AdS4

AdS4, short for four-dimensional anti-de Sitter space, is a maximally symmetric spacetime with constant negative curvature. It serves as a fundamental setting in theoretical physics, particularly in the study of quantum gravity, holography, and strongly coupled quantum field theories.

Geometrically, AdS4 is a four-dimensional manifold with negative curvature characterized by a single length scale, the AdS radius L. In Poincaré coordinates, its metric can be written as ds^2 = L^2 (dz^2 + dx^2 + dy^2 − dt^2) / z^2, with z > 0. Global coordinates provide an alternative description that highlights causal structure. AdS4 is an Einstein space with R_ab = −(3/L^2) g_ab, and the cosmological constant is Λ = −3/L^2. The isometry group is SO(3,2), reflecting maximal symmetry.

A defining feature of AdS4 is its conformal boundary, located at z → 0 in Poincaré coordinates. This boundary is three-dimensional and conformally equivalent to Minkowski space, making AdS4 a natural arena for the AdS/CFT correspondence: a quantum gravity or string theory in AdS4 is holographically related to a conformal field theory living on the 3D boundary.

In string and M-theory, AdS4 appears as near-horizon geometries of brane configurations. Notably, AdS4 × S^7 arises in eleven-dimensional supergravity as the near-horizon limit of M2-branes, while AdS4 × CP^3 appears in type IIA constructions related to the ABJM theory, a 3D N=6 superconformal Chern-Simons-matter theory. These realizations underpin holographic studies of strongly coupled quantum systems in three dimensions.