Partition Functions for Membrane Theories

TL;DR

The work develops a comprehensive framework to compute partition functions for M2-brane worldvolume theories at orbifold singularities with ${\cal N}=5$ or ${\cal N}=6$ supersymmetry, using the Plethystic Programme and Molien-type projections to organize protected spectra into $R$-symmetry representations. It provides explicit generating functions for the $N=1$ theory under abelian and non-abelian orbifolds, including the dihedral and exceptional ($\hat{D},\hat{E}$) families, and translates $Spin(8)$ content into $SU(4)$ or $Sp(2)$ language as appropriate. The paper also extends to higher $N$ via grand canonical constructions, analyzes the zero-baryon limit ($k\to\infty$) yielding KK spectra on $AdS_4\times\mathbb{P}^3$, and discusses the palindromic properties that reflect Calabi–Yau structure of the moduli spaces. Together, these results provide a detailed, representation-theoretic atlas of chiral operator spectra and moduli-space functions for M2-brane theories in diverse backgrounds, with implications for holography and geometric counting.

Abstract

Partition functions for M2-brane theories in various backgrounds are computed. We consider in particular configurations of membranes at orbifold singularities preserving N=5 or N=6 supersymmetry. The worldvolume membrane theory for some of these configurations has been recently constructed in terms of N=6 Chern-Simons theories. The detailed structure of the partition functions as well as their transformation rules under the R-symmetry are explicitly computed using the Plethystic Programme.