Exact overlaps for all integrable two-site boundary states of \( \mathfrak{gl} \)(N) symmetric spin chains

We find closed formulas for the overlaps of Bethe eigenstates of $\mathfrak{gl}$ \( \mathfrak{gl} \) (N) symmetric spin chains and integrable boundary states. We derive the general overlap formulas for $\mathfrak{gl}$ \( \mathfrak{gl} \) (M) ⊕ $\mathfrak{gl}$ \( \mathfrak{gl} \) (N − M) symmetric boundary states and give a well-established conjecture for the $\mathfrak{sp}$ \( \mathfrak{sp} \) (N) symmetric case. Combining these results with the previously derived $\mathfrak{so}$ \( \mathfrak{so} \) (N) symmetric formula, now we have the overlap functions for all integrable boundary states of the $\mathfrak{gl}$ \( \mathfrak{gl} \) (N) spin chains which are built from two-site states. The calculations are independent from the representations of the quantum space therefore our formulas can be applied for the SO(6) and the alternating SU(4) spin chains which describe the scalar sectors of $\mathcal{N}$ \( \mathcal{N} \) = 4 super Yang-Mills and ABJM theories which are important application areas of our results.