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

Nan Bai

TL;DR

This work develops a comprehensive fusion framework for open $\mathfrak{gl}(N)$ spin chains, constructing fused $R$-matrices, fused boundary $K^{\pm}$-matrices, and fused reflection equations, and identifies invariant subspaces carrying irreducible $\mathfrak{gl}(N)$ representations. The fused objects yield commuting families of open-chain transfer matrices and admit explicit evaluations on the full quantum space or on invariant subspaces, including a factorization into quantum determinants in reduced spaces. As a key application, the ABJM spin chain is shown to arise from an $\mathfrak{su}(4)$ fusion scheme involving fundamental and anti-fundamental representations, with explicit fused $R$- and $K^{-}$-matrix solutions on the anti-fundamental space. The results extend the fusion program to general open $\mathfrak{gl}(N)$ models and provide a concrete path to analyze open-chain spectra via boundary fusion techniques and their associated functional relations.

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 $\mathfrak{su}(4)$.

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

TL;DR

This work develops a comprehensive fusion framework for open spin chains, constructing fused -matrices, fused boundary -matrices, and fused reflection equations, and identifies invariant subspaces carrying irreducible representations. The fused objects yield commuting families of open-chain transfer matrices and admit explicit evaluations on the full quantum space or on invariant subspaces, including a factorization into quantum determinants in reduced spaces. As a key application, the ABJM spin chain is shown to arise from an fusion scheme involving fundamental and anti-fundamental representations, with explicit fused - and -matrix solutions on the anti-fundamental space. The results extend the fusion program to general open models and provide a concrete path to analyze open-chain spectra via boundary fusion techniques and their associated functional relations.

Abstract

We formulate a general fusion procedure for open 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 . 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 .

Paper Structure

This paper contains 31 sections, 146 equations, 26 figures.

Table of Contents

  1. Introduction
  2. Fusion procedure for closed spin chains
  3. Fused $R$-matrix and fused Yang-Baxter equation
  4. Some properties of the fused $R$-matrices
  5. Invariant subspaces of the fused $R$-matrices
  6. General intertwining relations
  7. Fusion operators
  8. Invariant subspaces for irreducible representations of $\mathfrak{gl}(N)$
  9. Fused transfer matrices for closed spin chains
  10. Fusion procedure for open spin chains
  11. Fused $K^{-}$-matrices and fused reflection equation
  12. Invariant subspaces of the fused $K^{-}$-matrices
  13. Fused $K^{+}$-matrices and fused dual reflection equation
  14. Invariant subspaces of the fused $K^{+}$-matrices
  15. Fused transfer matrices for open spin chains
  16. ...and 16 more sections

Figures (26)

  • Figure 1: (a) Fundamental $R$-matrix: $R_{ij}^{c_ic_j}(u-v)$; (b) YBE with inhomogeneity parameters.
  • Figure 2: Graphical representation for the fused $R$-matrix$: R_{(1\cdots n),(\underline{1}\cdots \underline{n}')}^{\mathbf{c}_{(1\cdots n)},\mathbf{c}_{(\underline{1}\cdots\underline{n}')}}(u-v)$.
  • Figure 3: Graphical representation of the fused YBE.
  • Figure 4: Braiding operator $B(1\cdots n\rightarrow i_1\cdots i_n)$.
  • Figure 5: Two specific 4-particle braiding operators: $(a): B(1234\rightarrow 4321)$$(b): B(1234\rightarrow 2413)$.
  • ...and 21 more figures

Theorems & Definitions (4)

  • Example 2.1
  • Example 2.2
  • Example 3.1
  • Example 3.2