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

TL;DR

This paper fully characterizes the solution space of a specific system of PDEs in conformal field theory and SLE, proving its dimension equals the Catalan number and constructing a basis using Coulomb gas formalism.

Contribution

It proves the dimension of the solution space matches the Catalan number and constructs an explicit basis using Coulomb gas solutions, extending previous bounds.

Findings

Solution space dimension equals Catalan number C_N

Constructed linearly independent Coulomb gas solutions

Connected results to CFT minimal models

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.