outer-automorphism symmetries that are explicitly worked out.


Introduction
Few years ago, a four dimensional (4D) topological gauge theory with complexified gauge symmetry G [1,2] has been proposed to be a mother theory of lower dimensional integrable systems such as quantum 1D integrable spin chains of statistical mechanics [4]- [5] and 2D integrable QFTs [6]- [8]. This 4D topological gauge theory is a tricky extension of the usual non abelian 3D Chern-Simons theory [3] with observables given by topological line defects such as the electrically charged Wilson lines and the magnetically charged 't Hooft lines. The 4D Chern-Simons (CS) theory has been linked with N = (1, 1) supersymmetric Yang Mills theory in 6D [9]- [11] and supersymmetric quiver gauge theories [12]- [15]. The observables of the 4D CS theory have been also realised in terms of M2/M5-branes and also in terms of intersecting NS5/Dp-branes in type II strings [16]- [23].
Recently, it has been shown in [24,25] that the coupling of a 't Hooft line with N perpendicular Wilson lines in the four-dimensional theory generates an integrable quantum spin chain [26]. The crossing matrix describing the coupling of an electrically charged Wilson and a magnetically charged 't Hooft line coincides precisely with the Lax-operator L of integrable systems [27]. In this 4D Chern-Simons (CS) setup, a beautiful formula for calculating the crossing matrix L has been derived for the sub-family of 't Hooft lines whose magnetic charges are given by minuscule coweights µ of the gauge symmetry G. In these regards, well known results from the literature of quantum spin chains [28,29] have been nicely recovered from the point of view of the 4D Chern-Simons theory and partial findings including QFT modelings and their interpretation have been completed [25]- [31]. In fact, the oscillator realisation of Lax matrices of A-and D-types derived from CS theory solving the RLL equations were shown to reproduce, up to a spectral parameter scaling, their homologue derived by using analytic Yangian based methods [32,33]. Moreover, new results such as the L-operators for minuscule representations of the exceptional symmetries E 6 and E 7 were generated from the CS gauge theory analysis in [34]. Notice that the derivation of Lax-matrices for the exceptional groups from the algebraic integrability methods is still missing in the quantum integrable spin chain literature using Yangian algebra.
In this paper, we contribute to the 4D Chern-Simons theory with two main objectives. The first objective aims at (i) the development of a suitable method for the explicit computation of the L-operator of integrable systems; and (ii) to complete partial results in the study of Lax operators in the framework of the CS theory with gauge symmetry G. A particular interest is also devoted to the missing analysis concerning the CS gauge theory with non simply laced B-and C-gauge symmetries and to the derivation of the associated harmonic oscillator representation of the Lax operators L B and L C . Moreover, by using the 4D CS approach, we identify the source behind similarities reported in the integrable spin chain literature between B-and D-Lax operators and between C-and A-types Lax operators. These links are shown to be due to discrete symmetries of the concerned L G 's whose cause is given by known outer-automorphism symmetries of the root system of the gauge symmetry G. We also show that our results based on CS theory are in perfect agreement with the quantum spin chains' calculations obtained recently using Yangian algebra [33]. The second objective is to calculate the complete list of all minuscule Lax operators L G . This list is presented into a unified set (see Tables 2, 3 and section 3) in order to provide to the interested reader a summary tool on the L G 's for easy use and also for a suitable parametrisation for further applications. In this regard, we recall that the minuscule Lax operators as formulated in the 4D Chern-Simons theory are classified by gauge symmetries G given by the series A N , B N , C N , D N , and the exceptional E 6 and E 7 . For a unified description of the operators L G , we revisit the explicit construction of the harmonic oscillator realisations of these L G 's and we investigate their properties and the relationships outlined in literature. The organisation of this paper is as follows: In section 2, we revisit aspects regarding the Lax-operator of integrable systems from two points of view: First, from the view of integrable spin chains method, as a matter of completeness; and second from the novel view of 4D CS theory. We also indicate the main way to follow in order to reach our goals. In section 3, we give the full list of minuscule L G -operators calculated from the 4D Chern-Simons theory with gauge symmetry G. This list concerns the minuscule A N , B N , C N , D N families and the exceptional E 6/7 with electrically charged Wilson lines taken in the fundamental vector-like representation. Line in spinor-like representation are investigated in the discussion section. For unified notations, we revisit all the calculations by using standard writings of Hilbert quantum states. In section 4, we give the explicit derivation of the L Boperator of the non simply laced SO 2N +1 gauge symmetry and give a comparison with the L D -operator of the simply laced SO 2N theory. In section 5, we derive the L C -operator of the non simply laced SP 2N series and comment on its links with the Loperators of the simply laced SO 2N -and SL 2N -gauge theories. Section 6 is devoted to conclusion and discussions.

Lax operator families in integrable systems
In this section, we revisit the construction of minuscule Lax operators L G for low dimensional integrable systems with symmetry Lie group G and refine aspects towards L G . We recall basic tools on ABCDE symmetries which are used in the forthcoming sections and also investigate the way to the minuscule L G 's. These L-operators are classified by the symmetry groups G having minuscule coweights. Their matrix representations have an interpretation as a matrix couplings of two topological lines as shown by the Figure 1. Figure 1: The operator L G describing the matrix coupling between a horizontal 't Hooft line (tH µ ξ 0 in red) at z=0 and a vertical Wilson line (W R ξ z in blue) at z with incoming i| and out going |j states. The spectral parameter z is interpreted in 4D CS theory as a point in the holomorphic plane.
As one of the main objective of this study is to give the full list of L G s, we think it interesting to: (1) describe here those basic quantities by using physical language and short paths for their properties. (2) emphasize the power of the QFT method of [1,2] compared to the standard algebraic approach based on the Yangian algebra representations [35]. (3) draw an overview on the minuscule operators L G and anticipate the construction by giving some results that will be derived rigorously later in this paper.

Families of minuscule L-operators
We begin by recalling that, in addition to the symmetry Lie group G, the minuscule Laxoperators L G of integrable 1D quantum spin chains and 2D integrable QFT systems are also classified by the minuscule coweights µ of finite dimensional Lie algebras. These minuscule coweights are quite well known in the mathematical literature on Lie group symmetries [36]; their useful properties for physical applications are as collected in the following Table 1. In Table 1: The list of the fundamental minuscule coweights µ of finite dimensional Lie algebras g. The Levi-decomposition of these algebras g with respect to the minuscule coweights is given by n − ⊕ l µ ⊕ n + . Here, the l µ refers to the Levi-subalgebra and n ± to the nilpotent subalgebras.
this classification table, we have also reported the Levi-subalgebra l µ associated with the Lie algebra g underlying the symmetry group G. As well, we have given the nilpotent n ± subalgebras accompanying l µ and which turn out play an important role in the construction of the L µ G 's. From this global table, we learn amongst others that there exist five families of minuscule operators L µ G denoted below as follows As exhibited on the Table 1, the four first operators L µ G constitute four infinite families L µ Gn labeled by the positive integer n (rank of G) and by the minuscule coweight µ of G. The fifth family in the Table 1 is finite, it concerns the exceptional L µ E 6 and L µ E 7 . Because a given family may have more than one minuscule coweight; say µ k labeled by an integer k, it results that the classification of the minuscule L-operators is given by two integers: (i) The rank n G of the gauge symmetry G n with Lie algebra g n . (ii) The number n µ of minuscule coweights µ k for each gauge symmetry G n . Moreover, knowing that the minuscule coweights µ i of finite Lie algebras are intimately related with their simple roots α i as shown by the following duality relation it follows that the L µ G 's can be put in correspondence with the Dynkin diagrams of the finite dimensional Lie algebras. In this regard, notice that as far as these Dynkin diagrams are concerned, those classical ones are given by infinite series A n , B n , C n , D n as depicted by the Figure 2. Similar comments can be said about the exceptional ones; especially the E 6 and E 7 which are relevant for the present study. These diagrams have nodes labeled by the simple roots α i of G; and links l ij given by their non trivial intersections α i .α j . Notice also Figure 2: Classification of minuscule Lax operators in terms of the Dynkin diagrams of simple groups with minuscule coweights. Here, we have given the families A n = sl n+1 , B n = so 2n+1 , C n = sp 2n and D n = so 2n . that formally speaking, the Dynkin diagrams considered here are graphic representations of the Cartan matrices K ij (G) of finite dimensional Lie algebras underlying gauge symmetries of QFT's. These intersection matrices have particular integer entries with det K > 0; it reads in terms of the simple α i 's and their co-roots α ∨ j as follows, where the scalar product α i .α i refers to the length of the root. It is equal to 2 for the simply laced ADE Lie algebras; thus leading to a symmetric matrix K ij . This feature does not hold for the non simply laced BC Lie algebras to be also investigated later.
Returning to the minuscule Lax operators L µ G ; they are associated to the Dynkin diagrams K ij (G) whose minuscule node µ is omitted. As illustration, we give in the Figure 3 four examples regarding the cutting of the minuscule node in the Dynkin diagram. The first example is given by the omission of the first node α 1 in the Dynkin diagram of sl (9). This omission breaks the diagram K (A 8 ) into two pieces given by with the isolated α 1 corresponding to K (A 1 ) and the seven others to K (A 7 ). The second graph in the Figure 3 represents the omission of the fourth node α 4 of the Dynkin diagram of sl (9). This cutting leads to the breaking of K (A 8 ) into three pieces with graphs as where the isolated α 4 corresponds to K (A 1 ) and the two other pieces to K (A 3 ) and K (A 4 ). These two ways of cutting the Dynkin diagram of K (A 8 ) describe two different Lax operators L µ 1 sl 9 and L µ 4 sl 9 in the sl 9 theory. Regarding the other two graphic examples in the Figure 3, they concern the orthogonal so(18) Lie algebra. The two cuttings correspond to the following graph decompositions The first preserving the orthogonal structure of the broken diagram as shown by the K (D 8 ) part. The second cutting destroys this feature since we have K (A 8 ). As for the two previous example, eqs(2.6-2.7) describe two different Lax operators in so(18) theory namely L µ 1 so 18 and L µ 9 so 18 . In conclusion, a general classification of the minuscule L µ G -operators describing Figure 3: Four examples of graphs describing minuscule Lax operators. The first example concerns cutting the first node in A 8 . The second example regards the cutting of the fourth minuscule node in A 8 . These two graphs describe two different Lax operators. The third and the fourth graphs deals with Lax operators classified by D 9 .
crossing Wilson and 't Hooft lines with unit electric and unit magnetic charges is given by the    Table 2. It should be noted here that these realisations of the L µ Gn 's are derived with details in section 3. Notice moreover, that some of the results reported in this table are completely new, in particular those regarding L µ B and L µ C which are further investigated in sections 4 and 5.
2.2 Two approaches for constructing the L µ G n 's There are two basic approaches to construct the minuscule-Lax matrices of lower dimensional integrable systems. (1) by using Yangian algebra representations [33]. (2) by following the 4D Chern-Simons gauge theory approach. To fix the ideas, we review below the main lines of these two methods.

Algebraic Yangian approach
Here, we describe how the Yangian algebra naturally arises in the framework of the Yang-Baxter equation and its RLL representation whose its solution leads to the Lax-operator we are interested in. We start by recalling that the Yangian algebra is an infinite-dimensional Hopf algebra giving a simple example for quantum groups. Below, we consider the case of homogeneous rational spin chains with A-type symmetry and we illustrate through a short way the key steps for building the Lax-matrix L gln (z) in this setup. We start from the usual Yang Baxter equation (YBE) given by [2] where the R-matrix R ij (z i − z j ) depends only on the spectral parameter z. This matrix acts on the tensor product of the two vector spaces V i ⊗ V j ; it describes the coupling of two particles' worldlines with inner spaces V i and V j . By setting u = u 1 − u 3 and v = u 2 − u 3 as well as u − v = u 1 − u 2 , we can bring the above YBE to the following form where the number of free parameters is reduced down: (u,v) instead of (z 1 , z 2 , z 3 ), A quasi-classical solution to this equation is given by R 12 (z) = z I + z P 12 [2] where P 12 is the permutation operator acting like P 12 (z 1 ⊗ z 2 ) = z 2 ⊗ z 1 . If we take two spaces in (2.9) as V 1 = C n and V 2 = C n and leave the third V 3 unspecified (auxiliary oscillator space), we recover the RLL relations that read as follows Here, L 13 (u) and L 23 (v) are the Lax-matrices we are interested in; these are n×n matrices constrained by the following commutation relations obtained after substituting with R 12 (z) = z I + z P 12 . [ To get the above L 13 and L 23 , we use the so-called monodromy matrix M(z) with the following properties. It is a n×n matrix operator function of the complex spectral parameter z that (i) satisfies the RLL relations (2.10-2.11) and (ii) allows to write down explicit commutation relations defining the Yangian algebra. By using the canonical matrix generator basis {e ab } with 1≤ a, b ≤n, we can expand M(z) as follows M ab (z) ⊗ e ab (2.12) with matrix elements M ab (z) analytic in the complex spectral parameter z. These matrix entries have the following Laurent expansion [35] M ab (z) = M with min(r, s) designating the minimal integer of the pair (r, s). In what follows, we briefly describe the construction of Lax matrices of A-type by using the Yangian algebra Y(gl n ). This is an algebraic method that yields the oscillator realisation of the A-type Lax operator L(z). We consider the particular case where the expansion (2.13) ends at the first order, the L(z) is therefore taken as ab B 2 where B 1 and B 2 are two invertible n × n matrices that act trivially in the quantum space V 3 . The diagonalM [0] ab reads in general like q aMaa and eventually can be taken as follows with label belonging 1 ≤ p ≤ n. Notice that this choice corresponds to M n coincides with the identity I n . To determine M [1] , we have to solve the two last relations of (2.16) with M [0] p = Π p . To that purpose, we split the label 1 ≤ a, b ≤ n like pairs (α,α) and (β,β) with 1 ≤ α, β ≤ p and p <α,β ≤ n; and then think of M [1] as follows such that A, B, C, and D are respectively p×p, p×(n − p) , (n − p)×p and (n − p)×(n − p) matrices. Next, we substitute (2.17) and (2.18) into the Yangian algebra representation (2.14), we obtain the following commutation relations constraining (2.18), and The last relations (2.22) show that the Dαβ's are central elements of the monodromy algebra. So, assuming det D = 0 we can use the remaining freedom, used in putting M ab B 2 = (Π p ) ab , to also put the Dαβ-matrix as given by δαβ. By substituting into (2.21), we end up with [B αβ , C˙γ δ ] = δ αγ δγβ that is convenient to rewrite like where we have set C ′γ δ = −C˙γ δ . In this way, the B αβ 's are interpreted as oscillator creators and C ′γ δ as the annihilators. Notice that this choice is due to the fact that the algebra (2.21) must admit a definition of a normalized trace over the oscillator algebras. This trace is different from the usual algebraic trace over gl(n); it is needed for the construction of transfer matrices and requires that det D = 0; see [35] for further details. Notice also that the Yangian algebra given by the commutation relations ( Here, the gl(p) algebra is generated by A αγ and H ⊗p(n−p) is the tensor product of p(n − p) copies of the oscillator algebra (2.23) generated by the quantum harmonic oscillators B αβ , C˙γ δ . The generators A αβ solving the constraint relations (2.19-2.20) are realized as follows withĴ αβ being the transpose of J αβ . Therefore the Lax matrix constructed within the Yangian approach reads in terms of these oscillators as This Lax operator serves as an elementary building block for other solutions via the fusion procedure. They are used to build the Baxter operator and the transfer matrices using quantum harmonic oscillators; for details see [35] and references therein.

4D Chern Simons theory
Here, we describe the main lines of the 4D Chern-Simons gauge theory invariant under gauge symmetry groups G with Lie algebra g having at least one minuscule coweight µ. The gauge symmetries considered below are as those given in the Table 2. First, we describe the gauge field action of the CS theory. Then, we give the RLL integrability equation formulated in terms of the Lax-operators and the R-matrix of Yang-Baxter equations.
Topological CS gauge field action: The 4D Chern-Simons gauge theory is a topological QFT that was first obtained in [1]. Its gauge invariant field action S [A] describing the dynamics of the gauge field A in the 4D space, that we take like M 4 = R 2 × CP 1 , reads as follows where Ω 3 is the CS 3-form To fix the ideas, we consider the family of gauge symmetry G = SL N , a similar description is valid for the gauge symmetries given in the Table 2. In this case, the 1-form gauge potential is a function of the variables (x, y; z) parameterising M 4 . It expands like A = t a A a with t a standing for the generators of the Lie algebra sl N and A a a partial gauge connection as The missing component dzA a z is killed by the factor dz∧ in the integral measure in (2.27). The equation of motion of the potential field A is given by the vanishing gauge curvature This flat curvature property agrees with the topological nature of the CS theory (2.27), it indicates that the 4D CS gauge system is in the ground state with zero energy.
Line defects and observables: To deform the topological ground state, we insert in the 4D CS gauge theory electrically charged Wilson and magnetically charged 'tHooft lines with magnetic charge given by the minuscule coweights. Interesting cases correspond to having interacting Wilson and 'tHooft lines. Examples of such couplings are given by crossing lines as depicted in the   The crossing matrix W R , tH µ describing the coupling of the two topological lines is given by the Lax operator L µ sl N we are interested in here. It is obtained by solving the so-called RLL integrability equations briefly described here below.

RLL equation satisfied by the Lax operators
An interesting way to deal with the properties of the phase space E ph of the classical Lax operator L is to use the graphic representation given by the Figures 4-(a) and (b). In the left picture of the this figure, we think of L µ sl N as a matrix operator L j i = i|L µ sl N |j describing the crossing of a horizontal 't Hoof line with a vertical Wilson line on which propagate i|-and |j -states. The symplectic structure of the phase space of two operators L r j (z) and L s l (w) , with spectral parameters z and w, is given by the RLL relations shown in the Figure 4. These integrability relations (which also hold at the quantum level) are due to the topological invariance of the 4D CS theory and read explicitly as follows [2], Generally speaking, this tensorial relation gives the quantum integrability conditions of the topological system. The four rank object R ik rs (z − w) is the usual R-operator used in the study of Yang-Baxter equation. At the leading order in the -expansion, we have with c ik jl standing for the double Casimir of the gauge symmetry. Putting back into 2.31, we obtain the following classical RLL relations where { * , * } P B stands for the Poisson Bracket. One of the key points about eqs(2.31) is that they are solved by the following minuscule Lax operators This solution first obtained in [25] concerns gauge symmetries G having minuscule coweight µ (here SL N ). The X and Y are special matrices having the typical expansions X = b α X α and Y = Y α c α where the X α 's and Y α 's are generators of the nilpotent subalgebras n + and n − involved in the Levi-decomposition of the Lie algebra of the gauge symmetry with respect to the minuscule coweight [37]. This decomposition is given in this case by, Another interesting point concerning (2.31) is that at the leading order in the -expansion of the R-matrix, one finds that the (b, c) parameters involved in the expansions b α X α and Y α c α satisfy the usual Poisson bracket {b α , c β } P B = δ α β of the symplectic geometry. Then, this classical limit teaches us that: (a) the (b α , c α ) are nothing but the Darboux coordinates (classical harmonic oscillators) of the phase space of the L-operator. (b) the particular solution (2.33) gives an oscillator realisation of the Lax operators like the ones obtained from the Yangian based method of integrable quantum spin chains.

Minuscule L G from CS theory: the full list
In this section, we introduce and develop a suitable operator basis (method of projectors) to deal with the explicit calculation of the Lax operator of integrable spin chain satisfying the RLL equation. We begin by describing the main steps of the derivation of the minuscule Lax operators L G for gauge symmetries G given by the A N , B N , C N , D N , E 6 , E 7 families. Then, we give the full list of the explicit oscillator realisations of these L G s. This list gives a unified description of all minuscule L G s and presents new results concerning the non simply laced families L µ 1 B and L µ N C . Other aspects regarding the relationships between the various L G 's and discrete symmetries are also studied in order to complete the investigation of minuscule L G s.

The calculation of L G : method of projectors
Here we give a suitable calculation method to work out the explicit matrix-representation of the minuscule Lax operator L G . We term this approach as the method of projectors; for the motivation behind this terminology, we refer to eqs(3.1-3.4) given below. We start by recalling that the minuscule Lax operator L G is given by (2.33) namely L µ G = e X z µ e Y . This formula involves three Lie algebraic objects that we comment below: • The adjoint action µ of the minuscule coweight; it belongs to the Lie sub-algebra l µ of Table 1. This l µ results from the Levi-decomposition of the Lie algebra g underlying the gauge symmetry which decomposes like g = n − ⊕ l µ ⊕ n + .
• The two X and Y operators are nilpotent matrices evaluated in the nilpotent subalgebras n + and n − following the Levi-decomposition. The intrinsic properties of X and Y carry data on the electric charge of the Wilson line and the magnetic charge of the 't Hooft line.
To determine the explicit expression of L G , we have to first find the matrix representations of these three objects; then use them to calculate e X z µ e Y . So, given a 4D Chern-Simons theory with gauge symmetry G, we can determine the expression of the minuscule L G solving the RLL equation (2.31) by using (2.33). This is done in three steps as described below: 1) Working out the adjoint action of µ The adjoint action of the minuscule coweight is a charge operator representation µ acting on the Hilbert space H of the Wilson line interacting with the 't Hooft defect. For concreteness, we denote the vector basis of the space H of the quantum states by kets |i as in the Figure  4-(a). These quantum states generating the representation R G of the gauge symmetry run along the Wilson line. This representation R G will be thought of below as given by the fundamental representation of the gauge symmetry.
Because of the Levi-decomposition of the gauge symmetry G, the representation R G splits in turns into a sum of representations R lµ a of the Levi-subalgebra l µ . Formally, we have The charge operator µ is effectively given by these representations; it explicitly reads in terms of projectors ̺ Ra : R G → R a as follows For a traceless representation R G , we get the following constraint relation on the Levicharges m Ra carried by a representation R 2) Solving the conditions of the Levi-decomposition Once the adjoint action µ is known, we move to determining the realisation of the X and Y matrices in the Hilbert space of quantum states of the Wilson/'t Hooft lines. They are obtained by (i) using the expansions X = b α X α and Y = c α Y α ; and (ii) solving the In these Levi-constraint relations, Each generator X α of n + carries a positive unit charge +1 and each generator Y α of n − carries a negative unit charge −1.
3) Calculating the L-operator L G as a bi-polynom in X and Y Because of the nilpotency property of the X α and Y α generators of n ± and commuting properties like [X α , X β ] = 0; the X and Y are as well nilpotent. So, there should exist an order n and an order m such that X k+1 = 0 and Y l+1 = 0. In fact the two orders are equal k = l because of the duality between X and Y . So, the exponentials e X and e Y have finite expansions of the form e Z = I + Z + ... 1 k! Z k . Putting this back into e X z µ e Y , we end up with the oscillator realisation of L G given by Finally, using the properties of the projectors ̺ Ra , we can present L G by a matrix L ab given by Here, we give an example to explain the above steps of the calculation of the Lax-matrix L G ab (3.8). We consider the case of a 4D Chern-Simons theory with gauge symmetry SL(N) in presence of an electrically charged Wilson line W with fundamental electric charge λ 1 as depicted by the Figure 4-(a). this W-line crosses a magnetically charged 't Hooft line with magnetic charge given by the minuscule coweight µ 1 . As far as the Lie algebra A N −1 ≃ sl N of the gauge symmetry SL(N) is concerned, it is interesting to recall some useful mathematical features that we collect in the following table (3.9).
These properties regard the Levi-decomposition of sl N with respect to the minuscule coweight µ 1 namely sl N → sl 1 ⊕ sl N −1 + n + ⊕ n − (3.10) The adjoint action of the minuscule coweights µ 1 is therefore given by The generators X i and Y i of the nilpotent sub-algebras n + and n − solving the Levi-constraint relations are given by The Lax matrix representing the L-operator reads as follows By multiplying this matrix with the factor z 1 N , we get the familiar Lax-matrix obtained in the quantum spin chain literature [27], namelyL

The full list of minuscule L-operators
The list of the full set M of minuscule Lax operators with unit charges is infinite. It contains five sub-families given by the four classical series A N , B N , C N , D N and the exceptional finite E 6 , E 7 . These sub-families o the set M of L-operators are revisited below separately in the given ordering.

A N −1 -type operators L A
First we give the general form of the Lax-matrices L µ k A N−1 for generic fundamentals coweights µ k with label 1 ≤ k ≤ N −1 and label N −1 being the rank of A N −1 . Then, we comment the exotic property of the A N −1 -family regarding the CS theory with gauge symmetry SL 2M ; that is N = 2M.
We start by recalling that the A N −1 Lie algebra series has N − 1 minuscule coweights µ k labeled by 1 ≤ k ≤ N − 1. These coweights are expressed in terms of the weight vector basis The unit vectors e i in the weight vector basis are now on denoted by the kets |i and their duals are represented by the bras i|. The Levi-decomposition of the sl(N) Lie algebra with respect to µ k is given by The adjoint action of the minuscule coweight is realised in terms of the |i weight vectors as For later use, we will also use the convenient notation where we have setī = N + 1 − i. Notice that i +ī = N + 1 such that The charge operator (3.18) is traceless, T r (µ k ) = 0. We often refer to µ k as the Levi-Charge operator. The k (N − k) operators generating the nilpotent algebras n ± are now denoted like X µ k ia and Y ia µ k . The explicit representation of these generators in the Hilbert space of the crossing lines is obtained by solving the Levi-constraint relations (3.5) reading as follows X with labels as 1 ≤ i ≤ k and k < a ≤ N or equivalentlyN ≤ā <k. The Lax-matrix associated with the coweight µ k is obtained by substituting the expansions Here, the b and c refer to the k(N − k) oscillators b iā and k(N − k) oscillators cā i given by We begin by noticing that the examination of eq(3.18) reveals a remarkable property of the coweights µ k . In fact, by making the following Z 2 discrete change, we learn that the corresponding operator charge operators transform as µ N −k → −µ k . This symmetry property indicates that eq(3.21) hides an interesting symmetry property explicitly exhibited by the case where N = 2M. This symmetry property has an interpretation in the language of A N −1 representations and their adjoint conjugates. It also allows to engineer new L-operators from (3.21) inspired by Dynkin diagram folding ideas. We will see later that the fixed point of the folding of (3.21) under the above Z 2 symmetry reading as gives precisely the L-operators L µ C M of the symplectic family SP (2M). In this regard, notice the following properties:   and The Lax-matrix (3.21) associated with the coweights µ M is given by (3.24). We will reconsider these relations when we study the construction of the Lax-matrix for 4D Chern-Simons theory with symplecticSP 2M gauge symmetry (see section 5).

B-type operators L B : new result
Now, we consider the case where the gauge symmetry of the 4D Chern-Simons is given by SO 2N +1 . As shown in Table 1, this gauge symmetry group has one minuscule coweight µ 1 dual to the simple root α 1 . To get more insight into the algebraic properties of L B , we recall some useful properties of the B N gauge symmetry. The SO 2N +1 has 2N 2 roots, half of them are positive and denoted as +α ± ij where i, j = 1, ..., N. The negative ones are given by the opposites that read as −α ± ij . These ±α ± ij 's have two lengths: N (2N − 1) of them have length 2 realised in terms of N weight vector basis {e i } like ± (e i ± e j ) with i = j. The remaining N others have length 1; they are given by ±α + ii and are realised as ±e i . The N simple roots α i of the Lie algebra of SO 2N +1 are given by: (a) α i = e i − e i+1 for i = 1, ..., N − 1 having length 2. (b) α N = e N having length 1. In this basis, the minuscule coweight is given by µ 1 = e 1 obeying µ 1 .α 1 = 1. The Levi-decomposition of the Lie algebra of SO 2N +1 reads as so 2 ⊕ so 2N −1 ⊕ n ± with nilpotent sector as n ± = (2N − 1) ± . It corresponds to cutting the first node α 1 in the Dynkin diagram of B N ≃ so 2N +1 as depicted by the Figure 6. The adjoint action of the minuscule coweight µ is given by where we have for commodity inserted the central terms although q = 0; this is because it contributes to z µ . The ̺ ± = |± ±| and the ̺ i = |i i| are projectors. The matrix realisation of the X i and Y i generators of the nilpotent algebras are as follows We also have the expansions X = b i X i and Y = c i Y i . These relations will be discussed further in section 4. The explicit matrix realisation of the Lax operator L µ 1 B is also derived in section 4; it reads as follows In the basis {|+ , |i , |− } , the entries of the X and Y are respectively given by (2N + 1) × 1 and 1 × (2N + 1) matrix oscillators given by b T = (0, b 1 , · · · , b 2N −1 , 0) and c T = (0, c 1 , · · · , c 2N −1 , 0) .

C-type operators L C : new result
In this case, the gauge symmetry of the 4D Chern-Simons is given by the symplectic group SP 2N having rank N and dimension 2N 2 + N thought of below as N 2 + N(N + 1). It has one minuscule coweight µ N dual to the simple root α N . In this regard, recall that SP 2N has 2N 2 roots, half of them are positive and the others are negative roots. These roots are realised in terms of the e i weight vectors as ±α ± ij = ±e i ± e j (1 ≤ i < j ≤ N) and ±2e i for 1 ≤ i ≤ N. The N simple roots α i of SP 2N are given by e i − e i+1 for i = 1, ..., N − 1 having length 2; and α N = 2e N with length 4. In this basis, the minuscule coweight is given by µ N = 1 2 (e 1 + ... + e N ); it describes the symplectic fundamental representation with dimension 2N. The Levi-decomposition of the Lie algebra of the symplectic gauge symmetry reads as so 2 ⊕sl N ⊕n ± with nilpotent algebras n ± = 1 2 N (N + 1) ± which are given by the symmetric representations of sl N and its conjugate. This corresponds to cutting the first node α 1 in the Dynkin diagram of B N ≃ so 2N +1 as depicted by the Figure 7. The adjoint action µ of Figure 7: Dynkin Diagram of C N illustrated for the example where N = 6. By cutting the first node α N , one ends with the Diagram associated with the Dynkin diagram of the A N −1 subalgebra and a free node α N capturing data on the SO 2 Levi-charge and the nilpotent subalgebras. the minuscule coweight and the matrix realisation of the X iī , X [ij] and Yī i , Y [īj] generators of the nilpotent algebras n ± are given by The Lax-matrix operator L µ N C is explicitly derived in section 5; it reads as follows This Lax-matrix must be compared with (3.24). The entries of X and Y matrices in the basis {|i , |ī } are given by Multiplying (3.33) by z 1/2 , we obtaiñ

D-type operators revisited
For the case of the 4D Chern-Simons with gauge symmetry SO 2N +1 ; we have three minuscule coweights µ 1 , µ N −1 and µ N as shown in Table 1. Therefore, we have three types of minuscule Lax operators. In this basis, the minuscule coweights, satisfying µ 1 .α 1 = 1 and µ N −1 .α N −1 = 1 as well as µ N .α N = 1, are given by The Levi-decompositions of the so 2N Lie algebra with respect to the three minuscule coweights are as follows; see also the Table 1, The three Lax-matrices (3.36) corresponding to these three minuscule coweights are described below.
I) Vectorial coweight: so 2N → so 2 ⊕ so 2N −2 ⊕ (2N − 2) ± In the decomposition of so 2N with respect to µ 1 corresponding to cutting the node α 1 in the Dynkin diagram of the Figure 8, the electric charge of Wilson line is given by the weight of the vector representation 2N which splits as 1 + ⊕ (2N − 2) ⊕ 1 − . Denoting the basis  where q = 0, it has been inserted for convenience. The oscillator realisation of the Lax operator L vect D in the fundamental 2N representation is given by In this relation, we have b T = (0, b 1 , ..., b 2N −2 , 0) and c T = (0, c 1 , ..., c 2N −2 , 0) where the (b i , c i ) are harmonic oscillators associated with the phase space of the vector like D Nsystem.

II) Spinorial coweight
In this case, the minuscule coweight is given by µ N . The Levi-decomposition is given by so 2 ⊕ sl N ⊕ N (N − 1) ± , it results from cutting the N-th node α N in the Dynkin diagram of D N ≃ so 2N as in the Figure 9, The Wilson charge representation 2N decomposes in this The calculation of the Lax matrix L In this relation, the N×N matrix B and theN×N matrix C are given by: In matrix notations, we have III) Cospinorial coweight In this case, the minuscule coweight is given by µ N −1 . The Levi-decomposition is given by it corresponds to cutting the node α N −1 in the Dynkin diagram of the Figure 10. decomposition using the coweight µ N considered above. In the present case, the sl ′ N results from cutting the N-th node α N −1 in the Dynkin diagram of so 2N while cutting the node α N yields sl N . As such, the simple root systems of the sl ′ N and sl N are isometric and are given by These two systems are related to each other by the outer-automorphism symmetry Z 2 acting by permutation of the simple roots α N −1 and α N as follows Notice that the above Z 2 discrete symmetry acts non trivially on the 2N quantum states |a , with label 1 ≤ a ≤ 2N, propagating along the Wilson line. As this electric line carries a vector-like charge given by the SO 2N representation; and by using the decomposition 2N = N +1/2 ⊕ N −1/2 respectively labeled by |i and |ī with 1 ≤ i ≤ N andī = 2N + 1 − i taking the values1 ≤ī ≤N, we end up with the following transformations Under this transformation, the charge operator (3.44) is preserved because the sums |i i| and |ī ī| are not affected by (3.51). The same invariance holds for the linear expansions

Exceptional Lax operators
As there is no minuscule coweight in the exceptional Lie algebra E 8 , we have minuscule Lax-operators only for the 4D exceptional Chern-Simons with gauge symmetries E 6 and E 7 . From the classification The Lax operator of E 6 -type for the coweight µ 1 is associated with the Levi-decomposition E 6 → so 10 ⊕ so 2 ⊕ 16 + ⊕ 16 − . In the diagrammatic language, this corresponds to omitting the first node in the Dynkin diagram of E 6 given by the Figure 11. This node, labeled by simple root α 1 , corresponds to the fundamental 27 representation of E 6 with quantum states |ϕ propagating along the Wilson line. These 27 quantum states decompose with respect to the Levi-subalgebra as follows So, we split the 27 states of the representation 27 like: (i) a state |0 denoting the singlet 1 −4/3 ; (ii) ten states {|i } 1≤i≤10 designating the ten-uplet 10 +2/3 and (iii) sixteen states {|β } 1≤β≤16 representing the 16 −1/3 . Using these states, we solve the Levi-constraint relations for generators of the nilpotent subalgebras 16 ± as where Γ i are so 10 Dirac-like matrices satisfying the euclidian Clifford algebra. Putting these relations back into L µ 1 E 6 = e X z µ e Y , we obtain after tedious but straightforward algebra the following expression, where we have defined the following quantities in terms of the oscillator degrees of freedom b α and c β .
The Lax operator L µ 5 E 6 associated to the coweight µ 5 is obtained in a similar way as before; but instead of cutting the first node α 1 in the Dynkin diagram of E 6 , we need to cut the fifth α 5 . Because this node corresponds to the anti-fundamental representation 27, it is realised in the same basis as in (3.54-3.55) but with µ replaced with the opposite −µ. This feature leads to the relationship The properties of the exceptional Lax operators L µ E 6 were briefly outlined in [25] and its detailed derivation can be found in [34].
The gauge symmetry group E 7 of the 4D Chern-Simons theory is characterized by one minuscule coweight µ 1 corresponding to the fundamental representation 56. As shown on the Tables 1-2, the Levi-subalgebra of E 7 contains its subalgebra E 6 and the nilpotent n ± are given by 27 ± . This corresponds to cutting the node α 6 as depicted by the Figure 3.

62.
The quantum states |a running on the electrically charged Wilson line are described by the representation 56 that decomposes as follows As such, the 56 states |a can be splitted as |0 + , β + , β − , |0 − where the labels take the values β + = 1, ..., 27 and β − =1, ..., 27. The generators of n ± solving the Levi-conditions are given by such that the Γ δ + γ − β andΓ β γ − δ + are tri-coupling objects of the E 6 representation theory [34]. The Levi-charge operator is given by The derivation of the Lax-matrix L µ 1 E 7 is a little bit technical and cumbersome, we choose to represent it in the basis |0 + , β + , β − , |0 − as follows The diagonal entries of this Lax-matrix are given by The other Lax-matrix entries are as listed below and as well as More details concerning the derivation of this operator from the 4D Chern-Simons theory can be found in [34].

B-type Lax operators
In this section, we calculate the minuscule Lax operator L B (z) from the 4D Chern-Simons theory with gauge symmetry SO 2N +1 . The L µ B is determined by using the formula e X z µ e Y . In this relation, the µ is the minuscule coweight of the underlying Lie algebra B N of the gauge symmetry. The X = b i X i and Y = c i X i are (2N + 1) × (2N + 1) matrices valued in the nilpotent algebras n − and n + appearing in the Levi-decomposition of B N ∼ so 2N +1 with respect to µ, namely Dimensions and ranks of the algebras appearing in (4.1) can be directly read from the following decompositions Recall that the finite dimensional Lie algebra B N has one minuscule coweight µ 1 = e 1 (denoted here as µ); it is the dual of the simple root α 1 = e 1 − e 2 (µ 1 .α i = δ 1i ); and corresponds to the first node of the Dynkin diagram of B N . For an illustration, see the Figure 6.
The nilpotent sub-algebras are generated by X i and Y i obeying the following commutation relations [µ, where µ stands for the adjoint action of the minuscule coweight. The explicit realisation of this algebra in terms of the harmonic oscillators of the phase space of the L-operators is investigated below.

Solving Levi-constraint relations
To solve (4.3), we need to define the Levi-decomposition of the vectorial representation 2N + 1 of the Lie algebra so 2N +1 and related objects. Vector states of 2N + 1 propagate on the Wilson line interacting with the 't Hooft line with magnetic charge µ. Under the Levi-decomposition, the real 2N + 1 representation of so 2N +1 splits as direct sum of representations of l µ = so 2 ⊕ so 2N −1 . In fact, we have 2N + 1 = 2 0 ⊕ (2N − 1) 0 where the zero label refers to the charge under the group SO (2) . By using the isomorphism SO (2) ∼ U(1), we can put this decomposition to the following form where we have substituted with 2 0 = 1 +1 ⊕ 1 −1 . For convenience, we use the kets (|+ , |i , |− ) with i = 1, ..., 2N − 1 and the bras ( +| , i| , −|) to denote the vector basis of the fundamental representation (2N + 1) and its dual. We have where |± refer to the two complex singlets 1 ±1 in the decomposition (4.4a) and |2N − 1 represents the 2N − 1 states {|i } 1≤i≤2N −1 . This basis is characterized by the following orthogonality relations +|− = +|i = i|− = 0 and +|+ = −|− = 1, as well as i|j = δ ij . The next step is to find the adjoint action µ of the minuscule coweight on the fundamental representation. This is a hermitian charge operator acting on the quantum states generated by (4.5). It can be represented like where ̺ ± are the projectors on the representation sub-spaces 1 ±1 appearing in (4.4a) and Π is the projector on (2N − 1) 0 . These projectors read in terms of the bras/kets as follows Notice here that the charge q vanishes (q = 0) because the |2N − 1 is chargeless. Now, we move to working out the explicit expressions of the generators X i and Y i of the nilpotent algebras n + and n − in the basis 4.5. They are obtained by solving the Levi-constraint relations (4.3). By taking X i and Y i like with x 1,2 and y 1,2 non vanishing arbitrary numbers, then using µ = ̺ + − ̺ − , we have [µ, X i ] = +X i and [µ, Y i ] = −Y i as well as [X i , Y i ] = δ i i µ provided the following conditions are satisfied: (i) x 2 y 2 = x 1 y 1 and (ii) x 1 y 1 = x 2 y 2 = 1. These conditions can be solved by taking x 1 y 1 = 1 , x 2 = y 2 = ±1 (4.10) Below, we take x 2 = y 2 = −1. With these generators, we can express the X and Y matrices appearing in the Lax operator that we want to calculate; we have In these expansions, the b i and the c i variables are the phase space coordinates of the L Boperator; they are treated here classically but they can be promoted to operators without ambiguity. This is because in the formula e X z µ e Y , the b i 's are in the left and the c i 's are in the right in agreement with the Wick theorem.

Building the operator L B
We begin by calculating the exponentials e X and e Y by using the expansion e A = A n /n!. Based on the eqs (4.9), we compute the powers of the X i and Y i generators. We find after some calculations that and From these relations and the expansions We also have e X = 1 + X + X 2 /2 and e Y = 1 + Y + Y 2 /2 because X 3 = Y 3 = 0. These features lead to the following polynomial-like expansion reading explicitly as Replacing z µ with z µ = z̺ + + Π + z −1 ̺ − (4. 16) and taking advantage of the properties X i ̺ + = 0 and ̺ + Y i = 0 as well as we obtain To determine the matrix representation of the L-operator, we use the following trick (projector basis) By substituting X = b i X i and Y = c i Y i , we end up with the following result where f T stands for the row vector (f 1 , ..., f 2N −1 ). Notice the two following features regarding eq(4.20).
(1) By multiplication of L µ 1 B by z, we recover the Lax-matrix of B-type obtained in [33] by using anti-dominant shifted Yangians.
(2) Eq(4.20 has a quite similar form to the Lax operator L vec D of the SO 2N family given by eq(3.42). The main difference concerns the number of (b, c) oscillators that appears in the middle block. For L vec D N+1 , we have 2N oscillators versus 2N − 1 oscillators for L B N . This property can be explained by the fact that the B N Dynkin diagram can be obtained from the folding the two spinorial-like nodes of the D N +1 Dynkin diagram as depicted by the Figure 13. In this folding, the vectorial minuscule coweight is preserved.

C-type Lax operators
In this section, we derive the minuscule Lax operator L C (z) from the 4D Chern-Simons theory with gauge symmetry SP 2N . The L µ C is determined by using the formula e X z µ e Y . Here, the µ is the minuscule coweight of the Lie algebra C N ∼ sp 2N and the X and Y are 2N × 2N matrices belonging to the nilpotent sub-algebras n − and n + appearing in the Levi-decomposition of sp 2N , namely The dimensions and the ranks of the algebras involved in this decomposition are as given below dim : N (2N + 1) = 1 + (N 2 − 1) + 1 2 N (N + 1) rank : Recall that the Lie algebra C N has one minuscule coweight reading in terms of the {e i } weight vector basis as µ = µ N = 1 2 (e 1 + ... + e N ). This minuscule coweight is the dual of the simple root α N = 2e N , it corresponds to the N-th node of the Dynkin diagram of C N given by the Figure 7.
Recall also that in (5.1), l µ is the Levi-subalgebra of sp 2N and the n ± are the nilpotent subalgebras having 1 2 N (N + 1) dimensions that split like 1 2 N (N − 1) + N. These subalgebras are generated by matrix generators denoted like X [ij] , X iī and Y [īj] , Yī i . They obey the following commutation relations where µ stands for the adjoint action of the minuscule coweight.

Solving Levi-constraints for C N
To solve the constraint relations (5.3), we need the Levi-decomposition of the fundamental representation 2N of the Lie algebra sp 2N given by where N +1/2 and N −1/2 are representations of sl N and the subscripts ±1/2 referring to the SO 2 charges. To proceed, we use (a) the 2N kets {|i , |ī } with i = 1, ..., N andī = 2N +1−i to represent the quantum basis states of the symplectic representation 2N. For convenience, we order theī-label likeī =N, ...,1. (b) the dual bras { i| , ī|} to denote the vector basis of (2N) T . Formally, we have The next step is to work out the adjoint action µ of the minuscule coweight on the fundamental representation 2N. It is given by with the projectors on N +1/2 and N −1/2 as follows In these relations, we have set Now, we move to the determination of explicit expressions of the matrices X [ij] , X iī and Y [īj] , Yī i generating the nilpotent algebras n + and n − . They are obtained by solving the Levi-constraint relations (5.3), we have found and They also obey other useful properties such as [X iī , X [ kl] ] = 0 and Yī i , Y [k l] = 0. With the generators (5.8-5.9), we can express the X and Y matrices appearing in the Lax operator.
We have In these expansions, the b iī , b [ij] and the cī i , c [īj] variables are the phase space coordinates of the L C -operator.

Building the operator L C
First, we use eqs (5.8-5.9) to determine the powers X and Y. Then, we calculate the exponentials e X and e Y appearing in the L-operator formula. Straightforward algebra leads to These properties show that the exponentials take simple forms e X = I 2N + X and e Y = I 2N + Y. Then, the Lax operator expands as follows reading explicitly as Replacing the charge operator z µ with (5.14) we get the following expression of the Lax operator for the symplectic family This relation can be simplified by taking advantage of properties of the X and Y matrices that descend from the generators realising (5.3). We have XΠ = 0 and ΠY = 0 as well XΠ = X andΠY = Y. By substituting, we end up with In the projector basis Π,Π , we have the following representation which also reads as In the vector basis {|i , |ī } , we have Finally, notice the two following features regarding (5.18): (1) Eq(5.18) is, up to multiplication by z, similar to the Lax-matrix of C-type obtained in [33] by using anti-dominant shifted Yangians.

Conclusion and comments
Four dimensional Chern-Simons gauge theory proposed in [1] has been shown to be a powerful QFT approach to deal with lower dimensional integrable systems. Several results on integrable 1D quantum spin chains such as the Lax operators of A-and D-types, obtained by using Bethe Ansatz formalism and standard statistical physics as well as algebraic methods, were nicely derived from the CS theory. The investigation given in this paper is a contribution to the topological 4D CS gauge theory and its applications. It essentially aims to complete some partial results obtained in literature and also to gather the explicit expressions of minuscule Lax operators L G and classify them according to algebraic properties of the gauge symmetry as given in the Tables 1-2 and the Table 3 given below. We recall that from the view point of the 4D Chern-Simons theory, the L G 's can be thought of as a matrix coupling an electrically charged Wilson line W R ξ z crossing a magnetically charged 't Hooft line tH µ ξ 0 . The study of this crossing yields a general formula that corresponds to the oscillator realisation of the Lax operator for an integrable XXX spin chain. This construction was introduced in this paper along with the mathematical tools needed to build our results. Among our contributions, we quote the three following: (1) We derived the non simply laced orthogonal B N -and the symplectic C N -families of Lax operators using the 4D CS theory method. These calculations have not been addressed before in the framework of the CS gauge theory. The Lax operators L B N and L C N were calculated in section 3 with regards the unified picture of all the L G 's. They were investigated with further details in sections 4 and 5 and they were shown to agree with recent expressions derived in the spin chain literature for the B N and C N symmetries.
(2) We gave an interpretation of the links between the B N -and C N -type Lax operators and their A N -and D N -homologue in terms of discrete symmetries and foldings with respect to discrete groups. We showed that these symmetries are nothing but the outer-automorphisms of Dynkin diagrams of A N and D N . These foldings were visualized in the Figures 13 and  14 showing the relationships between the B N /C N -types and D N /A N types.
(3) We built the set of the minuscule Lax operators labeled by a set parameters. In addition to the electric charge R of the Wilson line W R ξ z , these parameters are given by the rank of the Lie algebra of the gauge symmetry G and its minuscule coweights µ. The content of this set is given by the table 2. This basic set contains five subsets: four infinite given by the families A N , B N , C N , D N and one finite given by the exceptional E 6 and E 7 symmetries. We end this paper by giving brief comments concerning the L-operators of the SO 2N symmetry having a spinorial R representation with dimension R = 2 N , this corresponds to having a Wilson line W 2 N ξ z in the spinor representation crossing a 't Hooft line. The construction of the associated L-operator L µ D N R=2 N can be done by following the same analysis that we performed in the sub-subsection 3.3.4 to build L µ D N R=2N . In fact, both the fundamental L µ D N 2 N and L µ D N 2N are calculated by using the following formulas : V ect = 2N (6.2) However, though they look quite similar, the expressions of these two operators are completely different, the first (6.1) is realised by a ν × ν matrix with ν = 2 N , while the second (6.2) is given by a 2N × 2N matrix. So, the matrix realisations of the triplets (µ,X, Y ) used in (6.1) and in (6.2) are different. Below, we comment the matrix realisations of the triplet (µ,X,Y) involved in (6.1). The Levi-charge µ needed for the calculation of (6.1) is obtained by decomposing the 2 N representation as a direct sum of representations of the Levi -subalgebra l µ as in eqs(3.1-3.4). Because we have two types of l µ 's namely (i) so 2 ⊕ D N −1 and (ii) so 2 ⊕ sl N , we distinguish two kinds of reductions of 2 N with respect to l µ as shown in the Table 3 where the ̺ i 's are l µ eq(3.1) eq(3.2) b ij X ij /cī j Yī j (6.6-6.7) with magnetic charge µ crossing a Wilson line W R ξ z with representation R given by the spinorial 2 N of the SO 2N gauge symmetry.
projectors that can be thought of as |i i| . Notice that in the first row of the table, the projectors act like ̺ ± : 2 N → 2 N −1 ±1/2 while in the second row, they act as ̺ n : 2 N 1 → N ∧n qn . Notice also that the quantities N ∧n with powers 0 ≤ n ≤ N are the wedge product of n representations N of sl N . The dimension of each N ∧n is given by N!/n!(N − n)!. For example, the N ∧2 is given by N ∧ N with dimension 1 2 N (N − 1). The subscripts q n refer to the charges under so 2 and their trace must vanish as in (3.4); that is, N n=0 N ! n!(N −n)! q n = 0 reading also like [N/2] n=0 N! n!(N − n)! (q n + q N −n ) = 0 (6.3) which is solved by taking q N −n = −q n , in particular q N = q 0 and q N −1 = q 1 . The value of [N/2] depends on the parity of the integer N. For even N = 2M, the charge q M = 0. For the vector Levi-decomposition with Levi-subalgebra l µ = so 2 ⊕ D N −1 , the structure of the Lax matrix has 2 2 = 4 blocks: two diagonal and two off-diagonal ones. It reads as follows where B and C are η × η matrix oscillators with order η = 2 N −1 . They read in terms of the nilpotent generators X i = β + Γ β + γ − i γ − and Y i = γ − Γ i γ − β + β + solving eq(4.3) like B = b i X i and C = c i Y i . Here, the Γ i s are Gamma matrices of D N satisfying the Clifford algebra in 2N dimensions. Regarding the Levi-decomposition with l µ = so 2 ⊕ A N −1 , the structure of the associated Lax-matrix has (N + 1) 2 blocks; N + 1 of them are diagonal blocks; they correspond to the N + 1 terms involved in the following expansion 2 N = 1 q 0 ⊕ N q 1 ⊕ N ∧2 q 2 ⊕ · · · ⊕ N ∧n qn ⊕ · · · ⊕ N ∧N q K (6.5) For the example of N = 4 corresponding to a 4D CS gauge theory with SO 8 gauge symmetry in the presence of a Wilson line W R ξ z with R = 2 4 , the reduction of the spinor representation 2 4 = 16 with respect of the Levi subalgebra so 2 ⊕ sl 4 reads as 1 +2 + 4 +1 + 6 0 + 4 −1 + 1 −2 . From this reduction, we learn that the adjoint action of µ reads in terms of the 5 projectors as z ±2 ̺ 1 ± + z ±1 ̺ 4 ± + ̺ 6 0 . This indicates that the Lax-operator is 16×16 matrix having 5 diagonal blocls as follows where L refers to L µ N D N R=2 N . The Lax-matrix (6.6) reads in general as follows L mn = ̺ m .L.̺ n (6.7) It has N+1 diagonal blocks n × n with dimensions N ! n!(N −n)! given by the irreducible components in the expansion (6.5). Its explicit expression is obtained by starting from e X z µ e Y R=2 N and substituting X = b ij X ij and Y = cī j Yī j as well as z µ = z qn ̺ qn and charges q n = N 2 − n.