AdS5

AdS5, short for anti-de Sitter space in five dimensions, is a maximally symmetric Lorentzian manifold with constant negative curvature. It has five dimensions and a characteristic length scale L, the AdS radius. A common description embeds it as a hyperboloid in a flat space with signature (-,-,+,+,+,+): -X0^2 - X5^2 + Σ_{i=1}^4 Xi^2 = -L^2. This embedding makes its isometry group explicit as SO(2,4). The global spacetime contains a timelike direction and a conformal boundary at infinity, topologically S^1 x S^3; in a Poincaré patch, the boundary is four-dimensional Minkowski space.

In a convenient coordinate system, the metric is often written in Poincaré coordinates as ds^2 = L^2 (dz^2 + η_{μν} dx^μ dx^ν)/z^2, with z > 0. Global coordinates provide an alternative description that covers full AdS5. The geometry features a single timelike dimension and a boundary at infinity where a conformal field theory can be defined.

AdS5 frequently appears in the product space AdS5 x S^5, the near-horizon geometry of a large number of D3-branes in type IIB string theory. This background is central to the AdS/CFT correspondence, which posits a duality between string or gravitational theory in AdS5 x S^5 and a four-dimensional conformal field theory on the boundary. The dual theory is N=4 SU(N) super Yang-Mills theory in four dimensions in the large-N limit, with the SO(2,4) isometry of AdS5 matching the CFT’s conformal symmetry and the S^5 factor encoding R-symmetry.

In the holographic dictionary, bulk fields correspond to boundary operators, and bulk gravitational dynamics yield correlation functions of the CFT. AdS5 thus provides a framework for studying strongly coupled gauge theories via classical gravity in a higher-dimensional spacetime.