Discrete Holomorphicity at Two-Dimensional Critical Points

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).

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

References in corpus (2)

• 2D growth processes: SLE and Loewner chains
• Discretely Holomorphic Parafermions and Integrable Loop Models

Cited by in corpus (16)

• Free parafermions
• Parafermionic conformal field theory on the lattice
• Parafermionic excitations and critical exponents of random cluster and O(n) models
• Pattern Formation in Growing Sandpiles with Multiple Sources or Sinks
• Discrete holomorphicity and integrability in loop models with open boundaries
• Haldane relation for interacting dimers
• Integrability and braided tensor categories
• Integrability as a consequence of discrete holomorphicity: the Z_N model
• Integrability as a consequence of discrete holomorphicity: loop models
• Critical behaviour of the randomly stirred dynamical Potts model: Novel universality class and effects of compressibility
• The Ultraviolet Structure of Quantum Field Theories. Part 2: What is Quantum Field Theory?
• Divergence of the correlation length for critical planar FK percolation with $1\le q\le 4$ via parafermionic observables
• Conformal Field Theory, Vertex Operator Algebra and Stochastic Loewner Evolution in Ising Model
• Virasoro Generators in the Fibonacci Model Tensor Network -- Tackling Finite Size Effects
• Virasoro and Kac-Moody algebra in generic tensor network representations of 2d critical lattice partition functions
• Discrete Holomorphic Parafermions in the Eight Vertex Model