### 6.3 Semiclassical correspondence

It is an extended idea in theoretical physics that states in quantum mechanics carrying large charges can
be well approximated by a classical or semiclassical description. This idea gets realised in the AdS/CFT
correspondence too. Consider the worldsheet sigma model description of a fundamental string in
AdS_{5} × S^{5}. One expects its perturbative oscillations to be properly described by supergravity, whereas
solitons with large conformal dimension,
and the spectrum of their semiclassical excitations may approximate the spectrum of highly excited string
states in SYM. This is the approach followed in [270], where it was originally applied to rotating
folded strings carrying large bare spin charge.
To get an heuristic idea of the analytic power behind this technique, let me reproduce the spectrum of
large R-charge operators obtained in [70] using a worldsheet quantisation in the pp-wave background by
considering the bosonic part of the worldsheet action describing the AdS_{5} × S_{5} sigma model [270]

where is a unit vector describing S^{5}, is a hyperbolic unit vector describing AdS_{5}, the sigma model coupling
is
and I have ignored all fermionic and RR couplings.
Consider a solution to the classical equations of motion describing a collapsed rotating closed string at
the equator

where and are the polar and azimuthal angles on S^{2} in S^{5}. Its classical worldsheet energy is
Next, consider the harmonic fluctuations around this classical soliton. Focusing on the quadratic
oscillations,

one recognises the standard harmonic oscillator. Using its spectrum, one derives the corrections to the
classical energy
where is the excitation number of the n-th such oscillator. There is a similar contribution from the
AdS part of the action, obtained by the change to . Both contributions must satisfy the on-shell
condition
This is how one reproduces the spectrum derived in [70]
The method outlined above is far more general and it can be applied to study other operators.
For example, one can study the relation between conformal dimension and AdS_{5} spin, as done
in [270], by analysing the behaviour of solitonic closed strings rotating in AdS. Using global AdS_{5},

as the background where the bosonic string propagates and working in the gauge allows one to
identify the worldsheet energy with the conformal dimension in the dual CFT. Consider a closed string at
the equator of the 3-sphere while rotating in the azimuthal angle
For configurations , the Nambu–Goto bosonic action reduces to
where stands for the maximum radial coordinate and the factor of 4 arises because of the four string
segments stretching from 0 to determined by the condition
The energy and spin of the string are conserved charges given by
Notice the dependence of on is in parametric form since . One can obtain
approximate expressions in the limits where the string is much shorter or longer than the radius of
curvature of AdS_{5}.

##### Short strings:

For large , the maximal string stretching is . Thus, strings are shorter than the
radius of curvature . Calculations reduce to strings in flat space for which the parametric dependence
is [270]
Using the AdS/CFT correspondence, the conformal dimension equals the energy, i.e., .
Furthermore, for large . Thus,
for the leading closed string Regge trajectory, which reproduces the AdS/CFT result.

##### Long strings:

The opposite regime takes place when is close to one (from above)
so that the string is sensitive to the AdS boundary metric. The string energy and spin become
so that its difference approaches
This logarithmic asymptotics is qualitatively similar to the one appearing in perturbative gauge theories.
For a more thorough discussion on this point, see [270].
Applying semiclassical quantisation methods to these classical solitons [216], it was realised that one can
interpolate the results for to the weakly-coupled regime. It should be stressed that these techniques
allow one to explore the AdS/CFT correspondence in non-supersymmetric sectors [217], appealing
to the correspondence principle associated to large charges. It is also worth mentioning that
due to the seminal work on the integrability of planar SYM at one loop [393, 60],
much work has been devoted to using these semiclassical techniques in relation to integrability
properties [21]. Interested readers are encouraged to check the review [59] on integrability and references
therein.