### 3.3 The circular string:

For the string does not fold back onto itself and extends around the full circle of as
runs from to . The initial expression for (28) now changes only marginally to
Elementary transformations yield
Analogously (29) now becomes
The corresponding relations to (30) and (31) then take the form
From these, one deduces in complete analogy to (32) the two circular string equations
which encode the energy relation upon elimination of . In order to do this, we again
make an analytic ansatz in for and as we did in (33). This yields the following implicit
expression for in terms of the filling fraction
The first two energy terms in the expansion then take the form
which again gives a clean prediction for the dual gauge theory scaling dimensions at one-loop.
In Figure 3, we have plotted the energies of the folded and circular string solutions against the filling
fraction . As expected from the “string pendulum” picture of (20), the folded string solution has
lower energy for fixed filling fraction : The pendulum does not perform a full turn, but oscillates back
and forth. Note that the folded string approaches in the limit , which is the rotating
point-particle solution of Section 2.1. The quantum fluctuations about it correspond to the
plane-wave string domain. Note also that the simplest circular string solution with
discussed in Section 3.1 is the minimum () of the full circular string solution for
.

These classical string energies will be reproduced in a dual one-loop gauge theory computation in
Section 4.