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
.