2021/08/11 by Taha Ameen, Kalle Kytölä, Ameen, Taha +4
Mathematics · Physics and Astronomy · #Algebraic structures and combinatorial models #Boundary (topology) #Boundary conformal field theory #Conformal field theory #Conformal map #FOS: Mathematics #FOS: Physical sciences #Geometry #Holomorphic function #Ising model #Mathematical Physics (math-ph) #Mathematical analysis #Mathematics #Neumann boundary condition #Physics #Probability (math.PR) #Quantum mechanics #Random Matrices and Applications #Robin boundary condition #Scaling #Scaling limit #Theoretical and Computational Physics #math-ph #math.MP #math.PR
paper · pdf · doi:10.48550/arxiv.2108.05105
82 pages, 25 figures
arxiv created 2021/08/11 · openalex publication_date 2021/08/11 · arxiv updated 2021/08/12 · openalex created_date 2021/08/16 · openalex updated_date 2026/07/28
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.