The classical integrable structure of AdS/CFT Beno t Vicedo DESY, - - PowerPoint PPT Presentation

Motivation Constrained IS GS superstring Lie dialgebra Conclusions

The classical integrable structure of AdS/CFT

Benoˆ ıt Vicedo

DESY, Hamburg

Cambridge, UK Thursday 24-th February, 2011

Motivation Constrained IS GS superstring Lie dialgebra Conclusions

Motivation

λ L classical AdS strings

• perturb. gauge theory

?

S-matrix QISM TBA

TBA approach: Assumes integrability at finite λ, L.

• At L ≫ 1, factorizability of the S-matrix Fix 2-body

S-matrix using Yangian symmetry (universal R-matrix?)

• Zamolodchikov’s TBA trick Ground state energy E0(L).

Claim: Excited states described by solutions of Y-system (boundary & analyticity conditions?). [Frolov-Arutyunov,

Bombardelli-Tateo-Fioravanti, Gromov-Kazakov-Kozak-Vieira ’09]

Need to prove integrability ∀(λ, L) QISM.

Motivation Constrained IS GS superstring Lie dialgebra Conclusions

Green-Schwarz superstring on AdS-spaces

Described by σ-model on semi-symmetric spaces [Zarembo ’10] Σ ≡ R × S1 = − → super(AdSn × Y10−n) ≡ G/H Let g : Σ → G, A = −g−1dg ∈ Ω1(Σ, g), g = Lie G and impose

• Global left G-action: under g → Ug, U ∈ G, have A → A.
• Local right H-action: under g → gh, h : Σ → H have

A → h−1Ah − h−1dh, hence A(1,2,3) → h−1A(1,2,3)h, where A(n) ∈ gn. Lagrangian: LGS := − 1

2A(2) ∧ ∗A(2)

• kinetic

− 1

2A(1) ∧ A(3)

• WZW

+ Λ, dA − A2

• Maurer−Cartan

.

Motivation Constrained IS GS superstring Lie dialgebra Conclusions

Classical integrability (i)

Equations of motion for LGS: Can be neatly written as zero-curvature equation dJBPR(z) − JBPR(z) ∧ JBPR(z) = 0, for the following Lax connection [Bena-Polchinski-Roiban ’03] JBPR(z) := z−2A(2)

− + z−1A(3) + A(0) + z A(1) + z2A(2) +

where z ∈ C and A(2)

± := A(2) ± ∗A(2).

Integrals of motion:

˜ γ (σ′, τ′) (σ, τ) γ(σ′,τ′) γ(σ,τ)

Ω(z, σ, τ) := P← − − exp

• [γ(σ,τ)] JBPR(z)

Ω(z, σ′, τ ′) = T˜

γ(z) · Ω(z, σ, τ) · T˜ γ(z)−1,

⇒ ∂σ str Ω(z)m = ∂τ str Ω(z)m = 0.

Motivation Constrained IS GS superstring Lie dialgebra Conclusions

Classical integrability (ii)

Involution property: Integrability also requires {str Ω(z)m, str Ω(z′)n} = 0, ∀m, n ∈ N, z, z′ ∈ C. Writing JBPR(z) = LBPR(z)dσ + MBPR(z)dτ, this would follow provided [Maillet ’85]

• LBPR(z, σ)⊗

, LBPR(z′, σ′) ? =

• r(z, z′), LBPR(z, σ) ⊗ 1 + 1 ⊗ LBPR(z′, σ′)
• δ(σ, σ′)

−

• s(z, z′), LBPR(z, σ) ⊗ 1 − 1 ⊗ LBPR(z′, σ′)
• δ(σ, σ′)

− 2 s(z, z′)∂σδ(σ, σ′), for some r(z, z′), s(z, z′) taking values in g ⊗ g. Problem: JBPR(z) does not have this property! Rederive Lax connection within Hamiltonian formalism.

Motivation Constrained IS GS superstring Lie dialgebra Conclusions

Constrained Hamiltonian systems

Legendre transform: Given Lagrangian L ∈ C(TM), define DL : TM → T ∗M ≡ P, (q, ˙ q) → (q, p = ∂L/∂ ˙ q). If DL(TM) = Σ ⊂ P then L ∈ C(TM) H = p ˙ q − L ∈ C(Σ) Constraint surface: Σ ≡ {φA ≈ 0} ⊂ P φA = γa, χα γa : first class, {γa, φA} ≈ 0 χα : second class d.o.f. = (2n − m − 2p)

• dim Σ

−m

{·, γa} φA ≈ 0

Motivation Constrained IS GS superstring Lie dialgebra Conclusions

Extensions

Lemma

Let F, G ∈ C(P) be such that F ≈ G, i.e. F|Σ = G|Σ, then F = G +

• A

fAφA. Extension: Given F ∈ C(P), can ‘extend’ F F +

A fAφA

If F is first class, i.e. {F, φA} ≈ 0, then F F +

a faγa

e.g.: Extended Hamiltonian H HE ≡ H

• dyn.

+

• a

uaγa

• gauge tr.

+

• A,B

uA,BφAφB

• unphys.

.

Motivation Constrained IS GS superstring Lie dialgebra Conclusions

Symmetry

Moment map µ : P → R generates symmetry δF = {F, µ}. It preserves Σ if δφA = {φA, µ} ≈ 0, ∀A (⋆) i.e. provided µ is first class. Can ensure this by µ µ +

α mαχα.

Remark

(⋆) is preserved by: µ µ +

• a

maγa +

• A,B

mA,BφAφB

{·, γa} {·, µ} φA ≈ 0

Motivation Constrained IS GS superstring Lie dialgebra Conclusions

Constrained integrable systems

n-dimensional constrained Hamiltonian system (P, ω, H, φA), has 2(n − p − m) ≡ 2k indep. phase-space variables.

Definition

(P, ω, H, φA) is integrable if ∃µ1, . . . , µk ∈ C(P) s.t. {µi, φA} ≈ 0, ∀i, A (I)

• µi, µj
• = 0, ∀i, j

and dµ1 ∧ . . . ∧ dµk = 0 (II) and where H ≈ H(µ), with µ = (µ1, . . . , µk) : P → Rk.

Remark

Strong equality forbids extensions µi µi +

a mi,aγa + . . .

Apply these ideas to the Green-Schwarz superstring

Motivation Constrained IS GS superstring Lie dialgebra Conclusions

Lagrangian to Hamiltonian GS superstring

Start from the Lagrangian (dynamical variables A0, A1, Λ, hαβ) LGS = − 1

2[

√ −hhαβ str(A(2)

α A(2) β ) + ǫαβ str(A(1) α A(3) β )]

+ str Λ(∂0A1 − ∇1A0). Dirac’s consistency algorithm: Primary constraints: Π0 ≈ 0, Π1 ≈ Λ, ΠΛ ≈ 0, pαβ ≈ 0. Π1 − Λ is second class with ΠΛ Dirac bracket. Secondary constraints: ˙ Π0 ≈ 0 ⇒ C(0) ≈ C(1) ≈ C(2) ≈ C(3) ≈ 0; Π(2) and C(2) second class pair Eliminate. ˙ pαβ ≈ 0 ⇒ T± ≈ 0 (Virasoro constraints). Tertiary constraints: ˙ C(0,1,3) ≈ ˙ T± ≈ 0 ⇒ no new constraints. Partial gauge fixing conditions to eliminate Π(0,1,3) , pαβ.

Motivation Constrained IS GS superstring Lie dialgebra Conclusions

Hamiltonian GS superstring

In summary,

• Phase-space P parametrised by (A(0,1,2,3)

1

, Π(0,1,2,3)

1

) with {A(i)

11(σ), Π(4−i) 12

(σ′)}D.B. = C(i 4−i)

12

δ(σ − σ′).

• Total set of constraints {ΦA} = {T±, C(0,1,3)}.
• First class constraints {Γa} = {T±, C(0), K(1,3)}, where

T± ≡ T± ∓ str(A(1,3)

1

C(3,1)), K(1,3) ≡ 2 √ λ[A(2)

± , iC(1,3)]+.

• Extended Hamiltonian

HE = ρ+T+ + ρ−T−

• conformal tr.

− str(k(3)K(1)) − str(k(1)K(3))

• κ−symmetry

− str(µ(0)C(0))

• coset

.

Motivation Constrained IS GS superstring Lie dialgebra Conclusions

Hamiltonian Lax connection

Lax connection: Take a general linear combination L = A(0)

1

+ aA(1)

1

+ bA(2)

1

+ cA(3)

1

+ ρ(∇1Π1)(0) + γ(∇1Π1)(1) + β(∇1Π1)(2) + α(∇1Π1)(3), where a, b, c, α, β, γ, ρ ∈ C. Fix them by imposing: (I) str Ω(L)j are first class, i.e. {str Ω(L)j, Φ(σ)} ≈ 0, ∀Φ(σ) ∈ {T±, C(0,1,3)}. (II) L is the σ-component of a strongly flat connection, i.e. {L, P0} = ∂σM + [M, L], for some M, where P0 is the energy. Lax connection is then J(z) := L(z)dσ + M(z)dτ.

Motivation Constrained IS GS superstring Lie dialgebra Conclusions

Strong zero curvature equation

• σ-translation is unambiguously generated on C(P) by

P1 ≡ T+ − T− − str(A(1)

1 C(3)) − str(A(3) 1 C(1)) − str(A(0) 1 C(0)),

in the sense that {F(σ), P1} = ∂σF(σ) with P1 =

• dσ′P1(σ′).
• Generator of τ-translations is defined only on C(Σ) by

˜ P0 ≡ T+ + T− − str(A(1)

1 C(3)) + str(A(3) 1 C(1)) − str(A(0) 1 C(0)).

We are free to extend this definition as follows ˜ P0 P0 ≡ ˜ P0 + str (C(1)C(3)).

Motivation Constrained IS GS superstring Lie dialgebra Conclusions

Fixing parameters

Total of six independent constraints on seven parameters {a, b, c, α, β, γ, ρ}. Solution depends on one parameter z ∈ C: (I) First class monodromy fixes a = 1

4(z−3 + 3z),

α = 1

2(z−1 − z3)

b = 1

2(z−2 + z2),

β = 1

2(z−2 − z2)

c = 1

4(3z−1 + z3),

γ = 1

2(z−3 − z).

(II) Strong zero-curvature fixes ρ = 1

2(1 − z4).

Motivation Constrained IS GS superstring Lie dialgebra Conclusions

Hamiltonian Lax connection

Final result reads [BV ’09] L(z) = LBPR(z) + 1

2(1 − z4)

• C(0) + z−3C(1) + z−1C(3)

. This is an extension of the BPR Lax connection. Relation to pure spinors: Can also write L(z) = Lp.s.(z)

• ghosts=0 + 1

2(1 − z4) C(0).

Extension of the (matter part of) the pure spinor connection! In fact C(1,3) have second class parts, so L0(z) := Lp.s.(z)|ghosts=0 is a ‘Dirac’ extension of LBPR(z). Hint at a deeper connection between GS and PS? ...

Motivation Constrained IS GS superstring Lie dialgebra Conclusions

r/s-matrix algebra

Poisson algebra of the extended Lax matrix L reads [Magro ’08] {L1, L2} = [r12 − s12, L1]δσσ′ + [r12 + s12, L2]δσσ′ − 2s12δ′

σσ′.

where the r/s-matrices satisfy, so called ‘extended’ YBE: eYB(r, s) = [r + s, r − s] + [r + s, r + s] + [r + s, r + s] = 0.

Remark

This fails to be true for LBPR and is only weakly true for L0, {L01, L02} ≈ [r0

12 − s0 12, L1]δσσ′ + [r0 12 + s0 12, L2]δσσ′ − 2s0 12δ′ σσ′

where the r0/s0-matrices satisfy eYB(r0, s0) = ω0. The GS superstring on G/H is classically integrable.

Motivation Constrained IS GS superstring Lie dialgebra Conclusions

Quantum Inverse Scattering Method

Starting point for QISM: R12(u, v)Ln

1(u)Ln 2(v) = Ln 2(v)Ln 1(u)R12(u, v),

Ln

1(u)Lm 2 (v) = Lm 2 (v)Ln 1(u),

∀n = m. Defining monodromy M1(u) := LN

1 (u) . . . L1 1(u), we have

T(u) := tr1 M1(u), [T(u), T(v)] = 0, ∀u, v. Classical limit and CISM: Letting R12 = 1 + r12 + O(2) and Ln

1 = Ln 1 + O() we find {Ln 1, Lm 2 } = [r12, Ln 1Ln 2]δn m.

Continuum limit: Ln = P ← − − exp σn+1

σn

dσL(σ)

Ln

yields {L1, L2} = [r12, L1 + L2]δσσ′ Lie bialgebra structure.

Motivation Constrained IS GS superstring Lie dialgebra Conclusions

Algebraic level

A “= C(G)”

“Quantum group” →0

• ∆ : A → A ⊗ A

→0

• m : A ⊗ A → A

→0

• A0 = C(G)

Poisson-Hopf alg. “quantize”

• ∆0 = m∗

G : A0 → A0 ⊗ A0

• {·, ·} : A0 ⊗ A0 → A0

g→id∈G

• G

Poisson-Lie group “continuum”

• mG : G ⊗ G → G

g→id∈G

• g = Lie G

Lie bialgebra exp

• [·, ·] : g ∧ g → g

[·, ·]∗ : g∗ ∧ g∗ → g∗

Quasi-triangular A coboundary g: ∃R ∈ A ⊗ A s.t. ∆op

(a)R = R∆(a), ∀a ∈ A

= ⇒ ∃r ∈ End g s.t. [X, Y]∗ = 1

2([rX, Y] − [X, r∗Y]), ∀X, Y ∈ g.

Motivation Constrained IS GS superstring Lie dialgebra Conclusions

Dual of a Lie bialgebra

Let g be a Lie algebra. Dual g∗ is a Poisson manifold: Lie bracket on g ⇐ ⇒ Linear Poisson bracket on g∗. It is in this sense that the (linear) Poisson bracket {L1, L2} = [r12, L1 + L2]δσσ′ (b) with L ∈ C(S1, g∗), “comes from” the Lie bracket on g [X, Y]∗ = 1

2([rX, Y] − [X, r∗Y]).

(b∗) Note that (b∗), and hence also (b),

• is skew if

[X1 + X2, r12 + r21] = 0.

• satisfies Jacobi if

[r12, r13] + [r12, r23] + [r13, r23] = 0.

Motivation Constrained IS GS superstring Lie dialgebra Conclusions

Lie di algebras

Remark

If r∗ = −r, then we can write [X, Y]∗ = 1

2([rX, Y] + [X, rY]),

which is automatically skew.

Definition

A Lie dialgebra (g, R) is a Lie algebra g and R ∈ Endg s.t. [X, Y]R := 1

2([RX, Y] + [X, RY])

is a second Lie bracket on g. Only requirement for Jacobi is the mCYBE: [RX, RY] − R

• [RX, Y] + [X, RY]
• = −[X, Y],

∀X, Y ∈ g. In tensor form with ω(X, Y, Z) = [X, Y], Z, [R12, R13] + [R12, R23] + [R32, R13] = −ω123.

Motivation Constrained IS GS superstring Lie dialgebra Conclusions

Lie bialgebras vs Lie dialgebras

Lie bialgebras

(g, δ)

δr(X) = [X, r] R = r = −r ∗ R : g → g δ : g → g ∧ g

Lie dialgebras

(g, R) Lie dialgebra: R ∈ End g defines a second bracket on g, [x, y]R = 1

2

• [Rx, y] + [x, Ry]
• .

Lie bialgebra: δ∗ : g∗ ∧ g∗ → g∗ defines bracket on g∗. In coboundary case δ∗

r : (ξ, ξ′) → [ξ, ξ′]∗ = 1

2

• [rξ, ξ′] − [ξ, r∗ξ′]
• .

Motivation Constrained IS GS superstring Lie dialgebra Conclusions

Dual of a Lie dialgebra

A (linear) Poisson bracket of the type {L1, L2} = [r12, L1 + L2]δσσ′ − [s12, L1 − L2]δσσ′ − 2s12δ′

σσ′

with L ∈ C(S1, g∗), “comes from” the Lie bracket [·, ·]R with R = r + s, r∗ = −r, s∗ = s. In particular, Jacobi for {·, ·} ⇐ ⇒ mCYBE for R “⇐ ⇒” eYB(r, s) = 0. Non-ultralocality: A system is called ultralocal if {L1, L2} ∝ δσσ′ and non-ultralocal otherwise. Here, Non-ultralocality ⇐ ⇒ s = 0 ⇐ ⇒ R∗ = −R. What is the Lie dialgebra underlying superstring on G/H?

Motivation Constrained IS GS superstring Lie dialgebra Conclusions

Twisted loop algebra

Recall, L(z) = LBPR(z) +

1 2 √ λ(1 − z4)

• C(0) + z−3C(1) + z−1C(3)

, where JBPR(z) := z−2A(2)

− + z−1A(3) + A(0) + z A(1) + z2A(2) + .

So consider loop algebra Lg := gz, z−1 with decomposition Lg = gz ∔ z−1g[z−1] =: Lg+ ∔ Lg−. Z4-twist: All g = Lie G of interest admit Z4-automorphism Ω : g → g, Ω4 = id [Zarembo ’10]. Noticing Ω(L(z)) = L(iz), let ˆ Ω : Lg → Lg, ˆ Ω(X)(z) = Ω(X(−iz)). The twisted loop algebra is LgΩ := {X ∈ Lg | ˆ Ω(X) = X}. In particular LgΩ = LgΩ

+ ∔ LgΩ − and

L ∈ C∞(S1, LgΩ).

Motivation Constrained IS GS superstring Lie dialgebra Conclusions

Twisted inner product

Lax matrix can be rewritten as L = 4 φ(z)−1

∞

• k=1

zk k A(k)

1

+ 2 (∇1Π1)(k) , where φ(z) := 16z4 (1 − z4)2 . Introduce a twist in the standard inner product on LgΩ [BV ’10] (X, Y)φ :=

• dz

2πiz φ(z)X(z), Y(z) =

• du

2πi X(z), Y(z). The Zhukovsky variable u plays a central role in AdS/CFT, u = 21 + z4 1 − z4 .

Motivation Constrained IS GS superstring Lie dialgebra Conclusions

Standard R-matrix

With respect to the decomposition LgΩ = LgΩ

+ ∔ LgΩ −, let

R := π+ − π−, where π± : LgΩ → LgΩ

± are projections. It satisfies mCYBE,

[RX, RY] − R([RX, Y] + [X, RY]) = −[X, Y], ∀X, Y ∈ LgΩ, so that [X, Y]R := 1

2([RX, Y] + [X, RY]),

defines a second Lie bracket on LgΩ (dialgebra).

Remark

Due to the twist in the inner product, R is not skew: R∗ = −ϕ−1 ◦ R ◦ ϕ, ϕ(z) = φ(z)z−1.

Motivation Constrained IS GS superstring Lie dialgebra Conclusions

Conclusions & outlook (i)

• The Lax matrix LBPR admits an extension LBPR L which

(I) generates only first class integrals. (II) satisfies a strong zero curvature equation.

Leads to strong r/s-algebra, and hence integrability.

• How is algebraic curve of L affected by extension? What is

its significance? Dual gauge theory interpretation?

• Connection with pure spinors? BRST treatment of GS?
• Integrable structure of AdS/CFT at λ ≫ 1 is given by a Lie

dialgebra, with standard R-matrix but twisted inner product.

• Although loop algebra LgΩ is written in the z-variable, the

Zhukovsky map z → u enters naturally in inner product: (X, Y)φ =

• du

2πi X(z), Y(z). Zhukovsky twist intimately related to non-ultralocality!

Motivation Constrained IS GS superstring Lie dialgebra Conclusions

Conclusions & outlook (ii)

• Other Zm-gradings are also known to give rise to actions

admitting a Lax connection [Young ’05]. In this case twist and Zhukovsky map should be φ(z) = 4m zm (1 − zm)2 , u = 21 + zm 1 − zm

• How to quantize Lie dialgebas?

λ L

Lie dialgebra (ϕR∗ =−Rϕ) Lie bialgebra (r + r ∗ =t) Yangian (R∗ = R−1)

?

Yangian (R?)