Near-horizon geometries of supersymmetric branes

TL;DR

This work classifies near-horizon geometries of supersymmetric branes beyond the maximally symmetric AdS×S cases by framing the transverse space as a metric cone $C(M)$ over a compact Einstein manifold $M$. Supersymmetry imposes that $M$ admit Killing spinors, equivalently that $C(M)$ have parallel spinors, which is controlled by Berger holonomy; this yields a dictionary linking cone holonomy to geometric structures on $M$ (e.g., Sasaki–Einstein, 3–Sasaki, nearly Kähler, weak $G_2$, Spin(7)). Consequently, the allowed near-horizon geometries are $AdS_{p+2} \times M$ with $M$ drawn from well-characterized holonomy classes, giving explicit seven-, six-, and five-dimensional examples and abundant infinite families (Aloff–Wallach, toric Sasaki–Einstein, etc.). The results provide a geometric scaffold for potential holographic duals to less supersymmetric field theories and motivate further exploration of quotients, toric constructions, and duality effects on these generalized horizons.

Abstract

This is the written version of my talk at SUSY '98. It presents a geometric characterisation of the allowed near-horizon geometries of supersymmetric branes. We focus primarily on the M2-brane, but results for other branes (e.g., the D3-brane) are also presented. Some new examples are discussed.