Brane Brick Models, Toric Calabi-Yau 4-Folds and 2d (0,2) Quivers

TL;DR

<3-5 sentence high-level summary>Brane brick models provide a new Type IIA brane realization that directly links 2d $(0,2)$ gauge theories to toric Calabi–Yau 4-fold geometries, via a D4-NS5 configuration and a tropical (brick) skeleton on a $T^3$. The paper develops two complementary computational procedures: a fast inverse algorithm that builds brane brick models from geometry through phase boundaries, and a fast forward algorithm that extracts the CY$_4$ geometry from brane brick data using brick matchings, a novel combinatorial analogue of perfect matchings. These methods generalize brane tiling techniques to fourfolds, enable partial-resolution analyses, and extend constructions to CY$_3 imesC$ theories and beyond orbifolds, highlighting powerful connections between geometry, quivers, and combinatorics. The approaches have potential implications for classifying 2d $(0,2)$ theories, understanding their dualities, and providing tools for exploring toric CY$_4$ landscapes in a controlled, algorithmic manner.

Abstract

We introduce brane brick models, a novel type of Type IIA brane configurations consisting of D4-branes ending on an NS5-brane. Brane brick models are T-dual to D1-branes over singular toric Calabi-Yau 4-folds. They fully encode the infinite class of 2d (generically) N=(0,2) gauge theories on the worldvolume of the D1-branes and streamline their connection to the probed geometries. For this purpose, we also introduce new combinatorial procedures for deriving the Calabi-Yau associated to a given gauge theory and vice versa.

Figures (53)

• Figure 1: The four plaquettes $(\Lambda_{ij}, J_{ji}^{\pm})$ and $(\overline{\Lambda}_{ij}, E_{ij}^{\pm})$ corresponding to a Fermi field $\Lambda_{ij}$.
• Figure 2: A unit cell of the periodic quiver of $\mathbb{C}^4$.
• Figure 3: The standard and periodic quivers for $\mathcal{C}\times\mathbb{C}$.
• Figure 4: Brane box models. Schematic representation of the internal $(2,4,6)$ directions. The blue, red and green planes correspond to $\text{NS}$, $\text{NS}^\prime$ and $\text{NS}^{\prime\prime}$-branes that extend along $(24)$, $(26)$ and $(46)$ directions. D4-branes span the $(246)$ directions, filling the boxes. The geometric action of the dual abelian orbifold of $\mathbb{C}^4$ is translated into the periodicity conditions on $T^3$.
• Figure 5: Brane box models and periodic quivers. This figure presents the brane box model for $\mathbb{C}^4$, which has a single NS5-brane of each type. It also shows how the brane configuration gives rise to the corresponding periodic quiver. Orbifold models are obtained from this configuration by enlarging the unit cell.