M2-Branes and Quiver Chern-Simons: A Taxonomic Study

TL;DR

This paper systematically classifies 2+1 dimensional quiver gauge theories related to M2-branes, introduces a generating function for counting inequivalent theories, and explores their geometric origins and dualities.

Contribution

It develops a forward algorithm for classifying quiver theories, derives a generating function for counting them, and links these theories to toric Calabi-Yau 4-folds with new insights into toric duality.

Findings

Derived a generating function counting inequivalent theories

Explicitly obtained toric data for vacuum moduli spaces

Identified new instances of toric duality

Abstract

We initiate a systematic investigation of the space of 2+1 dimensional quiver gauge theories, emphasising a succinct "forward algorithm". Few "order parametres" are introduced such as the number of terms in the superpotential and the number of gauge groups. Starting with two terms in the superpotential, we find a generating function, with interesting geometric interpretation, which counts the number of inequivalent theories for a given number of gauge groups and fields. We demonstratively list these theories for some low numbers thereof. Furthermore, we show how these theories arise from M2-branes probing toric Calabi-Yau 4-folds by explicitly obtaining the toric data of the vacuum moduli space. By observing equivalences of the vacua between markedly different theories, we see a new emergence of "toric duality".