Complex D($2,1;\zeta $) and spin chain solutions from Chern-Simons theory

Using properties of OSp(4|2) and PSL(2|2), we investigate the super geometry of the parametric D($2,1;\zeta$) labeled by variable $\zeta$ belonging to $\mathbb{C}\backslash \{-1,0\}$ and we give applications in the study of integrable superspin chains. This $9|8$ dimensional Lie supergroup has three orthogonal isospins in its even part SL($2,\mathbb{C}$)$^{\otimes 3}$ assembled by the tri-fundamental $2^{\otimes 3}$ with odd parity. It undergoes contractions at $\zeta =-1,0$ where an SL($2,\mathbb{C}$) gets decompactified into commutative $\mathbb{C}^{3}$ interpreted in terms of three central extensions. By help of the obtained characteristic features of D($2,1;\zeta$) and their local structures at the special points $\zeta =\pm 1$, we calculate the Lax operator $\mathcal{L}_{\mathfrak{d}(2,1;\zeta )}^{(\mathfrak{\eta})}$ solving the RLL equation describing the integrability of the superspin chain $\mathfrak{d}$($2,1;\zeta$). We also complete missing results regarding the calculation of $\mathcal{L}_{psl(2|2)}^{(\mathfrak{\mu })}$ and $\mathcal{L}_{osp(4|2)}^{(\mathfrak{\mu})}$. Other features of the four super Dynkin diagrams $S\mathfrak{DD}_{\mathfrak{d}(2,1;\zeta )}^{(\mathfrak{\eta})}$ and weight graphs of $\mathfrak{d}$($2,1;\zeta$) as well as discrete automorphisms are also given.