Complete conformal field theory solution of a chiral six-point correlation function

Abstract

Using conformal field theory, we perform a complete analysis of the chiral six-point correlation function C(z)=< \phi_{1,2}\phi_{1,2} \Phi_{1/2,0}(z, \bar z) \phi_{1,2}\phi_{1,2} >, with the four \phi_{1,2} operators at the corners of an arbitrary rectangle, and the point z = x+iy in the interior. We calculate this for arbitrary central charge (equivalently, SLE parameter \kappa > 0). C is of physical interest because percolation (\kappa = 6) and many other two-dimensional critical points, it specifies the density at z of critical clusters conditioned to touch either or both vertical ends of the rectangle, with these ends `wired', i.e. constrained to be in a single cluster, and the horizontal ends free. The correlation function may be written as the product of an algebraic prefactor f and a conformal block G, where f = f(x,y,m), with m a cross-ratio specified by the corners (m determines the aspect ratio of the rectangle). By appropriate choice of f and using coordinates that respect the symmetry of the problem, the conformal block G is found to be independent of either y or x, and given by an Appell function.