String Theory and M-Theory

12.3 The AdS/CFT correspondence 643
of the operators generate the super-Poincaré subgroup, and the rest generate
other conformal transformations. In particular, 16 of the fermionic
operators generate linearly realized Poincaré supersymmetries and the
other 16 generate superconformal symmetries.
• The common radius R of the AdS5 and S 5 geometries is related to the ‘t
Hooft parameter λ = g 2 YM N of the gauge theory by R = λ1/4 ℓs.
Duality for M-branes
There are similar AdS/CFT conjectures for the two M-theory cases for which
extremal black-brane solutions were constructed in Section 12.1. However,
they have been explored in much less detail than the D3-brane case. There
are at least three reasons for this: (1) computations are much more difficult
in M-theory than in type IIB superstring theory; (2) the dual conformal field
theories are much more elusive than the N = 4 super Yang–Mills theory;
(3) there is great interest in using AdS/CFT dualities to learn more about
four-dimensional gauge theories.
The M2-brane conjecture
A stack of M2-branes has an AdS4 × S 7 near-horizon geometry, and Mtheory
for this geometry (with N units of ⋆F4 flux through the sphere) is
dual to a conformally invariant SU(N) gauge theory in three dimensions.
One significant difference from the type IIB superstring example, is that
the M-theory background does not contain a dilaton field, and therefore
there is no weak-coupling limit. Correspondingly, the three-dimensional
conformal field theory does not have an adjustable coupling constant, and
it is necessarily strongly coupled. As a result, it does not need to have a
classical Lagrangian description. In fact, there does not appear to be one.
Therefore, this three-dimensional CFT is much more difficult to analyze
than N = 4 super Yang–Mills theory.
CFT for the M2-brane case
One way of thinking about the three-dimensional CFT is as follows. The
low-energy effective world-volume theory on a collection of N coincident
D2-branes of type IIA superstring theory is a maximally supersymmetric
U(N) Yang–Mills theory in three dimensions. This theory is not conformal
because the Yang–Mills coupling in three dimensions is dimensionful and
introduces a scale. Recall that the type IIA coupling constant is proportional
to the radius of a circular eleventh dimension. When this coupling becomes
large, the gauge-theory coupling constant also becomes large. In view of