1303.7182 (Steven M. Flores et al.)

A solution space for a system of null-state partial differential
equations II    [PDF]

Steven M. Flores, Peter Kleban

In this second of two articles, we study a system of 2N+3 linear homogeneous second-order partial differential equations (PDEs) in 2N variables that arise in conformal field theory (CFT) and multiple Schramm-Loewner Evolution (SLE). In CFT, these are null-state equations and Ward identities. They govern partition functions central to the characterization 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 in the continuum limit. (SLE partition functions also satisfy these equations.) The partition functions for critical lattice models contained in a polygon P with 2N sides exhibiting a free/fixed side-alternating boundary condition are proportional to the CFT correlation function ^P, where the w_i are the vertices of P and psi_1^c is a one-leg corner operator. Partition functions conditioned on crossing events in which clusters join the fixed sides of P in some specified connectivity are also proportional to this correlation function. 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 second of two papers in which we completely characterize the space of all solutions for this system of PDEs that grow no faster than a power-law. In the first article, we proved, to within a precise conjecture, that the dimension of this solution space is no more than C_N, the Nth Catalan number. In this article, we use those results to prove that if this conjecture is true, then this solution space has dimension C_N and is spanned by solutions that can be constructed with the CFT Coulomb gas (contour integral) formalism.

View original: http://arxiv.org/abs/1303.7182