The four height variables of the Abelian sandpile model

We study the height one, two, three, and four variables in the Abelian sandpile model. We argue that correlation functions along closed boundaries, as well as general conformal field theory principles, show that the four variables are not represented by the same operator along closed boundaries, or in the bulk. Along open boundaries, we calculate all n-point correlations, and find that there, all height variables are represented by the same operator. We introduce dissipative defect points, and show that along open boundaries they are represented by the same operator as the height variables.