Folding QQ-relations and transfer matrix eigenvalues: towards a unified approach to Bethe ansatz for super spin chains

TL;DR

This paper develops a unified framework for deriving QQ-relations and transfer matrix eigenvalues in super spin chains, extending previous methods to a broad class of superalgebras and providing new explicit formulas.

Contribution

It introduces a generalized approach to QQ-relations and T-functions for various superalgebras, including twisted and untwisted quantum affine superalgebras, unifying and extending prior results.

Findings

Derived QQ-relations and T-functions for multiple superalgebras.

Reproduced known generating functions and tableau sum expressions.

Obtained Wronskian-type formulas and relations for spinorial representations.

Abstract

Extending the method proposed in [arXiv:1109.5524], we derive QQ-relations (functional relations among Baxter Q-functions) and T-functions (eigenvalues of transfer matrices) for fusion vertex models associated with the twisted quantum affine superalgebras $U_{q}(gl(2r+1|2s)^{(2)})$, $U_{q}(gl(2r|2s+1)^{(2)})$, $U_{q}(gl(2r|2s)^{(2)})$, $U_{q}(osp(2r|2s)^{(2)})$ and the untwisted quantum affine orthosymplectic superalgebras $U_{q}(osp(2r+1|2s)^{(1)})$ and $U_{q}(osp(2r|2s)^{(1)})$ (and their Yangian counterparts, $Y(osp(2r+1|2s))$ and $Y(osp(2r|2s))$) as reductions (a kind of folding) of those associated with $U_{q}(gl(M|N)^{(1)})$. In particular, we reproduce previously proposed generating functions (difference operators) of the T-functions for the symmetric or anti-symmetric representations, and tableau sum expressions for more general representations for orthosymplectic superalgebras [arXiv:0911.5393,arXiv:0911.5390], and obtain Wronskian-type expressions (analogues of Weyl-type character formulas) for them. T-functions for spinorial representations are related to reductions of those for asymptotic limits of typical representations of $U_{q}(gl(M|N)^{(1)})$.