### 3.2 The folded string:

In the folded case may be expressed in terms of an elliptic
integral
by substituting (using (20))
into (23) and performing some elementary transformations to find ()
where we only need to integrate over one quarter of the folded string due to symmetry considerations (see
Figure 2). Additionally, we have
The four Equations (21, 24, 28, 29) may then be used to eliminate the parameters of our solution ,
and . For this, rewrite (28) and (29) as ()
and use (24) to deduce
Then, the Virasoro constraint equation (21) and the identity yield the two folded
string equations
which implicitly define the sought after energy function upon further elimination of .
This is achieved by assuming an analytic behavior of and in the BMN type limit of large total
angular momentum , with held fixed
Plugging these expansions into (32) one can solve for the and iteratively. At leading order
(as it should from the dual gauge theory perspective) and is implicitly determined through
the “filling fraction”
The first non-trivial term in the energy is then expressed in terms of through
yielding a clear prediction for one-loop gauge theory. The higher order can then also be obtained. To
give a concrete example, let us evaluate for the first few orders in in the “half filled” case
:
which shows that the energy of the folded string configuration is smaller than the one of the closed
configuration of (26) for single winding .