Superspin Chains Solutions from 4D Chern-Simons Theory

As a generalisation of the correspondence linking 2D integrable systems with 4D Chern-Simons (CS) gauge theory, superspin chains are realized by means of crossing electric and magnetic super line defects in the 4D CS with super gauge symmetry. The oscillator realization of Lax operators solving the RLL relations of integrability is obtained in the gauge theory by extending the notion of Levi decomposition to Lie superalgebras. Based on particular 3-gradings of Lie superalgebras, we obtain graded oscillator Lax matrices for superspin chains with internal symmetries given by $A(m-1\mid n-1)$, $B(m\mid n)$, $C(n)$ and $D(m\mid n)$


Introduction
A newly discovered shortcut towards the realization and study of integrable systems is yielded by a four dimensional Chern-Simons gauge theory defined on the product of a topological real plane Σ and a holomorphic curve C, by the field action [1]- [5] This field theory is characterized by a complexified gauge symmetry G, and a partial gauge connection with three bosonic components as A = dxA x +dyA y +dzA z , valued in the Lie algebra g of the gauge symmetry.Endowing this topological field theory with crossing line defects allows to build two-dimensional solvable models in Σ, and recover solutions and conserved quantities of these lower dimensional models; thus opening the door for interestings findings enrishing the integrability literature [6]- [16].In particular, the R-matrix describing the scattering of two particles' worldlines [17]- [20] is calculated from the 4D CS as the crossing of two Wilson lines characterized by electrical charges given by highest weights of G.In this image, each Wilson line is represented in the topological plane by a line assimilated to the worldline of an electrically charged particle whose internal quantum states are valued in some representation R of g characterized by a highest weight λ R .Positions of these line defects in the complex C correspond to spectral parameters z i that play a major role in Yang-Baxter equation and in the RTT realization of Yangian representations [21], [22].
The integrable XXX spin chain [23], [24] emerges in the 4D CS theory defined on R 2 × C, as a set of parallel (vertical) Wilson lines sitting on the chain nodes and carrying degrees of freedom of the spins.The interaction between these spins is modelled by a horizontal 't Hooft line perpendicularly crossing the Wilson lines [7].The 't Hooft line defect is a disorder operator [25]- [27] characterized by a magnetic charge equivalent to a coweight µ of G; it acts like an auxiliary oscillatory space such that its intersection with a Wilson line at each node of the spin chain yields a Lax operator [29].This operator is a basic ingredient of the Bethe Ansatz approach [30]-[?]; it operates on the quantum spaces and is a solution to the RLL equation underlying the integrable spin chain.In the Gauge theory formulation, the Lax operator is computed as the parallel transport of gauge fields past the 't Hooft line.
This key result was demonstrated in [7] for the particular case where the magnetic charge is given by a minuscule coweight µ of G. There, the authors linked the Levi decomposition of the Lie algebra g to the dispersion of the gauge field bundles above and under the 't Hooft line due the Dirac-like singularity induced by the presence of this magnetic operator.The particularity of the minuscule coweight is that it acts on the roots of g with the eigenvalues 0, ±1 [33] which decomposes the Lie algebra g into three subspaces as n −1 ⊕ l µ ⊕ n +1 .This is a Levi decomposition of g where the Levi subalgebra l µ has charge 0 with respect to µ, and the n ±1 are nilpotent subspaces given by modules of l µ and carrying charges ±1 [35], [36].These algebraic features play a major role in this investigation because for any 't Hooft line with minuscule magnetic charge µ of G, the corresponding Lax operator can be simply computed by the general formula L µ (z) = e X z µ e Y , where X = X i b i and Y = Y i c i are elements of n + and n − respectively.The action of the coweight µ on the representation R carried by the Wilson line in question, can be deduced by the branching rule of R following from the Levi decomposition of g.The oscillator structure of the phase space of this L-operator follows from the Levi decomposition properties; the classical coordinates b i ∈ n +1 and c j ∈ n −1 verify the Poisson bracket {b i , c j } P B = δ i j which yields the commutator at the quantum level [7], [37].Based on this interpretation, the minuscule Lax operators for the simply laced A and D type bosonic spin chains were first realized using the CS gauge theory in [7], and then in [12,13], in agreement with solutions obtained using Yangian representations in [? ]- [39].Lax operators for bosonic spin chains with non simply laced B and C type symmetries were recovered from 4D CS in [13] and were compared with [40].The power of this 4D CS/ Integrability correspondence allowed also to build solutions for exceptional spin chains, with internal symmetries described by the simply laced e 6 and e 7 algebras, which were lacking in the spin chain literature [14].The missing exceptional e 8 , f 4 and g 2 symmetries in this bosonic list do not have minuscule coweights [33].Regarding superspin chains with internal symmetry described by Lie superalgebras, the generalization of the 4D CS/ Integrability correspondence requires the equipment of a 4D Chern-Simons theory having super gauge symmetry with super line defects carrying bosonic and fermionic degrees of freedom [45].This extension was motivated in [37] by uplifting from SL(m) to the SL(m|n) symmetry and by taking advantage of the resemblance of their algebraic structure.The super Lax operators characterizing the sl(m|n) superspin chain were calculated in the framework of the SL(m|n) 4D CS by using a generalized formula similar to the bosonic L µ (z) = e X z µ e Y .However, due to the lack of the notion of minuscule coweight and Levi decomposition in the superalgebras literature, a Dynkin diagram's node cutting method was used in order to generate 3-gradings of the sl(m|n) Lie superalgebra.These decompositions of the sl(m|n) family have similar properties to the Levi decomposition such that the role of the minuscule coweight is played by the cutted node.This approach allowed to construct explicit super L-operators in terms of bosonic and fermionic oscillators of the phase space, in agreement with the superspin chain literature solutions [41] obtained using degenerate solutions of the graded Yang-Baxter equation [42].In this paper, we follow a similar approach to [37] in order to build oscillator realizations of super Lax operators for integrable superspin chains classified by the basic ABCD Lie superalgebras.As for these Lie superalgebras, one has several super Dynkin diagrams depending on the number of fermionic roots and their ordering.Therefore, one distinguishes several varieties of the ABCD superspin chains due to their link to the super Dynkins [43].By considering a super Wilson line W R ξ z in a given super representation R and a super 't Hooft tH µ γ 0 with magnetic charge µ, we calculate the super Lax operator describing their crossing.For the sl(m|n) symmetry, we derive the super L-operators for any coweight µ of any super Dynkin diagram of the (m + n)!/m!n! possible graphs.We show that they agree with those calculated by using the super Yangian representations as a verification of our approach.For the B(m|n), C(n) and D(m|n) superspin chains, we give a family of solutions corresponding to specific coweights of the distinguished super Dynkin diagrams leading to Levi-like 3-gradings.The presentation is as follows: In section 2, we give basic tools of the 4D Chern-Simons theory with SL(m|n) gauge symmetry, and the realization of the sl(m|n) superspin chain by means of super line defects.We describe the super Lax operator construction for basic Lie superalgebras.In section 3, we use this construction to build solutions for the RLL equations of the sl(m|n) superspin chain, and compare with known results of the literature.Sections 4, 5 and 6 are respectively dedicated to the building of super Lax operators for superspin chains with B(m|n), C(n) and D(m|n) symmetries.We end with a conclusion and discussions.An appendix is added to this version as a verification of the new orthosymplectic solutions.

sl(m|n) superspin chain in 4D CS
In this section, we consider the standard A bf -family of superspin chains based on the sl(m|n) Lie superalgebra (m ̸ = n), to first introduce the basics of the present investigation, and to complete partial results in literature with regards to the A bf class [41].The label bf refers to chains with bosonic and fermionic degrees of freedom.This family of integrable super systems is realized in the framework of the 4D Chern Simons gauge theory having the SL(m|n) super gauge group.The superspin chain families B bf , C bf and D bf to be studied in the forthcoming sections are realized in a similar fashion; and as such, the algebraic basics of the line defects construction for all superchains are only detailed for the case of the A bf super chain.To this end, it is interesting to recall that the A bf special family of superchains generalizes the well known family of sl(n) spin chains termed below as the A bose family.The generalised A bf has two basic features: First, it is classified by the set of Lie superalgebras g bf given by the bi-integer series A m,n bf ≡ sl(m|n) including sl(n) and sl(m) as bosonic subsectors.Second, for sl(m|n) one distinguishes several types of superchains versus one ordinary sl(n) chain in the bosonic case.This is due to the Z 2 -grading of the bf gauge symmetry to be commented on later.For example, given two positive integers (m, n), one has varieties of sl(m|n) superchains.
To perform this study, we begin by briefly describing the corresponding four-dimensional gauge theory with SL(m|n) local symmetry, and its line defects that allow for the superspin chain realization.Then, we introduce a general formula for the computation of super Lax operators directly from the superalgebra 3-gradings.

4D CS gauge theory with SL(m|n) symmetry
The field action describing 4D Chern-Simons theory with super SL(m|n) gauge symmetry, living on the 4D manifold M 4 = R 2 × C, with R 2 the real plan parameterized by (x, y) and C an holomorphic curve parameterized by z, is written in terms of the supertrace of the CS 3-form [37] S The 1-form gauge potential A is given by A x dx+A y dy+A z dz where we have dropped the component A z dz because of the dz factor in the holomorphic volume form.It is valued in the sl(m|n) Lie superalgebra, and thus expands like where E ab are graded generators of sl(m|n) obeying the graded commutation relations with degree as The supertrace of the Chern-Simons 3-form in (2.2) is therefore written in terms of the graded metric g abcd = str(E ab E cd ) and the constant structures The field equation of the gauge connection A in absence of external charges is given by a vanishing 2-form field strength In order to realize lower dimensional integrable super systems, we need to introduce charges to the CS theory through super line defects, such as the super Wilson W m|n ξ z [3,37,44].This topological defect is represented by a line ξ z in R 2 , sitting in the position z in C along which propagate quantum super states.The W m|n ξ z can be imagined as an electrically charged line defect, characterized by the fundamental representation R = m|n of the superalgebra sl(m|n), such that the electric charge is given by the corresponding highest weight.The topological W m|n ξ z is defined as the supertrace of the holonomy of the gauge field along the line ξ z like , we can also introduce a magnetically charged super 't Hooft line [25]- [27], denoted here as tH µ γ z ′ .This is also a topological line defect, which is implemented in the 4D CS as a line γ z ′ extending in the space R 2 , and living at a point z ′ in C. The tH µ γ z ′ is a disorder operator that carries a magnetic charge given by a coweight µ of the SL(m|n) supergroup; and is identified with the parallel transport of gauge field bundles past the line.By taking γ z ′ as the x-axis in R 2 and z ′ = 0, we can write the 't Hooft line observable as where the transport of the gauge fields is measured from y < 0 to y > 0. Actually, just like in the bosonic case [7], the magnetically charged super 't Hooft line defect is classically defined such that its presence in a 4D CS with super gauge symmetry G bf , deforms the field action like Here, the field strength F is no longer flat (F ̸ = 0), and should look like a Dirac monopole with non trivial first Chern class where the surrounding of the line is conveniently viewed as a cylinder C instead of a sphere, due to the mixed nature of the theory [7], and the coweight µ : U (1) → G bf serves to map the U (1) Dirac monopole into a non-abelian G bf Dirac monopole.In this case, the gauge field defines a G bf − bundle on the surrounding of the 't Hooft line which is also defined as the mapping of the abelian U (1) bundle by the action of the magnetic graded coweight µ.This gauge configuration is described on three regions that can be identified with the bases and the contour of a cylinder C in (x, y, z).Above and under the tH µ γ 0 , we have trivialized bundles in the regions U I = {y ≤ 0, z ∼ 0} and U II = {y ≥ 0, z ∼ 0} , which are locally glued by a transition function (isomorphism) near the intersection U I ∩ U II = {y = 0, z ∼ 0} ≃ γ 0 .The trivial bundles in U I,II should be holomorphic G bf − valued functions that are regular at z = 0; they are given by gauge transformations g I (z), g II (z) with monopole charges.The transition function serves as a parallel transport from U I to U II , and is locally equal to the Dirac singularity z µ [28].The L-operator (2.8) measuring this gauge behaviour near tH µ γ 0 is therefore given by the formula Notice that the tH µ γ 0 is in fact sitting at the end of a Dirac string linking it to another 't Hooft line living at z = ∞, y = 0 and having the opposite magnetic charge −µ.The gauge behaviour near z = ∞ is equivalently evaluated as (2.12) g III (z −1 ) and g IV (z −1 ) are G bf − valued holomorphic functions that need to be regular and equal to the identity at z = ∞ [7].However, this detail will be omitted in the present inquiry, and we will be considering a 't Hooft line tH µ γ 0 which is to be understood as coupled to a tH −µ γ ∞ .The parallel transport of the gauge field in the presence of this double line having charges at z = 0 and z = ∞ is of the form where A(z) and B(z) are G bf − valued functions verifying the appropriate singularity constraints at zero and infinity.
In analogy to the parallel transport of a 't Hooft line in a bosonic Chern-Simons theory [7], we should be able to explicitly realize the functions of the formula (2.13) in terms of a 3-grading of the Lie superalgebra g bf .This was shown for the SL(m|n) symmetry in [37] (see Appendix A) and will be generalized in the next section for any superalgebra g bf .

Oscillator realization in 4D CS theory
We focus now on linking the graded gauge theory and its ingredients presented above to the integrable superchain systems we are concerned about here.We describe the super 4D CS/ superchain correspondence by focussing on the SL(m|n) symmetry, in order to introduce the super Lax operator as a solution to the RLL equation of integrability, and its interpretation and computation in the gauge theory.
To begin, the super 4D CS/ superchain correspondence we are considering here is an extension of the well known bosonic 4D CS/ Integrability linking a spin chain with bosonic internal symmetry sl(n) to 4D Chern-Simons theory with gauge group SL(n).This bosonic correspondence is also valid for other gauge symmetries G bose given by Cartan classification of finite dimensional Lie algebras including A n , B n , C n , D n and the exceptional ones.Alongside this interesting result, it was also conjectured that the bosonic correspondence extends to superspin chains where several checks were successfully carried out [45], [37].
In this Fermi/Bose generalisation, superchains are characterised by superspin representations of Lie superalgebras g bf splitting like with g0 a bosonic Lie algebra and g1 a module of it.As such, basic integrable superspin chains fall into families classified by the basic complexified Lie superalgebras g bf as listed below [50], [51] g bf even part g0 odd part g1 In general, an integrable superspin chain with intrinsic symmetry given by the superlagebra g bf is linked to the 4D Chern-Simons theory characterized by the corresponding super gauge group G bf , and a field action S gbf CS [A] as eq(2. 2) where the potential A = A x dx+A y dy +A z dz expands in terms of the generators of g bf as in eq(2.3).The extension of the bosonic gauge/ Integrability correspondence to graded symmetries is permitted by the implementation of the super line defects introduced previously.In the SL(m|n) generalisation, the super Wilson line W m|n ξ z carries degrees of freedom that allow to describe the quantum states of a super atom with superspin valued in the representation m|n [43], i.e. with sl(m|n) internal symmetry.Therefore, the integrable sl(m|n) superchain with N super atoms can be realised in the 4D CS theory by placing (i) L vertical Wilson lines W m|n ξ i z at every node ν i (1 ≤ i ≤ L) of the superspin chain such that 16) and (ii) an horizontal 't Hooft line tH µ γ 0 with γ 0 = (z 0 , R, y 0 ) that can be thought of as filling the x-axis (y 0 = 0) in R 2 and sitting at z 0 = 0 in CP 1 .These super line defects intersect in the topological plane R 2 (x, y) as depicted in Figure 1, where the 't Hooft line plays the role of a transfer matrix modeling the interactions between the electrically charged super atoms along the chain.Following this lattice system realization, each intersection of an electric W m|n ξ z carrying the vector quantum space m|n with the magnetic tH µ γ 0 carrying an auxiliary space, yields the super Lax operator for the corresponding node of the superspin chain.This coupling operator acts on the tensor product of End(m|n) of the Wilson and the algebra A of functions in the phase space of the 't Hooft line.It is nothing but the L-operator (2.8) which describes the paralell transport of gauge fields (given here by the fields in m|n, travelling along the Wilson line) past the 't Hooft line carrying the magnetic charge µ that acts on the space m|n; it can be simply labeled by the representation R = m|n and the coweight µ.The quantum integrability of this system is encoded in the RLL equation verified by the L-operator with matrix realisation L m n (z) obeying, where R ik rs (z − w) is the usual R-matrix of the Yang-Baxter equation.Notice here that the Wilson/ 't Hooft crossing was shown to verify this equivalence by field theory analysis [4,7].The RLL equation has a remarkable graphical representation given by Figure 2. It is diagrammatically verified thanks to the diffeomorphism invariance of the theory in 4D, where the 't Hooft line can be freely moved past the Wilsons' crossing since it sits at a different position z in C. Now, in order to explicitly compute the super Lax operators of a superspin chain in the framework of the 4D CS theory, we follow the approach of [7] which builds the oscillator realization for a Lax operator of a spin chain with internal symmetry g by considering the magnetic charge of the 't Hooft line as a minuscule coweight [33].The minuscule coweight µ acts on the algebra elements with eigenvalues 0, ±1 and induces a Levi decomposition of g as where l µ is the Levi subalgebra containing elements of charge 0 with respect to µ, and n ± are nilptent subspaces carrying charges ±1.These properties are described by the Levi constraints In this case, the splitting of gauge bundles in the L-operator formula as in eq(2.13) is identified on the Lie algebra level with the Levi decomposition with respect to the magnetic coweight µ.The Lax operator L µ (z) solving the RLL equation for a chain of spins in the representation R, associated to the node corresponding to the coweight µ in the Dynkin diagram can be directly computed using the expression The operator µ acts on the representation R by dividing it into subspaces carrying charges with respect to µ such that the trace on the total space R is zero, tr R µ = 0.These charges can also be deduced from the branching rules [34] of R following from the Levi decomposition (2.18).Now in order to compute the super Lax operators corresponding to integrable superspin chains, one may be tempted to just generalise the construction done for the integrable bosonic chain; but this poses a problem because the notion of Levi decomposition and minuscule coweight are not yet known for Lie superalgebras.We propose here to circumvent this difficulty by using superalgebra 3-gradings generated by the method of (extended) Dynkin diagram [49].This method consists of deducing the possible decompositions of a superalgebra by cutting nodes from a corresponding super Dynkin diagram.These decompositions are motivated by the bosonic case where the Levi decompositions of the ABCDE Lie algebras can also be read on the Dynkin diagram level as the cutting of specific nodes associated to minuscule coweights [13], [14], [12].The superalgebras decompositions we are interested in here are 3-gradings having similar properties to Levi decompositions, which allows for the oscillator realization of the auxiliary phase space.This type of 3-gradings act as Levi decompositions for superalgebras, and were used in [37] to recover super Lax operators for the sl(m|n) superchain from 4D CS, which were verified to agree with the literature.The Dynkin diagram cutting method is described in the series of papers [46]- [48] where the results are listed for all nodes of the Dynkin diagrams of basic superalgebras.Given a super Dynkin diagram of a superalgebra g bf , one can determine all regular subalgebras g 0 of g bf by consecutive nodes cutting from the Dynkin (and extended) diagrams.This yields decompositions of the superalgebra g bf having a general 5-grading form where g ±k with k = 1, 2 are g 0 -modules determined by means of representation techniques and where with w being the standard antilinear anti-involutive mapping of the Lie superalgebra g bf .For the purposes of our study, we are only interested in the case where g ±2 = 0, i.e. in decompositions of Lie superalgebras that look like These 3-gradings result from the cutting of specific nodes in the Dynkin diagram, and are analogous to Levi decompositions for bosonic Lie algebras because we have [46]- [48] [g j , g l ] = g j+l (2.24) with j, l = 0, ±1.In what follows, we will interpret these values 0, ±1 associated the subspaces of g bf , as Levi charges with respect to the coweight µ corresponding to the cut node and acting as a "minuscule" coweight of g bf .Now having a Levi-like decomposition of the superalgebra g bf of the form (2.23) that we write as, which is associated to a minuscule-like coweight µ with, we can show that the super Lax operator formula can be factorized and directly calculated from elements of this decomposition, just like the bosonic construction.
In this case, an element of the gauge group G bf is written as Notice here that elements of the graded nilpotents N + and N − expand in terms of bosonic as well as fermionic generators, linked respectively to bosonic, and fermionic coordinates that form graded oscillators; as we will see later on.By taking (2.27) into account, the functions A(z) ∈ G bf and B(z) ∈ G bf appearing in the formula (2.13) should be factorized as follows By imposing the singularity conditions at z = 0 and z = ∞ for these functions, we can bring the expression (2.13) into the form where X is valued in N + , and Y belongs to N − .This calculation is detailed for the sl(m|n) symmetry in Appendix A of [37]; it equivalently holds for any gauge super group G bf .In general, the super Lax operator construction for any Lie superalgebra g bf is described through the following steps : A) 3-grading of the superalgebra g bf We consider the Lie superalgebra g bf with a 3-grading obtained by a node cutting from a Dynkin diagram such that µ is the coweight of g bf associated to the deleted node.This coweight acts as a "super minuscule" coweight and the three algebraic blocks in (2.31) are as described below: • The l µ is a regular Lie sub-(super)algebra of g bf with elements carrying charge 0 with respect to the coweight µ; it plays the role of a Levi subalgebra.
[µ,l µ ] = 0 (2.32) In fact, the l µ is always given by a direct sum where Cµ is associated to the cutted node.This can be visualized in the example of the bosonic sl(p), where the cutting of the last node α • The remaining elements of the decomposition (2.31) (i.e: elements in N = g bf \l µ ) are nilpotent; they are given by l µ -modules that carry charges ±1 with respect to the µ.µ, The two graded subspaces N ± mutually supercommute They moreover verify the generalized Levi-like constraint B) Branching of representations of g bf Under the decomposition (2.31), a representation R of the g bf splits into a direct sum of irreducible representations R q k of the superalgebras in l µ = l µ ⊕ Cµ.These subspaces R q k carry charges q k with respect to µ, that can be identified in the bosonic case from known branching rules.In general, we write The action of the coweight µ on the representation R can be therefore defined as where Π k is the projector on the subspace R q k and q k is often termed as the Levicharge.In the superalgebra case, the charges q k verify in addition to, the super-traceless condition These constraints allow us to compute these charges in the absence of branching rules for representations of superalgebras in the literature.
The super L-operator describing the coupling of the representation R and the magnetic coweight µ acting on the superalgebra g bf as (2.31), is equal to Later on, it will be simply labeled as L µ since we will take R as the fundamental representation for every symmetry type.The z µ follows from the action of µ on R as given by (2.38).The X and Y are elements of N + and N − expand in general like where X i and Y i are bosonic generators, and (X α , Y α ) are fermionic ones.These graded root generators correspond to roots of g bf that are not contained in l µ = l µ ⊕ Cµ, they can be realized by using the following property: For a cutted node corresponding to a graded simple root β, the root system Φ gbf splits as where Φ N ± contains graded roots in Φ ′ gbf that depend on β, i.e.
The sign of roots in each nilpotent is defined by the condition (2.34), such that µ acts on generators of N ± with ±1.As an illustrative example in the bosonic linear algebra, the Levi decomposition obtained by the cutting of the node α 3 from the Dynkin diagram of A 3 having the simple roots { α 1 , α 2 , α 3 } reads as sl(4) → sl(3) ⊕ gl(1).
The root system of sl(4) containing the 12 roots splits as Here, the 6 roots independent of α 3 correspond to l µ 3 = sl(3), and the six roots depending on α 3 generate the subspaces n ± .
Finally, notice that the bosonic coefficients (b i , c i ) , and the fermionic (β α , γ α ) form graded oscillators verifying the Poisson brackets Their quantum versions obey the usual super Heisenberg algebra.These steps will be used below to complete missing results in literature concerning integrable superspin chains with underlying symmetries given by the Lie superalgebras like B bf , C bf and D bf .But before that, we begin by testing this approach by building the oscillator realizations of super L-operators for the sl(m|n) superspin chain, and comparing with equivalent super matrices in the literature.The possible 3-gradings of the type (2.31) that we will be using are classified for the family of complexified basic Lie superalgebras as follows [46] 3 Super L-operators for all sl(m|n) superspin chains In this section, we apply the formula (2.41) introduced in the previous section in order to build super L-operators for the sl(m|n) superspin chain.Thanks to the richness of this A-type supersymmetry, we will be able to generate all families of solutions L µ labeled by magnetic charges µ of SL(m|n).These coweights are in one to one with different nodes of different Dynkin diagrams D[sl(m|n)] (κ) of sl(m|n) labeled by positive integers κ.In these regards, recall that contrary to bosonic Lie algebras g bose , a superalgebra g bf has several Dynkin diagrams D[g bf ] (κ) ; this is due to the existence of two kinds of fundamental unit weight vectors : bosonic unit weights ε a with metric ⟨ε a , ε b ⟩ = δ ab , and fermionic δ a 's with ⟨δ a , δ b ⟩ = −δ ab .Hence, the graded simple roots α i have special properties depending on their realisations, which for sl(m|n) may be (i) fermionic of the form or (ii) bosonic having two possible forms like Recall also that, as for bosonic g bose , the set of the simple roots generate the graded root system Φ gbf ≡ {± α bf }; and because of the three possibilities, we distinguish different types of root systems for g bf labeled by κ and denoted like Φ As an illustration, we give in  (35) depending on the ordering of the fundamental unit weights δ a , ε a .The distinguished super Dynkin diagram having only one fermionic node is drawn for a general Lie superalgebra sl(m|n) in Figure 4; the weight basis associated to such diagram is also known as the distinguished basis.This multiplicity of Dynkin diagrams for a superalgebra results in different possible varieties of superspin chains for each symmetry g bf .For the sl(m|n) symmetry, we will consider all the possible superspin chain systems and build the general super Lax operator L µ sl(m|n) associated to a generic node of one of the (m + n)!/(m!n!) super Dynkin diagrams D[sl (m|n) ] (κ) .Notice that these general solutions include those derived in [37] using the same approach but only for the distinguished Dynkin diagram having one fermionic node given by α m = δ n − ε 1 (Figure 4).In order to proceed for the calculation, we begin by recalling that the Lie superalgebra A(m − 1|n − 1) = sl(m|n) with n ̸ = m has rank m + n − 1 and (m + n) 2 − 1 dimensions.It has two sectors: an even sector describing bosons; and an odd sector sl(m|n)1 given by the module m|n ⊕ m|n describing fermions, it will be denoted below as 2mn for short.The general 3-grading for the sl(m|n) superalgebra is written as This grading can correspond on the graphical level to the cutting of any of the m + n − 1 nodes (bosonic or fermionic) of an arbitrary super Dynkin diagram of the (m + n)!/(m!n!) possible diagrams D[sl (m|n) ] (κ) .This is a special property of the linear symmetry where all nodes act like minuscule coweights.Following this decomposition, the representations of sl(m|n) also get partitioned.In what concerns us, the fundamental representation m|n of sl(m|n) decomposes into irreps of sl(k|l) and sl(m − k|n − l) as follows The subscripts refer to the charges of these subspaces with respect to the GL(1) corresponding to the cut node, they are calculated using the conditions (2.39), (2.
We can now write the action of the coweight in terms of the four corresponding projectors as with vanishing super trace The nilpotent operators X and Y belonging to N + and N − (3.where summation on repeated indices is omitted.The (b ai , c ia ) and b αλ , c λα are couples of bosonic harmonic oscillators while β aλ , γ λa and β αi , γ iα form fermioinc oscillators.The L-operator is computed using the nilpotency properties X 2 = 0, Y 2 = 0 as well as It expands as yielding the matrix form 16) where we have set h = l−k m−n .This matrix is in agreement with the general solution obtained in the superspin chain literature; see eq (2.20) in [41].The special families of solutions corresponding to the nodes of the distinguished Dynkin diagram are calculated in details in [37], where the particular Lax matrix with purely fermionic oscillators is obtained by considering the only fermionic node of the distinguished diagram.

Super L-operators of B(m|n) type
In this section, we study the B bf -family of orthosymplectic integrable superspin chains with internal symmetry given by the Lie superalgebra series We focus on the family of distinguished superspin B(m|n) chain associated to the Distinguished Dynkin diagram, and calculate the Lax operator L µ n+1 B m|n by using the 3-grading of the orthosymplectic B(m|n) in the formula (2.41).To begin, recall that the B(m|n) superalgebra is a Z 2 -graded Lie algebra of rank r B m|n = m + n, and dimension dim B m|n = 2(m + n) 2 + m + 3n.It splits like B(m|n)0 ⊕ B(m|n)1 with even part as, and odd part B(m|n)1 generated by the bi-fundamental representation (2m + 1, 2n) of so(2m + 1) ⊕ sp (2n) .The root system Φ B m|n of the Lie superalgebra B(m|n) has 2(m + n) 2 + 2n elements; it does also split into an even part Φ0 and an odd part Φ1.By using the unit bosonic weight vectors {ε a } 1≤a≤m and the fermionic {δ a } 1≤a≤n , the content of Φ0 reads as with cardinal |Φ0| = 2m 2 + 2n 2 , and the roots of Φ1 read like Φ1 : ±δ a , ± (ε a ± δ a ) (4.4) with |Φ1| = 2n+2mn.Given the set Φ B m|n , a remarkable simple root basis generating it is given by the distinguished basis ( β a , γ, α a ) having one fermionic root γ = δ n − ε 1 with length γ 2 = 0 ; and m + n − 1 bosonic ones as with β 2 a = −2 and α 2 a = 2.The distinguished basis is characterised by the following ordering of the fundamental unit weight vectors for which the super Cartan matrix has the entries actually distinguish In what follows, we will consider the distinguished basis and construct the Lax operator for the corresponding super B-chain.In this regard, recall that by distinguished orthosymplectic super chain of B-type, we mean the two following: (1) an integrable superspin B m|n chain made of "super atoms" arranged along a straight line; and realised in the CS theory (2.2) in terms of a set of parallel super Wilson lines traversed by a horizontal 't Hooft line.Such a realisation by topological defects looks like the one investigated in [37] for the case of sl(m|n) superspin chain; the main difference is that here the super spins are of B-type instead of A-type.
( B n|m following from the 3-grading [46] B(m|n or equivalently with nilpotents The fundamental representation decomposes in this case as We further use the reducibility 2 0 = 1 + ⊕ 1 − to reveal the Levi-like charges under the GL(1) of the cut node.Thus, we can rewrite the above decomposition as follows Now, in order to realize the components of the super L-operator, we work in the graded basis where |±⟩ refer to the two singlets 1 ± , the states |i⟩ with 1 ≤ i ≤ 2m − 1 correspond to the 2m − 1 and the fermionic |α⟩ with 1 ≤ α ≤ 2n to the symplectic vector 2n.
In this basis, the coweight µ n+1 associated to the decomposition (4.10) is written as where q 1 = q 2 = 0 and the projectors on the subspaces of the fundamental representation are defined by The 2(2m + 2n − 1) elements of the N ± (eq.4.11) are realized here as where (b i , c i ) are bosonic oscillators and (β α , γ α ) are fermionic ones; the generators To substitute these realizations into the L-operator formula, we calculate where we set and The nilpotency properties X 3 = 0 and Y 3 = 0 yield e X = 1 + X + 1 2 X 2 and e Y = 1 + Y + 1 2 Y 2 .By substituting (4.15) into z µ n+1 , we also have So, the expression of the super L-operator reads as follows and expands like where we have used the properties In the projector basis introduced before, the super L-operator can be written in matrix language as where the various blocks are given in terms of oscillators of the 't Hooft line phase space by as well as its odd part is given by (2n − 2) ⊕ (2n − 2).It has rank n and its dimension is equal to 2n 2 + n − 2. The super Cartan matrix reads for the distinguished basis as follows The associated distinguished super Dynkin diagram is depicted in Figure 6.For this supersymmetry, we have two possible 3-gradings of the form (2.31); we can therefore construct two super Lax matrices solving the RLL equation for the distinguished superspin chain of C-type.In fact, the first one is associated to the only fermionic node α 1 = δ − ε 1 and the second one to the bosonic node α n = 2ε n−1 .We will begin by working out the first L-operator L µ 1 C(n) that we will label as fermionic since it only contains fermionic oscillators.

The super L-operator
The first 3-grading for the Lie superalgebra C(n) is given by where the l µ 1 is identified with the bosonic subalgebra sp(2n − 2) and The dimensions of the osp(2|2n − 2) split like This decomposition is actually obtained from the distinguished Dynkin diagram by the cutting of the fermionic node µ 1 dual to α 1 .In what concerns us here, the fundamental representation is decomposed to representations of the so(2)⊕sp(2n−2) Here as well, we split the representation 2 0 into two singlets 1 + ⊕1 − carrying opposite charges under the coweight, we work in the basis decomposed as where the states |i⟩ with 1 ≤ i ≤ 2n − 2 correspond to the subspace 2n − 2. In the same way as before, we write where q = 0, ϱ ± = |±⟩ ⟨±| and Π = 2n−2 i=1 |i⟩ ⟨i| .The X and Y matrices are realized here as with generators We calculate where we have set Eventually, we have The L operator formula (2.41) along with properties (5.16) and yielding This L-operator carries only bosonic oscillator degrees of freedom, which is expected since the cut node corresponds to the only fermionic node of the distinguished diagram.

The super L-operator
The second possible 3-grading for the orthosymplectic C(n) is associated to the node 2δ n−1 dual to the coweight µ n .It reads in Lie superalgebra language as with in agreement with the dimensions splitting In this case, the fundamental representation splits as which is thought of as, meaning that we can work in a basis of the form where 1 ≤ i ≤ n − 1 and n − 1 ≤ ī ≤ 1 with ī = 2n − 1 − i.We define projectors on these four subspaces as follows and therefore the action of the coweight reads as Each of the nilpotents N + and N − split as (n − 1) + (n − 1) + (n − 1)(n − 2)/2, they are generated by couples X ī, X [i j] and Y i , Y [īj] where X ī and Y i are simply realized as while the X [i j] and Y [īj] are anti-symmetric in i and j and are given by We eventually have as well as the nilpotency properties X 2 = Y 2 = 0 leading to e X = 1 + X and e Y = 1 + Y. Notice that the oscillators here (b i , c ī) are of fermionic nature, while b [i j] , c [īj] are bosonic.We further have and This simplifies the expression of the L-operator as follows 6 Super L-operators of D(m|n) type In this section, we focus on the basic Lie superalgebra of D(m|n) type in order to compute its corresponding super L-operators characterizing D-type superspin chains with the internal symmetry having 2(m + n) 2 − m + n dimensions and the rank r D(m|n) = m + n.It is defined by an even part D(m|n)0 reading as and an odd part D(m|n)1 generated by the bi-fundamental (2m, 2n) representation of D(m|n)0.The D(m|n) superalgebra has multiple graphical descriptions with m+n nodes represented by graded simple roots {α i } 1≤i≤m+n .These are generated in terms of m+n fundamental unit weights given by the bosonic {ε a } 1≤a≤m realising the roots of so(2m), and the fermionic {δ a } 1≤a≤n realising the roots of sp(2n); their mixing gives fermionic roots of D(m|n)1.In these regards, recall that the super root system Φ D(m|n) of the Lie superalgebra D(m|n) has 2(m + n) 2 − 2m roots that split into an even set Φ0 and an odd part Φ1 with content reading as where we have set A remarkable simple root basis generating the super root system Φ D(m|n) is given by the distinguished basis (β a , γ,α a ) having one fermionic root γ as given here below This simple root basis can be collectively denoted shortly as αi = ( β a , γ n , α a ) with super label i = 1, ..., m + n.Notice that this distinguished basis is characterised by the ordering of the set {δ a , ε a } of the fundamental unit weight vectors as follows leading in turn to an ordering of the set of graded simple root as ( β a , γ n , α a ), and consequently to the Distinguished Dynkin diagram depicted in Figure 7 where the graded simple roots are also reported.Notice that the basis in eq(6.6)gives a very particular ordering of the set {δ a , ε a } where all the δ a 's are put on the left and all the ε a 's are on the right.In the general case where the δ a 's and the ε a 's are mixed, there are possibilities of orderings of the type (6.6).This variety of orderings indicates that generally speaking the Lie superalgebra D(m|n) has N D(m|n) possible super Dynkin diagrams and eventually N D(m|n) of varieties of superspin chains of D-type.For the example of D(3|1) = osp(6|2), we have 4! 3!×1!= 4 possible Dynkin diagrams as in Table 1 of [51].For the present study, we use the 3-gradings in Table eq(2.47) in order to generate two types of Lax operators for the distinguished superspin chain with D(m|n) symmetry.7 which permutes the spinor and cospinor roots α m+n and α m+n−1 , we can deduce that the coweights µ m+n and µ m+n−1 act in the same way on the Lie superalgebra osp(2m|2n) and therefore we have similar super L-operators.In fact, the 3-grading obtained by the cutting of one of the these nodes in the distinguished super Dynkin diagram is the same.We have with the super special linear sl(m|n) as a sub-superalgebra.For this breaking pattern, we have the dimensions In terms of the super labels i and ī introduced in (6.13), we can rewrite these intervals in a short way like Using these super labels, we can construct the operators involved in the expression of the super Lax operators L µ m+n = e X z µ m+n e Y associated with the breaking pattern (6.9) of the distinguished diagram of the Figure 7.
First, from eq(6.11) we learn that the action of the coweight µ m+n is given by |ᾱ⟩ ⟨ᾱ| (6.17)Second, the matrix operators X and Y in the expression e X z µ m+n e Y belong respectively to the nilpotents N + and N − ; they can be expanded in terms of representations of sl m ⊕ sl (n) .This feature follows from the decomposition of dim N ± (6.10) involving the antisymmetric representation of sl(m), the symmetric representation of sl(n) and the bi-fundamental representation.Thus, an explicit realization of X is given by using the generators (X [i j] , X (α β) , X iᾱ ) and the graded Darboux coordinates (b [i j] , f (α β) , β iᾱ ) as follows The super L-operator is calculated by substituting in (2.41) with (6.17-6.21)and using Xz where we have set The matrix form after multiplying with the overall factor z 1 2 is given in the basis (|i⟩ , |α⟩ , |ᾱ⟩ , |ī⟩) by The Φ, Ψ in (6. with q = 0. Similarly, the 2 (m + n − 1) generators of the nilpotent superalgebras N ± expand like where the (b i , c i ) are bosonic Darboux coordinates and (β α , γ α ) are fermionic homologue.The realisation of the generators of N ± is given by they split like Using these expression, we compute the powers of X and Y ; the non vanishing ones are given by Substituting, the super L-operator L µ n+1 D m|n = e X z µ n+1 e Y expands as follows In the basis (6.33a) and in terms of bosonic (b i , c i ) and fermioinc (β α , γ α ) oscillators, we have 43) where we have multiplied by an overall factor z.This matrix has a very similar structure to (4.26), the only difference concerns the size of the block of the subspace |i⟩ which is of 2m − 2 dimensions here.

Conclusion and comments
The present investigation is an extension of the results of the 4D CS/ Integrability correspondence formulated in [1,3], and further complemented in [7].In the latter, XXX spin chains were linked to a construction of line defects in four dimensional Chern-Simons theory, and L-operators solving the RLL equation were interpreted as the parallel transport on the phase space of magnetic 't Hooft line defects.This correspondence yields a simple and direct formula for the computation of minuscule L-operators based on Levi decompositions of the bosonic symmetry algebra, which are in turns directly deduced by cutting minuscule nodes from the associated Dynkin diagram.This general formula allowed to explicitly realize oscillator Lax operators for the spin chains with bosonic ABCDE symmetries.Some of these solutions are new to the spin chain literature while the others perfectly agree with the results obtained from Yangian based techniques.The generalization of this correspondence to the super case was initially treated in [37] for the case of sl(m|n) superspin chains.In analogy to the aforementioned bosonic construction, the oscillator realizations of super Lax operators solving the RLL equation for a superspin chain are deduced from special decompositions of the Lie superalgebra.Following this rationale, we constructed in this paper a list of Lax operators for superspin chains with internal symmetries given by the ABCD Lie superalgebras.In this regard, notice that these solutions are obtained for specific nodes of the super Dynkin Diagrams that act like minuscule coweights, meaning that they lead to Levi-like decompositions of these superalgebras.Notice moreover that we focused on the fundamental representation for all the symmetries treated here, indicating that the superspin states of the super-atoms of the super chains are represented in the fundamental.The graded L-operators obtained here are to our knowledge, still missing in the superspin chain literature, except for the solutions of sl(m|n) chains that were computed using degenerate solutions of the graded Yang-Baxter equation, see eq(2.20) in [41].These matrices were rederived in [37] from the 4D Chern-Simons with SL(m|n) symmetry focusing on the distinguished Dynkin diagram and by extending features of the bosonic sl(m) spin chain.In this linear symmetry, all simple nodes are associated to minuscule coweights.Here, we gave a more general expression of these super Lax matrices for any node beyond the distinguished Dynkin diagram of sl(m|n); see (3.16) where the bosonic and fermionic oscillators are explicitly distinguished.Notice that the bosonic L-operators of the sl(m) spin chain [7,12] can be recovered as a special case of the graded distinguished solutions by simply taking n = 0. Unlike the A (m|n) superalgebra, the distinguished Dynkin diagram of the B (m|n) superalgebra only leads to one Levi-like decomposition associated to the distinguished Dynkin diagram given by Figure 5.The graded Lax operator of the osp(2m|2n) superspin chain was constructed for this specific case as presented in (4.26).This in [46]- [48].To end this conclusion, we collect in Table 1 the expressions of the super oscillator realisations of Lax operators for the families A(m − 1 | n − 1), B(m | n), C(n) and D(m | n) Lie superalgebras with 3-grading decompositions as g = l µ ⊕ N + ⊕ N − where l µ is Levi-like subalgebra and N ± nilpotent superalgebras.

Appendix: Explicit check of solutions
This appendix is added to the revised version of our paper in order to : • (i) Compare our orthosymplectic super Lax matrices computed from the 4D CS theory, to similar solutions that appeared in [52] shortly after the first version of this paper, and that are based on algebraic analysis.
• (ii) Comment the quantum versions of the classical orthogonal super Lax operators given here above.Notice that the quantum upgrading of the sl(m|n) super Lax operators calculated in 4D CS is detailed in [37].
• (iii) Describe the R-matrix and the RLL equation of integrability for an orthosymplectic superspin chain system, and refer to the explicit verification of the matrices L D m|n as appropriate solutions.
First, notice that the authors in [52]  D m|n have the same structure, and only differ in their dimensions; and the same for L µ n C(n) and L µ m+n D m|n .We will focus below on the comparison between Eq(5.54) of [52] with L µ n+1 D m|n , and Eq(4.12) of [52] with L µ m+n D m|n .But before that, we recall the explicit integrability condition for the orthosymplectic superspin chains.
• Orthosymplectic RLL equation To solve the RLL relations for the orthosymplectic families, we consider the orthosymplectic R-matrix reading for the superalgebra osp(N |M ) with pair M , as [53,54] R(z) = z(z + κ)I d + (z + κ) P − zQ (8.1) We have here 2κ = (N − M − 2), I d is the identiy operator, P is the permutation operator expressed in terms of the canonical matrix generators e xy as, • Eq(5.54) of [52] with L Although a quick look into the quadratic Lax matrix of [52] reveals the similarity with the structure of L µ n+1 B m|n and L µ n+1 D m|n , we think it is interesting to explicitly show this agreement by analysing the special notations used in [52] and linking them to ours.First, because [52] provides a computational check for the verification of the RLL equation, and second to discuss the quantum uplifting of these orthosymplectic solutions.For the D-type superalgebra that the authors in [52] note as osp(2n|2m), where the role of the integers n and m is opposite to our notation, the graded quadratic Lax matrix (Eq(5.54)) is given by In these relations, M is a graded 2 (n − 1 + m) × 2 (n − 1 + m) matrix given in Eq(5.63) [52].The complex u and its complex conjugate ū (adjoint conjugate u † for quantum version) are graded super-oscillators with 2 (n − 1 + m) components distributed into 2 (n − 1) bosonic oscillators (a i , āi ) , and 2m fermionic oscillators (c α , cα ).These components were ordered in [52] as follows u = (a 2 , ..., a n ; c n+1 , ..., c n+m+1 ; a n+2m+1 , ..., a 2n+2m−1 ) T ū = (ā 2 , ..., ān ; cn+1 , ..., cn+m+1 ; ān+2m+1 , ..., ā2n+2m−1 ) T (8.7) To make contact with our 4D CS based solution (6.43), we write the graded vectors (u, ū) as u = (b, β) , ū = (c, γ) (8.8) or equivalently by using contravariant u a and covariant ūa super labels like (u a ,ū a ) , u a = b i , β α , ūa = (c i , γ α ) (8.9) Notice that generally speaking, the complex variables ūa are not necessarily the complex conjugates of u a [i.e: ūa ̸ = (u a ) † ].Also, in (8.9), the (b i , c i ) are bosonic quantities and the (β α , γ α ) are fermionic; they are ordered as Using this new notation, we can identify the quadratic Lax matrix of (8.5) derived in [52], with our spinor-like matrix L µ n+1 D m|n given by eq(6.43), by using the following correspondence superoscillator bosonic fermionic which should be compared with (8.6).The metric G ab is equal to −δ ab , and G ab is its inverse.Notice here that the classical quantity c i b i + γ α β α reading also like the mean value This expression indicates that the number κ in the super Lax matrix (8.5) is just the quantity 2n − 2m which sometimes is termed as the Maxwell-Callading index.
• Eq(4.12) of [52] with L Regarding the identification of the orthosymplectic rational RLL solution given in matrix form in eq(4.12) of [52], with the spinor-like super Lax matrix L µ n+m D m|n , notice that they are equivalent by the multiplication of L µ n+m D m|n by the overall factor z −1 , which is permitted by the RLL equation symmetries.The two matrices are represented in similar bases dividing the 2m + 2n dimensions as (m + n, m + n), the difference is that the integers n and m play opposite roles in [52] where the superalgebra is taken as D(n|m) = osp(2n|2m) (instead of osp(2m|2n) for us).However, since our basis is ordered as (m, n, n, m) , and theirs as (n, m, m, n) , we have lookalike matrices where we have the correspondence L µ n+m D m|n from 4D CS Eq(6.27)Eq(4.12) of [52] Φ Finally, notice that the quadratic graded matrix (5.54) of [52] is proven as a solution of the RLL equation with the orthosymplectic R-matrix (8.1); which further verifies the matrices L µ n+1 D m|n and L µ n+1 B m|n as super Lax matrices for the orthosymplectic superspin chains of B and D type.Moreover, the elements of the matrix (4.12) of [52] verify the commutation relations (8.4), and thus provide a check for the solutions L C(n) is a special solution because it is purely bosonic.It should be verified by using the non-graded version of the orthosymplectic R-matrix, which is given by the orthogonal R-matrix Eq(2.1) in [38].Indeed, this matrix is equal to the verified Lax matrix of the orthogonal so(2n) spin chain given in eq(3.42) of [13]; see also eq(4.12) of [38].

Figure 1 .
Figure 1.Realization of an sl(m|n) superspin chain of L nodes in the fundamental representation using super line defects in the 4D Chern Simons.

Figure 2 .
Figure 2. Graphic representation of the RLL equation in terms of intersecting line defects in SL(m|n) 4D CS theory.

7 )Figure 5 .
Figure 5. Distinguished Dynkin diagram of the B(m|n) superalgebra having one fermionic simple root in Green color.

8 )
types of super Dynkin diagrams D[B m|n ] having m+n nodes represented by different forms of graded simple roots α.

)
The vertical super Wilson lines are run by graded quantum states (δ a |ε a ) ordered as in eq(4.6) and interpreted in terms of the distinguished Dynkin diagram DD B m|n given by Figure 5.Given this orthosymplectic superspin chain configuration and the associated super Dynkin diagram characterising each super Wilson line (a super-atom), we can calculate the distinguished super L-operator L µ n+1

Figure 6 .
Figure 6.Distinguished Dynkin diagram of the C(n) superalgebra, the only fermionic node is represented in green.

Figure 7 .
Figure 7. Distinguished Dynkin diagram of the D(m|n) superalgebra having one fermionic simple root in Green color.Here, we have ε 2 i = 1 and δ 2 i = −1.
The first one is referred to as the spinorial L-operator L µ m+n D m|n because it is linked to the (co)spinorial nodes of the super diagram 7. The second one is labeled as L µ n+1 D m|n since it concerns the coweight µ n+1 associated to ε 1 − ε 2 in 7. 6.1 The super L-operator L µ m+n D m|n Due to the Z 2 automorphism symmetry of the distinguished super Dynkin diagram of the Figure
15))due to graded commutativity, gets mapped at the quantum level into the sum of the operators (ĉ i bi + bi ĉi )/2+ (γ α βα + βα γα )/2.By using the graded commutation relations, one generates a vaccum contribution as shown below