## 3 Spinning String Solutions

We shall now look for a solution of the string with and , i.e. a string configuration rotating in the within and evolving only with the time coordinate of the space-time. This was first discussed by Frolov and Tseytlin [6163]. For this let us consider the following ansatz in the global coordinates of (5)
with the constant parameters , and the profile to be determined. The string action (4) then becomes
leading to an equation of motion for
We define , which we take to be positive without loss of generality. Integrating this equation once yields the “string pendulum” equation
where we have introduced an integration constant . Clearly, there are two qualitatively distinct situations for larger or smaller than one: For we have a folded string with ranging from to , where , and at the turning points where the string folds back onto itself (see Figure 2).
If, however, then never vanishes and we have a circular string configuration embracing a full circle on the : The energy stored in the system is large enough to let the pendulum overturn.

In addition, we have to fulfill the Virasoro constraint equations (6). One checks that our ansatz (17) satisfies the first constraint of (6), whereas the second constraint equation leads to

relating the integration constant to the parameters of our ansatz. Our goal is to compute the energy of these two solutions as a function of the commuting angular momenta and on the three sphere within . Upon using Equations (8) and inserting the ansatz (17) these are given by
From this we learn that