String Theory and M-Theory

638 Gauge theory/string theory dualities
sionless, it follows that the Lagrangian has dimension [L] = −1. From
the explicit form of the Lagrangian in Eq. (12.54), it then follows that
[R] = [g] = −3, [X i ] = −1. Also, [r] = [b] = −1 and [v] = −2. It follows
that g m v 2n+2 /r 3m+4n has dimension −4 as required. Therefore, these
dimensions lead to the expansion appearing in Table (12.76). Dimensional
analysis determines the entire v and r dependence of the effective actions at
each order in the perturbation expansion. Only the numerical coefficients
need to be computed by evaluating Feynman diagrams. ✷
12.3 The AdS/CFT correspondence
The basic idea of the AdS/CFT duality and its generalizations is that string
theory or M-theory in the near-horizon geometry of a collection of coincident
D-branes or M-branes is equivalent to the low-energy world-volume
theory of the corresponding branes. This section explains the AdS/CFT
correspondence.
The D3-brane case
The conjecture
The AdS/CFT conjecture (for the case of D3-branes) is that type IIB superstring
theory in the AdS5 × S 5 background described in Section 12.1 is
dual to N = 4, D = 4 super Yang–Mills theory with gauge group SU(N).
This string theory background corresponds to the ground state of the gauge
theory, and excitations and interactions in one description correspond to
excitations and interactions in the dual description.
D-brane world-volume theories were studied in considerable detail in Chapter
6. In the case of type II superstring theories we learned that the worldvolume
theory of N coincident BPS D-branes is a maximally supersymmetric
U(N) gauge theory. The formulas become complicated when terms that are
higher order in α ′ or nontrivial background fields are taken into account.
However, in the absence of background fields and at lowest order in α ′ , the
result is very simple: the low-energy effective action on the world volume
of N coincident Dp-branes is given by the dimensional reduction of supersymmetric
U(N) gauge theory in ten dimensions to p + 1 dimensions. This
theory is all that is required for the analysis that follows. The U(1) subgroup
of U(N) decouples as a free theory and does not participate in the