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3 Spinning String Solutions

We shall now look for a solution of the 5 AdS5 × S string with S1 = S2 = 0 and J3 = 0, i.e. a string configuration rotating in the 3 S within 5 S and evolving only with the time coordinate of the AdS5 space-time. This was first discussed by Frolov and Tseytlin [6163]. For this let us consider the following ansatz in the global coordinates of (5View Equation)
t = k t r = 0 g = p2 f3 = 0 f1 = w1 t f2 = w2 t y = y(s) , (17)
with the constant parameters k, w1,2 and the profile y(s) to be determined. The string action (4View Equation) then becomes
V~ -- integral integral 2p [ ] I = - --c- dt ds k2 + y'2 - cos2y w12 - sin2y w22 , (18) 4p 0
leading to an equation of motion for y(s)
'' 2 2 y + siny cosy (w2 - w1 ) = 0. (19)
We define w212 := w22 - w12, which we take to be positive without loss of generality. Integrating this equation once yields the “string pendulum” equation
V~ ---------- dy- 2 ds = w21 q- sin y , (20)
where we have introduced an integration constant q. Clearly, there are two qualitatively distinct situations for q larger or smaller than one: For q < 1 we have a folded string with y ranging from - y0 to y0, where q = sin2 y0, and y'= 0 at the turning points where the string folds back onto itself (see Figure 2View Image).
View Image

Figure 2: The folded string extending from y = -y0 to y = y0, where sin2 y0 := q.
If, however, q > 1 then ' y never vanishes and we have a circular string configuration embracing a full circle on the 3 S: The energy stored in the system is large enough to let the pendulum overturn.

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

k2 - w 2 q = ------1-- (w212 /= 0), (21) w212
relating the integration constant q to the parameters of our ansatz4. Our goal is to compute the energy E of these two solutions as a function of the commuting angular momenta J1 and J2 on the three sphere within S5. Upon using Equations (8View Equation) and inserting the ansatz (17View Equation) these are given by
V~ -- E = c , (22) V~ -- integral 2p V~ -- integral 2p J1 = c w1 ds-cos2 y(s) , J2 = c w2 ds sin2 y(s) . (23) 0 2p 0 2p
From this we learn that
V~ -- J1 J2 c = ---+ ---. (24) w1 w2

 3.1 The special case: w1 = w2
 3.2 The folded string: q < 1
 3.3 The circular string: q > 1
 3.4 Further Developments

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