2d (0,2) Quiver Gauge Theories and D-Branes

TL;DR

<3-5 sentence high-level summary>We study 2d $N=(0,2)$ quiver gauge theories realized on D1-branes probing toric Calabi–Yau 4-folds, establishing a systematic forward algorithm to extract the classical mesonic moduli space (the probed CY$_4$) and a Higgsing/partial-resolution program to generate gauge theories for arbitrary toric singularities. We develop the toric-geometry dictionary via the K- and P-matrices, GLSM fields, and the master space, and analyze anomaly cancellation and IR dynamics with an emphasis on abelian orbifolds and CY$_3 imesC$ relatives. The work introduces Brane Brick Models as the 2d analogue of brane tilings, providing a geometric–gauge bridge on $T^3$ and previewing a coamoeba-based construction; together these tools map a wide zoo of toric CY$_4$ geometries to explicit gauge theories. This framework paves the way for a deeper quantum understanding, including the role of triality and non-perturbative dynamics, and motivates a unified brane-brick picture of toric CY$_4$ physics.

Abstract

We initiate a systematic study of 2d (0,2) quiver gauge theories on the worldvolume of D1-branes probing singular toric Calabi-Yau 4-folds. We present an algorithm for efficiently calculating the classical mesonic moduli spaces of these theories, which correspond to the probed geometries. We also introduce a systematic procedure for constructing the gauge theories for arbitrary toric singularities by means of partial resolution, which translates to higgsing in the field theory. Finally, we introduce Brane Brick Models, a novel class of brane configurations that consist of D4-branes suspended from an NS5-brane wrapping a holomorphic surface, tessellating a 3-torus. Brane Brick Models are the 2d analogues of Brane Tilings and allow a direct connection between geometry and gauge theory.

Figures (51)

• Figure 1: A generic 1-loop diagram to compute anomalies in $2d$.
• Figure 2: Toric diagram of $\mathbb{C}^4$.
• Figure 3: The quiver diagram for $N$ D1-branes over $\mathbb{C}^4$. It consists of a single $U(N)$ gauge node, four adjoint chiral fields (shown in black) and three Fermi fields (shown in red).
• Figure 4: A unit cell of the periodic quiver of $\mathbb{C}^4$, which is periodically identified along the three axes.
• Figure 5: The local structure of the periodic quiver for an arbitrary $\mathbb{C}^4/\mathbb{Z}_{n}$ orbifold with action $(a_1,a_2,a_3,a_4)$, where $a_4 = -a_1-a_2-a_3$. All integer labels on the gauge nodes of the periodic quiver are considered modulo $n$. On the chiral field arrows, we indicate the shift in the node label between the two endpoints.