N=1 Type IIA brane configurations, Chirality and T-duality

TL;DR

This work establishes a detailed T-duality bridge between four-dimensional ${\cal N}=1$ theories realized in type IIA brane setups (NS-branes, D4-branes, D6$'$-branes, and O6$'$-planes) and type IIB constructions of D3-branes at orientifolds of $\mathbb{C}^2/\mathbb{Z}_N$. By explicitly constructing the IIB orientifolds and matching spectra, the authors show that several exotic IIA phenomena—such as flavour doubling, D6$'$- and O6$'$-splitting, and cosmological-constant effects—have natural, standard realizations in the IIB picture via D7-branes and orbifold projections. The analysis covers both odd and even $N$, with and without NS-branes on orientifold planes, and extends to more general D7-brane structures, with twisted-tadpole cancellation on the IIB side corresponding to anomaly cancellation and RR-charge conservation in the IIA frame. The results suggest a unified, holographically accessible framework for studying these chiral theories and their large-$N$ dynamics, connecting diverse string constructions through a common orientifold/duality language.

Abstract

We consider four-dimensional N=1 field theories realized by type IIA brane configurations of NS-branes and D4-branes, in the presence of orientifold six-planes and D6-branes. These configurations are known to present interesting effects associated to the appearance of chiral symmetries and chiral matter in the four-dimensional field theory. We center on models with one compact direction (elliptic models) and show that, under T-duality, the configurations are mapped to a set of type IIB D3-branes probing N=1 orientifolds of C^2/Z_N singularities. We explicitly construct these orientifolds, and show the field theories on the D3-brane probes indeed reproduces the field theories constructed using the IIA brane configurations. This T-duality map allows to understand the type IIB realization of several exotic brane dynamics effects on the type IIA side: Flavour doubling, the splitting of D6-branes and O6-planes in crossing a NS-brane and the effect of a non-zero type IIA cosmological constant turn out to have surprisingly standard type IIB counterparts.

Figures (7)

• Figure 1: Type IIA brane configuration corresponding to an ${\cal N}=2$ elliptic model with a set of D6$'$-branes introducing ${\cal N}=1$ chiral matter. In this picture, the crosses represent NS-branes and the thin vertical lines represent half D6$'$-branes. The D4-branes stretch between the NS-branes along the direction 6. For the sake of clarity, the D4-branes have been drawn at the origin in the Coulomb branch. This configuration presents the phenomenon of 'flavour doubling': each half D6$'$-branes provides two chiral fundamental flavours, one for each gauge factor bounded by the corresponding NS brane.
• Figure 2: Typical ${\cal N}=1$ brane configurations obtained by introducing O$6'$-planes in ${\cal N}=2$ elliptic models with odd number of NS-branes. Notice the fork configuration, where the O$6'$-plane changes sign when it crosses the NS-brane. The change of sign is depicted by a change of color. The eight half D6$'$-branes stuck on the negatively charged half of the orientifold are not shown in the picture.
• Figure 3: The ${\cal N}=1$ brane configurations with O$6'$-planes and an even number of NS-branes, in the case in which no NS-brane coincides with the orientifold planes.
• Figure 4: The ${\cal N}=1$ brane configurations with O$6'$-planes and an even number of NS branes, in the case with NS branes coinciding with the orientifold planes.
• Figure 5: The IIA brane configuration corresponding to the odd order IIB orientifold with a more general distribution of D7-branes. For clarity, in this picture we have depicted one copy of the interval and not its image by the orientifold $\bf Z_2$ symmetry. NS branes are indicated by crosses, D4 branes by horizontal lines, the O$6'$-planes by vertical dashed lines, and the half-D$6'$-branes by vertical continuous lines, with their number indicated by braces.