Boundary Integrability from the Fuzzy Three Sphere

TL;DR

The paper addresses integrable boundary states in an $SO(6)$-symmetric spin chain realized by fuzzy $S^3$-based MPS with $SO(4)$ symmetry. It constructs a boundary $K$-matrix that implements the boundary reflection algebra and satisfies the boundary Yang–Baxter equation, enabling an explicit overlap formula with Bethe states for arbitrary bond dimensions. The authors derive the overlap in terms of eigenvalues of Casimir-driven $F$-operators, framed in a Gelfand–Tsetlin basis, and a Gaudin determinant ratio encoding the achiral pairing structure. This work connects nonsupersymmetric defect configurations in $\\mathcal{N}=4$ SYM to integrable boundaries, with potential holographic interpretations and extensions to other $\\mathfrak{so}_{2n}$-type chains.

Abstract

We consider $\mathfrak{so}_4$ invariant matrix product states (MPS) in the $\mathfrak{so}_6$ 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 $\mathfrak{so}_6$ symmetric chains with the fuzzy three-sphere MPS for arbitrary bond dimensions.