Outline Introduction What are the cusp anomaly and generalized - - PowerPoint PPT Presentation

Scaling and integrability from AdS5 × S5

Riccardo Ricci

Imperial College London

DAMTP Cambridge

14th October Work in collaboration with S. Giombi, R.Roiban, A. Tseytlin and C.Vergu

Riccardo Ricci Scaling and integrability from AdS5 × S5

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

Riccardo Ricci Scaling and integrability from AdS5 × S5

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

Riccardo Ricci Scaling and integrability from AdS5 × S5

AdS/CFT

By the AdS/CFT correspondence, this gauge theory is believed to be dual to type IIB string theory on AdS5 × S5. α′ expansion ↔ 1 √ λ genus expansion ↔ 1 N expansion 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?

Riccardo Ricci Scaling and integrability from AdS5 × S5

Can we bridge the gap between weak and strong coupling regimes?

Riccardo Ricci Scaling and integrability from AdS5 × S5

...sometime we have “interpolating functions”. Simplest example: Half-BPS circular Wilson loop

x2 x1

W ∼ TrP exp

• A + Φ
• The VEV is non trivial and it is exactly computable for any λ!

W = 2 √ λ I1( √ λ) , N → ∞ Highly supersymmetric configuration

Riccardo Ricci Scaling and integrability from AdS5 × S5

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”

• perators.

Riccardo Ricci Scaling and integrability from AdS5 × S5

The simplest case is “twist two” (i.e. only two Z fields) O = Tr

• ZDS

+Z

• ,

Z = Φ1 + iΦ2 This is the N = 4 analogue of a QCD operator like ¯ q γ+DS

+ q ,

q = quark The conformal dimension can be read from the 2-point correlator O(x)O(y) ∼ 1 (x − y)2∆(S) ∆(S) = 2 + S + δ(S) For large spin we have a logarithmic scaling δ(S) = f(λ) log S , S → ∞

Riccardo Ricci Scaling and integrability from AdS5 × S5

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 AdS5 × S5 According to AdS/CFT duality it is the same computation Conformal dimension ∆(λ) = String energy E(α′) What is the string in AdS5 × S5 dual to the twist operator?

Riccardo Ricci Scaling and integrability from AdS5 × S5

Folded spinning string

It is a spinning closed string (“folded” on itself) in AdS3

Gubser Klebanov Polyakov

AdS boundary It is pointlike in S5

Riccardo Ricci Scaling and integrability from AdS5 × S5

Why this is the correct string dual? Eclassical = S + √ λ π log S , Gubser Klebanov Polyakov This logarithmic behaviour persists after including quantum corrections Eone loop = −3 log 2 π log S Frolov Tseytlin These results combined with the behaviour at weak coupling ∆ = S + (α1λ + α2λ2 + · · · ) log S support the idea that we have interpolation between weak and strong coupling ∆ = Estring = S + f(λ) log S

Riccardo Ricci Scaling and integrability from AdS5 × S5

Generalized scaling

A new interpolating function appears when we additionally turn

• n one (large) angular momentum J in S5.

This corresponds in gauge theory to the operator O ∼ Tr(DS

+ZJ)

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.

Riccardo Ricci Scaling and integrability from AdS5 × S5

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)

Riccardo Ricci Scaling and integrability from AdS5 × S5

The Bethe equations determine the allowed momenta for the S

• magnons. At one-loop (uk ∼ cot(pk/2)):

uk + i/2 uk − i/2 J =

S

• j=k

uk − uj − i uk − uj + i ,

S

• k=1

uk + i/2 uk − i/2 = 1 δ(J, S) = g2

S

• k=1

2 u2

k + 1/4

An all loop generalization for the Bethe equations exists

Beisert Eden Staudacher

Riccardo Ricci Scaling and integrability from AdS5 × S5

Solve them at strong coupling and compare with string prediction Equations simplify in the semiclassical scaling limit λ ≫ 1 S √ λ ≫ 1 J √ λ ≫ 1 ℓ ≡ πJ √ λ log S = fixed The anomalous dimension scales logarithmically in the spin δ(ℓ, S) = √ λ π f(ℓ, λ) log S f(ℓ, λ) is the generalized scaling dimension: new interpolating function

Belitsky Gorsky Korchemsky; Freyhult, Rej, Staudacher

Riccardo Ricci Scaling and integrability from AdS5 × S5

In the same regime also the string energy scales logarithmically: E − S = √ λ π f(ℓ, λ) log S It is a “long” string which lives in AdS3 × S1 ds2 = − cosh2 ρ dt2 + dρ2 + sinh2 ρ dθ2 + dϕ2 , t = κτ , ρ = µσ , θ = κτ , ϕ = ντ , κ, µ, ν ≫ 1 µ ≈ 1 π log S , ν = J √ λ , κ2 = µ2 + ν2 (Virasoro)

Riccardo Ricci Scaling and integrability from AdS5 × S5

The string is a folded segment passing through AdS center up to the boundary and rotating in S1 inside S5 θ t ρ ϕ Length of the string is controlled by the spin: log S

Riccardo Ricci Scaling and integrability from AdS5 × S5

Classically f0(ℓ) =

• 1 + ℓ2

Quantum corrections ∼ 1/( √ λ)n modify the classical result f(ℓ, λ) = f0(ℓ) + 1 √ λ f1(ℓ) + 1 λf2(ℓ) + · · · Up to one-loop string theory and Bethe-Ansatz agree

Frolov-Tirziu-Tseytlin; Casteill-Kristjansen; Belitsky

Riccardo Ricci Scaling and integrability from AdS5 × S5

Two-loops string theory computation f2(ℓ, λ) = −K + ℓ2

• 8 log2 ℓ − 6 log ℓ +

q0

• + O(ℓ4)

q0 = −3 2 log 2 + 7 4 − 2K , String (?)

Roiban-Tseytlin ‘07

Almost equal to the Bethe-Ansatz prediction q0 = −3 2 log 2 + 11 4 , Bethe Ansatz

Gromov ‘08

K =

∞

• k=0

(−1)k (2k + 1)2 ∼ 0.9159...

Riccardo Ricci Scaling and integrability from AdS5 × S5

Status so far: We compared the results at strong coupling for the conformal dimension of the operator O ∼ Tr(DSΦJ) 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”.

Riccardo Ricci Scaling and integrability from AdS5 × S5

A thermodynamical analogy

We need to compute E − S and J for our semiclassical string. Suppose we have a 2d σ-model with conserved charges Qi ˜ H2d = H2d +

• i

µiQi , Z(µi) = e−βΣ(µi) = Tre−β ˜

H

Averages Qi can be computed differentiating w.r.t. µi. In our context it is natural to consider ˜ H2d = H2d + κ(E − S) − νJ The parameters κ and ν explicitly appear in the string solution. They are not independent ˜ H2d = 0 → κ = κ(ν) Virasoro

Riccardo Ricci Scaling and integrability from AdS5 × S5

Thermodynamics (grand-canonical ensemble) − log Zgrand.can. = U −

• i

µiNi Here U = H2d , N1 = E − S , N2 = J Σ(ν) = H2d+κE −S−νJ , dΣ(ν) dν = dκ(ν) dν E −S−J We can solve these two equations to extract E − S and J.

Riccardo Ricci Scaling and integrability from AdS5 × S5

We obtain E − S =

• 1 + ν2
• Σ(ν) − ν dΣ(ν)

dν

• ,

Remember E − S = √ λ 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(ℓ, λ) =

• 1 + ν2
• F(ν) − ν dF(ν)

dν

• ,

F ∼ Σ/Vol

Riccardo Ricci Scaling and integrability from AdS5 × S5

AdS5 × S5 superstring

Riccardo Ricci Scaling and integrability from AdS5 × S5

It is based on the supercoset G H = PSU(2, 2|4) SO(2, 3)SO(5) ⊃ AdS5 × S5

Metsaev, Tseytlin

S ∼ √ λ

• GMN∂XM∂XN + ¯

θ(D + F5)θ∂X + · · · Classically integrable

L¨ uscher, Pohlmeyer; Bena, Polchinski, Roiban

Infinite tower of conserved charges: local (Noether)+non-local charges

Riccardo Ricci Scaling and integrability from AdS5 × S5

Performing a conformal transformation on the folded string we can do all computations in the Poincare patch which is technically easier ds2 = dxadxa + dzMdzM z2 M = 1, ..., 6 Poincare patch The induced metric on the worldsheet, after Euclidean rotation, is just flat 2d space. We still need to fix the worldsheet symmetries...

Riccardo Ricci Scaling and integrability from AdS5 × S5

Performing a conformal transformation on the folded string we can do all computations in the Poincare patch which is technically easier ds2 = dxadxa + dzMdzM z2 M = 1, ..., 6 Poincare patch The induced metric on the worldsheet, after Euclidean rotation, is just flat 2d space. We still need to fix the worldsheet symmetries...

Riccardo Ricci Scaling and integrability from AdS5 × S5

Light-cone

Superstring worldsheet symmetries 2d bosonic diffeomorphism σ → ˜ σ(σ, τ) , τ → ˜ τ(σ, τ) fermionic κ symmetry Fix these symmetries by suitably choosing a gauge.

Riccardo Ricci Scaling and integrability from AdS5 × S5

Let us consider κ symmetry: ΘI = (θ, η) θ ↔ Q , η ↔ S “S-gauge” η = 0 Conformal gauge and S-gauge lead to a simple quadratic fermionic action (after “T-duality”). ...but bosonic propagator is not simple. Disagreement with Bethe-Ansatz...!

Riccardo Ricci Scaling and integrability from AdS5 × S5

Let us consider κ symmetry: ΘI = (θ, η) θ ↔ Q , η ↔ S “S-gauge” η = 0 Conformal gauge and S-gauge lead to a simple quadratic fermionic action (after “T-duality”). ...but bosonic propagator is not simple. Disagreement with Bethe-Ansatz...!

Riccardo Ricci Scaling and integrability from AdS5 × S5

Other possibility: Γ+θI = 0 as for Green-Schwarz in flat space. Left with 8 θ’s and 8 η’s. Combine this with bosonic light-cone gauge x0 + x1 = x+ = τ , √−ggαβ = diag(−z2, 1/z2)

Metsaev Thorn Tseytlin

This “AdS light-cone” action was never tested at the quantum level

Riccardo Ricci Scaling and integrability from AdS5 × S5

Other possibility: Γ+θI = 0 as for Green-Schwarz in flat space. Left with 8 θ’s and 8 η’s. Combine this with bosonic light-cone gauge x0 + x1 = x+ = τ , √−ggαβ = diag(−z2, 1/z2)

Metsaev Thorn Tseytlin

This “AdS light-cone” action was never tested at the quantum level

Riccardo Ricci Scaling and integrability from AdS5 × S5

Very schematically LGS ∼ ˙ x ˙ x∗ + ˙ zM ˙ zM + 1 z4

• x′x′∗ + z′Mz′M

+η∂η + θ∂θ + (η2)2 + η∂θ Fermionic action is quadratic in θ and quartic η. Bosonic propagator is simple (almost diagonal). This is an encouraging property for higher loop computations. We have a 2d QFT theory problem Compute its partition function Zstring =

• D[x, z, θ, η] exp
• −
• LGS
• Riccardo Ricci

Scaling and integrability from AdS5 × S5

Expand bosonic fields around the semiclassical string solution x = x0 + δx , zM = zM

0 + δZM

Read vertices and compute bosonic and fermionic Feynman diagrams at a given loop order At one loop we simply need the fluctuation spectrum Fone loop =

• d2p log det K

Zone loop = e− V

2π Fone loop

Bosons and fermions conspire to give a finite answer

Riccardo Ricci Scaling and integrability from AdS5 × S5

At two-loops for the computation of F2loop ∼ log Zstring we need to consider all connected Feynman diagrams

Riccardo Ricci Scaling and integrability from AdS5 × S5

Possible integrals I a m2

• =
• d2p

(2π)2 1 (p2 + m2)a This integral is UV divergent for a = 1 (and IR divergent for m = 0). I a1 a2 a3 m2

1 m2 2 m2 3

• =

d2p d2q d2r (2π)4 δ(2)(p + q + r) (p2 + m2

1)a1 (q2 + m2 2)a2 (r2 + m2 3)a3

Catalan constant: I 1 1 1 1 1

2 1 2

• = K

Riccardo Ricci Scaling and integrability from AdS5 × S5

Summing up: F2loop(ℓ) = −K + ℓ2(· · · ) + ℓ4(· · · ) + O(ℓ6) All divergences cancel out! The quantum superstring in AdS light cone gauge is finite This proves the consistency of the light-cone superstring at the quantum level.

Riccardo Ricci Scaling and integrability from AdS5 × S5

From the partition function we can extract the generalized scaling: f2 = −K + ℓ2

• 8 log2 ℓ − 6 log ℓ − 3

2 log 2 + 11 4

• +

ℓ4

• −6 log2 ℓ − 7

6 log ℓ + 3 log 2 log ℓ − 9 8 log2 2 + 11 8 log 2 + 3 32K − 233 576

• + O(ℓ6)

In stupendous agreement with the Bethe-Ansatz prediction!

Riccardo Ricci Scaling and integrability from AdS5 × S5

Leading logarithms and all loops results

Summarising we have the following structure f2 = h2(ℓ) log2 ℓ + h1(ℓ) log ℓ + γ(ℓ)K + δ(ℓ) The coefficients h1(ℓ), h2(ℓ) and γ(ℓ) are known exactly, i.e. to any order in ℓ. At N-loop the expansion looks like fN = f(N)(ℓ) logN ℓ + f(N−1)(ℓ) logN−1 ℓ + · · · It is possible to reconstruct the leading logarithm coefficient at any loop order...

Riccardo Ricci Scaling and integrability from AdS5 × S5

log ℓ arise from fields with mass ℓ in one-loop integrals S5 fields ya have mass ∼ ℓ

ya fields in the loops φ in the legs (z = eφ)

Conjecture: Leading Logs come from “maximally disconnected” graphs L = 1 4 cosh(2φ) + e2φ(∂ty)2 + e−2φ(∂sy)2 + 1 4ℓ2e2φy2 The ya fields appear quadratically...integrate them out

Riccardo Ricci Scaling and integrability from AdS5 × S5

This leads to an effective action for φ Seff(φ) = √ λ 2π cosh(2φ) − e2φ 2π ℓ2 ln ℓ2 . = √ λ 2π F δSeff δφ = 0 → e2φ = 1

• 1 − 2 ℓ2

√ λ ln ℓ2 .

• Flead. log. =
• 1 + 2

√ λ F1 lead. log. , F1 lead. log. = −ℓ2 ln ℓ2 All loop-formula for the scaling function f(ℓ, √ λ)

• lead. log. =
• 1 +

ℓ2 1 +

2 √ λ ln ℓ2 .

Perfect agreement with asymptotic Bethe-Ansatz.

Riccardo Ricci Scaling and integrability from AdS5 × S5

Large ℓ expansion and non-renormalization

An expansion at large ℓ allows to make contact directly with perturbative gauge theory. j ≡ J log S = √ λ ℓ ≫ 1 E − S ln S = j+ λ j

• c10+ c11

j + c12 j2 +...

• + λ2

j3

• c20 + c21

j + c22(λ) j2 ...

• E − S =

√ λ (E − S)tree + (E − S)1−loop + 1 √ λ (E − S)2−loop c12 λ j3 = c12 1 √ λ 1 ℓ3 , c12 = 1 3π2 The same result was obtained in the one-loop spin-chain (Volin 08) confirming that c12 is indeed protected.

Riccardo Ricci Scaling and integrability from AdS5 × S5

Finite size corrections

The worldsheet is no longer infinite: L = ln S < ∞ E = S + 1 π ˜ f(λ, J) ln S + h(λ, J) + u(λ, J) ln S + ... In principle requires the use of the exact string solution with finite S... daunting!!! The worldsheet surface becomes a cylinder : R2 → R × S1 The S1 momentum p1 is quantized p1 = 2π L ,

• dp1 → 2π

L

• n

Shortcut: keep folded string solution but on cylinder!

Riccardo Ricci Scaling and integrability from AdS5 × S5

At one-loop for J = 0 this is enough

Beccaria, Dunne, Forini, Pawellek Tseytlin

Let us then use this prescription... Contributions come terms containing at least a massless field For example at one loop (L = ln S)

• dp0dp1

pm p2

0 + p2 1

→ +∞

−∞

dp0

• p1

pm p2

0 + p2 1

= π +∞

−∞

dp0 pm−1 coth 1 2Lp0

• = Div. + 1

Lm Finite

Riccardo Ricci Scaling and integrability from AdS5 × S5

We obtain: u1(ℓ) = − π 12 1 1 + ℓ2

• ne − loop

u2(ℓ = 0) = 0 ? two − loop The two-loop result is seemingly in agreement with gauge theory expectations

Fioravanti, Grinza, Rossi

Riccardo Ricci Scaling and integrability from AdS5 × S5

Future directions

All computations were actually done in a more general background ϕ = ντ + wσ , w = “winding” The solution with winding is a bended arc passing through AdS center

Riccardo Ricci Scaling and integrability from AdS5 × S5

The partition function now depends on both momentum and winding: F(ν, w) It is again finite at two-loops The exchange of ν and w is a symmetry: T-duality! We can extract a generalized scaling function in presence

• f winding

Prediction confirmed at leading order from Bethe-Ansatz.

Kruczenski-Tirziu

One-loop/two-loops?

Riccardo Ricci Scaling and integrability from AdS5 × S5

Going up...

q3 q2 q1 q1 q2 q3 q2 q1 q3 q1 q2 q3 q2 q3 q1 q1 q2 q3 q1 q2 q3 q1 q3 q2 (f) (g) (h) (e) (b) (a) (d) (c)

Riccardo Ricci Scaling and integrability from AdS5 × S5

The most feasible computation is the cusp anomaly (J = 0) f(λ) = √ λ

• 1 + a1

√ λ + a2 λ + a3 λ3/2 + a4 λ2 + · · ·

• Remarkable transcendentality properties

a1 = −3 log 2 , a2 = −K a3 = − 1 32 (27ζ(3) + 96 K log 2) a4 = − 1 16

• 84β(4) + 81ζ(3) log 2 + 32 K2 + 144 K log2 2
• Degree of transcedentality

[log 2] = 1 [K] = 2 [ζ(n)] = n [β(n)] = n; β(2) = K

Riccardo Ricci Scaling and integrability from AdS5 × S5

Conclusion

The light-cone superstring computation solves a long standing discrepancy between string and Bethe-Ansatz. It provides a highly non trivial check of quantum integrability beyond one-loop string semiclassical level. Viceversa it provides an important consistency check of all-loop Bethe-Ansatz (and BES phase). We have initiated a study of 2-loop finite size corrections.

Riccardo Ricci Scaling and integrability from AdS5 × S5