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

TL;DR

The paper develops a forward-algorithm–driven taxonomy of 2+1 dimensional quiver Chern-Simons theories associated with M2-branes probing toric Calabi–Yau 4-folds. It systematically classifies theories with two-term toric superpotentials across increasing numbers of nodes and fields, computes the corresponding toric moduli spaces, and uncovers toric dualities linking apparently different theories to the same CY$_4$ geometry. A dimer (2D tiling) description persists for these CS theories, while a novel generating function counts inequivalent quivers, revealing Hilbert-series structures tied to $ ext{C}^3/S_3$ and $ ext{C}^4/D_4$. The work exposes a rich algebraic structure governing the space of CS quivers and outlines future directions for extending the taxonomy, including higher-term superpotentials and broader dimensional generalizations.

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".

Figures (9)

• Figure 1: The quivers with 4 fields and 2 nodes. There are 2 solutions and the 2-term superpotentials are also given. The moduli space in both cases is just the trivial CY 4-fold $\mathbb{C}^4$.
• Figure 2: The quivers with 5 fields and 3 nodes. There are 3 good models and we call them (1), (3) and (4). We also write the corresponding 2-term superpotentials.
• Figure 3: The quivers with 6 fields and 4 nodes. There are 6 models. We also present the superpotential as well as the adjoint-fields where necessary.
• Figure 4: The two disconnected quivers with 6 fields and 4 nodes, together with their multi-trace superpotentials. Also Model (8) has completely vanishing superpotential.
• Figure 5: The two moduli spaces, drawn with explicit toric diagrams, for different choices of Chern-Simons levels (a) $(k_1,k_2)=(0,1)$ and (b) $(k_1,k_2)=(1,1)$ for Model (1) of the 5-edge, 3-noded theory with 2-term superpotential.