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

TL;DR

This paper provides a comprehensive conformal field theory analysis of a specific six-point correlation function in two-dimensional critical systems, applicable to models like percolation, and expresses the conformal block in terms of an Appell function.

Contribution

It offers a complete solution for the six-point correlation function in chiral conformal field theory for arbitrary central charge, including explicit formulas for the conformal blocks.

Findings

Conformal block expressed as an Appell function.

Correlation function relates to critical cluster densities in percolation.

Solution valid for any central charge, including percolation ().

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.