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 [87] and for the full superstring in [31]. The Lax
pair for the string was put to use in [74] 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 [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
-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 string: The near plane-wave spectrum
of the superstring of [46, 45, 44, 43], as well as the expected [68] 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 [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
(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 [6] can also be microscopically attributed to a
spin
chain model with long-range interactions up to (at least) order five in a small
expansion
[14].
The technically involved construction of algebraic curves solving the classical string
-model
has subsequently been generalized to larger sectors: In [17] to
(or SO(6) in gauge theory language)
configurations, in [75] to
(or SL(2)) string configurations, and finally in [18] 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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