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

Abstract

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 ${\mathcal{L}}_{\mathfrak{d}(2,1;\zeta )}^{(\eta )}$ 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 ${\mathcal{L}}_{psl(2|2)}^{(\mu )}$ and ${\mathcal{L}}_{osp(4|2)}^{(\mu )}$. Other features of the four super Dynkin diagrams $S{\mathfrak{DD}}_{\mathfrak{d}(2,1;\zeta )}^{(\eta )}$ and weight graphs of $\mathfrak{d}$($2,1;\zeta$) as well as discrete automorphisms are also given.

LaTeX, 72 pages, 18 figures

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