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

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.

Correlations of primary fields in the critical Ising model

“…Moreover, the only difference in the values of F at z a and z a actually comes from the difference between η za and η z b . Indeed, in the case (1) this is obvious since z…”

“…where the sum is taken over all pairs v ∼ v of nearest-neighbor vertices of C δ such that at least one of v, v belongs to Ω δ , and (vv ) ∩ {free} = ∅; the additional factor 1 2 is added to handle the two equal contributions σ v σ v = σ v * σ v * coming from the two sheets of Ω δ . As above, Z is the partition function that makes the total mass equal to one.…”

“…• are two endpoints of (the lift of) a free boundary arc ν, and z • a , z • b ∈ Ω δ are on (the lifts of) two wired boundary arcs on the same boundary component. The signs of η za and η z b in (2.9) are assumed to be related as follows: in the case (1), by rotating z a towards z b around z • a = z • b outside Ω δ ; in the case (2) by rotating z a towards z b inside Ω δ ; and in the case (3) by the extension along the path on C(Ω δ ) running along the free boundary arc outside Ω δ .…”

“…The critical scaling limit of the Ising model was one of the prime examples in the new theory as well, being described by a minimal CFT of central charge 1 2 . Already in [2], as an example of an application of the new theory, the four-point correlation function of the critical Ising model in the full plane was computed.…”

“…In [21], probabilities to find general "local patterns" of spins were considered. In a number of works [25,1,10], various aspects of the Ising CFT were elucidated on the lattice level.…”