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3.3 The circular string: q > 1

For q > 1 the string does not fold back onto itself and extends around the full circle of y (- [0,2p] as s runs from 0 to 2p. The initial expression for (28View Equation) now changes only marginally to
V~ -- integral 2p 2 J1 = --c-w1 dy --- V~ -cos-y-----. (37) 2p 0 w21 q- sin2 y
Elementary transformations yield
V~ - [ ] J = 2--c-w1- 1---q K(q -1) + V~ q-E(q -1) . (38) 1 p w21 V~ q-
Analogously (29View Equation) now becomes
integral 2p integral p/2 dy 4 2p = ds = 4 ---- V~ --------------- = ---- V~ -K(q - 1) . (39) 0 0 w21 sin2 y0 - sin2y w21 q
The corresponding relations to (30View Equation) and (31View Equation) then take the form
V~ -- V~ -- -1-- p-----q--- -w1- --p-- ----------q-J1--------- w = 2 K(q -1) , w = V~ - (1 - q) K(q -1) + qE(q) , 21 21 2 c -w2-= - V~ p- V~ --------J2---------- . (40) w21 2 c q[K(q - 1)- E(q -1)]
From these, one deduces in complete analogy to (32View Equation) the two circular string equations
4c- --E2--- ---------J12--------- p2 = K(q-1)2 - [(1- q)K(q-1)+qE(q-1)]2 4 qc 2 2 2 -----= [K(q-1J)-2 E(q-1)]2-- [(1-q)K(qq- 1J)1+qE(q-1)]2 , (41) p2
which encode the energy relation E = E(J1, J2) upon elimination of q. In order to do this, we again make an analytic ansatz in c/J 2 for q and E as we did in (33View Equation). This yields the following implicit expression for q0 in terms of the filling fraction J2/J
J2 ( E(q -1)) ---= q0 1- ---0-1-- , (42) J K(q 0 )
The first two energy terms in the c/J2 expansion then take the form
-2- -1 -1 E0 = 1 , E1 = p2 E(q0 )K(q 0 ), (43)
which again gives a clean prediction for the dual gauge theory scaling dimensions at one-loop.

In Figure 3View Image, we have plotted the energies of the folded and circular string solutions against the filling fraction J2/J. As expected from the “string pendulum” picture of (20View Equation), the folded string solution has lower energy for fixed filling fraction J2/J: The pendulum does not perform a full turn, but oscillates back and forth. Note that the folded string approaches E1 = 0 in the limit J2/J --> 0, 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 w1 = w2 discussed in Section 3.1 is the minimum (J2/J = 0.5) of the full circular string solution for n = 1.

View Image

Figure 3: The one-loop energies of the folded (dark) and circular (light) string solutions plotted against the filling fraction J2/J. The dashed curve is the mirrored folded string solution where one interchanges J1 <--> J2.
These classical string energies will be reproduced in a dual one-loop gauge theory computation in Section 4.
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