View metadata, citation and similarScale papers Vs. at core.ac.uk Conformal Invariance in the AdS/CFT Correspondence brought to you by CORE provided by CERN Document Server Adel M. Awad] and Clifford V. Johnson\

]Department of Physics and Astronomy, University of Kentucky, Lexington, KY 40506, U.S.A. and Department of Physics, Faculty of Science, Ain Shams University, Cairo 11566, Egypt

\Centre for Particle Theory, Department of Mathematical Sciences, University of Durham, Durham, DH1 3LE, U.K.

[email protected], [email protected]

(June 2000) We present two examples of non–trivial field theories which are scale invariant, but not confor- mally invariant. This is done by placing certain field theories, which are conformally invariant in flat space, onto curved backgrounds of a specific type. We define this using the AdS/CFT corre- spondence, which relates the physics of gravity in asymptotically Anti–de Sitter (AdS) spacetimes to that of a conformal field theory (CFT) in one dimension fewer. The AdS rotating (Kerr) black holes in five and seven dimensions provide us with the examples, since by the correspondence we are able to define and compute the action and stress tensor of four and six dimensional field theories residing on rotating Einstein universes, using the “boundary counterterm” method. The rotation breaks conformal but not scale invariance. The AdS/CFT framework is therefore a natural arena for generating such examples of non–trivial scale invariant theories which are not conformally invariant.

There is an often quoted piece of folklore in the sub- vanish, while the types B and D anomalies do. In this a ject that states that any non–trivial example of a field case, there will be an irremovable anomaly Ta , for which a theory which is scale invariant is automatically confor- Ta = 0, showing that scale invariance is preserved. mally invariant. Proofs of this statement only exist in S b In short, we must find a way to place a conformally certain specific situations, and in fact it is known to be R invariantb theory on a spacetime for which the Euler den- not generally applicable. Some early discussion can be sity is non–vanishing, but which is topologically trivial, found in refs. [1,2]. The focus in those cases is on finding so that the integral vanishes. Our spacetime must also field theory examples in flat space. Another way to vi- be conformally flat, thus not contributing to the Type B olate conformal invariance is of course to place the field anomaly, which would break scale invariance too. In this theory on a spacetime with non–zero curvature. Then letter, we show how to do this, and in this way find a there are conformal anomalies, and the non–vanishing new class of counterexamples to the folklore. trace of the stress–energy tensor, Tab, can be written in terms of various local curvature invariants of the back- New tools have appeared on the market for defining ground spacetime. The general formb of the anomaly in and studying conformally invariant field theories in in- dimension n (which is even, since we only have conformal teresting situations, often even at strong coupling. One anomalies in those cases) is given by (see e.g. ref. [3]): of these, the “AdS/CFT correspondence”, relates an (n+1)–dimensional theory of gravity on anti–de Sitter T a =c E + c I + J a . (1) a 0 n i i ∇a (AdS) spacetime (times a compact manifold) to a con- i X formal field theory (CFT) in n dimensions. This duality b Here, the c’s are constants, En is the Euler density, Ii are first arose as a result of investigating [4] N parallel D3– terms constructed from the Weyl tensor and its deriva- branes (reviewed in refs. [5]) in the context of the low tives, and the last term is a collection of total derivative energy, classical limit of type IIB superstring theory, — terms. The first type of term is called “type A”, the the supergravity limit— on five dimensional anti–de Sit- 5 next “type B”, and the last “type D”. It is important to ter spacetime times a five sphere (AdS5 S ). The dual note that the coefficients of all terms are regularisation CFT in this case is the four dimensional × =4 supersym- scheme independent except the type D anomaly. These metric SU(N) Yang–Mills theory for largeN N. For other latter terms can be removed by a suitable addition of lo- dimensions, the dual theories exist, but are less well un- cal counterterms to the action. The type A anomaly is derstood. For example eleven dimensional supergravity 4 only locally a total derivative, in general. on AdS7 S is dual to a six dimensional “(0,2)” CFT, In order to construct a non–trivial example of a scale the notation× denoting the number and chirality of the six invariant theory which is not conformally invariant, we dimensional supercharges. (See ref. [6] for a review.) A can simply place a conformally invariant theory on a precise statement of the AdS/CFT correspondence [7,8] spacetime for which the type A anomaly does not equates the partition functions: S

1 2 δI ZAdS(φi)=ZCFT(φ0,i) . (2) T ab = . (5) √ h δhab From the gravity–on–AdS point of view, φi is a bulk field − constrained to the values φ0,i on the AdS boundary, while For orientation, a metric on AdSn+1, in global coordi- from the CFT point of view, φ0,i are sources for pointlike nates is: operators, i, in the theory. In the low energy limit of O 2 2 the theory one can use the classical gravitational action 2 r 2 dr 2 2 ds = 1+ dt + + r dΩn 1 , (6) to calculate the partition function of the CFT “on the − l2 (1 + r2/l2) −   boundary”. This action has the form [9], 2 n 1 where dΩn 1 is the metric on a round S − . Equation (5) − 1 n+1 n(n 1) gives a definition of the action and stress–tensor on any Ibulk + Isurf = d x√ g R + − − 16πG − l2 region (radius r in the coordinates that we will choose ZM   1 later) bounding the interior of AdS . The AdS/CFT dnx√ hK. (3) n+1 relation equates these quantities to a dual conformal field − 8πG ∂ − Z M theory residing “on the boundary” at (r ). →∞ The first term is the Einstein–Hilbert action with nega- As stated before, the metric restricted to the boundary, tive cosmological constant (Λ= n(n 1)/2l2). The sec- − − hab, diverges due to an infinite conformal factor, which ond term is the Gibbons–Hawking boundary term. Here, is r2/l2. We take the background metric upon which the hab is the boundary metric and K isthetraceofthe dual field theory resides as extrinsic curvature Kab of the boundary. The theory on the boundary can be seen to obtain l2 its conformal invariance properties from two (related) γab = lim hab . (7) r r2 sources: First, the metric on the boundary of the the- →∞ ory is not uniquely defined, since the AdS metric has a and so the field theory’s stress–tensor, T ab, is related to double pole there. It is instead only defined up to a con- the one in (5) by the rescaling [19]: formal class of metrics. The double pole divergence of b the metric is precisely what allows the theory to inherit bc bc √ γγabT = lim √ hhabT . (8) r the properties of a conformal field theory, since the pole − →∞ − shows up as the correct behaviour of the operator prod- b ab uct expansion in the CFT [7]. Second, the SO(n+1, 2) This amounts to multiplying all expressions for T dis- n 2 played later by (r/l) − before taking the limit r . isometry of the AdSn+1 spacetime descends to the local →∞ conformal group of the field theory on the boundary [4]. For the AdSn+1 example (6), the n dimensional bound- The AdS/CFT correspondence is therefore a powerful ary upon which the theory resides is the Einstein uni- 2 2 2 2 way of studying properties of the dual conformal field verse, with metric ds = dt + l dΩn 1.Inthecaseof − − theory, by relating them to properties of the gravity the- n=4, the field theory stress tensor computed using the ory. Here, we will compute the action and stress tensor above methods [11] can be written (as for other n)inthe for certain gravity solutions and relate them to properties standard perfect fluid form [19] (ua=(1, 0, 0, 0)): of their dual field theories. To deal with the divergences 1 N 2 which appear in the gravitational action (arising from in- Tab = (4uaub + γab)= (4uaub + γab) . (9) tegrating over the infinite volume of spacetime), we shall 64πlG 32π2l4 use the “counterterm subtraction” method [11], which Hereb we used the dictionary [4] between gravity and regulates the action by the addition of certain boundary field theory quantities, G=l3π/2N 2. The total energy, counterterms which depend upon the geometrical prop- E= d3xT =3N 2/16l, is in fact [11] the Casimir energy erties of the boundary of the spacetime. They are chosen 00 of the =4 supersymmetric SU(N) Yang–Mills theory to diverge at the boundary in such a way as to cancel the on theR NS3. (See also ref. [20].) The conformal invari- bulk divergences [11,13] (see also refs. [14–18]): b a ance of the theory is evident in the fact that Ta =0. It is worth stressing that there should be no confusion about 1 n (n 1) l Ict = d x√ h − R + the fact that the theory is conformal while inb a box, S3, 8πG ∂ − " l − 2(n 2) Z M − which has a scale, l. Conformal invariance is preserved l3 n since this scale enters in a conformally invariant way in ab 2 . (4) 2 ab the metric itself. 2(n 4)(n 2) R R − 4(n 1)R # − −  −  This prescription gives a method for computing the Here and ab are the Ricci scalar and tensor for the stress tensor of a large class of field theories, which may boundaryR metricR h. Using these counterterms one can be obtained by studying spacetimes which are asymptot- construct a divergence–free stress tensor from the total ically locally AdS. We will now show that it also provides action I=Ibulk+Isurf +Ict by defining (see e.g. ref. [12]): a natural method for generating examples of field theories

2 which are scale but not conformally invariant, by placing action and define an “improved” [28] stress tensor (see a conformal field theory on a spacetime with just the cor- also refs. [29]). However, as pointed out in a note–in– rect properties we asked for earlier. We shall present two proof added to ref. [27], it was not possible to do this examples here, and will report more details and examples while preserving the values for the physical conserved in an extended publication [21]. quantities (for example) like angular momentum. The Our examples come from the Kerr–AdS spacetimes in point we stress here is that it is quite a special case that five and seven dimensions (with only one of the two ro- the Euler density happened to be proportional to 2 .In tation parameters non–zero [22]): general it cannot be written in this form. Our anomalyR is purely of type A, and as such its coefficient cannot 2 ∆ a sin2 θ ρ2 ρ2 be changed by adding counterterms. Instead, we accept ds2 = r dt dφ + dr2 + dθ2 2 − ρ − Ξ ∆r ∆θ the presence of the anomaly and give up conformal in-   variance; the rotation parameter a has broken conformal ∆ sin2 θ (r2 + a2) 2 + θ adt dφ invariance of the theory, but scale invariance is preserved. ρ2 − Ξ   We now turn to the case of Kerr–AdS7. We computed 2 2 2 +r cos θdΩn 3 , the non–vanishing components for the stress tensor at − 4 large r,toO(r− ) using eqns. (4) and (5): 2 2 2 2 2 2 Ξ=1 a /l ,ρ= r + a cos θ, 3 − l 2 2 2 2 4 4 4 2 2 2 2 4 n T = (1 + a /l )( 131a /l 423a /l cos θ) ∆ =(r + a )(1 + r /l ) 2MGr − , tt 4 r − 640πGr − − ∆ =1 (a2/l2)cos2 θ (10) +219(1 +a4/l4)a2/l2 cos2 θ + 25(1 + a6/l6) θ − +235a6/l6 cos6 θ + 400MG/l4 +492a4/l4 cos2 θ + , where n=4 or 6, and: ··· 3 2 l a sin θ 6 6 4 2 2  2 2 2 2 2 2 2 2 Ttφ = 231a /l cos θ +5+a /l dΩ1 = dψ ; dΩ3 = dψ +sin ψdη +cos ψdβ . (11) −640πGr4Ξ − +55a6/l6 cos6θ + 189a6/l6 cos2 θ 25a6/l6 Using the prescription (7), the metric on which the field − +168a4 cos2 θ/l4 51a4/l4 cos4 θ 101a4/l4 theory (either the =4 Yang–Mills theory for n=4 or the − − (0,2) CFT for n=6)N resides can be seen to be that of a 3a2/l2 cos2 θ + 400MG/l4 + , − ··· rotating Einstein universe [24] (see also refs. [23,25,26]): al5 sin2 θ T = 102a4/l4 +51a4/l4 cos4 θ φφ 640πGr4Ξ2 2a sin2 θ dθ2 ds2 = dt2 + dtdφ + l2 55a6Ξ/l6 cos 6 θ 171a4/l4 cos2 θ + 189a8l8 cos2 θ/ − Ξ ∆θ − − +3a2/l2 cos2 θ +4a2/l2 231a8/l8 cos4 θ sin2 θ 2 2 2 2 2 6 6 4 − 2 6 2 +l dφ + l cos θdΩn 3 . (12) 180a /l cos θ + 480MGa /l cos θ Ξ − +80MG/l4 21a6 cos2 θ/l6 76a6/l6 25a8/l8 The gauge theory stress tensor of the Kerr–AdS5 space- − − − +400a2MG/l6 5 + , time was computed in ref. [27], and the full expression can − ··· be found there. In particular the trace is: l5 T = 5a2Ξ /l2 80MG/l4 θθ −640πG∆ r4 − N 2a2 θ T a = [a2/l2(3 cos4 θ 2cos2 θ) cos 2θ] . (13) +3a2/l2 cos2 θ(1 + a4/l4)+66a4/l4 cos2 θ a − 4π2l6 − − 45a4/l4 cos4 θ(1 + a2/l2)+25a6/l6 cos6 + , bWhile it is non–zero, a quick computation shows − ··· l5 cos2 θ that this is a total derivative, and it is in fact T a = T = 5Ξ(1 a4/l4) 80MG/l4  a ψψ 4 2 2 −640πGr − − (N /π )E4, where the Euler density is: − 51a4/l4 cos 4 θ(1 + a2/l2) 3a2/l2 cos2 θ(1 + a4/l4) b − − 1 4 4 2 E µνκλ 4 µν + 2 . (14) +46a /l cos θ + , 4 64 µνκλ µν ··· ≡ R R − R R R T =sin2 ψT ,T =cos2 ψT . (15)   ηη ψψ ββ ψψ The coefficient is precisely the field theory value [14]. 4 a A computation shows that the action is again finite: Since d x√ γ Ta =0, (which fits that the action, I,is computed to− be finite [27,22]), this is our first example R 2π3(r2 + a2)r of having preservedb scale invariance while having broken I = + + 160(r6 /l6 4 2 2 2 2 2 + conformal invariance. This was considered as a possibil- GΞ 3r+/l +2r+(1 + a /l )+a ity in ref. [27]. However, it was noticed there that the 4 6 2 4 4 4  6 6 +r+/l a M/l )+5a /l + 50Ξ
+ a /l , (16) trace was also proportional to 2 .Soinfact,itwas − R  misidentified there with an anomaly of type D. A pro- where r+ is the location of the horizon, the largest root posal was made there to add an 2 counterterm to the of ∆ =0. R r

3 Taking the trace of the stress tensor yields: [1] C. M. Hull and P. K. Townsend, Nucl. Phys. B274, 349 (1986). a2N 3 [2] J. Polchinski, Nucl. Phys. B303, 226 (1988). T a = 5a4/l4 cos6 θ 8cos4 θa2/l2(1 + a2/l2) a −2π3l8 − [3]S.DeserandA.Schwimmer,Phys.Lett.B309 (1993) +3 cos2 θ(1 + a4/l4 +3a2/l2) 2(1 + a2/l2) (17) 279, hep-th/9302047. b − [4] J. Maldacena, Adv. Theor. Math. Phys. 2 231 (1998), where we used the relation [4] N 3=3π2l5/16G between hep-th/9711200. [5] J. Polchinski, S. Chaudhuri and C. V. Johnson, hep- the gauge theory and the gravity parameters. th/9602052; J. Polchinski, hep-th/9611050. Again, we see that this trace is a total derivative. Fur- [6] O. Aharony, S. S. Gubser, J. Maldacena, H. Ooguri and thermore, T a = (N 3/4508π3)E . (The Euler density a − 6 Y. Oz, Phys. Rept. 323, 183 (2000), hep-th/9905111. E6 is displayed in e.g. ref. [30], where the CFT is dis- [7] E. Witten, Adv. Theor. Math. Phys. 2 253 (1998), hep- cussed at weakb coupling). The coefficient matches the th/9802150. results in refs. [14,17,30]. (n.b., a typo in ref. [14] is cor- [8] S. S. Gubser, I. R. Klebanov and A. M. Polyakov, Phys. rected in ref. [30].) Just as in the four dimensional case, Lett. B428 105 (1998), hep-th/9802109. we see that for this special situation, the Euler density [9] G. W. Gibbons and S. W. Hawking, Phys. Rev D15, can be written in terms of type D quantities. In the no- 2752 (1977). 7 [10] S. W. Hawking and G. T. Horowitz, Class. Quant. Grav. tation of ref. [30], it is of the form i=1 diCi,withd5 and d zero, since they depend on the Weyl tensor, and 13, 1487 (1996) gr-qc/9501014. 7 [11] V. Balasubramanian and P. Kraus, Commun. Math. d ,d ,d ,d ,d = 0, 1/9, 1/72, 5P/12, 1/72 .(Ause- 1 2 3 4 6 Phys. 208, 413 (1999), hep-th/9902121. {ful parameterisation,} { but of course,− not unique.)} This [12] J. D. Brown and J. W. York, Phys. Rev. D47, 1407 completes our second counterexample to the folklore. (1993). In conclusion, we have used the AdS/CFT correspon- [13] R. Emparan, C. V. Johnson and R. C. Myers, Phys. Rev. dence to define known four and six dimensional confor- D60, 104001 (1999), hep-th/9903238. mal field theories on spacetimes with properties chosen [14] M. Henningson and K. Skenderis, J.H.E.P. 9807 023 so that conformal invariance, but not scale invariance, is (1998), hep-th/9806087. broken. The crucial point is that there are “many faces” [15] S. R. Lau, Phys. Rev. D60 (1999) 104034, gr-qc/9903038. [13] to anti–de Sitter spacetime, which can be found by [16] R. B. Mann, Phys.Rev. D60 (1999) 104047, hep- slicing it in different ways by various coordinate choices, th/9903229. and then picking the boundary for the field theory to live [17] P. Kraus, F. Larsen and R. Siebelink, Nucl. Phys. B563, 259 (1999), hep-th/9906127. on. The Kerr–AdS solutions give a particularly interest- [18] S. N. Solodukhin, hep-th/9909197. ing slicing, for our present purposes: The M=0 limit is [19] R.C. Myers, hep-th/9903203. just AdS in very non–standard coordinates. The bound- [20] G. T. Horowitz and R. C. Myers, Phys. Rev. D59, 026005 ary is related to the standard static Einstein universe (1999) hep-th/9808079. (which has vanishing Euler density) by a complicated [21] Adel M. Awad and C. V. Johnson, to appear. change of variables, and a conformal transformation [22]. [22] S. W. Hawking, C. J. Hunter and M. M. Taylor– This is why the Euler density can be globally written as a Robinson, Phys. Rev. D59 (1999) 064005, hep- total derivative, while the manifold remains conformally th/9811056. flat. It would be interesting to characterise such space- [23] D. Klemm, JHEP 9811, 019 (1998), hep-th/9811126. times further, since they give a straightforward means [24] M.J. Cassidy and S.W. Hawking, Phys. Rev. D57, 2372 of defining the sort of field theory examples discussed (1998) hep-th/9709066. [25] D. S. Berman and M. K. Parikh, Phys. Lett. B463, 168 herein. (1999), hep-th/9907003. [26] S. W. Hawking and H. S. Reall, Phys. Rev. D61, 024014 Acknowledgements: We would like to thank Vijay (2000), hep-th/9908109. Balasubramanian, Roberto Emparan, Juan Maldacena, [27] A. Awad and C. V. Johnson, Phys. Rev. D61 (2000) Rob Myers, Joe Polchinski, and Al Shapere for comments 084025, hep-th/9910040. and discussions, and especially Paul Townsend for raising [28] C.G. Callan, S. Coleman and R. Jackiw, Ann. Phys. 59, the main issue discussed in this paper. CVJ would like to 42 (1970). thank the Theory group at Harvard University for hospi- [29] J. Ho, hep-th/9910124 and hep-th/0005250. tality while this paper was written. AMA’s work was sup- [30] F. Bastianelli, S. Frolov and A. A. Tseytlin, JHEP 0002, ported by an NSF Career grant, #PHY–9733173. This 013 (2000), hep-th/0001041. paper is report #’s UK/00–02, and DTP/00/43. This paper was brought to you by the letters A, d, S, C, F, and T, and the numbers 4, 5, 6 and 7.

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