Baxter Q-operator for graded SL(2|1) spin chain

TL;DR

This work introduces a Baxter $Q$-operator framework for a noncompact graded spin chain with $SL(2|1)$ symmetry, relevant to the one-loop dilatation operator in $ ext{N}=1$ SYM where each site carries an infinite-dimensional representation. By factorizing the universal $R$-matrix into three components, the transfer matrix is written as a product of three $Q$-operators $Q_a(u)$, satisfying second-order TQ-relations for $Q_1$ and $Q_3$ and a first-order relation for $Q_2$, with a chiral limit linking $Q_3$ to the dilatation Hamiltonian via a logarithmic derivative. The paper develops integral representations for the $Q$-operators, derives a hierarchy of transfer matrices and their inter-relations, and shows how the nested Bethe Ansatz arises from these TQ-relations, while highlighting the method’s independence from a pseudovacuum. This formalism generalizes to higher-rank supergroups and provides a robust tool for analyzing noncompact superspin chains, with direct implications for gauge-theory spectra and potential applications to the separation of variables. The results bridge $Q$-operator techniques with nested Bethe Ansatz and offer new pathways to exact solutions in supersymmetric, noncompact integrable models.

Abstract

We study an integrable noncompact superspin chain model that emerged in recent studies of the dilatation operator in the N=1 super-Yang-Mills theory. It was found that the latter can be mapped into a homogeneous Heisenberg magnet with the quantum space in all sites corresponding to infinite-dimensional representations of the SL(2|1) group. We extend the method of the Baxter Q-operator to spin chains with supergroup symmetry and apply it to determine the eigenspectrum of the model. Our analysis relies on a factorization property of the R-operators acting on the tensor product of two generic infinite-dimensional SL(2|1) representations. It allows us to factorize an arbitrary transfer matrix into a product of three `elementary' transfer matrices which we identify as Baxter Q-operators. We establish functional relations between transfer matrices and use them to derive the TQ-relations for the Q-operators. The proposed construction can be generalized to integrable models based on supergroups of higher rank and, in distinction to the Bethe Ansatz, it is not sensitive to the existence of the pseudovacuum state in the quantum space of the model.

Figures (9)

• Figure 1: The structure of reducible indecomposable $SL(2|1)$ representations $[j,\bar{j}]$ for different values of the spins $j$ and $\bar{j}$.
• Figure 2: Diagrammatical representation of the integral operators $\mathcal{R}^{(1)}(u)$ and $\mathcal{R}^{(3)}(u)$, Eq. (\ref{['R13-kernels']}). The arrow line with the index $(\alpha,\bar{\alpha})$ and the end-points $\mathcal{W}$ and $\mathcal{Z}^*$ represents for the kernel $\mathcal{K}_{\alpha,\bar{\alpha}}(\mathcal{W},\mathcal{Z}^*)$.
• Figure 3: Diagrammatical representation of the operator $\mathcal{Q}_{3}(u+\bar{j}_q)$. The indices specify the values of spins $\boldsymbol{\alpha}_3=(-u,0)$ and $\boldsymbol{\beta}_3=(j_q+u,\bar{j}_q)$.
• Figure 4: Diagrammatical representation of the operator $\mathcal{Q}_{1}(u-j_q)$. The indices specify the values of spins $\boldsymbol{\alpha}_1=(j_q,\bar{j}_q)$ and $\boldsymbol{\beta}_1=(0,-u)$. The leftmost incoming horizontal line is a continuation of the rightmost outgoing horizontal line. An integration over the position of 'fat' vertices is implied with the measure (\ref{['susy-scal']}) for $j=j_q$ and $\bar{j}=\bar{j}_q-u$.
• Figure 5: Hierarchy of transfer matrices (see Table \ref{['Table1']}). The number above arrowed line refers to the equation number.