Classical Spectral Curve of the AdS_5 x S^5 Lambda Superstring

The classical spectral curve for the worldsheet theory of the AdS_5 x S^5 lambda superstring is constructed. The lambda string is interpreted as a regularized, non-abelian T dual of the AdS_5 x S^5 superstring with respect to full PSU(2,2|4) symmetry. The form of the curve is identified as the semi-classical limit of a set of Bethe ansatz equations for an XXZ type spin chain for the supergroup PSU(2,2|4) in contrast to the string in AdS_5 x S^5 which is XXX type.


Introduction
The notion of a spectral curve is one of the deep structures of an integrable system [1]. At the classical level, the moduli of the curve are the action variables of the system, whilst the angle variables are associated to the Jacobian of the curve. Each classical solution determines a curve and the conserved charges of the integrable system can be read off the behaviour of a certain function-the quasi-momentum-defined on the curve at special points. There is an inverse algorithm for reconstructing the classical solution from a curve. If integrability survives quantization then it is natural to expect there to be a Quantum Spectral Curve (QSC). The curve provides a way to calculate the charges of states in the theory. A fascinating integrable system is the world-sheet theory of the string in AdS 5 ×S 5 and the N = 4 gauge-gravity duality. 1 In this case, the form of the classical curve has been determined in [4]. The curve is determined by the solution of some auxiliary integral equations that can be viewed as a complicated Riemann-Hilbert problem. Early proposals were then made for the how the curve could be quantized leading to a set of rather novel Bethe Ansatz type equations, known as the Asymptotic Bethe Ansatz (ABA) [5]. The ABA can be viewed as a direct quantization of the classical spectral curve and it also had a direct connection to the dual gauge theory. The quantum spectral curve has been determined in [6,7] (see also [8]).
The AdS 5 ×S 5 string theory admits various kinds of deformation that preserve integrability. In particular, there are the "eta" [9][10][11], "beta" [12][13][14] and "lambda" deformations [15][16][17]. The QSC of the eta deformed string has been determined in [18,19]. Here, we focus on the lambda deformation that for the bosonic sector can be viewed as the non-abelian T-duality of the theory with respect to the full PSU(2, 2|4) with a particular kind of compactification of the non-compact geometry that result from the naïve duality. This particular deformation has the necessary kappa symmetries and is one-loop finite [20]. The notion of a compactified non-abelian T-dual is crucial here in order that the naïve non-abelian T-dual is a consistent string background. It involves lifting the algebra-valued Lagrange multiplier field into a group-valued field. The lambda string model, has an associated level k and in some sense, the naïve nonabelian T dual is recovered in the limit k → ∞. The resulting λ string is formulated as a gauged WZW model for the supergroup F = PSU (2, 2|4) where the full F (vector) symmetry is gauged but with a deformation that breaks the gauge symmetry to the bosonic subgroup G = Sp(2, 2)×Sp(4)( SO(1, 4)×SO (5)). The fact that the lambda deformation gives rise to a consistent background for the string has been investigated in [21][22][23][24][25], with explicit results for AdS n × S n with n = 2 [23] and 3 [24] and on general grounds for n = 5 in [25].
In this work, we will undertake the first necessary step in constructing the quantum spectral curve of the lambda string, that is to construct its classical spectral curve. The QSC will be described in a subsequent work [26]. We will find that the classical curve can be obtained as the semi-classical limit of a set of ABA equations for a spin chain. The fundamental difference is that the AdS 5 ×S 5 case leads to an XXX type spin chain while the lambda string leads to an XXZ type chain.
The organization of the paper is as follows. In section 2 we provide some background on Lie superalgebras and establish notation. In section 3 we provide a short review of the lambda string model paying particular attention to its integrability. Section 4 discusses the monodromy of the model from which the classical curve is constructed. We also argue how the charges extracted from the monodromy include the energy and momentum of the gauge-fixed worldsheet theory. In section 5, we construct the classical spectral curve, writing it in terms of roots and weights of the superalgebra. Section 6 is devoted to showing that the classical curve can be identified with the semi-classical limit of a set of Asymptotic Bethe Ansatz (ABA) equations. The details of this will be presented in the follow-up paper [26]. The appendix contains a discussion of the analogue of the light-cone gauge-fixed lambda superstring and in particular how the worldsheet Hamiltonian emerges.

Some preliminaries
In this section, we describe some aspects of the theory of Lie supergroups and algebras that we will need.
In the defining representation, the generators are matrices E AB with a 1 in position (A, B) and zeros elsewhere. We can label the generators in the same way in an arbitrary representation. Each of the labels A ∈ {1, 2, . . . , 8} can be assigned a p-parity p A ∈ {0, 1} and a c-parity c A ∈ {0, 1}. The p-parity determines the graded Lie bracket of 2 generators: (2.10) So generators with p A +p B even/odd are even/odd generators of the Lie superalgebra. In our case, we have p A = (1, 1, 1, 1, 0, 0, 0). The c-parity determines the reality condition on the generators by specifying an anti-linear conjugation of the form The real form is then defined by taking anti-Hermitian combinations with respect to this definition of the conjugate. Writing (2.3) as M = −M ‡ , it follows that in our case c A = (1, 1, 0, 0, 0, 0, 0, 0).

Roots and weights
The Cartan subspace of the superalgebra in the defining representation consist of the diagonal matrices E AA . These are all even generators. We can introduce a root/weight vector space spanned by a set of vectors In a Cartan-Weyl basis, we can think of the Cartan generators as a vector  14) and then, the odd generators and roots, A weight vector is then expressed as so that Since the vector space has a non-trivial metric, one can define a dual vector We can extend the action of the Z 4 automorphism σ − onto weight vectors, (2.20) Note that on this vector space σ − has order 2.
The Cartan subalgebra of psu(2, 2|4) is six dimensional because of two properties. Firstly the unimodular property, the s in psu, implies that it only includes 7 independent elements which, in the defining representation are a basis of supertraceless diagonal matrices. For example, in a general representation we can take elements for j = 1, 2, 3. There is a central term which commutes with the rest of the algebra. The projectivity property, the p in psu, implies the equivalence relation a + C ∼ a, for any element of the algebra a. It follows that in a particular representation, the central term must vanish which implies that weight vectors satisfy the constraint Of the remaining six elements in the Cartan subalgebra, four are in f (0) and two are in f (2) . The two elements in f (2) are associated to vectors that are odd under σ − , (2.24) The associated generators in the defining representation are (2.25) In a conventional Lie algebra, all choices of simple roots are identical up to the action of the Weyl group. In a Lie superalgebra, there are no Weyl reflections associated to the odd roots and so there are inequivalent choices of simple roots. For example, one particular choice that we will use extensively is associated to the ordering The simple roots are then the differences of adjacent vectors in the ordering and can be associated to the Kac-Dynkin-Vogan (KDV) diagram

.27)
A KDV diagram encodes the properties of the simple roots. In particular, the fermionic simple roots are indicated with a cross; these are null α 2 = 0. Bosonic roots have either α 2 = ±2. The double circles encode the non-compactness of the Lie supergroup in the sense that a simple root with a c-parity odd generator is indicated by a double circle on the KDV diagram. In the notation of weights and with our choice of simple roots, the p and c parities are which then defines the parities of roots, e.g. p δ 1 −e 1 = p δ 1 + p e 1 , etc.

The loop algebra
Associated to the Lie superalgebra psu(2, 2|4) is a twisted affine (loop) algebrâ where z is the spectral parameter. Taking into account (2.3) and (2.6), we see that its generators a(z) ∈f satisfy

The lambda string
In this section, we present a mini review of the sigma model and the lambda model in the conformal gauge, where the world-sheet metric is of the form h µν = e φ diag(1, −1). The sigma model involves a coset F/G, F = PSU(2, 2|4) and G = Sp(2, 2)×Sp(4), with a particular WZ term that is exact and so can be written in local form [28]. The field f (τ, σ) is valued in the Lie supergroup F and the current J µ = f −1 ∂ µ f takes values in its Lie superalgebra f. The action takes the form 3 Here, the superscripts correspond to the decomposition into the eigenspaces of the Z 4 automorphism of the Lie superalgebra, f = ⊕ 3 j=0 f (j) . We will define projectors P (j) onto these eigenspaces. The invariant subspace f (0) = g is the Lie algebra of the bosonic subgroup G. This action is invariant under global F L transformations and z is the arbitrary spectral parameter. Notice that the Lax connection L µ (z) takes values in the twisted loop algebra (2.29), with the spectral parameter z being the affine parameter.
The lambda model is based on a WZW model for an F -valued group field F along with a gauge field A µ valued in f [17]. Actually, only the zero graded component of A µ is a genuine gauge field while the other components are Gaussian auxiliary fields. The action takes the form is the gauged WZW action for F gauged by the action F → U FU −1 for U ∈ F [30][31][32], and Notice that Ω ± − 1 vanishes on f (0) manifesting the fact that the gauge symmetry of S λ corresponds to the bosonic subgroup G ⊂ F .
The equations of motion for A µ give the constraints They can be written in terms of the usual Kac-Moody currents of the gauge WZW model as follows The components of these constraints in f (1) ⊕ f (2) ⊕ f (3) are second class whereas the constraints in f (0) are first class. The equations of motion of the group field can be written as either as the following which are trivially equivalent by conjugation with F.
Since the Lagrangian action is quadratic in the fields A µ , imposing eqs. (3.8) is equivalent to integrate out those fields at the classical level. Then, the equations of motion of F can be written in the same Lax form as the sigma model (3.3) and (3.4) with (3.12) Integrating out the gauge fields A ± allows one to write them in terms of F as follows This gives rise to an effective sigma model for the field F that can be understood as a deformation of the non-abelian T-dual of the sigma model with respect its full PSU(2, 2|4) global symmetry group. It is important to notice that, at the quantum level, the super-determinant that arises in integrating out A ± produces a dilaton on the world-sheet.

Virasoro constraints
In both the sigma and lambda models, we need to impose the Virasoro constraints, str(J that are left over after fixing conformal gauge. These conditions are solved by requiring where µ ± are arbitrary functions, V ± are group elements such that the current components are valued in f (2) . In the above, we have defined where we define Note that we could equally as well as chosen the combination Λ 1 − Λ 2 but this is conjugate to our choice above.
There is a second type of solution of the form which takes values in the subalgebra su(2, 2) (2) , and gives rise to string configurations that stay entirely in AdS 5 [33]. In this paper we will only consider the configurations of the first type (3.17), and will not discuss this second type of solutions any further.

Gauge symmetries
The Lagrangian action of the sigma model and the lambda model exhibit three different gauge symmetries each. First of all, the two models are invariant under conformal world-sheet reparameterizations σ ± → σ ± (σ ± ), which are not fixed by the conformal gauge condition. In addition, the sigma model action (3.1) is invariant under (infinitesimal) G R gauge transformations and under kappa symmetry transformations where is of the form Similarly, the lambda model action is invariant under G V gauge transformations Moreover, it is also invariant under the kappa symmetry transformations [17] and (1) and (3) are of the form (3.21). 5

The wave function
The fact that the sigma and lambda models share the same Lax connection provides a map between the solutions to the equations of motion of the two models, including the Virasoro constraints. This map becomes explicit by introducing the "wave function" to which we now turn.
The zero-curvature condition (3.3) is the compatibility condition of the associated linear problem whose solution is the wave function Ψ(τ, σ; z) ≡ Ψ(z). It takes values in the loop group F associated to the twisted loop algebra (2.29), so it satisfies Notice that the linear problem (3.25) determines the wave function up to multiplication on the right by a constant element of F 6 The wave function is defined on-shell and contains all the information about the space of solutions to the equations of motion. In terms of Ψ(z), the Lax connection can be recovered from (3.28) Moreover, one can write the group valued fields of the sigma model and the lambda model in terms of Ψ(z). Namely, using (3.4) and the definition of J µ in terms of the sigma model field f , it follows that L µ (1) = f −1 ∂ µ f and so is an arbitrary integration constant. The freedom to choose V exhibits the invariance of the equations of motion, and of the Lagrangian action in this case, under global F L transformations. We shall partially fix the freedom (3.27) by imposing that Similarly, using the identities the equations of motion for A µ (3.8) become simply In this case, the freedom to choose V uncovers a global symmetry of the equations of motion. In general, this is not a manifest symmetry of the Lagrangian action; however, it was argued in [29] that the action of the lambda model in the conformal gauge exhibits off-shell global symmetries that are precisely of this form. In the following, we will fix V = 1 so that The wave function allows one to describe in a unified way the action of the gauge symmetries of the sigma and lambda model on the space of solutions to their equations of motion. Namely, the G R and G V gauge transformations (3.19) and (3.22) correspond to Moreover, in terms of the wave function, the kappa symmetry transformations (3.20) and (3.23) of the sigma and lambda models become simply

Monodromy and symplectic form
In this section, we define the monodromy of the integrable system underlying the sigma and lambda models. This object is key for going on to define the CSC.
We assume that −π ≤ σ ≤ π and define the monodromy of the Lax connection as This satisfies Hence, if we impose closed string boundary conditions on the Lax connection, the spectrum of the monodromy T (τ ; z) is conserved in time. Since z is a free parameter this means that there are an infinite number of conserved quantities as required in an integrable field theory. In particular, if either the sigma model field f or the lambda model field F are periodic functions of σ It is worth noticing that the monodromy is invariant under the transformation (3.27). Moreover, under the gauge transformations (3.34) and (3.35), which include the kappa symmetry transformations, it changes as follows This shows that the spectrum of T (τ ; z) is gauge invariant, provided that the local parameters u (0) , κ (1) , and κ (3) are also periodic.
Once we have the monodromy, we can define the Classical Spectral Curve (CSC), the algebraic curve defined by the characteristic equation of the monodromy: The eigenvalues e ip A (z) of T (z), A = 1, 2, . . . , 8, thought of as a function of z, define the quasi-momenta p A (z) and correspond to the branches of the function p(z) that touch at various branch points. In the following, we will label the eigenvalues of the monodromy and the quasi-momentum as SU (4) . (4.7) The quasi-momenta generate all the conserved quantities in the theory. Some of them are local in the fields of the theory, as we shall see, whereas the remainder are non-local. We will often think of the quasi-momenta as a Cartan vector defined so that to be consistent with (4.7). Note the dual here, a distinction that will be important.
The periodic boundary conditions (4.3) ensure that Ψ(τ, π; z) = Ψ(τ, −π; z)W(z), where W(z) is a constant element of the loop group. Since W(z) and T (τ ; z) have the same eigenvalues and, thus, one could equivalently define the CSC in terms of W(z).
The sigma model and the lambda model share the same Lax connection and, therefore, one can expect that both models have a related monodromy and CSC. However, their symplectic structures are different and this leads to different preferred coordinates defined on the CSC. In addition, note that the boundary conditions (4.3) do not imply that the Lagrangian fields f and F are periodic. Writing their boundary conditions in terms of conserved quantities distinguishes different "special points" in each model. For the sigma model, using (3.30), one can easily show that (4.11) Therefore, imposing closed string boundary conditions f (τ, π) = f (τ, −π) is equivalent to T (1) = 1. This is related to the well-known level-matching condition, which relates periodic boundary conditions to the vanishing of the total world-sheet momentum [3].
Similarly, using (3.33), the lambda model field satisfies Hence, in this case, imposing closed string boundary conditions on F, implies that the monodromy at the two special points T (λ 1/2 ) and T (λ −1/2 ) have the same spectrum of eigenvalues. This will be the lambda string version of the level matching condition: Here, P is a possible permutation that arises from the freedom afforded by the Weyl group acting on the Cartan subalgebra of the loop group. The allowed permutations, i.e. ones that preserve the grading, are generated by the five elements e 1 ↔ e 2 , e 3 ↔ e 4 , (e 1 ↔ e 3 , e 2 ↔ e 4 ), δ 1 ↔ δ 2 and δ 3 ↔ δ 4 . 7 The equality in (4.14) is understood to be mod 2π.

Symplectic form
The sigma and lambda models share the same Lax connection, which implies that they have the same space of classical solutions. However, they have different symplectic forms and this is crucial for understanding the difference in their CSCs.
We can write the symplectic form in terms of the wave function and, in particular, in terms of the one forms on the phase space In the above, du is a 1-form on the space of the spectral parameter z that can be expressed in terms of the twist function, (4. 16) In the limit λ → 1 and k → ∞, from (4.16) we recover the twist function of the sigma model.
The contour in (4.15) is chosen to encircle the poles of φ(z), i.e. the special points z 4 = λ ±2 . Note that these special points correspond to z = λ ±1/2 and their images under the Z 4 automorphism z → iz. The 1-form du and the associated coordinate u will play a key rôle for the CSC and subsequently for the QSC. In particular, the action coordinates on phase space are obtained from a generic CSC by integrating the quasi-momenta where C are cycles on the CSC and p corresponds to the quasi momenta viewed as a function over the spectral curve. The conjugate angles correspond to a point on the Jacobian of the CSC. 7 Note that no permutation can mix the {e i } with the {δ i } in a Lie supergroup. In addition, in the non-compact part SU (2,2) there is no element of the Weyl group that is the analogue of (e 1 ↔ e 3 , e 2 ↔ e 4 ) in the compact sector. 8 The phase space is identified with the space of classical solutions and Ψ is defined on-shell and so depends implicitly on a point in the space of classical solutions.

Local conserved charges
Integrable field theories like the present one, typically exhibit conserved charges that are both integrals of local functions of the fields and their derivatives as well as non-local charges. All these charges are encoded in the monodromy. In particular, in this section the focus will be on the local conserved charges. One of the motivations in constructing the local conserved charges is that they include the energy and momentum in the gauge fixed worldsheet theory. Having said that, we need to emphasis an important point here, the CSC and QSC are defined without the need for gauge fixing.
The solution of the Virasoro constraints imply that the Lax connection L µ (z) has poles at z = 0 and z = ∞ proportional to an element which is conjugate to the Cartan element Λ. Choosing z = ∞, the argument starts by establishing that around the pole of the Lax connection at ∞, one can construct a gauge transformation Φ 1 (z) commutes with its leading term proportional to Λ. This means that L is also block diagonal. Note that the boundary conditions, ensure that Φ (∞) (π; z) = Φ (∞) (−π; z).
We can repeat the analysis expanding around z = 0 constructing a gauge transformation Φ (0) (z) where the right-hand side commutes with Λ. It follows that the gauge transformed monodromy is also block diagonal in the same sense as T (∞) (z).
Following the argument in [4], let us take one of the blocks, indicated by the subscript s ∈ {1, 2}. The gauge transformed monodromy of a particular block is given by Taking the superdeterminant gives a quantity that can be written as an exponential integral but without path ordering. Hence, for the monodromy this yields a generator of local conserved quantities q (s) (z): In fact, there is only one independent generator q (1) (z) + q (2) (2) (z) = 1. We can express the q (s) (z) through the quasi-momenta (4.24) The overall supertraceless combination is The charge generator Q(z) corresponds to the current A similar story plays out for the the expansion around z = 0. It is important that analyticity ensures that the charge associated to this expansion is equal to the same function Q(z) as obtained from the expansion around z = ∞.
In appendix A, we argue that one of these charges, in the gauge fixed theory, is identified with the physical Hamiltonian and momentum and on the worldsheet. We should emphasize that it is only in the gauge-fixed theory that it makes sense to define an energy: the Hamiltonian vanishes before gauge fixing. This identifies the energy and momentum of a configuration as and Notice that the level matching condition (4.14) (with the identity permutation) in the lambda string implies the vanishing the momentum, up to some winding numbers, just as it does in the sigma model case.

Classical Spectral Curve
In this section, we construct an integral representation of the CSC of the lambda model. Since the two models share the same Lax connection and wave function, some of the discussion is similar to that of the sigma model. In particular, we will follow quite closely the approach of Beisert et al. [4], although we will write the curve in terms of Lie superalgebra roots which makes the subtle form of the curve more transparent.
Both the sigma and lambda model curves are defined as a multi-sheeted cover over a base curve naturally parameterized by a distinguished coordinate u. The latter is determined by the symplectic form and since this is different in the lambda model, the CSC will be different. In particular, the behaviour at specials points is different: the sigma model curve has a single special point at z = 1 while the lambda model curve has a pair of special points z = λ ±1/2 , up to the action of the automorphism.

Base curve
The CSC is the algebraic curve defined in (4.6). We can think of it as a branched covering over the base defined by complex plane of the spectral parameter z. In fact, in the discussion, it is useful to introduce two alternative spectral parameters, u and x. The first alternative, u is obtained by integrating the 1-form du in (4.16), where we define the constant u 0 via So we can think of the z plane as 4-fold cover of the complex u cylinder u ∼ u + ik.
The quasi-momenta take values in the even part of the superalgebra and so it is natural to focus on z 2 rather than z itself. In that case, we can think of the z 2 plane as being two copies of the of the u cylinder that we denote as Σ (±) , joined by the branch cut [−u 0 , u 0 ], as shown in Figure 1. The value of z 2 on Σ (±) are related by z 2 → −z 2 , consistent with the Z 4 structure of the underlying twisted affine algebra. The other spectral parameter is the "Zhukovsky" parameter x which is related to z 2 by a simple transformation So the two branches Σ (±) are related by x → 1/x, i.e. z 2 → −z 2 , which is generated by the Z 4 automorphism on the spectral parameter z → iz.
The relationship between x and u is then determined to be of the form where we have introduced a constant 0 ≤ ξ ≤ 1 defined via This is a generalization of the Zhukovsky map x + 1/x = u/g of the sigma model; indeed, in the sigma model limit k → ∞, λ → 1, with g = k(1 − λ)/2π fixed, it reduces precisely to this. This mapping first appeared in the context of the S-matrix theory of the excitations of the gauge-fixed lambda model [36]. Note that the branch points u = ±u 0 , correspond to x = ±1. A given point u on the cylinder has images x and 1/x, with |x| ≷ 1, on Σ (±) , respectively.

Special points
A particularly important rôle is played by the points at infinity in the u parameterization, u = ±∞. These correspond to z 4 = λ ±2 or x = ∞ and x = −1/ξ, respectively, on Σ (+) , and x = 0 and x = −ξ, respectively, on Σ (−) . These points are related directly to the conserved charges of the two Kac-Moody symmetries of the theory and these charges include the world-sheet energy in the gauge-fixed lambda model [29], although we should emphasize that the curve can be defined without any such gauge fixing.
Note that these special values are at the poles of the twist function defined in (4.16). Before discussing the special points in the lambda model, let us first remind ourselves what happens in the sigma model. In this case, the mapping (5.1) becomes and (5.4) reduces to the usual Zhukovsky map x + 1/x = u/g. There is now a single point at infinity, or "special" point, u = ∞, i.e. z 4 = 1. If we expand around z = 1, the Lax connection in the sigma model has the asymptotic form Making a gauge transformation L ± → L ± = f (∂ ± + L ± )f −1 , and so the zero curvature condition at order 1/u has the form of a conservation equation for the Noether current for the global PSU(2, 2|4) left symmetry f → U f of the sigma model. So L ± is related to the components of the current ±J L ± by a gauge transformation by f . But this gauge transformation does not affect the spectrum of the monodromy, which is invariant provided that f satisfies periodic boundary conditions. Hence, this means that as u → ∞, the quasi-momenta have the asymptotic behaviour p A (u) = 2gQ A /u + O(u −2 ). where Q A are the Noether charges for the global PSU(2, 2|4) left symmetry. In the gauge-fixed worldsheet theory, these charges include the energy of the states. Now let us turn to the lambda model. Here, there are two special points, z = λ ±1/2 (up to automorphism), or u = ∓∞. At these points, the spatial component of the Lax connection becomes proportional to the Kac-Moody currents (3.9): where So at the special points, the quasi-momenta are the non-abelian charges associated to the Kac-Moody symmetry.
Following the notation of [7] (and the excellent review [8]), we will write the quasimomenta p A = (p i ,p i ), for i ∈ {1, 2, 3, 4} associated to the su(4) and su(2, 2) bosonic sub-algebras of psu(2, 2|4), respectively. The charges are extracted from the quasimomenta at the special points z = λ ∓1/2 , i.e. u = ±∞. Using this notation, we will define the charges as where the (λ i ) are understood to be defined modulo k since only the eigenvalues of the monodromy exp(ip A ) are physically significant.

The charges define a pair of weight vectors
which are related by the boundary conditions. With closed string boundary conditions, the level matching conditions (4.14) state that 14) The equality in here is understood to be mod k. The CSC can be constructed with arbitrary weights ω (±) and the level matching condition can be imposed ex post facto.
The relation (5.13) deserves some comment. Weight vectors should satisfy the vanishing of the central charge condition (2.23). This condition is actually implied by the unimodularity condition on the quasi-momenta This is to be expected because the charges, or weights, and the quasi-momentum are dual under the supertrace. 9 The sigma model has a global PSU(2, 2|4) symmetry whereas we expect the lambda model to have a quantum group symmetry PSU q (2, 2|4), q k = −1. 10 Both the group and the quantum group share the same Cartan generators and so the structure of the charges can be expected to be the same. In particular, we can identify the charges of the quantum group as where J 1,2,3 and S 1,2 are associated to the quantum group versions of the compact subalgebras, su(4) and su(2) ⊕ su(2) ⊂ su(2, 2), respectively, and so are half integers, while ∆ is associated to the non-compact part of the subalgebra su(2, 2) and is not quantized. These relations are only valid in the classical limit when k and the charges scale like k and so the integer nature cannot be resolved.
The energy of the gauge-fixed theory are identified with the following combination of charges (4.28), (4.29). Choosing the trivial permutation in (5.14), this gives A z-dependent charge is an object Q q (z) = str(q p(z)) defined in terms of an algebra element q and the quasi-momentum p(z) = * p(z) · H. The charge of the central element q = C = i1 8 vanishes precisely because of the unimodularity constraint str(p(z)) = 0. 10 The quantum group is a deformation of the universal enveloping of psu(2, 2|4), a Drinfeld-Jimbo algebra, e.g. see the books [34,35]. The deformation parameter q is a root-of-unity and in that case, the quantum supergroup has additional central elements. We are interested in the restricted quantum group obtained by quotienting by these additional central elements. So by PSU q (2, 2|4) we mean the restricted quantum supergroup. just as in the AdS 5 ×S 5 string, while the level matching condition requires the vanishing of the momentum (mod k).

Engineering the curve
Now we turn to the CSC itself. Some of the details are similar to the sigma model case described at length [4] although here we will keep the relation with the root system of the Lie superalgebra more transparent. The CSC is the branched 8-sheeted covering of the base Σ defined by the eigenvalues exp[ip A (x)] which we associate to a weight vector p(x) as in (4.8), although here we find it more useful to think of it as a function of the spectral parameter x which covers both sheets Σ (±) . The quasi-momenta has various properties that we list below. Note that the curve is most naturally engineered in terms of its dual * p(x).
(5. 18) This means that the symmetry exchanges the values of the quasi-momenta on the second sheet Σ (−) with those on the physical sheet Σ (+) along with a permutation of the index. We can think of the integers m j as winding numbers.
(ii) The fact that we are working in PSU(2, 2|4) rather than U(2, 2|4) has the following implications. Firstly, unimodularity, the "S", means that the quasi-momenta are subject to the constraint (5.15). Projectivity, the "P", means that shifts by a central element are not physical: for an arbitrary function ζ(x), with ζ(1/x) = −ζ(x) so that the shift has the correct grade inf.
(iii) The Virasoro constraints (3.14), specify the poles of the quasi-momenta at z → 0 and z → ∞, i.e. x → ∓1, respectively. Since the Lax operator has the poles at z → 0, ∞ (3.15), a l,s ξ l + · · · , (5.21) for constants a s,l subject to the constraints In (5.21), the ellipsis is regular at x = ±1 and we have defined the functions Notice that P ± (x) are consistent with the symmetries of the curve (iv) There are square root branch cuts associated to even roots of the superalgebra α = e i − e j (or δ i − δ j ) that join a pair of sheets labelled by e i and e j (or δ i and δ j ) on the sheet Σ (+) of the base. Each such cut comes in a pair related by the symmetry x → 1/x, with a permuting of the sheets according to the action of the automorphism in (2.20). More specifically, the square root branch cuts are in the eigenvalues exp[ip A (x)] rather than the quasi-momenta themselves and this means that the latter can jump by multiples of 2π on crossing a cut. We can express this by saying that the averages of the quasi-momenta across a cut C that joins the two sheets, must differ by an integer multiple of 2π, where α is the associated even root. In the above, we have defined / p(x) = (p(x + ) + p(x − ))/2, where x ± lie infinitely close, but on either side, of a point x on the cut. The filling of a cut is defined using the symplectic 1-form du as where A in a contour that encircles the cut C on the sheet e i in a positive sense (or e j in a negative sense) and similarly for δ i − δ j .
So each cut has a filling K and an integer n. This data is the curved space analogue of the amplitude and the mode number of a string in flat space.
(v) Fermionic excitations are associated to fermionic roots α = e i − δ j , but instead of being associated to cuts they correspond to isolated poles with equal residues on each of the sheets e i and δ j in the pair. The poles also come in pairs associated to the transformation x → 1/x. The residue should be understood as the product of two Grassmann numbers. In this case, we define / p(x) as the regular part at the pole and then both (5.25) and (5.26) are still valid.
(vi) In the lambda model, the quasi-momenta at the special points determine the charges of the configuration, a pair of weight vectors ω (±) in (5.13). This data is determined by the fillings of the cuts and the integer data in a way that we will uncover. These quantum numbers include, in the gauge-fixed worldsheet theory, the energy of a configuration.
The CSC can be thought of as the solution of a Riemann-Hilbert problem for the quasi-momenta p defined over the base curve Σ with the analytic properties outlined above. The solution of this Riemann-Hilbert problem can be formulated in terms of seven independent "densities", ρ r (x), r = 1, 2, . . . , 7, that can naturally be associated to a set of simple roots {α r } of the superalgebra associated to some ordering of the sheets e A . A cut in the quasi-momentum associated to an even root α = r∈S α r for some S ⊂ {1, 2, . . . , 7}, corresponds to an equal contribution to the densities of ρ r (x), for r ∈ S, of a function with support along the image of the cut (an open contour). If the root is odd, then the function has delta function support.
We will need two kinds of resolvents defined in terms of the densities. The first is defined solely in terms of the Zhukovsky spectral parameter x and a suitable kernel g(x, y): The basic property of the resolvent is that it is an analytic function with cuts along the support of the density for even roots excitations and poles for fermionic excitations. The basic requirement, for the cuts, is that the discontinuity across a cut gives the density, along with a Jacobian: Hence, the kernel g(x, y) must have a pole at x = y with residue This does not completely determine g(x, y) so there is some freedom here that is ultimately just a matter of convention, but we make the following choice, based on hindsight afforded by the semi-classical limit of the QSC that we discuss in [26], Note that the defining property (5.28), ensures that an integral of the 1-form G r (x)du/2πi around a cycle A that encloses the cut C r,p , where p labels the cut, or surrounds a pole, gives the integral of the density, or the "filling". To see this, one shrinks A onto the cut and then expands at each point of the support x = y: · 1 x − y = Cr,p dy ρ r (y) = K r,p .

(5.31)
For later we will define the total filling of each density function ρ r (x) For an odd root, say α = δ i − e j , the cuts are replaced by isolated poles which we can also express via kernel g(x, y). Near x = x p , the position of the pole, (5.33) The residues κ p are products of two Grassmann numbers. The resolvent G r (x) for an odd simple root, is defined as a sum over the poles G r (x) = p κ r,p g(x, x r,p ) .
With these preliminaries, we can make an ansatz for the quasi-momenta that incorporates all the constraints and symmetries: where φ is the constant vector Note that we could add a trivial shift to φ in the form of a constant element of the Cartan subalgebra of f (0) .
Let us write the curve in a different way which will facilitate a simpler comparison with the semi-classical limit of the Bethe Ansatz equations in the companion work [26] (see also section 6), by introducing an inversion-symmetric resolvent naturally expressed in terms of the coordinate u: Using the above, we can write the curve as * p( We remark here, that 1 − σ − projects onto the component f (2) that lies in the Cartan subalgebra. This subspace is spanned by the two vectors ξ 1 and ξ 2 defined in (5.25), so we can write 11 where we have defined quantities, for j = 1, 2, The final expression for the curve is * p( Hence, we can write the conditions that determine the curve (5.25) as a set of integral equations for the densities where the general expression for the "driving terms" is Notice that F r (x) is analytic along the cuts C r,p . The way we have constructed the curve in terms of integral equations for densities is a characteristic way of solving a Riemann-Hilbert problem.
To summarize, we have defined an integral representation of the CSC in terms of the densities ρ r (x) which have to satisfy the auxiliary conditions (5.43). It is a simple matter to extract the charges of the configuration by looking at the behaviour of the quasi-momenta at the special points (5.16): In writing the above, we have used the fact that G r (0) = −G r (−ξ) and so β r (0) = −β r (−ξ).
We have defined the curve for general ω (±) and for closed strings we need to impose the level matching condition (4.14).

A particular choice of simple roots
In order to make contact with [4], let us consider one particular choice of simple roots, precisely those described by the KDV diagram in (2.27). In that case, we can write the components of the quasi-momentum, up to a shift (5.19), explicitly aŝ where we have defined ε i = (1, 1, −1, −1) and (5.47) In the above, (5.48) The "driving terms" F r (x) are, from (5.44), In order to make contact with the gauge-fixed theory, we must impose conditions on the residues of the curve at x = ±1; namely, a 1,+ = −a 2,+ and a l,− = 0 in (5.38). 12 These conditions are consistent with the fact that in the gauge fixed theory described in detail in appendix A, has poles of form In the gauge-fixed theory, the energy of a configuration is given by (4.28), The momentum is given by (4.29)

(5.53)
Note that it is the density ρ 4 (x) that contributes to the energy and momentum.

More general bases
It is interesting to construct the form of the CSC for four choices of simple roots, three additional ones to add to the choice in the last section. These are choices in [5] that correspond to four different ordering of the Riemann sheets with the following four KDV diagrams: The case η 1 = η 2 = −1 is the one considered in the last section.
What distinguishes all four of these choices is that the action of the automorphism on the simple roots is the same: (5.54) The dot products the roots with the Cartan vectors ξ j are In particular, (ξ 1 − ξ 2 ) · α r = −2δ r4 , so the energy and momentum in the gauge-fixed theory is always associated to the density ρ 4 (x) in all of the 4 bases.
From (5.54), we find that The driving terms are (5.57)

Discussion
In this work, we have constructed the spectral curve of the lambda string model at the classical level as a Riemann-Hilbert problem, i.e. in terms of some densities that satisfy the integral equations (5.43). The densities determine a set of resolvent functions which in term determine the quasi-momentum. An outstanding problem is to present the construction of the classical solution that corresponds to a particular curve.
Our motivation for presenting the classical curve is as a necessary step towards the construction of the quantum curve of the lambda string which we will present in a companion paper [26]. As an additional step towards this goal, we can point out that the integral equations of the classical curve (5.43) have the form of a classical limit of a set of Bethe Ansatz equations, the Asymptotic Bethe Ansatz (ABA) equations, for the supergroup PSU(2, 2|4). Importantly the ABA is of XXZ type rather than the XXX type familiar from the sigma model case.
Let us outline the steps that lead from the quantum spectral curve to the classical curve in the classical limit. The quantum curve is formulated in terms of a set of basic "Q functions", that define a multi-sheeted cover of the underlying base curve Σ. These Q functions are defined over the base curve Σ and subject to certain non-trivial conditions and have a prescribed analytic structure. In particular, the charges of a quantum state are extracted from the behaviour at the special points. There exists a certain asymptotic limit of the curve, where 2 of the charges, namely ∆ and J 1 are large but with ∆ − J 1 fixed. In this limit, one can identify a subset of these Q functions Q r (u) associated to particular choice of simple roots of the Lie superalgebra. Each of these basic Q functions has a set of zeros {u r,j }, j = 1, 2, . . . , K r , the set of Bethe roots. Consequently, one has where Q r (u) is a Baxter function of XXZ type, familiar from an XXZ type Heisenberg spin chain, where the {u r,j } are the Bethe roots for the r th simple root. The dressing factor Φ r (u) is rather non-trivial and will be described fully in [26]. The non-trivial relations of the Q, in the asymptotic limit, become the conditions for u ∈ {u r,j }. These equations take the form of the conventional nested Bethe Ansatz equations for the Baxter functions, for the supergroup PSU(2, 2|4), for each u ∈ {u r,j }, but with unconventional driving terms on the right-hand side. These driving terms are determined by the dressing factors Φ r as we determine in [26].
The classical limit involves taking g → ∞ and k → ∞ keeping the ratio g/k fixed, i.e. ξ and λ are fixed. In this limit, all the charges scale like k and we are naturally in the regime of the ABA so the number of Bethe roots K r scale like k. The Q functions have a WKB-like limit in terms of the quasi-momentum of the classical theory, where the {ω r } are dual to the simple roots ω r · α s = δ rs ; equivalently, In this limit, the Bethe roots associated to the even simple roots of the underlying Lie superalgebra "condense" along cuts described by densities ρ r (x): Kr j=1 · · · −→ dx ρ r (x) · · · . (6.7) The ABA equations can be written as integral equations over resolvents, in fact precisely the conditions of the classical curve (5.43). The first term on the left-hand side of (5.43) arise from the classical limit ∂ u Q r (u) −→ H r (x(u)), the resolvent in (5.37).
The gauge-fixed set of configurations are built on top of the reference/vacuum configuration with a gauge-fixed Lax connection [27] L + = γ −1 ∂ + γ + zψ + + µz 2 Λ , In the above the G-valued group field γ and ψ ± are a pair of Grassmann fields lying in f (1) and f (3) . There are residual constraints that γ −1 ∂ + γ, ∂ − γ γ −1 and ψ ± must lie in the image of ad Λ on the algebra. The degrees of freedom in γ, so constrained, are the analogue of the transverse coordinates X i of the flat space string.
Now we turn to the lambda model and follow the same logic. A suitable reference, or vacuum, solution in this case is obtained by taking compare (A.1). The group field of the lambda model takes the form which is also interpreted as a point-like closed string propagating along a null geodesic in the geometry. This corresponds to a different solution of the linear system compared with the sigma model case above, Hence, the gauge-fixed Lax connection is different by an important sign Before we continue, it is interesting to look at the plane wave limit in both the sigma and lambda models. Ignoring the fermions, the plane wave limit is obtained by linearizing around the identity for the field γ = e φ = 1 + φ + · · · . The equation of motion for the algebra field φ is the massive equation: The ∓ sign here corresponds to the sigma and lambda models, respectively. The mass terms for φ have the conventional sign for the sigma model case, corresponding to the fact that geodesics nearby the BMN geodesic are converging. However, in the lambda model, the mass terms have the "wrong" sign, corresponding to the fact that geodesics nearby the vacuum geodesic are diverging. The implication is that the plane wave limit in the lambda model around our chosen vacuum (A.5) is not self consistent: geodesics diverge away and the plane wave limit of the metric breaks down. Another characteristic of the lambda model vacuum (A.5) is that the classical solutions, the magnons, defined in the limit of large µ (the Hofmann-Maldacena limit) around the vacuum solutionnaïvely at least-are superluminal. 13 The problem is solved by realizing that magnons should be defined around a different reference configuration with γ = γ 0 = 1 (so lying within the gauge-fixed subspace): (A.10) So γ 0 is a constant group element that has the property γ −1 0 Λ 1 γ 0 = Λ 1 and γ −1 0 Λ 2 γ 0 = −Λ 2 . 14 Notice that the Virasoro constraints are still satisfied (as they are for all solutions in the gauge-fixed subspace).
With this choice, .11) and the group field of the reference solution has the form It describes a string stretched in the compact part of the group, rather than a pointlike string, propagating in the time direction. In this case, the magnon solutions are precisely those constructed in [29]. Ultimately the type of gauge fixing should not be physically significant, both reference solutions lie in the same gauge-fixed subspaceand so for simplicity in this work when we describe the gauge-fixed theory we will imply the choice with reference configuration (A.5).
With these preliminaries there are a series of steps to go through in order to identify the physical Hamiltonian and momentum of the gauge-fixed theory with the choice of vacuum solution (A.5). We start by defining the Kac-Moody currents of the lambda model formulated as a deformed gauge WZW model The equations of motion of the gauge field yield the constraints (3.8) which take the form 14) In the theory of constrained Hamiltonian systems, these constraints are partly first and partly second class constraints. Fortunately we will not need to describe this in detail and we will be able to avoid most of the resulting complications.
The Poisson brackets can also be formulated directly in terms of the Lax connection in much the same way as for the sigma model itself [38][39][40]. In fact, the formalism of [40] applies directly to the lambda theory by replacing the twist function by the λ dependent one (4.16). In order to define this Poisson bracket consistently, one needs to write the Lax connection in the Hamiltonian formalism. The key point is to find a formulation where the Lax connection is flat in the strong sense without imposing any constraints. This involves adding additional terms that vanish on the constraint surface. This is discussed in details in [38][39][40] in the sigma model case. Mirroring this in the lambda model, the Hamiltonian Lax operator, the spatial component can be written as where R = R + + R − and the algebra endomophisms are R +f =f ≥0 and R −f = −f <0 so that R + − R − = 1. Explicitly, for X(z) = n X n z n ∈f, R + (X(z)) = n≥0 X n z n , R − (X(z)) = − n<0 X n z n . (A.18) The natural setting for the Poisson structure of our theory, is the double loop algebra: that is elements of the loop algebra F that are functions of the spatial coordinate σ. We will need a twisted inner product on this object: dz 2πiz φ(z) str a(σ; z)b(σ; z) , (A. 19) where the contour integral picks up the poles of the twist function. Note that the twisted inner product involves the 1-form du in (4.16).
The key result here, is that (A.24) defines a Hamiltonian flow on the phase space. We now will identify a particular ψ[L 1 ] such that t is the physical time τ . This will identify the physical Hamiltonian.
We begin by performing the particular gauge transformation in the loop group of the spatial component of the Lax operator as in (4.18) such that [L Using this fact allows us to define a flow η associated to any function ζ(z) such that ζ(z)Λ is in the loop algebra, so ζ(iz) = −ζ(z). The flow can be written in two equivalent ways The flow can be written in terms of R = R + + R − as It follows that these flows on the Lax operator take the form of zero curvature condition 15 Implicitly we working on shell for the Virasoro constraints. The flows we define will be seen to lie within the constrained subspace of the phase space.