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3.4 Further Developments

The discussed explicit solutions of the bosonic string on a R × S3 background are the first simple elements of the general set of classical solutions of the bosonic AdS × S5 5 string considered in the literature. Earlier examples not discussed in the text are [6910291]. The bosonic part of the classical string action consists of a SO(6) and a SO(2,4) s-model augmented by the conformal gauge (Virasoro) constraints. The O(p,q) s-models have been known to be integrable for a long time [988511857] and one would expect this to remain true also once one imposes the conformal gauge constraints. And, indeed, in [57] more complicated solutions of the AdS × S5 5 string were constructed by reducing the system to the so-called Neumann integrable system through a suitable ansatz. The solutions of [57] involve non-vanishing values for all spins and angular momenta (S1,S2, J1,J2,J3). However, they do not generically display an analytic behavior in the effective coupling constant c'. This is true only for configurations with at least one large charge Ji on the S5.

In the context of the O(p, q) s-models, integrability is based on the existence of a Lax pair, a family of flat connections on the 2d string worldsheet, giving rise to an infinite number of conserved charges. These were first discussed in the context of the bosonic string in [87] and for the full superstring in [31]. The Lax pair for the string was put to use in [74] for string configurations on R × S3 - the sector we also considered in the above. These investigations led to the construction of an underlying algebraic curve parameterizing the solutions. This enabled the authors of [74] to write down an integral equation of Bethe type yielding the associated energies of the solutions. Very similar equations will appear below in our discussion in Section 4.2 on the thermodynamic limit of the gauge theory Bethe equations. The extraction of these integral equations from the string s-model then allows for a direct comparison to the gauge theory Bethe equations. On this level of formalization, there is no need to compare explicit solutions any longer - as we are doing here for pedagogical purposes. This construction based on an underlying algebraic curve makes full use of the technology of integrable systems and has been nicely reviewed by Zarembo in [119].

In the very interesting paper [6], these continuum string Bethe equations were boldly discretized leading to a conjectured set of Bethe equations for the quantum spectrum of the string. This proposal has been verified by comparing it to known quantum data of the AdS5 × S5 string: The near plane-wave spectrum of the superstring of [46454443], as well as the expected [68] generic scaling of the string energies with 1/4 c in the strong coupling limit, agree with the predictions of the quantum string Bethe equations. But there is more quantum data for the 5 AdS5 × S string available: In a series of papers by Tseytlin, Frolov and collaborators, one-loop corrections on the string worldsheet to the energies of various spinning string solutions have been computed [626495]. The one-loop correction for a circular string moving in AdS3 × S1 < AdS5 × S5 obtained in [95] was recently compared [106] to the result obtained from the proposed quantum string Bethe equations of [6]. The authors of [106] find agreement when they expand the results in c' (up to third order), but disagreements emerge in different limits (where c' is not small). The interpretation of this result is unclear at present. Finally, the proposed quantum string Bethe equations of [6] can also be microscopically attributed to a s = 1/2 spin chain model with long-range interactions up to (at least) order five in a small c expansion [14].

The technically involved construction of algebraic curves solving the classical R × S3 string s-model has subsequently been generalized to larger sectors: In [17] to 5 R × S (or SO(6) in gauge theory language) configurations, in [75] to 1 AdS3 × S (or SL(2)) string configurations, and finally in [18] to superstrings propagating in the full AdS5 × S5 space.

There also has been progress on a number of possible paths towards the true quantization of the classical integrable model of the AdS × S5 5 string in the works [1134293536], however, it is fair to say that this problem remains currently unsolved.

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