String Theory and M-Theory

12.5 Plane-wave space-times and their duals 677
The chiral superfields Ai and Bi each have R = 1/2. The chiral fermions
in these multiplets, which are coefficients of θα, therefore have R-charge
−1/2. Similarly, the chiral gluinos have R = 1. Thus, in the case of the
SU(M + N) the total contribution is
and in the case of SU(N)
K = 4N · (−1/2) + 2(M + N) · 1 = 2M
K = 4(M + N) · (−1/2) + 2N · 1 = −2M.
There are the required results. ✷
12.5 Plane-wave space-times and their duals
As was explained earlier, the tree-level approximation to the type IIB superstring
theory in an AdS5×S 5 background, with N units of R–R flux through
the five-sphere, corresponds to the planar approximation to the dual N = 4
super Yang–Mills theory with an SU(N) gauge group. Both sides of this
duality, even for the planar/tree-level approximation, are difficult. With
great effort, one can compute a few order in λ in the field theory and a few
orders in α ′ in the string theory. However, these are expansions in opposite
limits and cannot be compared.
Compared with superstring theory in flat space, there are two severe complications.
One is that the background geometry causes the world-sheet
theory to be a nonlinear system. Thus, solving classical string theory in
this geometry is mathematically the same as solving a complicated interacting
two-dimensional quantum field theory. The second difficulty is that
the background includes nonzero R–R gauge fields, specifically the self-dual
five-form field strength that threads the five-sphere with N units of flux.
The RNS formalism is not capable of handling R–R backgrounds, so one
is forced to use the GS formalism. This formalism is notoriously difficult
to quantize, especially if one wants to keep the symmetries of the geometry
manifest.
The type IIB plane-wave
There is a plane-wave limit of AdS5 × S 5 geometry that can be defined that
gives a space-time of intermediate complexity between AdS and flat spacetime,
which is also maximally supersymmetric. In this geometry it is more
difficult to define the duality, because there is not a well-defined dual gaugetheory.
Instead, one has to consider limits of appropriately defined families