Since then, there has been much work searching for
higher dimensional extensions of this result, in particular due to
possible connections to high-temperature superconductivity. The
clearest statement of the basic physical reasons to expect such an
extension are due to Misguich et. al, who argued that any such system
would either have long-range spin order, and hence have gapless spin
wave excitations, or else would have a class of topological excitations
with vanishing gap. Thus, showing this result in higher
dimensions would connect directly to recent ideas on topological order
in quantum systems. I will sketch my proof of this result,
emphasizing connections to these basic physical ideas, and discuss more
recent results on the stability of topological order. In the
process, I will derive various results about locality of correlation
functions in these systems.