LAUSR

In previous work, two of the authors determined, completely and rigorously, a solution space SN 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-Lo ̈wner evolution (SLEκ). The system comprises 2N null-state equations and three conformal Ward identities that govern CFT correlation functions of 2N one-leg boundary operators or SLEκ partition functions. M. Bauer et al. conjectured a formula, expressed in terms of “pure SLEκ partition functions,” for the probability that the growing curves of a multiple-SLEκ process join in a particular connectivity. In a previous article, we rigorously define certain elements of SN , which we call “connectivity weights,” argue that they are in fact pure SLEκ 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 SN for N ∈ {1, 2, 3, 4} (those with N ∈ {3, 4} are new) and for so-called “rainbow connectivity weights” in SN 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. Keywords: conformal field theory, Schramm-Lo ̈wner evolution, connectivity weights, crossing probability, pure SLEκ partition function