From Characters to Quantum (Super)Spin Chains via Fusion

TL;DR

This work provides an elementary, representation-theoretic proof of the Bazhanov–Reshetikhin determinant formula for twisted transfer matrices of $gl(K|M)$ quantum spin chains, including the supersymmetric case. It reformulates the monodromy with a left co-derivative acting on group (super)characters, yielding a quantum Jacobi–Trudi determinant for the transfer matrices up to a scalar factor $S(u)$, and proves the full multi-spin BR formula by establishing a key identity for group-derivative actions on generating functions. The authors extend the construction to supergroups, develop R-matrix fusion to arbitrary symmetric irreps with commutativity of T-matrices, and derive Hirota relations that connect to the discrete KdV hierarchy and QQ-relations, linking quantum integrability to classical tau-function structures. The approach provides a unifying framework for fusion, nested Bethe Ansatz, and potential generalizations to other algebras and types of R-matrices, with implications for both mathematical structure and integrable-model techniques.

Abstract

We give an elementary proof of the Bazhanov-Reshetikhin determinant formula for rational transfer matrices of the twisted quantum super-spin chains associated with the gl(K|M) algebra. This formula describes the most general fusion of transfer matrices in symmetric representations into arbitrary finite dimensional representations of the algebra and is at the heart of analytical Bethe ansatz approach. Our technique represents a systematic generalization of the usual Jacobi-Trudi formula for characters to its quantum analogue using certain group derivatives.

Figures (11)

• Figure 1: The central object of the paper, transfer-matrix $T_{\{\lambda\}}(u)={\rm tr~}_\lambda$R_N^λ(u-θ_N)… R_1^λ(u-θ_1)π_λ(g)$$: the individual $R$-matrices are multiplied along the auxiliary, horizontal space (solid circle) of an arbitrary finite dimensional representation $\lambda$, represented by its Young tableau, whereas the vertical lines represent the spaces on which individual spins in fundamental representations act. Each crossing corresponds to one $R$-matrix depending on a spectral parameter $u-\theta_k$. The twist matrix $g$ is also taken in the representation $\lambda$. The indices of the auxiliary space disappear when taking the trace but this object is still a complex operator in the quantum space with indices $T_{\{ \lambda \}}(u)^{i_1\dots i_N}_{j_1\dots j_N}$.
• Figure 2: A bold solid line from upper node $n$ to lower node $m$ represents a $$gz1-gz$^{i_n}_{j_m}$ factor whereas a dashed line corresponds to $$11-gz$^{i_n}_{j_m}$. In figure \ref{['fig:BR1']}a we represent the action of the left co-derivative on the symmetric generating function. In figure \ref{['fig:BR1']}b we add an extra derivative (living on a new quantum space represented by the empty balls at position $2$) which will yield two new terms corresponding to the action on the generating function and on the previously created line.
• Figure 3: The action of the left co-derivative, living on a new quantum space $n$, on a line going from upper $a$ to lower $b$ positions generates a dashed line going to left from upper position $n$ to lower position $b$ and a solid line going to the right from the upper position $a$ to the lower position $n$. The final result is independent on whether the original line is dashed or solid.
• Figure 4: To compute $\hat{D}^{\otimes N} w(z)^{i_1\dots i_N}_{j_1\dots j_N}$ we draw all $N!$ permutation diagrams, dash the lines going to the left and read the contribution of each term from the rule that a dashed (solid) line going from upper position $a$ to lower position $b$ represents a factor $$11-gz$^{i_a}_{j_b}$$$$\frac{gz}{1-gz}$^i_a_j_b$$ respectively.
• Figure 5: To compute $(1+\hat{D})^{\otimes N} w(z)^{i_1\dots i_N}_{j_1\dots j_N}$ we draw all $L!$ permutation diagrams, dash the vertical and left-going lines and read the contribution of each term from the rule that a dashed (solid) line going from upper position $a$ to lower position $b$ represents a factor $$11-gz$^{i_a}_{j_b}$$$$\frac{gz}{1-gz}$^i_a_j_b$$ respectively. In figures \ref{['fig:BR4']}a,b,c we represent the outcome for $N=1,2,3$.