  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.
