AdS7

AdS7, short for seven-dimensional anti-de Sitter space, is a maximally symmetric Lorentzian manifold with constant negative curvature. It has seven dimensions and an isometry group SO(6,2). As a solution of Einstein’s equations with negative cosmological constant, AdS7 serves as a gravitational background in high-energy theory and quantum gravity. The radius of curvature is L; the Ricci tensor satisfies R_{μν} = -(d-1)/L^2 g_{μν} with d=7, and the scalar curvature is R = -d(d-1)/L^2 = -42/L^2.

In Poincaré coordinates, the metric can be written ds^2 = L^2 (dz^2 + η_{μν} dx^μ dx^ν)/z^2, with μ,ν = 0,...,5 and z>0. A global coordinate system covers the entire space and reveals a conformal boundary at z→0, which has six dimensions. This boundary structure underpins the role of AdS7 in holographic dualities, where gravity in the bulk is related to a conformal field theory on the boundary.

A prominent context for AdS7 is in M-theory, where the near-horizon geometry of N coincident M5-branes is AdS7 × S^4 with N units of four-form flux on S^4. The dual six-dimensional theory is the (2,0) superconformal field theory, whose large-N dynamics exhibit degrees of freedom that scale as N^3. This AdS7 background thus occupies a central place in the AdS/CFT correspondence and in the study of six-dimensional conformal theories and M-theory compactifications. See also AdS/CFT, anti-de Sitter space, M-theory, M5-branes, and the (2,0) theory.