I'm trying to understand the elements of the AdS-CFT correspondence. The definition I have in mind is a correspondence between:
- (A) a Super String Theory of $D_p$-branes and closed strings in a $10$d flat bulk, with open strings living on the branes
- (B) a $II$B Supergravity Theory where closed strings live in a $10$d bulk that is asymptotically flat and tends to $AdS_5 \times S^5$ near the origin
and the two are linked because the curved spacetime of theory (B) is the effect that the $D_p$-branes of theory (A) would have on a flat spacetime. Then if one chooses $p=3$ fun things happen in the low-energy limit.
My question is, why is (A) called a SST and (B) is called a SUGRA? I mean, they both have closed strings. Is the only difference between the two theories (in general, not necessarily applied to the AdS-CFT correspondence) the absence of open strings in SUGRA?