Discrete Holomorphicity at Two-Dimensional Critical Points

Details Discrete Holomorphicity at Two-Dimensional Critical Points Discrete Holomorphicity at Two-Dimensional Critical Points

2009-07-23

arxiv.org/abs/0907.4070v1

J. Stat. Phys. 137, 814, 2009.

Physics

Condensed Matter

Statistical Mechanics

Invited talk at the 100th Statistical Mechanics Meeting, Rutgers, December 2008

Scientific paper

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).

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