Boundary Integrability from the Fuzzy Three Sphere

TL;DR

This paper demonstrates the integrability of certain matrix product states related to fuzzy three-sphere solutions in an $ ext{so}_6$ symmetric spin chain, providing explicit formulas for overlaps with Bethe states.

Contribution

It establishes the integrability of $ ext{so}_4$ invariant MPS in an $ ext{so}_6$ symmetric chain and derives formulas for overlaps with Bethe states for all bond dimensions.

Findings

Proves the integrability of fuzzy three-sphere MPS.

Derives boundary reflection algebra from fuzzy three-sphere generators.

Provides explicit overlap formulas for Bethe states.

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.