### 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.