AdS5×S5

AdS5×S5 is the product of five-dimensional anti-de Sitter space and a five-sphere. It is a maximally supersymmetric solution of type IIB supergravity, preserving 32 supercharges. The background supports a self-dual five-form flux with N units, threading the S5 and AdS5 factors.

Geometrically, AdS5 and S5 have equal radii R. The isometry group is SO(2,4)×SO(6), corresponding to the four-dimensional conformal symmetry and the R-symmetry of the dual theory. In string theory terms, the radius is related to the string coupling g_s, the string length α', and the ’t Hooft coupling λ by R^4 = λ α'^2, with λ = g_YM^2 N = 4π g_s N. In the limit of large N and large λ, the curvature is small and classical Type IIB supergravity provides a good description.

AdS5×S5 is the canonical background for the AdS/CFT correspondence, relating Type IIB string theory on AdS5×S5 to N=4 SU(N) Super Yang-Mills theory in four dimensions. The conformal boundary of AdS5 hosts the four-dimensional CFT, and the spectrum of fluctuations around AdS5×S5 maps to local gauge-invariant operators in the CFT, with operator dimensions related to energies of string states. The correspondence offers a powerful framework to study strong coupling phenomena in the gauge theory and, conversely, to probe quantum gravity in a controlled setting.

The background is central to many developments, including tests of holography, the study of integrability in AdS/CFT, and various deformations and extensions such as orbifolds and flux modifications. AdS5×S5 thus serves as a foundational, highly symmetric laboratory for gauge/gravity duality.