Review of AdS/CFT Integrability, Chapter IV.2: Deformations, Orbifolds and Open Boundaries

Revisiting the Y = 0 open spin chain at one loop

“…It can be shown that the requirement that the eigenvalues (43) be analytic at u = 1 2 , 3 2 (where d(u) has zeros) does not lead to further constraints. 6 The constraints (45), ( 48), (50), (52) do not uniquely determine the functions c 1 (u), . .…”

“…Much of the focus has been on the problem of computing the anomalous dimensions of single-trace operators of N = 4 SYM, which can be mapped to the problem of determining the eigenvalues of certain integrable closed spin-chain Hamiltonians, as observed in the seminal work of Minahan and Zarembo [2]. 1 However, progress has also been made on the problem of computing the anomalous dimensions of determinant-like operators, which can be mapped to the problem of determining the eigenvalues of certain open spin-chain Hamiltonians [6]. By the AdS/CFT correspondence, these two types of operators correspond to states of closed strings and open strings attached to D-branes, respectively.…”

“…The simplest and most-studied open string/chain example is the so-called Y = 0 maximal giant graviton brane [7]. The one-loop SO (6) scalar sector open spin chain Hamiltonian was found by Berenstein and Vázquez [8]. They also determined the (one-loop) boundary Smatrix, and showed that it satisfies the boundary Yang-Baxter equation (BYBE) [9,10,11].…”

Spin chains with boundary inhomogeneities

“…Integrable open spin chains with boundaries have been widely studied in a variety of contexts, see e.g. [1][2][3][4][5][6][7][8] and references therein. Sklyanin [3] provided a general recipe for constructing such models, based on solutions of the bulk [9] and boundary [10,11] Yang-Baxter equations (YBEs), to which we refer here as R-matrices and K-matrices, respectively.…”

Fusion for AdS/CFT boundary S-matrices

“…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.)…”