3 December 1998

Ž .Physics Letters B 442 1998 145–151

Yang-Mills instantons in the large-N limitand the AdSrCFT correspondence

Nicholas Dorey a,b, Valentin V. Khoze c, Michael P. Mattis d, Stefan Vandoren a

a Physics Department, UniÕersity of Wales Swansea, Swansea SA2 8PP, UKb Physics Department, UniÕersity of Washington, Seattle, WA 98195, USA

c Department of Physics, Centre for Particle Theory, UniÕersity of Durham, Durham DH1 3LE, UKd Theoretical DiÕision T-8, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

Received 30 August 1998Editor: M. Cvetic

Abstract

Ž .We examine a certain 16-fermion correlator in NNs4 supersymmetric SU N gauge theory in 4 dimensions.Ž .Generalizing recent SU 2 results of Bianchi, Green, Kovacs and Rossi, we calculate the exact N-dependence of the

effective 16-fermion vertex at the 1-instanton level, and find precise agreement in the large-N limit with the prediction of thetype IIB superstring on AdS =S5. This suggests that the string theory prediction for the 1-instanton amplitude considered5

here is not corrected by higher-order terms in the aX expansion. q 1998 Published by Elsevier Science B.V. All rights

reserved.

w x ŽIn interesting recent work by Maldacena 1 seew x .also 2 for relevant previous work by other authors ,

the large-N limit of NNs4 supersymmetric Yang-Ž .Mills theory SYM has been related to the low-en-

ergy behavior of type IIB superstrings on AdS =S5.5

In the conjectured correspondence, the gauge cou-pling g and vacuum angle u of the four-dimensionaltheory are given by

f Ž0.'gs 4p g s 4p e , us2p c 1Ž .( st

Here g is the string coupling while cŽ0. is thest

expectation value of the Ramond-Ramond scalar of

IIB string theory. Also N appears explicitly, throughthe relation

X2 2(L ra s g N , 2Ž .

where L is the radius of both the AdS and S55

factors of the background. One striking consequenceof these identifications is that the action of a Yang-Mills instanton in the gauge theory is equated to thatof a D-instanton in the string theory. The relationbetween these two seemingly different types of in-stantons has been investigated further by Bianchi,

Ž . w xGreen, Kovacs and Rossi BGKR 3 . These authors

0370-2693r98r$ - see front matter q 1998 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 98 01233-7

( )N. Dorey et al.rPhysics Letters B 442 1998 145–151146

compared the leading semiclassical contribution of asingle Yang-Mills instanton in the NNs4 theory

Ž .with gauge group SU 2 with that of a D-instanton inthe IIB theory on AdS =S5. Specifically, the gauge5

theory is at its conformal point where all the HiggsVEVs vanish and the instanton is an exact solutionof the field equations. BGKR found an interestingcorrespondence between the moduli-space and zeromodes of the two classical configurations and be-tween the resulting contributions to various correla-tion functions in their respective theories. A particu-larly attractive aspect of the correspondence is thatthe scale size of the Yang-Mills instanton is mappedonto the radial position of the D-instanton in AdS .5

This is consistent with the interpretation of Malda-cena’s conjecture in which the four-dimensional

w xgauge theory lives on the boundary of AdS 4,5 .5w xAn obvious puzzle about the results of Ref. 3 is

that such an agreement between weakly-coupledgauge theory and the low-energy effective field the-ory of the IIB string is found for gauge group

Ž .SU N with Ns2. In contrast, Maldacena’s conjec-ture only predicts such an agreement in the large-Nlimit. 1 In this letter we generalize to arbitrary N the

Ž .SU 2 calculation of the 1-instanton contribution to aw xsixteen fermion correlator considered in Ref. 3 . In

Ž .the SU N SYM theory a single instanton has a totalof 8 N adjoint fermion zero modes and, for Ns2,the resulting Grassmann integrations are saturated bythe sixteen fermion insertions of this correlator. ForN)2 there are additional fermion zero modes whichmust be lifted in order to obtain a non-zero result.Thus the main technical challenge in generalizing thecalculation of BGKR is to account correctly for this

w xlifting. As in several other cases in three 6 and fourw x7 dimensions, this can be accomplished by deter-mining the Grassmann quadrilinear term in the in-stanton action. Our final result for the instantoncontribution has a complicated algebraic dependenceon N. However, in the large-N limit, the dependenceon both g 2 and N is exactly that extracted from the

1 More accurately, as explained below, the low-energy IIBprediction should only hold when g 2N is large. On the otherhand, to justify using semiclassical methods we must also have g 2

small. These conditions together certainly require N to be large.

superstring by BGKR, and earlier by Banks andw xGreen 8 , on the basis of Maldacena’s conjecture.

It is important to emphasize that, although we areworking in the large-N limit, the agreement we havefound is still somewhat mysterious. Maldacena’sconjecture relates the ratio of the string length-scale,a

X, and the radius of curvature of the spacetime, L,Ž .to the gauge theory parameters via Eq. 2 . This

means that the aX expansion of the IIB theory, on

w x w xwhich the prediction of Refs. 3 and 8 relies, isonly valid in the regime of large g 2N. In particular,it is not obvious that stringy corrections to the

Žlow-energy IIB effective action i.e., higher orders inX .the a expansion can be neglected when comparing

to our semiclassical calculation. A similar situationarises for the calculation of an eight-fermion correla-tor in the three-dimensional theory with sixteen su-

w xpercharges 9,6 , where weak-coupling multi-instan-ton calculations agree exactly with the predictions

Ž . w xbased on the M atrix model of M-theory 9,10 ,which, strictly speaking, only apply in a strong-cou-pling limit. As emphasized by Banks and Green, ingeneral such an agreement would only be expected ifthe relevant correlator is constrained by a supersym-metric nonrenormalization theorem. Very recentlyw x11 , exactly such a non-renormalization theorem hasbeen proved for eight-fermion terms in the effectiveaction of the three-dimensional theory. Our presentresults suggest that a similar nonrenormalization the-orem should be at work in the four-dimensionalcontext. Specifically, it appears that the predictionfor the 16 fermion correlator extracted from the

w xlow-energy IIB action in Ref. 8,3 , is not modifiedby higher-order stringy corrections. On the otherhand, the exact N-dependence of our one-instantonresult given below includes an infinite series of 1rNcorrections which should correspond to quantum cor-rections on the IIB side.

We first briefly review the type IIB superstringw xprediction, closely following 3 . At leading order

beyond the Einstein-Hilbert term in the derivativeexpansion, the IIB effective action is expected tocontain a totally antisymmetric 16-dilatino effective

w xvertex of the form 12,8 ,

y1X 10 yf r2 16'a d X det g e f t ,t L qh.c.Ž . Ž .H 16

3Ž .

( )N. Dorey et al.rPhysics Letters B 442 1998 145–151 147

Ž .Here L is a complex chiral SO 9,1 spinor, and f16Ž .is a certain weight 12,y12 modular form under

Ž .SL 2,Z . In particular f has the following weak-16

coupling expansion:

`

y3 f r2 f r2 f r2f sa e qa e q GG e , 4Ž .Ý16 0 1 kks1

where

1 25r2yfGG s keŽ .Ýk 2ž /n<n k

= yf Ž0.exp y2p k e q ic 5Ž . Ž .

neglecting perturbative corrections; the summation inŽ .5 runs over the positive integral divisors of k.

Ž .Notice that with the conjectured correspondence 1to the couplings of 4D SYM theory, the expansionŽ .4 has the structure of a semiclassical expansion: thefirst two terms correspond to the tree and one-loop

Žpieces, respectively a and a are numerical con-0 1.stants , while the sum on k is interpretable as a sum

on Yang-Mills instanton number. In the IIB theory,these terms, which are non-perturbative in the stringcoupling come from D-instantons.

From this effective vertex one can constructŽ .Green’s functions for 16 fermions L x , 1F iF16,i

which live on the boundary of AdS :5

² :L x PPP L xŽ . Ž .1 16

y1X yf r2; a e f tŽ . 16 16

=

4 16d x dr0 FK x ,r ; x ,0 6Ž . Ž .ŁH 7r2 0 i5r is1

suppressing spinor indices. Here K F is the bulk-7r2

to-boundary propagator for a spin-1r2 Dirac fermionof mass msy3r2 L and scaling dimension Ds

w x7r2, 13,4,5,14 ,:

K F x ,r ; x ,0Ž .7r2 0

sK x ,r ; x ,0Ž .4 0

= r1r2g qry1r2 x yx g n , 7Ž . Ž .Ž .5 0 n

with

r 4

K x ,r ; x ,0 s . 8Ž . Ž .4 0 422r q xyxŽ .Ž .0

In these expressions the x are 4-dimensional space-i

time coordinates for the boundary of AdS while r5

is the fifth, radial, coordinate; we suppress the coor-dinates on S5 as the propagator does not depend on

Žthem save through an overall multiplicative factor. Ž . Žwhich we drop . The quantity t in Eq. 6 is in the16

.notation of BGKR a 16-index antisymmetric invari-ant tensor which enforces Fermi statistics and en-sures, inter alia, that precisely 8 factors of r1r2g5

and 8 factors of ry1r2g n are picked out in theproduct over K F .7r2

According to Maldacena’s conjecture, the correla-Ž .tor 6 in the IIB theory should correspond to a

certain 16-fermion correlator in 4D large-N SYM.The correspondence of operators in the two pictures

w xwas established in Refs. 4,5,15,16 . For presentpurposes, the fermion operator in the 4D SYM pic-ture with the right transformation properties is thegauge invariant composite operator

L A ss m n b Tr Õ l A 9Ž .Ž .a a N m n b

which is a spin-1r2 fermionic Noether current asso-ciated with a particular superconformal transforma-

Ž .tion. Here Õ is the SU N gauge field strengthm n

while the l A are the Weyl gauginos, with the indexb

As1,2,3,4 labeling the four supersymmetries. Thenumerical tensor s m n projects out the self-dualcomponent of the field strength, 2 so that only in-stantons rather than anti-instantons can contribute.

w xIn Ref. 3 , BGKR explicitly compared the formof the first term in the D-instanton expansion of the

Ž .16 dilatino correlator 6 with the 1-instanton contri-Ž .bution to a correlator in NNs4 SU 2 Yang-Mills

Ž .theory with 16 insertions of the operator 9 . Theseauthors noted that the two correlators agree up to anoverall normalization. In particular the integration

Ž .measure and integrand appearing in Eq. 6 exactlymatch their counterparts in the gauge theory calcula-tion. As reviewed below, technically this identifica-

2 w xWe use Wess and Bagger conventions throughout 17 .

( )N. Dorey et al.rPhysics Letters B 442 1998 145–151148

Ž .tion is possible because the expression 8 is propor-tional to the 1-instanton action density:

962 <tr Õ s K . 10Ž . Ž .1- i n stN m n 42g

Ž .As mentioned earlier, for gauge group SU 2 thesingle NNs4 superinstanton contains precisely 16adjoint fermion zero modes: a supersymmetric plus asuperconformal zero mode, each of which is a Weyl2-spinor, times four supersymmetries. A nonvanish-ing 16-fermion correlator is therefore obtained bysaturating each of the fermion insertions with adistinct such zero mode. In what follows we general-

Ž .ize the calculation of BGKR to SU N . Since forŽ .any value of N the relation 10 still works, the

functional similarity between the 16-fermion correla-tors in the two pictures noted by BGKR continues tohold. However, by correctly accounting for the 8 Ny16 lifted fermion modes, we will also extract theoverall multiplicative constant C which determinesN

the strength of the effective 16-fermion SYM vertex.As shown below, in the large-N limit, C scales likeN'N . We identify this behavior with the factor of

X Ž .1ra in front of the IIB effective vertex 3 , usingŽ .the dictionary 2 . The extraction of this overall

constant as a function of N is our chief result.In order to calculate the required 16-fermion cor-

relator in 4D SYM theory, one needs to understandŽ . Ž .i the collective coordinate integration measure, ii

Ž .the instanton action S , and iii the form of theinstŽ .fermionic insertions 9 . Let us discuss each of these,

in turn:( )i The measure. For general topological number

k, the collective coordinate integration measure inthe NNs1, NNs2 and NNs4 supersymmetric cases

w x 3was derived in Refs. 18–20 . The form of thesemeasures is uniquely fixed by supersymmetry, theindex theorem, renormalization group decoupling,and cluster decomposition. The collective coordi-nates used in these expressions are those of the

3 The one specific case of the NN s4 k-instanton measure forŽ .general SU N is not explicitly given in these references, but may

Ž .be written down by inspection, by generalizing the SU 2 expres-Ž . w x Ž . w xsion 7 of Ref. 19 to SU N by the methods of Ref. 20 . When

Ž .ks1, this measure reduces to Eq. 13 below.

w xADHM multi-instanton 21,22 , suitably supersym-w xmetrized as explained in Refs. 23,24,20 . In particu-

lar the bosonic and adjoint fermionic collective coor-dinates are encoded in ADHM matrices a and MM A,respectively, where the index A runs over the inde-

Ž .pendent supersymmetries. For gauge group SU NŽ .and topological number k, a is an Nq2k =2k

A Ž .complex-valued matrix, while MM is an Nq2kŽ=k matrix of complex Grassmann numbers see

w x . ARef. 20 for a review . The elements of a and MM

are subject to polynomial constraints and gauge-likeinvariances which reduce the number of independentcollective coordinates to the number required by theindex theorem. In the 1-instanton sector these matri-

w xces have the simple canonical form 20 :

° A ¶m1

0 0° ¶ PP P PP P P

AP P mA Ny2as , MM s . 11Ž .0 0 A14 irhr 0

A24 irh0 rAm¢ ß 4jyx s 10 mA¢ ß4j 2

We have chosen familiar notation whereby rgR

and x m gR4 denote the size and position of the0A A ainstanton, and j and h are the supersymmetrica

and superconformal fermion zero modes, respec-Ž .tively. Eq. 11 assumes the canonical ‘North pole’

Ž .embedding of the instanton within SU N ; moregenerally there is a manifold of equivalent instantons

Ž .obtained by acting on 11 by group generators V inthe coset space

U NŽ .Vg . 12Ž .

U Ny2 =U 1Ž . Ž .A Ž .The complex Grassmann coordinates m in Eq. 11i

Ž .which do not carry a Weyl spinor index may bethought of as the superpartners of the coset embed-

A A A AŽ .ding parameters 12 . Together, j , h , m and ma a i i˙constitute 8 N fermionic collective coordinates, asneeded.

In terms of these 1-instanton variables, the cor-rectly normalized NNs4 collective coordinate inte-

( )N. Dorey et al.rPhysics Letters B 442 1998 145–151 149

gration measure may be easily deduced from thew x 4early literature 25–27 :

24 Nq2p 4 Ny2 1 dr 4 N4d x m rŽ .H H0 04 N 5Ny1 ! Ny2 ! g rŽ . Ž .

=

424 g2 Ad jŁH 2ž /16p mAs1 0

=

424 g2 Ad hŁH 2 2ž /32p r mAs1 0

=

Ž .4 Ny224 Ny2 gA Adm dm . 13Ž .Ł ŁH i i 2ž /2p mAs1 is1 0

w x Ž .As usual in instanton calculations 25 , gsg m is0

evaluated in the Pauli-Villars scheme; since the NN

s4 model is a finite theory, the subtraction scale m0

should, and by inspection does, cancel out of Eq.Ž . Ž Ž .13 . The overall constant in front see Eq. 33 ofw x. Ž .26 comes from the volume of the coset space 12 ;this factor presupposes that the measure will be usedto integrate only gauge singlets, as we shall be doingŽsince all adjoint Higgs VEVs, which pick out spe-cial directions in the color space, will be set to zero

.in the present paper .( )ii The instanton action. For all instanton num-

bers k, the NNs4 instanton action was derived inw x Ž .Ref. 7 , for the gauge group SU 2 . Using the

w xmethods of Ref. 20 , that expression immediatelyŽ .generalizes to SU N for arbitrary N. In particular,

in the conformal case where all adjoint Higgs VEVsare set to zero, the instanton action has the form

8p 2 kS s qS 14Ž .inst quad2g

Here S is a particular fermion quadrilinear term,quad

with one fermion collective coordinate chosen from

4 A A 2 2The normalization factors of j and h , g r16p m and0

g 2r32p 2r 2m respectively, used in this measure, agree with Refs.0w x w x28,23 , but disagree, by factors of two, with Ref. 29 and much

w xof the subsequent literature; see Ref. 28 for a discussion. In theA A AADHM language, the relative values of the j , h and m

Ž .normalizations used in this measure follow from Eq. 11 togetherwith the fact that the inner product of MM A matrices is propor-

A AŽ . Ž . Ž .tional to Tr MM PP q1 MM where the Nq2k = Nq2k di-`

agonal matrix PP q1 has 2’s in the first N diagonal entries and`

w x1’s in the remaining 2k diagonal entries 22,23 .

each of the four gaugino sectors As1,2,3,4. In the1-instanton sector this term collapses to 5

p 2N A B N C DS s e L MM , MM L MM , MMŽ . Ž .quad A BC D f f2 22 r g

15Ž .where

Ny21N A B A B B AL MM , MM s m m ym m 16Ž . Ž .Ž .Ýf i i i i'2 2 is1

w xAs we explained in Ref. 7 , for all k, in the absenceof VEVs, S is a supersymmetric invariant quan-quad

tity that lifts all the fermion zero modes except forthe 16 supersymmetric and superconformal modes.For ks1 this latter property is obvious from Eq.Ž .16 , which explicitly depends only on the collective

A A A Acoordinates m and m and not on j or h .i i a a

( )iii The fermion insertions. As stated earlier, theŽ .16 explicit fermion insertions 9 are needed to

saturate the 16 global supersymmetric and supercon-formal zero modes, which are not otherwise lifted by

Ž .the action 15 . Accordingly we substitute for theA Ž .gaugino l in Eq. 9 :b

A Ag A gg m m k l˙l x sy j yh s P x yx sŽ . Ž .b g m 0 gbž /˙

= Õ xyx q PPP 17Ž . Ž .k l 0

Ž . Ž . w xas follows from Eqs. 4.3a and A.5 of 23 ; thedots stand for admixtures of the remaining fermion

Žmodes which we can neglect since these are satu-. Ž .rated by S . The field strengths in Eqs. 9 andinst

Ž .17 are to be evaluated on the instanton. With Eq.Ž .10 together with the identity

21 SDtr Õ Õ s PP tr Õ , 18Ž . Ž .N m n k l m n ,k l N p q3

where PPSD is the projector onto self-dual antisym-m n,k lŽ . Ž .metric tensors, Eqs. 9 and 17 reduce to

Ag Ag A gg m m˙L x sy j yh s P x yxŽ . Ž .g m 0ž /˙

=2

tr Õ xyxŽ .Ž .N p q 0

96Ag A gg m m˙s y j yh s P x yxŽ .g m 0ž /˙2g

=K x ,r ; x ,0 . 19Ž . Ž .4 0

Putting this all together, we now insert 16 copiesŽ .of the composite fermion 19 , at 16 spacetime points

5 Ž . Ž . Ž . w x Ž . Ž .See Eqs. 14 , 4 and 6 of 7 , and Eqs. 3.20 , 8.6 andŽ . w x8.7 of 20 .

( )N. Dorey et al.rPhysics Letters B 442 1998 145–151150

Ž .x , times exp yS , into the 1-instanton collectivei instŽ .coordinate measure 13 . Apart from the integrations

A A� 4over the lifted fermion modes m ,m and thei i

N-dependent constant in front of the measure, theŽresulting expression is identical up to some cor-

. Ž .rected factors of 2 to the SU 2 expression exam-ined by BGKR. Our task is precisely to evaluate theN-dependent effect of the lifted modes. On a formallevel, for general N, and general topological number

Ž .k, the integral of exp yS over the lifted fermionquad

modes is given by the Gauss-Bonnet-Chern theorem,and equals the Euler character of a certain quotient

Ž .space formed from the charge-k SU N ADHMw xmoduli space 6,30 . However, since much less is

known about such spaces than about the multi-mono-pole spaces which govern analogous instanton calcu-

w xlations in 3D 6 , here we will adopt an alternative,direct calculational approach. Specializing to ks1,

Ž .we define I to be the unnormalized contributionN

of the lifted modes to the correlator:

4 Ny2A A ySquadI s dm dm e . 20Ž .Ł ŁHN i i

As1 is1

By explicit computation we find

3p 4

I s 21Ž .3 4 4r g

and we also define I s1.2

To evaluate I for general N, it is helpful toN

rewrite the quadrilinear term S as a quadraticquad

form. To this end we introduce six independentauxiliary bosonic variables x syx , and sub-A B B A

Ž .stitute into Eq. 20 the integral representation

ir 6 g 6ySquad X Xe s dxŁH A B9 6 X X2 p 1FA -B F4

=r 2 g 2

exp e x xA BC D A B C D2ž 32p

1 N A Bq x L MM , MM 22Ž . Ž .A B f2 /where an appropriate analytic continuation of theintegration contours is understood. Our strategy is to

A Aperform only the integrations over m and mNy2 Ny2

Ž .in Eq. 20 , and thereby relate I to I . Accord-N Ny1

ingly we break out these terms from LN,:f

LN MM A , MM BŽ .f

sLNy1 MM A , MM BŽ .f

1A B B Aq m m ym m . 23Ž .Ž .Ny2 Ny2 Ny2 Ny2'2 2

A A� 4 Ž . Ž .The m ,m integration in Eqs. 20 and 22Ny2 Ny21brings down a factor of det x . Next we exploitA B64

the fact that the determinant of an even-dimensionalantisymmetric matrix is a perfect square:

21det x s e x x 24Ž .Ž .A B A BC D A B C D8

Ž .Since the right-hand side of Eq. 24 is proportionalŽ .to the square of the first term in Eq. 22 , the result

A A� 4of the m ,m integration can be rewritten as aNy2 Ny2

parametric second derivative relating I to I ,:N Ny1

E 2y31 4 6 6 2 2I s p r g r g I 25Ž .Ž .ž /N Ny14 22 2E r gŽ .

Ž Ž 2 2 .y3The insertion of r g under the parenthesesforces the derivatives to act on the exponent of Eq.Ž . .22 , and not on the prefactor. This recursion rela-

Ž .tion, combined with the initial condition 21 , givesfinally

2 Ny42p1I s 2 Ny2 ! . 26Ž . Ž .N 2 2 2ž /2 r g

Ž . Ž . Ž .Combining Eqs. 13 , 19 and 26 , we thereforefind for the 1-instanton contribution to the 16-ferm-ion correlator in NNs4 SYM theory:

² 1 4 :L x PPP L xŽ . Ž .a 1 a 161 16

4 4d x dr0 2 A 2 AsC d j d hŁH HN 5r As1

1 1 g m m˙= j yh s P x yxŽ .a g m a 1 0ž /˙1 1

=K x ,r ; x ,0 = PPPŽ .4 0 1

4 4 g m m˙= j yh s P x yxŽ .a g m a 16 0ž /˙16 16

=K x ,r ; x ,0 27Ž . Ž .4 0 16

( )N. Dorey et al.rPhysics Letters B 442 1998 145–151 151

where the overall constant C is given byN

2 Ny2 !Ž .y24 y2 Nq57 16 y10C sg 2 3 p .N Ny1 ! Ny2 !Ž . Ž .

28Ž .

Taking the N™` limit using Stirling’s formulagives

y24 55 16 y21r2'C ™ g N 2 3 p . 29Ž .N

This is in agreement with the IIB prediction, whichŽ . Ž .can be read off Eqs. 1 - 6 :

y1X yf r2 y24'<a e f ; g N , 30Ž . Ž .1- i n st16

up to an overall numerical constant. We reiterate thatthe structural agreement between the IIB and SYM

Ž . Ž .integrals 6 and 27 , and between the powers of gŽ . Ž . Ž .in Eqs. 28 and 30 , was already noted in the SU 2

analysis of BGKR; the new ingredient here is thenontrivial agreement in the N-dependence between

Ž . Ž .Eqs. 29 and 30 .

Acknowledgements

We are indebted to Tim Hollowood and GiancarloRossi for clarifying discussions, and to Tim Hol-lowood for useful comments on the draft. ND, VKand MM acknowledge a NATO Collaborative Re-search Grant, ND and VK acknowledge the TMRnetwork grant FMRX-CT96-0012 and SV acknowl-edges a PPARC Fellowship for support.

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