Journal of Statistical Physics

, 137:814 | Cite as

Discrete Holomorphicity at Two-Dimensional Critical Points

Article

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).

Integrable models Conformal field theory Schramm-Loewner evolution 

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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  1. 1.Rudolph Peierls Centre for Theoretical PhysicsOxfordUK
  2. 2.All Souls CollegeOxfordUK

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