M2-branes and plane partitions

TL;DR

This paper explores the relationship between M2-branes in certain geometries and plane partitions, providing combinatorial formulas for counting local operators and analyzing their asymptotic behaviors.

Contribution

It introduces explicit combinatorial expressions for indices counting local operators in M2-brane theories using plane partitions, extending understanding of their mathematical structure.

Findings

Derived combinatorial formulas for indices in terms of plane partitions.

Analyzed asymptotic behavior of grand potential at high temperature.

Studied scaling dimensions in the large N limit.

Abstract

There is a correspondence between the protected local operators in the 3d SCFTs describing the geometry $\mathbb{C}^2$ 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 $\mathbb{C}^2/\mathbb{Z}_k$ 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.