Conformal Field Theory, Vertex Operator Algebra and Stochastic Loewner Evolution in Ising Model

TL;DR

This paper explores the deep connections between conformal field theory, vertex operator algebras, and stochastic Loewner evolution in the context of the Ising model, providing a rigorous mathematical framework and explicit constructions.

Contribution

It establishes a rigorous CFT/SLE correspondence for the Ising model using Clifford VOA and fermionic Fock space, with explicit Fock space representations of SLE martingale generators.

Findings

Derived the scaling limit of Ising fermion correlation functions.

Constructed explicit Fock space models for SLE martingale generators.

Linked operator formalism in Clifford VOA to SLE observables.

Abstract

We review the algebraic and analytic aspects of the conformal field theory (CFT) and its relation to the stochastic Loewner evolution (SLE) in an example of the Ising model. We obtain the scaling limit of the correlation functions of Ising free fermions on an arbitrary simply connected two-dimensional domain $D$. Then, we study the analytic and algebraic aspects of the fermionic CFT on $D$, using the Fock space formalism of fields, and the Clifford vertex operator algebra (VOA). These constructions lead to the conformal field theory of the Fock space fields and the fermionic Fock space of states and their relations in case of the Ising free fermions. Furthermore, we investigate the conformal structure of the fermionic Fock space fields and the Clifford VOA, namely the operator product expansions, correlation functions and differential equations. Finally, by using the Clifford VOA and the fermionic CFT, we investigate a rigorous realization of the CFT/SLE correspondence in the Ising model. First, by studying the relation between the operator formalism in the Clifford VOA and the SLE martingale generators, we find an explicit Fock space for the SLE martingale generators. Second, we obtain a subset of the SLE martingale observables in terms of the correlation functions of fermionic Fock space fields which are constructed from the Clifford VOA.