Multiple-SLE κ connectivity weights for rectangles, hexagons, and octagons

Abstract

Abstract In previous work, two of the authors determined, completely and rigorously, a solution space S N for a homogeneous system of 2N + 3 linear partial differential equations (PDEs) in 2N variables that arises in conformal field theory (CFT) and multiple Schramm–Löwner evolution (SLE κ ). The system comprises 2N null-state equations and three conformal ward identities that govern CFT correlation functions of 2N one-leg boundary operators or SLE κ partition functions. M Bauer et al conjectured a formula, expressed in terms of ‘pure SLE κ partition functions,’ for the probability that the growing curves of a multiple-SLE κ process join in a particular connectivity. In a previous article, we rigorously define certain elements of S N , which we call ‘connectivity weights,’ argue that they are in fact pure SLE κ partition functions, and show how to find explicit formulas for them in terms of Coulomb gas contour integrals. Our formal definition of the connectivity weights immediately leads to a method for finding explicit expressions for them. However, this method gives very complicated formulas where simpler versions may be available, and it is not applicable for certain values of κ ∈ (0, 8) corresponding to well-known critical lattice models in statistical mechanics. In this article, we determine expressions for all connectivity weights in S N for N ∈ {1, 2, 3, 4} (those with N ∈ {3, 4} are new) and for so-called ‘rainbow connectivity weights’ in S N for all N ∈ Z + + 1 . We verify these formulas by explicitly showing that they satisfy the formal definition of a connectivity weight. In appendix B, we investigate logarithmic singularities of some of these expressions, appearing for certain values of κ predicted by logarithmic CFT.