Solutions of the T-system and Baxter equations for supersymmetric spin chains

TL;DR

This work develops a Wronskian-like determinant framework for the ${U}_{q}(\widehat{gl}(M|N))$-symmetric spin chains, providing determinant expressions for the ${\mathsf T}$- and Baxter ${\mathsf Q}$-functions that bound the matrix size by ${M+N}$. It shows these determinants satisfy the ${T}$-system and ${QQ}$-relations, yield finite-order Baxter equations, and connect to ${\mathsf q}$-characters and Backlund flows; it also extends to typical representations with two-parameter deformations and factorization formulas. The approach unifies transfer-matrix eigenvalues and ${\mathsf Q}$-functions in a single algebraic framework, enabling efficient treatment of large-dimensional auxiliary spaces and offering pathways to NLIE/TBA analyses and AdS/CFT applications. The results also suggest operator realizations via ${q}$-oscillator constructions and potential extensions to other quantum affine superalgebras and mixed representations.

Abstract

We propose Wronskian-like determinant formulae for the Baxter Q-functions and the eigenvalues of transfer matrices for spin chains related to the quantum affine superalgebra U_{q}(hat{gl}(M|N)). In contrast to the supersymmetric Bazhanov-Reshetikhin formula (the quantum supersymmetric Jacobi-Trudi formula) proposed in [Z. Tsuboi, J. Phys. A: Math. Gen. 30 (1997) 7975], the size of the matrices of these Wronskian-like formulae is less than or equal to M+N. Base on these formulae, we give new expressions of the solutions of the T-system (fusion relations for transfer matrices) for supersymmetric spin chains proposed in the abovementioned paper. Baxter equations also follow from the Wronskian-like formulae. They are finite order linear difference equations with respect to the Baxter Q-functions. Moreover, the Wronskian-like formulae also explicitly solve the functional relations for Backlund flows proposed in [V. Kazakov, A. Sorin, A. Zabrodin, Nucl. Phys. B790 (2008) 345 [arXiv:hep-th/0703147]].

Figures (6)

• Figure 1: The $a\times s$ rectangular Young diagram ($a,s \in {\mathbb Z}_{\ge 1}$) to label the ${\mathsf T}$-function ${\mathsf T}^{(a)}_{s}(x)$ of the $T$-system \ref{['ori-t-system1']}-\ref{['ori-t-sys-bc1']} has to be in the so-called $(M,N)$-hook.
• Figure 2: The hierarchy of the $2^{4}$ Baxter ${\mathsf Q}$-functions for $U_{q}(\widehat{gl}(2|2))$. The functions ${\mathsf Q}_{\emptyset}(x)$ and ${\mathsf Q}_{\{1,2,3,4\}}(x)$ are special on the point that they do not contain roots of the Bethe ansatz equation, and often normalized as just $1$. The index sets $I$ of the Baxter ${\mathsf Q}$-functions ${\mathsf Q}_{I}(x)$ form a Hasse diagram. ${\mathsf Q}_{I_{a}}(x)$ and ${\mathsf Q}_{I_{a-1}}(x)$ are connected by a thick line if $I_{a}\setminus I_{a-1} \subset {\mathfrak B}=\{1,2\}$, a thin line if $I_{a}\setminus I_{a-1} \subset {\mathfrak F}=\{3,4\}$. As a graph, this contains many cycles. Minimal non-trivial cycles are 4-cycles, which correspond to functional relations among the Baxter ${\mathsf Q}$-functions on the vertexes. 4-cycles containing only thick lines correspond to \ref{['QQ-rel1']} for $p_{i}=p_{j}=1$; only thin lines correspond to \ref{['QQ-rel1']} for $p_{i}=p_{j}=-1$; both thick lines and thin lines correspond to \ref{['QQ-rel2']}. In addition, the function ${\mathcal{X}}_{I_{a}}(x)$ (\ref{['boxes']} for ${\mathtt I}=I_{a}$) lives on the edge which connects two vertexes for ${\mathsf Q}_{I_{a}}(x)$ and ${\mathsf Q}_{I_{a-1}}(x)$. Thus the function ${\mathcal{F}}_{(1)}^{I_{a}}(x)$ (\ref{['DVF-tab1']} for $\lambda \subset \mu =(1)$, ${\mathtt I}=I_{a}$, $M=N=2$) lives on a path from ${\mathsf Q}_{I_{0}}(x)={\mathsf Q}_{\emptyset}(x)$ to ${\mathsf Q}_{I_{a}}(x)$; the function $\overline{\mathcal{F}}_{(1)}^{\overline{I}_{a}}(x)$ (cf. \ref{['DVF-tab2']} for $\lambda \subset \mu =(1)$, ${\mathtt I}=\overline{I}_{a}$, $M=N=2$) lives on a path from ${\mathsf Q}_{I_{a}}(x)$ to ${\mathsf Q}_{I_{4}}(x)={\mathsf Q}_{\{1,2,3,4\}}(x)$, where these paths must not contain more than two functions ${\mathsf Q}_{I}(x),{\mathsf Q}_{J}(x)$ of the same level ${\rm Card}(I)={\rm Card}(J)$. In particular, they do not depend on the paths.
• Figure 3: The Young diagram with shape $\mu=(4,3,2,1^2)$.
• Figure 4: The Young diagram for the partition $\mu^{\prime}=(5,3,2,1)$, conjugated to $\mu=(4,3,2,1^2)$ in Figure \ref{['Young1']}.
• Figure 5: The skew Young diagram $\lambda \subset \mu$ with $\lambda=(2,1)$ and $\mu=(4,3,2,1^2)$.

Theorems & Definitions (11)

• Lemma 3.1
• Theorem 3.2
• Theorem 3.3
• Theorem 3.4
• Conjecture 3.5
• Proposition 3.6
• Lemma 6.1
• Theorem 6.2
• Lemma 6.3
• Theorem 6.4