A general fusion procedure for open $\mathfrak{gl}(N)$ spin chains: Application to the ABJM spin chain

TL;DR

This paper develops a comprehensive fusion method for open $rak{gl}(N)$ spin chains, constructs fused boundary matrices, and applies the framework to derive boundary solutions for the ABJM spin chain, revealing new integrable boundary conditions.

Contribution

It introduces a general fusion procedure for open $rak{gl}(N)$ spin chains and applies it to the ABJM model, including explicit construction of fused reflection matrices and transfer matrices.

Findings

Constructed fused boundary reflection matrices and equations.

Identified invariant subspaces carrying irreducible representations.

Derived boundary reflection solutions for the ABJM spin chain.

Abstract

We formulate a general fusion procedure for open $\mathfrak{gl}(N)$ spin chains. We construct the fused boundary reflection matrices and the corresponding fused reflection equations. By using the intertwining relation between the fused reflection matrices and the fusion operator, we identify the invariant subspace of the fused reflection matrices carrying the irreducible representations of $\mathfrak{gl}(N)$. We also construct the fused transfer matrix and evaluate it explicitly in the total tensor product space and the invariant subspaces. Finally, we demonstrate that the ABJM spin chain model originates from such fusion procedure and derive three classes of boundary reflection matrices solutions on the anti-fundamental representation space of $\mathfrak{su}(4)$.