A solution space for a system of null-state partial differential equations 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 $⟨{\psi }_1^c(w_1){\psi }_1^c(w_2)\cdots {\psi }_1^c(w_{2N-1}){\psi }_1^c(w_{2N}){⟩}^{\mathcal{P}}$ 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.

Minor typos from v3 corrected, reference to Fig. 11 inserted into text

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Cited by in corpus (13)

• A solution space for a system of null-state partial differential equations 3
• 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
• 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
• Boundary correlations in planar LERW and UST
• The singularities of Selberg- and Dotsenko-Fateev-like integrals
• Numerical Study on a Crossing Probability for the Four-State Potts Model: Logarithmic Correction to the Finite-Size Scaling
• Uniform Spanning Tree in Topological Polygons, Partition Functions for SLE(8), and Correlations in $c=-2$ Logarithmic CFT
• The quantum group dual of the first-row subcategory for the generic Virasoro VOA