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3.2 The folded string: q < 1

In the folded case J1 may be expressed in terms of an elliptic integral5 by substituting (using (20View Equation))
dy ds = --- V~ -------2--- (27) w21 q- sin y
into (23View Equation) and performing some elementary transformations to find (q = sin2 y0)
V~ -- V~ -- cw1 integral y0 cos2y 2 cw1 2 J1 = ------4 dy ---- V~ ---2--------2-- = --------E(sin y0) , (28) 2p 0 w21 sin y0 - sin y p w21
where we only need to integrate over one quarter of the folded string due to symmetry considerations (see Figure 2View Image). Additionally, we have
integral integral 2p y0 dy 4 2 2p = ds = 4 --- V~ ---2--------2---= w--K(sin y0) . (29) 0 0 w21 sin y0- sin y 21
The four Equations (21View Equation, 24View Equation, 28View Equation, 29View Equation) may then be used to eliminate the parameters of our solution k, w1 and w 2. For this, rewrite (28View Equation) and (29View Equation) as (k = E/V ~ c-)
w p J k p E --1-= - V~ -----1- , ----= - V~ ------- , (30) w21 2 c E(q) w21 2 c K(q)
and use (24View Equation) to deduce
w p J --2-= - V~ ---------2-----. (31) w21 2 c K(q) - E(q)
Then, the Virasoro constraint equation (21View Equation) and the identity 1 = (w22 - w12)/w212 yield the two folded string equations
4 qc E2 J 2 4c J 2 J 2 --2--= -----2- --1-2-, -2-= -------2------2 - ---1-2 , (32) p K(q) E(q) p (K(q) - E(q)) E(q)
which implicitly define the sought after energy function E = E(J1, J2) upon further elimination of q. This is achieved by assuming an analytic behavior of q and E in the BMN type limit of large total angular momentum J := J1 + J2 --> oo, with 2 c/J held fixed
2 q = q0 +-c-q1 + c--q2 + ... J 2 J 4 ( -c- c2- ) E = J E0 + J 2 E1 + J4 E2 + ... . (33)
Plugging these expansions into (32View Equation) one can solve for the qi and Ei iteratively. At leading order E0 = 1 (as it should from the dual gauge theory perspective) and q0 is implicitly determined through the “filling fraction” J2/J
J2- -E(q0) J = 1 - K(q0) . (34)
The first non-trivial term in the energy is then expressed in terms of q0 through
( ) 2-- E1 = p2 K(q0) E(q0) - (1- q0) K(q0) (35)
yielding a clear prediction for one-loop gauge theory. The higher order E i can then also be obtained. To give a concrete example, let us evaluate E for the first few orders in 2 c/J in the “half filled” case J1 = J2:
2 E = 2J1 (1 + 0.71 -c--- 1.69 c---+ ...) (36) 8 J12 32 J14
which shows that the energy of the folded string configuration is smaller than the one of the closed configuration of (26View Equation) for single winding n = 1.
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