Branes at conical singularities and holography

TL;DR

The paper classifies supersymmetric vacua of the form $\mathrm{adS}_{p+2} \times X_d$ by constraining $X_d$ to geometries that admit Killing spinors: Einstein–Sasaki, 3–Sasaki, weak $G_2$, and nearly Kahler. It ties supersymmetry to the holonomy of the cone $C(X)$ and shows that branes at conical singularities interpolate between AdS near-horizon geometries and Minkowski space times the cone, implying a holographic duality to worldvolume superconformal field theories that emerge as cone-brane theories. The authors exhaustively analyze how holonomy determines the amount of preserved supersymmetry, present a wide range of explicit geometric examples (including non-homogeneous cases), and map these geometries to corresponding superconformal algebras in three, four, and six dimensions. They also propose that the dual SCFTs can be described using (hyper-)Kähler quotients whose moduli spaces reproduce the cones, offering a concrete bridge between geometric engineering and holographic dualities. The work broadens the landscape of AdS/CFT by linking cone geometry to a rich set of dual SCFTs with varying amounts of supersymmetry and R-symmetry structures.

Abstract

For supergavrity solutions which are the product of an anti-de Sitter space with an Einstein space X, we study the relation between the amount of supersymmetry preserved and the geometry of X. Depending on the dimension and the amount of supersymmetry, the following geometries for X are possible, in addition to the maximally supersymmetric spherical geometry: Einstein-Sasaki in dimension 2k+1, 3-Sasaki in dimension 4k+3, 7-dimensional manifolds of weak G_2 holonomy and 6-dimensional nearly Kaehler manifolds. Many new examples of such manifolds are presented which are not homogeneous and have escaped earlier classification efforts. String or M theory in these vacua are conjectured to be dual to superconformal field theories. The brane solutions interpolating between these anti-de Sitter near-horizon geometries and the product of Minkowski space with a cone over X lead to an interpretation of the dual superconformal field theory as the world-volume theory for branes at a conical singularity (cone branes). We propose a description of those field theories whose associated cones are obtained by (hyper-)Kaehler quotients.