On Integrable subsectors of AdS/CFT and LLM geometriesneo.phys.wits.ac.za/public/workshop_11/vanzyl.pdf · match, (iii) three-point functions vanish This is a conjecture with very

Motivation and Background Gauge Theory SU(2) strings SL(2) strings Outlook

On Integrable subsectors of AdS/CFT and LLMgeometries

Jaco van Zyl

Mandelstam Institute for Theoretical PhysicsUniversity of the Witwatersrand

6 September 2018, Kruger


Talk Layout

1 Motivation and Background

2 Gauge Theory

3 SU(2) strings

4 SL(2) strings

5 Outlook


LLM Magnons, R de Mello Koch, C Mathwin, JvZ,1601.06914

Integrable Subsectors from Holography, R de Mello Koch, MKim, JvZ, 1802.01367

Semi-Classical SL(2) string on LLM backgrounds, M Kim,JvZ, 1805.12460

Exciting LLM Geometries, R de Mello Koch, J Huang, LTribelhorn, 1806.06586


Motivation

Conjectured duality between type II B string theory onAdS5 × S5 ⇔ N = 4 super-Yang Mills gauge theory

One-to-one mapping between gauge invariant operators ⇔string theory states

Energies of string states ⇔ dimensions of operators

The conjecture has passed a multitude of non-trivial checks

Planar AdS/CFT ↔ integrability

More to be studied in the non-planar sector


LLM geometries [Lin, Lunin, Maldacena, hep-th/0409174]

ds2 =−h−2(dt +Vidx

i )2+ h2(dy2+ dx idx i )+ yeGdΩ3+ ye−GdΩ3

h−2 = 2y cosh(G ), z = 12 tanh(G )

y∂yVi = ǫij∂jz , y(∂iVj − ∂jVi ) = ǫij∂yz .

Laplace equation ∂i∂iz + y∂y∂y z

y= 0

12 BPS and regular geometries

Simple one-to-one map between Young diagrams and y = 0boundary condition

LLM plane coloring - z = 12 white and z = −1

2 black


Schur Polynomials

A basis for gauge invariant operators labelled by Youngdiagrams

Tr(Z 3) = χ (Z )− χ (Z ) + χ (Z )

〈χR(Z )χS(Z )〉 ∝ δRS

Lengths of sides of the Young diagram ↔ areas of rings onLLM plane

Vertical sides map to black rings, horizontal sides to whiterings


Localised excitations

Localised excitations with Schur polynomials [De Mello Koch, Mathwin,

HJRvZ, 1601.06914]

Background B =

Excitation r = χ, ,

(Z ,Y )



Localised excitations with Schur polynomials [De Mello Koch, Mathwin,

HJRvZ, 1601.06914]

Background B =

Excitation r = χ, ,

(Z ,Y )

rB = 1fBχ

, ,

(Z ,Y )

We have in mind much larger operators...



SU(2) sector - operators with Z ’s and Y ’s

D1 = Tr(

[Z ,Y ][ ddZ, ddY

])

[Beisert, hep-th/0407277]

D1χTr(Zn1YZn2Y ··· )B = 1) terms acting on the trace ⊕ 2)terms mixing the background and the trace

For exictations localised at a distant corner: Terms of type 2)scale like 1

N

λ→ λeff = λr20 is the only difference(

r0 =√

Neff

N

)



Localised excitations ↔ emergent gauge theory [ De Mello Koch,

Huang, Tribelhorn, 1806.06586]

Planar limit of this emergent gauge theory isomorphic toplanar N = 4 SYM

(i) Isomorphism between operators, (ii) scaling dimensionsmatch, (iii) three-point functions vanish



Localised excitations ↔ emergent gauge theory [ De Mello Koch,

Huang, Tribelhorn, 1806.06586]

Planar limit of this emergent gauge theory isomorphic toplanar N = 4 SYM

(i) Isomorphism between operators, (ii) scaling dimensionsmatch, (iii) three-point functions vanish

This is a conjecture with very clear predictions to test


Weak coupling S-matrix

Background of Z -fields with impurities - residual su(2|2)2symmetry [Beisert, hep-th/0511082], [Hofman, Maldacena, 0708.2272], [De Mello Koch,

Tahiridimbisoa, Mathwin, 1506.05224]

Exact dispersion for a single Y -impurity:

E =√

1 + λ4π2 (r

21 r

22 )− λ

2π2 g2r1r2 cos(p)

r1 = r2 ≡ r0 : E =

√

1 +λr20π2 sin2

(

p2

)


Weak coupling S-matrix

Background of Z -fields with impurities - residual su(2|2)2symmetry [Beisert, hep-th/0511082], [Hofman, Maldacena, 0708.2272], [De Mello Koch,

Tahiridimbisoa, Mathwin, 1506.05224]

Exact dispersion for a single Y -impurity:

E =√

1 + λ4π2 (r

21 r

22 )− λ

2π2 g2r1r2 cos(p)

r1 = r2 ≡ r0 : E =

√

1 +λr20π2 sin2

(

p2

)

Example:Tr(

Y ddV

Y ddW

)

Tr (VZn1YZn2)corner1 Tr (WZn3YZn4)corner2


The symmetry also fixes the kinematic part of the S-matrixuniquely, S = e iφDSkin

According to the proposal the rescaled coupling should also bepresent in the dynamical phase φD .

Asymptotic S-matrix∑

nm ψYX (l1, l2)eip1l1e ip2l2 · · ·ZZZZYZZ · · ·ZZXZZ · · ·

Dilatation operator ∼ spin chain Hamiltonian

Diagonalise to read of energies and S-matrix elements


Distant corners diagram - generalised spin chain

∑

nm ψYX (l1, l2)eip1l1e ip2l2 · · ·Z1Z1Z1Z1Y12Z2Z2 · · ·Z2Z2X23Z3Z3 · · ·

Weights of terms after the dilatation operator action arealtered - these can be computed (Swaps ∼ √

ri rj)

Two-loop dilatation operator on these spin chains may bediagonalised with ansatz for ψXY (l1, l2), ψYX (l1, l2), ψαβ(l1, l2)

Consistency check w.r.t symmetry argument → dispersionmatches; two-loop S-matrix matches

The one-loop dynamical phase thus matches


Point-like strings

Z -fields added to a corner are dual to point-like strings on a

t, r , φ, y = 0 subspace of an LLM background

ds2 =(

r2h2(r)− h−2(r)(1 + Vφ(r))2)

dt2 + 2(r2h2(r)

−2Vφh−2(r)(1 + Vφ(r)))dφdt

+(

r2h2(r)− h−2(r)Vφ(r)2)

dφ2 + h2(r)dr2

Vφ =∑

ri<r cir2i

r2−r2i

+∑

ri>r cir2

r2i−r

+ o(y2)

ci = ±1 for black and white rings respectively.

h−2 =

√

∣

∣

∣

2r∂rVφ

∣

∣

∣+ o(y2)


r = r0, φ = −t(τ, σ)

E − J = 0

Rotate along geodesic at the speed of light

Same solutions are found on the LLM backgrounds...

...only now there are several possible geodesics on the LLMplane

Whenever r0 represents a pole in Vφ(r)


LLM Giant Magnons

SU(2) sector [Hofman, Maldacena, hep-th/0604135], [De Mello Koch, Kim, HJRvZ, 1802.01367]

O ∼∑l eipl · · ·Z1 Z1 Z1 Y12 Z2 Z2 Z2 · · ·


LLM Giant Magnons

SU(2) sector [Hofman, Maldacena, hep-th/0604135], [De Mello Koch, Kim, HJRvZ, 1802.01367]

O ∼∑l eipl · · ·Z1 Z1 Z1 Y12 Z2 Z2 Z2 · · ·

Nambu Goto: SNG =√λ

2π

∫

√

(X X ′)2 − X 2X ′2dσdτ

Any LLM geometry r = c sec(φ(τ, σ) + t(τ, σ) + φ0)

Conserved charges E =√λ

2π

∫

∂LNG∂ t

, J =√λ

2π

∫

∂LNG∂φ

Giant LLM magnon: E − J =√λ

2π

√

r21 + r22 − 2r1r2 cos(∆φ)

Identify ∆φ ∼ p


Finite Size GM

Large but finite number of Z -fields, string segment endpointspulled perturbatively close to a geodesic [Arutyunov, Frolov, Zamaklar,

hep-th/0606126]

O ∼∑l eipl · · · (Z1)

l−1Y12 (Z2)

J+1−l · · ·We will first considerO ∼∑l e

ipl · · · (Z0)l−1 Y00 (Z0)

J+1−l · · ·


Finite Size GM

Large but finite number of Z -fields, string segment endpointspulled perturbatively close to a geodesic [Arutyunov, Frolov, Zamaklar,

hep-th/0606126]

O ∼∑l eipl · · · (Z1)

l−1Y12 (Z2)

J+1−l · · ·We will first considerO ∼∑l e

ipl · · · (Z0)l−1 Y00 (Z0)

J+1−l · · ·Ansatz: r = r(κφ(τ, σ) + t(τ, σ))

t = κτ ;φ = σ − τ ; r = r(σ)


Equation of motion are solved by

r ′(σ) =κr√

1− r2

C2

√

(1− κ)2h4(r)r2 − (κ− (1− κ)Vφ(r))2

∫ σmax

σminF (σ)dσ =

∫ rend1rmid

1r ′(σ)F (r)dr +

∫ rend2rmid

1r ′(σ)F (r)dr

rmid = C while rendi is any value for which r ′(σ) → ∞We recover the infinite size solution when κ = 1


Finite Size GM

It is useful to define

r = r0

√

1− C 2z2

C = r0

√

1− C 2

κ(f ) =1

1 +[

(r0√

1− C 2f 2)h2(z = f ) + Vφ(z = f )]−1

Vφ(z) =1

C2z2(1− C 2z2 + Vφ) ; Vφ(0) = 0

All integrals we now wish to evaluate look like∫ 1fG (f , z)dz


Finite Size GM

A systematic expansion, capturing all the log(f ) pieces, ofthese types of integrals can be performed


Finite Size GM


E−J =

√λ

πr0 sin

(p

2

)

−4

√λ

πr0 sin

3(p

2)e

−2

(

π(J0−V )√λr0 sin(

p2 )

+1

)

+o(f 4)


Finite Size GM


E−J =

√λ

πr0 sin

(p

2

)

−4

√λ

πr0 sin

3(p

2)e

−2

(

π(J0−V )√λr0 sin(

p2 )

+1

)

+o(f 4)

J0 →√λ

π

∫ r0

C

√r2 − C 2

rVφ(r) dr

J0 − V →√λ

π

∫ r0

C

√r2 − C 2

r

(

r2

r20 − r2

)

dr (1)


Finite Size GM

2π√λ(E − J)

=√

r21 + r22 − 2r1r2 cos(p)

+1

16(r21 + r22 − 2r1r2 cos(p))32

(

4(r21 + 3r21 r22 + r42 )

+r1r2(−15(r21 + r22 ) cos(p) + 12r1r2 cos(2p)

−(r21 + r22 ) cos(3p)))

f 2.

J0−V1−V2 = −√λ

2π

√

r21 + r22 − 2r1r2 cos(p)

(

1 + log

(

f

4

))


Dyonic LLM Magnons

O ∼∑l eipl · · ·Z1 Z1 Z1 (Y12)

J2 Z2 Z2 Z2 · · · [Harsuda, Suzuki,

0801.0747]


Dyonic LLM Magnons

O ∼∑l eipl · · ·Z1 Z1 Z1 (Y12)

J2 Z2 Z2 Z2 · · · [Harsuda, Suzuki,

0801.0747]

ds2 = ds2t,r ,φ + h2(r)dθ2

Infinite size dyonic giant magnon

φ = cos−2(

cr(σ,τ)

)

− t(σ, τ) ;

θ = a(t(σ, τ) + cF (r(σ, τ)))

F ′(r) =Vφ(r)

r√r2−c2

We have E , J and J2 =√λ

2π

∫

∂LNG∂θ

E − J =√

J22 + λ4π2 (r

21 + r22 )− λ

2π2 r1r2 cos(p)


Finite size Dyonic GM

t = κτ ;φ = σ − τ, r = r(σ), θ = a(τ + f (r(σ)))

Longer expressions for r ′(σ) and f ′(r)

Can tune the turning points of the solution with a similarprescription as before

E − J−√

J22 + r20λπ2 sin

2(

p2

)

= −4e−2K

√λ

πr0 sin

4( p2 )

√

J22

λ

π2 r20

+sin2( p2 )

+o(f 4)

K =J22+

λ

π2 r20 sin2( p

2 )J22+

λ

π2 r20 sin4( p

2 )

(

J0−V√

J22+r20λ

π2 sin2( p2 )

+ 1

)

sin2(

p2

)


SL(2) strings

Need a test away from the highly restrictive su(2|2)2symmetry [Frolov, Tseytlin, hep-th/0204226], [Kim, HJRvZ, 1805.12460]

Schematically O ∼ Tr (Zn1(D+Z )s1 · · ·Znk (D+Z )

sk )

White region of LLM plane

t = κτ, φ = ατ, θ = ωτ

Conformal gauge, (r ′(σ))2 =−(κ+ α)2r2 + e2G+(κ2 − ω2 + 2κ(κ+ α)Vφ + (κ+ α)2V 2

φ )

r ′(σ) = 0 at ring edges and can choose rmax(κ, α, ω).

ω2 = (κ+ (α+ κ)Vφ(rm))2 + 1

2(α+ κ)2V ′φ(rm) ; α 6= −κ

Conserved charges E , J, S


Classical string length

L(κ, α, ω) =

∫

dσr ′(σ)×√

−(κ+ α)2r2 + eG+(κ2 − ω2 + 2κ(κ+ α)Vφ + (κ+ α)2V 2φ )

Remarkably E = ∂κL, S = ∂ωL, J = ∂αL.


Classical string length

L(κ, α, ω) =

∫

dσr ′(σ)×√

−(κ+ α)2r2 + eG+(κ2 − ω2 + 2κ(κ+ α)Vφ + (κ+ α)2V 2φ )

Remarkably E = ∂κL, S = ∂ωL, J = ∂αL.

Short string limit: α = −κ+ κr0(rm − r0)β,

ω = −κ√1 + β + o(rm − r0)

E 2 = J2 + 2S√

J2 + n2

4 r20λ+ o(S2).


Outlook

Many more checks should still be performed: more generaltreatment of SL(2) strings, scattering solutions, multi-spinstring solutions, quantum corrections...

Can the integrable structure be made explicit?

Can we learn more for the case where the magnon stretchesbetween different edges?

D-branes in these backgrounds


Thank you for your attention!

Research supported by the Claude Leon Foundation

Documents

On Integrable subsectors of AdS/CFT and LLM geometriesneo.phys.wits.ac.za/public/workshop_11/vanzyl.pdf · match, (iii) three-point functions vanish This is a conjecture with very