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(Non-)perturbative tests of the AdSCFT correspondence

a r X i v :h e p -t h /0103112v 2 4 A p r 2001

1

(Non-)perturbative tests of the AdS/CFT correspondence

Massimo BIANCHI ?

Department of Applied Mathematics and Theoretical Physics,University of Cambridge,Centre for Mathematical Sciences,Wilberforce Road,CB30WA Cambridge,England

I summarize perturbative and non-perturbative ?eld theory tests of the holographic correspondence between type IIB superstring on AdS 5×S 5and N =4SYM theory.The holographic duality between D-instantons and YM instantons is brie?y described.Non renormalization of two-and three-point functions of CPO’s and their extremal and next-to-extremal correlators are then reviewed.Finally,partial non-renormalization of four-point functions of lowest CPO’s is analyzed in view of the interpretation of short distance logarithmic behaviours in terms of anomalous dimensions of unprotected operators.

The AdS/CFT correspondence [1]is an un-precedented tool in the study of the interplay be-tween gauge theory and gravity.In the simplest case it relates type IIB superstring on AdS 5×S 5to N =4supersymmetric Yang-Mills theory (SYM)with gauge group SU (N ).The correspon-dence is “holographic”in that the gauge theory lives on the boundary of AdS.The conjecture is motivated by the study of the low-energy dynam-ics of D-branes [2].Their open strings excita-tions include massless vector supermultiplets and a stack of N coincident D3-branes is governed by U (N )N =4SYM in D =4.The remark-able fact about this gauge theory is its exact su-perconformal invariance.This re?ects into the the dilaton being constant and the metric being nowhere singular for the D3-brane solution!The D3-brane can thus be viewed as a smooth soliton interpolating between maximally supersymmetric ?at Minkowski spacetime at in?nity and maxi-mally supersymmetric AdS 5×S 5near the hori-zon.The AdS scale L is related to the RR 5-form

?ux N by L 4=4πg s Nα′2

.Due to the red-shift,the geometry near the horizon captures the low-energy limit where bulk supergravity e?ectively decouples from the boundary dynamics.Open-closed string duality suggests a perfect equiva-

?s = χ =

θ

2

posal,Alexander Polyakov observed that one of

the drawbacks of previous attempts,i.e.the lack of zig-zag symmetry of the non-critical string, could be cured by assuming the?ow to a?xed point at vanishing Liouville?eldρ=0[4].Mal-dacena’s proposal then looks like what the Doctor orders in that it puts forward the existence of a ?fth coordinateρ(transverse to the boundary) that could be identi?ed with the Liouville mode or,possibly equivalently,with a renormalization scale.What sounds surprising is that N=4SYM theory is not con?ning and has no mass-gap in the superconformal phase.

The superisometry group of AdS5×S5, SU(2,2|4),acts by superconformal transforma-tions on the boundary CFT.Unitary irreducible representations of P SU(2,2|4)are labelled by the quantum numbers{?,j L,j R;[k,l,m]}.?is the scaling dimension,(j L,j R)denote the SU(2)L×SU(2)R spins,and[k,l,m]are the Dynkin labels of an irrep r of the SU(4)R-symmetry group. The fundamental SYM?elds{?i,λA,Fμν}de-scribing the lowest lying open-string excitations belong to the singleton representation and live on the boundary.Gauge invariant composite operators O?dual to type IIB bulk?eldsΦM form doubleton or long multiplets.In particu-lar,the“ultra-short”N=4supercurrent mul-tiplet is dual to the“massless”AdS supergrav-ity multiplet.Operators dual to higher KK exci-tations assemble into short multiplets with spin j=j L+j R≤2and Dynkin labels k,m≤2. Their lowest components are chiral primary op-erators(CPO’s)

= σT r[?σ(i1)?σ(i2)...?σ(i?)?...] Q i1i2...i?

[0,?,0]

of dimension?=?belonging to the?-fold trace-less symmetric product of the6of SU(4)≈SO(6).Other shortenings are possible for multi-trace operators.For scalar primaries,for in-stance,shortening occurs when r=[k,?,k]and ?=?+2k or r=[k+2n,?,k]and?=?+2k+3n [5].The spin ranges over6or7units respec-

tively.Operators dual to string excitations with

AdS masses of order1/

3

lation functions and their dual amplitudes [12].In

SYM it

corresponds to a chiral rotation ac-companied by a continuous electric-magnetic du-ality transformation.Its type IIB counterpart is the U (1)B anomalous chiral symmetry.When supergravity loops and higher derivative string corrections are negligible the “bonus”symmetry becomes a true symmetry.Independently of the coupling κand N ,all two-point correlation func-tions,three-point functions with at most one in-sertion of unprotected operators and four-point functions of single-trace protected operators seem to respect this symmetry [12].

In a superconformal ?eld theory,two-point functions of normalized primary operators O ?are completely speci?ed by their dimensions

O ??(x )O ?(y ) =

1

(ρ2+(x ?x ′)2)?

(7)

It is not at all a coincidence that K ?resembles a YM instanton form factor.Plugging K ?into the scalar ?eld equation one ?nds the following mass-to-dimension relation (ML )2=?(??4)(8)

and its inverse

?=2±

2The

other possible near-boundary behaviour Φ(ρ,x )→ρ?V (x )corresponds to turning on a VEV O =V for the operator dual to Φ.

is enforced [16].Only the positive branch is rel-evant for N =4SYM.Carefully computing the

quadratic on-shell type IIB action and di?erenti-ating wrt to the sources J exactly reproduce the ?eld theory result (4).

Three-point functions,although largely ?xed by superconformal invariance,encode the dynam-ics of the theory since one can in principle re-construct all correlation functions by factoriza-tion.A particularly interesting class of three-point functions are those of CPO’s Q ?1(x 1)Q ?2(x 2)Q ?3(x 3) =

C (?1,?2,?3)

(10)

where x ij =x i ?x j and Σ=?1+?2+?3.The trilinear couplings C (?1,?2,?3)can be easily com-puted at weak coupling for large N .In order to perform the dual AdS computation one has to go beyond the linearized approximation.Quadratic terms in the ?eld equations,or equivalently cu-bic terms in the action are necessary.These are in general very complicated to extract but the computation turns out to be feasible for CPO’s.Quite remarkably,one ?nds the same result as in free-?eld theory at large N [17].The exact match-ing suggests the validity of a non-renormalization theorem for any κand N .This has been tested at one-loop [18]and to two-loops [19,20].A judi-cious use of the “bonus”U (1)B symmetry [12]in the context of N =2harmonic superspace gives a demonstration of the non-renormalization of two-and three-point functions of CPO’s [21].The ex-tremal case,?1=?2+?3,is subtler.We will return to this issue after discussing non-perturbative ef-fects.

Using the AdS/CFT dictionary (1),the charge-k type IIB D-instanton action coincides with the action of a charge-k YM instanton.This strongly indicates a correspondence between these sources of non-perturbative e?ects [7,6].Moreover,it is known that the k =1SU (2)YM instanton moduli space coincides with Euclidean AdS 5and the same is true for a type IIB D-instanton on AdS 5.The ?fth radial coordinate transverse to the boundary plays the role of the YM instanton size ρ.The correspondence can be made more quantitative by noticing that the super-instanton

measure contains an overall factor g 8

Y M

that arises

4

from the combination of bosonic and fermionic

zero-mode norms and exactly matches the power expected on the basis of the AdS/CFT correspon-

dence.

The computation of the one-instanton contri-

bution to the SYM correlation function G16=

Λ(x1)...Λ(x16) ,whereΛA=T r(FμνσμνλA) is the fermionic composite operator dual to the

type IIB dilatino,and its comparison with the D-

instanton contribution to the dual type IIB am-plitude has given the?rst truly dynamical test of the correspondence[8].Correlation functions of this kind are almost completely determined by the systematics of fermionic zero-modes in the YM instanton background.Performing(broken) superconformal transformations on the instanton ?eld-strength

Fμν(ρ0,x0;x)=K2(ρ0,x0;x)σμν(11) yields the relevant gaugino zero-modes

1

λA=

5 As far as the instanton contributions are con-

cerned,it is easy to check that(14)cannot absorb

the relevant16zero-modes.The induced scalar

zero-modes read

1

?i=

(1?(17)

(x?y)2?(0)

γ2

γlog[μ2(x?y)2]+

C C

D K(z?w,?w)

(x?y)?A+?B??K

6

?K =?(0)

K +γK .Similarly C IJ K =C (0)

IJ K +ηIJ K .Indeed,although three-point functions of single-trace

CPO’s are not renormalised beyond tree

level

[18],

a priori nothing can be said concerning corrections to three-point functions also involving unprotected operators.

Neglecting descendants and keeping the lowest order terms in γand η

Q A (x )Q B (y )Q C (z )Q D (w ) (1)=(19)

K

O K (y )O K (w ) (0)

2

C (0)

AB K

C (0)

CD

K

log

(x ?y )2(z ?w )2

6

?k ?k ).

(20)

Due to the lack of a manifestly N =4o?-shell

super?eld formalism,perturbative computations have to be either performed in components or in one of the two available o?-shell super?eld for-malisms.Although the number of diagrams is typically larger in the N =1super?eld approach [32,20,40]its simplicity makes it more accessible than the less familiar N =2harmonic superpace [33,19].It is remarkable that up to some overall normalization factors depending on the (not al-ways standard)conventions adopted the two re-sults are in perfect quantitative agreement with one another and in qualitative agreement with the AdS predictions at strong coupling.

Instead of computing the most general four-point function of lowest CPO’s we simply display

(2π)12x 212x 234x 213x 224

B (r,s ),

where B (r,s )is a box-type integral that can be

expressed as a combination of logarithms and dilogarithms as follows B (r,s )=

1

p {ln r ln s ?ln

2

r +s ?1?√2

?2Li 2

2

p

(22)

?2Li 2

2

p

As indicated,B (r,s )depends only on the two in-dependent conformally invariant cross ratios

r =x 212x 234

x 213x 224

.(23)

and

p =1+r 2+s 2?2r ?2s ?2rs .

(24)

Thanks to crossing symmetry,the only other a priori independent four-point function of lowest CPO’s is

G V (x 1,x 1,x 3,x 4)=

(25)

(φ1)2(x 1)(φ?1)2(x 2)(φ1)2

(x 3)(φ?1)2(x 4)

The non-perturbative contributions,computed in [8,32],are quite involved and we refrain to display them.We simply notice that the relation [31]

(x 1?x 3)2(x 2?x 4)2G H (x 1,x 2,x 3,x 4)=(x 1?x 4)2(x 2?x 3)2G V (x 1,x 2,x 3,x 4)(26)can be easily derived from the systematics of the fermionic zero-modes [32].Some additional e?ort allows one to derive it in perturbation theory [33,19,32,20].The AdS computation is even more involved and the ?nal result is quite uninspiring [36].

7 In order to extract some physics one has to per-

form an OPE analysis.Restricting for brevity our

attention to the sectors1,20′,84,and105the

results can be summarized as follows4.

In the105one?nds only subdominant loga-

rithms,consistently with the expected absence

of any corrections to the dimension of protected

single-and double-trace operators of dimension

?=4in the105[32,20,38,35,30].

In the84channel,the dominant contribution

at one and two loops is purely logarithmic and

consistent with the exchange of the operator K84

in the Konishi multiplet.The absence of domi-

nant logarithmic terms in the instanton as well

as AdS results suggests con?rms the absence of

any corrections to the dimension and trilinear of

a protected operator?D84of dimension4,de?ned

by subtracting the Konishi scalar K84from the

projection on the84of the naive normal ordered

product of two Q20′[32,20,38,35,30].

In the20′sector,there is no dominant loga-

rithm suggesting a vanishing anomalous dimen-

sion for the unprotected operator:Q20′Q20′:20′

[38].This striking result seems to be a conse-

quence of the partial non-renormalization of four-

point functions of lowest CPO’s[31,39]summa-

rized by(26)that is valid not only at each order

in perturbation theory(beyond tree level!)but

also non-perturbatively and at strong coupling

(AdS).In order to disentangle the various scalar

operators of naive dimension4exchanged in this

channel it is necessary to compute other indepen-

dent four-point functions involving the insertions

of the lowest Konishi operator K1[40].

The analysis of the singlet channel is very

complicated by the presence of a large number

of operators.In perturbation theory one has

logarithmically-dressed double pole associated to

the exchange of K1with5

γ(1) K =3

g2

Y M

N

16π2

(27)

Non-perturbative and strong coupling results

8

REFERENCES

1.J.Maldacena,Adv.Theor.Math.Phys.2

(1998)231,[hep-th/9711200].

2.O.Aharony,S.S.Gubser,J.Maldacena,H.

Ooguri,Y.Oz,hep-th/9905111.

3.M.B.Green and M.Gutperle,Nucl.Phys.

B498(1997)195,[hep-th/9701093];JHEP 9801(1998)005,[hep-th/9711107].

4. A.M.Polyakov,Nucl.Phys.Proc.Suppl.68

(1998)1[hep-th/9711002].

5.L.Andrianopoli and S.Ferrara,Phys.Lett.B

430(1998)248[hep-th/9803171];Lett.Math.

Phys.46(1998)265[hep-th/9807150];Lett.

Math.Phys.48(1999)145[hep-th/9812067];

eidem, E.Sokatchev and B.Zupnik,hep-th/9912007.

6. E.Witten,JHEP9807(1998)006[hep-

th/9805112].

7.T.Banks and M.B.Green,JHEP9805

(1998)002,[hep-th/9804170].

8.M.Bianchi,M. B.Green,S.Kovacs and

G.Rossi,JHEP9808(1998)013[hep-

th/9807033].

9.M.Bianchi,G.Pradisi and A.Sagnotti,Nucl.

Phys.B376(1992)365;M.Bianchi,Nucl.

Phys.B528(1998)73[hep-th/9711201];

E.Witten,JHEP9802(1998)006[hep-

th/9712028].

10.M.Gunaydin, D.Minic and M.Zager-

mann,Nucl.Phys.B544(1999)737[hep-th/9810226];Nucl.Phys.B534(1998)96 [hep-th/9806042].

11.A.Bilal and C.Chu,Nucl.Phys.B

562(1999)181[hep-th/9907106];hep-th/0003129.

12.K.Intriligator,Nucl.Phys.B551(1999)575,

[hep-th/9811047];K.Intriligator,W.Skiba, hep-th/9905020.

13.S.S.Gubser,I.R.Klebanov and A.M.

Polyakov,Phys.Lett.B428(1998)10,[hep-th/9802109].

14.E.Witten,Adv.Theor.Math.Phys.2(1998)

253[hep-th/9802150].

15.D.Z.Freedman,S.D.Mathur,A.Matusis and

L.Rastelli,Nucl.Phys.B546(1999)96[hep-th/9804058].

16.P.Breitenlohner and D.Z.Freedman,Ann.

Phys.144(1982)249;Phys.Lett.B115 (1982)197.

17.S.Lee,S.Minwalla,M.Rangamani and

N.Seiberg,Adv.Theor.Math.Phys.2(1998) 697[hep-th/9806074].

18.E.D’Hoker,D.Z.Freedman and W.Skiba,

Phys.Rev.D59(1999)045008,[hep-th/9807098].

19.B.Eden,P.S.Howe, C.Schubert, E.

Sokatchev and P.C.West,Nucl.Phys.B581 (2000)523[hep-th/0001138].

20.M.Bianchi,S.Kovacs,G.Rossi and

Y.S.Stanev,Nucl.Phys.B584(2000)216 [hep-th/0003203].

21.P.S.Howe, C.Schubert, E.Sokatchev and

P.C.West,Phys.Lett.B444(1998)341[hep-th/9808162].

22.N.Dorey,V.V.Khoze,M.P.Mattis and S.

Vandoren,Phys.Lett.B442(1998)145, [hep-th/9808157];N.Dorey,T.J.Hollowood, V.V.Khoze,M.P.Mattis and S.Vandoren, hep-th/9810243.

23.M.B.Green and S.Kovacs,in preparation.

24.H.Liu and A.A.Tseytlin,“Dilaton-?xed

scalar correlators and AdS5x S5-SYM correspondence”,JHEP9910(1999)003 hep-th/9906151.

25.G.Arutyunov and S.Frolov,“Some cubic

couplings in type IIB supergravity on AdS5 x S5and three-point functions in SYM(4)at large N”,Phys.Rev.D61(2000)064009, hep-th/9907085.

26.E.D’Hoker,D.Z.Freedman,S.D.Mathur,

A.Matusis and L.Rastelli,hep-th/9908160.

27.M.Bianchi and S.Kovacs,Phys.Lett.B468

(1999)102[hep-th/9910016].

28.B.Eden,P.S.Howe, C.Schubert, E.

Sokatchev and P.C.West,Nucl.Phys.B571 (2000)71[hep-th/9910011].Phys.Lett.B 472(2000)323[hep-th/9910150].Phys.Lett.

B494(2000)141[hep-th/0004102].

29.J.Erdmenger and M.Perez-Victoria,Phys.

Rev.D62(2000)045008[hep-th/9912250].

30.E.D’Hoker,J.Erdmenger,D.Z.Freedman

and M.Perez-Victoria,Nucl.Phys.B589 (2000)3[hep-th/0003218].

31.B.Eden,A.C.Petkou,C.Schubert and E.

Sokatchev,“Partial non renormalisation of

9

the stress-tensor four-point function in N=4 SYM and AdS/CFT”,hep-th/0009106. 32.M.Bianchi,S.Kovacs,G.Rossi and

Y.S.Stanev,JHEP9908(1999)020[hep-th/9906188].

33.B.Eden,P.S.Howe, C.Schubert, E.

Sokatchev and P.C.West,Nucl.Phys.B557 (1999)355[hep-th/9811172];Phys.Lett.B 466(1999)20[hep-th/9906051].

34.H.Liu and A.A.Tseytlin,Phys.Rev.D59

(1999)086002[hep-th/9807097].

35.E.D’Hoker,D.Z.Freedman,S.D.Mathur,A.

Matusis and L.Rastelli,Nucl.Phys.B562 (1999)353[hep-th/9903196].

36.G.Arutyunov and S.Frolov,Phys.Rev.D62

(2000)064016[hep-th/0002170].

37.D.Anselmi, D.Z.Freedman,M.T.Grisaru

and A.A.Johansen,Phys.Lett.B394(1997) 329,[hep-th/9608125];Nucl.Phys.B526 (1998)543,[hep-th/9708042];eidem and J.

Erlich,Phys.Rev.D57(1998)7570,[hep-th/9711035];D.Anselmi,[hep-th/9809192].

38.G.Arutyunov,S.Frolov and A.Petkou,hep-

th/0010137;Nucl.Phys.B586(2000)547 [hep-th/0005182].

39.G.Arutyunov,B.Eden,A.C.Petkou and

E.Sokatchev,hep-th/0103230.

40.M.Bianchi,S.Kovacs,G.C.Rossi and Ya.S.

Stanev,“Properties of the Konishi multiplet in N=4SYM theory”,hep-th/0104016.

41.M.Bianchi and S.Kovacs,hep-th/9811060.

42.J.Distler and F.Zamora,JHEP0005(2000)

005[hep-th/9911040];Adv.Theor.Math.

Phys.2(1999)1405[hep-th/9810206];L.Gi-rardello,M.Petrini,M.Porrati and A.Zaf-faroni,Nucl.Phys.B569(2000)451[hep-th/9909047];JHEP9905(1999)026[hep-th/9903026];JHEP9812(1998)022[hep-th/9810126].

43.D.Z.Freedman,S.S.Gubser,K.Pilch and

N.P.Warner,hep-th/9904017;JHEP0007 (2000)038[hep-th/9906194];A.Brandhuber and K.Sfetsos,hep-th/9906201.

44.K.Pilch and N.P.Warner,hep-th/0006066;

Nucl.Phys.B594(2001)209.

45.M.Bianchi,O.DeWolfe, D.Z.Freedman

and K.Pilch,JHEP0101(2001)021[hep-th/0009156];A.Brandhuber and K.Sfetsos,

JHEP0012(2000)014[hep-th/0010048]. 46.C.Angelantonj and A.Armoni,Phys.Lett.B

482(2000)329[hep-th/0003050];M.Bianchi and J. F.Morales,JHEP0008(2000)035 [hep-th/0006176];hep-th/0101104.

47.Y.Cai and J.Polchinski,Nucl.Phys.B296

(1988)91.M.Bianchi and A.Sagnotti,Phys.

Lett.B247(1990)517;Nucl.Phys.B361 (1991)519;M.Bianchi and J.F.Morales, JHEP0003(2000)030[hep-th/0002149];J.

Schwarz and E.Witten,hep-th/0103099.

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