Scaling and integrability from AdS 5 × S 5 Riccardo Ricci Imperial College London DAMTP Cambridge 14th October Work in collaboration with S. Giombi, R.Roiban, A. Tseytlin and C.Vergu Scaling and integrability from AdS 5 × S 5 Riccardo Ricci
Outline Introduction What are the cusp anomaly and generalized scaling function? Superstring in the light cone gauge Scaling function from string theory and comparison to gauge theory Scaling and integrability from AdS 5 × S 5 Riccardo Ricci
Understand quantum gauge theories at any coupling. Make best use of symmetries by choosing simplest non trivial gauge theory. Ideal candidate: N = 4 SYM It has maximal symmetry It is conformal Infinite dimensional symmetry: Integrability It is dual to a superstring theory in a non-trivial background Scaling and integrability from AdS 5 × S 5 Riccardo Ricci
AdS/CFT By the AdS/CFT correspondence, this gauge theory is believed to be dual to type IIB string theory on AdS 5 × S 5 . 1 genus expansion ↔ 1 α ′ expansion ↔ √ N expansion λ This is a strong/weak duality which is in general very hard to test directly. How do we interpolate between weak ( λ ≪ 1 ) and strong coupling ( λ ≫ 1 ) regimes? Scaling and integrability from AdS 5 × S 5 Riccardo Ricci
Can we bridge the gap between weak and strong coupling regimes? Scaling and integrability from AdS 5 × S 5 Riccardo Ricci
...sometime we have “interpolating functions”. Simplest example: Half-BPS circular Wilson loop x 2 �� � W ∼ Tr P exp A + Φ x 1 The VEV is non trivial and it is exactly computable for any λ ! √ 2 � W � = √ I 1 ( λ ) , N → ∞ λ Highly supersymmetric configuration Scaling and integrability from AdS 5 × S 5 Riccardo Ricci
Non-supersymmetric example: cusp anomaly No simple formula, as for circular Wilson loop, but an integral equation exists. Beisert Eden Staudacher An exact solution is not known but one can extract a perturbative solution at any desired order. The cusp anomaly surprisingly appears in many different contexts: It governs the renormalization of a light-like Wilson loops with cusps Gluon scattering amplitudes It governs the logarithmic scaling of high spin “twist” operators. Scaling and integrability from AdS 5 × S 5 Riccardo Ricci
The simplest case is “twist two” (i.e. only two Z fields) Z = Φ 1 + i Φ 2 Z D S � � O = Tr + Z , This is the N = 4 analogue of a QCD operator like q γ + D S ¯ + q , q = quark The conformal dimension can be read from the 2-point correlator 1 �O ( x ) O ( y ) � ∼ ( x − y ) 2∆( S ) ∆( S ) = 2 + S + δ ( S ) For large spin we have a logarithmic scaling S → ∞ δ ( S ) = f ( λ ) log S , Scaling and integrability from AdS 5 × S 5 Riccardo Ricci
Comparing the spectra on the two sides of the duality: Gauge theory side: Compute the conformal dimension of local operators of planar ( N → ∞ ) N = 4 SYM AdS side: Compute the energy of free string in AdS 5 × S 5 According to AdS/CFT duality it is the same computation Conformal dimension ∆( λ ) = String energy E( α ′ ) What is the string in AdS 5 × S 5 dual to the twist operator? Scaling and integrability from AdS 5 × S 5 Riccardo Ricci
Folded spinning string It is a spinning closed string (“folded” on itself) in AdS 3 Gubser Klebanov Polyakov AdS boundary It is pointlike in S 5 Scaling and integrability from AdS 5 × S 5 Riccardo Ricci
Why this is the correct string dual? √ λ E classical = S + π log S , Gubser Klebanov Polyakov This logarithmic behaviour persists after including quantum corrections E one loop = − 3 log 2 log S Frolov Tseytlin π These results combined with the behaviour at weak coupling ∆ = S + ( α 1 λ + α 2 λ 2 + · · · ) log S support the idea that we have interpolation between weak and strong coupling ∆ = E string = S + f ( λ ) log S Scaling and integrability from AdS 5 × S 5 Riccardo Ricci
Generalized scaling A new interpolating function appears when we additionally turn on one (large) angular momentum J in S 5 . This corresponds in gauge theory to the operator O ∼ Tr( D S + Z J ) The conformal dimension is a non-trivial function √ ∆( S, J, λ ) It can be explored in various regimes of the parameters testing important features of gauge/string duality. Interpolation between BMN-like ( large J , small S ) and minimal twist ( J = 2 ) operators. Scaling and integrability from AdS 5 × S 5 Riccardo Ricci
Integrability allows to map the one-loop anomalous dimension into the energy of an SL(2) Heisenberg spin-chain. Tr ( Z ... Z ... Z ) ≡ | ↓ ... ↓ ... ↓� , spin chain vacuum � � Tr Z ... D + Z ... Z ≡ | ↓ ... ↑ ... ↓� excitation (magnon) Scaling and integrability from AdS 5 × S 5 Riccardo Ricci
The Bethe equations determine the allowed momenta for the S magnons. At one-loop ( u k ∼ cot( p k / 2) ): S S � J � u k + i/ 2 u k − u j − i u k + i/ 2 � � = u k − u j + i , u k − i/ 2 = 1 u k − i/ 2 j � = k k =1 S 2 δ ( J, S ) = g 2 � u 2 k + 1 / 4 k =1 An all loop generalization for the Bethe equations exists Beisert Eden Staudacher Scaling and integrability from AdS 5 × S 5 Riccardo Ricci
Solve them at strong coupling and compare with string prediction Equations simplify in the semiclassical scaling limit S J πJ √ √ √ λ ≫ 1 ≫ 1 ≫ 1 ℓ ≡ = fixed λ λ λ log S The anomalous dimension scales logarithmically in the spin √ λ δ ( ℓ, S ) = π f( ℓ, λ ) log S f( ℓ, λ ) is the generalized scaling dimension: new interpolating function Belitsky Gorsky Korchemsky; Freyhult, Rej, Staudacher Scaling and integrability from AdS 5 × S 5 Riccardo Ricci
In the same regime also the string energy scales logarithmically: √ λ E − S = π f( ℓ, λ ) log S It is a “long” string which lives in AdS 3 × S 1 ds 2 = − cosh 2 ρ dt 2 + dρ 2 + sinh 2 ρ dθ 2 + dϕ 2 , t = κτ , ρ = µσ , θ = κτ , ϕ = ντ , κ, µ, ν ≫ 1 µ ≈ 1 ν = J κ 2 = µ 2 + ν 2 π log S , √ , (Virasoro) λ Scaling and integrability from AdS 5 × S 5 Riccardo Ricci
The string is a folded segment passing through AdS center up to the boundary and rotating in S 1 inside S 5 t ϕ ρ θ Length of the string is controlled by the spin: log S Scaling and integrability from AdS 5 × S 5 Riccardo Ricci
Classically � 1 + ℓ 2 f 0 ( ℓ ) = √ λ ) n modify the classical result Quantum corrections ∼ 1 / ( f( ℓ, λ ) = f 0 ( ℓ ) + 1 f 1 ( ℓ ) + 1 √ λ f 2 ( ℓ ) + · · · λ Up to one-loop string theory and Bethe-Ansatz agree Frolov-Tirziu-Tseytlin; Casteill-Kristjansen; Belitsky Scaling and integrability from AdS 5 × S 5 Riccardo Ricci
Two-loops string theory computation � � 8 log 2 ℓ − 6 log ℓ + f 2 ( ℓ, λ ) = − K + ℓ 2 + O ( ℓ 4 ) q 0 q 0 = − 3 2 log 2 + 7 4 − 2K , String (?) Roiban-Tseytlin ‘07 Almost equal to the Bethe-Ansatz prediction q 0 = − 3 2 log 2 + 11 , Bethe Ansatz 4 Gromov ‘08 ∞ ( − 1) k � K = (2 k + 1) 2 ∼ 0 . 9159 ... k =0 Scaling and integrability from AdS 5 × S 5 Riccardo Ricci
Status so far: We compared the results at strong coupling for the conformal dimension of the operator O ∼ Tr( D S Φ J ) in gauge theory (Bethe-Ansatz) and in string theory. Disagreement emerges at 2-loops... Breakdown of integrability in string theory?? We will perform the string computation again, but this time using a different “ gauge ”. Scaling and integrability from AdS 5 × S 5 Riccardo Ricci
A thermodynamical analogy We need to compute E − S and J for our semiclassical string. Suppose we have a 2 d σ -model with conserved charges Q i Z ( µ i ) = e − β Σ( µ i ) = Tr e − β ˜ ˜ � H H 2 d = H 2 d + µ i Q i , i Averages � Q i � can be computed differentiating w.r.t. µ i . In our context it is natural to consider ˜ H 2 d = H 2 d + κ ( E − S ) − νJ The parameters κ and ν explicitly appear in the string solution. They are not independent ˜ H 2 d = 0 → κ = κ ( ν ) Virasoro Scaling and integrability from AdS 5 × S 5 Riccardo Ricci
Thermodynamics (grand-canonical ensemble) � − log Z grand . can . = � U � − µ i � N i � i Here � U � = � H 2d � , � N 1 � = � E − S � , � N 2 � = � J � d Σ( ν ) = dκ ( ν ) Σ( ν ) = � H 2d � + κ � E − S �− ν � J � , � E − S �−� J � dν dν We can solve these two equations to extract � E − S � and � J � . Scaling and integrability from AdS 5 × S 5 Riccardo Ricci
We obtain � Σ( ν ) − ν d Σ( ν ) � � 1 + ν 2 � E − S � = , dν Remember √ λ E − S = π f( ℓ, λ ) log S The logarithmic scaling of � E − S � is due to the effective action Σ which is proportional to the worldsheet volume: Σ ∝ Vol ∝ log S We can therefore read the generalized scaling � F ( ν ) − ν d F ( ν ) � � 1 + ν 2 f( ℓ, λ ) = , F ∼ Σ / Vol dν Scaling and integrability from AdS 5 × S 5 Riccardo Ricci
AdS 5 × S 5 superstring Scaling and integrability from AdS 5 × S 5 Riccardo Ricci
It is based on the supercoset G PSU (2 , 2 | 4) SO (2 , 3) SO (5) ⊃ AdS 5 × S 5 H = Metsaev, Tseytlin √ � G MN ∂X M ∂X N + ¯ S ∼ λ θ ( D + F 5 ) θ∂X + · · · Classically integrable L¨ uscher, Pohlmeyer; Bena, Polchinski, Roiban Infinite tower of conserved charges: local (Noether)+non-local charges Scaling and integrability from AdS 5 × S 5 Riccardo Ricci
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