Bethe ansatz for an AdS/CFT open spin chain with non-diagonal boundaries

We consider the integrable open-chain transfer matrix corresponding to a Y=0 brane at one boundary, and a Y_theta=0 brane (rotated with the respect to the former by an angle theta) at the other boundary. We determine the exact eigenvalues of this transfer matrix in terms of solutions of a corresponding set of Bethe equations.


Introduction
Two remarkable conjectures have commanded considerable attention for over a decade: the AdS 5 /CFT 4 correspondence [1,2], positing the equivalence of type IIB superstring theory on AdS 5 × S 5 [3] and N = 4 supersymmetric SU(N) Yang-Mills theory in 3+1 dimensions [4]; and the integrability of the spectral problem in planar AdS 5 /CFT 4 [5], positing that the energies of string states, or equivalently, the scaling dimensions of all local gauge-invariant operators in the planar limit of the dual gauge theory, are described by an integrable 1+1 dimensional model. 1 We shall refer to the latter as the AdS/CFT integrable model. This AdS/CFT integrable model is essentially the string world-sheet quantum field theory in a light-cone gauge. It has a centrally-extended su(2|2) symmetry 2 , and the spectrum includes four fundamental particles: two bosons, and two fermions. The exact (non-relativistic) dispersion relation is known, as are exact bulk and boundary world-sheet S-matrices.
The momentum quantization condition for a set of such particles on a ring (i.e., periodic boundary conditions) of finite length leads [8,9] to the all-loop asymptotic Bethe equations [10], which determine the energies of closed strings/scaling dimensions of single-trace operators in the dual gauge theory, up to wrapping (finite-size) corrections.
Similarly, the momentum quantization condition for a set of such particles on an interval of finite length leads to all-loop asymptotic Bethe equations that determine the energies of open strings/scaling dimensions of determinant-like operators in the dual gauge theory, again up to wrapping (finite-size) corrections. The detailed results depend on the specific boundary conditions at the two ends of the interval. Among the integrable cases that have been studied are Y = 0 branes [11,12] at both ends [13,14,15]; and Y = 0 at one end and Y = 0 at the other end [16]. (For a review of integrable boundary conditions in AdS/CFT, see [17].) The key technical step in deriving the various asymptotic Bethe equations is to determine the eigenvalues of the corresponding integrable inhomogeneous transfer matrices, which are constructed with the bulk and -for cases with boundaries -boundary S-matrices. The boundary S-matrices for Y = 0 andȲ = 0 branes are diagonal. However, the boundary Smatrix for a Y θ = 0 brane [16], which interpolates between them, is not diagonal. Hence, the problem of diagonalizing the transfer matrix constructed with the latter boundary S-matrix is nontrivial, and is the main goal of this paper. Our strategy is to exploit the unbroken u(1) symmetry by carrying out the first step of the nested algebraic Bethe ansatz, following [18,19]. This leads to a second-level open-chain spin-1/2 XXX transfer matrix with nondiagonal boundary terms, which we diagonalize by introducing an inhomogeneous term in its T-Q equation [20,21]. A similar strategy was employed to solve the open Hubbard [22] and supersymmetric t-J [23] models with non-diagonal boundary interactions; however, those works used coordinate Bethe ansatz (instead of nested algebraic Bethe ansatz) for the first step.
The paper is organized as follows. In section 2 we introduce our notations, recall the 1 Integrability is believed to appear also for AdS 4 /CFT 3 [6] and AdS 3 /CFT 2 [7]. However, we focus here on AdS 5 /CFT 4 , which is the simplest and best-understood case.
2 Actually, the symmetry consists of two copies of this algebra, but we focus here on just one copy.
relevant AdS/CFT bulk and boundary S-matrices, and review the construction of the corresponding integrable open-chain transfer matrix. In section 3 we determine the exact eigenvalues of this transfer matrix in terms of solutions of a corresponding set of Bethe equations. We then use the unbroken su(2) symmetry to derive formulas for the number of distinct eigenvalues (and hence, number of solutions of the Bethe equations) and their degeneracies. We check these results numerically for small system size. In section 4 we briefly discuss our results and note some remaining problems. In the appendix we propose a generating functional for the eigenvalues of transfer matrices whose auxiliary spaces belong to higherdimensional representations of su(2|2).

Construction of the transfer matrix
Here we introduce our notations, recall the relevant AdS/CFT bulk and boundary S-matrices, and review the construction of the corresponding integrable open-chain transfer matrix.

Parametrization
Following Arutyunov and Frolov [24], we use the elliptic parametrization for the momentum p and the parameters x ± 3 such that and (We shall often refrain from exhibiting the dependence of x ± and p on the uniformizing parameter z.) The parameter g is the coupling constant of the AdS/CFT integrable model (string tension), which is related to the 't Hooft coupling λ of the dual gauge theory by The two periods are given by where K(k) is the complete elliptic integral of the first kind. The crossing transformation is effectuated with a shift of z by the half-period ω 2 , is the energy. We note that z → −z corresponds to a reflection, We define u(z) by and therefore (2.10)

Transfer matrix
The open-chain transfer matrix for a single copy of su(2|2) is given by [29,31,32] where the monodromy matrices are given by the auxiliary space is denoted by a, and str denotes super trace. The {z i }, which correspond to the rapidities of the N particles on an interval, are to be regarded as fixed inhomogeneities. (To lighten the notation, we shall often suppress the dependence on these inhomogeneities.) By construction (see e.g. [29,31]), the transfer matrix has the fundamental commutativity property for arbitrary values of z and z ′ . For the boundary S-matrices (2.21) and (2.23) that we consider here, the transfer matrix also has the right su(2) symmetry (2.30)

Exact diagonalization of the transfer matrix
We turn now to the main task of deriving the eigenvalues of the transfer matrix (2.26) and obtaining the corresponding Bethe equations.

Nested algebraic Bethe ansatz
The transfer matrix has an unbroken u(1) ⊂ su(2) symmetry. In particular, the state with "all spins down" is an eigenstate of the transfer matrix. Therefore, using this state as the reference state, we can carry out the first step of the nested algebraic Bethe ansatz, following [18,19]. To this end, it is convenient to write the boundary S-matrices (2.21), (2.23) as We also write the monodromy matrices (2.27) as follows

The action of the transfer matrix on the reference state
We observe that the elements of the monodromy matrices have the following action on the reference state Here and below we use the following notations where a 1 , . . . , a 10 are given by (2.15).
The double-row monodromy matrix is defined as We obtain In order to obtain the actions of operators A ij (z) and D(z) on the reference state, we use exchange relations derived from the Yang-Baxter equation (2.18) After some algebra, we obtain We defineÃ

34)
From the Yang-Baxter relation (3.23), we also obtain Using the definition ofÃ ij (z), we obtain we obtainD The transfer matrix (2.26) can be expressed in terms of elements of the double-row monodromy matrix (3.16) It is now straightforward to verify from the above results that the reference state is an eigenstate of the transfer matrix, with eigenvalue a 5 (z, z k )a 6 (z k , −z) .  (3.43) and the definitions ofÃ ij (z) andD(z), we obtain -after lengthy computations -the following exchange relations: where with h 1 (z 1 , z 2 ) = a 1 (z 1 , z 2 ) + a 9 (z 1 , z 2 )a 10 (z 1 , z 2 ) a 3 (z 1 , z 2 ) , h 2 (z 1 , z 2 ) = a 12 (z 1 , z 2 ) ,
In terms of u(z) (2.9), we can now write where u j ≡ u(z j ), and R (2) (u) is the familiar spin-1/2 XXX R-matrix where I and Π are the 4 × 4 identity and permutation matrices, respectively.

Off-diagonal Bethe ansatz
In view of (3.59), in order to determine the eigenvalues of the transfer matrix (2.26), it now remains to diagonalize the nested transfer matrix t (2) (z). We recognize the latter as the transfer matrix of an open spin-1/2 XXX chain of length M with non-diagonal boundary terms. Therefore, using the off-diagonal Bethe ansatz [20,21] (see also [33]), we can immediately write down an expression for the corresponding eigenvalues.
Indeed, let us introduce the following functions According to the off-diagonal Bethe ansatz, the eigenvalues of t (2) (z) can be given by where the polynomial Q 2 (u) is parameterized by M Bethe roots {w j } One can recognize (after multiplying both sides by Q 2 (u)) that (3.68) is a T-Q equation with an additional inhomogeneous term.

Eigenvalues and Bethe equations
Combining the results (3.59) and (3.68), we conclude that the eigenvalues of the transfer matrix t(z) (2.26) are given by The requirement that Λ(z) should not have any poles leads to the following Bethe equations 7 These results can be reexpressed more succinctly by using the shorthand notation of [35,15] (3.73) 7 It should also be possible to obtain the Bethe equations from the cancellation of the unwanted terms that appear when the transfer matrix acts on an off-shell Bethe state; however, we have not determined the complete off-shell equation. Such an off-shell equation has been found recently for the XXZ chain [34].
For example, the expression R (−)− (z) should be understood to mean Similarly, . In terms of this notation, the eigenvalues (3.70) are given by The corresponding Bethe equations are (3.79) 8 We recall the definitions (3.58) and also note the identity The Bethe equations (3.77) and (3.78) are equivalent to (3.71) and (3.72), respectively.
Interestingly, for θ = π/2, the inhomogeneous term does not vanish, even though the boundary S-matrices are diagonal for this case (see (3.2), (3.3) and (3.64))! This is the price we pay for having an expression for Λ(z) that is analytic in θ. An alternative expression is 9 (For θ = π/4, either s = +1 or s = −1 can be chosen.) The inhomogeneous term in this expression does vanish for both θ = 0 and θ = π/2; and for θ = π/2, this result for the transfer matrix eigenvalue is consistent with the duality transformation of the result (3.20) in [16].

Degeneracy and multiplicity
The degeneracy of the transfer matrix eigenvalue (3.75) corresponding to a given solution of the Bethe equations (3.77)-(3.78), as well as the number of such solutions (multiplicity), can be inferred from the unbroken su(2) symmetry (2.29) of the transfer matrix. 10 Indeed, we expect (see e.g. [18,36]) that the Bethe states are su(2) lowest-weight states, with For one site, the decomposition of the 4-dimensional vector space into su(2) representations is given by 0 ⊕ 0 ⊕ 1 2 . For N sites, the decomposition of the space of states into a direct sum of su(2) irreducible representations can be easily determined using the Clebsch-Gordan theorem where n s is the multiplicity of spin s. With the help of the multinomial theorem, an explicit expression for n s can be derived
For N = 1, we expect according to (3.85) one solution with M = 0 (namely, the trivial solution with no Bethe roots, corresponding to the reference state (3.1)), and two solutions with M = 1. We indeed find these solutions, as shown in Table 1  Similarly, for N = 2, we expect (3.85) one solution with M = 0, four solutions with M = 1, and five solutions with M = 2. We indeed find these solutions, as shown in Table 2. The corresponding eigenvalues obtained from (3.75) match with the 16 eigenvalues obtained by direct diagonalization of the transfer matrix.
In short, we have verified that our Bethe ansatz solution correctly gives the complete set of eigenvalues of the transfer matrix for N = 1 and N = 2.

Discussion
For the transfer matrix (2.26) of the Y θ −Y system, we have determined the exact eigenvalues (3.75) in terms of solutions of a corresponding set of Bethe equations (3.77)-(3.78). We have checked this result numerically for small system size.
The Y θ = 0 boundary S-matrix (2.23) is one of the few known integrable AdS/CFT boundary S-matrices with a free parameter. (Other examples are discussed in [32,37].) The present work represents the first time in the AdS/CFT context that an open-chain transfer matrix with a non-diagonal boundary S-matrix is diagonalized.
We hope to use this result in a future publication to compute asymptotic energies and finite-size corrections for one-particle states, as a function of the angle θ. Such corrections have already been computed for the special (diagonal) cases Y − Y (θ = 0) andȲ − Y (θ = π/2) in [14,15] and [16], respectively. The latter system is noteworthy for the presence of tachyons in its spectrum.
We expect that similar techniques can also be used to analyze other integrable cases with non-diagonal boundary S-matrices.

A Generating functional for higher transfer matrices
In the body of this paper, we have focused on a transfer matrix t(z) (2.26) whose auxiliary space belongs to the fundamental (4-dimensional) representation of su(2|2). This transfer matrix is only the first member of an infinite hierarchy of commuting transfer matrices T a,s (with T 1,1 = t(z)) whose auxiliary spaces belong to rectangular representations of su(2|2), and which satisfy the Hirota equation [35] T + a,s T − a,s = T a+1,s T a−1,s + T a,s+1 T a,s−1 . (A.1) We propose here a generating functional for the eigenvalues of these transfer matrices (which we also denote by T a,s ), which are useful for computing finite-size corrections (see e.g. [35,15,16]). This generating functional is a generalization of the one proposed in [33] for the XXX chain with nondiagonal boundary terms.
In order to streamline the notation, we rewrite the eigenvalue result (3.75) as where h is a normalization factor We propose that the generating functional for antisymmetric representations is given by The generating functional (A.5) therefore reduces to which coincides with the result given by (E.13) and (E.17) in [16]. Moreover, for N = 0 but still M = 0, we obtain which coincides with (E.21) in [16].
The generating functional for symmetric representations is given by the inverse of (A.5),