Correlators on the wall and $\mathfrak{sl}_n$ spin chain

We study algebras and correlation functions of local operators at half-BPS interfaces engineered by the stacks of D5 or NS5 branes in the 4d $\mathcal{N}=4$ super Yang-Mills. The operator algebra in this sector is isomorphic to a truncation of the Yangian $\mathcal{Y}(\mathfrak{gl}_n)$. The correlators, encoded in a trace on the Yangian, are controlled by the inhomogeneous $\mathfrak{sl}_n$ spin chain, where $n$ is the number of fivebranes: they are given in terms of matrix elements of transfer matrices associated to Verma modules, or equivalently of products of Baxter's Q-operators. This can be viewed as a novel connection between the $\mathcal{N}=4$ super Yang-Mills and integrable spin chains. We also remark on analogous constructions involving half-BPS Wilson lines.