Algebro-Geometric approach for a centrally extended U_q[sl(2|2)] R-matrix

Abstract

In this paper we investigate the algebraic geometric nature of a solution of the Yang-Baxter equation based on the quantum deformation of the centrally extended $sl(2|2)$ superalgebra proposed by Beisert and Koroteev \cite{BEKO}. We derive an alternative representation for the $\mathrm{R}$-matrix in which the matrix elements are given in terms of rational functions depending on weights sited on a degree six surface. For generic gauge the weights geometry are governed by a genus one ruled surface while for a symmetric gauge choice the weights lie instead on a genus five curve. We have written down the polynomial identities satisfied by the $\mathrm{R}$-matrix entries needed to uncover the corresponding geometric properties. For arbitrary gauge the $\mathrm{R}$-matrix geometry is argued to be birational to the direct product $\mathbb{CP}^1 \times \mathbb{CP}^1 \times \mathrm{A}$ where $\mathrm{A}$ is an Abelian surface. For the symmetric gauge we present evidences that the geometric content is that of a surface of general type lying on the so-called Severi line with irregularity two and geometric genus nine. We discuss potential geometric degenerations when the two free couplings are restricted to certain one-dimensional subspaces.