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

Understand

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.