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

Abstract

In a previous article, we define "connectivity weights" to be functions with these two properties: 1) They solve the three conformal Ward identities of conformal field theory (CFT) and a system of $2N$ null-state differential equations governing a CFT $2N$-point function of $\phi_{1,2}$ or $\phi_{2,1}$ primary Kac operators. 2) They satisfy a certain "duality" condition. In that same article, we argue that these functions are in fact pure partition functions for a multiple-SLE$_\kappa$ process with $2N$ curves, and we show how to find explicit formulas for them in terms of Coulomb gas contour integrals. However, this method gives very complicated formulas where simpler versions may be available, and it is not applicable for certain values of $\kappa\in(0,8)$ corresponding to well-known critical lattice models in statistical mechanics. In this article, we determine expressions for all connectivity weights for $N\in\{1,2,3,4\}$ (those with $N\in\{3,4\}$ are new) and for so-called "rainbow connectivity weights" for all $N\in\mathbb{Z}^++1$. 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 $\kappa$ predicted by logarithmic CFT.