Abstract
We revisit D3-branes at toric CY3 singularities with orientifolds and their description in terms of dimer models. We classify orientifold actions on the dimer through smooth involutions of the torus. In particular, we describe new orientifold projections related to maps on the dimer without fixed points, leading to Klein bottles. These new orientifolds lead to novel mathcal{N} = 1 SCFT’s that resemble, in many aspects, non-orientifolded theories. For instance, we recover the presence of fractional branes and some of them trigger a cascading RG-flow à la Klebanov-Strassler. The remaining involutions lead to non-supersymmetric setups, thus exhausting the possible orientifolds on dimers.
Highlights
Only on the local features of the compactification [16,17,18]
While direct construction in string theory is in practice only feasible for some orbifold theories, they may be constructed directly in the dimer model [37] by identifying gauge groups and fields according to a suitable involution of the graph and possibly assigning some signs to the fixed loci in the dimer, corresponding to the different choices in the orientifold projection
It is clear that not all dimer models have the required symmetry, and in section 5 we provide a necessary condition for a given toric CY3 to admit a glide reflection directly from its toric diagram
Summary
A large class of quiver gauge theories can be engineered in String Theory by placing a bunch of D3-branes at the tip of a toric CY3 singularity. The dictionary between the bipartite graph elements of the brane tiling and the corresponding gauge theory is presented in table 1 This bipartite graph is physically realized by double T-duality on the D3-brane setup as a 5-brane web (or brane tiling) on T2, where D5-branes are suspended between NS5-branes, or rather the NS5-brane wraps a holomorphic cycle in the T2 wrapped by the D5. The quiver, dimer and toric diagrams for the conifold are shown in figure 1 Another ingredient that we will extensively use are the Zig-Zag paths (ZZPs)2 [28, 41,42,43]. The ZZPs of the conifold dimer are shown in figure 1b These form non self-intersecting closed loops on the torus with non-trivial homology around the two fundamental cycles. They are in correspondence with legs in the (p, q) web diagram obtained as the dual graph to the toric diagram, their (p, q) labels being exactly the homology charges of the ZZPs
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