Shear Viscosity and Chern-Simons Diffusion Rate from Hyperbolic Horizons
George Koutsoumbas, Eleftherios Papantonopoulos, George Siopsis
aa r X i v : . [ h e p - t h ] M a y Shear Viscosity and Chern-Simons Diffusion Rate from Hyperbolic Horizons
George Koutsoumbas , Eleftherios Papantonopoulos , George Siopsis Department of Physics, National Technical University of Athens, GR 157 73 Athens, Greece and Department of Physics and Astronomy, The University of Tennessee, Knoxville, TN 37996 - 1200, USA (Dated: November 2, 2018)We calculate the shear viscosity and anomalous baryon number violation rate in quantum fieldtheories at finite temperature having gravity duals with hyperbolic horizons. We find the explicitdependence of these quantities on the temperature. We show that the ratio of shear viscosity toentropy density is below 1 / (4 π ) at all temperatures and can be made arbitrarily small in the lowtemperature limit for hyperbolic surfaces of sufficiently high genus so that the hydrodynamic limitremains valid. In certain finite-temperature quantum field theorieshaving gravity duals with black brane solutions in higherspacetime dimensions, the hydrodynamic behaviour ofthe thermal field theory is identified with the hydrody-namic behaviour of the dual gravity theory [1]. It wasshown [2] that for these field theories, the ratio of theshear viscosity to the volume density of entropy has auniversal value η/s = 1 / (4 π ) and it was further conjec-tured that this is the lowest bound on the ratio η/s for alarge class of thermal quantum field theories.This conjecture was tested against a wide range of ther-mal field theories having gravity duals: in gauge theorieswith chemical potentials studying their R-charged blackhole duals [3], in field theories with stringy corrections [4]and also in field theories with gravity duals of Einstein-Born-Infeld gravity [5]. In all these theories it was foundthat the lower bound is satisfied. However, in confor-mal field theories dual to Einstein gravity with curvaturesquare corrections it was found that the bound is vio-lated [6] but the physical implication of the violation ofthe bound is still not clear.In the gravity sector of this gravity/gauge duality,maximally symmetric spaces naturally arise as the near-horizon region of black brane geometries [7]. Sphericallysymmetric spaces have been extensively investigated.Also hyperbolic geometries involving n -dimensional hy-perboloids H n or H n / Γ cosets, where Γ is a discrete sub-group of the isometry group of H n , arise naturally insupergravity as a result of string compactifications [8].However, the presence of the discrete group Γ introducesanother scale which breaks all supersymmetries. N = 0conformal field theories can be constructed having grav-ity duals with constant negative curvature [8, 9].The hydrodynamic properties of the boundary con-formal field theory can be inferred from the lowest fre-quency quasinormal modes of the gravity sector [10].The lowest-lying gravitational quasinormal modes for aSchwarzschild-AdS solution were numerically calculatedin four and five dimensions and were shown to be in agree-ment with hydrodynamic perturbations of the gauge the-ory plasma on the AdS boundary [11]. For AdS this wasunderstood as a finite “conformal soliton flow” after thespherical AdS boundary obtained in global coordinateswas conformally mapped to the physically relevant flat Minkowski spacetime. This study was extended to blackholes with a hyperbolic horizon. It was shown in [12]that the quasinormal modes obtained agreed with thefrequencies resulting from considering perturbations ofthe gauge theory fluid on the boundary.Recently, interesting features have shown up in thestudy of topological black holes (TBH). The spectrumof the quasinormal modes of TBH [13] has been studiedextensively [14]. For large black holes this spectrum issimilar to the Schwarzschild-AdS spectrum. For smallblack holes however the quasinormal modes spectrum isquite different. It was found [15] that there is a criticaltemperature, below which there is a phase transition ofthe TBH to AdS space. This has been attributed entirelyto the properties of the hyperbolic geometry.In this work we will show that the hyperbolic geom-etry allows us to calculate hydrodynamic transport co-efficients like shear viscocity and the Chern-Simons dif-fusion rate of the boundary thermal field theory at anytemperature under certain conditions. This should beconstrasted with the case of a spherical black hole wherelow temperature is invariably associated with small hori-zon area and therefore the hydrodynamic approximationbreaks down. In the hyperbolic case, the area of the hori-zon can be large even at low temperatures provided thehyperbolic surface is of high genus.Topological black holes are solutions of the Einsteinequations for vacuum AdS space. Consider the action I = 116 πG Z d d x √− g (cid:20) R + ( d − d − l (cid:21) , (1)where G is the Newton’s constant, R is the Ricci scalarand l is the AdS radius. The presence of a negative cos-mological constant (Λ = − ( d − d − l ) allows the exis-tence of black holes with topology R × Σ, where Σ is a( d − ds = − f ( r ) dt + 1 f ( r ) dr + r dσ f ( r ) = r − − Gµ/r , (2)where we have set the AdS radius l = 1, µ is a constantwhich is proportional to the mass and dσ is the line ele-ment of the two-dimensional manifold Σ, which is locallyisomorphic to the hyperbolic manifold H and of the formΣ = H / Γ , Γ ⊂ O (2 , , (3)where Γ is a freely acting discrete subgroup (i.e. withoutfixed points) of isometries. The line element dσ of Σ is dσ = dθ + sinh θdϕ , (4)with θ ≥ ≤ φ < π being the coordinates of thehyperbolic space H or pseudosphere, which is a non-compact two-dimensional space of constant negative cur-vature. This space becomes a compact space of constantnegative curvature with genus g ≥ θ = ϕ = 0of the pseudosphere [13, 17]. An octagon is the simplestsuch polygon, yielding a compact surface of genus g = 2under these identifications. Thus, the two-dimensionalmanifold Σ is a compact Riemann 2-surface of genusg ≥
2. The configuration (2) is an asymptotically locallyAdS spacetime.This construction can be generalized to higher dimen-sions and our aim in this work is to elucidate the effectof hyperbolic horizons on the gauge theory on the AdSboundary. In five spacetime dimensions the metric takesthe form ds = − f ( r ) dt + dr f ( r ) + r d Σ , f ( r ) = r − − µr , (5)where Σ = H / Γ. The horizon radius r + is found from2 µ = r (cid:18) − r (cid:19) . (6)The Hawking temperature is T = 2 r − πr + , (7)while the mass and entropy of the black hole are givenrespectively by M = 3 V πG r ( r − , S = V G r (8)where V is the volume of the hyperbolic space Σ . Notethat in the horizon radius range 1 / ≤ r ≤ E CF T = 3 V πG (cid:18) r − (cid:19) (9)which is shifted with respect to the black hole energy bya positive amount (Casimir energy due to countertermsone needs to add to the action to cancel infinities). No-tice that the minimum energy ( E CF T = 0) is at T = 0,therefore the energy of the CFT is never negative, unlikeits dual black hole.For the study of perturbations, we need the behaviourof harmonic functions on Σ . In general, they obey (cid:0) ∇ + k (cid:1) T = 0 . (10)Without identifications (i.e., in H ), the spectrum is con-tinuous. We obtain [14] k = ξ + 1 + δ (11)where ξ is arbitrary and δ = 0 , , ξ may be made as small as desired, i.e., zero is anaccumulation point of the spectrum of ξ [17]. We alsoobtain negative values of ξ . As ξ approaches its min-imum value, the complexity of the set of isometries Γincreases and the volume V of the hyperbolic space Σ can be made arbitrarily large (hence also the mass andentropy of the black hole).Using the harmonics on Σ , we may write the waveequation for gravitational perturbations in the generalSchr¨odinger-like form [18] − d Φ dr ∗ + V [ r ( r ∗ )]Φ = ω Φ , (12)in terms of the tortoise coordinate r ∗ defined by dr ∗ dr = f ( r ) where f ( r ) is defined in (5). The potential takesdifferent forms for different types of perturbation.To calculate the Chern-Simons diffusion rate one needsto solve the wave equation for a massless scalar field. Theradial wave equation is1 r ( r f ( r )Φ ′ ) ′ + ω f ( r ) Φ − k S r Φ = 0 . (13)By defining Φ = r − Ψ it can be cast into theSchr¨odinger-like form (12) with the potential given by V S ( r ) = f ( r ) (cid:26)
154 + k S − r + 9 µ r (cid:27) . (14)We may solve the wave equation in terms of a Heun func-tion and use the latter to determine the spectrum exactlyalbeit numerically [15]. However, such explicit expres-sions will not be needed for our purposes.If the hyperbolic space Σ is infinite, then k S ≥ is finite, then it is easy tosee that the minimum eigenvalue is k S = 0 . The corre-sponding hyperspherical harmonic is a constant. Above k S = 0, the spectrum is discrete.For the AdS/CFT correspondence, we need the flux F = N π √− gg rr Φ ∗ ∂ r Φ | Φ | (cid:12)(cid:12)(cid:12) r →∞ . (15)The imaginary part is independent of r (conserved flux).It is convenient to evaluate it at the horizon where thewavefunction behaves asΦ( r ) ≈ (cid:16) − r + r (cid:17) − iω πT . (16)We obtain √− gg rr ℑ (Φ ∗ ∂ r Φ) = − ωr (17)therefore ℑF = − N r ω π | Φ( ∞ ) | . (18)It is related to the imaginary part of the retarded Greenfunction, ℑ ˜ G R ( ω, k S ) = − ℑF (19)of some scalar operator O ( G ( x ) = hO ( x ) O (0) i , where x ∈ R × Σ ). We readily obtain ℑ ˜ G R ( ω, k S ) = N r ˆ ω π | Φ( ∞ ) | . (20)For O = F aµν ˜ F aµν , we may define the Chern-Simonsdiffusion rateΓ = (cid:18) g Y M π (cid:19) Z dt Z Σ d σ hO ( x ) O (0) i (21)Γ determines the rate of anomalous baryon number vio-lation at high temperatures in the Standard Model. Ex-panding G ( x ) in hyperspherical harmonics, the integralover Σ projects onto the lowest harmonic ( k S = 0). Theintegral over time then yields the Fourier transform at ω = 0. Using˜ G (0 , k S ) = − lim ω → Tω ℑ ˜ G R ( ω, k S ) (22)we deduceΓ = (cid:18) g Y M π (cid:19) ˜ G (0 ,
0) = ( g Y M N ) π T r | Φ( ∞ ) | (cid:12)(cid:12)(cid:12) ω =0 ,k S =0 (23)Evidently, Φ( r ) (cid:12)(cid:12)(cid:12) ω =0 ,k S =0 = 1 (24) at any temperature, thereforeΓ = ( g Y M N ) π r (cid:18) − r (cid:19) . (25)At high temperatures, Γ ∼ T whereas as T → ∼ T →
0, i.e., anomalous baryon number violationis suppressed at low temperatures.To calculate the shear viscosity, we need to discussvector gravitational perturbations. The lowest eigenvalueof the angular equation (10) for a vector harmonic V i ( ∇ i V i = 0) on a finite hyperbolic space Σ is found byobserving that ∇ j ( ∂ i V j − ∂ j V i ) = ( k V + 2) V i where we used R ij = − γ ij ( γ ij being the metric onΣ ). Therefore, we have a constant vector harmonic if k V + 2 = 0. The minimum eigenvalue is k V = −
2. Aboveit, we have a discrete spectrum of eigenvalues k V = − , ∆ ≥ . (26)The radial wave equation is of the form (12) with poten-tial V V ( r ) = f ( r ) (cid:26)
34 + k V − r − µ r (cid:27) . (27)We may solve the radial equation and obtain a solutionin terms of a Heun function. Since we are interested inthe hydrodynamic behaviour, we shall solve the radialequation only for small ω and ∆ using perturbation the-ory.More precisely, the hydrodynamic approximation isvalid provided ω , √ ∆ ≪ r + (28)(recall that we are working in units in which the AdSradius l = 1). At high temperatures, this constraint isequivalent to ω , √ ∆ ≪ T . Also, the area of the horizon( A + ∼ r ) is large and the constraint (28) is satisfiedfor eigenvalues ∆ ∼ O (1) because then ∆ ≪ A + . Thisis similar to the case of a sphere. In both cases, thehydrodynamic limit is valid at high temperature (largeblack hole) [11, 12].At low temperatures, in the case of a spherical horizon,its area becomes small. Even with A + ∼ O (1), it is nolonger possible to satisfy the constraint (28) because thelow-lying eigenvalues ∆ ∼ O (1) regardless of the size ofthe horizon. Thus, for a small spherical black hole thehydrodynamic approximation is invalid.For a hyperbolic horizon at low temperature, we have r + ∼ O (1) ( r + ≥ / √ of high genus canhave a large volume V ≫
1. The low lying eigenvaluesare √ ∆ ∼ V / (29)and therefore can be small ( √ ∆ < ∼ O (1)) if V is large.Thus, for topological black holes of high genus hyperbolichorizons the hydrodynamic approximation is valid evenin the low temperature (small horizon radius) limit owingto the complexity of the horizon surface.To solve the radial wave equation, it is convenient tointroduce the coordinate u = (cid:16) r + r (cid:17) . (30)In terms of the wavefunction F ( u ) defined byΨ( u ) = (1 − u ) − iω πT F ( u ) (31)we have A F ′′ + B F ′ + C F = 0 , (32)where A = u ˆ f , B = u ˆ f ′ + 32 ˆ f + iω πT u ˆ f − u , C = − ˆ V u ˆ f + iω πT u ˆ f ′ + ˆ f − u + iω πT u ˆ f (1 − u ) + O ( ω /T ) . (33)where prime denotes differentiation with respect to u andwe have definedˆ f ( u ) ≡ f ( r ) r = 1 u − r − µr u , ˆ V V ( u ) ≡ V V ( r ) r = ˆ f ( u ) (cid:26)
34 + ∆ − r u − µ r u (cid:27) . (34)We obtain the zeroth order equation by setting ω =0 , ∆ = 0. The acceptable solution is F = u / (35)independent of the temperature.Expanding the wavefunction, F = F + F + . . . , (36)at first order the wave equation reads H F = −H F , (37)where H F = iω πT (cid:26) u + 3 (cid:18) − r (cid:19)(cid:27) F − ∆4 r u F . (38)The solution may be written as F = F Z W F Z F H F AW , (39) where W = 1 / ( u / ˆ f ) is the Wronskian. The limits ofthe inner integral may be adjusted at will because thisamounts to adding an arbitrary amount of the unaccept-able zeroth-order wavefunction. To ensure regularity atthe horizon, we should choose one of the limits of integra-tion at u = 1. Then by demanding that the singularityvanish at the boundary ( u = 0), we arrive at the first-order constraint Z du F H F AW = 0 (40)After some straightforward algebra, this leads to the dis-persion relation ω = − i ∆4 r + (41)in agreement with ref. [12] at high temperatures andmatching numerical results at all temperatures (Fig. 1).From (41) we read off the diffusion coefficient D = 14 r + (42)which is related to the viscosity coefficient via D = ηǫ + p . (43)This is known to be valid in flat space. It is also validin our case, as can be seen by writing the hydrodynamicequations ∇ µ T µν = 0 for a static fluid of constant pres-sure perturbed by a small velocity field u i = e − iωt V i . The conservation law of the hydrodynamic equationsyields [12] − iωp + η ( k V + 2) = 0 . (44)Eq. (43) then follows if we use (41), (42) together with ǫ = 3 p which is valid for a confomal fluid. -0.15-0.14-0.13-0.12-0.11-0.1-0.09-0.08 0.9 1 1.1 1.2 1.3 1.4 1.5 I m ( o m ega ) rp FIG. 1: The imaginary part of the lowest, purely dissipative,mode versus r + . The continuous line and the points representthe perturbative and numerical results, respectively. From the expression for the energy (9), we obtain theenergy density ǫ = E CF T /V and the shear viscosity co-efficient η = 43 ǫD = 116 πGr + (cid:18) r − (cid:19) . (45)Dividing by the entropy density ( s = S/V , where theentropy is given by (8)), we obtain ηs = 14 π (cid:18) − r (cid:19) . (46)At high temperature (large r + ), ηs ≈ π . As T → ηs ∼ T →
0. At all temperatures, the ratio is below π .In flat spacetime, one also obtains the viscosity coeffi-cient from the Kubo formula [1], which agrees with theresult obtained via the diffusion coefficient. The formeris derived by considering tensor perturbations. This is not possible in our case, because tensor perturbations ofthe static fluid in hyperbolic space do not exist due totheir being traceless and divergenceless [12]. In conclusion
Using the AdS/CFT correspondence wehave calculated the anomalous baryon number violationrate and the ratio of shear viscosity to entropy density inthermal field theories having gravity duals with hyper-bolic horizons. We found the explicit temperature de-pendence of the anomalous baryon number violation andwe showed that it is suppressed at low temperatures. Forhigh genus hyperbolic spaces the hydrodynamic approx-imation is valid at low temperatures, and the ratio ofshear viscosity to entropy density is found to be below1 / (4 π ) at all temperatures. It can be made arbitrarilysmall in the low temperature limit. Acknowledgments
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