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

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 $\dim {\mathcal{S}}_N\le C_N$, 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.

Minor typos from v1 corrected

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Cited by in corpus (12)

• Global and Local Multiple SLEs for $\kappa \le 4$ and Connection Probabilities for Level Lines of GFF
• Pure partition functions of multiple SLEs
• A formula for crossing probabilities of critical systems inside polygons
• A solution space for a system of null-state partial differential equations 4
• Towards a conformal field theory for Schramm-Loewner evolutions
• Exact logarithmic four-point functions in the critical two-dimensional Ising model
• Four-point boundary connectivities in critical two-dimensional percolation from conformal invariance
• Crossing Probabilities of Multiple Ising Interfaces
• Connection Probabilities of Multiple FK-Ising Interfaces
• Numerical Study on a Crossing Probability for the Four-State Potts Model: Logarithmic Correction to the Finite-Size Scaling
• The singularities of Selberg- and Dotsenko-Fateev-like integrals
• Bosonization of primary fields for the critical Ising model on multiply connected planar domains