Tilings, Chern-Simons Theories and M2 Branes

TL;DR

The paper constructs an infinite class of 2+1D ${\cal N}=2$ Chern-Simons theories from brane tilings, whose abelian moduli space is a four-fold toric Calabi–Yau cone $X$ and whose non-abelian moduli space is the $N$-fold symmetric product ${\rm Sym}^N X$. It extends 3+1D tiling methods to 2+1D by a modified D-term analysis with CS levels, computing the master space and Hilbert series for abelian cases using Molien integrals. A suite of explicit models is analyzed, including modified conifold, modified $\mathbb{C}^2/\mathbb{Z}_2\times\mathbb{C}$, SPP, $\mathbb{C}^3/\mathbb{Z}_3$, and $\mathbb{F}_0$, with extensions to chiral theories and orientifolds; the work connects to ABJM-like structures, AdS$_4$/CFT$_3$, and Type IIA descriptions. The authors outline a program to classify 2+1D CS theories dual to toric and non-toric CY4 singularities and propose methods to compare gravity and field theory data, including a surrogate for a-maximization in 2+1D.

Abstract

A new infinite class of Chern-Simons theories is presented using brane tilings. The new class reproduces all known cases so far and introduces many new models that are dual to M2 brane theories which probe a toric non-compact CY 4-fold. The master space of the quiver theory is used as a tool to construct the moduli space for this class and the Hilbert Series is computed for a selected set of examples.