Boundary integrability from the fuzzy three sphere

We consider so 4 invariant matrix product states (MPS) in the so 6 symmetric integrable spin chain and prove their integrability. These MPS appear as fuzzy three-sphere solutions of matrix models with Yang-Mills-type interactions, and in particular they correspond to scalar defect sectors of N = 4 SYM. We find that the algebra formed by the fuzzy three-sphere generators naturally leads to a boundary reflection algebra and hence a solution to the boundary Yang-Baxter equation for every representation of the fuzzy three-sphere. This allows us to find closed formula for the overlaps of Bethe states of so 6 symmetric chains with the fuzzy three-sphere MPS for arbitrary bond dimensions.