Brane Tilings and M2 Branes

TL;DR

<3-5 sentence high-level summary>

Abstract

Brane tilings are efficient mnemonics for Lagrangians of N=2 Chern-Simons-matter theories. Such theories are conjectured to arise on M2-branes probing singular toric Calabi-Yau fourfolds. In this paper, a simple modification of the Kasteleyn technique is described which is conjectured to compute the three dimensional toric diagram of the non-compact moduli space of a single probe. The Hilbert Series is used to compute the spectrum of non-trivial scaling dimensions for a selected set of examples.

Figures (23)

• Figure 1: (i) Brane tiling for $\mathbb{C}^4$ (and $\mathbb{C}^4/\mathbb{Z}_k$). The fundamental domain is shown in red. The green arrows indicate the direction of the bifundamental fields based on the convention that the black node is on the left-hand side. (ii) The corresponding quiver.
• Figure 2: Fundamental cell of the $\mathbb{C}^4$ brane tiling. The weights of the four edges are shown in blue.
• Figure 3: (i) Toric diagram for the conifold. (ii) Introducing the level translates the points as shown by the arrows. (iii) The resulting toric diagram is that of $\mathbb{C}^4$.
• Figure 4: Brane tiling for $\mathcal{C} \times \mathbb{C}$ with edge weights around the two black nodes.
• Figure 5: (i) The 2d toric diagram for $\mathbb{C}_2 / \mathbb{Z}_2 \times \mathbb{C}$, denoted below by ${\cal T}_2$. (ii) The 3d toric diagram for $\mathcal{C} \times \mathbb{C}$, denoted below by ${\cal T}_3$. The internal point of multiplicity 2 in ${\cal T}_2$ splits into two external points in ${\cal T}_3$.