Algebraic Geometry methods associated to the one-dimensional Hubbard model

TL;DR

This paper explores the algebraic geometry underlying the one-dimensional Hubbard model, revealing the geometric structures of the Lax operator and R-matrix, and explaining their spectral parameter properties.

Contribution

It demonstrates that the Lax operator resides on a genus one curve isogenous to the AdS/CFT curve and shows the R-matrix weights lie on an Abelian surface, clarifying spectral parameter issues.

Findings

Lax operator is on a genus one curve isogenous to the AdS/CFT curve

R-matrix weights lie on an Abelian surface birational to a product of elliptic curves

The geometric structures explain why the R-matrix cannot be expressed solely in terms of spectral parameter differences

Abstract

In this paper we study the covering vertex model of the one-dimensional Hubbard Hamiltonian constructed by Shastry in the realm of algebraic geometry. We show that the Lax operator sits in a genus one curve which is not isomorphic but only isogenous to the curve suitable for the AdS/CFT context. We provide an uniformization of the Lax operator in terms of ratios of theta functions allowing us to establish relativistic like properties such as crossing and unitarity. We show that the respective $\mathrm{R}$-matrix weights lie on an Abelian surface being birational to the product of two elliptic curves with distinct $\mathrm{J}$-invariants. One of the curves is isomorphic to that of the Lax operator but the other is solely fourfold isogenous. These results clarify the reason the $\mathrm{R}$-matrix can not be written using only difference of spectral parameters of the Lax operator.