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

Abstract

Extending the method proposed in [1], 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 $U_q\left(gl{(2r+1|2s)}^{\left(2\right)}\right)$, $U_q\left(gl{\left(2r\right|2s+1)}^{\left(2\right)}\right)$, $U_q\left(gl{\left(2r\right|2s)}^{\left(2\right)}\right)$, $U_q\left(osp{\left(2r\right|2s)}^{\left(2\right)}\right)$ and the untwisted quantum affine orthosymplectic superalgebras $U_q\left(osp{(2r+1|2s)}^{\left(1\right)}\right)$ and $U_q\left(osp{\left(2r\right|2s)}^{\left(1\right)}\right)$ (and their Yangian counterparts, $Y\left(osp\right(2r+1\left|2s\right))$ and $Y\left(osp\right(2r\left|2s\right))$) as reductions (a kind of folding) of those associated with $U_q\left(gl{\left(M\right|N)}^{\left(1\right)}\right)$. 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 [2], [3], 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 $U_q\left(gl{\left(M\right|N)}^{\left(1\right)}\right)$.