Logarithmic operator intervals in the boundary theory of critical percolation

Abstract

We consider the sub-sector of the $c=0$ logarithmic conformal field theory (LCFT) generated by the boundary condition changing (bcc) operator in two dimensional critical percolation. This operator is the zero weight Kac operator ${\varphi }_{1,2}$, identified with the growing hull of the SLE${}_6$ process. We identify percolation configurations with the significant operators in the theory. We consider operators from the first four bcc operator fusions: the identity and bcc operator; the stress tensor and its logarithmic partner; the derivative of the bcc operator and its logarithmic partner; and the pre-logarithmic operator ${\varphi }_{1,3}$. We construct several intervals in the percolation model, each associated to one of the LCFT operators we consider, allowing us to calculate crossing probabilities and expectation values of crossing cluster numbers. We review the Coulomb gas, which we use as a method of calculating these quantities when the number of bcc operator makes a direct solution to the system of differential equations intractable. Finally we discuss the case of the six-point correlation function, which applies to crossing probabilities between the sides of a conformal hexagon. Specifically we introduce an integral result that allows one to identify the probability that a single percolation cluster touches three alternating sides a hexagon with free boundaries. We give results of the numerical integration for the case of a regular hexagon.

32 pages, 20 figures

References in corpus (6)

• From Percolation to Logarithmic Conformal Field Theory
• Indecomposability parameters in chiral Logarithmic Conformal Field Theory
• Logarithmic observables in critical percolation
• The puzzle of bulk conformal field theories at central charge c=0
• Percolation Crossing Formulas and Conformal Field Theory
• SLE local martingales in logarithmic representations

Cited by in corpus (6)

• A solution space for a system of null-state partial differential equations 3
• Pure partition functions of multiple SLEs
• A solution space for a system of null-state partial differential equations 1
• A solution space for a system of null-state partial differential equations 4
• Exact logarithmic four-point functions in the critical two-dimensional Ising model
• Two-dimensional non-abelian BF theory in Lorenz gauge as a solvable logarithmic TCFT