Integrability in AdS/CFT: open problems D. Serban, IPhT Saclay - - PowerPoint PPT Presentation

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Integrability in AdS/CFT: open problems D. Serban, IPhT Saclay - - PowerPoint PPT Presentation

Sep 08, 2022 509 likes •685 views

Integrability in AdS/CFT: open problems D. Serban, IPhT Saclay Miniworkshop on integrability in string theory, Galileo Galilei Institute Florence, 29-30 October 2008 Summary the AdS/CFT corespondence arguments for integrability

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SLIDE 1

Integrability in AdS/CFT:

  • pen problems
  • D. Serban,

IPhT Saclay

Miniworkshop on integrability in string theory, Galileo Galilei Institute Florence, 29-30 October 2008

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SLIDE 2

Summary

  • the AdS/CFT corespondence
  • arguments for integrability
  • conjectured Bethe Ansatz equations
  • spin chain vs. sigma model features
  • connection with the Hubbard model
  • TBA and finite size effects
  • integrability and the amplitudes
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SLIDE 3

N=4 gauge theory: superconformal symmetry PSL(2,2|4)

conformal group SO(4,2)≅SU(2,2) R-group SO(6)≅SU(4)

Field content SU(N) matrices:

Type IIB string theory on AdS5 x S5: sigma model on PSL(2,2|4)/SO(4,1)xSO(5)

[Maldacena 97] [Witten 98] [Gubser, Klebanov, Polyakov 98] [Metsaev, Tseytlin 98]

AdS/CFT correspondence

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SLIDE 4

Number of colors

N g

planar limit strong coupling

‘t Hooft coupling

  • h g2 =

g2

YM N

16 π2

String tension

T = 2g

String coupling

free strings classical strings

N gs = g N

Local operators

Scaling dimension R-charges

∆(g)

String states

Energy of the string

!"# S E

Angular momenta Ja

Tr (ΦI1ΦI2...ΦIL)

AdS/CFT correspondence

E(g), S1, S2, J1, J2, J3

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SLIDE 5

Integrability

One loop dilatation operator = integrable spin chain

tr ZZZW W ZZZW W W ZW ZZZZ . . .

ˆ D1 = 2

L

  • l=1

(1 − Pl,l+1)

s Z = Φ1 + iΦ2 d W = Φ3 + iΦ4: [Minahan, Zarembo, 02] [Lipatov, 98]

String sigma model is classically integrable

[Bena, Polchinski, Roiban, 02] [Kazakov, Marshakov, Minahan, Zarembo, 04]

solution of the classical sigma model in terms of an algebraic curve

[Bena, Polchinski, Roiban, 02] solution in terms of Bethe Ansatz equations string solution, e.g. [Frolov, Tseytlin, 02]

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SLIDE 6

Integrability

extends to the whole PSL(2,2|4) group survives at higher loops

[Beisert, Staudacher 03] [Beisert, Kristjansen, Staudacher 03] [Beisert 03-04]

There exists a model which is integrable for any value of the coupling constant g spin chain at g → 0

survives at higher loops survives at higher loops

sigma model at g → ∞

[conjecture]

?

perturbative N=4 SYM perturbative string theory on AdS5 x S5

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SLIDE 7

, x± + 1 x± = 1 g

  • u ± i

2

  • 1

=

K2

  • j=1

u1,k − u2,j + i

2

u1,k − u2,j − i

2 K4

  • j=1

1 − 1/x1,kx+

4,j

1 − 1/x1,kx−

4,j

, 1 =

K2

  • j=k

u2,k − u2,j − i u2,k − u2,j + i

K3

  • j=1

u2,k − u3,j + i

2

u2,k − u3,j − i

2 K1

  • j=1

u2,k − u1,j + i

2

u2,k − u1,j − i

2

, 1 =

K2

  • j=1

u3,k − u2,j + i

2

u3,k − u2,j − i

2 K4

  • j=1

x3,k − x+

4,j

x3,k − x−

4,j

,

  • −

−

  • −
  • x+

4,k

x−

4,k

L =

K4

  • j=k

u4,k − u4,j + i u4,k − u4,j − i σ2(x4,k, x4,j)

  • −

− ×

K1

  • j=1

1 − 1/x−

4,kx1,j

1 − 1/x+

4,kx1,j K3

  • j=1

x−

4,k − x3,j

x+

4,k − x3,j K5

  • j=1

x−

4,k − x5,j

x+

4,k − x5,j K7

  • j=1

1 − 1/x−

4,kx7,j

1 − 1/x+

4,kx7,j

,

  • j=1

−

  • j=1

−

  • j=1

−

  • j=1

− 1 =

K6

  • j=1

u5,k − u6,j + i

2

u5,k − u6,j − i

2 K4

  • j=1

x5,k − x+

4,j

x5,k − x−

4,j

, 1 =

K6

  • j=k

u6,k − u6,j − i u6,k − u6,j + i

K5

  • j=1

u6,k − u5,j + i

2

u6,k − u5,j − i

2 K7

  • j=1

u6,k − u7,j + i

2

u6,k − u7,j − i

2

, 1 =

K6

  • j=1

u7,k − u6,j + i

2

u7,k − u6,j − i

2 K4

  • j=1

1 − 1/x7,kx+

4,j

1 − 1/x7,kx−

4,j

.

Dressing factor

[Janik’06; Beisert-Hernandez-Lopez’06; Beisert-Eden-Staudacher’06]

map x + 1 x = u g psu(2,2|4) u1 u3 u2 u4 u7 u5 u6

[Beisert, Staudacher, 05] [Beisert, 05] [Arutynov, Frolov, Zamaklar, 06]

The all-loop Bethe Ansatz equations

magnon symmetry: centrally extended [su(2|2)]^2

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SLIDE 8

Particular features of the N=4 SYM integrable model

  • ne-parametrer integrable super-spin chain
  • long-range interaction (∼ Inozemtsev or Hubbard at half filling)
  • length-changing interactions
  • BAE only asymptotic (L → ∞)
  • crossing-like symmetry (particle/antiparticle) → dressing phase
  • not (exactly) relativistically invariant
  • scattering matrix does not depend on rapidity difference

∼ g2n

⇒ n + 1 spins

g3

E(p) = ±

  • 1 + 16g2 sin2 p/2 − 1

spin chain sigma model

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SLIDE 9

Connection(s) with the Hubbard model

2 seemingly unrelated connections with the 1d Hubbard model

  • su(2) sector reproducible from the Hubbard model at half filling

(except for the dressing phase)

  • Beisert’s su(2|2) symmetric S-matrix ∼ Hubbard Shastry’s R-matrix

⇒ hidden supersymmetry in the Hubbard model

[Beisert, 06] [Rej, Serban, Staudacher, 06]

eipkL =

M

  • j=1

j=k

uk − uj + i uk − uj − i , k

u(p) = 1 2 cot p 2

  • 1 + 16g2 sin2 p

2 ,

[Beisert, Dippel, Staudacher, 04]

E(p) =

  • 1 + 16g2 sin2 p

2 − 1 .

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SLIDE 10

H = 1 2 g

L

  • i=1
  • σ=↑,↓
  • eiφσ c†

i,σci+1,σ + e−iφσ c† i+1,σci,σ

  • − 1

2g2

L

  • i=1

c†

i,↑ci,↑c† i,↓ci,↓

The BDS model from Hubbard at half-filling

  • itinerant fermions with onsite repulsion (U=1/g ):
  • Hilbert space: 4 states per site:
  • at half filling N=L and g → 0 the spin states decouple:

singly-occupied states

E

→ 1/g2

| ↑↓↑↑ ... ↑>

  • fluctuations ~

g2

XXX model spin permutation

φ = π(L + 1) 2L

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SLIDE 11

The BDS model from Hubbard at half-filling

h2 = (1 − Pi,i+1) , h4 = −2(1 − Pi,i+1) + 1 2(1 − Pi,i+2) , h6 = 15 2 (1 − Pi,i+1) − 3(1 − Pi,i+2) + 1 2(1 − Pi,i+3) −1 2(1 − Pi,i+3)(1 − Pi+1,i+2) +1 2(1 − Pi,i+2)(1 − Pi+1,i+3) . higher orders: itinerant fermion making 2n step walks on the lattice

< > > <

g4

  • coincides with the dilatation operator up to three loops
  • corrects a result from [Klein, Seitz, 73]
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SLIDE 12

eie

qnL = M

  • j=1

uj − 2g sin( qn + φ) − i/2 uj − 2g sin( qn + φ) + i/2 , n = 1, . . . , L

L

  • n=1

uk − 2g sin( qn + φ) + i/2 uk − 2g sin( qn + φ) − i/2 =

M

  • j=1

j=k

uk − uj + i uk − uj − i , k = 1, . . . , M eiqnL =

M

  • j=1

uj − 2g sin(qn − φ) − i/2 uj − 2g sin(qn − φ) + i/2 , n = 1, . . ., 2M

2M

  • n=1

uk − 2g sin(qn − φ) + i/2 uk − 2g sin(qn − φ) − i/2 = −

M

  • j=1

j=k

uk − uj + i uk − uj − i , k = 1, . . . , M

The BDS model from Hubbard at half-filling

Lieb-Wu equations (half filling): [Lieb, Wu, 68] L large, M small

  • H(g; φ, φ) → −H(−g; π − φ, φ) − M

2g2

Shiba (particle/hole) transformation: Dual Lieb-Wu equations:

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SLIDE 13

u ± i/2 = 2g cos p 2 ∓ iβ

  • u(p) = 1

2 cot p 2

  • 1 + 16g2 sin2 p

2

The BDS model from Hubbard at half-filling

Bound-state solutions (strings): [Takahashi, 72]

  • 2g sin(q1 − φ)

2g sin(q2 − φ)

u

q1 − φ = π 2 + p 2 + i β , q2 − φ = π 2 + p 2 − i β

complex momenta: L → ∞

E(p) = 1 2g2

  • 1 + 16g2 sin2 p

2 − 1

  • eipkL =

M

  • j=1

j=k

uk − uj + i uk − uj − i , k Lieb-Wu eq. → BDS eq.

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SLIDE 14

But the Hubbard construction:

  • does not extends to other sectors than su(2), e.g. su(1|1)
  • does not take into account the dressing phase
  • is it possible to define an all-loop hamiltonian?
  • how to take into account the fine-size effects?

Remark: the su(2|2) symmetric S-matrix is also an essential ingredient of the new AdS4 x CP3 duality [Aharony, Bergman, Jafferis Maldacena, 08]

cf [Gromov, Vieira 08]

OSp(2,2|6)

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SLIDE 15

TBA program

Use the field-theoretical methods to compute finite-size corrections:

  • Lüscher terms [Janik, Lukowski 07,...]
  • put the theory on the cylinder and make a “double Wick rotation” 1/T → R

[Arutynov, Frolov 07; Bajnok, Janik,08]

  • difficulty: the rotated theory is not equivalent to the original one (“mirror theory”)

[Ambjorn, Janik, Kristjansen 05] [Bajnok, Janik,08]: from TBA [Fiamberti, Santambrogio, Sieg, Zanon ,08]: perturbative computation in N=4 SYM

simplest wrapping correction: the four loop L=4 (Konishi operator)

=

> > > > 1/T R

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SLIDE 16

The origin of integrability?

There is more in N=4 SYM than the dilatation operator...

  • the multigluon amplitudes have a particular structure at higher loops - >

BDS conjecture [Bern, Dixon, Smirnov 05] (fails for n>5)

  • this structure was checked at strong coupling for 4 (and many) gluons

[Alday, Maldacena 07]

  • dual superconformal symmetry [Drummond, Henn, Korchemsky, Sokatchev, 07-08]

(and duality between multigluon amplitudes and the Wilson loops with lightlike cusps)

Is there any connection between this structure and the integrability?

[Berkovits, Maldacena, 08] [Beisert, Ricci, Tseytlin, Wolf, 08]

Integrable open spin chain for gluon amplitudes [Lipatov, 08]

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SLIDE 17

Conclusion

  • the AdS/CFT correspondence provides an unusual integrable structure
  • it puts together many known integrable models into a highly symmetric structure
  • the complete definition still not under control ( what is the hamiltonian?)
  • what are the consequences of integrability on the overall structure of N=4 SYM?
  • which are the other (integrable) dualities?