Open spinning strings and AdS/dCFT duality

Abstract

We consider open spinning string solutions on an AdS 4 ×S 2 -brane (D5-brane) in the bulk AdS 5 × S 5 background. By taking account of the breaking of SO(6) R to SO(3) H × SO(3) V due to the presence of the AdS-brane, the open rotating string ansatz is discussed. We construct the elliptic folded/circular open string solutions in the SU (2) and the SL(2) sectors, so that they satisfy the appropriate boundary conditions. On the other hand, in the SU (2) sector of the gauge theory, we compute the matrix of anomalous dimension of the defect operator, which turns out to be the Hamiltonian of an open integrable spin chain. Then we consider the coordinate Bethe ansatz with arbitrary number of impurities, and compare the boundary condition of the Bethe wavefunction with that of the corresponding open string solution. We also discuss the Bethe ansatz for the open SL(2) spin chain with several supports from the string theory side. Then, in both SU (2) and SL(2) sectors, we analyze the Bethe equations in the thermodynamic limit and formulate the 'doubling trick' on the Riemann surface associated with the gauge theory.10 The one-loop correction to the energy of closed spinning strings in the SU (2) and the SL(2) sectors are calculated in [59] and [60] respectively.

One-point functions in AdS/dCFT from matrix product states

“…The brane intersection introduces a domain wall that separates the vacua with respectively unbroken SU(N ) and SU(N − k) gauge symmetry in the N = 4 supersymmetric Yang-Mills (SYM) theory, with additional degrees of freedom living on the defect [2,3]. One-point functions in this dCFT were studied in [2,[4][5][6] whereas two-point functions of defect operators were considered in [2,[7][8][9][10][11], where integrability of the underlying N = 4 SYM proved particularly useful. The defect operators are mapped to spin chains with open boundary conditions and are dual to open strings attached to the probe D5-brane.…”

The Bethe ansatz equations for reflecting magnons

“…Previous works studying open strings with a variety of boundary conditions and the corresponding gauge theory open spin chain also include, for instance, Refs. [14][15][16][17][18][19][20].…”

Higher loop Bethe ansatz for open spin-chains in AdS/CFT

“…Here L is a large quantum number representing the total spin of a string on one side, and the R-charge of an operator super Yang-Mills (SYM) theory on the other. Such open string sectors are realized by inserting some D-branes or orientifolds into the bulk AdS 5 × S 5 , and there are many literatures dealing with the open string issues [21][22][23][24][25][26][27][28][29][30][31][32]. There a spin-chain has no longer a periodic boundary condition of a closed spin-chain case; instead of the periodicity, it has boundaries imposed by the D-brane setup.…”

“…It is then not clear whether the integrability still holds or not with the boundaries. In some particular cases, the boundaries were shown to be integrable at least at the one-loop level [21,23,24,26,27], i.e., the system can be solved by the Bethe ansatz technique. When we consider higher-loop contributions to the boundaries, however, the problem gets more complicated and whether the integrable structures still remains or not is unclear.…”