Orientifolding the conifold

TL;DR

This work develops a framework to study N=1 supersymmetric field theories from type IIB D3-branes at orientifolds of non-orbifold singularities, notably the conifold and suspended pinch point, by leveraging T-duality with type IIA brane setups. It introduces a constructive approach: since direct IIB orientifold techniques are hard for non-orbifolds, the authors generate models through partial resolutions of orientifolds of orbifold spaces, translating geometric blow-ups into Higgsing in the D3-brane gauge theories. They demonstrate a consistent map between IIB orientifolds of $xy=zw$ and $xy=zw^2$ and their IIA duals, including detailed tadpole checks and spectra, and extend the analysis to orientifolds of the conifold via ${f C^3/({f Z_2 imes Z_3})}$. The results establish a coherent, toric-geometric view of chiral theories from branes at non-orbifold singularities and reveal how different branches (Higgsing, blow-ups) connect diverse chiral models across dual descriptions. Overall, the paper provides a unified method to construct and validate orientifolds of non-orbifold singularities and clarifies their IIA/IIB dual relationships and geometric interpretations.

Abstract

In this paper we study the N=1 supersymmetric field theories realized on the world-volume of type IIB D3-branes sitting at orientifolds of non-orbifold singularities (conifold and generalizations). Several chiral models belong to this family of theories. These field theories have a T-dual realization in terms of type IIA configurations of relatively rotated NS fivebranes, D4-branes and orientifold six-planes, with a compact $x^6$ direction, along which the D4-branes have finite extent. We compute the spectrum on the D3-branes directly in the type IIB picture and match the resulting field theories with those obtained in the type IIA setup, thus providing a non-trivial check of this T-duality. Since the usual techniques to compute the spectrum of the model and check the cancellation of tadpoles, cannot be applied to the case orientifolds of non-orbifold singularities, we use a different approach, and construct the models by partially blowing-up orientifolds of C^3/(Z_2 x Z_2) and C^3/(Z_2 x Z_3) orbifolds.

Figures (7)

• Figure 1: Examples of ${\cal N}=1$ elliptic models. Figure a) is T-dual to a set of D3-branes at a conifold singularity $xy=zw$. Figure b) is T-dual to D3-branes at the 'suspended pinch point' singularity, $xy=zw^2$.
• Figure 2: The toric diagrams corresponding to a) the $\bf C^3/(\bf Z_2\times\bf Z_2)$ orbifold, $xyz=w^2$, b) the suspended pinchpoint singularity, $xy=zw^2$, c) the conifold, $xy=zw$, and d) the $\bf Z_2$ orbifold, $xy=w^2$.
• Figure 3: Non-chiral orientifolds of the IIA brane configuration T-dual to the suspended pinch point singularity.
• Figure 4: Chiral orientifolds of the IIA brane configuration T-dual to the suspended pinch point singularity. Note that O6$'$-planes extend along 7 (but not 89) and NS$'$-branes extend along 89 (but not 7).
• Figure 5: The introduction of O6$'$-planes in the IIA brane configuration T-dual to the conifold singularity. The picture appears confusing since we have tried to show too many dimensions in it. The only point to keep in mind is that a NS$'$-brane does not split an O6$'$-plane in two halves.