(0,2) dynamics from four dimensions

We study (0,2) supersymmetric two-dimensional theories obtained by compactifying four-dimensional $N=1$ supersymmetric theories on a two-torus, with a magnetic field for a global $U(1)$ symmetry, and present evidence that Seiberg duality in four dimensions leads to an identification of different models of this type.

Figure 1

The brane system that realizes $N=1$ SQCD in type IIA string theory consists of $N_c$ $D4$-branes stretched in the $x^6$ direction between two differently oriented $NS5$-branes. $N_f$ $D6$-branes intersect the $D4$-branes at a particular $x^6$.

Figure 2

Turning on the $D$ field gives rise to a configuration in which the $D6$-branes are rotated in the $(x^6,x^7)$ plane.

Figure 3

The brane system that realizes the Seiberg dual of $N=1$ SQCD includes $N_f-N_c$ color $D4$-branes connecting two $NS5$-branes, and $N_f$ flavor $D4$-branes, each connecting the $N{S}^{\prime }$-brane to one of $N_f$ $D6$-branes.

Figure 4

The effect of the $D$ field on the magnetic brane system.

Figure 5

Quantization of the magnetic field on the two-torus corresponds in the IIB language to quantization of the angle that the $D5$-brane makes with the $x^1$ axis. Plotted is a $D5$-brane with $e=2$ in the double covering space of the torus (parallel dashed lines are identified). As the $D5$-brane wraps the $x^1$ circle once, it wraps the $x^2$ circle twice.

Figure 6

Two views of the brane configuration describing compact three-dimensional $N=2$ SQCD in a background global $U(1)$ under which half of the flavors have charge $+1$ and half $-1$. (a) The configuration in the (12) plane; the flavor branes wind once around the $x^2$ circle as they wind once around the $x^1$ one. (b) The configuration in the (67) plane; turning on the $D$ field for the flavor symmetry corresponds to a rotation of the flavor branes from $x^7$ to $x^6$.

Figure 7

Turning on $B$ and $D$ fields for a global symmetry that acts on the dynamical fields in the same way as the $U(1)$ part of the gauge group leads to the brane configuration (a), which is equivalent to the configuration (b) obtained by turning on an FI term and scalar field ${\varphi }_2(x^1)$ for that $U(1)$.

Figure 8

(a) At a generic point on the Coulomb branch, the $N_c$ $D3$-branes are separated in $x^2$. (b) At special points on the Coulomb branch, the $Q$’s and $\tilde{Q}$’s are localized at the same points in $x^1$.

Figure 9

The magnetic brane configuration for general $N_f$, $N_c$. Vacua are labeled by an integer $\widehat{N}$ that runs over the range described in the text. For each $\widehat{N}$ the theory splits into two decoupled theories at the intersections.