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

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]].

💡 Research Summary

The paper presents a new set of determinant formulas of Wronskian type for the Baxter Q‑functions and the eigenvalues of transfer matrices (the T‑functions) of spin chains whose symmetry is described by the quantum affine superalgebra U₍q₎(Ĝl(M|N)). The authors start by recalling the T‑system – a set of Hirota‑type fusion relations (equations 1.1‑1.4) – which governs the commuting family of transfer matrices for a wide class of graded integrable lattice models. In earlier works the supersymmetric Bazhanov‑Reshetikhin (or supersymmetric Jacobi‑Trudi) formula expressed T‑functions as determinants whose size grows with the dimension of the auxiliary representation, making calculations cumbersome for large Young diagrams.

To overcome this limitation, the authors introduce a Wronskian‑like determinant whose size never exceeds M+N, the total number of bosonic plus fermionic degrees of freedom. The construction relies on the 2^{M+N} Baxter Q‑functions Q_{I}(x) labelled by subsets I⊂{1,…,M+N}. By fixing a grading (bosonic indices B={1,…,M}, fermionic indices F={M+1,…,M+N}) and choosing a permutation I_{M+N}=(i₁,…,i_{M+N}), they define a chain of subsets I_{a} (a=0,…,M+N). The Q‑functions satisfy functional relations (2.17‑2.18) which reduce the independent set to exactly M+N functions.

The main results are Theorem 3.2 and Theorem 3.3. Theorem 3.2 gives a Wronskian determinant expression for the “empty” Young diagram, i.e. for the Q‑functions themselves, while Theorem 3.3 shows that for any rectangular Young diagram of height a and width s (with a+s≤M+N) the corresponding T‑function can be written as

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