Slowly Varying Dilaton Cosmologies and their Field Theory Duals
Adel Awad, Sumit R. Das, Archisman Ghosh, Jae-Hyuk Oh, Sandip Trivedi
aa r X i v : . [ h e p - t h ] D ec UK/09-03, SITP/09-28, SLAC-PUB-13683,TIFR-TH/09-16
Slowly Varying Dilaton Cosmologiesand their Field Theory Duals
Adel Awad a,b
Sumit R. Das c , Archisman Ghosh c Jae-Hyuk Oh c , Sandip P. Trivedi d,e,f Center for Theoretical Physics, British University of EgyptSherouk City 11837, P.O. Box 43, EGYPT a Department of Physics, Faculty of Science,Ain Shams University, Cairo, 11566, EGYPT b Department of Physics and Astronomy,University of Kentucky, Lexington, KY 40506
USA c Tata Institute of Fundamental Research, Mumbai 400005,
INDIA d Stanford Institute of Theoretical Physics, Stanford CA 94305 USA e SLAC, Stanford University, Stanford, CA 94309 USA f [email protected], [email protected], [email protected]@uky.edu, [email protected] Abstract
We consider a deformation of the
AdS × S solution of IIB supergravity obtained by takingthe boundary value of the dilaton to be time dependent. The time dependence is taken to beslowly varying on the AdS scale thereby introducing a small parameter ǫ . The boundary dilatonhas a profile which asymptotes to a constant in the far past and future and attains a minimumvalue at intermediate times. We construct the sugra solution to first non-trivial order in ǫ , andfind that it is smooth, horizon free, and asymptotically AdS × S in the far future. Whenthe intermediate values of the dilaton becomes small enough the curvature becomes of orderthe string scale and the sugra approximation breaks down. The resulting dynamics is analysedin the dual SU ( N ) gauge theory on S with a time dependent coupling constant which variesslowly. When N ǫ ≪
1, we find that a quantum adiabatic approximation is applicable, and useit to argue that at late times the geometry becomes smooth
AdS × S again. When N ǫ ≫ N limit. For large values of the ’tHooft coupling this reproduces the supergravityresults. For small ’tHooft coupling the coherent state calculations become involved and wecannot reach a definite conclusion. We argue that the final state should have a dual descriptionwhich is mostly smooth AdS space with the possible presence of a small black hole. Introduction
The AdS/CFT correspondence [1, 2, 3] provides us with a non-perturbative formulation ofquantum gravity. One hopes that it will shed some light on the deep mysteries of quantumgravity, in particular on the question of singularity resolution.Motivated by this hope we consider a class of time dependent solutions in this paper whichcan be viewed as deformations of the
AdS × S background in IIB string theory. Thesesolutions are obtained by taking the boundary value of the dilaton in AdS space to becometime dependent . We are free to take the boundary value of the dilaton to be any timedependent function. To keep the solutions under analytical control though we take the rateof time variation of the dilaton to be small compared to the radius of AdS space, R AdS . Thisintroduces a small parameter ǫ and we construct the bulk solution in perturbation theory in ǫ .The resulting solutions are found to be well behaved. In particular one finds that no black holehorizon forms in the course of time evolution. The metric and dilaton respond on a time scaleof order R AdS which is nearly instantaneous compared to the much slower time scale at whichthe boundary value of the dilaton varies. For dilaton profiles which asymptote to a constantin the far future one finds that all the energy that is sent in comes back out and the geometrysettles down eventually to that of
AdS space. What makes these solutions non-trivial is thatby waiting for a long enough time, of order R AdS ǫ , a big change in the boundary dilaton canoccur. The solutions probe the response of the bulk to such big changes.Consider an example of this type where the boundary dilaton undergoes a big change makingthe ’tHooft coupling of order unity or smaller at intermediate times, λ ≡ g s N ≤ O (1) , (1)when t ≃
0, before becoming large again in the far future. As was mentioned above, the bulkresponds rapidly to the changing boundary conditions and within a time of order R AdS thedilaton everywhere in the bulk then becomes small and meets the condition, eq.(1). Now thesupergravity solution receives α ′ corrections in string theory, these are important when R AdS becomes of order the string scale. Using the well known relation, R AdS /l s ∼ ( g s N ) (2)we then find that once eq.(1) is met the curvature becomes of order the string scale everywherealong a space-like slice which intersects the boundary. As a result the supergravity approxima-tion breaks down along this slice and the higher derivative corrections becomes important for It is important in the subsequent discussion that we work in global
AdS with the boundary S × R . When we refer to the ’tHooft coupling we have the gauge theory in mind and accordingly by the dilaton inthis context we will always mean its boundary value. Here N is the number of units of flux in the bulk and the rank of the gauge group in the boundary theory. l P l (or the 5-dim Planck scale)remains small for all time. The radius R AdS in l P l units is given by, R AdS /l P l ∼ N (3)We keep N to be fixed and large throughout the evolution, this then keeps the curvature smallin Planck units . The solutions we consider can therefore be viewed in the following manner:the curvature in Planck units in these solutions stays small for all time, but for a dilaton profilewhich meets the condition eq.(1) the string scale in length grows and becomes of order thecurvature scale at intermediate times. At this stage the geometry gets highly curved on thestring scale. We are interested in whether a smooth spacetime geometry can emerge again inthe future in such situations.It is worth relating this difference in the behaviour of the curvature as measured in stringand Planck scales to another fact. We saw that when the curvature becomes of order the stringscale α ′ corrections become important. The second source of corrections to the supergravityapproximation are quantum loop corrections. Their importance is determined by the parameter1 /N . Since N is kept fixed and large these corrections are always small. From eq.(3) we seethat this ties into the fact that the AdS radius stays large in Planck units.To understand the evolution of the system once the curvature gets to be of order the stringscale we turn to the dual gauge theory. The gauge theory lives on an S of radius R and theslowly varying dilaton maps to a Yang-Mills coupling which varies slowly compared to R . Sincethese are the only two length scales in the system the slow time variation suggests that one canunderstand the resulting dynamics in terms of an adiabatic approximation.In fact we find it useful to consider two different adiabatic perturbation theories. Thefirst, which we call quantum adiabatic perturbation theory is a good approximation when theparameter ǫ satisfies the condition, N ǫ ≪ . (4)Once this condition is met the rate of change of the Hamiltonian is much smaller than theenergy gap between the ground state and the first excited state in the gauge theory. As a resultthe standard text book adiabatic approximation in quantum mechanics applies and the systemat any time is, to good approximation, in the ground state of the instantaneous Hamiltonian.In the far future, when the time dependence turns off, the state settles into the ground stateof the resulting N = 4 SYM theory, and admits a dual description as a smooth AdS space.Note that this argument holds even when the ’tHooft coupling at intermediate times becomesof order unity or smaller. The fact that the states of the time independent N = 4 SYM theory The backreaction corrects the curvature but these corrections are suppressed in ǫ . /R for all values of the Yang Mills coupling, [4], see also, [5], [6]. Thus as longas eq.(4) is met the conditions for this perturbation theory apply. As a result, we learn thatfor very slowly varying dilaton profiles which meet the condition, eq.(4), the geometry afterbecoming of order the string scale at intermediate times, again opens out into a smooth AdS space in the far future.The supergravity solutions we construct are controlled in the approximation, ǫ ≪ . (5)This is different, and much less restrictive, than the condition stated above in eq.(4) for thevalidity of the quantum adiabatic perturbation theory. In fact one finds that a different per-turbation theory can also be formulated in the gauge theory. This applies when the conditions, N ǫ ≫ , ǫ ≪ O and the modes which arise fromit. The time varying boundary dilaton results in a driving force for these oscillators. When N ǫ ≫
1, these oscillators are excited by the driving force into a coherent state with a largemean occupation number of quanta, of order
N ǫ , and therefore behave classically. This is areflection of the fact that at large N , the system behaves classically : coherent states of theseoscillators correspond to classical configurations (see e.g. Ref [7]).Usually a reformulation of the boundary theory in terms of such oscillators is not very useful,since these oscillators would have a nontrivial operator algebra which would signify that thebulk modes are interacting. Simplifications happen in low dimensional situations like MatrixQuantum Mechanics [8] where one is led to a collective field theory in 1 + 1 dimensions asan explicit construction of the holographic map [9]. Even in this situation, the collective fieldtheory is a nontrivial interacting theory, i.e. the oscillators are coupled. In our case there arean infinite number of collective fields which would seem to make the situation hopeless.3n our setup, however, the slowness of the driving force simplifies the situation drastically.The source couples directly to the dilaton in the bulk, and when ǫ ≪
1, to lowest order theresponse of the dilaton as well as the other fields is linear and independent of each other. Thiswill be clear in the supergravity solutions we present below. This implies that to lowest orderin ǫ , the oscillators which are dual to these modes are really harmonic oscillators which aredecoupled from each other.The resulting dynamics is then well approximated by the classical adiabatic perturbationtheory, which we refer to as the LNCAPT as mentioned above. The criterion for its applicabilityis that the driving force varies on a time scale much slower than the frequency of each oscillator.In particular if the frequency of the driving force is of order that of the oscillators one would beclose to resonance and the perturbation theory would break down. In our case this conditionfor the driving force to vary slowly compared to the frequency of the oscillators, becomes eq.(5).When this condition is met, the adiabatic approximation is valid for all modes - even thosewith the lowest frequency. The expectation value of the energy and the operator dual to thedilaton, ˆ O , can then be calculated in the resulting perturbation theory and we find that theleading order answers in ǫ agree with the supergravity calculations .Having understood the supergravity solutions in the gauge theory language we turn to askingwhat happens if the ’tHooft coupling becomes of order unity or smaller at intermediate times(while still staying in the parametric regime eq.(6)). The new complication is that additionaloscillators now enter the analysis. These oscillators correspond to string modes in the bulk.When the ’tHooft coupling becomes of order unity their frequencies can become small andcomparable to the oscillators which are dual to supergravity modes.At first sight one is tempted to conclude that these additional oscillators do not change thedynamics in any significant manner and the system continues to be well approximated by thelarge N classical adiabatic approximation. The following arguments support this conclusion.First, the anharmonic terms continue to be of order ǫ and thus are small, so that the oscillatorsare approximately decoupled. Second, the existence of a gap of order 1 /R for all values ofthe ’tHooft coupling, which we referred to above, ensures that the driving force varies muchmore slowly than the frequency of the additional oscillators, thus keeping the system far fromresonance. Finally, one still expects that in the parametric regime, eq.(6), an O ( N ǫ ) numberof quanta are produced keeping the system classical. These arguments suggest that the systemshould continue to be well approximated by the LNCAPT. In fact, since the additional oscil-lators do not directly couple to the driving force produced by the time dependent dilaton, but More precisely, both the supergravity and the forced oscillator calculations need to be renormalised to getfinite answer. One finds that after the counter terms are chosen to get agreement for the standard two pointfunction ( which measures the response for a small amplitude dilaton perturbation) the expectation value ofthe energy and ˆ O , agree. ǫ , their effectsshould be well controlled in an ǫ expansion. If these arguments are correct the energy which ispumped into the system initially should then get completely pumped back out and the systemshould settle into the ground state of the final N = 4 theory in the far future. The dualdescription in the far future would then be a smooth AdS space-time.However, further thought suggests another possibility for the resulting dynamics which is ofa qualitatively different kind. This possibility arises because, as was mentioned above, when the’tHooft coupling becomes of order unity string modes can get as light as supergravity modes.This means that the frequency of some of the oscillators dual to string modes can becomecomparable to oscillators dual to supergravity modes, and thus the string mode oscillators canget activated. Now there are many more string mode oscillators than there are supergravitymode oscillators, since the supergravity modes correspond to chiral operators in the gauge the-ory which are only O (1) in number, while the string modes correspond to non-chiral operatorswhich are O ( N ) in number. Thus once string mode oscillators can get activated there is thepossibility that many new degrees of freedom enter the dynamics.With so many degrees of freedom available the system could thermalise at least in the large N limit. In this case the energy which is initially present in the oscillators that directly coupleto the dilaton would get equi-partitioned among all the degrees of freedom. The subsequentevolution would be dissipative and this energy would not be recovered in the far future. At latetimes, when the ’tHooft coupling becomes big again, the gravity description of the dissipativebehaviour depends on how small is ǫ . From the calculations done in the supergravity regimeone knows that the total energy that is produced is of order N ǫ . When N ǫ ≫
1, but ǫ ≪ ( g Y M N ) − / the result is likely to be a gas of string modes. However if ǫ > ( g Y M N ) − / ,the energy is sufficient to form a small black hole (with horizon radius smaller than R AdS ). Abig black hole cannot form since this would require an energy of the order of N , and ǫ ≪ N R AdS .It is difficult for us to settle here which of the two possibilities discussed above, eitheradiabatic non-dissipative behaviour well described by the LNCAPT, or dissipative behaviourwith organised energy being lost in heat, is the correct one. One complication is that the rateof time variation which is set by ǫ is also the strength of the anharmonic couplings betweenthe oscillators. In thermodynamics, working in the microcanonical ensemble, it is well knownthat with energy of order N ǫ the configuration which entropically dominates is a small black5ole . This suggests that if the time variation in the problem were much smaller than theanharmonic terms a small black hole would form. However, in our case their being comparablemakes it a more difficult question to decide. One should emphasise that regardless of whichpossibility is borne out our conclusion is that most of the space time in the far future is smoothAdS, with the possible presence of a small black hole.Let us end with some comments on related work. The spirit of our investigation is closeto the work on AdS cosmologies in [10] and related work in [11] - [14]. See also [15], [16], [17]for additional work. Discussion of cosmological singularities in the context of Matrix Theoryappears in [18].The supergravity analysis we describe is closely related to the strategy which was used in thepaper [19], for finding forced fluid dynamics solutions; in that case one worked with an infinitebrane at temperature T and the small parameter was the rate of variation of the dilaton (ormetric) compared to T . Our regime of interest is complementary to that in [20] where thedilaton was chosen to be small in amplitude, but with arbitrary time dependence and whichleads to formation of black holes in supergravity for a suitable regime of parameters.This paper is organised as follows. In section § O in §
3. The quantum adiabatic perturbation theory is discussed in §
4. A forced harmonicoscillator is discussed in §
5. This simple system helps illustrate the difference between the twokinds of perturbation theory and sets the stage for the discussion of the The Large N classicaladiabatic approximation in §
6. Conclusions and future directions are discussed in §
7. Thereare three appendices which contains details of derivation of some of the formulae in the maintext.
In this section we will calculate the deformation of the supergravity solution in the presence ofa slowly varying time dependent but spatially homogeneous dilaton specified on the boundary.This will be a reliable description of the time evolution of the system so long as e Φ( t ) neverbecomes small. At least when the ’tHooft coupling is big enough so that supergravity can be trusted. .1 Some General Considerations IIB supergravity in the presence of the RR five form flux is well known to have an
AdS × S solution. In global coordinates this takes the form, ds = − (1 + r R AdS ) dt + dr r R AdS + r d Ω + R AdS d Ω . (7)Here R AdS is given by, R AdS = (4 πg s N ) / l s ∼ N / l pl (8)where l s is the string scale and l pl ∼ g / s l s is the ten dimensional Planck scale. g s is the valueof the dilaton, which is constant and does not vary with time or spatial position, e Φ = g s . (9)In the time dependent situations we consider below N will be held fixed. Let us discusssome of our convention es before proceeding. We will find it convenient to work in the 10-dim.Einstein frame. Usually one fixes l P l to be of order unity in this frame. Instead for our purposesit will be convenient to set R AdS = 1 . (10)From eq.(8) this means setting l P l ∼ /N / . The AdS × S solution then becomes, ds = − (1 + r ) dt + 1(1 + r ) dr + r d Ω + d Ω , (11)for any constant value of the dilaton, eq.(9). Let us also mention that when we turn to theboundary gauge theory we will set the radius R of the S on which it lives to also be unity.The essential idea in finding the solutions we describe is the following. Consider a situationwhere Φ varies with time slowly compared to R AdS . Since the solution above exists for anyvalue of g s and the dilaton varies slowly one expects that the resulting metric at any time t iswell approximated by the AdS × S metric given in eq.(11). This zeroth order metric will becorrected due to the varying dilaton which provides an additional source of stress energy in theEinstein equations. However these changes should be small for a slowly varying dilaton andshould therefore be calculable order by order in perturbation theory.Let us make this more precise. Consider as the starting point of this perturbation theorythe AdS metric given in eq.(11) and a dilaton profile,Φ = Φ ( t ) (12)which is a function of time alone. We take Φ ( t ) to be of the form,Φ = f ( ǫtR AdS ) (13)7here f ( ǫtR AdS ) is dimensionless function of time and ǫ is a small parameter, ǫ ≪ . (14)The function f satisfies the property that f ′ ( ǫtR AdS ) ∼ O (1) (15)where prime indicates derivative with respect to the argument of f .When ǫ = 0, the dilaton is a constant and the solution reduces to AdS × S . When ǫ issmall, d Φ dt = ǫR AdS f ′ ( ǫtR AdS ) ∼ ǫR AdS (16)so that the dilaton is varying slowly on the scale R AdS , and the contribution that the dilatonmakes to the stress tensor is parametrically suppressed . In such a situation the back reactioncan be calculated order by order in ǫ . The time dependent solutions we consider will be of thistype and ǫ will play the role of the small parameter in which we carry out the perturbationtheory. A simple rule to count powers of ǫ is that every time derivative of Φ comes with afactor of ǫ .The profile for the dilaton we have considered in eq.(12) is S symmetric. It is consistentto assume that the back reacted metric will also be S symmetric with the radius of the S being equal to R AdS . The interesting time dependence will then unfold in the remaining fivedirections of
AdS space and we will focus on them in the following analysis.The zeroth order metric in these directions is given by, ds = − (1 + r ) dt + 1(1 + r ) dr + r d Ω . (17)And the zeroth order dilaton is given by eq.(12),Φ = f ( ǫt ) . (18)We can now calculate the corrections to this solution order by order in ǫ .Let us make two more points at this stage. First, we will consider a dilaton profile Φ whichapproaches a constant as t → −∞ . This means that in the far past the corrections to themetric and the dilaton which arise as a response to the time variation of the dilaton must alsovanish. Second, the perturbation theory we have described above is a derivative expansion.The solutions we find can only describe slowly varying situations. This stills allows for a bigchange in the amplitude of the dilaton and the metric though, as long as such changes accruegradually. It is this fact that makes the solutions non-trivial. The more precise statement for the slowly varying nature of the dilaton, as will be discussed in a footnotebefore eq.(84), is that its Fourier transform has support at frequencies much smaller than 1 /R AdS . .2 Corrections to the Dilaton Let us first calculate the corrections to the dilaton. We can expand the dilaton as,Φ( t ) = Φ ( t ) + Φ ( r, t ) + Φ ( r, t ) · · · , (19)where Φ is the zeroth order profile we start with, given in eq.(13). Φ is of order ǫ , Φ is oforder ǫ and so on. The metric can be expanded as, g ab = g (0) ab + g (1) ab + g (2) ab + · · · (20)where g (0) ab is the zeroth order metric given in eq.(17) and g (1) ab , g (2) ab ... are the first order, secondorder etc corrections.The dilaton satisfies the equation, ∇ Φ = 0 . (21)Expanding this we find that to order ǫ , ∇ Φ + ∇ Φ + ∇ Φ + ∇ Φ + ∇ Φ = 0 . (22)Here ∇ is the Laplacian which arises from the zeroth order metric, and ∇ , ∇ are the cor-rections to the Laplacian to order ǫ, ǫ respectively, which arise due to the corrections in themetric. The first term on the left hand side is of order ǫ , since it involves two time derivativesacting on Φ . The second term is of order ǫ , and so is the third term. However, we see in § O ( ǫ ) correction to the metric and thus ∇ vanishes. So the second term is theonly one of O ( ǫ ) and we learn that Φ = 0 . (23)The first correction to the dilaton therefore arises at O ( ǫ ). Eq.(22) now becomes, ∇ Φ + ∇ Φ = 0 . (24)Since Φ preserves the S symmetry of AdS , Φ will also be S symmetric and must thereforeonly be a function of t, r . Further since Φ is O ( ǫ ) any time derivative on it would be of higherorder and can be dropped. Solving eq.(24) then gives,Φ ( r, t ) = Z r dr ′ ( r ′ ) (1 + ( r ′ ) ) "Z r ′ y y dy ¨Φ ( t ) + a ( t ) + a ( t ) . (25)Here a ( t ) , a ( t ) are two functions of time which arise as integration “constants”. It is easy to see that Φ , if non-vanishing, must depend on the radial coordinate, this makes ∇ Φ of order ǫ . Φ would be r dependent for the same reason that Φ in eq(25) is. ( r, t ) = 14 ¨Φ ( t ) (cid:20) r log(1 + r ) −
12 (log(1 + r )) − dilog(1 + r ) (cid:21) + a ( t ) 12 (cid:20) log(1 + r ) − r − r (cid:21) + a ( t ) . (26)The first term in Φ is regular at r = 0, while the term multiplying a ( t ) diverges here. Tofind a self-consistent solution in perturbation theory Φ must be small compared to Φ for allvalues of r , we therefore set a = 0. The first term in Φ ( r, t ) has the following expansion forlarge values of r , ¨Φ ( t ) " π − r + (cid:18)
316 + 14 log r (cid:19) r + · · · . (27)Since we are solving for the dilaton with a specified boundary value Φ ( t ), Φ ( r, t ) should vanishat the boundary. This determines a ( t ) to be, a ( t ) = − π
24 ¨Φ ( t ) , (28)leading to the final solutionΦ ( r, t ) = 14 ¨Φ ( t ) " r log(1 + r ) −
12 (log(1 + r )) − dilog(1 + r ) − π . (29)The solution is regular everywhere. Since Lim t →−∞ ˙Φ ( t ) , ¨Φ ( t ) = 0, the correction vanishes inthe far past, as required. The time varying dilaton provides an additional source of stress energy. The lowest ordercontribution due to this stress energy is O ( ǫ ) as we will see below. It then follows, after asuitable coordinate transformation if necessary, that the O ( ǫ ) corrections to the metric vanishand the first non-vanishing corrections to it arise at order ǫ . The essential point here is thatany O ( ǫ ) correction to the metric must be r dependent and thus would lead to a contribution tothe Einstein tensor of order ǫ , which is not allowed. This is illustrated by the dilaton calculationabove, where a similar argument lead to the O ( ǫ ) contribution, Φ , vanishing. In this subsectionwe calculate the leading O ( ǫ ) corrections to the metric.Before we proceed it is worth discussing the boundary conditions which must be imposed onthe metric. As was discussed in the previous subsection we consider a dilaton source, Φ , whichapproaches a constant value in the far past, t → −∞ . The corrections to the metric that arisefrom such a source should also vanish in the far past. Thus we see that as t → −∞ the metricshould approach that of AdS space-time. Also the solutions we are interested in correspondto the gauge theory living on a time independent S × R space-time in the presence of a time10ependent Yang Mills coupling (dilaton). This means the leading behaviour of the metric forlarge r should be that of AdS space. Changing this behaviour corresponds to turning on anon-normalisable component of the metric and is dual to changing the metric of the space-timeon which the gauge theory lives.We expect that these boundary conditions, which specify both the behaviour as t → −∞ and as r → ∞ should lead to a unique solution to the super gravity equations. The formerdetermine the normalisable modes and the latter the non-normalisable modes. This is dual tothe fact that in the gauge theory the response should be uniquely determined once the timedependent Lagrangian is known (this corresponds to the fixing the non-normalisable modes)and the state of the system is known in the far past(this corresponds to fixing the normalisablemodes).Since Φ is S symmetric, we can consistently assume that the corrections to the metric willalso preserve the S symmetry. The resulting metric can then be written as, ds = − g tt ( t, r ) dt + g rr ( t, r ) dr + 2 g tr ( t, r ) dtdr + R d Ω . (30)Now as is discussed in Appendix A upto O ( ǫ ) we can consistently set g tr = 0. In addition wecan to this order set R = r . Below we also use the notation, g tt ≡ e A ( t,r ) , (31) g rr ≡ e B ( t,r ) . (32)The metric then takes the form, ds = − e A ( t,r ) dt + e B ( t,r ) dr + r d Ω . (33)The trace reversed Einstein equation are: R AB = Λ g AB + 12 ∂ A Φ ∂ B Φ . (34)In our conventions, Λ = − . (35)To order ǫ we can set Φ = Φ in the second term on the rhs.A few simple observations make the task of computing the curvature components to O ( ǫ )much simpler. As we mentioned above the first corrections to the metric should arise at O ( ǫ ).To order ǫ the metric is then g ab ( t, r ) = g (0) ab ( r ) + g (2) ab ( t, r ) . (36)Now the zeroth order metric, g (0) ab , is time independent. The time derivatives of g (2) ab are non-vanishing but of order ǫ and thus can be neglected for calculating the curvature tensor to this11rder. As a result for calculating the curvature components to order ǫ we can neglect all timederivatives of the metric, eq.(36).Before proceeding we note that the comments above imply that the equations determiningthe second order metric components schematically take the form,ˆ O ( r ) g (2) ab = f ab ( r ) ˙Φ (37)where ˆ O ( r ) is a second order differential operator in the radial variable, r . As a result thesolution will be of the form, g (2) ab = F ( r ) ab ˙Φ , (38)where F ( r ) are functions of r which arise by inverting ˆ O ( r ). We see that the corrections to themetric at time t are determined by the dilaton source Φ at the same instant of time time t .Note also that since we are only considering a dilaton source Φ which vanishes in the far past,the solution eq.(38) correctly imposes the boundary condition that g (2) ab vanishes in far past andthe metric becomes that of AdS .Bearing in mind the discussion above, the curvature components are now easy to calculate.The t − t component of eq.(34) gives,( A ′ e ( A − B ) ) ′ e ( A + B ) + 3 A ′ e − B r = ˙Φ e − A + 4 . (39)The r − r component gives, − ( A ′ e ( A − B ) ) ′ e ( A + B ) + 3 B ′ e − B r = − . (40)The component with legs along the S gives, B ′ − A ′ e B r + 2 r (1 − e − B ) = − . (41)In these equations primes indicates derivative with respect to r and dot indicates derivativewith respect to time.Adding the t − t and r − r equations gives,3( A ′ + B ′ ) e − B r = ˙Φ e − A . (42)Eq.(41) and eq.(42) then lead to2 B ′ e − B r −
16 ˙Φ e − A + 2 r (1 − e − B ) = − . (43)This is a first order equation in B . Integrating we get to order ǫ , e − B = 1 + r + c r −
16 ˙Φ r [ Z r e − A r dr ] . (44)12ere c is an integration constant and e A = 1 + r is the zeroth order value of e A . We requirethat the metric become that of AdS space as t → −∞ this sets c = 0 . A negative value of c would mean starting with a black hole in AdS in the far past.The integral within the square brackets on the rhs in eq.(44) is given by, Z r e − A r dr = 12 [ r − ln(1 + r ) + d ] . (45)This gives, e − B = 1 + r −
112 ˙Φ r [ r − ln(1 + r ) + d ] . (46)A solution which is regular for all values of r , is obtained by setting d to vanish. This gives, e − B = 1 + r −
112 ˙Φ [1 − r ln(1 + r )] . (47)We can obtain e A from eq.(42). To second order in ǫ this equation becomes, A ′ = 16 r ˙Φ e − A − B ) − B ′ , (48)which gives, A = − B + 112 ˙Φ [ −
11 + r + d ] , (49)with d being a general function of time. Eq.(49) and eq.(47) leads to e A = 1 + r + ˙Φ [ −
14 + 112 ln(1 + r ) r + d r )] . (50)The last term on the right hand side changes the leading behaviour of e A as r → ∞ , if d doesnot vanish, and therefore corresponds to turning on a non-normalisable mode of the metric. Aswas discussed above we want solutions where this mode is not turned on, and we therefore set d to vanish.This gives finally, e A = 1 + r −
14 ˙Φ + 112 ˙Φ ln(1 + r ) r . (51)Eq.(47), (51) are the solutions to the metric, eq.(33), to second order. Note that the Einsteinequations gives rise to three equations, eq.(39), eq.(40), eq.(41). We have used only two linearcombinations out of of these to find A, B . One can show that the remaining equation is alsosolved by the solution given above.In summary we note that the Einstein equations can be solved consistently to second orderin ǫ . The resulting solution is horizon-free and regular for all values of the radial coordinate Note that c could be a function of time and still solve eq.(43), recall though that the equations abovewere derived by neglecting all time derivatives of the metric, eq.(36). Only a time independent constant c is consistent with this assumption. A similar argument will also apply to the other integration constants weobtain as we proceed. ǫ compared to the leading term for all values of r ,thereby making the perturbation theory self consistent.Let us end by commenting on the choice of integration constants made in obtaining thesolution above. The boundary conditions, as t → −∞ and r → ∞ , determine most of theintegration constants. One integration constant d which appears in the solution for e B , eq.(46)is fixed by regularity at r → . For d = 0 the second order correction is small compared tothe leading term, and the use of perturbation theory is self-consistent. Moreover we expect thatthe boundary conditions imposed here lead to a unique solution to the supergravity equations,as was discussed at the beginning of this subsection. Thus the solution obtained by setting d = 0 should be the correct one.The solution above is regular and has no horizon. It has these properties due to the slowlyvarying nature of the boundary dilaton. The dual field theory in this case is in a non-dissipativephase. Once the dilaton begins to change sufficiently rapidly with time we expect that a blackhole is formed, corresponding to the formation of a strongly dissipative phase in the dual fieldtheory. In [20] the effect of a small amplitude time dependent dilaton with arbitrary timedependence was studied. Indeed it was found that when the time variation is fast enough thereare no regular horizon-free solutions and a black hole is formed.Finally, the analysis of this section holds when e Φ is large enough to ensure applicabilityof supergravity. The fact that a black hole is not formed in this regime does not precludeformation of black holes from stringy effects when e Φ becomes small enough. In fact we willargue in later sections that the latter is a distinct possibility. An important feature of the lowest order calculation of this section is that the perturbations ofthe dilaton and the metric are essentially linear and do not couple to each other. To this order,the dilaton perturbation is simply a solution of the linear d’Alembertian equation in
AdS .Similarly the metric perturbations also satisfy the linearized equations of motion in AdS , albeitin the presence of a source provided by the energy momentum tensor of the dilaton. This is afeature present only in the leading order calculation. As explained above, this arises because ofthe smallness of the parameter ǫ . We will use this feature to compare leading order supergravityresults with gauge theory calculations in a later section. Similarly in solving for the dilaton perturbation the integration constant a is fixed by requiring regularityat r = 0, eq.(25). Calculation of Stress Tensor and Other Operators
In this section we calculate the boundary stress tensor and the expectation value of the operatordual to the dilaton, staying in the supergravity approximation. This will be done using standardtechniques of holographic renormalization group [21, 22, 23, 24, 25, 26, 27, 28].
The metric is of the form, eq.(33), eq.(47), eq.(51). For calculating the stress tensor a boundaryis introduced at large and finite radial location, r = r . The induced metric on the boundaryis, ds B ≡ h µν dx µ dx ν = − e A dt + r ( dθ + sin θdφ + sin θ sin φdψ ) . (52)The 5 dim. action is given by S = 116 πG Z M d x √− g ( R + 12 −
12 ( ∇ Φ) ) − πG Z r = r d x √− h Θ . (53)Here h µν is the induced metric on the boundary , and Θ is the trace of the extrinsic curvatureof the boundary. In our conventions, with R AdS = 1, G = π N . (54)A counter term needs to be added, it is, S ct = − πG Z ∂M d x √− h [3 + R −
18 ( ∇ Φ) − log ( r ) a (4) ] . (55)The last term is needed to cancel logarithmic divergences which arise in the action, it iswell known and is discussed in e.g. [21, 27]. From eq.(24) of [27] we have that a (4) = 18 R µν R µν − R − R µν ∂ µ Φ ∂ ν Φ + 124 Rh µν ∂ µ Φ ∂ ν Φ + 116 ( ∇ Φ) + 148 { ( ∇ Φ) } . (56)Here ∇ is a covariant derivative with respect to the metric h µν .Varying the total action S T = S + S ct gives the stress energy, T µν = 2 √− h δS T δh µν (57)= 18 πG [Θ µν − Θ h µν − h µν + 12 G µν − ∇ µ Φ ∇ ν Φ + 18 h µν ( ∇ Φ) + · · · ] . Here G µν is the Einstein tensor with respect to the metric h µν . The ellipses stand for extraterms obtained by varying the last term in eq.(55) proportional to a (4) . While these terms arenot explicitly written down in eq.(57), we do include them in the calculations below. Note that our definition of the dilaton Φ is related to φ (0) in [27] by φ (0) = Φ / < T µν > = r T µν (58)Carrying out the calculation gives a finite answer, < T tt > = N π [ − − ˙Φ
16 ] < T θθ > = < T ψψ > = < T φφ > = N π [ 18 − ˙Φ
16 ] (59)where we have used eq.(54). We remind the reader that in our conventions the radius of the S on which the boundary gauge theory lives has been set equal to unity. The first term onthe right hand side of (59) arises due to the Casimir effect. The second term is the additionalcontribution due to the varying Yang Mills coupling.From eq.(59) the total energy in the boundary theory can be calculated. We get, E = − < T tt > V S = 3 N
16 + N ˙Φ . (60)where V S = 2 π is the volume of a unit three-sphere. Note that the varying dilaton gives rise toa positive contribution to the mass, as one would expect. Moreover this additional contributionvanishes when the ˙Φ vanishes. In particular for a dilaton profile which in the far future, as t → ∞ , again approaches a constant value (which could be different from the starting value ithad at t → −∞ ) the net energy produced due to the varying dilaton vanishes. The operator dual to the dilaton has been discussed explicitly in [3], [29], [10].It’s expectation value is given by, < ˆ O l =0 > = δS T δ Φ B | Φ B → (61)Here S T is the total action including the boundary terms, eqn. (55). Since Φ B is a function of t alone the lhs is the l = 0 component of the operator dual to the dilaton which we denote by,ˆ O l =0 .The steps involved are analogous to those above for the stress tensor and yield, < ˆ O l =0 > = − N
16 ¨Φ (62)Note that the lhs refers to the expectation value for the dual operator integrated over theboundary S . In obtaining eq.(62) we have removed all the divergent terms and only kept thefinite piece. A quadratically divergent piece is removed by the third term in eq.(55) proportionalto ( ∇ Φ) , and a log divergence is removed by a contribution from the last term in eq.(55)proportional to a (4) . 16 .3 Additional Comments Let us end this section with a few comments.The only source for time dependence in the boundary theory is the varying Yang Millscoupling. A simple extension of the usual Noether procedure for the energy, now in the presenceof this time dependence, tells us that dEdt = − ˙Φ < ˆ O l =0 > . (63)It is easy to see that the answers obtained above in eq.(60), eq.(62) satisfy this relation. Therelation eq.(63) is a special case of a more general relation which applies for a dilaton varyingboth in space and time, this was discussed in Appendix A of [19].In general, for a slowly varying dilaton one can expand < ˆ O l =0 > in a power series in ˙Φ .For constant dilaton, the solution is AdS where one knows that the < ˆ O l =0 > vanishes. Thusone can write, < ˆ O l =0 > = c ˙Φ + c ¨Φ + c ( ˙Φ ) · · · (64)where the ellipses stand for higher powers of derivatives of the dilaton. Comparing with theanswer in eq.(62) one sees that in the supergravity limit c and c vanish. As a result dEdt is atotal derivative, and as was discussed above if the dilaton asymptotes to a constant in the farfuture there is no net gain in energy.It is useful to contrast this with what happens in the case of an infinite black brane attemperature T subjected to a time dependent dilaton which is slowly varying compared to thetemperature T . This situation was analysed extensively in [19]. In that case (see eq.(2.13),eq.(3.20) and section 7.2 of the paper) the leading term in eq.(64) proportional to ˙Φ does notvanish. The temperature then satisfies an equation, dTdt = 112 π ˙Φ (65)As a result any variation in the dilaton leads to a net increase in the temperature, and theenergy density. Note the first term in eq.(64) contains only one derivative with respect to timeand breaks time reversal invariance. It can only arise in a dissipative system. In the case ofa black hole the formation of a horizon breaks time reversal invariance and turns the systemdissipative allowing this term to arise. In the solution we construct no horizon forms andconsistent with that the first term is absent.We see in the solution discussed above that the second order corrections to the dilatonand metric arise in an instantaneous manner - at some time t , and for all values of r , they aredetermined by the boundary value of the dilaton at the same instant of time t . This might seema little puzzling at first since one would have expected the effects of the changing boundary17onditions to be felt in a retarded manner. Note though that in AdS space a light ray canreach any point in the bulk from the boundary within a time of order R AdS . When ǫ ≪ O l =0 it follows that in thefar future the system settles down into an AdS solution again. The near instantaneous natureof the solution means that this happens quickly on the times scale of order R AdS . This agreeswith general expectations. The supergravity modes carry an energy of order 1 /R AdS and shouldgive rise to a response time of order R AdS .Also note that in our units, where R AdS = 1, each supergravity mode carries an energyof order unity. The total energy at intermediate times is of order N ǫ , so we see that an O ( N ǫ ) number of quanta are excited by the time varying boundary dilaton. This can be abig number when N ǫ ≫
1. In fact the energy is really carried by the various dilaton modes.The metric perturbations are S symmetric and thus contain no gravitons (in the sense ofgenuine propagating modes). One can think of this energy as being stored in a spatial regionof order R AdS in size located at the center of AdS space. This is what one would expect, sincethe supergravity modes which are produced by the time varying boundary dilaton have a sizeof order R AdS and their gravitational redshift is biggest at the center of AdS space .In summary, the response in the bulk to the time varying boundary dilaton is characteristicof a non-dissipative adiabatic system which is being driven much more slowly than its own fastinternal time scale of response. We now turn to analysing the behaviour of the system in the dual field theory. The motivationbehind this is to be able to extend our understanding to situations in which the ’tHooft couplingat intermediate time becomes of order one or smaller, so that the geometry in the bulk becomesof order the string scale. In such situations the supergravity calculation presented in theprevious section breaks down and higher derivative corrections become important. The gaugetheory description continues to be valid, however. Using this description one can then hopeto answer how the system evolves in the region of string scale curvature, and in particularwhether by waiting for enough time a smooth geometry with small curvature emerges again onthe gravity side. AdS is of course a homogeneous space-time, but our boundary conditions pick out a particular notion oftime. The center of AdS, where the energy is concentrated, is the region as mentioned above where the redshiftin the corresponding energy is the biggest.
18e saw in the previous subsection that the bulk response was characteristic of an adiabaticsystem which was being driven slowly compared to the time scale of its own internal response.This suggests that in the gauge theory also an adiabatic perturbation theory should be validand should prove useful in understanding the response. A related observation is the following.The bulk solutions we have considered correspond to keeping the radius R of the S on whichthe gauge theory lives to be constant and independent of time. We will choose conventions inwhich R = R AdS = 1. The Yang Mills theory is related to the boundary dilaton by, g Y M = e Φ ( t ) . (66)The dilaton profile eq.(18) also means that Yang Mills coupling in the gauge theory variesslowly compared to the radius R . Since this is the only other scale in the system, this alsosuggests that an adiabatic approximation should be valid in the boundary theory.We will discuss two different kinds of adiabatic perturbation theory below. The first, whichwe call Quantum adiabatic perturbation theory, is studied in this section. This is the adiabaticperturbation theory one finds discussed in a standard text book of quantum mechanics, see[30],[31]. Its validity, we will see below requires the condition, N ǫ ≪
1, to be met. We willargue that once this condition is met the gauge theory analysis allows us to conclude that, evenin situations where the curvature becomes of order the string scale at intermediate times, adual smooth
AdS geometry emerges as a good approximation in the far future.The supergravity calculations, however, required only the condition ǫ ≪
1, which is muchless restrictive than the condition
N ǫ ≪
1. Understanding the supergravity regime on the gaugetheory side leads us to formulate another perturbation theory, which we call “Large N ClassicalAdiabatic Perturbation Theory” (LNCAPT). To explain this we find it useful to first discussthe example of a driven harmonic oscillator, as considered in §
5. Following this, we discussLNCAPT in the gauge theory in §
6. We find that its validity requires that the conditionseq.(6) are met. Using it we will get agreement with the supergravity calculations of sections § §
3, when the ’tHooft coupling remains large for all times.Towards the end of §
6, we discuss what happens in the gauge theory when conditions eq.(6)are met but with the ’tHooft coupling becoming small at intermediate times. Two qualitativelydifferent behaviours are possible, and we will not be able to decide between them here. Eitherway, at late times a mostly smooth AdS description becomes good on the gravity side, with thepossible presence of a small black hole.In the discussion below we will consider the following type of profile for the boundarydilaton: it asymptotes in the far past and future to constant values such that the initial andfinal values of the ’tHooft coupling, λ , are big, and attains its minimum value near t = 0. If thisminimum value of λ ≤ N = 4 theory, on S the spectrum19f the gauge theory is gapped and this state is well defined. It is well known that the spectrum of the N = 4 theory on S has a gap between the energyof the lowest state and the first excited state. This gap is of order 1 /R and thus is of orderunity in our conventions. The existence of this gap follows very generally just from the factthat the spectrum must provide a unitary representation of the conformal group, [4], and thegap is therefore present for all values of the Yang Mills coupling constant. In the supergravityapproximation the spectrum can be calculated using the gravity description and is consistentwith the gap, the lowest lying states have an energy E = 2. This is also true at very weak’tHooft coupling.Now for a slowly varying dilaton eq.(18) we see that the Yang Mills coupling and thereforethe externally imposed time dependence varies slowly compared to this gap. There is a wellknown adiabatic approximation which is known to work in such situations, see e.g. [30],[31]whose treatment we closely follow. We will refer to this as the quantum adiabatic approximationbelow and study the Yang Mills theory in this approximation.The essential idea behind this approximation is that when a system is subjected to a timedependence which is slow compared to its internal response time, the system can adjust itselfvery quickly and as a result to good approximation stays in the ground state of the instantaneousHamiltonian.More precisely, consider a time dependent Hamiltonian H ( ζ ( t )), where ζ ( t ) is the timevarying parameter. Now consider the one parameter family of time independent Hamiltoniansgiven by H ( ζ ). To make our notation clear, a different value of ζ corresponds to a differentHamiltonian in this family, but each Hamiltonian is time independent. Let | φ m ( ζ ) > be acomplete set of eigenstates of the Hamiltonian H ( ζ ) satisfying, H ( ζ ) | φ m ( ζ ) > = E m ( ζ ) | φ m ( ζ ) >, (67)in particular let the ground state of H ( ζ ) be given by | φ ( ζ ) > . We take | φ m ( ζ ) > to have unitnorm. Then the adiabatic theorem states that if ζ → ζ in the far past, and we start with thestate | φ > which is the ground state of H ( ζ ) in the far past, the state at any time t is wellapproximated by, | ψ ( t ) > ≃ | φ ( ζ ) > e − i R t −∞ E ( ζ ) dt . (68)Here | φ ( ζ ) > is the ground state of the time independent Hamiltonian corresponding to thevalue ζ = ζ ( t ). Similarly in the phase factor E ( ζ ) is the value of the ground state energy for ζ = ζ ( t ). 20orrections can be calculated by expanding the state at time t in a basis of energy eigenstatesat the instantaneous value of the parameter ζ . The first corrections take the form, | ψ ( t ) > = X n =0 a n ( t ) | φ n ( ζ ) > e − i R t −∞ E n dt (69)where the coefficient a n ( t ) is, a n ( t ) = − Z t −∞ dt ′ < φ n ( ζ ) | ∂H∂ζ | φ ( ζ ) >E − E n ˙ ζ e − i R t ′−∞ ( E − E n ) dt ′ (70)In the formula above on the rhs | φ n ( ζ ) >, ∂H∂ζ , E n ( ζ ) , are all functions of time, through the timedependence of ζ . For the adiabatic approximation to be good the first corrections must be small. To ensure thiswe impose the condition, | < φ n | ∂H∂ζ | φ > ˙ ζ | ≪ ( E − E ) (71)where ( E − E ) is the energy gap between the ground state and the first excited state and | φ n > is any excited state. (This would then imply that the lhs in eq.(71) is smaller than( E n − E ) for all n .) This condition is imposed for all time for the adiabatic approximationto be valid .In our case the role of the parameter ζ is played by the dilaton Φ (with the gauge coupling g Y M = e Φ ). Thus eq.(71) takes the form, | < φ n | ∂H∂ Φ | φ > ˙Φ | ≪ ( E − E ) . (72)Now, as we will see below in subsection § ∂H∂ Φ is, up to a sign, exactly the operator ˆ O l =0 which is dual to the modes of the dilaton which are spherically symmetric on the S . Thereforeeq.(72) becomes | < φ n | ˆ O l =0 | φ > ˙Φ | ≪ ( E − E ) . (73)We have argued above that the rhs is of order unity in our conventions due to the existenceof a robust gap. On the lhs, ˙Φ ∼ O ( ǫ ), and as we will argue below the matrix element, | < φ n | ˆ O l =0 | φ > ∼ O ( N ). Thus eq.(73) becomes, N ǫ ≪ . (74) The actual condition is that the corrections to | ψ > must be small. This means that at first order < ψ | ψ > should be small. When eq.(71) is met | a n | is small, but in some cases that might not be enoughand the requirement that the sum P | a n | is small imposes extra restrictions. There could also be additionalconditions which arise at second order etc. .2 Highly Curved Geometries Eq.(74) is the required condition then for the applicability of quantum perturbation theory.When this condition is met, we can continue to trust the quatum adiabatic approximationin the gauge theory even when the ’tHooft coupling becomes of order unity or smaller atintermediate times. All the conditions which are required for the validity of this approximationcontinue to be hold in this case. First, as was discussed above the gap of order unity continuesto exist. Second, the matrix elements which enter are in fact independent of λ since theycorrespond to the two-point function of dilaton which is a chiral operator. Thus the systemcontinues to be well described in the quantum adiabatic approximation so long as eq.(74) ismet. It follows then that in the far future the state of the system to good approximation is theground state of the N = 4 theory. This implies that the dual description in the far future is asmooth AdS geometry.There is one important caveat to the above conclusion. It is possible that at λ ∼ O (1) thereare several states in the spectrum, scaling as a positive power of N , which accumulate near thefirst excited state. This does not happen for λ ≫ λ ≪ N . This is a question which can be settled in principle once the spectrumof the N = 4 theory is known for all λ . Similarly, the possibility for unexpected surprisesat higher orders can also be examined once enough is known about the N = 4 theory. Thepoint is simply that in this approximation all matrix elements and conditions can be phrasedas statements in the time independent N = 4 theory. As our knowledge of the N = 4 theorygrows we will be able to check for any such unexpected surprises.Let us also mention before proceeding that when the condition eq.(74) is met and fora dilaton profile where the ’tHooft coupling stays large for all time, the metric is to goodapproximation smooth AdS for all time. However the small corrections to this metric anddilaton cannot be calculated reliably in the classical approximation used in section 2. Thisis because in this regime it is very difficult to even produce one supergravity quantum as anexcitation above the adiabatic vaccum. Therefore quantum effects are important in calculatingthese corrections. We close this section by discussion two points relevant to the analysis leading up to condition,eq.(74).First, let us argue why ∂H∂ Φ = − ˆ O l =0 . The argument is sketched out below, more details22an be found in [10]. The action of the N = 4 theory is given by, S = Z dt d Ω √− g ( − e Φ ) T rF µν F µν + · · · (75)where the ellipses indicate extra terms coming from scalars and fermions. Varying with respectto Φ gives us the operator dual to the dilaton,ˆ O = √− g ( 14 e Φ ) T rF µν F µν + · · · (76)where the ellipses denote extra terms which arise from the terms left out in eq.(75). Henceforth,to emphasise the key argument we neglect the additional terms coming from the ellipses.Working in A = 0 gauge, the Hamiltonian density H is given by, H = e Φ π i π i e − Φ F ij F ij (77)where π i = e − Φ ∂ A i (78)is the momentum conjugate to A i . Varying with respect to Φ gives, ∂ H ∂ Φ = π i π i e Φ − e − Φ F ij F ij . (79)Substituting from eq.(78) one sees that this agrees (up to a sign) with the operator ˆ O given ineq.(76). When the dilaton depends on time alone we can integrate the above equations over S , which leads to the relation ∂H∂ Φ = − ˆ O l =0 , where H now stands for the hamiltonian (ratherthan the hamiltonian density).Second, we estimate how the matrix element, < φ n | ˆ O l =0 | φ > , which appears in eq.(73),scales with N . It is useful to first recall that the N = 4 theory, which is conformally invariant,has an operator state correspondence. The states | φ n > can be thought of as being createdfrom the vacuum by the insertion of a local operator. This makes it clear that the only stateshaving a non-zero matrix element, < φ n | ˆ O l =0 | φ > , are those which can be created from thevacuum by inserting ˆ O l =0 , since the only operator with which ˆ O l =0 has a non-zero two pointfunction is ˆ O l =0 itself.Now in terms of powers of N the two-point function scales like, < ˆ O l =0 ˆ O l =0 > ∼ N . (80)The state | φ n > which appears in the matrix element in eq.(73) has unit norm and is thereforecreated from the vacuum by the operator, | φ n > ∼ N ˆ O l =0 | > (81)23rom eq.(80), eq.(81), we then see that the matrix element scales like, < φ n | ˆ O l =0 | φ > ∼ N (82)as was mentioned above.Our discussion leading up to the estimate of the matrix element has been imprecise in somerespects. First, strictly speaking the operator state correspondence we used is a property ofthe Euclidean theory on R , where as we are interested in the Minkowski theory on S × R .However, this is a technicality which can be taken care of by first relating the matrix elementin the Minkowski theory to that in Euclidean S × R space and then relating the latter to thaton R by a conformal transformation.More importantly, the state created by ˆ O l =0 is not an eigenstate of energy, but is in facta sum over an infinite number of states labelled by an integer n with energies ω n = 4 + 2 n .This can be understood as follows. The operator ˆ O can be expanded into positive and negativefrequency modes, A n , A † n respectively, for an infinite set n , and acting with any of the A † n ’sgives a state, | ϕ n > ≃ A † n | > . (83)One must therefore worry about the dependence on the mode number n in the matrix elementand the effects of summing up the contributions for all these modes. We will return to addressthis issue in more detail in subsections 6.2 and 6.3, when we describe the operators A n , A † n more explicitly and discuss renormalization. For now, let us state that after the more carefultreatment we will find that the condition for the quantum adiabatic approximation eq.(74)goes through unchanged. The physical reason is simply this: we are interested here in the verylow-frequency response of the system and its very high frequency modes are not relevant forthis. The supergravity calculations required the condition ǫ ≪
1. To understand this regime in thedual gauge theory it is first useful to consider a quantum mechanical Harmonic oscillator withfrequency ω driven by a time dependent source J ( t ) . We will see that in this case a classicaladiabatic perturbation theory becomes valid when ¨ J ˙ J ω ≪ , (84) Eq.(84), (85), clearly cannot hold when ˙ J vanishes. The more precise versions of these conditions are asfollows. Eq.(84) is really the requirement that J is slowly varying. By this one means that the fourier transformof J has support, up to say exponentially small corrections, only for small frequencies compared to ω . Eq.(85)is the requirement that the coherent state parameter, λ ( t ) given in eq.(99), is large. J ≫ ω / . (85)Having understood this system we then return to the gauge theory in the following subsection.The Hamiltonian is given by H = 12 ˙ X + 12 ω ( X + J ( t ) ω ) . (86)In the quantum adiabatic approximation one considers the instantaneous Hamiltonian. At time t this is given by, H = 12 ˙ X + 12 ω ( X + J ( t ) ω ) (87)where J ( t ) is to regarded as a time independent constant in H .The ground state of H is a coherent state. Define, X = a + a † √ ω , P = − i √ ω ( a − a † √ P = ˙ X (89)is the conjugate momentum. The ground state is | φ > = N α e αa † | > . (90)Here N α is a normalisation constant, determined by requiring that < φ | φ > = 1. The state | > is the vacuum annihilated by a , i.e., a | > = 0 , (91)and α = − J q ω . (92)The ground state energy is E = 12 ω , (93)it is independent of time.A quick way to derive these results is to work with the shifted creation and destructionoperators, ˜ a = a − α, ˜ a † = a † − α (94)where α is given in eq.(92). The Hamiltonian takes the form, H = ω (˜ a † ˜ a ) + 12 ω (95)25t is clear then that the ground state is annihilated by ˜ a , leading to eq.(90) and the groundstate energy is eq.(93).For the quantum adiabatic theorem to be valid, the condition in eq.(71) must hold. For theharmonic oscillator it is easy to see that this gives,˙ J ≪ ω / . (96)In fact the time evolution in this case can be exactly solved. We consider the case where J ( t ) → , t → −∞ . Starting with the state | > in the far past, which is the vacuum of theHamiltonian in the far past, we then find that the state at any time t is given by, | ψ ( t ) > = N ( t ) e λ ( t ) a † | φ > (97)where | φ > is the adiabatic vaccum given in eq.(90), N ( t ) is a normalisation constant and thecoherent state parameter is λ ( t ). Imposing Schrodinger equation one gets i ˙ λ = i ˙ J q ω + ω λ. (98)The solution for λ ( t ) with initial condition λ ( −∞ ) = 0 is given by, λ ( t ) = e − iω t q ω Z t −∞ ˙ J ( t ′ ) e iω t ′ dt ′ . (99)Some details leading to eq. (98) are given in Appendix B. This state will behave like a classicalstate when the coherent state parameter is big in magnitude, i.e., when | λ | ≫ . (100)The integral on the rhs of eq.(99) can be done by parts (we set J ( −∞ ) = 0), Z t −∞ dt ′ ˙ J e iω t ′ = ˙ J ( t ) e iω t iω − Z t −∞ dt ′ ¨ J e iω t ′ iω . (101)Subsequent iterations obtained by further integrations by parts gives rise to a series expansion for λ in terms of higher derivatives of J . The higher order terms are small if J is slowly varyingcompared to the frequency of the oscillator ω . Evaluating the second term which arises in hisexpansion for example and requiring it to be smaller than the first term in eq.(101) gives,¨ J ˙ J ω ≪ In general one expects this to be an asymptotic rather than convergent series.
26e assume now that J is slowly varying and the first term on the rhs of eq.(101) is a goodapproximation to the integral. This tells us that for eq.(100) to be true the condition whichmust be met is, ˙ J ≫ ω / . (103)Note that this condition is opposite to the one needed for the quantum adiabatic theorem toapply eq.(96).The answer for the < X > can be easily obtained by inserting the expression for λ obtainedin eq.(99) in the wave function, eq.(97). Let us obtain it here in a slightly different manner.When eq.(100) is true the system behaves classically and its response to the driving force canbe obtained by solving the classical equation of motion for the forced oscillator. In terms ofthe fourier transform of J this gives, X ( t ) = Z J ( ω ) ω − ω e − iωt dω (104)The correct pole prescription on the rhs is that for a retarded propagator.When the source is slowly varying compared to ω , the denominator ω − ω in eq.(104)can be expanded in a power series in ω ω and the resulting fourier transforms can be expressedas time derivatives of J . The first two terms give, X = − J ( t ) ω + ¨ Jω + · · · (105)The first term on the rhs is the location of the instantaneous minimum. The second term isthe first correction due to the time dependent source. Subsequent corrections are small if thesource is slowly varying and condition eq.(102) is met. It is useful to express this result as, X + J ( t ) ω = ¨ Jω + · · · . (106)The left hand side is the expectation value of X after adding a shift to account for the instanta-neous minimum of the potential. The right hand side we see now only contains time derivativesof J . Before proceeding let us note that the expanding the denominator in eq.(104) in a powerseries in ω ω gives a good approximation only if J ( ω ) has most of its support for ω ≪ ω . Thisis how the more precise condition mentioned in the footnote before eq.(84) arises.It is also useful to discuss the energy. From eq.(105) and the Hamiltonian we see that theleading contribution comes from the Kinetic energy term and is given to leading order by, E = 12 ˙ J ω (107)(strictly speaking this is the energy above the ground state energy).27he external source driving the oscillator changes its energy. Noether’s argument in thepresence of the time dependent source leads to the conclusion that ∂H∂t = ˙ J ( X + Jω ) (108)(this also directly follow from the Hamiltonian, eq.(86)). From eq.(106) and eq.(107) we seethat this condition is indeed true. Let us also note that the rate of change in energy can beexpressed in terms of the shifted operators, eq.(94), as, ∂H∂t = ˙ J ( ˜ a + ˜ a † √ ω ) , (109)this form will be useful in our discussion below.To summarise, we find that when the conditions eq.(103), eq.(102), are met the drivenharmonic oscillator behaves like a classical system. Its response, for example, < X > , and theenergy, E , can be calculated in an expansion in time derivatives of J , which is controlled wheneq.(102) is valid and the source is slowly varying. We will refer to this perturbation expansionas the classical adiabatic perturbation approximation below. Note that the condition, eq.(103)is opposite to the one required for the quantum adiabatic perturbation theory to hold. In thenext subsection we will discuss how a similar classical adiabatic approximation arises in thegauge theory. We now return to the gauge theory and formulate a large N classical adiabatic approximationbased on coherent states in this theory. This will allow us to obtain results in the gauge theorywhich agree with those obtained using supergravity in § § The supergravity solution in § classical solutions rather than states which containa small number of bulk particles. The AdS/CFT correspondence implies that bulk classicalsolutions corresponds to coherent states in the boundary gauge theory with a large numberof particles in which operators like ˆ O have nontrivial expectation values. On the other hand,states obtained by the action of a few factors of ˆ O on the vaccum are few-particle states inthe bulk. The quantum adiabatic approximation described in § §
2. 28e, therefore, need to formulate an adiabatic approximation in terms of coherent statesof gauge invariant operators in the boundary theory to try and understand the supergravitysolutions of § N → ∞ limit. (See e.g. [7]). Consider a complete (usually overcomplete)set of gauge invariant operators in the Schrodinger picture, ˆ O I . A general coherent state is ofthe form | Ψ( t ) > = exp " iχ ( t ) + X I λ I ( t ) ˆ O I (+) | > A . (110)Here ˆ O I (+) denotes the creation part of the operator and | > A denotes the adiabatic vacuumcorresponding to some instantaneous value of the dilation Φ , H [Φ ] | > A = E Φ | > A (111)with the ground state energy E Φ .The algebra of operators ˆ O I , together with the Schrodinger equation then leads to a dif-ferential equation which determines the time evolution of the coherent state parameters λ I ( t )in terms of the time dependent source Φ ( t ). The idea is then to solve this equation in anexpansion in time derivatives of Φ ( t ). This is the coherent state adiabatic approximation weare seeking.In general it is almost impossible to implement this program practically, since the operatorsˆ O I have a non-trivial operator algebra which mixes all of them. The coherent state (110) isin the co-adjoint orbit of this algebra [7]. The resulting theory of fields conjugate to theseoperators would be in fact the full interacting string field theory in the bulk. In our case,however, the situation drastically simplifies for large ’t Hooft coupling at the lowest order of anexpansion in ˙Φ . This is because these various operators decouple and their algebra essentiallyreduces to free oscillator algebras.We have already found this decoupling in our supergravity calculation. The departure ofthe solution from AdS × S is due to the time-dependence of the boundary value of the dilaton,and are small when the time variations are small, controlled by the parameter ǫ . To lowest orderin ǫ (which is O ( ǫ )) the deformation of the bulk dilaton in fact satisfied a linear equation inthe AdS background in the presence of a source provided by the boundary value Φ ( t ). Thisequation does not involve the deformation of the metric. Similarly, the equation for metricdeformation does not involve the dilaton deformation to lowest order.This allows us to treat each supergravity field and its dual operator separately. With thisunderstanding we will now consider the coherent state (110) with only the operator dual tothe dilaton, ˆ O . Since our source is spherically symmetric and higher point functions of theoperators are not important in this lowest order calculation, we can restrict this operator to itsspherically symmetric part. 29 .2 Large N Classical Adiabatic Perturbation Theory (LNCAPT) Let us now elaborate in more detail on the LNCAPT.The linearised approximation in the gravity theory means that only the two point functionis non-trivial and all connected higher point functions vanish. The non-linear terms correspondto nontrivial higher order correlations. In this approximation the gauge theory simplifies agreat deal. Each gauge invariant operator- which is dual to a bulk mode- gives rise to a towerof harmonic oscillators. The response of the gauge theory can be understood from the responseof these oscillators.In fact in the quadratic approximation the only oscillators which are excited are those whichcouple directly to the dilaton and so we only have to discuss their dynamics. We have alreadydiscussed the operator dual to the dilaton in section § S symmetric and correspondingly the only modes of ˆ O which are excited are S symmetric.Here we denote these by ˆ O l =0 .In the Heisenberg picture ˆ O l =0 can be expanded in terms of time dependent modes, this isdual to the fact that the S symmetric dilaton can be expanded in terms of modes with differentradial and related time dependence in the bulk. One finds, as is discussed in Appendix C, thatonly even integer frequencies appear in the time dependence giving,ˆ O l =0 = N ∞ X n =1 F (2 n )[ A n e − i nt + A † n e i nt ] . (112)Here A n , A † n are canonically normalised creation and destruction operators satisfying therelations, [ A m , A n ] = [ A † m , A † n ] = 0 [ A m , A † n ] = δ m,n . (113)Their commutators with the gauge theory hamiltonian are[ H, A † n ] = (2 n ) A † n [ H, A n ] = − (2 n ) A n (114)The normalization factor F (2 n ) may be computed by comparing with the standard the 2-pointfunction as is detailed in Appendix C. The result is | F (2 n ) | = Aπ n ( n −
1) (115)for n ≥ F (0) and F (2) vanish, so this means that the sum in eq.(112) receives its firstcontribution at n = 2. It also means that the lowest energy state which can be created byacting with ˆ O l =0 on the vacuum has energy equal to 4. This is what we expect on generalgrounds, since the energies of states created by an operator with conformal dimension ∆ aregiven by ω ( n, l ) = ∆ + 2 n + l ( l + 2) n = 0 , , · · · (116)30he constant A in eq.(115) is the normalization of the 2-point function which may bedetermined e.g. from a bulk calculation. Before proceeding let us also note that F (2 n ) growslike F (2 n ) ∼ n , eq.(115), for large mode number n . This enhances the coupling of the higherfrequency modes to the dilaton and will be important in our discussion of renormalisationbelow.From now onwards we will find it convenient to work in the Schrodinger representation, inwhich operators are time independent. The operator ˆ O l =0 in this representation is given by,ˆ O l =0 = N X n F (2 n )[ A n + A † n ] . (117)From eq.(114) it follows that the Hamiltonian for A n , A † n modes can be written as, H = X n nA † n A n . (118)Note this Hamiltonian measures the energy above that of the ground state.The operators, A † n , A n create and destroy a single quantum of excitation when acting onthe vaccum of the N = 4 theory with the instantaneous value of g Y M = e Φ . Thus they arethe analogue of the shifted creation and destruction operators we had defined in the harmonicoscillator case, ˜ a, ˜ a † . The Hamiltonian, eq.(118), is the analogue of the Hamiltonian, eq.(95) inthe harmonic oscillator case.The time dependence of the Hamiltonian due to the varying dilaton can be expressed asfollows, ∂H∂t = ∂H∂ Φ ˙Φ = − ˆ O l =0 ˙Φ (119)leading to, ∂H∂t = − ˆ O l =0 ˙Φ = − N X n F (2 n )[ A n + A † n ] ˙Φ , (120)where we have used eq.(117). It is useful to write this as ∂H∂t = − N X F (2 n ) √ n ˙Φ [ A n + A † n √ n ] , (121)which is analogous to the time dependence in the forced oscillator system, eq.(109).So we see that the gauge theory, in the quadratic approximation maps to a tower of os-cillators, with frequencies, ω n = 2 n . Comparing with eq.(109) we see that the oscillator withenergy 2 n couples to a source, ˙ J n = − N F (2 n ) √ n ˙Φ . (122)The analysis of the harmonic oscillator now directly applies. The resulting state is a coherentstate, | ψ > = ˆ N ( t ) e ( P n λ n A † n ) | φ > . (123)31ere | φ > is the adiabatic vacuum, which in is the ground state of the N = 4 theory withcoupling g Y M = e Φ . ˆ N ( t ) is a normalisation constant and the coherent state parameter λ n isgiven from eq.(99) by, λ n = e − iω n t q ω n Z t −∞ ˙ J n ( t ′ ) e iω n t ′ dt ′ . (124)The condition that the source is varying slowly, eq.(102), becomes, | ¨Φ n ˙Φ | ≪ ∀ n. (125)It is clearly sufficient to satisfy this condition for n = 1, | ¨Φ ˙Φ | ∼ ǫ ≪ . (126)This condition is met for the dilaton profile we have under consideration . When thiscondition is true λ n can be evaluated by keeping the first term in eq.(101). The condition thatthe state is classical, is that λ n ≫
1, this gives , | N F (2 n ) √ n ˙Φ | ≫ (2 n ) / . (127)Noting from eq.(115) that F (2 n ) ∼ n for large n we see that the factors of n cancell outon both sides, leading to the conclusion that when, | N ˙Φ | ∼ N ǫ ≫ ǫ ≪ , N ǫ ≫ ǫ as a system of harmonicoscillators. The oscillators which couple to the dilaton are excited by it and are in a classicalstate.This description can be used to calculate the resulting expectation value of operators. Thecalculation for < A n + A † n √ n > is analogous to that for < X + Jω > in the harmonic oscillator This condition is analogous to eq.(84) for the driven harmonic oscillator. As discussed in that context inthe footnote before eq.(84) there is a more precise version of this condition. It is the statement that for allmodes, n , the fourier transform of J n must have essentially all its support at frequencies much smaller than theoscillator frequency, 2 n . The more precise condition is simply that λ n ≫ , ∀ n . This gives, eq.(127) provided that the integral ineq.(124) can be approximated by the first term of the derivative approximation. A n , A † n are analogous to the shifted operators, ˜ a, ˜ a † eq.(94)). From eq.(106)and eq.(122) we get that to leading order in ǫ , < A n + A † n √ n > = − N F (2 n ) √ n (2 n ) ¨Φ . (130)Substituting in eq.(117) next gives, < ˆ O l =0 > = − CN ¨Φ (131)where C is C = X F (2 n ) n . (132)The functional dependence on Φ and N in eq.(130) agrees with what we found in the super-gravity calculation, eq.(62). The constant of proportionality C is in fact quadratically divergent.This follows from noting that for large n , F (2 n ) ∼ n .A little thought tells us that the divergence should in fact have been expected. The super-gravity calculation also had a divergence and the finite answer in eq.(62) was obtained onlyafter regulating this divergence and renormalising. Therefore it is only to be expected that asimilar divergence will also appear in the description in terms of the oscillators. In the subsec-tion which follows we will discuss the issue of renormalisation in more detail. The bottom lineis that counter terms can be chosen so that the coefficient in eq.(62) agrees with that in thesupergravity calculation.It is also important to discuss how the energy behaves. From eq.(107) and eq.(122) we seethat the energy above the ground state is < E > − E gnd = 12 CN ˙Φ (133)We note that the functional dependence on ˙Φ , N match with those obtained in the supergravitycalculations, eq.(60). The constant of proportionality which is obtained by summing over theoscillator modes in the case of the energy is the same as C defined above, eq.(132). It is alsotherefore quadratically divergent.The fact that the two constants of proportionality in eq.(133) and eq.(131) are the samefollows on general grounds. Noether’s argument in the presence of the time dependence meansthat each oscillator satisfies the relation, eq.(108). On summing over all of them we then getthe relation < dEdt > = − ˙Φ < ˆ O l =0 > (134)leading to the equality of the two constants. Earlier we had also seen that the supergravitycalculation satisfies this relation, eq.(63). It follows from these observations that if after renor-malisation the answer for < ˆ O l =0 > agrees between the supergravity theory and the oscillatordescription developed here, then the expectation value for E will also agree in the two cases.33ere we have analysed the gauge theory to leading order in ǫ . Going to higher orders in-troduces anharmonic couplings between the different oscillators. These couplings arise becauseof connected three-point and higher point correlations in the gauge theory. The three pointfunction for example is suppressed by 1 /N , the four point function by 1 /N and so on. Forcomputations in the ground state these would therefore be suppressed in the large N limit.However as we have seen here the time dependence results in a coherent state which contains O ( N ǫ ) quanta being produced. The 3- pt function in such a state is suppressed by O ( ǫ ) andnot by O (1 /N ). Since ǫ ≪
1, this is still enough though to justify our neglect of the cubic termsto leading order in ǫ . Similarly the effect of 4-pt correlators in the coherent state are suppressedby O ( ǫ ) etc. This is in agreement with the supergravity calculation, where the cubic terms inthe equations of motion are suppressed by O ( ǫ ) etc.To go to higher orders in ǫ using the oscillator description the effect of the anharmoniccouplings induced by the higher order correlations would have to be introduced. In addition onewould have to keep the contributions from the quadratic approximation to the required order in ǫ . As long as the ’tHooft coupling stays big for all times and the supergravity approximation isvalid, there is no reason to believe that these effects will be significant and the behaviour of thesystem should be well described by the leading harmonic oscillator description, in agreementwith what we saw in supergravity. When the ’tHooft coupling begins to get small though theanharmonic couplings could potentially significantly change the behaviour of the system, as wewill discuss in section 6.4. Let us now return to the constant C eq.(132). One would like to know if it can be made toagree with the supergravity answer eq.(62). Since the mode sum in C diverges, at first sightit would seem that by suitably removing the infinities this can always be done. To be explicit,imposing a cutoff on the mode sum in C one gets from eq.(132), C = X F (2 n ) n = c n max + c ln( n max ) + finite term (135)(A term linear in n max can always be removed by shifting n max ). Removing the infinities wouldmean removing the first two terms, but by changing n max by a finite amount the finite termleft over will clearly change and can be made equal to any answer we want.However this seems too superficial an answer. One would like to ensure that the freedom toadjust C corresponds to the freedom to add local counterterms in the theory, and also that oncethe counter terms are chosen so that C agrees no other discrepancy appears with supergravity.This is in fact true and can be easily seen by relating the calculation for < ˆ O > in eq.(131)to the two-point function for the dilaton. In fact we will only need the two point function of the34-wave dilaton which is equal to the two-point function of < ˆ O l =0 ˆ O l =0 > in the gauge theory.Since the S-wave dilaton couples directly to ˆ O l =0 , we have < ˆ O l =0 ( t ) > = Z dt ′ < ˆ O l =0 ( t ) ˆ O l =0 ( t ′ ) > Φ( t ′ ) (136)Using eq.(112) we find that < ˆ O l =0 ( t ) ˆ O l =0 ( t ′ ) > = N X n F (2 n ) (4 n ) Z dω πi e − i ( t − t ′ ) ω ( ω − (2 n ) ) (137)where we have expressed the answer in terms of a fourier transform in frequency space. Weare not being explicit about the pole prescription here, this will determine which propagator(Feynman, Retarded etc) one requires. From eq.(137) the propagator in frequency space canbe read off to be, G ( ω ) = N X n F (2 n ) (4 n )( ω − (2 n ) ) (138)Since F (2 n ) ∼ n the sum over modes on the rhs is quartically divergent.For purposes of comparing with the adiabatic approximation we expand this propagator inpowers in ω . This gives, G ( ω ) N = − X F (2 n ) (4 n )(2 n ) − ω X F (2 n ) (4 n )((2 n ) ) − ω X F (2 n ) (4 n )((2 n ) ) + · · · (139)The terms within the ellipses contain powers higher than ω and are not divergent. The firstterm on the rhs must be set to zero after renormalisation to preserve conformal invariance,otherwise the vacuum expectation value for < ˆ O > in the N = 4 theory with constant couplingwould not vanish. The leading contribution to < ˆ O > in the adiabatic approximation thenarises from the second term which is quadratically divergent. After fourier transforming the ω dependence of this term gives rise to the second derivative with respect to the time of thedilaton. And the sum over modes is the same as that in C , eq.(132).Now the point is that all divergences in the two-point function can be removed by localcounterterms since they correspond to contact terms. In fact the gravity calculation also neededcounterterms and from our discussion in § ∇ Φ) cancelsthe quadratic divergence while the last term in eq.(55), a (4) , contains terms which cancel thesubleading logarithmic divergence. Also once the counter terms are chosen so that C agreesno other discrepancy can appear. The point here is that the leading order in ǫ calculations areonly sensitive to the two-point function. And the finite terms in the two-point function arewell known to agree between the gravity and gauge theory sides. In fact the finite two pointfunction is just determined by conformal invariance and since the anomalous dimension of ˆ O does not get renormalised, it can be calculated in the free field limit itself.35he bottom line then is that using the freedom to adjust the counter terms, C can be madeto agree with the supergravity calculations in § C , eq.(62) is, C sugra = 116 (140)which means that the effect of renormalisation is to only include the contributions of modeswith mode number n ∼ O (1). This makes good physical sense, we are dealing with the lowfrequency response of the system here, and the high frequency modes should not be relevantfor this purpose.This last comment also has a bearing on our discussion in § | φ n > = A † n | > containing any one single oscillator excitation is small. However there are aninfinite number of such single excitation states, corresponding to the infinite number of valuesthat n takes, and one might be worried that this condition is not sufficient. Even though theamplitude to excite the system into any given state | φ n > is small the sum of these amplitudes,more correctly the norm of the first order correction of the wave function < ψ | ψ > , eq.(69), isstill be large and in fact would diverge when summed over all the modes. This would invalidatethe approximation. The reason this concern does not arise is tied to our discussion above.After renormalisation only a few low frequency modes contribute to the response of the systemand one is only interested in how the wave function changes for these modes. For this purposethe condition in eq.(74) is enough and we see that when it is met the quantum adiabaticapproximation is indeed valid. So far we have considered what happens in the parametric regime, eq.(129), when the ’tHooftcoupling stays big all times. In this case the supergravity description is always valid. Wesaw above that the gauge theory can be described in this regime in terms of approximatelydecoupled classical harmonic oscillators and this reproduces the supergravity results.Now let us consider what happens when the dilaton takes a larger excursion so that the’tHooft coupling at intermediate times becomes of order unity or even smaller. Some of theresulting discussion is already contained in the introduction above.A natural expectation is that description in terms of classical adiabatic system of weaklycoupled oscillators should continue to apply even when the ’tHooft coupling becomes small.There are several reasons to believe this. First, anharmonic terms continue to be of order ǫ andthus are small. The leading anharmonic terms arise from three -point correlations, < ˆ O ˆ O ˆ O > .36n the vaccum these go like 1 /N . In the coherent state produced by the time dependence thesego like ǫ . The enhancement by N ǫ arises because the coherent state contains O (( N ǫ ) ) quanta,so that the probability goes as ( N ǫ ) /N ∼ ǫ . Four-point functions give rise to terms goinglike O ( ǫ ) and so on, these are even smaller. In the absence of anharmonic terms the theoryshould reduce to a system of oscillators. Second, the existence of a gap of order 1 /R meansthat for each oscillator the time dependence is slow compared to its frequency. Therefore thesystem continues to be very far from resonance and should evolve adiabatically. Finally, in theparametric regime, eq.(129) the analysis of the previous subsections should then apply leadingto the conclusion that an O ( N ǫ ) ≫ N = 4 theory in the far future and should have a good description in terms of smoothAdS space then.However, as discussed in the introduction, there are reasons to worry that this expectationis not borne out. New features could enter the dynamics when the ’tHooft coupling becomessmall at intermediate times, and these could change the qualitative behaviour of the system.These new features have to do with the fact that string modes can start getting excited in thebulk when the curvature becomes of order the string scale. These modes correspond to non-chiral operators in the gauge theory and the corresponding oscillators have a time dependentfrequency. When the ’tHooft coupling is big these frequencies are much bigger than those of thesupergravity modes and as a result the string mode oscillators are not excited. But when the’tHooft coupling becomes of order unity some of the frequencies of these string modes becomeof order the supergravity modes and hence these oscillators can begin to get excited . In factthe string modes are many more in number than the supergravity modes, since there are anorder unity worth of chiral operators in the gauge theory and an O ( N ) worth of non-chiralones.The worry then is that if a significant fraction of these string oscillators get excited thecorrect picture which could describe the ensuing dynamics is one of thermalisation rather thanclassical adiabatic evolution. In this case the energy pumped into the system initially would getequipartitioned among all the different degrees of freedom. Subsequent evolution would thenbe dissipative, and the energy would increases in a monotonic manner, as it does for a largeblack hole, eq.(65).Due to the dissipative behaviour the energy which is initially pumped in would not be The probability | < φ | ˆ O ˆ O ˆ O| φ > | is proportional to N ( N ǫ ) , with each factor of N ǫ as an estimateof the contribution for each of the operators ˆ O . The contribution of the 2-pt function | < φ | ˆ O ˆ O| φ > | is justproportional to ( N ǫ ) , resulting in a relative suppression of O ( ǫ ). The primary reason for them getting excited are the anharmonic terms which couple them to the modesdual to the dilaton. N ǫ remains in the system. The gravity descriptionof the resulting thermalized state depends on the value of ǫ relative to λ ≡ g Y M N and N .In this late time regime of large ’t Hooft coupling, the various possibilities can be figuredout from entropic considerations in supergravity ( see e.g. section 3.4 of [6]). The resultin our case is the following. For ǫ ≪ ( g Y M N ) / /N a gas of supergravity modes is favored.For ( g Y M N ) / /N < ǫ ≪ ( g Y M N ) − / one would have a gas of massive string modes. For( g Y M N ) − / < ǫ ≪ R AdS . A big black hole requires O ( N ) energy which is parametrically much larger. Thus,the strongest departure from AdS space-time in the far future would be presence of small blackholes. Such black holes would eventually evaporate by emitting Hawking radiation. Howeverthis takes an O ( N R AdS ) amount of time which is much longer than the time scale O ( R AdS /ǫ )on which the ’tHooft coupling evolves. As a result for a long time after the ’tHooft couplinghas become big again the gravity description would be that of a small black hole in AdS space.An important complication in deciding between these two possibilities is that the rate of timevariation is ǫ which is also the strength of the anharmonic couplings between the supergravityoscillators and string oscillators. If the rate of time variation could have been made muchsmaller, thermodynamics would become a good guide for how the system evolves. In themicrocanonical ensemble, which is the correct one to use for our purpose, with energy N ǫ theentropically dominant configurations are as discussed in the previous paragraph, and this wouldsuggest that dissipation would indeed set in. However, as emphasised above this conclusion isfar from obvious here since the time variation is parametrically identical to the strength of theanharmonic couplings.In fact we know that the guidance from thermodynamics is misleading in the supergravityregime, where the ’tHooft coupling stays large for all times. In this case we have explicitlyfound the solution in §
2. It does not contain a black hole. Moreover, it does not suffer fromany tachyonic instability - since it is a small correction from AdS space which does not haveany tachyonic instability . The only way a black hole could form is due to a tunneling processbut this would be highly suppressed in the supergravity regime.One reason for this suppression is that the energy in the supergravity solution discussed in § R AdS . This energy would haveto be concentrated in much smaller region of order the small black hole’s horizon to form theblack hole and this is difficult to do. In contrast, away from the supergravity regime this couldhappen more easily. When the ’tHooft coupling becomes small at intermediate times, stringsbecome large and floppy, of order R AdS , at intermediate times. If a significant fraction of the Note that we are working on S here. AdS space, with the possible presence of a small blackhole. Hopefully, the framework developed here will be useful to think about this issue further.
In this paper we examined the behaviour of the
AdS × S solution of IIB supergravity when itis subjected to a time dependent boundary dilaton. This is dual to the behaviour of the N = 4Super Yang -Mills theory subjected to a time dependent gauge coupling. The AdS solutionwas studied in global coordinates and the dual field theory lives on an S of fixed radius R .We worked in units where R AdS = R = 1. Three parameters are relevant for describing theresulting dynamics:1. N - which is the number of units of flux and is dual to the rank of the gauge group. Thiswas held fixed during the evolution.2. λ = e Φ( t ) N - which determines the value of R AdS in string units is the ’tHooft coupling inthe gauge theory. Especially relevant is its minimum value λ min during the time evolution.When λ min ≫ λ min ≤ O (1)supergravity breaks down at intermediate times.3. ǫ ∼ ˙Φ - which determines the rate of change of the boundary dilaton in units of R AdS .Throughout the analysis we worked in the slowly varying regime where ǫ ≪ • When
N ǫ ≪ AdS spacetime.This is true even when λ min ≤ § • When
N ǫ ≫ λ min ≫
1, the system is well described by a supergravity solution,which consists of
AdS spacetime with corrections which are suppressed in ǫ . The gaugetheory provides an alternate description in terms of weakly coupled harmonic oscillatorswhich are modes of gauge invariant operators dual to supergravity modes. These oscilla-tors are subjected to a driving force that is slowly varying compared to their frequency.39 classical adiabatic perturbation theory, the LNCAPT, describes the dynamics of thesystem. This dual description reproduces the supergravity answers for the energy and < ˆ O > , as discussed in § § • When
N ǫ ≫
1, and λ min ≤ O (1), supergravity breaks down. In this case we do nothave a clean conclusion for the final state of the system. Additional oscillators whichcorrespond to string modes can now get activated. There are two possibilities : either thedescription in terms of classical adiabatic dynamics for the oscillators continues to apply,or a qualitative new feature of thermalisation sets in. In the former case spacetime inthe far future is well approximated by smooth AdS space. In the latter case the gravitydescription depends on the value of ǫ and may consist of a string gas or small black holes.This is discussed in § • We have not addressed here what happens when the dilaton begins to vary more rapidlyand ǫ becomes ∼ O (1). It is natural to speculate that a black hole forms eventually inthis case. The oscillators in the gauge theory now become strongly coupled with O (1)anharmonic couplings.If λ min ≫ ǫ ≪ § ǫ increases the natural expectationis that eventually a black hole should begin to form at some critical value. The size of thisblack hole should then grow with ǫ , leading to a big black hole with radius bigger thanAdS scale. Very preliminary indications for this come from the calculations in § ǫ increases the value of | g tt | becomes smaller at the center of AdS eq.(51),suggesting that a horizon would eventually form at ǫ ∼ O (1). Better evidence comesfrom studying a region of parameter space where ǫ ≫ one finds that a boundary variation ofthe dilaton, which is sufficiently fast compared to its amplitude, always produces a blackhole.When λ min ≤ O (1), and ǫ becomes ∼ O (1), supergravity breaks down at intermediatetimes. If thermalisation has already set in in the parametric regime, N ǫ ≫ , ǫ ≪
1, asdiscussed above, then one expects that the small black hole which has formed for ǫ ≪ ǫ ≥ O (1). If thermalisationdoes not set in when ǫ ≪
1, then at some critical value ǫ ∼ O (1) one would expect thatthis does happen leading to the formation of a black hole whose mass then grows as ǫ further increases.It will be interesting to try and analyse this regime further in subsequent work. The results reported in [20] are for the case of
AdS d +1 spacetimes with d odd. Finally one can consider a regime where ǫ → ∞ at time t →
0. This regime was consideredin [10] where the dilaton was taken to vanishes like e Φ ∼ ( t ) p as t → , leading to adiverging value for ˙Φ. In a toy quantum mechanics model it was argued that the responseof the system in this case is singular, suggesting that this singularity is a genuine pathologywhich is not smoothened out. However the conclusions for the toy model do not directlyapply to the field theory. Important questions regarding the renormalisation of this timedependent field theory remain and could invalidate this conclusion.One is hesitant to try and draw general conclusions about the possibility of emergence ofa smooth spacetime from string scale curved regions on the basis of the very limited analysispresented here. One lesson which has emerged is that, at least for the kind of time dependencestudied in this paper, AdS space has a tendency to form a black hole . This fate can be avoided(as in the case when N ǫ ≪
1) but it requires slow time variation or perhaps more generallyrather finally tuned conditions. To understand in greater detail when this fate of black holeformation can be avoided requires a deeper understanding of the process of thermalisation inthe dual field theory.In this paper we analysed the effects of a time dependent dilaton. It will be interestingto extend this to other supergravity modes as well by making their boundary values timedependent - e.g, making the radius of the S on which the gauge theory lives time dependentor introducing time dependence along the other exactly flat directions in the N = 4 theorybesides the dilaton. Also, we have kept the parameter N fixed in this work. As was discussedin the introduction N measures the strength of quantum corrections and is also the value of R AdS in Planck units eq.(3). It would be interesting to consider cases where N changes andbecome smaller thereby increasing the strength of quantum effects and making the curvature oforder l P l . One way to do this might be by introducing time dependence that moves the systemonto the Coulomb branch. This could reduce the effective value of N in the interior. For recentinteresting work see, [32], also the related earlier work, [33], [34]. Finally, a length scale wasintroduced in the gauge theory by working on S here. Instead one could consider a confininggauge theory like the Klebanov-Strassler kind , [35], which has a mass gap on R . In thiscase one could consider the response of the system to time dependence slow compared to theconfining scale and hope to use an adiabatic approximation to understand this response. AdS space is of course homogeneous so the reader might be puzzled about where the black hole forms.The point is that the time dependence imposed on the boundary picks out a particular notion of time and theblack hole forms where the redshift factor for this time is smallest, this is the “center of AdS space” in globalcoordinates. We thank M. Mulligan for a related discussion. Acknowledgments
We would like to thank Ian Ellwood, Gary Horowitz, Shamit Kachru, Per Kraus, Steve Shenker,Eva Silverstein, Spenta Wadia and especially Shiraz Minwalla for discussions, and K. Narayanfor discussions and collaboration at the early stages of this work. The work of A.A. is par-tially supported by ICTP grant Proj-30 and the Egyptian Academy for Scientific Researchand Technology. A.A. and A.G. would like to thank the Chennai Mathematical Institute (andthe organizers of Indian Strings Meeting 2008) for hospitality. S.R.D. would like to thank theInternational Center for Theoretical Sciences, Tata Institute of Fundamental Research and theorganizers of “String Theory and Fundamental Physics” for hospitality. The work of S.R.D,A.G and J.O is supported in part by a National Science Foundation (USA) grant PHY-0555444.S.T. thanks the organisers of the Monsoon Workshop, at TIFR, for the stimulating meetingduring which some of this work was initiated. S.T. is on a sabbatical visit to Stanford Uni-versity and SLAC National Accelerator Laboratory for the period Oct. 2008-Sept. 2009 andthanks his hosts for their kind hospitality and support. Most of all he thanks the people ofIndia for generously supporting research in string theory.
A Comments on Metric to O ( ǫ ) We are interested in calculating the back reaction on the metric to O ( ǫ ) that arises due tothe dilaton Φ . Without loss of generality we can assume that the metric is S symmetric andtherefore of form, ds = − g tt dt + g rr dr + 2 g tr dtdr + R d Ω (141)where the metric coefficients are functions of r, t . The zeroth order metric is that of AdS ,eq.(7). We argued above that the backreaction to the dilaton source arises at order ǫ . Thus g tr in eq.(141) is of order ǫ .We now show that by doing a suitable coordinate transformation, the mixed component g tr can be set to vanish up to order ǫ . The coordinate transformation is, from ( t, r ) to ( t, ˜ r ),where, r = ˜ r − g tr g rr t, (142)which leads to dr = d ˜ r − ( g tr g rr ) ′ td ˜ r − g tr g rr dt + O ( ǫ ) . (143)Prime above indicates derivatives with respect to r , We can drop the ǫ terms for our purpose,these originate from additional time derivatives on the metric components. Substituting ineq.(141) we see that in the new coordinates the g t ˜ r components of the metric vanish upto O ( ǫ )42orrections which we are neglecting anyways. To avoid clutter we will henceforth drop the tildeon the r coordinate and write the metric as ds = − g tt dt + g rr dr + R d Ω (144)Next we show that up to O ( ǫ ) we can set R equal to the coordinate r without reintroducingthe mixed components. First define, ¯ r = R (145)leading to, d ¯ r = R ′ dr + ˙ Rdt (146)where dot indicates a time derivative. Now any time dependence in R arises only due to thedilaton and therefore is of order ǫ . This means that ˙ R is O ( ǫ ) and can be neglected. So upto O ( ǫ ) no mixed components arise in the metric due to this coordinate transformation. Wenow drop the bar on the radial coordinate and write the final metric as, ds = − g tt dt + g rr dr + r d Ω . (147) B More on the Driven Harmonic Oscillator
In this appendix we provide the steps leading to (98) and (99). The time derivative of the statevector | ψ ( t ) > in (97) is i ∂∂t | ψ ( t ) > = i ( ˙ λ + ˙ α ) a † | ψ ( t ) > + i ( ˙ NN + ˙ N α N α ) | ψ ( t ) > (148)where we have used the expression for | φ > in (90). The action of the hamiltonian H on thestate is easily obtained by noting that[ H, e λa † ] = ω λa † + J λ √ ω ! e λa † . (149)This leads to H | ψ ( t ) > = ω λa † + J λ √ ω ! | ψ ( t ) > + ω | ψ ( t ) > . (150)It may easily be checked that the states | ψ ( t ) > and a † | ψ ( t ) > are linearly independent.Equating the coefficients of a † | ψ ( t ) > in eq.(148) and (150) and using eq.(92) then leads toeq.(98). Equating the coefficients of | ψ ( t ) > in eq.(148) and (150) gives an equation thatdetermines N ( t ). Note that | N ( t ) | is determined directly from the requirement that < ψ | ψ > =1. 43 The normalization factor F (2 n ) In computing the normalization F (2 n ) in (115) it is best to first continue to euclidean signatureand then perform a conformal transformation from R × S to R . The radial coordinate on the R is given by r = e τ . where τ is the euclidean time in R × S . Then the Heisenberg pictureoperator on R is given by ˆ O l =0 = ∞ X m = −∞ O m r m +4 (151)The factor of r m +4 in the denominator reflects the fact that the operator ˆ O l =0 has dimension4. The conformally invariant vacuum satisfies O m | > = 0 m ≥ − < |O m = 0 m ≤ < ˆ O l =0 ( r ) ˆ O l =0 ( r ′ ) > = ∞ X m =4 − X n = −∞ < |O m O n | >r m +4 ( r ′ ) n +4 (153)The 2 point function only involves the central term in the operator algebra. This means wecan write O m = N F ( m ) A m ( m > O − m = N F ⋆ ( m ) A † m ( m >
0) (154)where the operators A m , A † satisfies an operator algebra and F ( m ) is a normalization[ A m , A n ] = [ A † m , A † n ] = 0 [ A m , A † n ] = δ mn (155)Note that because of (153) only terms for n ≥ < ˆ O l =0 ( r ) ˆ O l =0 ( r ′ ) > = N r ∞ X m =4 | F ( m ) | r ′ r ! m − (156)On the other hand since the dimension of the operator ˆ O Φ ( r, Ω ) is 4 we know the 2 pointfunction on R . This is given by < ˆ O ( r, Ω ) ˆ O ( r ′ , Ω ′ ) > = AN | ~r − ~r ′ | (157)where A is a order one numerical constant. Here ~r = ( r, Ω) etc., is the location of the operatoron R . Integrating over Ω , Ω ′ we get Z d Ω Z d Ω ′ < ˆ O ( r, Ω ) ˆ O ( r ′ , Ω ′ ) > = AN (8 π ) Z π sin θ dθ ( r + ( r ′ ) − rr ′ cos θ ) (158)44he integral can be performed. The result is, for r > r ′ Z d Ω Z d Ω ′ < ˆ O ( r, Ω ) ˆ O ( r ′ , Ω ′ ) > = N Aπ r (cid:16) r ′ r (cid:17) + 1(1 − (cid:16) r ′ r (cid:17) ) (159)Using the power series expansion1 + x (1 − x ) = ∞ X m =0
112 ( m + 1)( m + 2) ( m + 3) x m (160)we finally get Z d Ω Z d Ω ′ < ˆ O ( r, Ω ) ˆ O ( r ′ , Ω ′ ) > = N Aπ r ∞ X m =0 ( m + 1)( m + 2) ( m + 3) r ′ r ! m (161)The result clearly shows that only operators with even mode numbers are present in the ex-pansion (151). Comparing (161) and (156) we get F (2 m + 1) = 0 | F (2 m ) | = Aπ m ( m −
1) (162)which is the result in equation (115).
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