On the Percolation BCFT and the Crossing Probability of Watts

TL;DR

This paper explores the logarithmic conformal field theory of critical percolation, identifying a key boundary operator with a shifted Kac table, leading to an extended theory with new modules and consistency checks.

Contribution

It introduces a new boundary condition changing operator within the logarithmic CFT of percolation, expanding the theory with a shifted Kac table and novel indecomposable modules.

Findings

Identification of Watts' boundary operator with a primary field outside the extended Kac table

Development of an augmented logarithmic theory with new rank-2 modules

No observed rank-3 Jordan cells or Gurarie-Ludwig inconsistencies

Abstract

The logarithmic conformal field theory describing critical percolation is further explored using Watts' determination of the probability that there exists a cluster connecting both horizontal and vertical edges. The boundary condition changing operator which governs Watts' computation is identified with a primary field which does not fit naturally within the extended Kac table. Instead a "shifted" extended Kac table is shown to be relevant. Augmenting the previously known logarithmic theory based on Cardy's crossing probability by this field, a larger theory is obtained, in which new classes of indecomposable rank-2 modules are present. No rank-3 Jordan cells are yet observed. A highly non-trivial check of the identification of Watts' field is that no Gurarie-Ludwig-type inconsistencies are observed in this augmentation. The article concludes with an extended discussion of various topics related to extending these results including projectivity, boundary sectors and inconsistency loopholes.