In the context of the -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  and for the full superstring in . The Lax pair for the string was put to use in  for string configurations on - 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  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 -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 .
In the very interesting paper , 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 string: The near plane-wave spectrum of the superstring of [46, 45, 44, 43], as well as the expected  generic scaling of the string energies with in the strong coupling limit, agree with the predictions of the quantum string Bethe equations. But there is more quantum data for the 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 [62, 64, 95]. The one-loop correction for a circular string moving in obtained in  was recently compared  to the result obtained from the proposed quantum string Bethe equations of . The authors of  find agreement when they expand the results in (up to third order), but disagreements emerge in different limits (where is not small). The interpretation of this result is unclear at present. Finally, the proposed quantum string Bethe equations of  can also be microscopically attributed to a spin chain model with long-range interactions up to (at least) order five in a small expansion .
The technically involved construction of algebraic curves solving the classical string -model has subsequently been generalized to larger sectors: In  to (or SO(6) in gauge theory language) configurations, in  to (or SL(2)) string configurations, and finally in  to superstrings propagating in the full space.
There also has been progress on a number of possible paths towards the true quantization of the classical integrable model of the string in the works [113, 4, 2, 9, 35, 36], however, it is fair to say that this problem remains currently unsolved.
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