Discrete holomorphicity and quantized affine algebras

TL;DR

This paper links non-local currents in quantized affine algebras to discretely holomorphic observables in lattice models, revealing their conservation laws and establishing a connection between algebraic structures and discrete complex analysis.

Contribution

It demonstrates that non-local currents in specific quantum affine algebras correspond to discretely holomorphic loop observables in associated lattice models, unifying algebraic and analytic perspectives.

Findings

Non-local currents are identified with discretely holomorphic loop observables.

Bulk and boundary discrete holomorphicity relations are equivalent to conservation laws.

The work connects quantum algebra structures with discrete complex analysis in statistical models.

Abstract

We consider non-local currents in the context of quantized affine algebras, following the construction introduced by Bernard and Felder. In the case of $U_q(A_1^{(1)})$ and $U_q(A_2^{(2)})$, these currents can be identified with configurations in the six-vertex and Izergin--Korepin nineteen-vertex models. Mapping these to their corresponding Temperley--Lieb loop models, we directly identify non-local currents with discretely holomorphic loop observables. In particular, we show that the bulk discrete holomorphicity relation and its recently derived boundary analogue are equivalent to conservation laws for non-local currents.