Lower dimensional AdS/CFT correspondencesand their integrability

Alessandro Torrielli

University of Surrey, UK

Selected Topics in Theoretical High Energy PhysicsTbilisi, Georgia

September 23, 2015

Reviews −→ arXiv: 1012.3982 1012.4005 1104.2474with Beisert et al.

Recent −→ arXiv: 1211.1952 1303.5995 1306.2512AdS3 with Borsato, Ohlsson Sax, Sfondrini and Stefanski

−→ arXiv: 1406.2840 1505.06767with Pittelli, Prinsloo, Regelskis and Wolf

Recent −→ arXiv: 1407.0303 and to appearAdS2 with Hoare, Pittelli

Summary

1. Integrable string backgrounds

2. Three dimensional case: the AdS3 × S3 × T 4 story

3. Remarkable similarities, and some difference

4. Massless integrability

5. Two dimensional case: AdS2 × S2 × T 6 and long reps

STRING BACKGROUNDSMetsaev-Tseytlin 9805028

Classification by Zarembo 1003.0465 - see also Wulff 1505.03525

SEMI-SYMMETRIC SUPERSPACES

Supergroup cosets with Z4 symmetry for integrability (Lax pair)

In order to be consistent string theories:

• Beta function vanishes

• Sigma-model central charge equals 26 to balance bc ghosts

[one loop in α′ - planar (genus 0) analysis]

−→

Vanishing Killing form: psu(n|n), osp(2n + 2|2n) - and d(2, 1;α)

FIVE DIMENSIONS: AdS5/CFT4

AdS5 × S5 dual to N = 4 Super Yang Mills in 4D

Superalgebra from psu family: psu(2, 2|4)

32 supersymmetries

• Coset: PSU(2, 2|4)/[SO(4, 1)× SO(5)]

• Relation among parameters: gs = g2YM , λ = g2

YMN ∝ R4/α′2

• Central charge equals 26

• Ramond-Ramond 5-form ∼ N (rank of SYM U(N) gauge)

−→

• Spectral problem ”solved” (significant portion of people sitting all around you)

FOUR DIMENSIONS: AdS4/CFT3

AdS4 × CP3 dual to N = 6 Chern-Simons in 3D

Superalgebra from osp family: osp(6|4)

24 supersymmetries

• Coset: OSP(6|4)/[SO(3, 1)× U(3)]

• Relation among params: RCP3 = 2RAdS , λ = N/k ∝ R4/α′2

• Central charge equals 26

• R-R: 4-form ∼ N (U(N)× U(N) gauge), 2-form ∼ k (level)

−→

• Spectral problem “similar” (in a way) to five-dimensions - h(λ)

THREE DIMENSIONS: AdS3/CFT2

1. AdS3 × S3 × S3 × S1 dual to mystery CFT2 [Tong 1402.5135]

Superalgebra from osp family (deformation): d(2, 1;α)2

16 supersymmetries: large N = (4, 4)

• Coset: D(2, 1;α)2/[SU(1, 1)× SU(2)× SU(2)]

• Relation among parameters: R2AdS/R

2S+

= α = 1− R2AdS/R

2S−

• Central charge < 26, needs extra S1 CFT

• Can mix R-R & NS-NS Cagnazzo-Zarembo 1209.4049 Hoare-Tseytlin 1303.1037

−→ Babichenko-Stefanski-Zarembo 0912.1723

• Massless modes appear

THREE DIMENSIONS: AdS3/CFT2

2. AdS3 × S3 × T 4 dual to CFT2 [Ohlsson Sax, Sfondrini, Stefanski 1411.3676]

Superalgebra from psu family: psu(1, 1|2)2

16 supersymmetries: small N = (4, 4)

• Coset: PSU(1, 1|2)2/[SU(1, 1)× SU(2)]

• Relation among parameters: RAdS = RS , λ ∝ R4/α′2

• Central charge < 26, needs extra T 4 CFT

• Mixed flux

−→ Symmetric orbifold, Higher spins

• More massless modes

TWO DIMENSIONS: AdS2/CFT1

AdS2× S2×T 6 dual to mystery sup conf quantum mechanics

Superalgebra from psu family: psu(1, 1|2)

8 supersymmetries

• Coset: PSU(1, 1|2)/U(1)2

• Relation among parameters: RAdS = RS , λ ∝ R4/α′2

• Central charge < 26, needs extra T 6 CFT

• Massless modes Sorokin-Tseytlin-Wulff-Zarembo 1104.1793 Murugan-Sundin-Wulff 1209.6062

• Recently conjectured integrable S-matrix Hoare-Pittelli-AT 1407.0303

There are many more, with and without time in the coset

STANDARD PICTURE

String modes as particles −→ Toy model for QFT

2D INTEGRABLE SYSTEMS

Exact quantisation of solitons

AdS/CFT CORRESPONDENCE

Strings in 10D

Fields at same holographic point “Spins” on a chain

tr[A(x)B(x) d

dx iC(x)...

]−→ |AB C ′...〉

Example: su(2) sector - one loop

Z = φ1 + iφ2 W = φ3 + iφ4

O{α}(x) = tr[Z ...ZW ...WZ ...W ] −→ ↓ ... ↓ ↑...↑ ↓ ...↑

Corresponding Hamiltonian −→ su(2) Heisenberg spin-chain(’isotropic’)

WHAT THIS PRODUCES FOR US

Spectrum of an interacting CFT via 2D solvable model

Necessity of taking ’t Hooft limit (planar Feynman graphs)

COORDINATE BETHE ANSATZ [Bethe ’31] (Subseq. version: Algebraic BA)

Infinite spin-chain limit: 2-particle state

|ψ >=∑n1<n2

ψ(n1, n2) |Z ...Z V Z ...Z WZ ...〉

n1↑ n2

↑

ψ(n1, n2) = e ip1n1 + ip2n2 + S(p1, p2) e ip2n1 + ip1n2

S(p1, p2) S-matrix: magnon scattering

↓ ... ↓ ⇑ ↓ ... ↓ ... ↓ ... ↓ ⇑ ↓ ... ↓

Periodicity restored by Bethe Equations

AdS/CFT [Arutyunov-Frolov-Staudacher ’04; Beisert-Staudacher ’05]

RESIDUAL ALGEBRAS

Spin-chain sites: reps of the superconformal algebra

After fixing (BMN) vacuum the magnon (spin-wave) excitationscarry reps of smaller subalgebras

Reduced scattering problem

• Five dimensions: psu(2|2)2

• Four dimensions: psu(2|2)

• Three dim: psu(1|1)L × psu(1|1)R in S3 × S1Borsato, Ohlsson Sax, Sfondrini

[psu(1|1)L × psu(1|1)R ]2 in T 4and with Stefanski, AT

• Two dimensions: psu(1|1)2Abbott-Murugan-Sundin-Wulff 1308.1370

−→ all centrally extended [as first shown by Beisert in 5D]

• Quantum inverse scattering method

Massless magnons

New feature in three and two dimensions:integrability with massless particles

Separate story: to be discussed later

Apart from this, remarkable universality of structures

• Zhukovsky variables

• Centrally-extended superalgebras (also with vanishing Killing form)

• Braided coproduct

• Yangian-like symmetry firmly ubiquitous

• Secret (bonus) symmetry review de Leeuw et al 1204.2366

−→ exotic quantum group underlying all the cases

AdS3/CFT2: The T 4 casereview Sfondrini 1406.2971

Start from: AdS3 × S3 × S3 × S1︸ ︷︷ ︸ , 1R2

++ 1

R2−

= 1R2

AdS3

limiting caseα→1 T 4

Superconformal algebra: D(2, 1;α)× D(2, 1;α)︸ ︷︷ ︸×u(1)

Inonu-Wigner α→1 psu(1, 1|2)× psu(1, 1|2)× u(1)3 × u(1)

BMN spectrum: 8b + 8f masses: (1, α, 1− α, 0)× 2b + f → (1, 0)× 4b + f

BMN ground state: point-like string on light-like geodesic in AdS3 × S3

su(1|1)2: algebra relations and fund rep (1 copy) ad − bc = 1

{QL,GL} = HL QL|φ〉 = a |ψ〉 QR |φ〉 = 0{QR ,GR} = HR QL|ψ〉 = 0 QR |ψ〉 = c |φZ+〉{QL,QR} = C GL|φ〉 = 0 GR |φ〉 = b |ψZ−〉{GL,GR} = C† GL|ψ〉 = d |φ〉 GR |ψ〉 = 0

and barred

• Action on states: Dynamical Spin-Chain

(HL + HR) |p〉 = ε(p) |p〉 ε(p) =

√1 + 16 h2 sin2 p

2

C† |p〉 = c(p) |p Z−〉 C |p〉 = c(p) |p Z+〉

Z+(−): one site of the chain is added (removed)h algebr. freedom→ coupling

Interpreted as a non-trivial Hopf algebra:

E.g. C† ⊗ 1 |p1〉 ⊗ |p2〉 =

C† ⊗ 1∑

n1<<n2

e i p1 n1 + i p2 n2 | · · ·Z Z V Z · · ·Z︸ ︷︷ ︸n2−n1−1

W Z · · · 〉

(rescale n2) = c(p1) e ip2 |p1〉 ⊗ |p2〉

S ∆(C†) = S [C† ⊗ 1 + 1⊗ C†] = S [e ip2C†local ⊗ 1 + 1⊗ C†local ]

∆(C†local) = C†local ⊗ e ip + 1⊗ C†local[Gomez-Hernandez ’06; Plefka-Spill-AT ’06]

CROSSING as shown by Janik in 5D

Connects L with R, e.g.

Σ(QL(p)

)= C−1

[QR

]str(p) C

Σ −→ antipode and str −→ supertranspose

Crossing equations can be derived purely from Hopf algebra

Antipode crossing compatible with level 1 Yangian:

S-matrix is projection of universal R-matrix of Yangian of u(1|1)

Zhukovsky variables crossing: x± → 1x±

mass 1: x+ + 1x+ − x− − 1

x−= i

h

Crossing symmetry of R-matrix

(C−1 ⊗ 1)[RL,R

]str1( 1

x±1, x±2

)(C ⊗ 1) RL,L(x±1 , x

±2 ) = 1⊗ 1 etc.

RL,Lpq |φ〉 ⊗ |φ〉 = ρL,Lpq

x+q − x−p

x−q − x+p

e i(p−q)

4 |φ〉 ⊗ |φ〉

RL,Lpq |φ〉 ⊗ |ψ〉 = ρL,Lpq

x+q − x+

p

x−q − x+p

e−i(p+q)

4 |φ〉 ⊗ |ψ〉+ ρL,Lpq

x+q − x−q

x−q − x+p

ηpηq|ψ〉 ⊗ |φ〉

RL,Lpq |ψ〉 ⊗ |φ〉 = ρL,Lpq

x−q − x−p

x−q − x+p

e i(p+q)

4 |ψ〉 ⊗ |φ〉+ ρL,Lpq

x+p − x−p

x−q − x+p

ηqηp|φ〉 ⊗ |ψ〉

RL,Lpq |ψ〉 ⊗ |ψ〉 = ρL,Lpq e−i

(p−q)4 |ψ〉 ⊗ |ψ〉 etc.

Analogs of Janik’s equation (recalling the two copies)

ρL,Lpq ρR,Lpq =

(x+q − x+

p

x+q − x−p

)2

ρL,Rpq ρR,Rpq =

((x−q x+

p − 1)

(x+q x+

p − 1)

x+q

x−q

)2

etc .

Borsato, Ohlsson-Sax, Sfondrini, Stefanski, AT 1306.2512 solution

Perturbation theory: massive OK (massless not ok)Abbott, Aniceto, Beccaria, Bianchi, Forini, Hoare, Levkovich-Maslyuk, Macorini, Murugan, Roiban, Rughoonauth,

Tseytlin, Sundin, Wulff

Yangian: L-R encompassing version by Regelskis 1503.03799

AdS2: a different central extension of su(1|1)2Hoare-Pittelli-AT 1407.0303

su(1|1): algebra relations and fund rep (1 copy) Kac module

{Q,G} = H Q|φ〉 = a |ψ〉 Q|φ〉 = b |φ〉{Q,Q} = C G|φ〉 = c |ψ〉 G|φ〉 = d |ψ〉{G,G} = C†

Does not contradict Iohara-Koga theorem - psu(1|1) not simple

No constraint on H2 − CC † [For other AdS’s, = mass ∈ Z]

−→ long reps [short reps]

Shortening condition happens for H2 − CC † = 0 massless 1D indecomp block

Yangian and long representations Hoare-Pittelli-AT to appear

(cf. AdS5: de Leeuw, Arutyunov, AT 0912.0209)

EXACT S-MATRICES

{for reviews} [P. Dorey ’98; Arutyunov-Frolov ’09]

2D integrable massive S-matrices

• No particle production/annihilation

• Equality of initial and final sets of momenta

• Factorisation: SM−→M =∏

S 2−→2

(all info in 2-body processes)

Extrapolate from relativistic case

S 2−→2 = S(u1 − u2) ≡ S(u) [Ei = mi cosh ui , pi = mi sinh ui ]

Crossing symmetry S12(u) = S 21(iπ − u)

Yang-Baxter Equation (YBE) S12 S13 S23 = S23 S13 S12

Bootstrap S3B(u) = S32

(u + iθ1

)S31

(u + iθ2

)[Zamolodchikov-Zamolodchikov ’79]

• S-matrix real analytic: S(s∗) = S∗(s)Mandelstam s = 2m2(1 + cosh u) for equal masses

• S-matrix simple poles ↔ bound states

0 < s < 4m2 ↔ u = iϑ ↔ p = m sinh u = im sinϑ

DRESSING FACTOR

S(u) = Φ(u) S(u)

S(u) acts as 1 on highest weight state

• Dressing factor Φ(u) not fixed by symmetry, matrix S(u) yes

• Dressing factor Φ(u) constrained by crossing S12(u) = S−112

(u − iπ)

• Dressing factor Φ(u) essential for pole structure

S-MATRIX and HOPF ALGEBRA

{for review} [Delius ’95]

Algebraic treatment −→ relativistic & non (spin chains)

R : V1 ⊗ V2 −→ V1 ⊗ V2 R is called Universal R-matrix

Vi carries a representation of symmetry algebra A

Symmetry on ‘in’ states: coproduct

∆ : A −→ A⊗ A

[∆(a),∆(b)] = ∆([a, b]) (homomorphism)

(P∆)R = R ∆

P (graded) permutation P∆ ‘opposite’ coproduct ∆op (‘out’)

Lie (super)algebras can have ‘trivial’ coproduct

∆op(Q) = ∆(Q) = Q ⊗ 1 + 1⊗ Q ∀Q ∈ A

non trivial → quantum groups

Hopf algebra: coproduct + extra algebraic structurese.g. antipode (antiparticles) + list of axioms

The Yangian is an ∞-dim non-abelian Hopf algebra

{books } [Chari-Pressley ’94; Kassel ’95; Etingof-Schiffmann ’98]

(Σ⊗ 1)R = R−1 = (1⊗ Σ−1)R

R12R13R23 = R23R13R12

(∆⊗ 1)R = R13 R23 ; (1⊗ ∆)R = R13 R12

Universal R-matrix is abstract object whichgenerates all S-matrices by projecting into irreps

R −→fundam⊗ fundam Sfund1,fund2

−→b.state⊗ b.state Sbound1,bound2

−→inf .dim⊗ inf .dim Sinf .1,inf .2

Inclusive of (minimal) dressing factors

MASSLESS LIMIT[Zamolodchikov-Zamolodchikov ’92]

E = m cosh u p = m sinh u E 2 − p2 = m2

Lorentz boost shifts u: the two branches are connected

- 3 - 2 - 1 1 2 3p

0.5

1.0

1.5

2.0

2.5

3.0

E H p L

SEND m→ 0 E = m2

(eu + e−u

)p = m

2

(eu − e−u

)u = u0 + ν

m

2e |u0| = M = finite u0 → ±∞

Two branches of E =√p2

• left moving u0 → +∞

E = Meν+ p = Meν+ E = p ν+ ∈ (−∞,∞)

• right moving u0 → −∞

E = Me−ν− p = −Me−ν− E = −p ν− ∈ (−∞,∞)

- 3 - 2 - 1 1 2 3p

0.5

1.0

1.5

2.0

2.5

3.0

E H p L

• Left-left and right-right scattering does not occur (v = c)

• Many theorems of integrability do not hold

• Nevertheless, ∃ notion of Yang-Baxter: limiting S-matrices are solutions

Left-left and right-right S-matrix does not depend on M:

u1 − u2 = |u0|+ ν1,+ − (|u0|+ ν2,+) = ν1,+ − ν2,+

Left-right S-matrix does, and degenerates in the limit:

u1 − u2 = |u0|+ ν1,+ − (−|u0|+ ν2,−) = 2|u0|+ ν1,+ − ν2,−

• SLL and SRR do not depend on M

−→ removing M describes a CFT via L and R Yang-Baxter

Thank you very much