| Abstrakt: |
This article is the last of four that completely and rigorously characterize a solution space $${{\mathcal{S}}_N}$$ for a homogeneous system of 2 N + 3 linear partial differential equations in 2 N variables that arises in conformal field theory (CFT) and multiple Schramm-Löwner evolution (SLE $${_\kappa}$$ ). The system comprises 2 N null-state equations and three conformal Ward identities that govern CFT correlation functions of 2 N one-leg boundary operators. In the first two articles (Flores and Kleban in Commun Math Phys, ; Flores and Kleban, in Commun Math Phys, ), we use methods of analysis and linear algebra to prove that dim $${{\mathcal{S}}_N \leq C_N}$$ , with C the Nth Catalan number. Using these results in the third article (Flores and Kleban, in Commun Math Phys, ), we prove that dim $${{\mathcal{S}}_N=C_N}$$ and $${{\mathcal{S}}_N}$$ is spanned by (real-valued) solutions constructed with the Coulomb gas (contour integral) formalism of CFT. In this article, we use these results to prove some facts concerning the solution space $${{\mathcal{S}}_N}$$ . First, we show that each of its elements equals a sum of at most two distinct Frobenius series in powers of the difference between two adjacent points (unless $${8/\kappa}$$ is odd, in which case a logarithmic term may appear). This establishes an important element in the operator product expansion for one-leg boundary operators, assumed in CFT. We also identify particular elements of $${{\mathcal{S}}_N}$$ , which we call connectivity weights, and exploit their special properties to conjecture a formula for the probability that the curves of a multiple-SLE $${_\kappa}$$ process join in a particular connectivity. This leads to new formulas for crossing probabilities of critical lattice models inside polygons with a free/fixed side-alternating boundary condition, which we derive in Flores et al. (Partition functions and crossing probabilities for critical systems inside polygons, in preparation). Finally, we propose a reason for why the exceptional speeds [certain $${\kappa}$$ values that appeared in the analysis of the Coulomb gas solutions in Flores and Kleban (Commun Math Phys, )] and the minimal models of CFT are connected. [ABSTRACT FROM AUTHOR] |
