3.1 The special case: 
It is instructive to first discuss the particularly simple special case of a
circular string, where
and (21) does not apply. This will turn out to be a limiting case of the
scenario. For
the equation of motion for
(19) immediately yields
with
being the integer winding number of this circular string
. In this case,
the Virasoro constraints (6) yield
. A little bit of algebra quickly shows that the energy
may be reexpressed as a function of
and reads
which is analytic in
! This amounts to an all-loop prediction for the dual gauge theory scaling
dimension in the BMN limit
with
fixed, quite similar to the result for the plane-wave
superstring discussed above.
Let us now discuss the folded and circular string solutions in turn.