Discrete Holomorphicity at Two-Dimensional Critical Points

TL;DR

This paper reviews recent discoveries of discretely holomorphic observables in 2D critical lattice models, highlighting their connection to integrability, conformal field theory, and potential implications for understanding scaling limits via SLE.

Contribution

It introduces the concept of discretely holomorphic observables in lattice models and explores their deep relation with integrability and conformal invariance.

Findings

Discretely holomorphic observables are found in critical 2D lattice models.

These observables relate to the integrability of the models.

They potentially link lattice models to Schramm-Loewner evolution (SLE).

Abstract

After a brief review of the historical role of analyticity in the study of critical phenomena, an account is given of recent discoveries of discretely holomorphic observables in critical two-dimensional lattice models. These are objects whose correlation functions satisfy a discrete version of the Cauchy-Riemann relations. Their existence appears to have a deep relation with the integrability of the model, and they are presumably the lattice versions of the truly holomorphic observables appearing in the conformal field theory (CFT) describing the continuum limit. This hypothesis sheds light on the connection between CFT and integrability, and, if verified, can also be used to prove that the scaling limit of certain discrete curves in these models is described by Schramm-Loewner evolution (SLE).