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We analyze the two-dimensional Abelian sandpile model, and demonstrate thatthe four height variables have different field identifications in the bulk, andalong closed boundaries, but become identical, up to rescaling, along openboundaries. We consider two-point boundary correlations in detail, and discussa number of complications that arise in the mapping from sandpile correlationsto spanning tree correlations; the structure of our results suggests aconjecture that could greatly simplify future calculations. We find a number ofthree-point functions along closed boundaries, and propose closed boundaryfield identifications for the height variables. We analyze the effects ofdissipative defect sites, at which the number of grains is not conserved, andshow that dissipative defects along closed boundaries, and in the bulk, have noeffect on any weakly allowed cluster variables, or on their correlations. Alongopen boundaries, we find a particularly simple field structure; we calculateall $n$-point correlations, for any combinations of height variables anddissipative defect sites, and find that all heights and defects are representedby the same field operator.Comment: 19 pages, 10 figures. Error in field identification fixed. Referee  suggestions incorporated, including new appendix. To be published in Phys.  Rev.

Four height variables, boundary correlations, and dissipative defects in the Abelian sandpile model