Boundary states, overlaps, nesting and bootstrapping AdS/dCFT

Integrable boundary states can be built up from pair annihilation amplitudes called $K$-matrices. These amplitudes are related to mirror reflections and they both satisfy Yang Baxter equations, which can be twisted or untwisted. We relate these two notions to each other and show how they are fixed by the unbroken symmetries, which, together with the full symmetry, must form symmetric pairs. We show that the twisted nature of the $K$-matrix implies specific selection rules for the overlaps. If the Bethe roots of the same type are paired the overlap is called chiral, otherwise it is achiral and they correspond to untwisted and twisted $K$-matrices, respectively. We use these findings to develop a nesting procedure for $K$-matrices, which provides the factorizing overlaps for higher rank algebras automatically. We apply these methods for the calculation of the simplest asymptotic all-loop 1-point functions in AdS/dCFT. In doing so we classify the solutions of the YBE for the $K$-matrices with centrally extended $\mathfrak{su}(2|2)_{c}$ symmetry and calculate the generic overlaps in terms of Bethe roots and ratio of Gaudin determinants.