On the Central Charge of Spacetime Current Algebras and Correlators in String Theory on AdS3
aa r X i v : . [ h e p - t h ] M a y On the Central Charge of Spacetime Current Algebrasand Correlators in String Theory on
AdS a and Massimo Porrati a,ba Center for Cosmology and Particle Physics,Department of Physics, New York University,4 Washington Place, New York, NY 10003, USA b School of Natural Sciences, Institute for Advanced StudyPrinceton NJ USA 08540 Abstract
Spacetime Virasoro and affine Lie algebras for strings propagating in
AdS are knownto all orders in α ′ . The central extension of such algebras is a string vertex, whoseexpectation value can depend on the number of long strings present in the backgroundbut should be otherwise state-independent. In hep-th/0106004, on the other hand, astate-dependent expectation value was found. Another puzzling feature of the theory islack of cluster decomposition property in certain connected correlators. This note showsthat both problems can be removed by defining the free energy of the spacetime boundaryconformal field theory as the Legendre transform of the formula proposed in the literature.This corresponds to pass from a canonical ensemble, where the number of fundamentalstrings that create the background can fluctuate, to a microcanonical one, where it isfixed. Member until May 2015, on sabbatical leave from NYU. osonic strings and superstrings compactified on
AdS × M with a nonzero Kalb-Ramondfield strength on AdS allow for a world-sheet description exact in α ′ . For simplicity, we willwork here with the bosonic string, but our argument works equally for superstrings. The confor-mal field theory living on the target space M is unitary while the AdS part of the backgroundis a Wess-Zumino-Witten SL (2 , R ) model, which is exactly soluble. Many properties of themodel are known, including its complete spectrum [1], its thermal partition function [2] andcorrelation functions [3].The spacetime affine Lie algebra and spacetime Virasoro generators were found in [4] inthe free field Wakimoto representation [5] of the SL (2 , R ) WZW model. Such representationis adequate to study spacetime algebras, since their generators are non-normalizable verticessupported in the near-boundary, weakly-coupled region of AdS . The exact form of thesevertices was found in [6]. In both representations, a non-vanishing central charge was found.Such central charge is exact to all orders in α ′ and exists already at tree level in the stringcoupling parameter, g S . It is the string theory generalization of the classical central chargefound in asymptotically AdS General Relativity by Brown and Henneaux [7].Instead of asymptotic in- or out- states, labeled by momenta and other quantum numbers,(Euclidean)
AdS possesses local operators, which define a boundary CFT. They are labeledby the boundary coordinate x ∈ C . In the string worldsheet description, local operators arerepresented by vertices, also labeled by x . A natural set of operators has form (see [6] fornotations and more details on the formalism) V ( x, ¯ x, h, I ) = Z d z Φ h ( x, ¯ x | z, ¯ z ) O I . (1)The operator Φ h ( x, ¯ x | z, ¯ z ) is a worldsheet Virasoro and SL (2 , R ) affine Lie primary, withworldsheet conformal weight ∆ = ¯∆ = − h ( h − / ( k − k is the level of the world-sheet SL (2 , R ) affine Lie algebra (not to be confused with the spacetime affine Lie algebra).The operators O I belong to the conformal theory on M . They have conformal dimension∆ I = ¯∆ I = 1 + h ( h − / ( k − SL (2 , R ) currents define the operator J ( x | z ) = 2 xJ ( z ) − J + ( z ) − x J − ( z ). The current algebra Operator Product Expansion (OPE)writes compactly as J ( x | z ) J ( y | w ) = k ( y − x ) ( z − w ) + 1 z − w [( y − x ) ∂ y − y − x )] J ( y | w ) + regular terms . (2)1he spacetime central charge is proportional to the vertex I = 1 k Z d zJ ( x | z ) ¯ J (¯ x | ¯ z )Φ ( x, ¯ x | z, ¯ z ) . (3)The SL (2 , R ) current algebra level k is the ratio l /α ′ , with l the radius of AdS . As provenin [6], I is independent of x, ¯ x . The I vertex can be written in terms of the ¯ ∂ derivativeof an operator Λ( x, ¯ x | z, ¯ z ). It is nevertheless nonzero because Λ is not a good observable;in particular, its two-point function is logarithmic in z, ¯ z [6]. In the near-boundary, weaklycoupled region, it is conveniently written in terms of Wakimoto variables [5, 6] ( β, γ, φ ) aslim φ →∞ Λ = ( x − γ ) − . The operators Λ and Φ are related by [6]¯ J Φ = kπ ∂ ¯ z Λ , (4)so the identity vertex is I = 1 k Z d zJ ( x | z ) ¯ J (¯ x | ¯ z )Φ ( x, ¯ x | z, ¯ z ) = − πik I dzJ Λ . (5)When inserted into correlation functions of vertices (1), the integral does not vanish because theoperator product expansion of J Λ with Φ h has poles and because J Λ transforms anomalouslyunder coordinate transformations [8].The first property follows from the OPEs [6] ( ∼ denotes equality up to regular terms)Λ( x, ¯ x | z, ¯ z )Φ h ( y, ¯ y | w, ¯ w ) ∼ x − y Φ h ( y, ¯ y | w, ¯ w ) ,J ( x | z )Φ h ( y, ¯ y | w, ¯ w ) ∼ z − w [( y − x ) ∂ y + 2 h ( y − x )]Φ h ( y, ¯ y | w, ¯ w ) . (6)The second one follows because the OPE of J ( x ) with Λ( x ) is singular [6] J ( x | z )Λ( x, ¯ x | w, ¯ w ) ∼ − z − w . (7)So, even though J and Λ transform under holomorphic changes of coordinates as tensors ofweight one and zero respectively , the normal ordered product : J ( x | z )Λ( x, z ): ≡ J Λ( x | z ) trans-forms anomalously as T ( z ) J Λ( x | w ) ∼ z − w ) J Λ( x | w ) + 1 z − w ∂ w [ J Λ( x | w )] − z − w ) . (8) This can be seen most easily using the Wakimoto representation. Notice that Λ is nevertheless a badobservable because, among other things, its two-point function contains logarithmic terms that need an IRregularization.
2o under an infinitesimal diffeomorphism ǫ , J Λ transforms as δJ Λ( x | z ) = ∂ z ǫ ( z ) J Λ( x | z ) + ǫ ( z ) ∂ z J Λ( x | z ) − ∂ z ǫ ( z ) . (9)Under a finite change of coordinates z → z ′ = φ ( z ) eq. (9) integrates to J Λ( x | z ) → ( J Λ) ′ ( x | z ′ ) = ∂ z φ ( z ) (cid:20) J Λ( x | z ) + 12 ∂ φ z∂ φ z (cid:21) . (10)Put together, eqs. (6,10) give rise to a puzzling result found in [8]: the operator I is notproportional to the identity. We can prove this by considering the correlator h I Y i Φ h i ( x i , ¯ x i | z i , ¯ z i ) i = 1 k h Z d zJ ( x | z ) ¯ J (¯ x | ¯ z )Φ ( x, ¯ x | z, ¯ z ) Y i Φ h i ( x i , ¯ x i | z i , ¯ z i ) i . (11)In ref. [8] it was evaluated on a genus zero surface, but the computation can be done for arbitrarygenus using the Schottky parametrization of Riemann surfaces. In such parametrization, agenus g surface is represented as the region of the complex plane outside a set of 2 g circlesˆ C n , n = 1 , .., g . The circles are identified pairwise by SL (2 , C ) transformations, z → ( az + b ) / ( cz + d ), ad − bc = 1, c = 0, that map the outside of one circle in the pair to the inside ofthe other. Using equation (5), correlator (11) can then be written as − πik I C dz h J Λ( x | z ) Y i Φ h i ( x i , ¯ x i | z i , ¯ z i ) i , (12)where the contour C is the union of small circles C i surrounding the operator insertion points z i , plus the circles ˆ C n , plus the limit for R → ∞ of a circle C R at radius | z | = R . The OPEs(6) give [8] − πi I C i dzJ Λ( x | z )Φ h i ( x i , ¯ x i | z i , ¯ z i ) = ( h i − h i ( x i , ¯ x i | z i , ¯ z i ) . (13)To find the contribution of the circles ˆ C n we apply eq. (10) to the the transformations thatidentify such circles pairwise. Under the map z → z ′ = ( az + b ) / ( cz + d ) the integrals of thehomogenous term in (10) cancel and one is left with g integral − − (2 πi ) − H ˆ C ′ n dz ′ [2 c/ ( − cz ′ + a )].Since the point z ′ = a/c is mapped to the point z = ∞ , it is inside the circle ˆ C ′ n , so that theintegral gives a contribution −
1. The sign is − because, under the SL (2 , C ) map, the image We take them to be all of finite radius so that the point z = ∞ lies outside all circles. g such integrals, we get − πi I ∪ j C j ∪ n ˆ C n dz h J Λ( x | z ) Y i Φ h i ( x i , ¯ x i | z i , ¯ z i ) i = [ X j ( h j − − g ] h J Λ( x | z ) Y i Φ h i ( x i , ¯ x i | z i , ¯ z i ) i . (14)Finally, the integral on C R is evaluated by performing the conformal inversion z → z ′ = − /z . Thanks to eq. (10), the integral becomeslim R →∞ πi I C R dzJ Λ( x | z ) = − lim R →∞ πi I | z | =1 /R dz [ J Λ( x | z ) − /z ] = 1 . (15)The sign here is + because the integral over C R is performed clockwise. We thus get anall-genera version of the g = 0 result of [8] h I Y i Φ h i ( x i , ¯ x i | z i , ¯ z i ) i = 1 k [1 − g + X j ( h j − h Y i Φ h i ( x i , ¯ x i | z i , ¯ z i ) i ; (16)So, the “identity” I is not constant but instead assumes different values on different irreduciblerepresentations of the Virasoro algebra.This result is quite disastrous, because it contradicts the fact that in the field theory limit α ′ →
0, the Brown-Henneaux calculation shows a unique central charge for all the irreduciblerepresentations corresponding to light fields. Among them there are many for which one coulduse instead eq. (16). More generally, a Hilbert space that decomposes into a sum of irreduciblerepresentations of the Virasoro algebra, each one with a different central charge, is incompatiblewith having a local 2D CFT on the
AdS boundary. Notice that the operator I can (and does)take different values on sectors containing a different number of long strings [6, 4].In this note, we point out that the problem has a solution. It does not involve any subtlety inthe calculation of ref. [8], though the latter assumes that the OPE of the “bad” operator Λ withΦ h has no logarithmic branch cuts. Rather, the solution is that formulas for the AdS/CFTcorrespondence proposed in the literature [9] do not specify how to treat the operator I . Astandard generalization would treat the identity I as any other vertex, add a source for it,and define a “free energy.” We will show that instead the correct generalization is a Legendretransform of the free energy. This change corresponds to pass from a canonical ensemble,where the number of fundamental strings that produce the background can fluctuate, to amicrocanonical one, where it is held fixed. Our proposal also resolves another puzzle of thestandard definition, namely the lack of cluster decomposition in some connected correlators.To begin with, recall that the spacetime correlators contain contributions from disconnected This is true in the large- φ limit, as can be checked using the Wakimoto representation. Z , is the exponential of the string partition function W = ∞ X g =0 g g − S h exp (cid:20)Z d xJ ( x, ¯ x, h, I ) V ( x, ¯ x, h, I ) + Z d xλ ( x ) I (cid:21) i g ,Z = C exp( W ) . (17)The expectation value h ... i is computed by performing the functional integral of the worldsheetaction over connected closed Riemann surfaces of genus g ; g S is the string coupling constant;the constant C is arbitrary. We recognize in W the generator of connected correlators for thespacetime CFT, that is the free energy. Even though we introduced a local source λ ( x ) for thevertex I , the free energy depends only on λ ≡ R d xλ ( x ), since I is independent of x, ¯ x .Next, consider the correlators hh N Y i =1 Z d z i Φ h i ( x i , ¯ x i | z i , ¯ z i ) O I ii ≡ N Y i =1 δδJ ( x i , ¯ x , h i , I i ) Z (cid:12)(cid:12)(cid:12)(cid:12) J =0 , (18) hh I N Y i =1 Z d z i Φ h i ( x i , ¯ x i | z i , ¯ z i ) O I ii ≡ N Y i =1 δδJ ( x i , ¯ x i , h i , I i ) δδλ ( x, ¯ x ) Z (cid:12)(cid:12)(cid:12)(cid:12) J,λ =0 , (19) hh ii ≡ Z | J = λ =0 . (20)For simplicity, assume that the disconnected components of correlator (18) vanish. Then, theexpectation value (19) is the sum of two pieces [9] hh I N Y i =1 Z d z i Φ h i ( x i , ¯ x i | z i , ¯ z i ) O I ii = h I ihh N Y i =1 Z d z i Φ h i ( x i , ¯ x i | z i , ¯ z i ) O I ii + hh ii X g k [1 − g + X i ( h i − g g − S h N Y i =1 Z d z i Φ h i ( x i , ¯ x i | z i , ¯ z i ) O I i g . (21)The expectation value over connected components includes here a sum over genera so that e.g. dW/dλ ≡ h I i ≡ P ∞ g =0 g g − S h I i g and the leading term in h I i is O ( g − S ). Since the last term ineq. (21) comes from connected VEVs, it would be absent if I were truly proportional to theidentity. In [6] it was argued that a connected component I ( g S ) h Q Ni =1 R d z i Φ h i ( x i , ¯ x i | z i , ¯ z i ) O I i g is permissible, as long as I ( g S ) is the same for all correlators containing at least an insertion ofeither I or Φ h , but this is not compatible with I being the identity as the following argument An easy way to see this is to notice that the string partition function contains only one integration over thezero modes of spacetime coordinate fields per each connected component of the worldsheet. hh I n ii for arbitrary integer n ≥
1. If I were the identity then hh I n ii = AB n , for some constants A and B . On the other hand, from the definition of connectedcorrelators we have W = D exp[ λ I ( g S )], with D a constant; thus we arrive at the recursionrelation hh I n ii = hh ii I ( g S ) n P n ( x ) | x = D , hh I n +1 ii = hh ii I ( g S ) n +1 (cid:20) xP n ( x ) + x ddx P n ( x ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) x = D , (22)with P n [ x ] a polynomial of degree n in x such that P = 1. By computing the VEVs for n = 1 , , AB n .The fact that I has nonzero connected correlators with physical vertices has another trou-bling consequence. In fact, as shown in [3], the four point function of operators with weight h , .., h in the spacetime CFT factorizes (when h i + h j < ( k + 1) /
2) on operators belonging tothe discrete series 1 / < h < ( k − /
2, as well as on other operators. Among the former is theoperator I , so the four point function factorization is W ( x , x , x , x ) = W λ ( x , x ) 1 W λλ W λ ( x , x ) + 1 ↔ ↔ .... (23)Here .... means a sum over other factorization channels and we used the shorthand W i = δW/δJ ( x i , ¯ x i , h i ), W λ = δW/δλ ( x ) etc. Since W ijλ ( x i , x j , x ) and W λλ ( x ) are independent of x , the connected correlator W does not obey the cluster decomposition property.So we must kill in a fell swoop all connected correlators containing the operator I . Tosee how to achieve this, we must recall first that spacetime Virasoro and affine-Lie algebracurrents are also represented by vertices, T xx ( x ) and K a x ( x ), whose explicit form is given in [6].The sources for these vertices are: g ¯ x ¯ x ( x, ¯ x ), transforming as the boundary 2D metric, for theVirasoro vertex and A a ¯ x ( x, ¯ x ), transforming as a 2D gauge field, for the affine-Lie algebra vertex.We will deal here with the affine Lie algebra Ward identity; the Virasoro Ward identity canbe treated in a similar fashion.By denoting with δ ǫ A ¯ x = D a ¯ x ǫ a the gauge variation of the source A ¯ x and with δ ǫ J I thevariation of the sources of vertices (1), we can write the Ward identity for the free energy W as G ǫ W [ A ] = 0 , G ǫ = Z d xδ ǫ A ( x, ¯ x ) δδA ( x, ¯ x ) + δ ǫ J I ( x, ¯ x ) δδJ I ( x, ¯ x ) . (24)Actually, this equation is wrong because the spacetime current algebra contains a central term, Eq. (23) was derived in [3] to lowest order in the g S expansion, so to compare our results with knownformulas one must truncate them and keep the O ( g S ) term only. h ...K a ( x ) K b ( y ) ... i = h ... x − y ) k G I + 1 x − y f abc K c ( y ) ... i . (25)It generates an anomaly in the conservation law of the current sourced by A ¯ x [10]. So, thedefinition of G ǫ must be modified as follows: we make λ ( x ), the source of the “identity” vertex I , change under gauge transformations as δ ǫ λ ( x, ¯ x ) = − πk G ǫ a ( x ) ∂ x A a ¯ x ( x ) . (26)Thus, the Ward identity generator G ǫ changes into G ǫ → G ′ ǫ = G ǫ + Z d xδ ǫ λ ( x, ¯ x ) δδλ ( x, ¯ x ) . (27)Because of its transformation law (26), λ is a Green-Schwarz [11] field, which cancels theanomaly; therefore, the Ward identity is G ′ ǫ W = 0. In fact, an anomalous term in the Wardidentity would be G ′ ǫ W = ∆( ǫ ), with ∆( ǫ ) a local functional of the background gauge field A ¯ x only, which obeys the standard Wess-Zumino consistency conditions [12]. Such term is canceledby adding to W a term linear in λ .The free energy W obeys another identity: thanks to eqs. (16,17), we have δWδλ ( x, ¯ x ) = − k g S ∂W∂g S + X h,I ( h − k Z d yJ ( y, ¯ y, h, I ) δWδJ ( y, ¯ y, h, I ) . (28)The solution to this linear equation is W [ λ, g S , A ( x ) , J ( x, ¯ x, h, I )] = W (cid:2) , e − λ / k g S , A ( x ) , e ( h − λ /k J ( x, ¯ x, h, I ) (cid:3) . (29)If I were a central term, the generating functional would obey G ǫ W = Kδλ , with K the(numerical) coefficient of the gauge anomaly. Instead we have G ǫ W = − Z d xδ ǫ λ ( x, ¯ x ) δWδλ ( x, ¯ x ) = − Z d xδ ǫ λ ( x, ¯ x ) dWdλ . (30)The observables we are interested in are the correlators of the vertices (1); the source λ isjust a convenient trick to write a simple Ward identity. In fact, an object at least as natural as W ( λ, J ) is a functional that depends on the VEV of I instead of λ : the Legendre transform of7 , that we call the “effective action”Γ[ h I i , J ] = W [ λ , J ] − λ h I i , computed at dWdλ = h I i . (31)Now the Ward identity on Γ has the correct form G ǫ Γ = − Z d xδλ ( x ) h I i . (32)The VEV h I i is essentially the total number of fundamental strings creating the AdS back-ground. At tree level each additional long string state adds +1 to the VEV while a short stringstate adds a ”fraction of a long string” equal to ( h − /k [8]. Legendre transforming in λ corresponds to defining Γ in a microcanonical ensemble where the string number is fixed. Thefree energy W [ λ ] is instead defined in a canonical ensemble where such number can fluctuatewhile the “chemical potential” λ is held fixed. Clearly, we can expect a standard CFT onlywhen the central charge (proportional to the number of fundamental strings) is fixed, not whenit fluctuates. Related issues were discussed in the context of precision counting of black holemicrostates in [13]. Given the similarity between the operator I in the Wakimoto representa-tion and the area operator of Liouville (see e.g. eq. (3.1) in ref. [8]), our definition is analogousto defining Liouville theory at fixed area. Besides the anomaly equation, the connected correlators of vertices (1) also change, becausethey are now defined by varying Γ with respect to the sources J at fixed h I i . Using the definitionof the Legendre transform (31), the same shorthand notation as before and the fact that thespacetime CFT has vanishing one-point functions, we can expand W − λ h I i around J = 0, λ = 0 asΓ = W [0 , J ] + X ij W λij [0 , J i J j λ + 12 W λλ [0 , λ + O ( J λ , J λ ) , computed at dWdλ = h I i . (33)This formula shows that the two and three-point correlators of vertices (1) are unchanged. Thefour-point function changes asΓ ( x , x , x , x ) = W ( x , x , x , x ) − W λ ( x , x ) 1 W λλ W λ ( x , x ) − ↔ − ↔ . (34)Comparing with eq. (23) we see that the non-clustering term cancels out. This cancelationholds in general. In fact, by construction W generates tree level connected correlators of I , We thank J. Maldacena for this remark and for bringing to our attention reference [13]. This analogy was pointed out to us by D. Kutasov.
8o that its Legendre transform Γ generates 1PI irreducible correlators, containing no internallines for the field I . This is an important check of our proposal: it not only solves the “identityproblem” but also takes care of the breakdown of cluster property in the spacetime CFT. Ofcourse these problems are related, they both originate from the fact that I has non-vanishingconnected correlators with physical vertices.So, finally we can write a partition function that obeys all standard properties of a spacetimeCFT living on the boundary of AdS as Z = C exp (Γ[ h I i , J ]) . (35)We conclude with two comments on this formula.The first one is that one must compute the effective action Γ[Φ , J ] at Φ = h I i . ComputingΓ[Φ , J ] at Φ = h I i results in an unphysical theory, with the wrong value for the anomaly andwithout cluster decomposition property. In fact the cancelation between dangerous terms ineqs. (23,34) holds only for Φ = h I i .The second is that our prescription is valid for k >
1. For k <
1, the identity is nota physical operator so it does not appear in eq. (23); therefore, the right definition for theboundary spacetime CFT may not involve a Legendre transform after all. This fact may playa role in explaining some of the unusual properties of strings on
AdS at k < Acknowledgements
We thank A. Giveon, D. Kutasov,J. Maldacena, N. Seiberg and E. Witten for useful commentsand suggestions. J-H.K. is supported by an NYU JAGA fellowship. M.P. is supported in partby NSF grant PHY-1316452.
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