Complex D($2,1;ζ$) and spin chain solutions from Chern-Simons theory
Fuente:
arXiv
Saved in:
| Main Author: | |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866916653103579136 |
|---|---|
| author | Saidi, El Hassan |
| author_facet | Saidi, El Hassan |
| contents | Using properties of OSp(4|2) and PSL(2|2), we investigate the super geometry of the parametric D($2,1;ζ$) labeled by variable $ζ$ 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 $ζ=-1,0$ 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($2,1;ζ$) and their local structures at the special points $ζ=\pm 1$, we calculate the Lax operator $\mathcal{L}_{\mathfrak{d}(2,1;ζ)}^{(\mathfrakη)}$ solving the RLL equation describing the integrability of the superspin chain $\mathfrak{d}$($2,1;ζ$). We also complete missing results regarding the calculation of $\mathcal{L}_{psl(2|2)}^{(\mathfrak{μ})}$ and $\mathcal{L}_{osp(4|2)}^{(\mathfrakμ)}$. Other features of the four super Dynkin diagrams $S\mathfrak{DD}_{\mathfrak{d}(2,1;ζ)}^{(\mathfrakη)}$ and weight graphs of $\mathfrak{d}$($2,1;ζ$) as well as discrete automorphisms are also given. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_14862 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Complex D($2,1;ζ$) and spin chain solutions from Chern-Simons theory Saidi, El Hassan High Energy Physics - Theory Using properties of OSp(4|2) and PSL(2|2), we investigate the super geometry of the parametric D($2,1;ζ$) labeled by variable $ζ$ 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 $ζ=-1,0$ 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($2,1;ζ$) and their local structures at the special points $ζ=\pm 1$, we calculate the Lax operator $\mathcal{L}_{\mathfrak{d}(2,1;ζ)}^{(\mathfrakη)}$ solving the RLL equation describing the integrability of the superspin chain $\mathfrak{d}$($2,1;ζ$). We also complete missing results regarding the calculation of $\mathcal{L}_{psl(2|2)}^{(\mathfrak{μ})}$ and $\mathcal{L}_{osp(4|2)}^{(\mathfrakμ)}$. Other features of the four super Dynkin diagrams $S\mathfrak{DD}_{\mathfrak{d}(2,1;ζ)}^{(\mathfrakη)}$ and weight graphs of $\mathfrak{d}$($2,1;ζ$) as well as discrete automorphisms are also given. |
| title | Complex D($2,1;ζ$) and spin chain solutions from Chern-Simons theory |
| topic | High Energy Physics - Theory |
| url | https://arxiv.org/abs/2409.14862 |