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 Yθ = 0 brane (rotated with the respect to the former by an angle θ) 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.

S-matrices
As already noted, there are four fundamental particles. Let us denote the corresponding Zamolodchikov-Faddeev operators by A † i (z) , i = 1, 2, 3, 4, where i = 1, 2 are bosonic and i = 3, 4 are fermionic. The matrix elements of the bulk S-matrix are defined by which can be arranged into a 16 × 16 matrix as follows We work with a graded version of Beisert's su(2|2) S-matrix [25]. Specifically, following Arutyunov and Frolov [26], we take where the matrices Λ 1 , . . . , Λ 10 are given in terms of quantities E kilj defined by (2.14) Hence, S(z 1 , z 2 ) has the following matrix structure

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

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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

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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

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From the Yang-Baxter relation (3.23), we also obtain Using the definition ofÃ ij (z), we obtain The transfer matrix (2.26) can be expressed in terms of elements of the double-row monodromy matrix (3.16)

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It is now straightforward to verify from the above results that the reference state is an eigenstate of the transfer matrix, with eigenvalue +2 cos(2θ) cos 2 p(z) 2 N k=1 a 5 (z, z k )a 6 (z k , −z) .
and the definitions ofÃ ij (z) andD(z), we obtain -after lengthy computations -the following exchange relations: where

48)
and "u.t." denotes so-called unwanted terms, which we do not explicitly write.

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Let us define By using the exchange relations (3.44)-(3.46) and the values ofÃ ij (z),D(z) and B(z) when acting on the reference state, we obtain We remind the reader that u = u(z) is given by (2.9), and therefore . (3.65)

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

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According to the off-diagonal Bethe ansatz, the eigenvalues of t (2) (z) can be given by 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 equa- 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].

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These results can be reexpressed more succinctly by using the shorthand notation of [15,35] For example, the expression R (−)− (z) should be understood to mean Similarly, . In terms of this notation, the eigenvalues (3.70) are given by where The corresponding Bethe equations are The Bethe equations (3.77) and (3.78) are equivalent to (3.71) and (3.72), respectively. For θ = 0, the last ("inhomogeneous") term in (3.75) vanishes; and we see (using ρ 2 ρ 1 = u + u − ) that our result (3.75) for the transfer matrix eigenvalue is consistent with the sl(2) grading result (C.8) in [15].
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 8 We recall the definitions (3.58) and also note the identity

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Indeed, we expect (see e.g. [18,36]) that the Bethe states are su(2) lowest-weight states, with 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. The corresponding eigenvalues obtained from (3.75) match with the 4 eigenvalues obtained by direct diagonalization of the transfer matrix (2.26).
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.

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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.

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We propose that the generating functional for antisymmetric representations is given by Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.