Fusion for AdS/CFT boundary S-matrices

We propose a fusion formula for AdS/CFT worldsheet boundary S-matrices. We show that, starting from the fundamental Y=0 boundary S-matrix, this formula correctly reproduces the two-particle bound-state boundary S-matrices.


Introduction
The formation of bound states ("fusion") is a ubiquitous phenomenon in quantum field theory. If the theory is integrable [1], then the factorized bulk S-matrices of the bound-state particles can be determined in terms of the corresponding S-matrices of the fundamental particles [2]. This phenomenon was abstracted in [3,4] into a general "fusion procedure" for constructing higher-dimensional R-matrices (solutions of the Yang-Baxter equation) starting from a fundamental R-matrix. For boundary S-matrices/K-matrices, i.e. solutions of the boundary Yang-Baxter equation [5,6,7], an analogous fusion procedure was formulated in [8,9,10].
In order to carry out the fusion procedure [3], the fundamental R-matrix should satisfy a certain technical requirement: namely, it must degenerate into a projection operator for some value(s) of the spectral parameter. Many R-matrices fulfill this requirement; and this fusion procedure has proved to be very useful: it not only generates new solutions of the Yang-Baxter equation, but it also leads to a hierarchy of commuting transfer matrices that can be used to solve the corresponding integrable models (see e.g. [11]).
However, the AdS/CFT worldsheet bulk S-matrix [12,13], which plays a key role in the understanding of integrability in AdS/CFT [14], does not satisfy this requirement. This apparent failure of the fusion procedure has been quite puzzling, since bound states do form in this model [15,16], and their bulk S-matrices have been determined [17,18], albeit by other means.
This puzzle was recently resolved by Beisert, de Leeuw and Nag [19], who showed that one can relax the requirement that the R-matrix degenerates into a projector. In particular, they proposed a new bulk fusion formula, which generates a bound-state AdS/CFT bulk S-matrix [17] from the fundamental one. As a bonus, the resulting fused matrix automatically has the correct dimensions -the additional similarity transformation and subsequent elimination of null rows and columns that are implicit in the original approach [3] are not needed. (A similar fusion formula was proposed for the XXX R-matrix in [20].) Factorized boundary S-matrices also play an interesting role in AdS/CFT (see e.g. [21,22]); and AdS/CFT boundary S-matrices for bound states have also been determined [23,24,25]. (See also [26,27,28,29,30,31] and references therein.) The main purpose of this note is to propose a new fusion formula for boundary S-matrices, which generates two-particle bound-state AdS/CFT boundary S-matrices from the fundamental one.
The outline of this paper is as follows. In Section 2 we briefly review the new bulk fusion procedure formulated in [19]. However, we work with different conventions, which we find more convenient. In Section 3 we present the corresponding boundary fusion formula, whose proof is relegated to an appendix. We then show that, starting from the fundamental Y = 0 boundary S-matrix [21], this formula correctly reproduces the bound-state boundary Smatrices found in [24] and [23], respectively. We conclude in Section 4 with a brief discussion of our results.

Bulk fusion
We begin by briefly reviewing the new fusion procedure proposed in [19]. We consider an R-matrix R(z 1 , z 2 ) which is a solution of the (graded) Yang-Baxter equation and P denotes the (graded) permutation matrix where ǫ a ∈ {0, 1} are the gradings.
We use an elliptic parametrization for the momentum p and the parameters x ± for Mparticle bound states [16,17] such that and where g > 0 is the coupling constant. However, we henceforth reserve p and x ± for the momentum and parameters of the fundamental particles (M = 1), and P and y ± for the corresponding quantities of the two-particle bound states (M = 2).
Consider a pair of fundamental particles with parameters x ± i = x ± (z i ), i = 1, 2. These particles form a bound state when [15,16] Indeed, adding the two constraint equations (2.19) imposing the fusion condition (2.20), and making the identifications we arrive at the two-particle bound-state constraint Note that the momentum of the bound state is indeed the sum of the momenta of its constituents, since where p i = p(z i ). This bound state lies in the 8-dimensional symmetric representation of su(2|2) [32].
When the fusion condition (2.20) is satisfied, the rank of R(z 1 , z 2 ) drops from 16 to 8. By determining the normalized eigenvectors corresponding to the nonzero eigenvalues, we obtain the decomposition (2.3) with (2.25) 1 We also assume [16] that |x ± i | > 1 and 26) and where we have definedF (z 1 , z 2 ) =Ẽ T (z 1 , z 2 ). Finally, Performing the similarity transformation (2.11) with the matrix we obtain where we have defined the new matrices U and V , which evidently have the same matrix structure as W , but have different matrix elements By explicit computation we obtain the following results for these matrix elements where η(z, M) is defined in (A.5), and the rapidity z 12 is defined such that as in (2.22). Remarkably, the square roots in u i and v i (recall the definitions of n i and w i given in (2.27) and (2.30)) have all disappeared.
Using these results to evaluate (2.31), we have verified numerically that this fused R- A similar result was argued in [19].

AdS/CFT bulk S-matrix: antisymmetric representation
We now proceed to construct the complementary fused S-matrix (2.15), which corresponds to the antisymmetric representation of su(2|2) [32], which is also 8-dimensional. The required complementary operatorsĒ andF can be obtained by considering the "opposite" fusion condition . When the fusion condition (2.36) is satisfied, the rank ofR(z 1 , z 2 ) indeed drops from 16 to 8, and we obtain the decomposition
We propose that the fused K-matrix is given by (cf. Eq. (3.5) in [9]) Indeed, we show in Appendix B that this object satisfies the fused boundary Yang-Baxter equation Using the complementary operatorsĒ andF satisfying (2.12) and (2.13), complementary fused boundary K-matrices can be constructed in a similar manner The proof of this result is sketched in Appendix B.

AdS/CFT boundary S-matrix: symmetric representation
Let us illustrate the boundary fusion formula (3.4) with the simplest AdS/CFT boundary S-matrix corresponding to a Y = 0 brane [21]. Using our previous expressions for E and F (2.25), (2.28), we obtain (cf. (2.31)) where U is defined in (2.32), and T is the following new matrix which also has the same matrix structure as W , but has different matrix elements. We find that these matrix elements are given by where v 1 and v 2 are given in (2.33). Using these results to evaluate (3.7), we have verified that this fused K-matrix coincides (up to an overall scalar factor) with the bound-state Y = 0 boundary S-matrix R B in [24], While the verification of some of the matrix elements is straightforward (e.g., r 3 requires just (2.24), and r 2 requires use of (2.20) and (2.21)), those involving η's are much more complicated. Nevertheless, by using the expression for η in terms of a square root (A.5) and using PowerExpand in Mathematica, we managed to explicitly check all of the matrix elements.

Discussion
We have found a fusion formula (3.2) that is applicable to AdS/CFT boundary S-matrices, many examples of which are now known. We have focused on the Y = 0 example only for simplicity. Although we have used the fusion formula to obtain only the M = 2 bound-state Y = 0 boundary S-matrices, we expect that a further generalization (along the lines of [10]) is possible for recovering the higher (M > 2) bound-state boundary S-matrices found in [23] and [25] for antisymmetric and symmetric representations, respectively.
We have noticed that the expressions generated by both the bulk and boundary fusion formulas are generally very complicated, and require considerable effort to simplify, particularly in the symmetric representation. It would be interesting to find a more efficient way of writing the basic elements (R, E and F ) that lead directly to simpler results for the fused quantities.

Acknowledgments
The work of RN was supported in part by the National Science Foundation under Grant PHY-1212337, and by a Cooper fellowship. RP thanks the Sao Paulo Research Foundation (FAPESP), grants # 2014/00453-8 and # 2014/20364-0, for financial support.

A Fundamental bulk S-matrix
The graded bulk S-matrix for a pair of particles in the fundamental (4-dimensional) representation is given by [17] where the 16 × 16 matrices Λ 1 , . . . , Λ 10 are given in terms of quantities E kilj defined by with indices that run from 1 to 4.