Continuum Limit of gl(M/N) Spin Chains

We study the spectrum of an integrable antiferromagnetic Hamiltonian of the gl(M|N) spin chain of alternating fundamental and dual representations. After extensive numerical analysis, we identify the vacuum and low lying excitations and with this knowledge perform the continuum limit, while keeping a finite gap. All gl(n+N|N) spin chains with n,N>0 are shown to possess in the continuum limit 2n-2 multiplets of massive particles which scatter with gl(n) Gross-Neveu like S-matrices, namely their eigenvalues do not depend on N. We argue that the continuum theory is the gl(M|N) Gross-Neveu model. We then look for remaining particles in the gl(2m|1) chains. The results suggest there is a continuum of such particles, which in order to be fully understood require finite volume calculations.

e α (e β ) = δ αβ , then E αβ are the standard generators of gl(M |N ) acting in V as E αβ · e γ = δ βγ e α and in V * as E αβ · e γ = −(−1) (|α|+|β|)|α| δ αγ e β . Let us label the V factors of the spin chain C(L) = (V ⊗ V * ) ⊗L from left to right by a subscript 1, . . . , L. Similarly, we label the V * factors by a subscript1, . . . ,L. For E ∈ End V we denote by E k ∈ End C(L) the endomorphisms acting as E on V k and trivially, up to grading signs, everywhere else in the chain. Similarly, for E ∈ End V * , Ek ∈ End C(L) will act as E on V * k and trivially, up to grading signs, everywhere else.
The C(L)-endomorphisms P kl = (−1) |β| ρ k (E αβ )ρ l (E βα ) provide a representation for the symmetric group acting on the V factors of C(L). Similarly, Pkl = (−1) |β|ρk (E αβ )ρl(E βα ) generate a representation for the symmetric group acting on the V * factors of C(L). On the other hand, Q kl = −(−1) |β| ρ k (E αβ )ρl(E βα ) generate a representation of the Temperley-Lieb algebra T 2L (n) with loop weight n = M − N . Together, the P 's and Q's generate the gl(M |N )-centralizer algebra of C(L), that is the set of all endomorphisms of the spin chain that commute with the gl(M |N ) action [Ser01]. This centralizer algebra is a representation of the walled Brauer algebra B L,L (n), which can be viewed as a subalgebra of the Brauer algebra B 2L (n). We shall discuss in detail the algebra B L,L (n) and its representations in sec. 3.1.
(2.4) Yang-Baxter relations (2.1) imply a Yangian structure given by the two relations R ab (u − v + n/2)T a (u)Tb(v) =Tb(v)T a (u)R ab (u − v + n/2) (2.6) and their duals. The following commutation relations immediately follow The nested algebraic Bethe ansatz for the most general gl(M |N ) spin chain was considered in [BR08]. The Bethe ansatz equations and the spectrum of t(u) formally depend on the nesting order, that is an ordering of a basis of V and the induced ordering of the dual basis of V * . If the basis {e α } M+N α=1 of V diagonalizes the Cartan subalgebra, then, without loss of generality, we label the basis vectors so that the total ordering reads e 1 > e 2 > · · · > e M+N , (2.7) where, however, we keep unspecified the grading of the basis vectors. Let wt(e α ) = ǫ α denote the weights of basis vectors called fundamental weights. The ordering (2.7) induces a weight space ordering ǫ 1 > · · · > ǫ M+N which fixes the simple root system ∆ 0 = {α i := . The choice of grading, which we denote by Σ = {σ α = |e α | = |α|} M+N α=1 , is equivalent to the choice of a Cartan matrix, or a Dynkin diagram. Changing the grading can be equivalently seen as changing the total ordering (2.7). As a result, the simple root system, the positive root system and the Borel subalgebra changes with Σ. So, keep in mind that the notion of highest weight always depends on the grading choice and changes when Σ changes.
Define the operator matrix elements of the monodromies (2.2, 2.3) as T = E ij ⊗T ij , where T ij ∈ End C(L) and T = T a , Tā. Choosing the reference state Ω to be the highest weight state of C(L), the eigenvalues of t(u) can be written in terms of polynomials (T a ) ii (u)Ω = Λ i (u)Ω Λ i (u) =      (u + (−1) |1| ) L (u + n/2) L , i = 1 u L (u + n/2) L , 2 ≤ i ≤ r u L (u + n/2 − (−1) |M+N | ) L , i = M + N and simple root Q-polynomials as follows where the u (k) j , k = 1, . . . , r = M + N − 1 are the Bethe roots appearing at the k-th step of the nesting. There are solutions of the following system of nested Bethe ansatz equations , (2.9) ensuring the analyticity of eigenvalues (2.8), where k = 1, . . . , r and j = 1, . . . , ν (k) and it has to be understood that Q 0 (u) = Q M+N (u) = 1. We stress that the BAE (2.9) are equivalent to the analyticity requirement of the transfer matrix (2.8) if and only if none of the Bethe roots of the same type coincide, which is an essential requirement in the algebraic Bethe ansatz construction.
The weight of the reference state is wt(Ω) = L k=1 Λ k + L k=1 Λk = L(ǫ 1 − ǫ M+N ). Bethe vectors ω described by the Bethe roots (2.10) are highest weight vectors of weight wt(ω) = wt(Ω) − r k=1 α k ν (k) . (2.12) We define the dynamics of the spin chain by the momentum and Hamiltonian operators . (2.13) In order to have explicit expression for the spectrum of these operators, the eigenvalues (2.8) of t(u) are not enough. One also needs to evaluate the eigenvalue oft(u) on a given Bethe eigenvector of t(u). A fundamental difference w.r.t. gl(N ) spin chains is that one cannot solve this problem by fusion. This is because tensor products of the fundamental representation V of sl(M |N ) will never generate the dual representation V * as a direct summand, nor even as a subquotient. To develop a clear idea about how this should be done, let us recall that a set of Bethe vectors for t(u) can be constructed in the framework of the algebraic Bethe ansatz (ABA) by using only the commutation relations (2.5). Then, the eigenvalue oft(u) on such a Bethe vector can be calculated, in principle, by using the second type of commutation relations (2.6). We shall not pursue this rather tedious route. Instead, we guess the eigenvalue oft(u) on a given Bethe vector of t(u) as follows.
First, notice that a different set of Bethe vectors can be obtained by performing the ABA fort(u), that is by using the commutation relations dual to eq. (2.5). We perform the nesting by ordering the dual basis vectors {e α } M+N α=1 of the auxiliary space V * according to their weights wt(e α ) = −ǫ α e M+N > · · · > e 2 > e 1 . (2.14) We denote the Bethe roots appearing at step k of the nested ABA byū , because the simple root which must be associated to them is wt( Then, the eigenvalues oft(u) can be written in term of polynomials ( (2.15) Analyticity conditions forΛ(u) take the same form as eqs. (2.10) in terms of variables and parametersν (k) . A Bethe vector constructed in this way, which we denote byω, has weight At this point, the eigenvalues of t(u) w.r.t. the second set of Bethe vectorsω is not known. If we can match Bethe vectors ω with Bethe vectorsω then the problem is solved.
Due to the subtle completeness issue of ABA for super spin chains, it is not clear at all if the matching can actually be performed. We assume it can and we do it as follows: a Bethe vector ω of t(u) is identified with a Bethe vectorω oft(u) if (2.17) The first condition in the braces means wt(ω) = wt(ω). Eqs. (2.8, 2.15, 2.11, 2.16) allow us to compute both the eigenvalues of t(u) andt(u) on a Bethe vector ω =ω in (2.17).
After this long detour, we come back to the spectrum of the operators (2.13), which can now be explicitly computed where θ t (u) = i log e t (u) + π = 2 tan −1 2u t with some fixed branch. First thing to be noticed is the explicit dependence of the definitions (2.13) on the grading. Therefore, it looks like the type of the chain -ferromagnetic or antiferromagnetic -also depends on it, which is also suggested by the grading signs in (2.18). However, the charges H and P can be written explicitly as a representation of an element of the periodic walled Brauer algebra where all (un)barred indices are defined modulo L and the affine generators are expressed in terms of non-periodic walled Brauer algebra elements P L1 := P 1L = P 12 . . . P L−1L . . . P 12 , P L1 := P 1L = P 12 . . . P L−1L . . . P 12 and QL 1 := Q 1L = P 1L Q LL P 1L . The following general formulas have been used The latter can be derived using only the cyclicity of the supertrace, which holds for a graded tensor product when even endomorphisms are considered, relations of the form P ij R jk (u) = R ik (u)P ij , P ij R jk (u) = R ik (u)P ij , str a P aj = ½ and their duals. We see from eq. (2.20) that the grading enters in the definition of the Hamiltonian only as a shift, therefore having nothing to do with ferro or antiferromagnetism. The equivalence of the solutions of BAE in different gradings, and therefore of spec H, is at present somewhat understood in terms of particle hole transformations [Tsu98], although the relationship between Bethe vectors in different gradings not at all.
A relation between the spectrum of H and −H can be constructed, but one has to consider different chains. Let H M|N denote the integrable Hamiltonian of the gl(M |N ) spin chain.
Then on has the following relation (2.21) To prove it one uses the algebra homomorphism between the walled Brauer algebras B L,L (n) and B L,L (−n) provided by P → −P and Q → −Q. This automorphism is realized in the spin chain representation by the shift of the grading function |i| → |i|+ 1, which maps gl(M |N ) → gl(N |M ). As we shall see in the next section, the sign in front of our Hamiltonian (2.20) ensures that we are dealing with antiferromagnetic spin chains for n = M −N > 0. Eq. (2.21) allows to fold back the gl(N |M ) spin chains with N < M to gl(M |N ) spin chains with n > 0 by changing the sign of the Hamiltonian (2.20), but now they will be ferromagnetic. · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · in other representations where its spectrum is much less degenerate compared to that in the gl(n + N |N ) spin chain, namely it does not depend on N .

Walled Brauer algebra and its standard modules
The walled Brauer algebra B L,L (n) can be conveniently viewed as a subalgebra of the Brauer algebra B 2L (n). The defining relations of B 2L (n) can be found in [GCA98]. The words of  replacing every loop generated in this process by n. The walled Brauer algebra B L,L (n) is the subalgebra of B 2L (n) generated by the elements Q iī := E 2i−1 , Qī i+1 := E 2i , P i,i+1 := P 2i−1 P 2i P 2i−1 , P ıı+1 := P 2i P 2i+1 P 2i . The generators P i P i+1 P i are represented on the right in fig. 1. For every diagram spanning B 2L (n) with vertices labeled as in fig. 1, imagine moving all odd vertices to the left of a wall, all the even ones to the right, while keeping the connectivity unchanged. Then B L,L (n) is spanned by the set of B 2L (n) diagrams, such that only strictly horizontal edge cross the wall. In this representation of B L,L (n) we label the L up and L down vertices on the left of the wall by the set 1, . . . , L from left to right and similarly those on the right by the set1, . . . ,L. The P ii+1 generators act on the left of the wall, the P ıı+1 act on the right, while the generators Q iī and Qī i+1 act across the wall.
Next we give a brief description of a set of modules of B L,L (n) over which we actually numerically diagonalize the algebraic Hamiltonian H. These modules shall be related in the following to gl(n + N |N ) traceless tensors of fixed co-and contravariant shapes. For λ and µ partitions of the non-negative integers |λ|, |µ| = f ≤ L, we denote by ∆ L,L (λ, µ) the standard modules of B L,L (n). There are constructed in the following way. Let Sym(f ) denote the symmetric group on f objects and S(λ), S(µ) denote its simple modules labeled by the corresponding partitions. Then, ∆ L,L (λ, µ) has as basis the tensor products between the set of some diagrams on 2L points, with L of them on the either side of the wall, and some basis of S(λ) and S(µ). The diagrams are such that every point is either free or belongs to a horizontal edge crossing the wall. The number of edges is fixed to L − f . We give a rough idea about how the action of the walled Brauer algebra can be constructed by diagrammatic multiplication in in fig. 2: i) in a first step, before the diagrammatic multiplication, assign labels to the free points of the diagrams on each side of the wall; ii) in a second step, after the diagrammatic multiplication, apply a surjective homomorphism from labeled diagrams to the tensor product of unlabeled diagrams with S(λ) and S(µ), that is to ∆ L,L (λ, µ). All labeled diagrams with more then L − f horizontal edges that can appear as a result of the diagrammatic multiplication belong to the kernel of this homomorphism. In particular, ∆ f,f (λ, µ) ≃ S(λ) × S(µ). The detailed definitions can be found in [CVDM08].
We implement the action of B L,L (n) in ∆ L,L (λ, µ) on a computer and investigate the spectrum of the algebraic Hamiltonian H. Before presenting and discussing the numerical results of sec. 3.3 one should explain how to extract from these data the spectrum of the original spin chain Hamiltonian H.

Traceless tensors
Consider the gl(M |N ) tensor (V ⊗ V * ) ⊗f . There is an obvious action of Sym(f ) × Sym(f ) on the V and V * factors. Let λ, µ be partitions of f , which we symbolically write as λ, µ ⊢ f .
One can apply a Young symmetrizer of shape λ to the V factors and a Young symmetrizer of shape µ to the V * factors to get a tensor of shape t(λ, µ) [Wey53]. We say that t(λ, µ) has rank (f, f ), covariant shape λ and contravariant shape µ or, shortly, shape (λ, µ). The number of Young symmetrizers of shape λ is equal to the number of standard Young tableau of shape λ, which is also equal to dim S(λ). Therefore, the number of tensors of shape (λ, µ) is dim S(λ) × dim S(µ). Every tensor t(λ, µ) is a gl(M |N ) module, which appears as a direct summand in (V ⊗ V * ) ⊗f . The symmetric group Sym(f ) × Sym(f ) acts in the space of tensors of the same shape (λ, µ), transforming them into each other. The latter subspace is isomorphic to a direct sum of S(λ) ⊗ S(µ) modules of Sym(f ) × Sym(f ). These statements can be compactly written as follows Consider now the subspace t 0 (λ, µ) ⊂ t(λ, µ) of traceless tensors. Notice that this is a gl(M |N ) submodule, which will not necessarily be a direct summand of t(λ, µ). More generally, one can consider the gl(M |N ) submodules t n−1 (λ, µ) ⊂ t(λ, µ) composed of tensors whose all contractions of n covariant indices with n contravariant indices vanish. These provide a filtration of the tensor t(λ, µ) It is very important to observe that the subquotients t n (λ, µ)/t n−1 (λ, µ) of this filtration are isomorphic to traceless tensors of lower rank (f − n, f − n). For instance, taking a trace of t 1 (λ, µ) one gets a traceless tensor of rank f − 1 because, by definition, all double contractions of t 1 (λ, µ) must vanish. Now, the preimage of single traces of t 1 (λ, µ) modulo the kernel t 0 (λ, µ) of the single trace homomorphisms is precisely to t 1 (λ, µ)/t 0 (λ, µ). Therefore, t 1 (λ, µ)/t 0 (λ, µ) is isomorphic to traceless tensors of rank (f − 1, f − 1). We see that traceless tensors t 0 (λ, µ) have a fundamental role -all direct summands of the spin chain (V ⊗ V * ) ⊗L can be built out of them. As a consequence, the full spectrum of the spin chain Hamiltonian can be reconstructed from the spectra in the subspaces of traceless tensors of shape λ, µ ⊢ f , f = 0, 1, . . . , L. In fact, not all of these shapes are possible. We shall determine the class of shapes for which the traceless tensors do not vanish later. The traceless tensors are not necessarily irreducible. One way to build submodules of a traceless tensor t 0 (λ, µ) is by embedding into it quotients of traceless tensors of lower rank as follows. Let t 0 (λ ′ , µ ′ ) be a traceless tensor, λ ′ , µ ′ ⊢ f − k, λ ′ ⊂ λ, µ ′ ⊂ µ and e λ , e µ denote some Young symmetrizers of shape λ, µ. The tensor might have an intersection with a non-trivial submodule of t 0 (λ, µ). The latter will generally be isomorphic to only a quotient of t 0 (λ ′ , µ ′ ), because the Young symmetrizers e λ , e µ are projectors. An illustrative example is the indecomposable gl(N |N ) tensor t(1, 1) = V ⊗ V * . The traceless subspace t 0 (1, 1), isomorphic to the adjoint representation, is spanned by elements of the form G i j e i ⊗ e j subject to the constraint str is the dual basis. The quotient t 1 (1, 1)/t 0 (1, 1) is one dimensional. A coset representative for this quotient is, for instance, (−1) |i| e i ⊗ e i . The traceless tensor t 0 (1, 1) is also indecomposable, but reducible. It has a unique proper non-trivial submodule spanned by the gl(N |N ) traceless invariant e i ⊗ e i .
Represent now the covariant part of every tensor t 0 (λ, µ) of fixed shape (λ, µ) and ranks (f, f ) by f dots on the left of an imaginary wall and the contravariant part by f dots on the right of that wall. Then the walled Brauer algebra generators P ii+1 will act on the left of the wall as in S(λ) and the P ıı+1 generators will act on the right as in S(µ) by transforming dim S(λ)×dim S(µ) different traceless tensors of shape (λ, µ) into each other. The generators Q iī and Qī i+1 will act across the wall by contracting a covariant index with a contravariant one. In view of the tracelessness condition this action is trivial. Thus, the space of all traceless

the walled Brauer algebra. A very important observation is the
triviality of the centralizer of a traceless tensor This is a generalization of the Schur lemma for gl(n) traceless tensors, which are irreducible.
The statement (3.2) follows immediately from the action of the walled Brauer algebra in the space of traceless tensors of shape (λ, µ) that we have just described. It means that traceless tensors are a very special type of indecomposables, namely any gl(M |N ) Casimir is diagonalizable and proportional to the identity in a traceless tensor. It should be noticed that this is typical of highest weight or Kac modules [Kac77b,Kac77a].
Assumption 1 Traceless tensors are highest weight modules.
This means that there is a Borel subalgebra b of gl(M |N ) ≃ b ⊕ n − and a b-highest weight vector v ∈ t 0 (λ, µ) such that the full tensor t 0 (λ, µ) can be generated from v by repeated action of n − . We shall see later how to choose b for given t 0 (λ, µ). Consider now the vector space δ L,L (λ, µ) of all possible embeddings of traceless tensors of shape (λ, µ) and ranks (f, f ) into the spin chain (V ⊗ V * ) ⊗L . It consists of tensor products which can be written as e i ⊗ e i . Representing this invariant by an edge connecting two vertices across the wall and the traceless tensors t 0 (λ, µ) as we did before, we can visualize δ L,L (λ, µ) as a diagram with L vertices on each side of the wall and L − f edges connecting pairs of vertices across the wall. Thus, we reconstruct the same diagrammatic representation of δ L,L (λ, µ) as for ∆ L,L (λ, µ). This proves that all the relations between the generators of B L,L (n) satisfied in ∆ L,L (λ, µ) will be satisfied in δ L,L (λ, µ) as well. The converse is generally not true, meaning that δ L,L (λ, µ) is generally only a quotient of (dim t 0 (λ, µ) direct sums of) ∆ L,L (λ, µ). We stress that neither δ L,L (λ, µ) nor ∆ L,L (λ, µ) are necessarily simple B L,L (n) modules and, therefore, the quotient can be non-trivial. The ABA in some grading Σ provides gl(M |N ) Bethe eigenvectors of highest weight with respect to the Borel subalgebra b Σ determined by Σ and the ordering (2.7). Therefore, in order to match the numerical spectrum of H in ∆ L,L (λ, µ) with the exact spectrum of H by the ABA we need to know the highest weight of a traceless tensor t 0 (λ, µ) at least in one grading Σ. We explain below how to evaluate it. Consider the Young diagrams corresponding to the shape (λ, µ) of a full tensor t(λ, µ).
The basis vectors in the tensor subspace of co(ntra)variant shape λ (µ) can be represented by co(ntra)variant Young supertableaux of shape λ (µ), that is Young diagrams of shape λ (µ) with a fundamental weight ǫ i (−ǫ j ) inscribed in every box. The pattern of weights within the Young diagrams must satisfy the supersymmetrization rules, that is i) in the same row the bosonic (fermionic) weights are weakly (strongly) ordered w.r.t. each other, ii) in the same column the bosonic (fermionic) weights are strongly (weakly) ordered w.r.t. each other and iii) bosonic weights are weakly (strongly) ordered w.r.t. fermionic weight in the same row (column). The weight of a supertableau is the sum of all weights it carries in its boxes.
Both λ and µ must fit into a hook whose horizontal (vertical) arm is of width M (N ), otherwise t(λ, µ) vanishes identically because it is not possible to fill in the Young diagrams and get Young supertableaux compatible with the supersymmetrization rules. The highest weight of a supertableau depends on the grading. The choice of grading is a splitting of the set of basis vectors into two sets B (F ) = {ǫ i | |i| ≡ 0 (1)} < which are ordered w.r.t. the total ordering (2.7). Equivalently, it can be represented by paths connecting the two corners of the (M, N )-hooks as represented in fig. 3. Fix these paths and consider Young diagrams λ, µ with rows λ i , µ i and columns λ ′ i , µ ′ i . Then, the highest weight of t(λ, µ) can be written in the following form where b(i) and f (i) are the elements at position i in the ordered sets B and F ,b(i) andf (i) are the elements at position i in the ordered sets −B and −F , /2) are number of boxes in a row or column of λ or µ overpassing the grading paths as represented in fig. 3.
The highest weight component of a tensor t(λ, µ) w.r.t. the grading Σ will belong to the traceless subspace t 0 (λ, µ) if the highest weight Young supertableau of shape (λ, µ) does not contain some fundamental weight ±ǫ i in both λ and µ. Otherwise, the corresponding highest weight component of t(λ, µ) w.r.t. the grading Σ will either i) not belong to t 0 (λ, µ) or ii) generate a submodule of t 0 (λ, µ) isomorphic to the embedding of a quotient of a lower Figure 3: Covariant Young tableau λ and contravariant Young tableau µ of (5, 4)-hook shape for gl(5|4). Bosonic fundamental weights belong to B = {ǫ 2 , ǫ 3 , ǫ 5 , ǫ 6 , ǫ 9 }, while fermionic rank tensor t 0 (λ ′ , µ ′ ). If the latter case holds, then the possible Young diagrams (λ ′ , µ ′ ) are obtained from the Young diagrams (λ, µ) by removing pairs of boxes from the highest weight Young supertableau of shape (λ, µ): a box of λ carrying some weights ǫ i together with a box of µ carrying the opposite weight −ǫ i . Moreover, in the case ii) the highest weight of the top 1 of t 0 (λ, µ) will be smaller then the highest weight of the submodule t 0 (λ ′ , µ ′ ). Therefore, t 0 (λ, µ) cannot be a Kac module w.r.t. b Σ .
We call gl(M |N )-admissible the shapes (λ, µ) for which the gl(M |N ) traceless tensors t 0 (λ, µ) neither vanish nor are isomorphic to lower rank traceless tensors. We are now ready to answer the very important question: what are the admissible shapes of traceless tensors? According to the previous discussion on the highest weight component of a tensor t(λ, µ), a shape (λ, µ) is admissible if there exists a grading Σ such that the highest weight Young supertableau of shape (λ, µ) does not contain any fundamental weight ±ǫ i both in λ and in µ. Consequently, for a traceless tensor t 0 (λ, µ) of admissible shape there is a grading Σ and a corresponding highest weight (3.3) such that no cancellation between the r i andr i or c i andc i terms can occur. If v Σ (λ),v Σ (µ) denote the number of nonzero "reduced rows" r i ,r i and h Σ (λ),h Σ (µ) denote the number of non-zero "reduced" columns c i ,c i , then one must for a gl(M |N )-admissible shape (λ, µ), as represented in fig. 3 These admissible shapes nicely reduce to hook shapes for purely covariant or contravariant gl(M |N ) tensors and to staircases for gl(n) traceless tensors [BCH + 94].
We say that a shape (λ, µ) of a traceless tensor t 0 (λ, µ) is Σ-admissible if the inequalities in eq. (3.4) are satisfied. Let K Σ (Λ) be the Kac module of highest weight Λ w.r.t. b Σ . We can make now assumption 3.2 more precise.
Assumption 1 ′ The following isomorphism holds for Σ-admissible shapes (λ, µ) This assumption in combination with the general theory of Kac modules [Kac77b,Kac77a] is very useful for counting highest weight vectors. Namely, if (λ, µ) is Σ-admissible then the number of highest weight vectors in t 0 (λ, µ) w.r.t. b Σ is equal to the number of irreducible subquotients. Noticing that a highest weight vector cannot generate more then a highest weight module, we prove the following claim in app. A.
To sum up, we have explained the connexion between traceless tensors t 0 (λ, µ) of admissible shapes (λ, µ) and standard modules ∆ L, Evaluating the highest weight of t 0 (λ, µ) w.r.t. arbitrary Borel subalgebras is more delicate, mainly because of indecomposability issues. Finally, claim 1 indicates a natural relationship between traceless tensors and Bethe vectors, which have the highest weight property, constructed in the framework of ABA.

Spectrum
We present the spectrum of H in various ∆ L,L (λ, µ) standard modules of B L,L (n) in tab. 3.3.
At a first glance, it appears that the vacuum always lies in ∆ L,L (∅, ∅). To check more thoroughly this vacuum hypothesis we need an additional assumption on the spectrum. Consider the spectral sets of H defined as   can prove the following relationship between the spectral sets spec H N +n|N with n fixed This "embedding of spectra" is a very interesting and general feature of supergroup spin chains and one might wonder how does it carry on to the field theory description of the continuum limit. In this respect, two scenarios are possible. The first possibility is that spec H n+N ′ |N ′ becomes a very excited subset within spec H n+N ′′ |N ′′ , where N ′ < N ′′ , and decouples from it in the thermodynamic limit L → ∞. This means that the vacuum energy per site of H n+N ′ |N ′ is higher then the vacuum energy per site of H n+N ′′ |N ′′ in the thermodynamic limit. The second, more interesting, possibility is that the vacuum energies per site for both Hamiltonians coincide in the thermodynamic limit. The first possibility occurs, for instance, in the V ⊗L chains with Hamiltonian ± P i,i+1 . In next section we describe the mechanism which is behind the embedding of spectra (3.5) at the level of BAE. The answer to the question of how excited spec H n+N ′ |N ′ is within spec H n+N ′′ |N ′′ for N ′ < N ′′ will have to wait until sec. 4. . We shall restrict to Bethe vectors whose weights are given by highest weight Young supertableaux of Σ-admissible shapes (λ, µ) according to eq. (3.3). Due to fundamental rôle of traceless tensors explained in sec. 3.2 and claim 1, it is clear that with the imposed restriction, one must consider the ABA in all the gradings in order to recover the full spectrum of the spin chain Hamiltonian H. For a fixed grading and a corresponding simple root system, fig. 3 and eq. (3.3) implies that we are restricting to Bethe vectors of weight w = M+N i=1 w i ǫ i , w i ∈ Z such that for every simple bosonic root α j = ǫ j − ǫ j+1 one has w i ≥ w i+1 , while for every simple fermionic root one has the following implications

Restriction and lift of BAE
The main reason for introducing this restrictions and working with BAE in multiple gradings is the bounds on the number of Bethe roots resulting from eqs. (4.1).

Restriction
We wish to consider the BAE in the form (2.9) corresponding to a simple root α k such that the following assumptions hold A1 α k is odd, that is σ k σ k+1 = −1 A2 α k has no source terms or, equivalently, k = 1, r With these assumptions, we have ν (k) BAE for α k−1 of the form and ν (k+1) equations for α k+1 . (4.5) R2 multiplying the BAE for α k−1 and α k+1 corresponding to, say u ) . (4.6) Indeed, eq. (4.3) is trivially satisfied because according to R1 one has Q k−1 (u) = Q k+1 (u).
Furthermore, multiplying BAE according to R2, the Bethe roots {u drop off yielding a BAE of the form (4.2) with k replaced by k + 2. We call a restriction of BAE the procedure

R1-R2.
The restriction procedure can be given an algebraic meaning and, therefore, partially explained as follows. As described in detail in [CCMS10], choose to be the odd gl(M |N ) element that squares to zero and defines the gl to yield a non-trivial Q-cohomology is for it to be in the kernel of Q and, therefore, one must have

Lift
To resume, assuming A1-A2 holds for the BAE of the spin chain (V M|N ⊗ V * M|N ) ⊗L written w.r.t. a simple root system ∆  with Q as in eq. (4.7). We must show that there is a solution {u which restricts to the given one. First of all, eq. (4.8) implies ν (k+1) = ν (k−1) = P . After defining u (k+1) j = u (k−1) j for j = 1, . . . , P the task is reduced to finding a solution {u (k) j } p j=1 to either eq. (4.2) or eq. (4.4), where we have set p = ν (k) . If p ≥ P , such a solution obviously exists.
More then that, for p > P there is a continuum of such solutions! However, recall that for a fixed root system and corresponding BAE we have restricted to Bethe vectors such that their weights satisfy the constraints (4.1). If the solution we are looking for exists then the weight of the corresponding Bethe vector ω can be written as wt(ω) = · · · + (P − p)(ǫ k − ǫ k+1 ) + . . . . The constraints (4.1) imply p ≤ P . So, for p = P the solution always exists. In our numerical investigations we have observed that solutions might exist even for p < P . We call the solutions of gl(M |N ) BAE constructed in this way from solutions of gl(M − 1|N − 1) BAE lifted solutions. We discard lifted solutions with p < P for the following reason. We prove the claim in appendix B.
Next, we show that the lift is unique. So, we have seen that lifted Bethe vectors correspond to lifted solutions with ν (k) = ν (k±1) = P . Let us now prove that the P equations, say, (4.2) regarded as a constraint on the unknowns {u (k) j } P j=1 admit a unique solution. We start by multiplying both sides with Q k (u (k−1) j ) and then expand in {u (k) j } P j=1 . What we get is a system of P linear equations for the P unknowns
There are important cases when one can compute the roots {u (k) j } P j=1 explicitly in terms of the other roots. This happens if C 1.1) α k±1 is odd and sourceless where we meant that either we choose the plus sign or the minus sign overall. In the case C 1 eq. (4.2) or eq. (4.4) can be rewritten in terms of variables (2.11) as We immediately read off the obvious solution which we already know is unique. Another important case is C 2.1) k = 2, α 1 is odd and P = L C 2.2) k = r − 1, α r is odd and P = L.
In the case C 2.1 the one has where we have used eqs. (2.10) with non-zero arbitrary inhomogeneities. Again, we read off the unique solutions (4.10) It is important to notice that inhomogeneities are essential to lift the degeneracy of solutions in the case C 2.

Generalizations
We have argued that the restriction and lift of BAE have an algebraic origin. From this viewpoint, assumption A2 seems unnecessary. Indeed, all of the above constructions can be appropriately modified to accommodate the boundary case corresponding to A1 and A2 ′ k = 1, r.
The necessary condition Subsectors in integrable systems are not a new phenomenon. For example the homogeneous gl(2n + 1) spin chain (V ⊗ V * ) L of sec. 2 contains a subsector corresponding to the 2L-th tensor power of the fundamental representation of osp(1|2n), see [SWK02]. We shall provide more examples in sec. 5. However, all these examples lack by far the generality of the restriction and lifting procedures we have described. This is because in our case there is an algebraic mechanism behind which relates the representation theories of {gl(n + N |N )} N ∈Z + Lie superalgebras. The same cohomological reduction mechanism exists for any Lie superalgebra except osp(1|2n). Therefore there is no doubt that the same restriction and lift phenomena occur also for general osp(R|2S) integrable spin chains. Moreover, the embedding of spectra (3.5) was already observed on a (non-integrable) osp(2S + 2|2S) spin chain in [CS09a]. We come back now to the spin chain of sec. 2 and answer the question about how excited w.r.t. each other are the subsectors (3.5).

Vacuum and low lying excitations
In this section we present numerical evidence showing that all integrable gl(n + N |N ) spin chains (2.20) with n fixed have the same vacuum energy. In view of the embedding of spectra (3.5), we introduce the notion of degree of an excitation, which is the smallest value of N for which it appears in spec H n+N |N . We then proceed to classify the excitations of degree 0 and 1, and present the form of the numerical solutions of BAE reproducing them.

Bosonic lift
From ( vector ω is the lift of a gl(n) Bethe vector, that is ω has degree 0, then it should be possible (in some grading) to write its weight in the form wt(ω) = M+N i=1 w i ǫ i with no fermionic components and at most n bosonic components. Equivalently, the weight of a Bethe vector ω of degree 0 must be representable (in some grading) by highest weight Young supertableaux of gl(n)-admissible shape (λ, µ)  where λ ′ 1 ≤ m ≤ n − µ ′ 1 . There are many choices for the nilpotent element to define the restriction and lift in the grading (5.2). For instance, the two possibilities are represented in fig. 5. Notice that for a fixed Q the lift is unique, although it is different for different Q's. Multiple gradings of the type (5.2) must be considered in order to recover the full bosonic lift.
To conclude, we write down explicitly the vacuum solution in the grading [ where the roots {x

Classification
Fix an odd element Q ∈ gl(n + N |N ) that squares to zero, has rank N − 1 and defines the gl(n + 1|1) spin chain (V ⊗ V * ) ⊗L as the Q-cohomology of the gl(n + N |N ) spin chain (V ⊗ V * ) ⊗L . Excitations of degree 1 correspond to gl(n + N |N ) Bethe vectors which: i) are non-reducible to gl(n) Bethe vectors and ii) are lifted gl(n + 1|1) Bethe vectors. We shall call the corresponding Bethe vectors also of degree 1. According to sec. 4, the weight of a Bethe vector ω of degree 1 admits a representation (in some grading) of the form wt(ω) = M+N i=1 w i ǫ i with at most n + 1 bosonic components and at most one fermionic component. Notice that due to the claim 2 of sec. 4, there are also excitations of degree 1 with the same weight as excitations of degree 0, that is no fermionic components and at most n bosonic components. Taking into account sec. 3.2 as well, the weight of Bethe vectors of 1 ≤ m ≤ n + 1 − µ ′ 2 . On the right: root numbers for the lowest lying degree 1 excitation. degree 1 must be representable (in some grading) by highest weight Young supertableaux of gl(n + 1|1)-admissible shapes (λ, µ), that is at least one of the two conditions holds Shapes (5.5) are Σ-admissible and most obviously gl(n + 1|1) reducible w.r.t. the gradings Shapes (5.6) are Σ-admissible and most obviously gl(n + 1|1) reducible w.r.t. the gradings [(+) m (−+) 2N (+) n−m ], such that λ ′ 1 ≤ m ≤ n + 1 − µ ′ 2 , and the choice Q ∈ CE m+3,m+2 ⊕ · · · ⊕ E m+2N −1,m+2N −2 . The corresponding root configurations are represented in fig. 6 on the left.

Low lying excitations
Assumption 2 of sec. 3.3 and tab. 3.3 clearly indicates that the lowest lying excitation of the B L,L (n) algebraic Hamiltonian which is of degree higher then 0 lies in ∆ L,L (1 k , 1 k ) with k = [ n 2 ] + 1. Equivalently, the lowest lying excitation of the gl(n + N |N ) spin chain Hamiltonian which is of degree higher then 0 is a traceless tensor t 0 (1 k , 1 k ). Eqs. (5.5, 5.6) imply this excitation is of degree 1. To reproduce it from gl(n + 1|1) BAE we have chosen the following gradings The corresponding root numbers are represented in fig. 6  are always complex. We recall that all roots are real for the vacuum solution (5.4). Thus, this excitation is not constructed as usual by making "minor" modifications to the vacuum solution.
Another strange feature is the non-selfconjugacy of the solutions, at least in general. This is a consequence of the non-hermiticity of the Hamiltonian (2.20). 5 Such solutions have been investigated already in the study [EFS05] of the gl(2|1) spin chain, but also encountered in different contexts dealing with non-hermitian Hamiltonians [SWK00]. Instead, all lowest lying degree 1 solutions are invariant w.r.t. a modified conjugation symmetry The symmetry (5.9) looks surprising, because the Dynkin diagram in fig. 6 is not symmetric w.r.t. the transposition α k → α n+1−k . However, notice that the roots {x also solve the BAE corresponding to the transposed Dynkin diagram of fig. 6, see [Tsu98] for details. For ν (m+1) even, the solutions are (exceptionally) selfconjugate The following picture of the distribution of roots in the complex plane in the thermodynamic limit is emerging from our numerical analysis gl(2m|1) : The gl(2m|1) complex root configurations where called ±-strange strings in [EFS05]. We see that the lowest degree 1 excitation looks pretty complicated. What about other degree 1 excitations? We have verified that properties (5.8-5.9) hold for all low lying excitations with the same root numbers as in fig. 6 on the right and, more then that, for all low lying solutions with symmetric configurations of the root numbers ν (k) = ν (n+1−k) . Many of these excitations seem to tend to the form (5.11, 5.12) in the thermodynamic limit. These are the best understood excitations for which we shall give a continuum description in the next section. We have also noticed that among the low lying excitations with ν (m+1) even, there is always a subset of selfconjugate solutions of the type (5.10). These have interesting properties, as we shall see in a moment.
To complete the overview of low lying degree 1 excitations, let us mention that we have also identified solutions with symmetrical configurations of the root numbers, which do not 5 See [Vla86] for a proof of selfconjugacy in the case of hermitian Lie algebra spin chain Hamiltonians.
seem to tend to the form (5.11, 5.12) in the thermodynamic limit. For these solutions we have nothing to say, although their study might prove crucial in constructing an Smatrix description of the continuum theory in large volume. The situation is even worse for solutions with non symmetrical configurations of the root numbers, because ±-strange string configurations cannot be clearly defined.

Selfconjugate solutions
As we have said, for ν (m+1) even and ν (k) = ν (n+1−k) , there is always a subset of selfconjugate solutions of the type (5.10). It is straightforward to show that the gl(2m|1) BAE (2.10) subject to the constraints (5.8, 5.10) are equivalent to the BAE of the so(2m+1) fundamental spin chain of length L with periodic boundary conditions for the first m − 1 types of roots and antiperiodic boundary conditions for the spinorial roots. This is the generalization of the Takhtajan-Babujian subsector of the gl(2|1) spin chain of [EFS05]. The eigenvalues of the integrable Hamiltonian for this so(2m + 1) spin chain are given by eq. (2.18) subject to the constraint (5.10). While this correspondence between solutions of gl(2m|1) and so(2m + 1) BAE is quite curious, the remarkable thing is that it extends to a correspondence between weakly excited solutions on both sides. In particular, the vacuum solution of the so(2m + 1) fundamental chain in the thermodynamic limit [Mar91] (plus holes) is compatible with the form (5.11) when the constraint (5.10) is taken into account. We shall see how this correspondence can be used effectively in the next section. The situation is quite different for gl(2m + 1|1). Notice that assumptions A1-A2 and R1 of sec. 4 hold for solutions of gl(2m+1|1) BAE (2.10) satisfying (5.9, 5.10) and, therefore, these can be restricted to solutions of gl(2m) BAE. However, the latter are very special, because in addition to satisfying the gl(2m) BAE they must also admit multiple gl(2m + 1|1) lifts. According to the claim 2 of sec. 4, gl(2m + 1|1) lifted Bethe vectors correspond exclusively to gl(2m) solutions that do not admit multiple lifts. Presumably, what happens is that the additional conditions satisfied by these gl(2m) solutions obtained by restriction ensure the vanishing of the corresponding gl(2m) Bethe vectors. We have checked that there are many other low lying solutions satisfying the constraints (5.9, 5.10). However, it is most confusing that many low lying solutions that do not satisfy the constraints (5.9, 5.10) in finite volume seem to have a thermodynamic limit (5.12) that does satisfy the constraints (5.9, 5.10). We are not sure how to interpret this behavior, although we are tempted to believe this indicates that the large volume limit is a subtle issue. Therefore, we concentrate in the next section solely on the continuum description of gl(2m|1) degree 1 excitations of the type (5.8, 5.11).

Continuous limit
In this section we shall consider the spectrum of the spin chain of sec. 2 in the thermodynamic limit L → ∞. The homogeneous spin chains are gapless and the only thermodynamic quantity, as far as the spectrum is concerned, is the vacuum energy per site. The latter does not provide any insight into the the continuous limit of the chain, which is expected to be governed by a conformal field theory (CFT). Probing this CFT requires computing scaling corrections to the spectrum [Car86]. However, these are much harder to compute, e.g.
see [KP91], then thermodynamic quantities. Therefore, gaining insight into the CFT solely from the lattice is quite non-trivial. To avoid such complications, one can introduce a smooth gap in the spin chain. The continuous limit is then expected to be a massive deformation of the CFT. Many interesting quantities, such as the β-function, the particle spectrum, Smatrices, can now be computed in the thermodynamic limit. The standard way to generate a gap in the homogeneous spin chains is by introducing an alternating inhomogeneity Λ in the monodromies (2.2, 2.3), called staggering [Fad96]. The source terms of the homogeneous BAE (2.10) then change to while the eigenvalues of the new energy and momentum operators become where σ 1 = (−1) |1| , σ M+N = (−1) |M+N | and we have assumed L to be even. These expressions reduce the the previous ones 2.18 in the limit Λ → 0.
In this section we define and study the continuous limit of the staggered gl(n + N |N ) chains of sec. 2. First we show that the bosonic lift of the staggered chain is described in the continuous limit by the gl(n) Gross-Neveu (GN) model [AL80a,AL80b,AL79]. This strongly suggests that the continuous limit of the staggered spin chain coincides with the gl(n + N |N ) GN model, because the latter contains the gl(n) GN model as a cohomological subsector [CCMS10]. The identification of the continuous limit is the main result of the paper. In the second subsection we explore the particle content of the gl(2m|1) GN model which does not lie in the gl(2m − 1) GN model lift, that is the bosonic lift.  dynamical degrees of freedom (associated to holes) are determined by the even roots. The odd roots behave as auxiliary quantities useful for defining lifted Bethe vectors. So, we conclude that the BAE of the gl(n) spin chain describe entirely the bosonic lift of the gl(n + N |N ) spin chain. We recall some of the old results on the continuum limit of gl(n) spin chains and then reinterpret them in the gl(n + N |N ) context.
There is one important remark. Multiparticle states have zero total u(1) = gl(n)/ sl(n) charge, because the spin chain (V ⊗ V * ) ⊗L has zero u(1) charge. Therefore, it is not possible to fix the u(1) charge of the individual particles in the spin chain. The allowable multiparticle configurations in the spin chain make the value of this charge irrelevant. However, the field theories with u(1) charged and uncharged particles (6.4) are clearly different: the first one has 2n − 2 irreducible multiplets of particles, while the latter, which is the GN model, has n − 1 irreducible multiplets.
We now reinterpret the results of the above calculations in the framework of the bosonic lift of the gl(n+N |N ) spin chain. The continuous limit of the gl(n+N |N ) spin chain is defined by the same eq. (6.3). It already implies that the β-function of the continuous theory does not depend on N . The n − 1 excitation branches of the bosonic lift are interpreted again as relativistic particles of mass (6.4). Eq. (6.5) gives now the energy of a scattering eigenstate corresponding to a gl(n + N |N ) traceless tensor t 0 (λ, µ) of gl(n) admissible shape (5.1). However now, it suggests that the bosonic lift contains 2n − 2 distinct multiplets of particles corresponding to the gl(n + N |N ) covariant antisymmetric tensors 1, 1 2 , . . . , 1 n−1 and their antiparticles, that is the contravariant antisymmetric tensors1,1 2 , . . . ,1 n−1 . Notice that the number of particles does not depend on whether they carry or not a u(1) charge, which again cannot be fixed in the spin chain. This is because the sl(n + N |N ) antisymmetric tensors 1 k and 1 n−k are not isomorphic anymore.
The suggested gl(n+N |N ) symmetry of the particles in the bosonic lift must be confirmed by an S-matrix calculation. While we do not know the full S-matrix, one can compute its restriction to the bosonic lift. The calculation is formally the same as for the gl(n) spin chain, because the BAE are the same. Thus, the eigenvalues of the gl(n + N |N ) spin chain S-matrix restricted to the bosonic lift coincide with the eigenvalues of the gl(n) spin chain Smatrix. More precisely, the eigenvalue of the gl(n+ N |N ) spin chain S-matrix on a scattering eigenstate corresponding to a gl(n + N |N ) traceless tensor t 0 (λ, µ) of gl(n)-admissible shape does not depend on N . Due to the fact that the Q-cohomology of a tensor product of gl(n+N |N ) representations is the tensor product of their Q-cohomologies [CCMS10] all these eigenvalues are compatible with our earlier assumption that the particles of the bosonic lift are co-and contravariant antisymmetric tensors of gl(n)-admissible shape.
Finally, the scattering of particles in the bosonic lift can be computed exactly when no scattering eigenstates outside the bosonic lift are generated. Thus, particles 1 and1 scatter with the same S-matrix (6.6), where this time P and Q act on tensor products of gl(n+N |N ) representations. Notice that we did not assume any crossing or unitarity to derive S 11 from S 11 . The question now is what field theory can reproduce these S-matrices? Given its formal similarity with the gl(n) GN S-matrix, the most obvious candidate is the gl(n + N |N ) GN model. There are deeper reasons to believe this. First of all, the Q-cohomology of the gl(n + N |N ) GN model is the gl(n) GN model. In particular, their β-functions coincide [CCMS10]. Secondly, there is no doubt that the perturbative 1/n calculations of the gl(n) GN S-matrix can be generalized. While we did not carry out an honest perturbative S-matrix calculation as in [KKS79], the result should certainly be (6.6). This is because the Feynman rules are gl(n + N |N ) invariant tensors and their algebra, generated by tensor multiplication and contraction, is a representation of the walled Brauer algebra B L,L (n) of sec. 3.1 and 3.2. Thus, as long as one computes an invariant tensor, such as the S-matrix, starting from other invariant tensors, such as the Feynman rules, the result cannot depend on N . Thirdly, the detailed analysis of [EFS05,SS07] suggests that the continuum limit of the homogeneous gl(2|1) spin chain is the sl(2|1) 1 WZNW model, that is the gl(2|1) GN model at zero coupling. Finally, we mention the similarity with the situation for osp(R|2S) GN models [BL00,SWK02,SP10]. Therefore, we conjecture that the continuous limit of the gl(n + N |N ) spin chain (V ⊗ V * ) ⊗L is the gl(n + N |N ) GN model.

Degree one excitation
In this section we investigate degree 1 excitations of the gl(2m|1) spin chain which are of the type (5.8) and tend to the form (5.11) in the thermodynamic limit. As explained in sec. 5.2, the lowest lying excitation of degree 1 is precisely of this type. It corresponds to the antisymmetric tensor t 0 (1 m , 1 m ) and its root numbers are represented in fig. 6.
It is certainly reassuring that degree 1 excitations have the same mass scale m as degree 0 excitations. The doubling of masses (6.12) w.r.t. the masses of particles in the bosonic lift can be understood as follows. We have observed in sec. 5.2.2 that the imaginary parts of the odd roots tend to vanish for large L and, therefore, we have neglected them in the thermodynamic limit. However, due to the conjugation symmetry (5.8) of the solutions, this approximation leads to an identification (6.7) of distinct roots. Therefore, the positions of holes in the distribution of even roots {x have also been identified. So, the excitation (6.12) is in fact a state of two particles 1 k and1 k each of mass m k and rapidity θ. These are the old particles from the bosonic lift. On the other hand, the particles (6.13) are new.
The artificial binding of holes (6.12) induced by the thermodynamic limit does not allow to derive meaningful S-matrices from eqs. (6.9) Most probably, to correct the approach and recover the degrees of freedom lost in the thermodynamic limit one needs to distinguish between the two types of roots Im x (k) j > 0 and Im x (k) j < 0, exactly as we did for the strange ±-strings (5.11). However, a finite volume treatment will be required, because the imaginary parts of even roots vanish in the thermodynamic limit.
and, therefore, they always come in pairs. The Fourier transform (6.14) is singular and clearly must be regularized, because one has to satisfy the constraint where b is a fixed real number parametrizing the state. On a lattice of length L the the momentum can take a minimal value of p ∼ 1/L. Using this value as a regulator one gets for a pair of ±-holes from eqs. (6.14 6.15) which clearly shows that for b = 0 the deviations of strange ±-strings from the form (5.11) of the solution in the thermodynamic limit have to be taken into account. Notice that the energy does not depend on the continuous parameter (6.15) parametrizing the state. Therefore, it is tempting to conclude that there is a continuum of new particles (6.13).

Conclusions and Outlook
We have put on firm grounds the relationship between gl(n + N |N ) integrable spin chains with n fixed. This allowed us to prove that all gl(n + N |N ) spin chains (V ⊗ V * ) ⊗L with n, N > 0 possess in the continuum limit 2n − 2 multiplets of massive particles which scatter with gl(n) Gross-Neveu like S-matrices, namely their eigenvalues do not depend on N . We concluded that the continuum theory is the gl(M |N ) Gross-Neveu model. Evidence that the massive spectrum is much richer, possibly continuous, was established on the example of gl(2m|1) chain. Finally, our analysis of the thermodynamic limit strongly suggests that understanding the nature of new particles requires a finite volume treatment.
The question that begs the quickest answer is how to close the fusion of S-matrices (6.6) of the gl(M |N ) Gross-Neveu model starting with just the vector multiplet and its antiparticles. The gl(N |N ) spin chains require a separate treatment, which we hope to report on later.