Wind on the boundary for the Abelian sandpile model

TL;DR

This paper explores new boundary conditions in the 2D Abelian sandpile model within a logarithmic conformal field theory framework, revealing novel boundary phenomena and confirming fusion rule conjectures.

Contribution

Introduces two novel boundary conditions with orientation dependence and non-uniformity, expanding the understanding of boundary effects in the model.

Findings

Seven new boundary condition changing fields identified

Fusion algebra matches Gaberdiel and Kausch's conjectured rules

Boundary conditions involve indecomposable representations with Jordan cells

Abstract

We continue our investigation of the two-dimensional Abelian sandpile model in terms of a logarithmic conformal field theory with central charge c=-2, by introducing two new boundary conditions. These have two unusual features: they carry an intrinsic orientation, and, more strangely, they cannot be imposed uniformly on a whole boundary (like the edge of a cylinder). They lead to seven new boundary condition changing fields, some of them being in highest weight representations (weights -1/8, 0 and 3/8), some others belonging to indecomposable representations with rank 2 Jordan cells (lowest weights 0 and 1). Their fusion algebra appears to be in full agreement with the fusion rules conjectured by Gaberdiel and Kausch.