Non-spherical horizons. I
The authors formulate an extension of Maldacena's AdS/CFT conjectures to the case of branes located at singular points in the ambient transverse space. For singularities which occur at finite distance in the moduli space of M or \(F\) theory models with spacetime-filling branes, the conjectures identify the worldvolume theory on the \(p\)-branes with a compactification of M or IIB theory on \(\text{AdS}_{p+2}\times H^{D-p-2}.\) Extending the AdS/CFT correspondence to describe branes at singularities is required for a complete description of the duality. Since configurations with branes at singularities occur at a finite distance in moduli space, the modifications to the extreme low-energy theory at these points should be captured by an appropriate modification of the AdS compactification. It is shown how the singularity determines the horizon \(H\), and demonstrate the relationship between global symmetries on the worldvolume and gauge symmetries in the AdS model. The modification in question is to replace the spherical horizon with a (non-spherical) horizon manifold determined by the local geometry at the singular point. As a first application, some singularities relevant to the D3-branes required in four-dimensional \(F\)-theory are studied. For these the low-energy field theory on the worldvolume is explicitly derived and its properties are compared to predictions from the dual AdS model. In particular, the baryon spectra of the models and the fate of the abelian factors in the gauge group are examined. The plan of the paper is as follows. The authors review the scaling limit in detail and explain how it must be modified in the present context. Some generic features of the singularities are also discussed. A geometric characterization of the amount of supersymmetry carried by the limiting conformal field theories is found. Using existing mathematical theorems, it is shown that the \(R\)-symmetry group of the conformal field theory acts by isometries on the corresponding horizon manifold \(H.\) This discussion is closely related to the differential geometry of \(H.\) A very brief discussion of these theories for M2-branes and M5-branes is presented. In the case of M2-branes, a few examples are merely given. A detailed classification of holonomy groups for M5-branes by considering the component \(\text{Hol}^0\) of the holonomy group of the cone \(C(H)\) is obtained. An extensive analysis of the singularities occurring in D3-brane theories is given, focusing on a few particular classes which exhibit many of the general features. The tool is algebraic geometry, and a nice correspondence with some of the results obtained earlier via differential geometry is found. A particular singularity -- a \(\mathbb Z_2\times\mathbb Z_2\) orbifold -- in considerable detail is studied. This theory has been studied before in the literature, but here the focus is on giving more detail and in particular in extracting geometric interpretations for field theory parameters and moduli. This is then applied in a number of new examples, all derived in some way from the \(\mathbb Z_2\times\mathbb Z_2\) orbifold. From the Contents: The scaling limit. Supersymmetry and holonomy. Detailed classification of holonomy groups. M2-branes and M5-branes. M2-branes at orbifolds. M2-branes and exotic spheres. Gorenstein canonical singularities in three complex dimensions. Singularities resolved by only one blowup. Toric singularities. Partial resolution of the \(\mathbb Z_2\times\mathbb Z_2.\) Complex cones over del Pezzo surfaces. Horizon manifold. Orbifolds. An orbifold model. Orbifolds and the AdS/CFT correspondence. (Non-)Isolated singularities. Branes at a \(\mathbb Z_2\times\mathbb Z_2\) singularities. The AdS compactification. New examples in four dimensions. The \(\mathbb Z_2\) quotient singularity. The conifold. The suspended pinch point singularity. The complex cone over \(\mathbb F_0.\) Discussion. Appendix: Sasaki and \(G_2\) structures; \(D\)-term equation and moduli spaces. References.
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