AdS/CFT Correspondence - Maldacena's Example

Maldacena's Example

Comparisons between a stack of D3-branes in type IIB string theory and the extremal charged 3-brane metric background were carried out by Klebanov and later sharpened by Maldacena. One starts with the observation that a stack of N D3-branes in type IIB string theory has massless brane fields residing on it. With respect to the brane, they form Yang–Mills supermultiplets transforming under N = 4 SUSY in 3+1D. The vector hypermultiplets form a gauge group U(N) ≅ SU(N) × U(1). This isn't quite a conformal field theory, even though it runs to one in the infrared once gravitation and string dynamics decouple. In the infrared, the U(1) hypermultiplet decouples, but the SU(N) hypermultiplets remain interacting as the beta function is zero. The metric background is given by an extremal 3-brane black hole. The event horizon is infinitely far away; the distance to it diverges logarithmically. The near horizon geometry is approximately AdS₅ × S5 with the approximation becoming more and more exact closer to the horizon. Now, take the scaling limit as the string scale goes to zero with the string coupling kept fixed. All the string and gravitational dynamics decouple, and the U(1) hypermultiplet too. We are left with a bona fide N = 4 superconformal field theory. If we take the limit in which we are always in the near horizon region, the geometry becomes exactly AdS₅ × S5. A D3-brane has a self-dual charge under the self-dual NS 5-form flux. A stack of N of them gives rise to an integral flux of N over S5