A solution space for a system of null-state partial differential equations 3

Abstract

This article is the third 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-Lowner 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 previous two articles (parts I and II), 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. Extending these results, we prove in this article that $\dim\mathcal{S}_N=C_N$ and $\mathcal{S}_N$ entirely consists of (real-valued) solutions constructed with the CFT Coulomb gas (contour integral) formalism. In order to prove this claim, we show that a certain set of $C_N$ such solutions is linearly independent. Because the formulas for these solutions are complicated, we prove linear independence indirectly. We use the linear injective map of lemma 15 in part I to send each solution of the mentioned set to a vector in $\mathbb{R}^{C_N}$, whose components we find as inner products of elements in a Temperley-Lieb algebra. We gather these vectors together as columns of a symmetric $C_N$ by $C_N$ matrix, with the form of a meander matrix. If the determinant of this matrix does not vanish, then the set of $C_N$ Coulomb gas solutions is linearly independent. And if this determinant does vanish, then we construct an alternative set of $C_N$ Coulomb gas solutions and follow a similar procedure to show that this set is linearly independent. The latter situation is closely related to CFT minimal models.