Integrability and the AdS/CFT correspondence'physics.ipm.ac.ir/conferences/iss2009/lecture notes/rej.pdfAdam Rej (Imperial College London) 09.04.2009 1 / 29 Overview Introduction

"Integrability and the AdS/CFT correspondence"

Adam Rej

Theoretical Physics Group

Imperial College London

Talk at IPM String School (ISS2009), Teheran

09.04.2009

Adam Rej (Imperial College London) 09.04.2009 1 / 29

Overview

IntroductionThe N = 4 SYM and asymptotic integrabilityString theory on AdS5 × S5

The AdS/CFT correspondenceScaling functions of the AdS/CFT

The BES scaling function and its generalizationComparison with string theory

Twist operators and the BFKL equation

The Y-system conjecture

Conclusions


Overview






Conclusions


Overview






Conclusions


Overview






Conclusions


Overview






Conclusions


Overview






Conclusions


Overview






Conclusions


Overview






Conclusions


Overview






Conclusions


Overview






Conclusions


The N = 4 super Yang-Mills theory

The N = 4 SYM is a four-dimensional gauge theory with fourdifferent supersymmetry generators.

The building blocks of the theory are: Φm, Ψαa, Ψ̇aα̇ und Aµ.

The gauge group is SU(N) and the fields transform in the adjointrepresentation (U(x) ∈ SU(N)){

Φ; Ψ; Ψ̇}7→ U

{Φ; Ψ; Ψ̇

}U−1, Aµ 7→ UAµU−1−ig−1 ∂µU U−1 .

The Lagrangian reads

LYM = Tr(

14FµνFµν +

12DµΦnDµΦn −

14

g2[Φm,Φn][Φm,Φn]

+Ψ̇aα̇σ

α̇βµ DµΨβa −

12

igΨαaσabm εαβ[Φm,Ψβb]

−12

igΨ̇aα̇σ

mabε

α̇β̇[Φm, Ψ̇bβ̇

]

).






Φ; Ψ; Ψ̇}7→ U

{Φ; Ψ; Ψ̇

}U−1, Aµ 7→ UAµU−1−ig−1 ∂µU U−1 .


LYM = Tr(

14FµνFµν +

12DµΦnDµΦn −

14


+Ψ̇aα̇σ


12


−12

igΨ̇aα̇σ

mabε


]

).






Φ; Ψ; Ψ̇}7→ U

{Φ; Ψ; Ψ̇

}U−1, Aµ 7→ UAµU−1−ig−1 ∂µU U−1 .


LYM = Tr(

14FµνFµν +

12DµΦnDµΦn −

14


+Ψ̇aα̇σ


12


−12

igΨ̇aα̇σ

mabε


]

).






Φ; Ψ; Ψ̇}7→ U

{Φ; Ψ; Ψ̇

}U−1, Aµ 7→ UAµU−1−ig−1 ∂µU U−1 .


LYM = Tr(

14FµνFµν +

12DµΦnDµΦn −

14


+Ψ̇aα̇σ


12


−12

igΨ̇aα̇σ

mabε


]

).


Due to a large amount of supersymmetries the beta functionvanishes and the theory exhibits superconformal symmetry also atthe quantum level. The global symmetry algebra gets extendedso(1,3)⊕ so(6)→ psu(2,2|4).

There are no asymptotical distances and thus the physicalS-matrix cannot be defined. Correlation functions are well defined.Interesting observables are AD of the composed operators

O(x) = Tr (Φ Ψ ∗ ∗ . . .) ,

which receive quantum contributions

∆(g) = ∆0 + γ(g) .

The full dimensions are eigenvalues of the dilatation operator

DO(x) = ∆O(x)(g)O(x) .




O(x) = Tr (Φ Ψ ∗ ∗ . . .) ,


∆(g) = ∆0 + γ(g) .


DO(x) = ∆O(x)(g)O(x) .




O(x) = Tr (Φ Ψ ∗ ∗ . . .) ,


∆(g) = ∆0 + γ(g) .


DO(x) = ∆O(x)(g)O(x) .


Huge mixing problem!

O(x) = c1 Tr (Φ Ψ ∗ ∗ . . .) + c2 Tr (Ψ Φ ∗ ∗ . . .) + . . . .

There exist close subsectors with respect to the action of D. Thesu(2) is the simplest example

Tr(XM ZL−M

)+ all inequivalent permutations of X and Z .

The usual perturbative expansion applies

D =∑

n

D2n(N)g2n .

Even more symmetries appear in the planar limit(N →∞ ,g2 =

g2YMN

16π2 = const)

psu(2,2|4)→ psu(2,2|4) n u(1)∞ .



O(x) = c1 Tr (Φ Ψ ∗ ∗ . . .) + c2 Tr (Ψ Φ ∗ ∗ . . .) + . . . .


Tr(XM ZL−M



D =∑

n

D2n(N)g2n .


g2YMN

16π2 = const)

psu(2,2|4)→ psu(2,2|4) n u(1)∞ .



O(x) = c1 Tr (Φ Ψ ∗ ∗ . . .) + c2 Tr (Ψ Φ ∗ ∗ . . .) + . . . .


Tr(XM ZL−M



D =∑

n

D2n(N)g2n .


g2YMN

16π2 = const)

psu(2,2|4)→ psu(2,2|4) n u(1)∞ .



O(x) = c1 Tr (Φ Ψ ∗ ∗ . . .) + c2 Tr (Ψ Φ ∗ ∗ . . .) + . . . .


Tr(XM ZL−M



D =∑

n

D2n(N)g2n .


g2YMN

16π2 = const)

psu(2,2|4)→ psu(2,2|4) n u(1)∞ .


More precisely, the dilatation operator is a member of an infinitefamily of commuting charges.

This was rigorously proven for few first orders of perturbation theoryand some subsectors in the asymptotic region ( ` < L ) .

[N.Beisert ’03], [N.Beisert, V.Dippel, M.Staudacher ’04],[B.Zwiebel ’05], ...



This was rigorously proven for few first orders of perturbation theoryand some subsectors in the asymptotic region ( ` < L ) .



No interactions

Tr (X Z Z Z X Z X Z)


One-loop

Dsu(2)2 =

∑i

12

(1− ~σi ~σi+1)


Two-loop

Dsu(2)4 =

∑i

(−(1− ~σi ~σi+1) +14

(1− ~σi ~σi+2))


Wrapping!

Dsu(2)16 = ???



This proven rigorously for few first orders of perturbation theory andsome subsectors in the asymptotic region ( ` < L ) .


Whether this feature persists for arbitrary operators is still unclear.

The mixing problem in the asymptotic region can be solved bymeans of the methods of solid state physics

dilatation operator of the planarN = 4 SYM = Hamiltonian of an integrable spin chain .

The corresponding spin chain exhibits many novel features, whencompared to the usual spin chains considered in the literature:long-rangeness of the interactions increases with the order of theperturbation theory, length fluctuations, ...... but this spin chain is still integrable and can be solved bymeans of the Bethe ansatz:



























The excitation numbers Ki , i = 1, . . . ,7 are uniquely specified bythe eigenvalues of the elements of the Cartan algebra ofpsu(2,2|4).

The x±(u) variables are defined by: [N.Beisert, V.Dippel, M.Staudacher ’04]

x(u) =u2

(1 +

√1− 4 g2

u2

), x± = x(u ± i

2).

The eigenvalues of the higher conserved charges are given by

Qr =i

r − 1

K4∑j=1

(1

(x+(uj ))r−1 −1

(x−(uj ))r−1

).

The second of these charges Q2 corresponds to the eigenvalue ofthe dilatation operator (D − D0)

γ(g) = 2 g2 Q2 .




x(u) =u2

(1 +

√1− 4 g2

u2

), x± = x(u ± i

2).


Qr =i

r − 1

K4∑j=1

(1

(x+(uj ))r−1 −1

(x−(uj ))r−1

).


γ(g) = 2 g2 Q2 .




x(u) =u2

(1 +

√1− 4 g2

u2

), x± = x(u ± i

2).


Qr =i

r − 1

K4∑j=1

(1

(x+(uj ))r−1 −1

(x−(uj ))r−1

).


γ(g) = 2 g2 Q2 .




x(u) =u2

(1 +

√1− 4 g2

u2

), x± = x(u ± i

2).


Qr =i

r − 1

K4∑j=1

(1

(x+(uj ))r−1 −1

(x−(uj ))r−1

).


γ(g) = 2 g2 Q2 .


String theory on AdS5 × S5 product space

The IIB string theory on the super coset space PSU(2,2|4)SO(5)×SO(4,1) has

the same symmetry group as the N = 4 SYM theory.

The bosonic subspace is AdS5 × S5. The radii of the bothcomponents are equal (R).

The integrability of the classical equations of motion has beenproven rigorously. [I.Bena, J.Polchinski, R.Roiban ’03]

The quantization of the theory is, however, not understood.

In some limits semiclassical quantization can be applied.[S.Frolov, A.Tseytlin ’02]


































The AdS/CFT correspondence

According to the proposal of J. Maldacena (1997) the boththeories are equivalent under following identification of theparameters

gs=4πg2

N,

R2

α′=4πg . (1)

In particular: ∆ = E !There exist various formulations of the AdS/CFT duality

The strongest claims the equivalence of the both theories forarbitrary values of the parameters in (1).

⇒ In the weaker formulation the equivalence is expected in the planarlimit (N →∞ , g = const) only.Yet a different possibility is the coincidence of the asymptoticexpansion in 1

N of both theories but not of the non-perturbativecorrections...




gs=4πg2

N,

R2

α′=4πg . (1)








gs=4πg2

N,

R2

α′=4πg . (1)








gs=4πg2

N,

R2

α′=4πg . (1)








gs=4πg2

N,

R2

α′=4πg . (1)








gs=4πg2

N,

R2

α′=4πg . (1)






Scaling functions of the AdS/CFT

In the limit L→∞ the ABE constitute the exact all-loop solution tothe spectral problem (wrapping corrections may be neglected).Comparison with string theory!

A suitable subsector to compare is the sl(2) sector

O = Tr(DMZL

)+ . . . .

The Bethe equations for this sector can be reduced to[M.Staudacher ’04; N.Beisert, M.Staudacher ’05](

x+k

x−k

)L

=M∏

j 6=k

x−k − x+j

x+k − x−j

1− g2/x+k x−j

1− g2/x−k x+jσ2(uk ,uj ) .

and should correctly determine the AD up to O(g2L+4

).

In this subsector the AD of the ground states in the limitM →∞, L = j log M scales as follows:

γ(g) = f (g, j) log M +O(M0).





O = Tr(DMZL

)+ . . . .


x+k

x−k

)L

=M∏

j 6=k

x−k − x+j

x+k − x−j

1− g2/x+k x−j

1− g2/x−k x+jσ2(uk ,uj ) .


).


γ(g) = f (g, j) log M +O(M0).





O = Tr(DMZL

)+ . . . .


x+k

x−k

)L

=M∏

j 6=k

x−k − x+j

x+k − x−j

1− g2/x+k x−j

1− g2/x−k x+jσ2(uk ,uj ) .


).


γ(g) = f (g, j) log M +O(M0).





O = Tr(DMZL

)+ . . . .


x+k

x−k

)L

=M∏

j 6=k

x−k − x+j

x+k − x−j

1− g2/x+k x−j

1− g2/x−k x+jσ2(uk ,uj ) .


).


γ(g) = f (g, j) log M +O(M0).


The BES scaling function and its generalization

The case of j → 0 is known as the BES limit. First interpolatingfunction. [B.Eden, M.Staudacher ’06; N.Beisert, B.Eden, M.Staudacher ’06]

The j = 0 scaling function is uniquely specified by the solution ofthe BES equation

f ABA(g) =

(8 g2 − 64 g2

∫ ∞0

dt ′K̂ (2gt ,2gt ′) σ̂BES(t ′)).

The quantity, as follows from ABE, σ̂BES(t) satisfies a linearFredholm integral equations of the second kind.

The BES equation cannot be solved at arbitrary g!

Weak coupling expansion coincides up to four-loops with theknown field theory result

fABA(g) = 8 g2−83π2g4+

8845

π4g6−16(

73630

π6 + 4 ζ(3)2)

g8±. . . .





f ABA(g) =

(8 g2 − 64 g2

∫ ∞0

dt ′K̂ (2gt ,2gt ′) σ̂BES(t ′)).




fABA(g) = 8 g2−83π2g4+

8845

π4g6−16(

73630

π6 + 4 ζ(3)2)

g8±. . . .





f ABA(g) =

(8 g2 − 64 g2

∫ ∞0

dt ′K̂ (2gt ,2gt ′) σ̂BES(t ′)).




fABA(g) = 8 g2−83π2g4+

8845

π4g6−16(

73630

π6 + 4 ζ(3)2)

g8±. . . .





f ABA(g) =

(8 g2 − 64 g2

∫ ∞0

dt ′K̂ (2gt ,2gt ′) σ̂BES(t ′)).




fABA(g) = 8 g2−83π2g4+

8845

π4g6−16(

73630

π6 + 4 ζ(3)2)

g8±. . . .





f ABA(g) =

(8 g2 − 64 g2

∫ ∞0

dt ′K̂ (2gt ,2gt ′) σ̂BES(t ′)).




fABA(g) = 8 g2−83π2g4+

8845

π4g6−16(

73630

π6 + 4 ζ(3)2)

g8±. . . .


Comparison with string theory

The strong coupling expansion was understood many monthsafter the BES equation was published:

f ABA(g) = 4 g − 3 log 2π

− K4π2

1g− . . . .

On the other hand the sl(2) operators correspond to a foldedstring with spin M on the AdS5 and angular momentum J on S5.The limit M →∞ can be using the semiclassical methods. To firstfew orders:

E −M =

(4 g − 3 log 2

π− K

4π21g− . . .

)log M .



The strong coupling expansion was understood many monthsafter the BES equation was published:

f ABA(g) = 4 g − 3 log 2π

− K4π2

1g− . . . .

On the other hand the sl(2) operators correspond to a foldedstring with spin M on the AdS5 and angular momentum J on S5.The limit M →∞ can be using the semiclassical methods. To firstfew orders:

E −M =

(4 g − 3 log 2

π− K

4π21g− . . .

)log M .


The ABE also allow to find a closed integral equation also forf (g, j 6= 0)! [L.Freyhult, A.R., M.Staudacher ’07]

f (g, j) = 16(σ̂(0) + j

16

),

The fluctuation density σ̂(t) obeys again a linear integral equation.Its form, however, is significantly more complicated then in thej = 0 case.

The semiclassical arguments allow also to consider this limit onthe string theory side

[Frolov, Tseytlin, ’02], [Frolov, Tirziu, Tseytlin, ’06], [Alday, Maldacena, ’07]

E −M = h(g, j) log M +O(

M0).



f (g, j) = 16(σ̂(0) + j

16

),





M0).



f (g, j) = 16(σ̂(0) + j

16

),





M0).



Is f (g, j) = h(g, j) in this much more complicated case?

The methods of semiclassical quantization also enable to studythe j 6= 0 case! Moreover a conjecture has been put forward,according to which in the case of j � g the string theory shouldreduce to the O(6) sigma model.


The equivalence of f (g, j) and the energy density of the groundstate of the O(6) sigma model has been recently rigorouslyproven!

[Basso, Korchemsky, ’08]
















The analytic properties of sl(2) operators

The aforementioned asymptotic all-loop equations wereformulated in the asymptotic region. Are they correct beyond thewrapping order?

The twist-two operators (in the sl(2) twist equals the length) helpto answer this question

O = Tr(DMZ2

)+ . . . .

Interestingly enough closed expressions (as function of M) of theAD can be found to first few orders. At one-loop

γ(g) = 8 g2 S1(M) .

The harmonic sum S1 is the simplest of the so called nestedharmonic sums

Sa(M) =M∑

i=1

(sgn(a))i

i |a|,Sa1,...,an (M) =

M∑i=1

(sgn(a1))i

i |a1|Sa2,...,an (i) .





O = Tr(DMZ2

)+ . . . .


γ(g) = 8 g2 S1(M) .


Sa(M) =M∑

i=1

(sgn(a))i

i |a|,Sa1,...,an (M) =

M∑i=1

(sgn(a1))i

i |a1|Sa2,...,an (i) .





O = Tr(DMZ2

)+ . . . .


γ(g) = 8 g2 S1(M) .


Sa(M) =M∑

i=1

(sgn(a))i

i |a|,Sa1,...,an (M) =

M∑i=1

(sgn(a1))i

i |a1|Sa2,...,an (i) .





O = Tr(DMZ2

)+ . . . .


γ(g) = 8 g2 S1(M) .


Sa(M) =M∑

i=1

(sgn(a))i

i |a|,Sa1,...,an (M) =

M∑i=1

(sgn(a1))i

i |a1|Sa2,...,an (i) .




The BFKL equation

The harmonic sums can be analytically continued in M in thewhole complex space, e.g. [A. Kotikov, V. Velizhanin, 2005]

S1(M) =M∑

i=1

1i

= ψ(M + 1)− ψ(1) .

This is of fundamental importance, since the analytic structure(residues+order of the pole) at the points M = −1,−2, . . . can bepredicted by the methods of high-energy physics!

Consequently the correctness of the four-loop correction (asderived from ABE) can be tested, without performing four-loopcomputations in field theory!!


The BFKL equation


S1(M) =M∑

i=1

1i

= ψ(M + 1)− ψ(1) .




The BFKL equation


S1(M) =M∑

i=1

1i

= ψ(M + 1)− ψ(1) .




The BFKL equation, which predicts the leading poles atM = −1 + ω, originates from the analysis of the QCD amplitudesin the so called Regge kinematics

s � −t ∼ M2 .

The BFKL equation allows to compute the leading log scontribution to any 2→ 2 Amplitude

A(s, t) = α

∞∑j=0

aj (α log s)j +O(

1α

).

The BFKL equation in the momentum space is a Fredholmintegral equation of the second kind. The Mellin transformation(s → ω) allows to find the eigenfunctions of the BFKL kernel andthus to solve the equation.



s � −t ∼ M2 .


A(s, t) = α

∞∑j=0

aj (α log s)j +O(

1α

).




s � −t ∼ M2 .


A(s, t) = α

∞∑j=0

aj (α log s)j +O(

1α

).



For N = 4 SYM the eigenvalue of the BFKL kernel takes thefollowing form

ω

−4g2 = Ψ(−γ

2

)+ Ψ

(1 +

γ

2

)− 2Ψ(1) (2)

This equation determines the structure of the leading singularities.Using the perturbative expansion of the AD γ(g) one finds from(2), the following expansion at M = −1 + ω

γ = 2(−4 g2

ω

)−0

(−4 g2

ω

)2

+0(−4 g2

ω

)3

−4 ζ(3)

(−4 g2

ω

)4

±. . .

On the other hand the one-, two-, three- and four-loop corrections,as derived from ABE, give

γABA = 2(−4 g2

ω

)−0

(−4 g2

ω

)2

+0(−4 g2

ω

)3

−2(−4 g2)4

ω7 ± . . .



ω

−4g2 = Ψ(−γ

2

)+ Ψ

(1 +

γ

2

)− 2Ψ(1) (2)


γ = 2(−4 g2

ω

)−0

(−4 g2

ω

)2

+0(−4 g2

ω

)3

−4 ζ(3)

(−4 g2

ω

)4

±. . .


γABA = 2(−4 g2

ω

)−0

(−4 g2

ω

)2

+0(−4 g2

ω

)3

−2(−4 g2)4

ω7 ± . . .



ω

−4g2 = Ψ(−γ

2

)+ Ψ

(1 +

γ

2

)− 2Ψ(1) (2)


γ = 2(−4 g2

ω

)−0

(−4 g2

ω

)2

+0(−4 g2

ω

)3

−4 ζ(3)

(−4 g2

ω

)4

±. . .


γABA = 2(−4 g2

ω

)−0

(−4 g2

ω

)2

+0(−4 g2

ω

)3

−2(−4 g2)4

ω7 ± . . .


This proves unequivocally that the sl(2) ABE are not valid in thenon-asymptotic region: O

(g2k) , k > L + 2.

The reason are the wrapping interactions, which range exceedsthe length of the spin chain.

Due to their complexity it is not fully understood how to incorporatethem, though a robust proposal has been made recently...



(g2k) , k > L + 2.





(g2k) , k > L + 2.




...the Y-system conjecture

Gromov, Kazakov and Vieira have recently conjectured a systemof spectral equations, which should be valid to all orders in theperturbation theory and for operators of arbitrary length!

Subsequently a set of TBA equations has proposed by differentgroups.[D.Bombardielli, D. Fioravanti, R. Tateo ’09; N.Gromov, V.Kazakov, A.Kozak, P.Vieira ’09; G.Arutyunov, S.Frolov ’09]

This conjecture extensively exploits another conjecture - the mirrortheory of the psu(2,2|4) spin chain...

[S. Frolov, G. Arutyunov ’07; S.Frolov, G.Arutyunov ’08]

The final equations are conjectured to be TBA equations, withABE being large parameter (L) solution.

They allow to compute the wrapping correction to the four-loop ADof twist-two operators, which reproduces the BFKL pole structure!

Still a conjecture...















































Conclusions

The planar N = 4 SYM and the free IIB superstring theory onAdS5 × S5 seem to be two visages of the same novel integrablemodel!

The integrability in the asymptotic region is confirmed to be on afirm ground.

The full solution will be probably given (is given) by some kind ofY-system or TBA equations.

The underlying integrable model must be revealed! Manyunanswered questions (e.g. why is L discrete?)...

Dynamically developing field!


Conclusions







Conclusions







Conclusions







Conclusions







Documents

Integrability and the AdS/CFT correspondence'physics.ipm.ac.ir/conferences/iss2009/lecture notes/rej.pdfAdam Rej (Imperial College London) 09.04.2009 1 / 29 Overview Introduction