Brane Box Realization of Chiral Gauge Theories in Two Dimensions

We study type IIA configurations of D4 branes and three kinds of NS fivebranes. The D4 brane world-volume has finite extent in three directions, giving rise to a two-dimensional low-energy field theory. The models have generically $(0,2)$ supersymmetry. We determine the rules to read off the spectrum and interactions of the field theory from the brane box configuration data. We discuss the construction of theories with enhanced $(0,4)$, $(0,6)$ and $(0,8)$ supersymmetry. Using T-duality along the directions in which the D4 branes are finite, the configuration can be mapped to D1 branes at $\IC^4/\Gamma$ singularities, with $\Gamma$ an abelian subgroup of SU(4). This provides a rederivation of the rules in the brane box model. The enhancement of supersymmetry has a nice geometrical interpretation in the singularity picture in terms of the holonomy group of the four-fold singularity.