Testing the Ads/CFT Correspondence Beyond Large N

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Quantum Aspects of Gauge Theories, Supersymmetry and Uniﬁcation

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Testing the AdS/CFT correspondence beyond large N

Adel Bilal and Chong-Sun Chu Institute of Physics, University of Neuchˆatel, CH-2000 Neuchˆatel, Switzerland E-mail: [email protected] , [email protected]

Abstract: According to the AdS/CFT correspondence, the d =4,N =4SU(N) super Yang-Mills 5 theory is dual to the type IIB string theory compactified on AdS5 × S . Most of the tests performed so far are confined to the leading order at large N or equivalently string tree-level. To probe the 1 correspondence beyond this leading order and obtain N 2 corrections is difficult since string one-loop 5 computations on an AdS5 × S background generally are beyond feasibility. However, we will show that the chiral SU(4)R anomaly of the super YM theory provides an ideal testing ground to go beyond leading order in N. In this paper, we review and develop further our previous results [1] that the 1/N 2 corrections to the chiral anomaly on the super YM side can be exactly accounted for by the supergravity/string effective action induced at one loop.

Keywords: Duality in Gauge Field Theories, Anomaly, D-branes.

1. Introduction lutions of the supergravity field equations, namely 1 ds2 = √ −dt2 +dx2 +dx2 +dx2 f 1 2 3 p We begin by briefly reviewing some relevant basic 2 2 2 + f dr +r dΩ5 , aspects of the AdS/CFT correspondence [2, 3, 4], R4 see in particular [5]. f =1+ ,R4=4πg α02N. (1.1) r4 s Consider type IIB string theory with a num- Here t, x1,x2,x3 are the longitudinal coordinates ber N of D3 branes. There are open strings end- (on the D3) while r and Ω5 describe the trans- ing on these D3 branes and closed strings in the verse space. N is the number of (coincident) D3 bulk. The effective low energy action consists branes. There is also a five-form field strength F of IIB supergravity describing the bulk closed which is proportional to N. Again, one wants to N strings, and = 4 supersymmetric U(N) gauge study the low energy excitations in this descrip- theory (SYM) describing the open strings ending tion and compare with the first one. Due to the on the D3 branes. The gauge group of the latter non-trivial function f(r) in the metric there is a actually is SU(N) since the U(1) decouples. Con- red-shift factor between energies measured at r 0 sidering low energies at fixed α is equivalent to and energies measured at r = ∞: 0 → fixed energy and taking the α 0 limit. In this ∼ 02 4 −1/4 limit the gravitational coupling κ gsα van- −1/4 R E∞ = f E = 1+ E . (1.2) ishes and the interactions between the branes and r r4 r the bulk can be neglected, as well as all higher One sees that if r → 0, E∞ → 0 for any finite derivative terms in the brane action. Only free E . So there are two types of low energy ex- bulk supergravity and pure N =4SU(N)SYM r citations: low energy at finite r (bulk) or finite in d = 4 dimensions remain. The latter is a con- energy near the horizon (r =0),andthetwo formal field theory (CFT). types decouple yielding (free) bulk supergravity There is a different way to describe the same and near horizon supergravity. Concentrate on → ∼ R4 physics. D3 branes may be viewed as certain so- the near horizon limit: as r 0 one has f r4 . Quantum Aspects of Gauge Theories, Supersymmetry and Unification Adel Bilal and Chong-Sun Chu

Since also α0 → 0, it is convenient to introduce if the ’t Hooft coupling λ is small. Supergravity, the finite quantity u = r/α0 so that the near rather than string theory, should be a good de- horizon metric becomes scription if the radius of curvature of AdS5 and 5 2 0 1 S is large, meaning R α or λ large. The two ds2 = λ−1/2u2 −dt2 +dx2 +dx2 +dx2 α0 1 2 3 regimes are opposite as is often the case with du- du2 alities. This of course avoids the obvious contra- +λ1/2 +λ1/2dΩ2 , (1.3) u2 5 diction that both descriptions look so different. At fixed ’t Hooft coupling λ, the second re- where we introduced the finite quantity 1 lation (1.6) tells us that N 2 corresponds to the R4 string loop-counting parameter g2, so that the λ =4πg N = . (1.4) s s α02 large N limit of SYM corresponds to classical string theory (or classical supergravity if also λ The metric (1.3) is the metric of AdS × S5. 5 1), and 1 corrections should correspond to one- Comparing both descriptions of the same phy- N 2 loop effects in string theory. sics one then is led to identify the conformally invariant, N =4SU(N) SYM theory in d =4 One now has various possible conjectures: with the supergravity or small α0 limit of IIB 1.) The weakest one is: SYM is dual to AdS 5 →∞ string theory on AdS5 × S . We will be more supergravity only for λ , but the full string precise shortly. theory is different. This version would not be How do the different parameters on the SYM very useful. 2.) The SYM theory is dual to string side compare to those of the supergravity/string theory on AdS for finite λ but only as N →∞or → 0 theory? In the former we have the coupling gYM equivalently gs 0. This includes α corrections and the integer N determining the gauge group beyond the supergravity approximation, but no SU(N). On the supergravity/string side we have string loops. 3.) This is the strongest version, α0 and R (with only the dimensionless ratio R2/α0 generally referred to as the Maldacena conjec- being a relevant parameter) and the coupling gs. ture: the SYM theory is dual to string theory for 4 02 A first relation is already obtained in eq. (1.1), all λ and all N, i.e. all R /α and gs. 4 02 namely R =4πgsα N or equivalently eq. (1.4). While there is now reasonable evidence for A second relation is obtained from the D3 brane version 2.) of the conjecture (see e.g. [5]) the action from which one reads the YM coupling in strong version 3.) is hard to test since standard terms of the string coupling. Thus string or supergravity loop computations on an AdS × S5 background are difficult, to say the 1 1 1 R4 5 = and N = (1.5) least, if not unfeasible, with the present state of g2 4πg 4πg α02 YM s s the art. expresses the SYM parameters gYM and N in Many successful tests are group theoretic in terms of the string/supergravity parameters gs nature. Some are not restricted to tree-level or 0 and R2/α and vice versa. In large N gauge the- even perturbation theory, but on the other hand ories the relevant loop-counting parameter is the they do not really provide any real test at one- 2 2 ’t Hooft coupling gYMN rather than gYM.Com- loop. Examples are global symmetries (disre- 2 R4 bining both eqs (1.5) yields gYMN = α02 which garding possible anomalies for the moment): AdS5 by eq (1.4) is just the quantity called λ.Itis space-time has an SO(4, 2) symmetry which also useful to rewrite the relations between the pa- is the conformal group of the N = 4 SYM the- rameters of the two descriptions as ory. The “internal” symmetry is SO(6) ' SU(4): this is the isometry of the S5 as well as the R- R4 N 1 λ = , = (1.6) symmetry of the SYM theory. The latter actu- α02 λ 4πg s ally is anomalous which will be important for us. with λ now meaning the ’t Hooft coupling. Both theories have the same amount of super- Let us first comment on the first relation: symmetry, the full supergroup being SU(2, 2|4) ⊃ perturbative SYM theory is a good description SO(4, 2) × SU(4)R. Also the duality symmetry

2 Quantum Aspects of Gauge Theories, Supersymmetry and Uniﬁcation Adel Bilal and Chong-Sun Chu

SL(2, Z) is the same as it acts on and there is an SU(4)R gauge group with gauge fields Aã (x, z), µ =0,1,...4anda=1,...15 = 4πi θ i χ µ τ = + = + . (1.7) dim SU(4). This gauge group is of course not g2 2π g 2π YM s to be confused with the SU(N) of the confor- A promising arena for performing tests be- mal SYM theory. Note also that in the latter, yond the large N limit or string tree-level is to SU(4)R is a global symmetry, hence there are a look at certain anomalies. On the SYM side SU(4)R currents Jµ (x), µ =0,1,2,3, but no as- anomaly coefficients are easily established one- sociated gauge fields. We can nevertheless cou- a loop effects in λ that are protected against higher ple these currents to external gauge fields Aµ(x), order corrections. Typically such an anomaly co- µ =0,1,2,3 which act as sources for these cur- efficient will depend on the number of fermion rents. Then by a standard argument, the non- fields in the SYM theory, i.e. on N. The goal invariance of the one-loop effective action Γ[Aµ] then is to reproduce the exact N-dependence from under gauge transformations of these external the string theory including subleading terms ∼ gauge fields is equivalent to the covariant non- 1 N 2 coming from string loops. If we want to conservation of the currents: let δv be such a have any chance to be able to do this calcula- gauge transformation with parameter v,then tion, the relevant quantity to compute in string Z Z a δΓ a a,µ theory should be of a topological nature, like a δvΓ[Aµ]= δvA = δvA J µδAa µ Chern-Simons term, so that the actual metric on Z Zµ a a,µ a µ a AdS5 is irrelevant. = (Dµv) J = − v (DµJ ) (2.1)

with 2. The chiral SU(4)R anomaly in the µ a ∼− 2− abc µνρσ b c N =4SYM (DµJ ) (N 1)d ∂µAν ∂ρAσ + ... (2.2) The anomaly we will consider is the chiral SU(4)R the precise numerical coefficient being given be- anomaly. As already mentioned, SU(4)R is a low. (classical) global symmetry of the N =4SYM There is a standard prescription [4] in the theory. Due to the presence of chiral fermions AdS/CFT correspondence how to compute cor- transforming in complex conjugate representa- relation functions: we will give this prescription tions of SU(4)R this symmetry is spoiled at one for the case of present interest. For any (SU(N) loop and there is an anomaly: the one-loop ef- gauge-invariant) operator O(x) like the currents fective action in the presence of external SU(4) a R Jµ (x) of the SYM theory, introduce a source φ0(x) gauge fields is no longer invariant under SU(4) a R like the Aµ(x). Then the generating functional and the non-invariance is proportional to the num- a for correlators of Jµ is just ber of fermions. Since they are also in the adjoint R 4 a a,µ −Γ[A] − d xAµ(x)J (x) representation of the “true” gauge group SU(N) e ≡he iSYM . (2.3) there are N 2 − 1 of them, and the anomaly is In AdS string theory there is a field φ(x, z)such proportional to N 2 − 1. As we recall below, the 5 that at the boundary z =0ofAdS (note that leading term ∼ N 2 is accounted for by tree-level 5 z =1/u) which is just the four-dimensional space supergravity [6]. It is the −1 correction which of the SYM theory one has φ(x, z =0)=φ (x). should originate from a string/supergravity loop 0 In our case this is just Aã (x, z =0)=Aa(x)(for effect, and it indeed does as we showed in [1] and µ µ µ=0,1,2,3 only) where the Aã are the gauge explain in the remainder of this paper. µ fields of the gauged supergravity. The prescrip- Before explaining the string/supergravity loop tion [4] then is correction let us review how the leading N 2 term −Γ[A] is obtained in the string/supergravity descrip- e = Zstring , (2.4) ã a tion. Here SU(4) ' SO(6) acts as an isometry Aµ(x,z=0)=Aµ(x) 5 on the S . As a consequence, the AdS5 super- meaning that the string partition function should gravity is actually a gauged supergravity [7, 8, 9] be evaluated subject to the boundary condition

3 Quantum Aspects of Gauge Theories, Supersymmetry and Uniﬁcation Adel Bilal and Chong-Sun Chu

ã a R Aµ(x, z =0)=Aµ(x)forµ=0,1,2,3. Writing where A( ) is the anomaly coefficient defined by the ratio of the d-symbols taken in the represen- − cl − 1−loop− − eff Sstring S ... S Zstring =e string ≡ e string (2.5) tation R and in the fundamental representation. In general 2n or 2n + 1 dimensions, since the d- eq (2.4) together with eq (2.1) implies that, if the symbol is given by a symmetrized trace of n +1 AdS/CFT correspondence is correct, we should Lie algebra generators, it is easy to show that have the complex conjugate representation R∗ has an eff δvS = δvΓ[A] anomaly coefficient string ã a Aµ(x,zZ=0)=Aµ(x) ∗ n+1 a µ a A(R )=(−1) A(R). (2.10) = − v (DµJ ) (2.6) Due to the connection of anomalies and Chern- which is non-vanishing according to (2.2). Thus Simons actions in one higher dimension, it is nat- the SU(4) gauge variation of Seff should di- R string ural to expect that the four-dimensional field the- rectly reproduce the SYM chiral SU(4) anomaly. R ory anomaly is dual to a Chern-Simons action in Actually for the purpose of reproducing the lead- the gauged AdS5 supergravity. This is indeed ing N 2 part of the anomaly it is enough to con- the case as was first pointed out in [4]. The tree sider the classical supergravity action [6]. level supergravity action on AdS5 contains the Let us now determine the exact anomaly co- following terms [4, 6, 7, 9] N efficient of the = 4 SYM theory in 4 dimen- Z Z √ sions. This theory has four complex Weyl fermions 1 5 a µνa Scl[A]= 2 d x gFµν F + k ω5. λ in the fundamental representation of SU(4)R 4gSG AdS5 with the chirality part (0,1/2) in 4 and (1/2,0) (2.11) in 4∗ (see for example [10]. Our conventions Note that here F is the field strength associated ˜ here are equivalent to those of [10].) Moreover, with the five-dimensional gauge field Aµ.The 1 exact values of the coefficients 2 and k will all fields are also in the adjoint representation 4gSG of the “true” gauge group SU(N) which acts as be important for us. Their ratio is fixed by su- a “flavour” group with respect to the SU(4)R. persymmetry [7, 9]. They may be obtained by Thus there are actually N 2 − 1 complex Weyl dimensional reduction of the ten-dimensional IIB fermions λ in the 4,resp.4∗. The correctly nor- supergravity on S5 using the fact that the radius 5 4 02 malised R-symmetry anomaly is given by of S is given by eq. (1.4) as R /α =4πgsN. Z Then it is easy to determine the normalization of 2− 1 δvΓ[A]=(N 1) ω4(v, A). (2.7) the gauge kinetic energy term and one finds S4 16π2 The differential forms g2 = ,k=N2. (2.12) SG N 2 1 1 ω1(v, A)= Tr [vd(AdA + A3)], 4 24π2 2 Note that the action (2.11) with the values (2.12) 1 3 3 ω (A)= Tr [A(dA)2 + A3dA + A5] has been used to compute the 2-point and 3-point 5 2 a 24π 2 5 correlators of the currents Jµ in the SYM theory (2.8) [6]. To leading order in N this gives the correct

1 3 result. satisfy the descent equations dω5 = 24π2 Tr F , 1 2 a a In usual considerations of supergravity on and δvω5 = dω with F = dA + A , A = A T 4 AdS, one considers gauge conﬁgurations A˜ that and v = vaT a as usual, the T a being the gener- µ vanish at the boundary and so the Chern-Simons ators of SU(4) in the fundamental 4 representa- term is gauge invariant since δω = dω1 and the tion. For later use we note that for T a in a gen- 5 4 integral vanishes. For the considerations of the eral representation R of SU(4), the correspond- AdS/CFT correspondence however, we precisely ing quantities with the trace taken in R are want nonvanishing boundary values for A˜µ as ex- 1 R R 1 R R ω2n = A( ) ω2n ,ω2n+1 = A( ) ω2n+1, plained above, cf eq (2.4). Then under a gauge (2.9) variation δvA˜, the variation of the Chern-Simons

4 Quantum Aspects of Gauge Theories, Supersymmetry and Unification Adel Bilal and Chong-Sun Chu term is a boundary term where R is the representation of the Dirac fermion. Z The sign depends on the regularization and can 1 δvScl = k ω4(v, A) . (2.13) often be fixed within a specific context. S4 This result was originally [11, 12] obtained (We take the SYM theory to be defined on com- for fermions coupled to gauge fields in a flat space- pactified Euclidean space, i.e. on S4.) Now by time and has been extended to full generality for eff → arbitrary curved backgrounds and any odd di- eq (2.6), approximating Sstring Scl and using mension d =2n+ 1 [13]. The induced parity vio- (2.13) one can read off the SU(4)R anomaly obtained from the supergravity action (2.11). It is lating terms are given (up to a normalization fac- Z tor) by the secondary characteristic class Q(A, ω) 2 1 satisfying δvΓ[A]=δvScl = N ω4(v, A) , (2.14) S4 dQ(A, ω)=Aˆ(R)ch(F )|2n+2, (3.2) which agrees with the gauge theory computation where ω is the gravitational connection. Since (2.7) to leading order in N. Aˆ(R)=1+O(R2)andTrF = 0 for SU-groups, it We thus see that the IIB supergravity action is clear that for n =2(d= 5) there are no mixed contains a Chern-Simons term at tree level which gauge/gravitational terms. Also, there can be no can account for the chiral anomaly of the gauge purely gravitational term since it would corre- theory to leading order in N. But there is also a spond to a gravitational anomaly in four dimen- 2 mismatch of “-1” which is of order 1/N relative sions which is not possible. Hence for the present to the leading term. As discussed above, this case of SU(4) with n = 2, (3.2) simply reduces should correspond to a 1-loop effect in IIB string to dQ = ch(F )|6 giving rise to the Chern-Simons theory. Thus we are lead to examine the string action upon descent, which does not depend on one-loop effective action. the geometry of AdS5 at all! Hence the result of (3.1) for a Dirac fermion in flat space(-time) 3. One-loop induced Chern-Simons remains valid on AdS5. Now we need the particle spectrum of the action 5 type IIB string theory on AdS5 × S . The only explicitly known states are the KK states coming Loop effects in AdS supergravity are technically 5 from the compactification of the 10 dimensional very diffciult to compute due to the complicated IIB supergravity multiplet [14]. So we will exam- propagators in AdS geometry. Here however, ine them first. We will argue in the discussion this is possible due to the topological character section that the other string states are not likely of the Chern-Simons action. to modify the result. Fermionic contributions Particles in AdS5 are classified by unitary Consider a Dirac fermion ψ in odd dimen- irreducible representations of SO(2, 4). SO(2, 4) sions (flat) minimally coupled to vector bosons has the maximal compact subgroup SO(2)×SU(2) A of a group G. At the quantum level, a regu- µ ×SU(2) and so its irreducible representations are larization needs to be introduced to make sense labelled by the quantum numbers (E0,J1,J2). of the theory and one cannot preserve both the The complete KK spectrum of the IIB supergrav- gauge symmetry (small and large) and the par- 5 ity on AdS5 ×S was obtained in [14, 8] together ity at the same time [11, 12]. If one chooses to with information on the representation content preserve the gauge symmetry by doing a Pauli- under SU(4)R. We reproduce these results in Villars regularization, then there will be an in- the table below. Actually, all fermions are sym- duced Chern-Simons term generated at one loop. plectic Majorana, giving half the anomaly of a The result is independent of the fermion mass. In Dirac fermion. But there also is a mirror ta- our notation, the induced Chern-Simons term is ble with conjugate SU(4) representations and Z Z R 1 1 SU(2)×SU(2) quantum numbers exchanged (op- ∆Γ = ωR = A(R) ω , (3.1) 2 2n+1 2 2n+1 posite chiralities). So these “mirror” fermions

5 Quantum Aspects of Gauge Theories, Supersymmetry and Unification Adel Bilal and Chong-Sun Chu give the same anomaly as those in the table and Fourier expansion. This can be fixed however by the net effect is that we may restrict ourselves to imposing a certain condition on the variation the fermions of the table treating them as if they i were Dirac fermions. δψ = D + γ (3.5) α α 2R α × SU(2) SU(2) SU(4)R of the gravitino. Denote the local coordinates of 4 20 ··· ← ψµ (1, 1/2) , , AdS × S5 by (xµ,yα). A general spinor has the 4∗ 20∗ ··· (1, 1/2) , , decomposition ∗ λ (1/2, 0) 20 , ··· ← X (1/2, 0) 4, 20, ··· (3.3) = I,(x)ΞI,(y) (3.6) λ0 (1/2, 0) 4∗, 20∗, ··· ← (1/2, 0) 4, 20, ··· where ΞI, (y) are the spinor spherical harmonics 00 5 α λ (1/2, 0) 36, 140, ··· on S and satisfy (D/ y = γ Dα,α=5,...9) (1/2, 0) 36∗, 140∗, ··· I, 5 1 I, D/ yΞ = ∓i(k + ) Ξ (3.7) Notice that the fermion towers always come 2 R in pairs with conjugate representation content, where k = I ≥ 0andΞI, can be written in ∗ except for a missing 4 state in the first tower terms of the killing spinors ηI, on S5. Subsitut- of λ. As a result [1], all contributions cancel two ing (3.6) into (3.5) and using (3.7) we get by two except for the contribution from the un- X paired 4 of the λ tower. The net resulting in- α i 5 5 I, I, δ(γ ψα)= ∓(k+ )+ (x)Ξ (y) . duced Chern Simons action is R 2 2 Z (3.8) 1 ∆Γ = − ω5. (3.4) So one finds that only the component correspond- 2 0,+ AdS5 ing to Ξ is gauge invariant while all other com- a While this is almost what we want, it is only half ponents of γ ψα can be gauged away. Thus we of the desired result. However this is not the arrive at the gravitino gauge fixing condition whole story. α I0 I0 ,+ γ ψα(x, y) ∼ χ (x)η (y) (3.9) Doubleton multiplet There are similar “missing states” in the boso- where χI0 (x) are some arbitrary spacetime spinor nic towers. Together they are identified with the fields. We refer the reader to [14] for more details. doubleton multiplet of SU(2, 2|4) which consists Therefore we see that (3.9) is the closest one can α of a gauge potential, six scalars and four complex get to the gauge condition γ ψα =0.Onecan spinors. These are nonpropagating modes in the also rewrite this condition as bulk of AdS5 and can be gauged away completely I0 α I0,+ [14, 15], which is the reason why they don’t ap- ψα = ψ(α) + χ (x)γ η (y) (3.10) pear in the physical spectrum. These modes are where the part ψ satisfies γαψ =0.The exactly dual to the U(1) factor of the U(N)SYM (α) (α) other Killing spinors η− have been gauged away. living on the boundary [5]. We will now show The coefficient of η− would be a field in the 4∗ of that the other half of the induced Chern-Simons SU(4) and is precisely the doubleton spinors we action is due to the corresponding Faddeev-Popov are after. Now the constraint can be taken care ghosts. of in the functional approach by introducing in Let us recall that the doubleton multiplet the path integral the factor is absent because it has been gauged away [14] Z R by imposing the gravitino gauge fixing condition 1 5 ¯ dbd¯b e d xbMb (3.11) (see also [15] for the gauging in the case of AdS × 7 detXM S4 case). The basic idea is that upon compact- α I I,+ I I,− δ(γ ψα − χ η − b (x)η (y)) · δ(h.c.) ifying on S5, the original supersymmetyries in 10 dimensions decompose into an infinite tower where b(x) is a complex fermionic field in the ∗ of (unwanted) supersymmetries according to the 4 of SU(4) and M = D/ x. Integrating over b,¯b

6 Quantum Aspects of Gauge Theories, Supersymmetry and Uniﬁcation Adel Bilal and Chong-Sun Chu results in the gauged ﬁxed lagrangian. The factor of regularization and a better way to determine (det M)−1 can be handled by intoducing ghost the shift is called for. One possibility might be

fields c, c¯, which are bosonic spinor fields on AdS5 to do a string theory calculation by embedding and are in the 4∗ of SU(4). Thus the Chern-Simons action in a string setting and Z R to determine the quantum loop effects from the 1 − 5 = dcdce¯ d xcMc¯ . (3.12) string loop effects. Since string theory is free det M from divergences, no regularization related am- and so give rises to another -1/2 contribution to buigities should occur and a definite answer can the induced Chern-Simons action. So altogether be expected. we get a total induced Chern-Simons term of -1, Z 4. Discussion ∆Γ = − ω5, (3.13) 2 AdS5 We have reproduced the correct shift N → 2 − which is exactly the desired result. Notice that N 1 of the anomaly coefficient as a one-loop ef- × the induced Chern-Simons action (coming with a fect in IIB supergravity/string theory on AdS5 5 constant integer coefficient) is independent of the S . This shift is entirely due to the towers of radius R and this is consistent with the AdS/CFT Kaluza-Klein states. No massive string states proposal since the anomaly and its corrections need to be invoked. It is indeed likely that the are independent of λ. latter play no role at all since anomalies are usu- Bosonic contributions ally due to massless fields only. Note however that we need the full towers of Kaluza-Klein states There is another interesting effect related to to get the correct result. A truncation to five- the Chern-Simons action. It is known that in three dimensions, the gluons at one loop can mod- dimensional AdS supergravity alone would not give the desired result. Also the AdS supergrav- ify the coefficient of the Chern-Simons action by 5 ity Chern-Simons term originates from compact- an integer shift. It has been argued that [4, 1] ifying the full IIB supergravity. This is another there is no such shift for the present case. There- indication that string states beyond the Kaluza- fore only spinor loops contribute to the induced Chern-Simons action and we find that at finite Klein towers are unlikely to modify our result. We have been able to obtain a non-trivial N, the coefficient k is shifted by one-loop result within a particularly favourable k → k − 1orN2→N2−1 (3.14) case. In general, one-loop calculations in AdS5 are very difficult - already tree computations are due to the quantum effects of the full set of Kaluza- quite non-trivial! Of course, the anomaly co- Klein states. efficient ∼ N 2 − 1 should be exact and there A few comments about the absence of a shift cannot be any further higher-loop corrections ∼ 2 1 due to gluon loops are in order. The bosonic N N 4 . Again, since the induced Chern-Simons shift in pure Chern-Simons theory was first com- term in 5 dimensions is closely related to anoma- puted in [16] using a saddle point approximation. lies in 4 dimensions, we expect some sort of non- Later calculations trying to reproduce this re- renormalisation theorem to be at work, although sults from the perturbative point of view revealed it would be nice to have a proof of this statement. that the precise shift depends on the choice of Finally, we would like to make some com- regularization scheme 1. In the present case of ments on more or less related situations. There 5-dimensions, one may try to employ a regular- are other dualities like those involving AdS7 × ization scheme and do a 1-loop perturbative cal- S4 where one can expect similar Chern-Simons culation to determine the possible shift. How- terms and doubleton multiplets. The issue of the ever, like in the 3-dimensional case, it can be ex- trace-anomaly in AdS5 should be closely related pected that the result will depend on the choice to the present study. Already the leading N be- 1We thank R. Stora for a useful discussion about the haviour of this conformal anomaly is non-trivial issues of regularization. to establish [17] and to explicitly obtain the sub-

7 Quantum Aspects of Gauge Theories, Supersymmetry and Uniﬁcation Adel Bilal and Chong-Sun Chu

leading terms might well turn out to be more dif- [8] M. Gunaydin, L.J. Romans, N.P. Warner, ficult than for the chiral anomaly studied here. Compact and Non-compact Gauged Supergravity Effects that are of lower order than N 2 have also Theories in Five Dimensions,Nucl.Phys.B272 been considered in [18] which essentially studies (1986) 598. situations where the leading effect corresponds to [9] M. Pernici, K. Pilch, P. van Nieuwenhuizen, open strings at tree level and hence comes with Gauged N =8,d=5Supergarvity,Nucl.Phys. B259 just one power of N. A somewhat related discus- (1985) 460. sion is [19]. [10] S. Ferrara, C. Fronsdal, A. Zaffaroni, On N=8 It will also be interesting to investigate these Supergravity on AdS5 and N=4 Superconformal B532 anomaly issues within the non-commutative ver- Yang-Mills theory,Nucl.Phys. (1998) 153, hep-th/9802203. sion of the AdS/CFT correspondence [20] to see the origin of the higher derivative Chern-Simons [11] A.N. Redlich, Gauge Noninvariance and Par- terms on the supergravity side. ity Nonconservation of Three-dimensional Fermions, Phys. Rev. Lett. 52 (1984) 18; Parity Violation and Gauge Noninvariance Acknowledgments of the Effective Gauge Field Action in Three Dimensions, Phys. Rev. D29 (1984) 2366. This work was partially supported by the Swiss [12] A.J. Niemi, G.W. Semenoff, Axial-Anomaly- National Science Foundation, by the European Induced Fermion Fractionization and Effec- Union under TMR contract ERBFMRX-CT96- tive Gauge Theory Actions in Odd-Dimensional 0045. Spacetimes, Phys. Rev. Lett. 51 (1983) 2077. [13] L. Alvarez-Gaume, S. Della Pietra, G. Moore, References Anomalies and Odd Dimensions, Ann. Phys. 163 (1985) 288. [1] A. Bilal, C.-S. Chu, A Note on the Chiral [14] H.J. Kim, L.J. Romans, P. van Nieuwenhuizen, Anomaly in the AdS/CFT Correspondence and Mass Spectrum of Chiral Ten-dimensional N = 2 1/N Correction,Nucl.Phys.B562 (1999) 181, 2 Supergravity on S5, Phys. Rev. D32 (1985) hep-th/9907106. 389. [2] J. Maldacena, The Large N Limit of Su- [15] M. Gunaydin, P. van Nieuwenhuizen, N.P. perconformal Field Theories and Supergravity, Warner, Nucl. Phys. B255 (1985) 63. 2 Adv. Theor. Math. Phys. (1998) 231, hep- [16] E. Witten, Quantum Field Theory and Jones th/9711200. Polynomial, Commun. Math. Phys. 121 (1989) [3] S.S. Gubser, I.R. Klebanov, A.M. Polyakov, 351 . Gauge Theory Correlators from Non-Critical [17] M. Henningson, K. Skenderis, The Holographic String Theory, Phys. Lett. B428 (1998) 105, Weyl Anomaly, JHEP 9807 (1998) 023, hep- hep-th/9802109. th/9806087. [4] E. Witten, Anti de Sitter Space and Hologra- [18] O. Aharony, J. Pawelczyk, S. Theisen, S. phy, Adv. Theor. Math. Phys. 2 (1998) 253, hep- Yankielowicz, ANoteonAnomaliesinthe th/9802150. AdS/CFT Correspondence, hep-th/9901134. [5] For a review, see O. Aharony, S. S. Gub- [19] M. Blau, E. Gava, K.S. Narain, On Subleading ser, J. Maldacena, H. Ooguri, Y. Oz, Large N Contributions to the AdS/CFT Trace Anomaly, Field Theories, String Theory and Gravity,hep- hep-th/9904179. th/9905111. S.Nojiri,S.D.Odintsov,On the conformal anomaly from higher derivative gravity in [6] D. Z. Freedman, S. D. Mathur, A. Ma- AdS/CFT correspondence, hep-th/9903033. tusis, L. Rastelli, Correlation Functions in the CFT(d)/AdS(d+1) Correpondence,hep- [20] J. M. Maldacena and J. G. Russo, Large N th/9804058. Limit of Non-Commutative Gauge Theories, hep-th/9908134. C.-S. Chu, Induced Chern- [7] M. Gunaydin, L.J. Romans, N.P. Warner, Simons and WZW action in noncommutative Gauged N =8Supergravity in Five Dimensions, spacetime, hep-th/0003007. Phys. Lett. B154 (1985) 268.