Question

I am getting confused as I study the AdS/CFT correspondence, so I ask this question. CFT is given on the conformal boundary of AdS, which can be derived from Poincare coordinate patch to AdS. Would this mean that CFT has its AdS dual only on the subspace covered by this Poincare coordinate patch? Or does CFT has its AdS dual on the entire AdS space?

Answer 1

The CFT in the AdS/CFT correspondence is dual to the entire AdS space, including but not limited to the part covered by the Poincare coordinate patch. The Poincare patch's conformal boundary can be seen as a restriction of the CFT defined on the global conformal boundary.

Step-by-step explanation:

The AdS/CFT correspondence, also known as the Maldacena conjecture or gauge/gravity duality, posits a relationship between a type of ten-dimensional gravitational theory on anti-de Sitter space (AdS) and a Conformal Field Theory (CFT) on its nine-dimensional boundary.

The confusion arises from the use of Poincare coordinates, which only cover part of the AdS space. Importantly, the CFT defined on the conformal boundary of AdS is actually dual to the entire AdS space, not just the subspace covered by the Poincare patch. Both the global AdS space and the Poincare patch have their own conformal boundary where the dual CFT resides, but the CFT on the Poincare patch conformal boundary can be thought of as a restriction of the CFT on the global boundary. In this sense, the complete AdS space is necessary to fully describe the dual CFT.