Embedding Integrable Superspin Chain in String Theory

Using results on topological line defects of 4D Chern-Simons theory and the algebra/cycle homology correspondence in complex surfaces $\mathcal{S}$ with ADE singularities, we study the graded properties of the $sl(m|n)$ chain and its embedding in string theory. Because of the $\mathbb{Z}_{2}$-grading of $ sl(m|n)$, we show that the $\left( m+n\right) !/m!n!$ varieties of superspin chains with underlying super geometries have different cycle homologies. We investigate the algebraic and homological features of these integrable quantum chains and give a link between graded 2-cycles and genus-g Rieman surfaces $\Sigma _{g}$. Moreover, using homology language, we yield the brane realisation of the $sl(m|n)$ chain in type IIA string and its uplift to M-theory. Other \textrm{aspects} like graded complex surfaces with $sl(m|n)$ singularity as well as super magnons are also described.


Introduction
String theory represents a rich framework to investigate the quantum field theories in diverse dimensions and quantum integrable models as well as the connection between them [1]- [4]. This link, commonly termed as the Gauge/Integrability (G/I) correspondence [5,6,7,8,9], relates two areas of quantum physics and can be manifested in the framework of string theory using brane realisations and dualities [10]- [16].
First hints of the G/I correspondence linked Bethe Anzats soluble 2D integrable systems to 2D N = 2 supersymmetric gauge theories [5,6]. But more recently, these integrable models were shown to make a natural appearance in the 4D bosonic Chern Simons (CS) theory equipped with line and surface defects that generate lattice models [17]- [23]. In this context, the XXX Heisenberg spin chain is realised using a set of Wilson and 't Hooft lines such that their gauge configuration reproduces the degrees of freedom of the spin chain and allows to recover solutions of this integrable system such as the Yang-Baxter R-matrix and the Lax operator [24]- [31].
In string theory language, the 4D CS describing integrable lattice models can be realised via stacks of branes like D4s ending on NS5-brane in type IIA string theory; while Yang-Baxter equation and the Yangian can be obtained via the Hilbert space of a 6D gauge theory by applying T-and S-duality [25]. Linking CS gauge theory to maximally supersymmetric and topologically twisted Yang Mills theories; and applying dualities in the string theory context, allows for the unification of integrability in supersymmetric gauge theories [26]. For example, quantum integrable systems such as the eight-vertex model and the XYZ spin chain [32] were shown to appear in different field theory setups, all related by dualities in string theory.
All of these results represent building blocks for the study of the integrable sl(m|n) superspin chain that we are concerned about here. As a generalisation of the sl(m) bosonic chain, the sl(m|n) integrable system should be endowed with an extended G/I correspondence (super-G/I below). Indeed, in [33], closed spin chains characterized by the super Yangian of gl(m|n) were linked to a class of N = (2, 2) supersymmetric gauge theories in 2D; such that the supersymmetric vacua were identified with the Bethe states of the superspin chain. On the other side, these integrable models are also identified with a 4D Chern Simons theory having a super gauge symmetry and equipped with super line defects. In [34], the super-G/I allowed us to recover the Lax operator of the sl(m|n) spin chain by studying the coupling of Wilson and 't Hooft super lines and properties of the gauge field bundles in the CS theory with SL(m|n) symmetry. Moreover, in the recent inspiring work [16], a brane realisation of the superspin chain of gl(m|n) Yangian was employed to manifest the equivalence of the gauge theories appearing in the gauge side of the super-G/I and the dualities relating them in string theory [35,36]. Another result of [16] concerns the brane and supersymmetric quiver interpretations of magnons for the family of sl(m|n) superspin chains. These super magnons are generalisations of the standard sl (m) magnons which are quasi-particles of the integrable quantum spin chains given by excitation modes of the ground state |Ω Λ labeled by a highest weight vector Λ of the symmetry group. For the SU (2) spin, the magnons are quantised spin waves that was extensively studied in physical literature with regards to the magnetic properties of materials and the entanglement [37]- [40]. In particular, the superspin chain models that we are interested in here are of particular utility due to the special structure of super algebras and the rich structure they hold. Thanks to super-G/I, these interesting algebraic features lead to exotic phenomenons to discover on the string theory side and at the level of gauge field configurations as well as in realisations using topological line defects of the super CS theory. In fact, it was shown in [41], that for the psu(2, 2|4) integrable super spin chain underlying the AdS/CFT correspondence, the different Dynkin diagrams and Cartan matrices of the super Lie algebra are related via fermionic dualities. In [42], the authors encoded the possible sets of Bethe equations of the SU(m|n) spin chain arising from different Dynkin diagrams in the QQ-system, and described the transformation properties of this system under fermionic and bosonic dualities allowing to move between two different descriptions of the same spin chain.
In this paper, we make a further step towards the understanding of the super-G/I correspondence with insertion in string theory. We focuss on the specific aspects of the sl (m|n) superspin chain family arising from the Z 2 -grading of the super algebra and the multiplicity of its Dynkin super diagrams; and investigate connections with super line defects of the CS theory. We also study the embedding of the super chain in M-theory and in type II strings with two main objectives.  Secondly, we take advantage of results on complex surfaces with ADE singularities used in string compactifications [43]- [49] to study their generalisation to complex manifolds with super singularity given by the sl(m|n) gauge symmetry. We derive the Super Algebra/Homology correspondence (Super-A/H) in complex surfaces with super singularity extending the Du Val geometries of the Table 5. We show that, unlike ADE singularities involving 2-cyclesC with self intersectionsC 2 = −2, the super geometries have three kinds of 2-cycles C with self intersections C 2 = 0, ±2 including the usual C 2 = −2 in addition the two exotic C 2 = 0, +2. Regarding the embedding of the closed super chain R t × S 1 θ in M-theory, we construct the structure of the 11D space time M 1,10 and show that it has the fibration R t × S 1 θ × S 1 M × Y 4 with complex 4D manifold as depicted by the graphs of Figure 1 giving a lattice description of Y 4 . The S 1 M is the usual circle of M- The presentation is as follows. In section 2, we describe the degrees of freedom of the sl(m|n) chain and propose a short path for representing this integrable chain in terms of line defects of 4D Chern-Simons theory and branes in type II strings and in M-theory.
In section 3, we show that there are (m + n)!/m!n! varieties of sl(m|n) superspin chains and give their properties. In section 4, we study the bridge between the bosonic sl(m+n) and the graded sl(m|n) by using their Cartan matrices A sl m+n and K sl m|n . We introduce colored Dynkin super diagrams and use them to describe the 2-cycle homology of complex manifolds with sl(m|n) singularity. In section 5, we use results obtained above to embed the sl(m|n) chain in type II strings and in M-theory. In section 6, we construct the geometry of the M-theory manifold M 1,10 hosting the M-brane system realising the SL(m|n) chain with flavor symmetry U(N f ). In section 7, we give a conclusion and comments. In the appendix, we describe the super magnons by using the super Yangian algebraic method. The study concerning magnons was reported in the appendix in order to preserve the sequence of ideas in the presentation of our work.

Superspin chain: Degrees of freedom and defects
In this section, we investigate the degrees of freedom of the sl(m|n) superspin chain family and employ them to model this integrable chain in terms of line defects of the 4D Chern-Simons (CS) theory. We use the algebraic properties of this superspin chain to (i) introduce the symmetries of the underlying topological field theory; in particular the SL(m|n) gauge invariance of the CS action and the U (N f ) flavor symmetry of the line defects. (ii) motivate the brane realisations in type II strings and in M-theory.
We will use convenient terminologies like for example super atoms and super chains for homogeneous chains with sites having the same superspin; also the sl(m|n) super Cartan matrix associated to Dynkin super diagrams of sl(m|n).

sl(m|n) superspin chain revisited
We begin by considering a finite integrable sl(m|n) superspin chain with L sites Υ l to which we refer as the superspin "atoms" or simply super atoms. We assume that the super chain has the following specific properties: (α) It is closed with length L; that is having L super atoms Υ 1 , Υ 2 , ..., Υ L .
(β) The closure (Υ L+1 ≡ Υ 1 ) is realised by twisted boundary conditions as described in [16]; we refer to the space dimension of this chain by the circle S 1 θ .
(γ) It is an homogenous chain in the sense that all of its super atoms Υ l are identical; thus having the same values for the superspin states.
Graphically, the super chain S 1 θ is represented by the Figure 2 with red cross points referring to the Υ l 's. The superspin atoms of the chain have two kinds of degrees of freedom: "extrinsic" degrees and "intrinsic" ones. We describe them below with some details by first considering a representative super atom Υ, say the first super atom Υ 1 , and then turning to the chain S 1 θ = {Υ l } 1≤l≤L . (a) The λ η 's are weight vectors describing the superspin states for the atom Υ; they form a highest weight (HW) representation of sl(m|n). For the interesting case where the states of Υ sit in the fundamental representation, the weight vectors are given by the (m + n) unit weight vectors ǫ A , ..., ǫ m+n . For this natural choice, we have λ η ≡ λ A with However, for generic sl(m|n) representations, the weight vectors λ η are given by linear combinations like λ ηA ǫ A . Below, we will mainly focuss on the (m+n)-dimensional representation (2.1) to which we refer as R (ω 1 ) because the HW vector ǫ 1 is given by the fundamental coweight ω 1 ; i.e In this regard, notice that sl(m|n) has (m + n − 1) fundamental coweights ω A ; i.e: these are duals of the simple roots α A to be described in details later. We have (m + n − 1) simple roots The spectral parameter z termed sometimes as rapidity of the atom, with reference to the spectral parameter of scattering theory [13,51,52]. This complex z is also identified with the usual complex parameter introduced in the realisation of representations of the Yangian algebra Y [sl(m|n)] in relation to the study of quantum integrable spin chains.
Recall that the Yangian superalgebra is generated by the transfer matrix T AB (z) obeying the famous RTT equation [53] R with R (z 1 − z 2 ) being the R-matrix and T (z) = E AB ⊕T AB (z). Notice that the spectral parameter z is also used as a complex coordinate giving the position of a line defect L (z) in the holomorphic plane of 4D Chern-Simons gauge theory as formulated in [17].

B) Degrees of freedom of the super chain
Before approaching the super chain, notice first that from the above description, a given super atom Υ is characterised by the degrees of freedom (ξ; λ ω 1 , z) that describe the position ξ, the HW vector λ ω 1 and the spectral parameter z. For convenience, we rewrite these degrees like where we exhibited the super label A of (2.1). The A = 1, ..., m + n is the label of the fundamental coweight ω 1 . Eq(2.6) means that at the quantum level, we can think of the quantum state |Υ describing the ground state of the super atom Υ as follows To fix the ideas and for later use, we give in Table 1 the classical positions ξ 1 , ξ 2 , ..., ξ L super atom real position quantum state The ground state of the super chain minimizing the energy of the chain model is given In what follows, we will refer to this |Ψ [̺] state as the pseudo-vacuum of the super chain.

C) Symmetries and Super Magnons
To smooth the path towards the interpretation in terms of line defects and the brane realisation of super chains in type II string theory and M-theory, we need to replace the classical variable ξ l by a one suitable for this purpose. As one of the roles of the classical ξ l s is to discriminate the L super atoms of the super chain, we can use instead a flavor symmetry group which turns out to be given by the unitary where for convenience we have set L = N f . In this way of doing, the N f positions {ξ l } are replaced by the weight vectors {e l } of the fundamental representation of U (N f ) ; that is by substituting with ξ l → e l (2.11) in eq(2.8). Therefore, we obtain the following quantum description of the pseudo-vacuum To deal with these super atomic states, we will think about them in terms of the factori- Notice the following three features: (i) the three blocks |e l , λ l A and |z l are respectively related with the flavor U (N f ), the gauge SL (m|n) and the Yangian Y SL(m|n) . (ii) The properties of the composite bloc e l ; λ l A = |e l ⊗ λ l A are associated with the finite dimensional symmetry, while the |e l ⊗ Ω l A is described by the infinite dimensional invariance We focus here on the sector e l ; λ l A with finite symmetry (2.15); and report to the Appendix the analysis of the sector |e l ⊗ Ω l A with full symmetry (2.16). The full sector (2.13) is useful for the study of super magnon states (7.3-7.7) that we give here Notice that to fix the ideas and as an anticipation of the sector |z l ; one can think about the spectral parameters z l in (2.8) as the zeros of the following holomorphic polynomial to be further discussed later in details; see eqs(6.8,6.14) and (6. with abelian charge operator h l . As such, the N f phases of the super chain can be combined into the following compact operator These h l charge operators generate an abelian U (1) N f symmetry which is isomorphic to the Cartan subgroup of the non abelian U (N f ) flavor symmetry with group elements as standing for the N 2 f generators of U (N f ). The use of the flavor symmetry to deal with the features of the superspin chain has interesting consequences. In particular, the super chain has the larger symmetry Moreover, the presence of the complex z l 's and the flavor invariance in the formulation of degrees of freedom of the super chain is very suggestive. The z l 's are interpreted below as the positions of N f parallel (vertical) topological line defects γ z l in the holomorphic plane of the 4D Chern-Simons theory. We will think about these N f topological lines as describing electrically charged Wilson lines that we denote as such that the electric charges are given by highest weight vectors λ l of the superspin representations R λ l of sl(m|n). Regarding the brane realisation of these super chain The coupling between the Wilsons is mediated by a horizontal 't Hooft line.
states, the topological lines are imagined as living at the intersection of two or more branes [50]. From the type IIA string view, the super atoms, thought of as line defects W l , are represented by N f D4 branes intersecting NS5 and D2. In type IIB string, these W l 's involve D5 branes; and in M-theory, they involve M5 branes. These representations in the brane language are summarised in the Table 2. In these regards, notice that graphically speaking, the interaction between the super atoms of the chain is insured by a horizontal line tH[γ z ] crossing all the W l 's.
As a result of this description, the superspin chain of the Figure 2 gets mapped to the line defect system given by the Figure 3 and the stringy realisation with intersecting brane as in the Table 2. For the example of the M-brane picture, see Figure 7.

Topological field description
Here, we build the bridge between the super chain degrees of freedom described above and the field modeling of the 4D Chern-Simons (CS) theory. This bridge is based on the line defects W γ /tH l and the spectral parameter z that appears in (2.19). The complex z is the coordinate variable of the complex line C in the CS theory.
Recall that the action S [A] of the topological CS gauge theory reads in absence of external sources as follows [17], In this gauge field action, C is the complex holomorphic line; and the R 2 (x, y) is the real topological plane where live the external line defects such as the electrically charged Wilson lines and the magnetically charged 't Hooft line.
Moreover, the 1-form gauge potential A in (2.27) is a function of the variables (x, y; z) parameterising R 2 ×C, it is a Z 2 -graded matrix. For the case of 3D CS with supergroups, see [18].
This potential also expands like A = t a A a where t a generate the Lie superalgebra sl (m|n) and A a is a partial gauge connection given by In the absence of external line defects, the gauge action is as in eq(2.27), and the field equation of A following from the variation δS [A] = 0 reads as in agreement with the topological nature of the 4D Chern-Simons theory without external sources.
In the presence of 't Hooft lines as in Figure 2, the gauge curvature F is no longer trivial and looks like the curvature of a Dirac monopole of 4D Yang Mills theory [27,54,55,56]. In this situation, the gauge connection A on the complement of the lines defines a topologically non trivial bundle on the 2-spheres surrounding these lines.
We end this section by giving some useful algebraic tools regarding the sl(m|n) chain with super atoms Υ l carrying generic representation charges with HW vectors λ l . First, recall that the HW vectors λ l are expanded in terms of the basis vectors ǫ A (2.1) like, As the ǫ A 's can be also expressed in terms of the (m + n − 1) fundamental coweights ω A of sl(m|n), or equivalently in terms of its simple roots α A ; we can rewrite (2.31) either as a sum over the ω A 's or as a sum over the α A 's. By using the ω A 's, we have with the λ l 's defining HW representations R λ l of the Lie superalgebra sl(m|n). As the quantum state Ψ l A of the super atoms given in (2.12) transforms in the representation R λ l ≡ R l , the full state spectrum of the super chain transforms in the tensor product representation that is reducible as sum over irreducible representations of sl(m|n). By using the weight vector basis ǫ A of the superalgebra sl(m|n) with inner product ǫ A , ǫ B = (−) |A| δ AB where |A| refers to the two possible degrees0 and1, we can expand the HW vectors λ l s as follows To concretize our investigation, we think it interesting to consider a particular Lie superalgebra and realize the objectives of this study on that example. In what follows, we will focus on the sl(3|2) superspin model and comment on the extension of the results to the sl(m|n) family with m > n.
3 sl(3|2) superspin models: algebraic set up In this section, we study the algebraic setup of the superspin chain with length N f in order to (i) classify the varieties of such type of integrable super chains; and (ii) give a front matter towards their embedding in type II strings that will be investigated in sections 5 and 6.
First, we show that there are 10 varieties of sl(3|2) superspin chains and explore their properties. In general, for the generic sl(m|n) chain with m > n, there are varieties of sl(m|n) graded chains. This diversity constitutes a special feature of superspin chains due to the Z 2 -grading of Lie superalgebras that does not occur in the bosonic sl(m) chain. We also show that among the ten sl(3|2) superspin models, four of them are somehow redundant; thus leaving six basic sl(3|2) super chains. One of these super chains is very special; it is termed as the distinguished chain, and will be the subject of a detailed study. Related ideas regarding the study of the gl(n|m) spin chain has been developed in [57]. the additional 4 are related to their homologues by a mirror symmetry.

Ten varieties of sl(3|2) chains
We start by recalling that the sl(3|2) chain given by the Figure 2 has N f super atoms with degrees of freedom as in (2.8-2.12). Each super atom Υ is specified by a set of quantum charges; in particular the five sl(3|2) ones indicating that |Υ is generated by the following quantum states These weight states |ǫ A describe the superspin representation of sl(3|2) with highest weight vector ǫ 1 . In representation theory language, this HW vector corresponds to the fundamental coweight ω 1 of sl(3|2); this means that ǫ 1 = ω 1 is as in (2.2) while the other four in (3.2) descend from it. In this regard, recall that, as far as the algebraic fundamentals of sl(3|2) are concerned, the ω 1 is the dual of the simple root α 1 of the Lie superalgebra sl(3|2). Obviously, this is not the unique fundamental coweight as sl(3|2) has other fundamental coweights ω A dual to the other simple roots α A in the sense that they obey ω A .α B = δ AB . This duality relation is remarkably solved by AB is the super Cartan matrix of sl(3|2). However, because of the Z 2 -grading implying the decomposition, AB is not uniquely defined. In fact, the sl(3|2) superalgebra has (i) 10 possible Dynkin-like super diagrams (DSD) as shown in Figure 4; and then (ii) 10 associated Figure 4: The ten varieties of Dynkin super diagrams for the sl(3|2) Lie superalgebra.
They can be organised like (4 + + 1 0 + 1 0 ) + 4 − with regards to reflection acting as N q → N −q given by eq(3.6). So, the four 4 − of the 10 DSDs are redundant as they can be recovered by reflection of the four 4 + .
super Cartan matrices that we denote like Since the roots, weights and super Cartan matrices K sl 3|2 are highly involved in the study of the quantum properties of superspins, we end up with ten classes of sl(3|2) super chains. However, not all of them are different because of the mirror symmetry exhibited by the Figure 4 and corresponding to the reflection To get more insight into this special feature of superspin chains compared to the usual bosonic-like chains, notice that simple roots α A of sl(3|2) are realised in terms of graded unit weight vectors as follows just like for sl (N) algebras. In particular, here N = m + n = 5; therefore this relation extends the usual realisation of simple roots for the bosonic Lie algebra sl (N) to the sl(m|n) superalgebra with the difference that now (3.7) depends on the grading of the weight vectors ǫ A . In fact, depending on the ordering of the ǫ A 's, we distinguish 5! 3!2! = 10 (3.8) possibilities given by the permutations of the weight basis vectors (3.2). Notice that this basis is made of three even Notice also that a direct consequence of the 10 possible realisations of α A is the existence of 10 varieties of the super Cartan matrix defined as By using (3.3), we deduce that as for the simple roots and K sl 3|2 AB ; there are also 10 ways to realise the fundamental coweights ω A .
In conclusion, the finite dimensional Lie superalgebra sl(3|2) has apparently 10 different DSDs. These super diagrams are characterised, amongst others, by the number n f of fermionic roots (1 ≤ n f ≤ 4). We give in Table 3 the interesting six DSDs with fermionic nodes represented by the green color. We also give the corresponding weight vector bases and the "lengths" of the simple roots. The first DSD in the Table 3 has three bosonic basis (e 1 , e 2 , e 3 , e 4 , e 5 ) α 2 a = 2 α 2 a = −2 α 2 a = 0 Dynkin diagram Table 3: Six of the ten Dynkin super diagrams of the Lie superalgebra sl(3|2). This list is ordered according to the number of green nodes.
nodes (two reds and one blue); and one fermionic green node. The other DSDs have more than one fermionic (green) node.

Building the super Cartan matrices (3.5)
Here, we construct the explicit expressions of the super Cartan matrices associated with the six different DSDs listed in the Table 3. We also give the expression of their inverses as they are important for the determination of the coweights ω A in terms of the roots α A .
• Super Cartan matrix K I sl 3|2 In this case, the simple roots are realised as The corresponding super Cartan matrix K I sl 3|2 describing the first DSD I in the table 3 reads as follows It has det K I sl 3|2 = 1. The associated DSD I has three bosonic simple roots α 1 , α 2 , α 4 and one fermionic α 3 ; they are as follows: (a) two simple roots with length α 2 1 = α 2 2 = 2, they correspond to the simple roots of sl (3) and are given by the two red nodes in the first row of Table 3.
(b) one bosonic simple root given by the blue node, it corresponds to sl (2) but with α 2 4 = −2 making its geometrical interpretation very interesting; a proposal using the Euler characteristic χ will be given later on.
These three bosonic simple roots indicate that K I sl 3|2 concerns the bosonic subsymmetry (c) One fermionic-like simple root α 3 given by the green node having the remarkable property α 2 3 = 0; it is the unique odd simple root in the root system of sl(3|2) with super Cartan given by K I sl 3|2 . This root can be interpreted in terms of the following Lie sub-superalgebra of sl(3|2), sl (1|1) (3.13) This is the fermionic homologue of the bosonic sl (2). A geometric interpretation of this root in terms of 2-cycles will be given later.
The inverse of (3.11) is given by it has negative entries.
The simple roots are realised as The Cartan matrix K II sl 3|2 corresponding to the second DSD II and its inverse read as follows, We have det K II sl 3|2 = 1. The DSD II has four nodes: two bosonic and two fermionic. The two bosonic roots are given by α 1 and α 3 ; the first obeys α 2 1 = 2 while the second has α 2 3 = −2. These roots indicate that the underlying bosonic symmetry of sl(3|2) with super The two fermionic nodes are given by the simple roots α 2 and α 4 ; they have vanishing lengths α 2 2 = α 2 4 = 0 and correspond to The simple roots are realised as The Cartan matrix K III sl 3|2 describing the third DSD III and its inverse are given by, with det K III sl 3|2 = 1. This DSD III has two bosonic simple roots with α 2 2 = α 2 3 = 2 describing the bosonic symmetry The DSD III has also two fermionic simple roots with α 2 1 = α 2 4 = 0 corresponding to The simple roots are realised by The Cartan matrix K IV sl 3|2 for the fourth DSD IV and its inverse read as follows, The DSD IV has one bosonic-like simple root with α 2 1 = 2 underlying the bosonic subsymmetry The DSD IV also has three fermionic-like simple roots with α 2 2 = α 2 3 = α 2 4 = 0.
The simple roots are now realised by The Cartan matrix K V sl 3|2 for the fifth DSD V and its inverse are given by, with det K IV sl 3|2 = 1. The DSD V has one bosonic-like simple root with α 2 2 = 2 and three fermionic-like simple roots with α 2 The simple roots read in this case like This super Cartan matrix is very special as it is associated to the purely fermionic super Dynkin diagram with all nodes odd.
The super K V I sl 3|2 and its inverse are given by, In what follows, we will focuss on the distinguished sl(3|2) chain model with DSD I as an example. This super chain is also imagined as a representative of the distinguished class of the sl(m|n) family (m > n). The corresponding Distinguished Dynkin Super-Diagram (for short DDSD) has one fermionic simple root given by the green node in DSD I of eq(3). The DDSD is described by the generalised Cartan matrix with graded simple roots realised as in eq(3.7), namely In matrix representation, the K sl 3|2 AB reads like in eq(3.11) with inverse as in (3.14).

Distinguished superspin chain
Here, we investigate the quantum properties of the distinguished sl(3|2) super chain by comparing it with the bosonic sl(5) chain. The distinguished sl(3|2) and the sl(5) share some basic features that we will use to build the algebraic geometry interpretation. An example of the shared properties, besides rank and dimension, are given by the bosonic sl(3) ⊕ sl(2) which appears as the maximal Lie subalgebra.

Bosonic sl(5) spin chain
We begin by recalling that for the bosonic-like sl(5) spin chain, the quantum states of the atoms are generated by five weight vectorsεā labeled byā = 1, 2, 3, 4, 5. This is the same number as for the five ǫ A 's regarding the super atoms of the distinguished sl(3|2) superspin chain. The sl(5) atomic weight charges generate a 5D HW representation of sl(5) with basis vectors as By using the four fundamental coweightsωā of the bosonic sl (5), the five weight vectors εā are realised as followsε obeying the sl (5) traceless conditioñ requiring in turns the following relationship with the simple rootsαā of sl(5) Notice that the highest weight vectorε 1 is precisely the fundamental coweightω 1 which is dual to the simple rootα 1 =ε 1 −ε 2 ; that is In this context, recall that the Lie algebra sl(5) has four simple rootsαā =εā −εā +1 with intersection matrix A sl 5 ab =αā.αb given by Its determinant is equal to 5 and its inverse is given by In what follows, we will use properties of the sl(5) spin chain captured by (3.32) to unveil characteristics of the distinguished sl(3|2) super chain. Notice that sl(5) and the distinguished sl(3|2) have the same rank 4 and the same dimension 24. Moreover, they both belong to the finite dimensional special linear "sl" family manifested by a linear Dynkin diagram.

Super atoms in the distinguished sl(3|2) chain
Using the algebraic properties of the bosonic sl (5) described above, we construct below those homologous properties in the distinguished sl(3|2) which are satisfied by the super weights ǫ A . This approach gives a short way towards the interpretation of exotic properties regarding the super atoms and the distinguished sl(3|2) superspin chain. By the word exotic, we mean the special values of the elements of the Cartan matrices A sl 5 and Now, we think about the invertible A sl 5 and the invertible K sl 3|2 as two cousin matrices that offer a bridge between the sl(5) spin chain and the sl(3|2) super chain. This link implies that the quantum states (3.2) namely |ǫ 1 , |ǫ 2 , |ǫ 3 , |ǫ 4 , |ǫ 5 can be put in correspondence with the |ε 1 , |ε 2 , |ε 3 , |ε 4 , |ε 5 satisfying (3.32-3.36); i.e: As theεā of the sl(5) are nicely related to the simple rootsαā and the simple coweights ωā, we investigate below the extension of this feature to the ǫ A 's of sl(3|2).
A) root/weight duality in sl(3|2) As for sl (5), the four fundamental coweights ω 1 , ω 2 , ω 3 , ω 4 of the distinguished sl(3|2) are dual to the four simple roots α 1 , α 2 , α 3 , α 4 . These two basic quantities are related by the super Cartan matrix like α A = K sl 3|2 AB ω B . By substituting, we have Moreover, using (3.3) with the inverse K −1 sl 3|2 as in (3.14), we also have B) simple roots and coweights in terms of ǫ A 's The expressions of the simple roots α A in terms of the basic weights ǫ A are given by Putting these relations into (3.42), and using the super traceless condition, we end up with the following expressions Below, we take advantage of this formal similarity between K sl 3|2 and A sl 5 to pave the way AB in terms of intersecting 2-cycles as a shortcut to brane realisation using algebraic geometry and methods of singularity theory [43,44,45,46,49].

Distinguished DSD and super geometry
In this section, we develop the study of the bridge between sl(5) and sl(3|2) by using their Cartan matrices A sl 5 and K sl 3|2 . Then, we use this bridging to construct the 2-cycle homology associated with the distinguished sl(3|2) singularity and propose a complex super geometry to use later for embedding the superspin chain into type II strings and M-theory compactified on Y 4 , as represented by the Figure 1.

Colored Dynkin diagrams
The Dynkin diagrams associated to the Cartan matrix A sl 5 and its super homologue K sl 3|2 look very close to each other; a property that deserves to be examined more closely. The K sl 3|2 and A sl 5 are graphically given by the pictures of Figure 5. They have four nodes with the same linear shape; but with different colors. The bosonic A sl 5 has a unified color, say the red color as in the right picture of the Figure 5. The K sl 3|2 involves three different colors as in Table 3 and the left picture of Figure 5. The purpose for using these colors is to depict their differences while emphasizing their similarities.

Bridging
Compared to the usual Cartan matrix of the bosonic Lie algebra sl (5) given by withα 2 a = +2 andα a .α a+1 < 0, the super K sl 3|2 has exotic values; in particular the three following This feature makes the geometric engineering method of the super chain in type II strings on local manifolds somehow special. In this regard, it is interesting to recall that in the singularity theory 1 of the complex ALE surfacesS with sl (5) geometry, bosonic Cartan matrices like A sl 5 ab of (3.14) have an interpretation in terms of intersecting 2-cyclesC a with intersection matrix [43]- [49] precisely given by I sl 5 ab = −A sl 5 ab ; that is From this intersection matrix of 2-cycles, we learn the following features indicating that theC a 's are complex projective lines CP 1 a (2-spheres) intersecting transversally according to the Dynkin diagram of sl(5) [22]. So the four simple rootsα a = ε a −ε a+1 (a = 1, 2, 3, 4) of the Lie algebra sl (5) are put in correspondence with four 2-cyclesC a in the resolved ALE surfaceS sl 5 . Later on, we will use the Algebra/Homology  and Lie algebra sl(5) homology of ALE surfacesS sl 5 1 To fix the ideas; see the list of ADE surfaces collected in the Table of eq(5).
to deal with the geometrisation of the symmetries and therefore build brane realisations of the superspin chain. As our super chain has U (N f ) × SL (3|2) invariance, we also apply the correspondence (4.6-4.7) to the two symmetry factors while taking into account some specificities as described below.
As su (N f ) is a bosonic Lie algebra contained into sl (N f ), the correspondence (4.6-4.7) given for sl (5) also applies to su (N f ). In this case, we have: labeled by a = 1, ..., N f − 1 with self intersection of these cyclesC 2 a = −2 (4.8) • Being associated with a flavor symmetry and not a gauge invariance, these 2-cycles C a must have large volumes. The solution of this constraint will be given later.

B) graded SL(3|2) invariance
As the singular surfaces with SL(m|n) gauge symmetries have not been explored in the stringy literature, we use below the relationship between the Dynkin diagrams of  The simple roots α A generating the distinguished root system Φ sl 3|2 play an important role in our construction. They are realised as α A = ǫ A −ǫ A+1 where the five unit weight vectors (ǫ 1 , ǫ 2 , ǫ 3 , ǫ 4 , ǫ 5 ) are distinguishably ordered like with bosonic ε a .ε b = δ ab and fermionic δ i .δ j = −δ ij normalisation. These weights are graphically represented as depicted by the picture of Table 4. To make the correspon- Bottom: the four simple roots α A = ǫ A − ǫ A+1 with the fermionic one in blue.
dence with the bosonic sl(5) more transparent, we give below the homologue of eq(4.13).

From
AB to homology of SL(3|2) singularity First, we give the typical structure of the complex surfaces with isolated singularities classified by Dynkin diagrams of finite dimensional ADE Lie algebras. Then, we present a proposal for the extension of sl(n) geometries to sl (m|n) while focussing on sl (3|2).  Table 5.

Du Val singularities
In the Du Val list, the BD 4n−8 and the BT 24 are respectively the Binary-Dihedral and Binary-Tetrahedral groups [59,60]. The BO is the Binary-Octahedral group and BIcosa is the Binary Icosahedral. For the A 1 example, we have describing a vanishing sphere at (x, y, z) = (0, 0, 0) . By setting u = i (x + iy) , v = i (x − iy), we can express the above SU (2) singularity like uv = z 2 corresponding to the leading element of the SU (n) family uv = z n (4.20) Geometry f (x, y, z) Γ Cartan Resolution graph and 2-cycles  In the language of weightsε a and rootsα a =ε a −ε a+1 , the resolution of the A N −1 singularity is captured by the relationε 1 −ε N = α a . For the example of sl (5) , this reads as followsε 1 −ε 5 =α 1 +α 2 +α 3 +α 4 (4.21) By using A/H correspondence (4.10), we also have the homological equation [22],

Surface with sl (3|2) geometry
To our knowledge, the extension of the classification ( Table 5)  • Algebraically, in terms of super weights ǫ A and super roots α A like where we have inserted ǫ 2 , ǫ 3 , ǫ 4 as interpreted in terms of blowing up the singularity.
• Homologically in terms of the graded divisors E A and graded cycles C A as follows The intersection of these graded cycles within the graded surface S sl 3|2 having an SL(3|2) geometry is given by By using where the metric g AB is as in (4.16), we end up with Comparing the intersection matrix (4.28) to the bosonic intersection matrix (4.4), we learn that the self intersections of the 2-cycles of the graded S sl 3|2 have different values and different signs namely A way to think about these self intersections is by mimicking the 2-cycle homology associated with Du Val singularities involving four 2-spheres intersecting transversally according to the ADE Dynkin diagrams. By extending this property to the graded surface S sl 3|2 , we imagine the graded 2-cycles in (4.29) in terms of closed Rieman surfaces Σ g without boundary. The topology of the Σ g 's is given by Euler characteristics reading as [61,62], where the positive integer g refers to the genus of Σ g . Notice that the topological χ g

Brane realisation of the super chain
In this section, we use results obtained above to embed the distinguished sl(3|2) superspin chain in type II strings and in M-theory. First, we study the embedding of the super chain in type IIA string. Then, we give the uplift to the M-theory realisation. and Ω-deformation, see [26] and [63,64,65,66].

Ground state of the super chain
We begin by recalling that the distinguished super chain of Figure 2 has an SL(3|2) × U (N f ) symmetry. The SL(3|2) is the superspin group appearing as a gauge symmetry in the 4D Chern-Simons gauge theory while the U (N f ) is the flavor symmetry distinguishing the N f atoms of the chain.

A) HW states and vacuum
Algebraically speaking, the atomic states Ψ l A of the super chain have the tensor structure For the purpose of the brane construction we are interested in here, we think it would be useful to give some technical details regarding the ǫ A 's and the e l 's. To avoid confusion with the previous section, we will use the following notation These five ǫ A 's obey the sl(3|2) super-traceless condition The relationship between ǫ A and the simple roots α A is obtained by using (3.42).
• the states |e l and is solved as B) Transitions |ǫ A → |ǫ B and |e l → |e l ′ As far as sl(3|2) is concerned 2 , the transition from the quantum |ǫ A to its neighbour |ǫ A+1 is generated by the step operator E −α A of sl(3|2) as follows, By using standard notations of Lie algebra representations, we can also present |ǫ A+1 like |ǫ A − α A . For convenience, we use the relation E −α A = |ǫ A+1 ǫ A | and the formal identification E −α A ∼ α A to think about the root α A as follows For the generic transition |ǫ A → |ǫ B , the involved root is given by A similar description holds for the flavor symmetry sector. The homologue of (5.8-5.9) reads as follows |e l+1 =Ẽ −α l |e l α l ∼ |e l+1 e l | (5.10) C) More on the factorisation eq(5.1) The factorisation |ǫ A ⊗|e l of the wave function (5.1) is justified by the fact that SL(3|2) and U (N f ) commute, The transition from |ǫ A ⊗ |e l to the state |ǫ B ⊗ |e l ′ is given by the tensor product E −α AB ⊗Ẽ −α ll ′ . As an example, the jumping from |ǫ A ⊗ |e l to |ǫ A+1 ⊗ |e l+1 is insured by E −α A ⊗Ẽ −α l ∼ α A ⊗α l (5.12)

Type IIA brane realisation
In type II superstrings, the atomic states ǫ A , e l of the super chain are represented by a system of branes whose directions expand in the 10D string spacetime dimensions denoted as M 1,9 . Recall that in 10D type IIA superstring, we have three pairs of pbranes namely: (1) the F1 string and the associated NS5. (2) the electric D0 and the magnetic D6. (3) the electric D2 and the D4 dual.
The brane system realising the states Ψ l A of the super chain can be derived using the dictionary of Table 6 giving the algorithm for the embedding of the sl (3|2) chain in type 2 In these kinds of transitions, we have ignored the effect of the spectral parameter z of line defects.
A rigourous description with magnons requires the implementation of this parameter. This issue needs considering the super Yangian Y sl(3|2) ; it is developed in the appendix.
super chain type IIA brane as well as similar relations for the sl (N f ) flavor sector. In these relations K AB is the distinguished sl (3|2) Cartan matrix.

A) Brane system and intersection
From the correspondence of Table 6 and the transitions between the quantum states |ǫ A → |ǫ A+1 and |e l → |e l+1 given by (5.8-5.10), we have the following brane system: (i) The transition between two neighboring NS5 A and NS5 A+1 is mediated by the brane D2 AA+1 stretching between them. The transition between D6 l and D6 l+1 is insured by the flavored D2 l,l+1 brane. For the generic transition between NS5 A and NS5 B , the natural mediator is given by D2 AB ; and the transition between D6 l and D6 l ′ is realised by the flavored D2 ll ′ .
(ii) Analogously, the transitions between neighboring D4 l A and D4 l A+1 is insured by D2 AA+1 while the transition between D4 l A and D4 l+1 A is given by D2 ll+1 . This indicates that the transition from D4 l A and D4 l+1 A+1 is reached in two steps as D2 AA+1 D2 ll+1 . This similarity between NS5 A and D4 l A becomes transparent in M-theory as both NS5 and D4 are mapped to an M5 brane.
(iii) The transition between the gauge NS5 A and the flavored D6 l is given by the D4 l A brane. General pictures involving multi-steps may be also drawn, for example through two types of branes like D2 AB D4 l B .
Notice that besides the transitions described by eqs (5.8-5.10), there are other types of quantum excitations of the super chain which are given by super magnons. The rigorous description of these quasi-particles goes beyond the sl (3|2) gauge symmetry since they are given by Verma modules of the Y sl 3|2 Yangian superalgebra (2.5). The building of these quantum excitations, that turn out to be described by kets like |M 1 , M 2 , M 3 , M 4 with M A being positive integers, is highly technical; it is reported in the Appendix.
However, to fix ideas, we give here below their typical structure where we have set β A = M A α A and where T −β A are generators of Y sl 3|2 . For further details, see the Appendix and also [39].

B) the super chain in type IIA
Regarding the embedding of the superchain in the type IIA string, the brane configuration realising the Ψ l A states is given by two stacks of branes; a first stack {A} labeled by the subscript A, and second stack {l} labeled by the upperscript l = 1, ..., N f . Below, we give a type IIA realisation where these stacks are given by {NS5 A } and {D6 l }; and due to Hanany-Witten transition, this brane system can be promoted to NS5 A , D2 AA+1 , D6 l and D4 l A as well as strings. In this regard, notice that the implementation of magnons requires also F1 strings stretching between n A stacks of D2 AA+1 denoted like; (D2 AA+1 ) n A (5.15) they will be hidden below. So by restricting to NS5 A , D2 AA+1 , D6 l and D4 l A , we have the following intersections of branes within the 10D spacetime directions of the type IIA string where the D6 l branes will play a secondary role because the D4 l A 's should be semi-infinite due to the flavor symmetry U(N f ). Notice that the four dimension euclidianR 4 with coordinates (X 6 , X 7 , X 8 , X 9 ) in (5.16) factorises likeR 2 0 ×R 2 1 with0 and1 referring to the Z 2 -grading degree of |A|. In addition to the orderings ξ 4 A < ξ 4 A+1 and ξ 5 A < ̺ 5 l as well as ̺ 5 l < ̺ 5 l+1 , we have used the following notations: (i) The cross (x) means that the dimension of type IIA is filled while the other boxes generate the transverse spaces of the branes; they give precisely the position degrees of freedom of the branes interpreted as scalar fields in super QFT at low energies.
(ii) The singleton ξ 4 A (resp. {̺ 5 l }) means that the NS5 A brane (resp. D6 l ) is located on the fourth-axis at the point X 4 = ξ 4 A (resp. the fifth-axis A , ξ 4 A+1 ] belongs to the fourth-axis, it is filled by D2 AA+1 stretching between NS5 A and NS5 A+1 ; the lengths |ξ 4 A − ξ 4 A′ | give the masses of the propagating quantum states between the NS5 branes. Similarly, the interval [ξ 5 A , ̺ 5 l ] belongs to the fifth-axis and is filled by D4 l A stretching between NS5 A and D6 l . However, because the D6 l s are flavor branes with symmetry U(N f ) containing the diagonal U (1) N f , the positions of the ̺ 5 l 's must be pushed far away (say to infinity) from the ξ 4 A s. As such, the intervals [ξ 5 A , ̺ 5 l ] of the D4 l A branes should be thought of as [ξ 5 A , +∞[ in agreement with the flavor symmetry requirement and in accord with the realisation given in [16]. Below, we hide the D6 l branes in (5.16), thus reducing the brane system to the following Using the dictionary of the Table 6, we learn that the p-branes NS5 A , D2 AA+1 , D6 l and D4 l A carry, in addition to quantum charges under sl(3|2)0, extra Z 2 grading charges given by |ε a | =0 and |δ i | =1 as in eq(5.18). Below, we describe the brane configuration in terms of these charges. 3 The SL(1) symmetry group is the complexification of the usual unimodular phase group U (1) with element e iθQ . TheQ is the generator of abelien u(1), the Lie algebra of U (1), acting on complex wave functions as Q , ψ = qψ. Similarly, elements of SL(1, C) are given by λQ with complex parameter λ = e ρ+iθ ∈ C * . Here, theQ is the generator of sl(1, C), the Lie algebra of the abelian group SL(1, C).
For example, it acts on homogeneous coordinates Z i of complex projective CP n as Q , Z i = qZ i .

• the sl(3) sector
Here, we have three bosonic weight vectors ε a and two bosonic simple roots α 1 , α 2 . The type IIA brane system for this sector is given by The p-brane worldvolumes of this system are as follows: First, the three NS5 a are bosonic like; they expand in 5 space directions, in particular in R 0 ×R 1 and the complex C = R 2 + iR 3 as well as R 6 × R 7 . The two D2 branes are given by D2 a,a+1 with a = 1, 2; they are bosonic-like and expand in R 0 × R 1 and in [ξ 4 a , ξ 4 a+1 ] representing the 1D space between the NS5 a and NS5 a+1 branes with positions in the 4-th directions ξ 4 a and ξ 4 a+1 .
For the 3N f bosonic-like D4 branes labeled as D4 l a , they expand in The p-brane worldvolumes in this sector correspond to the odd sector of Z 2 . It is described by two graded weight vectors δ 1 and δ 2 with a bosonic-like simple root α 4 . The p-brane system in this sector is given by The D2 45 brane interpolates between the odd NS5 4 and NS5 5 and sits at the points {z 4 } and ξ 5 4 . • the sl(1) sector This sector gives the link between the sl(3) and the sl(2) sectors. This bridging is given by the D2 34 brane which is a fermionic-like brane; it interpolates between the bosoniclike NS5 3 and the fermionic NS5 4 . where M 1,9 is as in type IIA string and where S 1 M ∼ R M e iϑ is the M-theory circle describing the eleventh direction.

Uplifting Table(6) to M-branes
As the NS5-and the D4-branes of type IIA string merge into M5-branes of M-theory and the D2 is mapped into the M2, the M-brane system realising the super chain can be obtained by promoting Table( in terms of M-theory compactified on the A N f −1 ALE singularity. This real 4D geometry can be described in different but equivalent ways; for example in terms of the orbifold C 2 /Z N f with discrete group Z N f acting on the two complex coordinates z and z ′ like We will think of these complex variables z and z ′ in terms of the spacetime variables X 2 + iX 3 and exp(−X 5 + iR M ϑ) respectively. Another interesting realisation of this singular complex surface is given in terms of its embedding in C 3 [u, v, w] where it is defined by the algebraic equation uv = w N . More interestingly, the deformation of the A N −1 singularity is described by uv = N l=1 (w − ζ l ) which is generated by the N f complex parameters ζ l . In the language of 2-cycle homology, the topology of A N −1 may be imagined in terms of the fibration S N = Σ N × C with Σ N being a complex curve (2-cycle) given by the intersection of generating 2-cycles C l with complexified Kahler moduli t l = ζ l − ζ l+1 . In terms of these objects, the M-brane description of the type IIA realisation (5.17) is given by Table (5.17). Moreover, the complex curve Σ l A is a complex half line parameterised by a complex variable ζ with "left end" value given by ζ = z ′ A and "right end" ζ l = e X 5 l +iϑ l . Here, the X 5 l = ̺ 5 l is as in (5.17), but due to the flavor nature of U(N f ), its absolute value |ζ l | = e ̺ 5 l is pushed to infinity; that |ζ l | → ∞. From this description, it follows that the correspondence given by Table 6 with regards to type IIA generalises in M-theory as in the Table 7 giving the algorithm for the embedding super chain type IIA M-branes Table 7: Lie superalgebra/M-brane correspondence. Simple roots are of sl(m|n) associated with graded M2-branes stretching between graded M5 pairs. of the sl (3|2) chain in M-theory. The brane intersections of the M-branes in the Table   7 follow from (5.13) for the sl (3|2) superspin sector and for flavor U (N f ). We havẽ is the Cartan matrix of sl (N f ) . From the correspondence in Table 7 and the transitions between the quantum states in the pseudo vacuum, we obtain the following M-brane candidates to build the embedding of the super chain in M theory: • Five sl 3|2 super M5 branes denoted like M5 A .
• M-theory on the singularity geometry A N −1 requiring by the promotion of D6.
A graphical illustration of this interacting M5-M2 brane system is depicted by the Figure 7 where the five basic M5 A stacks are represented by 5 vertical sheets and the M2 AA+1 brane messengers by 4 horizontal stacks.

Graded M-branes
Using the dictionary of Table 6, we learn that the M-branes M5 A , M2 AA+1 and M5 l A carry, in addition to quantum numbers under sl(3|2)0, extra Z 2 charges given by |ε a | =0 and |δ i | =1. Below, we give the brane system in terms of these charges.

• the sl(3) sector
The M-brane system for this bosonic-like sector is given by where the complex line Σ a is given by C × {z a } with z a = X 2 a + iX 3 a describing the loci of the M5 a in the orbifold singularity and where z ′ A is given by e ξ 5 a +iϑa . • the sl (2) sector The p-brane system in this sector is given by • the sl(1) sector This sector gives the bridge between the sl(3) and the sl (2) sectors. This bridging is given by the M2 34 which is a fermionic-like brane; it interpolates between the bosoniclike M5 3 and the fermionic M5 4 . U(N f ) superspin chain. We show that the brane system is induced by M5/M2 sitting in the singularity of a complex 4D manifold Y 4 that we want to construct. Under resolution of the singularity, the Y 4 has a real 4-cycle C 4 given by the fibratioñ that is a 2-cycle fibred over another 2-cycle. A graphical representation of C 4 ∼C was sketched for N f = 4 by the Figure 1; other equivalent representations will be given below. In this regard, notice that the 2-cycle base C  To fix the ideas, we give the cycle homology of the 4D manifold Y 4 in the Figure 8 as the cross product of the two Dynkin diagrams. withC l stretching between two divisorsẼ l andẼ l+1 as As an illustration, we give in the Figure 9 a graphical description of the cycleC N f 2 . Using The complex surfaceS N f is a complex ALE surface with a resolved SU(N f ) singularity; its resolution is as illustrated by Figure 10. This graph, and the Figure 9 are related by the 2D space duality. The defining equation ofS N f embedded in C 3 with complex local Figure 10: On the left, an ALE surface uv = z 6 having an SU (7) singularity at the origin of C 3 . On the right, its complete resolution using 6 transversally intersecting complex projective lines. Here we have set N f =7.
coordinates (u, v, z) is given by the following holomorphic where c is a complex number. The N f moduli µ l are the zeros ofS N f ; they can be put in correspondence with the weight vectors e l of the SU (N f ) flavor symmetry (5.3) and the Table 6; and by the A/H with theẼ l divisors. So, we have e l ↔ µ l ↔Ẽ l (6.9) By using the Table ?? linking SU (N f ) weights with M-branes, we deduce that the µ l 's can be interpreted in type IIA string as the loci where sit the D6 l branes inS N f .
Similarly, the difference is put in correspondence with the rootsα ll ′ = e l − e l ′ . By the A/H, theα ll ′ describe the 2-cyclesC ll ′ inS N f ; and by Table ??, the rootsα ll ′ and cyclesC ll ′ characterise the branes D2 ll ′ stretching between D6 l and D6 l ′ .
At the end of this construction, notice the following: and is regular.

Flavor symmetry constraints on (6.3)
The real 2-cycleC we have to impose on the moduli µ l in (6.8), we proceed as follows.
We start from eq(6.8) defining the non compact surfaceS N f in terms of the µ l 's, then we sit on the C 3 patch u = 1 , v =W (z) (6.13) with two free variables z andW . Substituting these values into (6.8), we obtain the complex curveW where we have set p = c 1/N f . Notice that for the particular situation N f = 1, the abovẽ W N f takes its simplest formw 1 = p (z − µ 1 ). Using this property, we can rewrite (6.14) as followsW with the holomorphic curvesC l given byw l −w l+1 . By replacingw l = p (z − µ l ), we end up with p µ l − µ l+1 characterising the irreducibleC l that generate the compactC N f 2 . So, the condition for U(N f ) to be a flavor symmetry is given by

Geometry of the 11D space time
In this subsection, we use (6.2) to construct the 11D space time M 1,10 (5.22) needed for embedding the sl(3|2) superspin chain in M theory. As there is a lack in the literature with regards to the properties of singularities based on supergroups (super singularities), we develop in the following an attempt to tackle this problem and give partial results in this matter. For that purpose, we will think about the space time M 1,10 as follows where M 1,9 is the space time of type IIA string and S 1 M is the M-theory circle.

The real 8-manifold M 8
In the fibration (6.17), the R t × S 1 θ is just the 2D space time of the superspin chain 4 while the real 8D space M 8 is thought of as a 4D complex manifold Y 4 given by the following This factorisation is motivated by the two symmetry factors of the superspin chain, namely U (N f ) × SL(3|2). The fiberS N f 2 is given by (6.2) with large volume cycles as in (6.3-6.7). The base surface S sl 3|2 2 is a graded complex 2-manifold given by Using the Super A/H (4.9), the compact 2-cycle C sl 3|2 2 is given by the sum of four intersecting graded 2-cycles with the irreducible graded C A stretching between two graded divisors E A and E A+1 as Eq(6.20) describes the blowing up of the graded singularity of C sl 3|2 2 corresponding to the vanishing of the graded C A cycles. From the homology view, the blowing of this super singularity is expressed as follows

Graded 2-cycles
Due to the Z 2 -grading of the SL (3|2) gauge symmetry, the graphical representation of is given by a colored Dynkin super diagram as depicted by Figure 11.
This Figure should be compared with the DDSD given by the Figure 5 where the two

M5 wrapping 4-cycle C 4
In this subsection, we study the geometry describing the wrapping of M5 over the compact 4-cycle (6.
where the 4-cycle is fibred as follows By wrapping M5-brane over the 4-cycle (6.29), we obtain the 5N f branes M5 l A representing the super chain given by the wrapped branes The wrapping of M5-brane over the 3-cycles C 3 A = S 1 M ×C A gives the four M2 AA+1 branes stretching between the M5 A and M5 A+1 because

The defining equation of Y 4
We realise the complex fourfold Y 4 as a 4D sub-manifold living in the ambient complex , we can think about the equation defining Y 4 inside C 5 as follows 5 with h and c A two functions as We give below four interesting features of the Y 4 hypersurface of C 5 .
(1) If we set the function h = 0 and hide the (x, y) coordinate variables of the base surface S sl 3|2 2 by thinking of the c A 's as constants, the eq(6.32) reduces to the particular given by the blowingS is to (i) embed it in C 6 with coordinates (u 1 , v 1 , z 1 , u 2 , v 2 , z 2 ) and (ii) use two equations as follows to reduce the number of variables down to 4.
(2) For a function h vanishing at (x, y, z, u, v) = x, y, µ l A , 0, 0 , and functions c A (x, y) taking constant values respecting SL (3|2) invariance; say for instance c A = (−) |A| c; we recover the flavor SU (N f ) singularity at the points (x, y) = (x A , y A ) solving h x, y, µ l A , 0, 0 = 0 , c A (x, y) = (−) |A| c (6.36) At the loci (x A , y A ) , the eq(6.32) has zeros at (u, v) = (0, 0) and along the z-direction at z = µ l A . The zeros at µ l A give the positions of the M5 l A branes in the surface fiberS 5N f 2 . In terms of these 5N f moduli µ l A , we can calculate the volume of 2-cycles and 4-cycles of Y 4 . In particular, we are interested in the volumes υ l AA+1 and υ ll+1 A of the 2-cycles associated with the SL(3|2) gauge symmetry and the U(N f ) flavor invariance: (i) The holomorphic volumes υ l AA+1 of the M2 AA+1 branes stretching between M5 A and M5 A+1 read as Because SL(3|2) is a gauge symmetry, these holomorphic volumes are required to take small values; thus constraining the µ l A 's like where |A| refers to the degree of |c A |.
(4) By setting u = 1 and v = W A (z) in (6.32); we get Following [16], a configuration of n A parallel (horizontal) M2 n A AA+1 branes suspended between the vertical M5 A and M5 A+1 , and located at z = σ n A A preserves supersymmetry if and only if the following constraint holds Solving this condition, we end up with that has an interpretation in the Bethe equation description [16].

Conclusion and comments
In this paper, we investigated the embedding of integrable sl(m|n) superspin chains and super magnons in M-theory and type II strings with two main objectives. the three self intersection values C 2 A = 0, ±2. On the right hand side, we have SL(5) cycles (in red) withC 2 l = −2. To our knowledge, super geometries based on SL(m|n) symmetry have not been studied enough in the stringy literature and the associated super singularities are still an open problem in algebraic geometry [67]. The investigation given in this paper constitutes a contribution to this matter.
To undertake this study, we started by investigating the degrees of freedom of the super chain by using representation group language and revisited the algebraic set up of properties of the super atoms making the chain. This analysis showed that there are (m + n)!/m!n! varieties of super chains classified by the (m + n)!/m!n! possible Dynkin super diagrams of sl(m|n). Recall that contrary to the bosonic Lie algebras, finite dimensional Lie superalgebras like sl(m|n) have several Dynkin-like diagrams termed in this paper as colored Dynkin diagrams. These colored diagrams have graded roots that enrich the study of superspin chains thanks to the different possible varieties and the exotic properties resulting from the three possible "lengths" α 2 A = 0, ±2. To exhibit these interesting features, we studied in details the 10 super sl(3|2) chains by using algebra and homology tools. Then, we focused on the distinguished sl(3|2) chain as representative of sl(m|n) and studied the geometrisation of properties of the ground state of the super chain in connection with exotic singularities due to SL(3|2) symmetry. After that, we extended the Algebra/Homology correspondence regarding Du Val surfaces with SU (N f ) singularity to the case of super groups and used this super A/H to approach geometries with SL(m|n) super singularity.
Using the obtained results concerning the properties of quantum states of the super chain as well as the super A/H, we investigated the embedding of the super chain in type II strings and M-theory. In the type IIA string, the brane realisation involves varieties of NS5-, D2-and flavored D4-branes as well as F1 strings. Because of the graded symmetry SL(m|n) × SU (N f ) of the super chain, we distinguished the brane sets NS5 A , D6 l , D4 l A and D2 AA+1 as listed in the Table 2. To complete this brane system, notice that there are moreover F1 strings stretching between stacks of N A gauge-like D2 AA+1 branes denoted as (D2 AA+1 ) N A ; they describe the super magnons. Although not explicitly elaborated, a quite similar construction can be performed for type IIB strings.
In M-theory, the brane realisation of the super chain uses various kinds of M5-and M2branes as given by the Table 7. There, we distinguished the brane sets M5 A , M5 l A and the gauge-like M2 AA+1 and their wrappings on compact cycles. The super magnons are described by strings M2/S 1 M stretching between stacks of N A gauge-like M2 AA+1 branes given by (M2 AA+1 ) N A .
After that, we constructed the 11D space time M 1,10 = S 1 M ×M 1,9 where the super chain is embedded. This space has the structure S 1 M × (R t × S 1 θ × Y 4 ) where the 4D complex manifold Y 4 is given by the fibrationS N f 2 × S sl 3|2 2 with complex flavor surfaceS N f 2 of Du Val type and gauge surface S sl 3|2 2 . This gauge surface has a super geometry with compact 2-cycles given by the colored Dynkin diagram of Figure 13.
We end these comments by mentioning that the brane realisation of super magnons is to be extended for the families of superspin chains; in particulat the sl(m|n) and the orthosymplectic osp(m|2n) chain. The first step towards the construction of these follows from the super Yangian representation as reported in the Appendix. The next step regards their brane realisations and demands some technical details concerning the brane engineering and orientifolds. Progress in this direction will be reported in a future occasion.

Appendix: Magnons in Yangian formalism
In this appendix, we use the Yangian superalgebra Y sl(m|n) to construct the magnons in the sl (m|n) superspin chain while focussing on the distinguished sl (3|2) . To that purpose, we introduce the Y sl(3|2) , the Bethe vectors and the Bethe roots. Then, we build the super magnons for the closed sl (3|2) chain and investigate their algebraic properties. Their brane realisation will be elsewhere.
• Pseudo-vacuum Verma modules of Y sl(3|2) (highest weight representations) are characterised by a HW state |Ω Λ often termed as a pseudo-vacuum of the integrable chain. This ground state is constrained as T AA (z, ζ) |Ω Λ = b λ A (z, ζ) |Ω Λ T AB (z, ζ) |Ω Λ = 0, A < B (7.3) with the graded labels A, B = 1, ...5. In these relations, we have Λ = l λ l ; that is the total superspin of the chain assumed to have L sites with positions x l (l = 1, ..., L). Λ = λ 1 + λ 2 + ... + λ L (7.4) Using the weight vector basis (ε 1 , ε 2 , ε 3 , δ 1 , δ 2 ), we can express (7.4) like Λ A ǫ A with For convenience, we use two useful notations: (i) we set below |Ω Λ ≡ |Λ or equivalently showing that the ground state |Ω Λ is characterised by L × 5 quantum numbers λ l A . (ii) We represent graphically the ground state (7.6) as shown in Figure 15. In the horizontal axis, we have the unit ǫ A weight vectors. In the vertical axis, we have the direction of the x l -chain indexed by the label 1 ≤ l ≤ L.

Elementary and composite magnons
Excitations of the ground state |Ω Λ of the superspin chain (basis vectors of the Y sl(3|2) Verma module) are obtained by acting with the T − AB (z) creators as follows In addition to the total weight Λ and the simple root strings β = A N A α A (not necessary roots of sl(3|2)), the excitation states are characterised by the complex z i s and ζ i s; they can be either elementary magnons or composites.

• Elementary super-magnons
Particular states of such excitations are given by the four following T 12 (z 1 , ζ 1 ) |Λ , T 23 (z 2 , ζ 2 ) |Λ , T 34 (z 3 , ζ 3 ) |Λ , T 45 (z 4 , ζ 4 ) |Λ (7.8) Using the four simple roots α A = ǫ A − ǫ A+1 of the superalgebra sl (2|3), we can present these four states as  with β N = N A α A and weight vector Λ N = Λ − β N reading explicitly as where Λ A is as in (7.5). As for the pseudo-vacuum (7.3), the states (7.11) are also eigenstates of the Bethe generators T AA (z, ζ) . Notice that from the weight vectors Λ N , we can compute the intersection Λ N .α B = Q B reading as Q B = Λ.α B − β N .α B (7.13) and having two contributions Q 0 B = Λ.α B and Q ′ B = β N .α B . The Q 0 B has an interpretation in quiver gauge theories as describing fundamental matter while Q ′ B is interpreted in terms of adjoint and bi-fundamental matter. By substituting Λ = Λ A ǫ A , we obtain with K AB as in (3.14) and the generalised matrices G AB = (ǫ A .α B ) given by (1) The magnon diagram has intrinsic symmetries due to the indistinguishable property of elementary magnons of the same nature. This symmetry factorizes like G g × G f with (a) G g standing for gauge symmetry due to internal magnons (stretching between two neighboring vertical lines). (b) G f referring to a flavor symmetry concerning the L × 5 external lines (in Green color). For the magnon diagram of Figure 17, we have the following symmetries G g = U (2) × U (1) × U (2) G f = U (L) 5 (7.18) (2) The magnon diagram has a dual representation which looks like the well known diagrams of quiver gauge theories. Using 1D duality mapping lines to points and points to lines, it is easy to see that the dual of the magnon diagram 17 is given by the quiver graph of the Figure 18. This description done for Y sl 3|2 extends straightforwardly to Y sl m|n .