14 The Statement of AdS/CFT

Transcription

1 14 The Statement of AdS/CFT 14.1 The Dctonary Choose coordnates ds 2 = `2 z 2 (dz2 + dx 2 ) (14.1) on Eucldean AdS d+1,wherex s a coordnate on R d.theboundarysatz =0. We showed above that scatterng problems n gravty map to correlaton functons n CFT. In ths relaton the boundary value of the bulk feld acted as a source for a CFT operator. correspondence: Z grav [ Ths s generalzed by the followng statement of the AdS/CFT = * exp Ths s called the GKPW dctonary. 65 X Z d d x 0(x)O (x)!+ CFT (14.2) The ndex runs over all the lght felds n the bulk e ectve feld theory, and correspondngly over all the low-dmenson local operators n CFT. The left-hand sde The lhs of (14.2) s the gravtatonal partton functon n asymptotcally AdS space. It s formally computed by the same path ntegral that we dscussed n the context of black hole thermodynamcs. Snce AdS has a boundary, we must provde boundary condtons to defne ths path ntegral. The boundary condtons on bulk scalars are (z,x) =z d 0(x)+subleadngas z! 0. (14.3) where the mass of the bulk scalar s related to the scalng dmenson of the CFT operator by m 2 = (d ), = d r d m2`2. (14.4) 65 After hep-th/ by Gubser, Klebanov and Polyakov and hep-th/ by Wtten. I hghly recommend readng Wtten s paper. 128

2 We wll see below that (14.3) stheleadngsolutonofthewaveequatonforabulk scalar of mass m. Smlar statements apply to all bulk felds, ncludng the metrc, though the boundary condton and formula for the dmenson s slghtly modfed for felds wth spn. The boundary condtons on the metrc nvolve a choce of topology as well as the actual metrc, whch s why we ve ndcated explctly that Z grav depends on the boundary The rght-hand sde The rhs of (14.2) s the generatng functonal of correlators n a CFT. In ths equaton the 0 (x) aresources,andtheo (x) are CFT operators. Denotng the rhs of (14.2) by Z cft [ 0], correlaton functons are computed n the usual way, ho 1 (x 1 ) O n (x n ) CFT n 1 n 0(x 1 ) 0 (x n ) Z cft[ 0]. (14.5) 0 =0 The mappng Each lght feld n gravty corresponds to a local operator n CFT. The spn of the bulk feld s equal to the spn of the CFT operator; the mass of the bulk feld fxes the scalng dmenson of the CFT operator. Here are some examples: Scalar: Abulkscalarfeld (z,x) s dual to a scalar operator n CFT. The boundary value of acts as a source n CFT. Ths s exactly the relatonshp we used n our dervaton of the absorpton cross secton of the black strng. Gravton: Every theory of gravty has a massless spn-2 partcle, the gravton g µ.ths s dual the stress tensor T µ n CFT. Ths makes sense snce every CFT has a stress tensor. The fact that the gravton s massless corresponds to the fact that the CFT stress tensor s conserved. It also fxes the scalng dmenson to T = d. We wll see ths n more detal later. Vector: If our theory of gravty has a spn-1 vector feld A µ, then the dual CFT has a spn-1 operator J µ.ifa µ s massless, then J = d 1andJ µ s a conserved current. Otherwse, J >d 1andthecurrentsnotconserved. 129

3 Ths llustrated a general and mportant feature of AdS/CFT: gauge symmetres n the bulk correspond to global symmetres n the CFT. Ths s UV complete. Note that CFTs are UV complete. Therefore (14.2) sanon-perturbatveformulatonof a UV complete theory of quantum gravty. Shockngly, t s a defnton of gravty from a QFT wthout gravty. Ths s very powerful because we understand QFT relatvely well Example: IIB Strngs and N =4Super-Yang-Mlls In some sense, t s beleved that the AdS/CFT correspondence as summarzed by (14.2) holds for any theory of gravty and and CFT. That s, gven a theory of gravty we can use t to defne a CFT va (14.2), and (perhaps) vce-versa. But asde from certan examples, the correspondence s well defned and useful only n certan lmts. To llustrate ths we turn to a specfc example where AdS/CFT s understood n great detal. Ths s the dualty between IIB strng theory and supersymmetrc gauge theory: IIB strngs on AdS 5 S 5 = Yang-Mlls n 4d wth N =4supersymmetry. The gravty sde The strng theory has adjustable scales ` `AdS, the Planck scale `P,andthestrng scale `s. We do not need to use any detals of strng theory except to say that at low energes, the e ectve acton s Ensten + Matter + hgher curvature correctons suppressed by the strng scale: 66 S IIB 1 G N Z pg R + Lmatter + `4 sr 4 + (14.6) The strngy states have masses of order 1/`2s, soatenergesbelow1/`2s t s just an ordnary e ectve feld theory lke we dscussed at the begnnng of the course. 66 There are no R 2 correctons allowed wth ths amount of supersymmetry, but there are smlar examples wth non-zero `2sR 2 terms. 130

4 The CFT sde N =4Super-Yang-Mllssahghlysupersymmetrcgaugetheoryn4d. Itsmatter context s fxed unquely by supersymmetry. It s just an SU(N) gaugefeldplusall the matter felds requred by supersymmetry, whch nclude matrx-valued scalar felds transformng the adjont representaton of SU(N) (unlkethefundamentalrepresentatons we usually encounter n, say, QCD). The gauge theory has two dmensonless parameters, N (e the sze of SU(N)) and the Yang-Mlls couplng constant g YM.Defnethecombnaton = g 2 YMN. (14.7) Ths s called the t Hooft couplng. It turns out that gauge theory at large N s most naturally organzed as an expanson n and 1/N, rather than g YM and 1/N. Ths s roughly because there are N felds runnng n loops, whch changes the expanson parameter from gym 2 to. The mappng The mappng from strng theory parameters to CFT parameters s `AdS `strng 4 (14.8) and `d 1 AdS G N `AdS `P d 1 N 2. (14.9) (wth known coe cents). We wll see where ths partcular scalng comes from below n more generalty. For now we just want to note that ths s a strong/weak dualty: when one sde s easy, the other s (usually) hard. For example to have semclasscal Ensten gravty, both loops and hgher curvature correctons must be suppressed on the gravty sde. Ths means N 1 and 1 so the CFT s very strongly coupled. On the other hand f we consder a weakly coupled CFT, then `s `AdS so strngy/hgher curvature correctons are not suppressed on the gravty sde and ths presumably behaves nothng lke ordnary gravty. (Ths s related to so-called hgher spn gravty or Vaslev gravty.) 131

5 nteger spn: etc. (14.10) 14.3 General requrements Returnng to AdS/CFT n general, we can make some smlar observatons about when t produces a nce semclasscal theory of gravty. Ths requres as least two thngs: 1. Strongly coupled CFT. If the CFT s weakly coupled, then there are too many operators. For example, a free scalar feld leads to conserved currents of every On the gravty sde, ths would requre lots of massless or very lght hgh-spn states. Ths s somethng we expect n strng theory at hgh enough energes but not n our low energy e ectve feld theory. So we must requre that the CFT has a sparse spectrum of low-dmenson operators. Ths s sometmes called a large gap n the spectrum, meanng a gap between the low-energy felds and the strngy stu. Ths can only happen at strong couplng, although there can also be strongly coupled theores wth no gap whch therefore do not have nce gravty duals. 2. Large N dof. In the super-yang-mlls example, we sad G N 1/N 2 so that the large number of degrees of freedom s requred for gravty to be weakly coupled. Ths s true n general, too. There are two ways to see ths, both of whch we wll dscuss n more detal later. I wll purposely be a lttle vague about the defnton of N dof snce there are several reasonable ways to defne t, and they are all d erent. Frst, note that black hole entropy s S / 1/G N,whchsverylarge. Snce entropy s the log of the densty of states, ths means holographc CFTs must have an enormous degeneracy of states at hgh energy. Ths means there are lots of degrees of freedom. For example, a 2d CFT consstng of N b free bosons has S(E) / p N b E. Second, we can roughly measure the degrees of freedom by lookng at the stresstensor 2pt functon. Ths s fxed by conformal nvarance up to a sngle coe - 67 Ths s schematc, you must add correctons to these operators for them to be conserved by the EOM. 132

6 cent: ht µ (x)t (y) = c (known functon of x, y). (14.11) The coe cent c s a measure of degrees of freedom. 68 Consder agan lots of free felds: the stress tensors add, so the total stress tensor wll have a very bg 2pt functon. On the gravty sde, the stress tensor s dual to the gravton. We wll see n detal below how to calculate corelators, but for now su ce t to say that htt cft wll be related to a gravton scatterng experment hgg gravty 1/G N.Thusc 1/G N and we see agan that weakly coupled gravty requres an enormous number of degrees of freedom. Clear there s a tenson between requrements (1) and (2). We want lots of degrees of freedom, and lots of states at hgh energes, but very few states at low energes. Roughly speakng, you can thnk of ths as the requrement that the CFT by confnng: t has lots of states at hgh energes, but very few at low energes where quarks are confned. Later we wll see a very drect lnk between black hole thermodynamcs and confnement The Holographc Prncple Many years ago Bekensten conjectured that the maxmum entropy you can ft nto a regon of space s equal to the entropy of the correspondng black hole: S max = area 4G N. (14.12) Ths s called the Bekensten bound. The argument s smple.if you have lots of stu n a regon and S stuff >S blackhole,thenyoucanthrownsomemorestu andforma black hole. In dong so, the entropy of the system decreases! Therefore the second law requres a bound lke (14.12). 69 Ths bound nspred t Hooft (n 93) and later Sussknd (n 94) to argue that a theory 68 But not an entrely satsfactory one. For example, t can ncrease under RG flow. 69 In the last few years ths bound has been understood much better usng entanglement entropy. See for example and references theren. 133

7 of quantum gravty must secretly lve n fewer dmensons than our observed spacetme. Ths prncple s realzed concretely by AdS/CFT. 134