1212.2301 (Steven M. Flores et al.)

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

Steven M. Flores, Peter Kleban

In this article, 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 Schramm-Loewner Evolution (SLE). In CFT, these are null-state equations and Ward identities. They are satisfied by 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. (Certain SLE partition functions also satisfy these equations.) The partition functions for critical lattice models contained in a polygon P with 2N sides exhibiting free/fixed side-alternating boundary conditions 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, this correlation function satisfies the system of PDEs that we consider. This article is the first of two in which we completely characterize the space of all solutions for this system of PDEs that grow no faster than a power law. In this article, we prove, to within a precise technical conjecture, that the dimension of this solution space is no more than C_N, the Nth Catalan number. In the appendices, we propose a method for proving the conjecture mentioned, and we posit that all solutions for this system of PDEs indeed grow no faster than a power law. In the second article, we use these results to prove that the solution space has dimension C_N and is spanned by solutions constructed with the Coulomb gas formalism.

View original: http://arxiv.org/abs/1212.2301