Slit-strip Ising boundary conformal field theory 2: Scaling limits of fusion coefficients

TL;DR

This paper investigates the scaling limits of fusion coefficients in the critical Ising model within slit-strip geometries, employing discrete holomorphic functions to connect lattice models with boundary conformal field theory structures.

Contribution

It introduces a novel technique associating operator valued discrete 1-forms with discrete holomorphic functions to analyze fusion coefficients in boundary Ising models.

Findings

Rederived the diagonalization of the Ising transfer matrix using boundary CFT analogy.

Characterized fusion coefficients via recursion relations involving discrete holomorphic functions.

Proved convergence of lattice fusion coefficients to their conformal field theory counterparts.

Abstract

This is the second in a series of three 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 study the fusion coefficients of the Ising model in the lattice slit-strip, with locally monochromatic boundary conditions. The fusion coefficients are certain renormalized limits of boundary correlation functions at the three extremities of the truncated lattice slit-strips, in a basis of random variables whose correlation functions have an essentially exponential dependence on the truncation heights. The key technique is to associate operator valued discrete 1-forms to certain discrete holomorphic functions. This provides a direct analogy with currents in boundary conformal field theory. For two specific applications of this technique, we use distinguished discrete holomorphic functions from the first article of the series. First, we rederive the known diagonalization of the Ising transfer matrix in a form that parallels boundary conformal field theory. Second, we characterize the Ising model fusion coefficients by a recursion written purely in terms of inner products of the distinguished discrete holomorphic functions. The convergence result for the discrete holomorphic functions proven in the first part can then be used to derive the convergence of the fusion coefficients in the scaling limit. In the third article of the series, it will be shown that up to a transformation that accounts for our chosen slit-strip geometry, the scaling limits of the fusion coefficients become the structure constants of the vertex operator algebra of a fermionic conformal field theory.