Brane Brick Models and 2d (0,2) Triality

TL;DR

<3-5 sentence high-level summary>We present a brane brick model realization of $2d$ $(0,2)$ Gadde–Gukov–Putrov triality, identifying triality with local cube-like moves in Type IIA brane configurations dual to D1-branes on toric CY$_4$'s. The classical mesonic moduli space, i.e. the probed CY$_4$, is shown to be invariant across all toric phases connected by triality, with explicit networks constructed for examples such as $Q^{1,1,1}$ and its $Z_2$ orbifold. A fast forward algorithm based on brick matchings and phase boundaries provides a concrete way to verify the moduli space invariance and to connect triality to geometric data. The study also connects triality to phase-boundary dynamics and hints at deeper integrable structures underlying these dualities. Overall, the paper offers a practical, geometry-driven framework for analyzing dual $2d$ $(0,2)$ gauge theories in string theory.

Abstract

We provide a brane realization of 2d (0,2) Gadde-Gukov-Putrov triality in terms of brane brick models. These are Type IIA brane configurations that are T-dual to D1-branes over singular toric Calabi-Yau 4-folds. Triality translates into a local transformation of brane brick models, whose simplest representative is a cube move. We present explicit examples and construct their triality networks. We also argue that the classical mesonic moduli space of brane brick model theories, which corresponds to the probed Calabi-Yau 4-fold, is invariant under triality. Finally, we discuss triality in terms of phase boundaries, which play a central role in connecting Calabi-Yau 4-folds to brane brick models.

Figures (50)

• Figure 1: The four plaquettes $(\Lambda_{ij}, J^{\pm}_{ji})$ and $(\overline\Lambda_{ij}, E^\pm_{ij})$ associated to a Fermi field $\Lambda_{ij}$. The $J$- and $E$-terms are $J_{ji}=J^{+}_{ji}- J^{-}_{ji}=0$ and $E_{ij}=E_{ij}^{+}- E_{ij}^{-}=0$, respectively, where $J_{ji}^\pm$ and $E_{ij}^{\pm}$ are holomorphic monomials in chiral fields.
• Figure 2: Phase boundaries are in one-to-one correspondence with edges of the toric diagram of the Calabi-Yau 4-fold. Certain point intersections of phase boundaries give rise to chiral or Fermi fields, depending on whether they are oriented or alternating.
• Figure 3: The quiver diagram for $2d$$(0,2)$ SQCD (original theory $D$). Square nodes indicate flavor symmetry groups.
• Figure 4: The quiver diagram for the dual of $2d$$(0,2)$ SQCD (dual theory $D^\prime$).
• Figure 5: The triality loop for $2d$$(0,2)$ SQCD.