Search

In previous work, two of the authors determined, completely and rigorously, a solution space S N for a homogeneous system of 2 N + 3 linear partial differential equations (PDEs) in 2 N variables that arises in conformal field theory (CFT) and multiple Schrammâ€"Löwner evolution (SLE Îş ). 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 or SLE Îş partition functions. M Bauer et al conjectured a formula, expressed in terms of ‘pure SLE Îş partition functions,’ for the probability that the growing curves of a multiple-SLE Îş 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 Îş 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 Îş ∈ (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, we investigate logarithmic singularities of some of these expressions, appearing for certain values of Îş predicted by logarithmic CFT. [ABSTRACT FROM AUTHOR]

We consider the sub-sector of the c = 0 logarithmic conformal field theory (LCFT) generated by the boundary condition changing (bcc) operator in two dimensional critical percolation. This operator is the zero weight Kac operator Φ1,2 =: Φ, identified with the growing hull of the SLE6 process. We identify percolation configurations with the significant operators in the theory. We consider operators from the first four bcc operator fusions: the identity and Φ; the stress-tensor and its logarithmic partner; ∂Φ and its logarithmic partner; and the pre-logarithmic operator Φ1,3. We construct several intervals in the percolation model, each associated to one of the LCFT operators we consider, allowing us to calculate crossing probabilities and expectation values of crossing cluster numbers. We review the CG, which we use as a method of calculating these quantities when the number of bcc operator makes a direct solution to the system of differential equations intractable. Finally we discuss the case of the six-point correlation function, which applies to crossing probabilities between the sides of a conformal hexagon. Specifically we introduce an integral result that allows one to identify the probability that a single percolation cluster touches three alternating sides a hexagon with free boundaries. We give results of the numerical integration for the case of a regular hexagon. [ABSTRACT FROM AUTHOR]