A solution space for a system of null-state partial differential equations 4

TL;DR

This paper characterizes the solution space of a system of PDEs from conformal field theory and SLE, proving its dimension equals a Catalan number, and explores its structure and implications for curve connectivities.

Contribution

It proves the dimension of the solution space matches the Catalan number and identifies basis solutions using Coulomb gas formalism, advancing understanding of CFT and SLE connections.

Findings

Solution elements are sums of at most two Frobenius series.

Identification of connectivity weights for multiple-SLE processes.

Conjecture relating exceptional speeds to minimal models of CFT.

Abstract

This article is the last of four that completely characterize a solution space $\mathcal{S}_N$ 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-Loewner evolution (SLE). The system comprises $2N$ null-state equations and three conformal Ward identities that govern CFT correlation functions of $2N$ one-leg boundary operators. In the first two articles, we use methods of analysis and linear algebra to prove that $\dim\mathcal{S}_N\leq C_N$, with $C_N$ the $N$th Catalan number. Building on these results in the third article, we prove that $\dim\mathcal{S}_N=C_N$ and $\mathcal{S}_N$ is spanned by (real-valued) solutions constructed with the Coulomb gas (contour integral) formalism of CFT. In this article, we use these results to prove some facts concerning the solution space $\mathcal{S}_N$. First, we show that each of its elements equals a sum of at most two distinct Frobenius series in powers of the difference between two adjacent points (unless $8/\kappa$ is odd, in which case a logarithmic term may appear). This establishes an important element in the operator product expansion (OPE) for one-leg boundary operators, assumed in CFT. We also identify particular elements of $\mathcal{S}_N$, which we call connectivity weights, and exploit their special properties to conjecture a formula for the probability that the curves of a multiple-SLE$_\kappa$ process join in a particular connectivity. Finally, we propose a reason for why the "exceptional speeds" (certain $\kappa$ values that appeared in the analysis of the Coulomb gas solutions in the third article) and the minimal models of CFT are connected.