Posted: May 26, 2011 11:57 am
by twistor59
Post #9B The AdS/CFT Correspondence

The AdS/CFT Correspondence

Dp-branes, being massive objects, have a back-reaction on the spacetime geometry. When N is large, N coincident branes will leave a significant imprint. If Tμν is the energy momentum tensor of the D-brane source, the Einstein equations lead to the geometry:

Rμν-(1/2)Rgμν = 8π G Tμν

If gs is the closed string coupling constant, then the Newton constant G is proportional to gs2 and, since the brane tension goes as 1/ gs, the brane energy momentum tensor is proportional to (N/ gs). Hence the Einstein tensor Rμν-(1/2)Rgμν is proportional to N gs = λ. For small λ there is a negligible backreaction, and the we have string theory in roughly flat spacetime.

As N increases, a “throat” appears:


AdSThroat.jpg (14.52 KiB) Viewed 2727 times

In the AdS/CFT scenario, we’re interested in the case where N D3-branes are present. The D3 branes are thought of as filling our perceived “large” 3 spatial dimensions, and of course sweep out a 4 dimensional world volume. If we decompose the 10 dimensional spacetime into a 3+1 Minkowski signature part and an ℝ6 part with coordinates y1, ..y6, then the distance from the brane is just r2 = (y1)2+..+(y6)2

Far from the branes (large r), the metric is approximately flat 9+1 space. Near the branes, (low r), it looks like AdS5xS5. Now somehow (I don’t really understand how this works), “taking the low energy limit”, the throat (which has the geometry of AdS5xS5) can be considered in its own right, i.e. as if it was decoupled from the rest of the spacetime. I suspect it might mean that we’re talking about low energy in the sense that we don’t have enough energy to get far from the brane, but I’m not sure.

Anyway, taking a large λ scenario and taking the low energy limit puts us in the AdS5xS5 geometry. Taking a small λ scenario (flat spacetime, no throat) gives us a 4D SYM theory.


AdSCFTCorr1.jpg (36.85 KiB) Viewed 2727 times

Comparing these two low energy limits of the theory, Maldacena conjectured that 4 dimensional 𝒩=4 SYM is equivalent to type IIB string theory on AdS5xS5. This is the famous AdS/CFT correspondence and is what led Ed Witten to his work on twistor string theory (which I’ll get to shortly).