Ad$S_5$ with two boundaries and holography of $\cal{N}=$4 SYM theory

According to the AdS/CFT correspondence, the ${\cal N}=4$ supersymmetric Yang-Mills (SYM) theory is studied through its gravity dual whose configuration has two boundaries at the opposite sides of the fifth coordinate. At these boundaries, in general, the four dimensional (4D) metrics are different, then we expect different properties for the theory living in two boundaries. It is studied how these two different properties of the theory are obtained from a common 5D bulk manifold in terms of the holographic method. We could show in our case that the two theories on the different boundaries are described by the Ad$S_5$, which is separated into two regions by a domain wall. This domain wall is given by a special point of the fifth coordinate. Some issues of the entanglement entropy related to this bulk configuration are also discussed.

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