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

作者：Flores Steven M^*; Kleban Peter

来源：Communications in Mathematical Physics, 2015, 333(1): 389-434.

DOI：10.1007/s00220-014-2189-4

This article is the first of four that completely and rigorously characterize a solution space 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). 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-gas 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 with its 2N sides exhibiting a free/fixed side-alternating boundary condition , this partition function is proportional to the CFT correlation function where the w (i) are the vertices of and where is a one-leg corner operator. (Partition functions for "crossing events" in which clusters join the fixed sides of in some specified connectivity are linear combinations of such correlation functions.) When conformally mapped onto the upper half-plane, methods of CFT show that this correlation function satisfies the system of PDEs that we consider.
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. While our motivations are based in CFT, our proofs are completely rigorous. This proof is contained entirely within this article, except for the proof of Lemma 14, which constitutes the second article (Flores and Kleban, in Commun Math Phys, arXiv:1404.0035, 2014). In the third article (Flores and Kleban, in Commun Math Phys, arXiv:1303.7182, 2013), 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 (Flores and Kleban, in Commun Math Phys, arXiv:1405.2747, 2014), we prove further CFT-related properties about these solutions, some useful for calculating cluster-crossing probabilities of critical lattice models in polygons.

• 出版日期2015-1