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 [1]. We derive an alternative representation for the 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 R-matrix entries needed to uncover the corresponding geometric properties. For arbitrary gauge the R-matrix geometry is argued to be birational to the direct product CP1×CP1×A where 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.
Highlights
A large variety of two-dimensional statistical mechanical models are known to be soluble on the basis of commuting transfer matrices method devised in the early 1970’s by Baxter [2]
We have derived a formulation for the R-matrix based on a q-deformation of the centrally extended sl(2|2) superalgebra towards the view of algebraic geometry
This made it possible to uncover the geometric properties of the elementary weights as well as of the corresponding R-matrix
Summary
A large variety of two-dimensional statistical mechanical models are known to be soluble on the basis of commuting transfer matrices method devised in the early 1970’s by Baxter [2]. Where the subscript indices ij denote the two-dimensional subspace in which a given operator is acting on At this point we emphasize that the geometrical properties associated to the R-matrix can not in general be read directly from that of the Boltzmann weights. Having at hand a solution of the Yang-Baxter equation it is of interest to uncover the geometric content of both varieties X and Y This is specially relevant in situations where the elements of the R-matrix are not all expressed in terms of rational functions. In section we derive an alternative representation for the R-matrix such that the matrix elements are rational functions of certain elementary weights sited on a degree six surface
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