Quiver Chern-Simons theories and crystals

TL;DR

This work develops a concrete framework to relate ${\cal N}=2$ quiver Chern-Simons theories, described by brane tilings, to toric Calabi-Yau $4$-fold moduli spaces and to corresponding brane crystals. It provides a practical method to extract toric data from tilings using perfect matchings and GLSM fields, and introduces a gauge-invariant operator basis that includes both mesonic and baryonic directions. The authors establish a precise mapping between gauge-invariant operators and closed 2-cycles in the crystal, and show how CS levels arise naturally from boundary terms in the brane picture. They also connect the tiling description to brane crystals, demonstrating that crystals encode the same toric data and can illuminate dualities among CS theories. Overall, the paper offers a toolkit for constructing and analyzing M2-brane setups in toric backgrounds and clarifies the bridge between tilings and crystals in three-dimensional CS contexts.

Abstract

We consider N=2 quiver Chern-Simons theories described by brane tilings, whose moduli spaces are toric Calabi-Yau 4-folds. Simple prescriptions to obtain toric data of the moduli space and a corresponding brane crystal from a brane tiling are proposed.

Figures (7)

• Figure 1: The tiling for the ABJM model at level $(k,-k)$. The arrow represents the flow ${\bm s}$ defined in (\ref{['ks']}).
• Figure 2: For a set of numbers $f_I$ assigned to links we define two flows. (a) is a normal flow $\bm f$ and (b) is a tangential flow $\bm f^*$. These two flows are related by the $\pi/2$ rotation of arrows.
• Figure 3: Examples of cycles $\bm\alpha$, $\bm\beta$ and ${\bm\gamma}_a$ for the ABJM tiling are shown.
• Figure 4: The four perfect matchings of the ABJM tiling.
• Figure 5: The toric diagram of the orbifold ${\bf C}^4/{\bf Z}_k$.