A Solution Space for a System of Null-State Partial Differential Equations: Part 1

Abstract

In this first of four articles, we study 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). In CFT, these are null-state equations and conformal Ward identities. They govern partition functions for the continuum limit of a statistical cluster or loop model, such as percolation, or more generally the Potts models and O$(n)$ models, at the statistical mechanical critical point. (SLE partition functions also satisfy these equations.) For such a lattice model in a polygon $\mathcal{P}$ with its $2N$ sides exhibiting a free/fixed side-alternating boundary condition, this partition function is proportional to the CFT correlation function $\langle\psi_1^c(w_1)\psi_1^c(w_2)\dotsm\psi_1^c(w_{2N-1})\psi_1^c(w_{2N})\rangle^{\mathcal{P}}$ where the $w_i$ are the vertices of $\mathcal{P}$ and where $\psi_1^c$ is a one-leg corner operator. When conformally mapped onto the upper half-plane, methods of CFT show that this correlation function satisfies the system of PDEs that we consider. This article is the first of four that completely and rigorously characterize the space of all solutions for this system of PDEs that grow no faster than a power law. In this first article, we use methods of analysis to prove that the dimension of this solution space is no more than $C_N$, the $N$th Catalan number. This proof is contained entirely within this article, except for the proof of lemma 14, which constitutes the second article ("part II"). In the third article ("part III"), we use the results of this article to prove that the solution space of this system of PDEs has dimension $C_N$ and is spanned by solutions constructed with the CFT Coulomb gas (contour integral) formalism. In the fourth article ("part IV"), we prove further CFT-related properties about these solutions.