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

2012/12/31 by Steven M. Flores, Peter Kleban
Mathematics · Physics and Astronomy · #Combinatorics #Conformal field theory #Conformal map #Ising model #Mathematical analysis #Mathematical physics #Mathematics #Partition function (quantum field theory) #Physics #Potts model #Quantum mechanics #Random Matrices and Applications #Stochastic processes and statistical mechanics #Theoretical and Computational Physics #cond-mat.stat-mech #hep-th #math-ph #math.AP #math.MP

paper · pdf · doi:10.1007/s00220-014-2189-4

published as Commun. Math. Phys., January 2015, Vol. 333, Issue 1, pp 389-434 · Minor typos from v3 corrected, reference to Fig. 11 inserted into text

openalex publication_date 2014/11/12 · arxiv created 2015/02/04 · arxiv updated 2015/02/06 · openalex created_date 2022/05/12 · openalex updated_date 2026/08/05

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 P with its 2N sides exhibiting a free/fixed side-alternating boundary condition, this partition function is proportional to the CFT correlation function ⟨ψ₁^c(w₁)ψ₁^c(w₂)\dotsmψ₁^c(w_2N-1)ψ₁^c(w_2N)⟩^P where the w_i are the vertices of P and where ψ₁^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 Nth 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.

Citations

• Elliptic Partial Differential Equations of Second Order
• Stochastic geometry of critical curves, Schramm–Loewner evolutions and conformal field theory
• Infinite conformal symmetry in two-dimensional quantum field theory
• Critical curves in conformally invariant statistical systems
• Critical percolation in finite geometries
• Maximum Principles in Differential Equations
• A self-avoiding random walk
• From percolation to logarithmic conformal field theory
• A Guide to Stochastic Löwner Evolution and Its Applications
• Critical percolation in the plane: conformal invariance, Cardy's formula, scaling limits
• Conformal invariance of planar loop-erased random walks and uniform spanning trees
• Conformal Field Theory