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 AdS5 × S5. One expects its perturbative oscillations to be properly described by supergravity, whereas solitons with large conformal dimension,
1 Δ ∼ √---, λ = g2YMN = gsN (442 ) λ
and the spectrum of their semiclassical excitations may approximate the spectrum of highly excited string states in š’© = 4 SYM. This is the approach followed in [270Jump To The Next Citation Point], 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 [70Jump To The Next Citation Point] using a worldsheet quantisation in the pp-wave background by considering the bosonic part of the worldsheet action describing the AdS5 × S5 sigma model [270Jump To The Next Citation Point]

1 ∫ 2 √ -- 2 2 S = 2α- d σ g((∇ αn) + (∇ αK ) ) + ..., (443 )
where n is a unit vector describing S5, K is a hyperbolic unit vector describing AdS5, the sigma model coupling α is α = √1- λ43 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

šœƒ = 0 , ψ = ωτ , (444 )
where šœƒ and ψ are the polar and azimuthal angles on S2 in S5. Its classical worldsheet energy is
1 α ω E = ---ω2 = -J 2 where J = --. (445 ) 2α 2 α

Next, consider the harmonic fluctuations around this classical soliton. Focusing on the quadratic šœƒ oscillations,

1-[ 2 2 2 ] 1-[ 2 2 2 2] αL = 2 (∇ šœƒ) + ω cos šœƒ ā‰ƒ 2 (∇ šœƒ) − ω šœƒ + ω , (446 )
one recognises the standard harmonic oscillator. Using its spectrum, one derives the corrections to the classical energy
δ = α-J2 + ∑ N √n2--+-α2J-2, (447 ) 2 n n
where Nn 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 α to − α. Both contributions must satisfy the on-shell condition
5 δ(S ) + δ(AdS5 ) ≈ 0. (448 )
This is how one reproduces the spectrum derived in [70]
āˆ˜ -------- ∑∞ λn2 Δ = J + Nn 1 + --2- . (449 ) n=−∞ J

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 AdS5 spin, as done in [270Jump To The Next Citation Point], by analysing the behaviour of solitonic closed strings rotating in AdS. Using global AdS5,

2 [ 2 2 2 2 ( 2 2 2 2 2)] ds5 = L − cosh ρ dt + d ρ + sinh ρ dšœƒ + sin šœƒd Ļ• + cos šœƒdψ , (450 )
as the background where the bosonic string propagates and working in the gauge τ = t 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
Ļ• = ωt. (451 )
For configurations ρ = ρ(σ), the Nambu–Goto bosonic action reduces to
2 ∫ ∫ ρ0 āˆ˜ --------------------- Sstring = − 4-L--- dt dρ cosh2 ρ − ( Ė™Ļ•)2sinh2ρ , (452 ) 2πα ′ 0
where ρ0 stands for the maximum radial coordinate and the factor of 4 arises because of the four string segments stretching from 0 to ρ 0 determined by the condition
2 coth ρ0 = ω2. (453 )
The energy E and spin S of the string are conserved charges given by
2 ∫ ρ 2 E = 4 -L--- 0dρ āˆ˜------cosh-ρ--------, (454 ) 2πα ′ 0 cosh2ρ − ω2 sinh2 ρ ∫ -L2-- ρ0 ------ω-sinh2-ρ------- S = 4 2πα ′ 0 dρ āˆ˜ ----2------2----2--. (455 ) cosh ρ − ω sinh ρ
Notice the dependence of √-- Eāˆ• λ on √ -- Sāˆ• λ is in parametric form since L4 = λα′2. One can obtain approximate expressions in the limits where the string is much shorter or longer than the radius of curvature L of AdS5.

Short strings:
For large ω, the maximal string stretching is ρ0 ≈ 1 āˆ•ω. Thus, strings are shorter than the radius of curvature L. Calculations reduce to strings in flat space for which the parametric dependence is [270Jump To The Next Citation Point]

L2 L2 2 22S E = -′--, S = ---′-2 , = ⇒ E = L --′ . (456 ) α ω 2α ω α
Using the AdS/CFT correspondence, the conformal dimension equals the energy, i.e., Δ = E. Furthermore, √ -- S ā‰Ŗ λ for large ω. Thus,
Δ2 ≈ m2L2 , where m2 = 2(S-−-2)- (457 ) α′
for the leading closed string Regge trajectory, which reproduces the AdS/CFT result.

Long strings:
The opposite regime takes place when ω is close to one (from above)

1 1 √-- ω = 1 + 2η , η ā‰Ŗ 1 =⇒ ρ0 → -log --, S ā‰« λ , (458 ) 2 η
so that the string is sensitive to the AdS boundary metric. The string energy and spin become
L2 ( 1 1 ) E = ----- --+ log --+ ... , (459 ) 2πα ′ η η L2 ( 1 1 ) S = ----′ --− log --+ ... , (460 ) 2πα η η
so that its difference approaches
√ -- E − S = --λ-log √S--+ ... (461 ) π λ
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 E − S 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 š’© = 4 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.
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