Open semiclassical strings and long defect operators in AdS/dCFT correspondence

Abstract

We consider defect composite operators in a defect superconformal field theory obtained by inserting an ${\mathrm{AdS}}_{4}\ifmmode\times\else\texttimes\fi{}{\mathrm{S}}^{2}$-brane in the ${\mathrm{AdS}}_{5}\ifmmode\times\else\texttimes\fi{}{\mathrm{S}}^{5}$ background. The one-loop dilatation operator for the scalar sector is represented by an integrable open spin chain. We give a description to construct coherent states for the open spin chain. Then, by evaluating the expectation value of the Hamiltonian with the coherent states in a long operator limit, a Landau-Lifshitz type of sigma model action is obtained. This action is also derived from the string action and hence we find a complete agreement in both super Yang-Mills and string sides. We see that an $\mathrm{SO}(3{)}_{\mathrm{H}}$ pulsating string solution is included in the action and its energy completely agrees with the result calculated in a different method. In addition, we argue that our procedure would be applicable to other AdS-brane cases.