A Solution Space for a System of Null-State Partial Differential Equations: Part 2

This article is the second of four that completely and rigorously characterize a solution space S N for a homogeneous system of 2 N + 3 linear partial differential equations in 2 N variables that arises in conformal field theory (CFT) and multiple Schramm–Löwner evolution (SLE κ ). The system comprises 2 N null-state equations and three conformal Ward identities which govern CFT correlation functions of 2 N one-leg boundary operators. In the first article (Flores and Kleban, Commun Math Phys, arXiv:1212.2301 , 2012 ), we use methods of analysis and linear algebra to prove that dim S N ≤ C N , with C N the N th Catalan number. The analysis of that article is complete except for the proof of a lemma that it invokes. The purpose of this article is to provide that proof. The lemma states that if every interval among ( x 2 , x 3 ), ( x 3 , x 4 ),…,( x 2 N -1 , x 2 N ) is a two-leg interval of F ∈ S N (defined in Flores and Kleban, Commun Math Phys, arXiv:1212.2301 , 2012 ), then F vanishes. Proving this lemma by contradiction, we show that the existence of such a nonzero function implies the existence of a non-vanishing CFT two-point function involving primary operators with different conformal weights, an impossibility. This proof (which is rigorous in spite of our occasional reference to CFT) involves two different types of estimates, those that give the asymptotic behavior of F as the length of one interval vanishes, and those that give this behavior as the lengths of two intervals vanish simultaneously. We derive these estimates by using Green functions to rewrite certain null-state PDEs as integral equations, combining other null-state PDEs to obtain Schauder interior estimates, and then repeatedly integrating the integral equations with these estimates until we obtain optimal bounds. Estimates in which two interval lengths vanish simultaneously divide into two cases: two adjacent intervals and two non-adjacent intervals. The analysis of the latter case is similar to that for one vanishing interval length. In contrast, the analysis of the former case is more complicated, involving a Green function that contains the Jacobi heat kernel as its essential ingredient.