M2-branes and plane partitions

Abstract

There is a correspondence between the protected local operators in the 3d SCFTs describing the geometry ℂ2 probed by a stack of N M2-branes and plane partitions of trace N. We give combinatorial expressions of the indices which count the local operators parametrizing ℂ2/ℤk probed by N M2-branes in the canonical and grand canonical ensembles in terms of generating functions for plane partitions. We derive the asymptotic behaviors of the grand potential in the high-temperature limit and the scaling dimension in the large N limit.