aa r X i v : . [ h e p - t h ] M a y UWO-TH-08/7
Shear viscosity of CFT plasma at finite coupling
Alex Buchel
Department of Applied MathematicsUniversity of Western OntarioLondon, Ontario N6A 5B7, CanadaPerimeter Institute for Theoretical PhysicsWaterloo, Ontario N2J 2W9, Canada
Abstract
We present evidence for the universality of the shear viscosity of conformal gaugetheory plasmas beyond infinite coupling. We comment of subtleties of computing theshear viscosity in effective models of gauge/gravity correspondence rather than in stringtheory.April 2008
Introduction
Various universal features of transport properties of strong coupled gauge theory plas-mas where discovered within the framework of gauge theory/string theory correspon-dence of Maldacena [1,2]. For example, all gauge theory plasmas (various gauge groups,matter content, with or without chemical potentials for conserved U (1) charges, withnon-commutative spatial directions) that allow for a dual supergravity description havea universal value of the shear viscosity at infinite ’t Hooft coupling [3–8] . Similarly,while not universal, the bulk viscosity of non-conformal gauge theory plasmas (againat infinitely strong coupling and provided the dual holographic description is available)appear to satisfy a universal bound [9].Much less is known about viscosity of gauge theory plasma at finite coupling: theshear viscosity of N = 4 supersymmetric Yang-Mills plasma at finite coupling wascomputed in [10–13] . The difficulties of extending the analysis of [10–13] to otherexamples of gauge/string correspondence are both technical and conceptual. On atechnical side, one deals with a daunting task of studying quasinormal modes of near-extremal black branes in full ten dimensional type IIB supergravity, including (atleast to leading order) higher derivative O ( α ′ )-corrections coming from integratingout massive modes of type IIB string. Conceptually, the problem is that the full setof such O ( α ′ )-corrections is not known presently. Typically, in studies of black branethermodynamics [16–18] and hydrodynamics [10–13] at O ( α ′ ) one takes into accountonly curvature corrections to type IIB supergravity [19–22]. What is known is thatadditional terms must be present [23–25] and that they might affect the hydrodynamicsof dual gauge theory plasma. Thus, the available results [10, 12] for the shear viscosityof N = 4 SYM plasma can receive further corrections from Ramond-Ramond fluxesin the dual string theory description. We emphasize though that such corrections willbe additive (to leading order in α ′ ), and contribution of O ( α ′ ) terms due to fluxescan be computed on top of type IIB supergravity, excluding purely higher-derivativecurvature corrections. Here, we consider only higher-derivative curvature correctionsto type IIB supergravity — as a result our conclusions are subject to caveat alluded toabove.In this letter we present evidence for the universal features of conformal gaugetheory plasmas at finite ’t Hooft coupling. A claim of this type can make sense only It was also computed in some phenomenological models of gauge/gravity correspondence in [14,15].
2f a prescription is given as to how to compare different gauge theories . For example,for SU ( N c ) N = 4 SYM plasma at ’t Hooft coupling λ N =4 α ′ L = 1 √ λ N =4 , (1.1)where L is the radius of AdS . Now, for a general CFT (with a dual string theorydescription) we have [26] α ′ L = 1 √ λ CF T r a N =4 a CF T , (1.2)where a N =4 and a CF T are the central charges of the N = 4 SYM and a given CFTcorrespondingly. We identify two CFT gauge theory plasmas if according to (1.2) theyhave the same holographic string tension, i.e. , λ CF T a CF T = λ CF T a CF T . (1.3)Our evidence for the universality of the shear viscosity of CFT plasma at finite cou-pling comes from analysis of a shear viscosity of N = 1 superconformal gauge theoryplasma of Klebanov-Witten (KW) [27]. Specifically, we show that the full spectrumof shear quasinormal modes of KW plasma at order O ( α ′ ) ∼ O (cid:16) λ − / KW (cid:17) is identicalto that of the N = 4 plasma at order O (cid:16) λ − / N =4 (cid:17) . For a specific case of the lowestquasinormal modes, this immediately implies that the ratio of the shear viscosity tothe entropy density is the same in KW and N = 4 SYM plasmas. Note that followingour convention (1.3), we set λ KW λ N =4 = 1627 . (1.4)Gauge theory plasma transport analysis in [10–12] and in this letter are done in thethe full ten dimensional type IIB supergravity including higher-derivative curvaturecorrections [19–22]. As a separate computation, we show that similar analysis done inlower dimensional higher-derivative effective action (obtained without proper Kaluza-Klein reduction) could be misleading. While a specific example we present gives thesame value of the shear viscosity as in its higher dimensional counterpart, the agreementis accidental and the two models have different spectra of quasinormal modes as wellas different higher order hydrodynamic corrections for the dispersion relation of thelowest quasinormal mode. In view of this, it would be interesting to re-examine theclaims in [14, 15]. This was not the issue in either shear viscosity universality or bulk viscosity bound at infinitecoupling. KW plasma at finite temperature and ’t Hooft coupling
A string theory dual to KW gauge theory is a near-horizon limit of D3 brane on theconifold [27]. We begin with computing leading order α ′ corrections to near-extremalD3 branes at the conifold, dual to finite temperature KW plasma at order O (cid:0) λ − / (cid:1) in the ’t Hooft coupling.Type IIB supergravity effective action with only higher-derivative curvature correc-tions takes form I = 116 πG Z d x √− g (cid:20) R −
12 ( ∂φ ) − ·
5! ( F ) + ... + γ e − φ W + ... (cid:21) , (2.1) γ = 18 ζ (3)( α ′ ) , where W = C hmnk C pmnq C rsph C qrsk + 12 C hkmn C pqmn C rsph C qrsk . (2.2)In (2.1) ellipses stand for O ( α ′ ) contribution of Ramond-Ramond fluxes that we ne-glect here.We represent ten dimensional background geometry describing γ -corrected black3-branes on the conifold by the following ansatz ds = g (0) µν dx µ dx ν + c e ψ + c
25 2 X a =1 (cid:0) e θ a + e φ a (cid:1) ≡ − c dt + c (cid:0) dx + dy + dz (cid:1) + c dr + c e ψ + c
25 2 X a =1 (cid:0) e θ a + e φ a (cid:1) , (2.3)where c i = c i ( r ). The metric ansatz (2.3) is the most general one, given the U (1) × SU (2) × SU (2) symmetries of the KW plasma [27]. The frames { e θ a , e φ a } are definedso that the metric on a unit T , (the angular part of the conifold) is (cid:0) dT , (cid:1) = e ψ + X a =1 (cid:0) e θ a + e φ a (cid:1) . (2.4)For the dilaton we assume φ = φ ( r ) and for the five-form F = F + ⋆ F , F = − dvol T , . (2.5) See [28] for explicit expressions.
4n (2.5) the 5-form flux is chosen in such a way that γ = 0 solution corresponds to c = c = 1. To leading order in γ we further parameterize c = r (cid:18) − r r (cid:19) / e − ν (1 + a + 4 b ) ,c = re − ν ,c = 1 r (cid:16) − r r (cid:17) / e − ν (1 + b ) ,c = e ν − ν ,c = e ν + ν , (2.6)where the asymptotic normalizable components of gravitational modes ν and ν aredual to KW plasma operators of dimension-eight and dimension-six correspondingly[28].Equation of motion for ν determined to order O ( γ ) from (2.1) takes form0 = ν ′′ + 5 r − r r ( r − r ) ν ′ − r r − r ν . (2.7)The absence of non-normalizable mode asymptotically as r → ∞ and the regularity atthe horizon r → r determines unique solution to (2.7): ν ( r ) = 0 . (2.8)Vanishing of ν implies that the remaining modes { a, b, ν } are exactly the same asfor the near-extremal D3 branes [16] a = − γ r r (cid:18) r r − r r + 25 (cid:19) ,b = γ r r (cid:18) r r − r r + 5 (cid:19) ,ν = γ r r (cid:18) r r (cid:19) . (2.9)The dilaton φ also receives γ corrections, φ ∝ γ [16]. It is easy to see that to order O ( γ ) gravitational perturbations do not mix with the dilaton perturbation; moreoverto study gravitational perturbations we can consistency set φ = 0. The Hawkingtemperature corresponding to the metric (2.3) is [16] T = T (1 + 15 γ ) ≡ r π (1 + 15 γ ) . (2.10) We explicitly verified this. Shear viscosity of KW plasma at finite ’t Hooft coupling
A relation between hydrodynamics of gauge theory plasmas and the spectrum of quasi-normal modes in dual near-extremal string theory backgrounds is explained in [29].Computation we need to perform here literally repeats the analysis done in [11] —the only difference being that here we have a slightly more complicated backgroundgeometry (2.3). Thus, we move directly to the results of the analysis.A shear quasinormal mode Z shear = Z shear, + γ Z shear, + O ( γ ) (3.1)of frequency and momentum ( ω, q = | ~q | ) in the background geometry (2.3) satisfiesthe following equations0 = Z ′′ shear, + x q + w x ( w − x q ) Z ′ shear, + w − x q x (1 − x ) / Z shear, , Z ′′ shear, + x q + w x ( w − x q ) Z ′ shear, + w − x q x (1 − x ) / Z shear, + J KWshear, , (3.2)where the source J KWshear, is a functional of the zero’s order shear mode Z shear, J KWshear, = C (4) shear,KW d Z shear, dx + C (3) shear,KW d Z shear, dx + C (2) shear,KW d Z shear, dx + C (1) shear,KW dZ shear, dx + C (0) shear,KW Z shear, ≡ ˆ C (1) shear,KW dZ shear, dx + ˆ C (0) shear,KW Z shear, , (3.3)where in the second equality we used the first equation in (3.2). As in [11], w = ω πT , q = q πT , (3.4)and instead of the radial coordinate r we used x ≡ (cid:18) − r r (cid:19) / . (3.5)The connection coefficients C ( i ) shear,KW are given explicitly in Appendix A.Introducing Z shear ( x ) = x − i w (1 − γ ) z shear ( x ) , (3.6) The incoming wave boundary conditions must be imposed as explained in [13]. z shear (cid:12)(cid:12)(cid:12)(cid:12) x → + = 1 , z shear (cid:12)(cid:12)(cid:12)(cid:12) x → − = 0 . (3.7)Superficially, the quasinormal modes of KW plasma are different from those of the N = 4 SYM plasma. Indeed, given that [11] C (4) shear, N =4 =45(1 − x ) (3.8)is different from C (4) shear,KW (see (A.1)), as well as all the other coefficients, one is temptedto conclude that J KWshear, = J N =4 shear, . This is incorrect however. The source term in (3.2) must be evaluated on the zero’sorder shear mode Z shear, . Using the first equation in (3.2) (which is a supergravityapproximation and thus is the same for both KW and N = 4 SYM plasmas) it isstraightforward to show that even though C ( i ) shear,KW = C ( i ) shear, N =4 , J KWshear, ≡ J N =4 shear, (3.9)for all ( w , q ).Eq.(3.9) is our main result. It establishes that the full spectrum of shear quasi-normal modes is the same in KW and N = 4 SYM plasmas . To some extend theequivalence of the source terms for the KW and the N = 4 CFTs could have beenexpected given that the scalar mode ν deforming the U (1) fiber inside the T , of thedual KW geometry is not excited at O ( γ ) order, see (2.8). Thus, it is reasonable toexpect that the Kaluza-Klein reduction of type IIB string theory on S would be thesame as for the rigid T , (apart from the obvious rescaling of the five-dimensionalgravitational coupling) .In order to relate to the analysis of the next section we reproduce from [11, 13] thelowest quasinormal mode of (3.2) and its dispersion relation: z shear, = z (0) shear, + i q z (1) shear, + O ( q ) ,z shear, = z (0) shear, + i q z (1) shear, + O ( q ) , (3.10) We believe the same statement is correct for the spectrum of sound quasinormal modes as well. We expect a similar argument to apply to more general examples of
AdS × SE /CF T corre-spondence. (0) shear, =1 , z (1) shear, = 12 qw x ,z (0) shear, = 2516 x (cid:0) x − x + 5 (cid:1) ,z (1) shear, = − qw x (cid:18) q (cid:0) − − x − x + 695 x (cid:1) + 16 w (cid:0) − x + 43 x (cid:1)(cid:19) − q w x . (3.11)Imposing the Dirichlet condition on z shear, at the boundary determines the lowestshear quasinormal frequency w = − i Γ η q + O ( q ) , Γ η = 12 + 1052 γ + O ( γ ) . (3.12) Computation of the quasinormal modes in near extremal backgrounds of ten dimen-sional type IIB supergravity including O ( α ′ ) higher-derivative corrections is techni-cally rather involved. One difficulty is a high dimensionality of the background space-time. For the latter reason it is desirable to do the computations in lower dimensionaleffective description. Of course, had a relevant Kaluza-Klein (KK) reduction of thehigher-derivative type IIB supergravity been know, both ways to do the computa-tions are equivalent. In the absence of a consistent KK reduction one usually resortsto guesses as to what the lower dimensional effective action would look like. In thissection we present one example which illustrates pitfalls of such an approach.Suppose we would like to guess an effective action corresponding to (2.1) KK re-duced on S (or T , ). As a minimal requirement we would ask that the near extremalbackground of our effective action faithfully reproduces the thermodynamics of the fullten dimensional background. A natural guess (satisfying the minimal requirement)would then be I = 116 πG Z d x √− g (cid:20) R + 12 + γ W (cid:21) , (4.1)where W is given again by (2.2). Note that (4.1) is precisely the effective action usedin [16] to describe α ′ corrections of the near extremal D3 branes.Repeating the analysis of the shear quasinormal modes in this case we find the same8et of equations (3.2), albeit with the different source term: J effshear, = C (4) shear,eff d Z shear, dx + C (3) shear,eff d Z shear, dx + C (2) shear,eff d Z shear, dx + C (1) shear,eff dZ shear, dx + C (0) shear,eff Z shear, ≡ ˆ C (1) shear,eff dZ shear, dx + ˆ C (0) shear,eff Z shear, . (4.2)The connection coefficients C ( i ) shear,eff are given explicitly in Appendix B. Here, J effshear, is actually different from J KWshear, . Indeed, using the first equation in (3.2) we find0 = J KWshear, − J effshear, = − x (3 x − x + 5) dZ shear, dx − q x − w ) (cid:18) q x (3 x − − w (9 x − x + 5) (cid:19) Z shear, . (4.3)We can explicitly compute the lowest quasinormal mode: z (0) shear, =1 , z (1) shear, = 12 qw x ,z (0) shear, =0 ,z (1) shear, = − x qw (cid:18) q (45 x − x − x −
15) + w ( − x + 43 x + 594) (cid:19) − q w x . (4.4)While (4.4) is clearly different from (3.11), nonetheless, it has the same dispersionrelation (3.12).Given (4.3), the full shear quasinormal spectrum of N = 4 SYM plasma is differentfrom the one corresponding to the purported holographic dual to (4.1). The differ-ence exists even for the lowest quasinormal mode, although in higher orders in thehydrodynamic approximation. In this letter we conjectured universality of the shear viscosity to entropy ratio ofdifferent CFT plasmas at finite ’t Hooft coupling. While we presented only one examplefor a comparison with the N = 4 SYM plasma, namely that of Klebanov-Witten plasmaand only to leading order in the inverse ’t Hooft coupling, the fact that the full shearquasinormal spectra of both plasmas agree suggests that the agreement extends to9ther CFT plasmas as well, and probably also beyond leading inverse ’t Hooft couplingcorrection.Proving our conjecture to all orders in α ′ (or the inverse ’t Hooft coupling) is unlikelyto be possible. We believe that at least to order O (cid:0) λ − / (cid:1) the universality can weestablished by carefully analyzing Kaluza-Klein reduction of (2.1) on five dimensionalEinstein-Sasaki manifolds. A progress on this issue can also be made by studying CFTplasmas in a boost-invariant setting, as in [12]. We hope to report on this elsewhere.Within the supergravity approximation (equivalently at infinite ’t Hooft coupling)the universality of the shear viscosity to the entropy density ratio extends to gaugetheory plasmas in various dimensions. We conjectured here that the finite ’t Hooftcoupling corrections are universal for all four dimensional CFTs, which allow for adual string theory description. It would be interesting to explore such corrections forconformal gauge theory plasmas in spacetime dimensions other than four. Acknowledgments
I would like to thank the organizers of “String Theory - From Theory to Experiment”workshop at Hebrew University of Jerusalem for hospitality where this work started.I would also like to thank the Galileo Galilei Institute for Theoretical Physics for thehospitality and the INFN for partial support during the completion of this work. Myresearch at Perimeter Institute is supported in part by the Government of Canadathrough NSERC and by the Province of Ontario through MRI. I gratefully acknowl-edges further support by an NSERC Discovery grant and support through the EarlyResearcher Award program by the Province of Ontario.
AppendixA Coefficients of J KWshear, C (4) shear,KW = (1 − x ) q x − w (cid:18) q (45 x − x + 57 x ) − w (37 x − x + 49) (cid:19) (A.1)10 (3) shear,KW = 2( x − q x − w ) x (cid:18) q (315 x − x + 261 x + 57 x ) − x w q (85 x − x + 103 x −
3) + w (333 x − x + 467 x − (cid:19) (A.2) C (2) shear,KW = − x ( q x − w ) √ − x (cid:18) q (83 x − x + 237 x − x )+ 2 q w x ( x − x − x + 37) + w (1 − x )(37 x − x + 49) (cid:19) + 18( q x − w ) x (cid:18) q x (14015 x + 3122 x − x + 5203 x + 1368) − x w q (16711 x − x + 10243 x + 5594 x − x q w ( − x − x + 75993 x − x ) − w (22327 x − x + 38179 x − − x ) (cid:19) (A.3) C (1) shear,KW = − x ( q x − w ) √ − x (cid:18) q x (332 x − x + 166 x + 71) − x q w (188 x − x + 454 x − − x q w (100 x − x + 10 x + 57)+ w (148 x − x + 74 x + 49) (cid:19) − q x − w ) x (cid:18) q x (8333 x + 3122 x + 1368 − x + 5525 x ) − x q w (8582 x − − x + 25791 x + 9983 x ) + 4 x q w ( − − x + 32523 x − x + 4943 x ) + 2 x q w (816 + 17556 x − x + 11286 x + 2935 x ) − w ( − x + 2217 x + 234 x + 392 + 7105 x ) (cid:19) (A.4)11 (0) shear,KW = 18 x ( q x − w )(1 − x ) / (cid:18) q x (5621 x − − x + 3385 x ) − x q w ( − x + 3917 x + 1726 x − x ) + w ( − − x + 4709 x + 982 x + 2665 x ) (cid:19) + 14 x (1 − x )( q x − w ) (cid:18) − q x (19 x − x + 7) + 4 x q w ( x + 5)( x − x q (25 x ( x − x −
4) + 2 w (91 + 55 x − x )) − w (4 w (37 x − x + 49) + 25 x ( x − x − x + 5)) (cid:19) (A.5) B Coefficients of J ef fshear, C (4) shear,eff = 32(1 − x ) (5 q x − w )3( q x − w ) (B.1) C (3) shear,eff = 64( x − x ( q x − w ) (cid:18) q x (7 x + 1) − x q w (35 x −
3) + 3 w (9 x − (cid:19) (B.2) C (2) shear,eff = − x − q x − w ) x √ − x (cid:18) q x + 5 x q w − w (cid:19) + x − x ( q x − w ) (cid:18) q x ( x − x + 107 x + 24) − x q w (21653 x − x − x + 2080) + 2 x q w (12379 x − x + 2585 x + 304) − w (2399 x − x + 733 x + 32) (cid:19) (B.3)12 (1) shear,eff = − − x ) / q x − w ) x (cid:18) q x (4 x + 1) − x q w (23 x − − x q w (3 x + 1) + 3 w (4 x + 1) (cid:19) − x ( q x − w ) (cid:18) q x ( x − x − x − x − − x q w (7103 x − − x + 21582 x + 9781 x ) + x q w ( − − x + 56106 x − x + 7587 x ) + x q w (448 + 4172 x − x + 4875 x + 5385 x ) − w ( − x − x + 32 + 67 x + 665 x ) (cid:19) (B.4) C (0) shear,eff = − q x − w ) x √ − x (cid:18) q x (59 x + 33 − x ) − x q w (925 x + 384 − x − x ) + 3 w (101 x + 375 x + 128 − x ) (cid:19) + 32( x − q x − w )( q x + 3 w )3 x (B.5) References [1] J. M. Maldacena, Adv. Theor. Math. Phys. , 231 (1998) [Int. J. Theor. Phys. ,1113 (1999)] [arXiv:hep-th/9711200].[2] O. Aharony, S. S. Gubser, J. M. Maldacena, H. Ooguri and Y. Oz, Phys. Rept. , 183 (2000) [arXiv:hep-th/9905111].[3] A. Buchel and J. T. Liu, Phys. Rev. Lett. , 090602 (2004)[arXiv:hep-th/0311175].[4] P. Kovtun, D. T. Son and A. O. Starinets, Phys. Rev. Lett. , 111601 (2005)[arXiv:hep-th/0405231].[5] A. Buchel, Phys. Lett. B , 392 (2005) [arXiv:hep-th/0408095].[6] P. Benincasa, A. Buchel and R. Naryshkin, Phys. Lett. B , 309 (2007)[arXiv:hep-th/0610145]. 137] D. Mateos, R. C. Myers and R. M. Thomson, Phys. Rev. Lett. , 101601 (2007)[arXiv:hep-th/0610184].[8] K. Landsteiner and J. Mas, JHEP , 088 (2007) [arXiv:0706.0411 [hep-th]].[9] A. Buchel, “Bulk viscosity of gauge theory plasma at strong coupling,”arXiv:0708.3459 [hep-th].[10] A. Buchel, J. T. Liu and A. O. Starinets, Nucl. Phys. B , 56 (2005)[arXiv:hep-th/0406264].[11] P. Benincasa and A. Buchel, JHEP , 103 (2006) [arXiv:hep-th/0510041].[12] A. Buchel, “Shear viscosity of boost invariant plasma at finite coupling,”arXiv:0801.4421 [hep-th].[13] A. Buchel, “Resolving disagreement for eta/s in a CFT plasma at finite coupling,”arXiv:0805.2683 [hep-th].[14] Y. Kats and P. Petrov, “Effect of curvature squared corrections in AdS on theviscosity of the dual gauge theory,” arXiv:0712.0743 [hep-th].[15] M. Brigante, H. Liu, R. C. Myers, S. Shenker and S. Yaida, “Viscosity BoundViolation in Higher Derivative Gravity,” arXiv:0712.0805 [hep-th].[16] S. S. Gubser, I. R. Klebanov and A. A. Tseytlin, Nucl. Phys. B , 202 (1998)[arXiv:hep-th/9805156].[17] J. Pawelczyk and S. Theisen, JHEP , 010 (1998) [arXiv:hep-th/9808126].[18] A. Buchel, Nucl. Phys. B , 45 (2006) [arXiv:hep-th/0604167].[19] M. T. Grisaru and D. Zanon, Phys. Lett. B , 347 (1986).[20] M. D. Freeman, C. N. Pope, M. F. Sohnius and K. S. Stelle, Phys. Lett. B ,199 (1986).[21] Q. H. Park and D. Zanon, Phys. Rev. D , 4038 (1987).[22] D. J. Gross and E. Witten, Nucl. Phys. B , 1 (1986).1423] S. de Haro, A. Sinkovics and K. Skenderis, Phys. Rev. D , 066001 (2003)[arXiv:hep-th/0302136].[24] G. Policastro and D. Tsimpis, Class. Quant. Grav. , 4753 (2006)[arXiv:hep-th/0603165].[25] M. F. Paulos, “Higher derivative terms including the Ramond-Ramond five-form,”arXiv:0804.0763 [hep-th].[26] H. Liu, K. Rajagopal and U. A. Wiedemann, JHEP , 066 (2007)[arXiv:hep-ph/0612168].[27] I. R. Klebanov and E. Witten, Nucl. Phys. B , 199 (1998)[arXiv:hep-th/9807080].[28] O. Aharony, A. Buchel and A. Yarom, Phys. Rev. D , 066003 (2005)[arXiv:hep-th/0506002].[29] P. K. Kovtun and A. O. Starinets, Phys. Rev. D72