Slit-strip Ising boundary conformal field theory 1: Discrete and continuous function spaces

TL;DR

This paper introduces a framework connecting discrete and continuum holomorphic functions in slit-strip geometries to recover the algebraic structure of boundary conformal field theories from the critical Ising model.

Contribution

It establishes convergence results of discrete holomorphic functions to continuum ones, enabling the recovery of boundary CFT algebraic structures from lattice models.

Findings

Discrete holomorphic functions converge to continuum functions

Distinguished functions characterized by singular behavior

Framework for recovering boundary CFT from Ising model

Abstract

This is the first in a series of articles about recovering the full algebraic structure of a boundary conformal field theory (CFT) from the scaling limit of the critical Ising model in slit-strip geometry. Here, we introduce spaces of holomorphic functions in continuum domains as well as corresponding spaces of discrete holomorphic functions in lattice domains. We find distinguished sets of functions characterized by their singular behavior in the three infinite directions in the slit-strip domains. We prove convergence results of the distinguished discrete holomorphic functions to the continuum ones. In the subsequent articles, the discrete holomorphic functions will be used for the calculation of the Ising model fusion coefficients (as well as for the diagonalization of the Ising transfer matrix), and the convergence of the functions is used to prove the convergence of the fusion coefficients. It will also be shown that the vertex operator algebra of the boundary conformal field theory can be recovered from the limit of the fusion coefficients via geometric transformations involving the distinguished continuum functions.