VANISH - International Magic Magazine - VANISH MAGIC MAGAZINE EDITION 1
On Integrable subsectors of AdS/CFT and LLM...
Transcript of On Integrable subsectors of AdS/CFT and LLM...
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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
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Motivation and Background Gauge Theory SU(2) strings SL(2) strings Outlook
Talk Layout
1 Motivation and Background
2 Gauge Theory
3 SU(2) strings
4 SL(2) strings
5 Outlook
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Motivation and Background Gauge Theory SU(2) strings SL(2) strings 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
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Motivation and Background Gauge Theory SU(2) strings SL(2) strings Outlook
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
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Motivation and Background Gauge Theory SU(2) strings SL(2) strings Outlook
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
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Motivation and Background Gauge Theory SU(2) strings SL(2) strings Outlook
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
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Motivation and Background Gauge Theory SU(2) strings SL(2) strings Outlook
Localised excitations
Localised excitations with Schur polynomials [De Mello Koch, Mathwin,
HJRvZ, 1601.06914]
Background B =
Excitation r = χ, ,
(Z ,Y )
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Motivation and Background Gauge Theory SU(2) strings SL(2) strings Outlook
Localised excitations
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...
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Motivation and Background Gauge Theory SU(2) strings SL(2) strings Outlook
Localised excitations
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
)
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Motivation and Background Gauge Theory SU(2) strings SL(2) strings Outlook
Localised excitations
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
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Motivation and Background Gauge Theory SU(2) strings SL(2) strings Outlook
Localised excitations
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
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Motivation and Background Gauge Theory SU(2) strings SL(2) strings Outlook
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
)
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Motivation and Background Gauge Theory SU(2) strings SL(2) strings Outlook
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
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Motivation and Background Gauge Theory SU(2) strings SL(2) strings Outlook
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
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Motivation and Background Gauge Theory SU(2) strings SL(2) strings Outlook
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
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Motivation and Background Gauge Theory SU(2) strings SL(2) strings Outlook
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)
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Motivation and Background Gauge Theory SU(2) strings SL(2) strings Outlook
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)
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Motivation and Background Gauge Theory SU(2) strings SL(2) strings Outlook
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 · · ·
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Motivation and Background Gauge Theory SU(2) strings SL(2) strings Outlook
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
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Motivation and Background Gauge Theory SU(2) strings SL(2) strings Outlook
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 · · ·
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Motivation and Background Gauge Theory SU(2) strings SL(2) strings Outlook
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)
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Motivation and Background Gauge Theory SU(2) strings SL(2) strings Outlook
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
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Motivation and Background Gauge Theory SU(2) strings SL(2) strings Outlook
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
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Motivation and Background Gauge Theory SU(2) strings SL(2) strings Outlook
Finite Size GM
A systematic expansion, capturing all the log(f ) pieces, ofthese types of integrals can be performed
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Motivation and Background Gauge Theory SU(2) strings SL(2) strings Outlook
Finite Size GM
A systematic expansion, capturing all the log(f ) pieces, ofthese types of integrals can be performed
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)
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Motivation and Background Gauge Theory SU(2) strings SL(2) strings Outlook
Finite Size GM
A systematic expansion, capturing all the log(f ) pieces, ofthese types of integrals can be performed
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)
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Motivation and Background Gauge Theory SU(2) strings SL(2) strings Outlook
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
))
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Motivation and Background Gauge Theory SU(2) strings SL(2) strings Outlook
Dyonic LLM Magnons
O ∼∑l eipl · · ·Z1 Z1 Z1 (Y12)
J2 Z2 Z2 Z2 · · · [Harsuda, Suzuki,
0801.0747]
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Motivation and Background Gauge Theory SU(2) strings SL(2) strings Outlook
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)
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Motivation and Background Gauge Theory SU(2) strings SL(2) strings Outlook
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
)
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Motivation and Background Gauge Theory SU(2) strings SL(2) strings Outlook
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
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Motivation and Background Gauge Theory SU(2) strings SL(2) strings Outlook
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.
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Motivation and Background Gauge Theory SU(2) strings SL(2) strings Outlook
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).
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Motivation and Background Gauge Theory SU(2) strings SL(2) strings Outlook
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
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Motivation and Background Gauge Theory SU(2) strings SL(2) strings Outlook
Thank you for your attention!
Research supported by the Claude Leon Foundation