Relaxation in conformal field theory, Hawking-Page transition, and quasinormal or normal modes

In (1+1)-dimensional conformal field theory at finite temperature, when the inverse temperature is large, relaxation is oscillatory with characteristic period set by the system size.

“On the other hand, when the inverse temperature is large, the relaxation is oscillatory with characteristic period set by the size of the system.”

In the strongly coupled large k conformal field theory dual to string theory on (2+1)-dimensional anti-de Sitter spacetime, in the k = ∞ limit there is a sharp transition between exponential decay and oscillatory regimes.

“In the latter case, there is a sharp transition between the two regimes in the k = ∞ limit”

In (1+1)-dimensional conformal field theory at finite temperature, when the system size is much larger than the inverse temperature, perturbations decay exponentially with time.

“When the size of the system is much larger than the inverse temperature, perturbations decay exponentially with time.”

The Fourier transform of the spectral function for (23) is again given by (10), with k restricted to the discrete values 2π/L m.

“It is clear that the Fourier transform of the spectral function for (23) is again given by (10), with k restricted to the discrete values 2 π L m .”

For {h,hbar} ∈ 1/2 N, the Fourier transform of the commutator ρ has an infinite set of simple poles on either side of the real line.

“From (10), we see that ρ ( ω, k ) has an infinite set of simple poles on either side of the real line.”

It represents a periodic sequence of singular peaks at t ± σ = nL.

“Clearly, it represents a periodic sequence of singular peaks at t ± σ = nL .”

In both cases, the boundary of the three-dimensional space is a torus with periods L and β.

“In both cases, the boundary of the three-dimensional space is a torus with periods L and β .”

For free fermions on the torus, the real time correlation function is a periodic function of t with period L.

“The resulting real time correlator is thus a periodic function of t with period L .”

The correction to the leading behaviour of the free fermion correlation function is governed by e^{-πL/β} e^{π(t+σ)/β}.

“we note that the correction to the leading behaviour is governed by e -πL/β e π ( t + σ ) /β .”

The transition between the two regimes occurs at β = L.

“The transition between the two regimes occurs at β = L .”

The spectral function for higher values of h takes a form similar to (14), with additional prefactors such that the pole structure is given by (15).

“The spectral function for higher values of h takes a form similar to (14), with additional prefactors designed such that the pole structure is given by (15).”

The poles of D_R are given by (11) with k = 2π/L m.

“Thus, the poles of D R are given by (11) with k = 2 π L m .”

The thermal AdS correlator (24) is periodic in t with period L.

“which is periodic in t with period L .”

In the limit L/β → ∞, the free fermion correlation function (16) approaches (9), which exponentially decays with time.

“In the limit L/β →∞ , the correlation function (16) approaches (9) (actually the left-moving part of (9) with h = 1 / 2) which exponentially decays with time.”

The two terms in (20) have drastically different behaviour as functions of real time.

“The two terms in (20) have drastically different behaviour as functions of real time.”

In the opposite limit L/β → 0, the free fermion correlation function (16) approaches the oscillating function (12).

“In the opposite limit, when L/β → 0, it approaches the oscillating function (12).”

When size L is finite and inverse temperature β is taken to infinity, the real time correlation function shows oscillatory behaviour.

“This clearly shows the expected oscillatory behaviour in real time.”

When temperature is finite and size L is taken to infinity, the real time correlation function decays exponentially for large time t; for example, D±(t,0) ∼ e^{-2πT(h+hbar)t} when T_L = T_R = T.

“Clearly, the correlation function decays exponentially for large time t ; for example, D ± ( t, 0) ∼ e -2 πT ( h +¯ h ) t , when T L = T R = T .”

For real time t, the free fermion correlation function (16) is a sequence of singular peaks located at (t+σ) = nL.

“Therefore, for real time t , the correlation function (16) is a sequence of singular peaks located at ( t + σ ) = nL .”

In the strict k = ∞ limit, the sole contribution to the correlator is given by the BTZ black hole.

“In the strict k = ∞ limit, the sole contribution to the correlator is given by the BTZ black hole.”

This is exactly the behaviour seen in the free fermion correlation function (16): at finite L it is oscillating with period L.

“This is exactly the behaviour seen in the free fermion correlation function (16): at finite L it is oscilating with period L .”

The inclusion of the contribution from thermal AdS gives a periodic contribution, whose frequencies are the same as the normal modes of AdS.

“We found that this gives a periodic contribution, whose frequencies are the same as the normal modes of AdS.”

Finite temperature introduces a new scale, so conformal Ward identities no longer determine the Green's functions completely.

“However, finite temperature introduces a new scale and consequently the conformal Ward identities no longer determine the Green's functions completely.”

The behaviour of the bulk AdS perturbation is mirrored precisely by the behaviour of the linear response of the CFT.

“Thus, the behaviour of the bulk AdS perturbation is mirrored precisely by the behaviour of the linear response of the CFT.”

The correspondence between retarded CFT correlators and quasinormal modes was verified explicitly in the high temperature regime of the 2-dimensional CFT dual to supergravity on AdS3.

“This correspondence was verified explicitly in the high temperature regime of the 2-dimensional CFT, dual to supergravity on AdS 3 [11].”

• cites [11] [11] D. Birmingham, I. Sachs and S.N. Solodukhin, Phys. Rev. Lett. 88 , 151301 (2002) [arXiv:hep-th/0112055]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The poles of retarded CFT correlators are given by the quasinormal modes in AdS.

“Furthermore, the poles of the retarded CFT correlators are given by the quasinormal modes in AdS [8, 11].”

• cites [8] [8] U. H. Danielsson, E. Keski-Vakkuri and M. Kruczenski, Nucl. Phys. B 563 (1999) 279 [arXiv:hep-th/9905227]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [11] [11] D. Birmingham, I. Sachs and S.N. Solodukhin, Phys. Rev. Lett. 88 , 151301 (2002) [arXiv:hep-th/0112055]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

There is a direct connection between quasinormal and normal modes of the gravity system and the decaying and oscillating behaviour of the conformal field theory.

“we establish a direct connection between quasinormal and normal modes of the gravity system, and the decaying and oscillating behaviour of the conformal field theory.”

For a system with discrete energy levels, the linear response will show oscillatory behaviour.

“Correspondingly, as can be seen from (6), the linear response will show oscillatory behaviour.”

If the energy levels of the system are discrete and the ensemble sum converges, then D_R(ω,k) is a meromorphic function of ω in the lower half plane, with simple poles on the real axis.

“If the energy levels of the system are discrete and the ensemble sum converges, then we conclude that D R ( ω, k ) is a meromorphic function of ω in the lower half plane, with simple poles on the real axis.”

The sharp transition in the k = ∞ limit is a manifestation of the gravitational Hawking-Page phase transition.

“which is a manifestation of the gravitational Hawking-Page phase transition.”

The AdS/CFT duality predicts that retarded CFT correlation functions are in 1-1 correspondence with Green's functions on anti-de Sitter space with appropriate boundary conditions.

“the AdS/CFT [4, 5, 6, 7] duality predicts that the retarded CFT correlation functions are in 1-1 correspondence with Green's functions on anti-de Sitter space with appropriate boundary conditions [8, 9, 10, 11, 12].”

• cites [8] [8] U. H. Danielsson, E. Keski-Vakkuri and M. Kruczenski, Nucl. Phys. B 563 (1999) 279 [arXiv:hep-th/9905227]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [9] [9] S. Kalyana Rama and B. Sathiapalan, Mod. Phys. Lett. A 14 (1999) 2635 [arXiv:hep-th/9905219]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [10] [10] G. T. Horowitz and V. E. Hubeny, Phys. Rev. D 62 (2000) 024027 [arXiv:hep-th/9909056]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [11] [11] D. Birmingham, I. Sachs and S.N. Solodukhin, Phys. Rev. Lett. 88 , 151301 (2002) [arXiv:hep-th/0112055]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [12] [12] D. T. Son and A. O. Starinets, JHEP 0209 (2002) 042 [arXiv:hep-th/0205051]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The regime of exponential decay is governed by the quasinormal modes of the BTZ black hole.

“the regime of exponential decay is governed by the quasinormal modes of the BTZ black hole”

The regime of oscillation is governed by the normal modes of thermal AdS.

“while the regime of oscillation is governed by the normal modes of thermal AdS.”

In two limiting cases, when the dimensionless parameter LT is infinite or zero, conformal invariance completely determines the correlation function, independent of the details of the theory.

“In two limiting cases, when the dimensionless parameter LT is infinite or zero, conformal invariance completely determines the correlation function, independent of the details of the theory in question.”

According to AdS/CFT, each supergravity perturbation Φ(m,s) of mass m and spin s propagating on AdS3 has a corresponding operator O(h,hbar) in the dual conformal field theory.

“According to this duality, each supergravity perturbation Φ ( m,s ) of mass m and spin s propagating on AdS 3 has a corresponding operator O ( h, ¯ h ) in the dual conformal field theory.”

Both the BTZ black hole and thermal AdS can be represented as a quotient of three-dimensional hyperbolic space H3.

“Both spaces can be represented [23] as a quotient of three-dimensional hyperbolic space H 3 , with line element”

• cites [23] [23] S. Carlip and C. Teitelboim, Phys. Rev. D 51 , 622 (1995) [arXiv:gr-qc/9405070]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

There is an SL(2,Z) family of AdS3 spaces which approach the torus asymptotically.

“There is an SL (2 , Z ) family of AdS 3 spaces which approach the torus asymptotically [15, 22].”

• cites [15] [15] J. Maldacena and A. Strominger, JHEP 9812 , 005 (1998) [arXiv:hep-th/9804085]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [22] [22] R. Dijkgraaf, J. Maldacena, G. Moore, and E. Verlinde, arXiv:hep-th/0005003. - sourceNot checked yet

  Not checked in the free run.

Depending on the ratio L/β, one of the two terms in (20) dominates.

“Depending on the ratio L/β , one of the two terms in (20) dominates [15].”

• cites [15] [15] J. Maldacena and A. Strominger, JHEP 9812 , 005 (1998) [arXiv:hep-th/9804085]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

In terms of gravitational physics, this corresponds to the Hawking-Page phase transition.

“In terms of the gravitational physics, this corresponds to the Hawking-Page phase transition [26].”

• cites [26] [26] S.W. Hawking and D.N. Page, Commun. Math. Phys. 87 , 577 (1983). -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

In [11], the correspondence was shown for L/β = ∞.

“In [11], the correspondence was shown for L/β = ∞ .”

• cites [11] [11] D. Birmingham, I. Sachs and S.N. Solodukhin, Phys. Rev. Lett. 88 , 151301 (2002) [arXiv:hep-th/0112055]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The Euclidean actions of the BTZ black hole and thermal AdS3 are given by (21).

“are the Euclidean actions of the BTZ black hole and thermal AdS 3 , respectively [15].”

• cites [15] [15] J. Maldacena and A. Strominger, JHEP 9812 , 005 (1998) [arXiv:hep-th/9804085]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The resolution of the apparent contradiction lies in the peculiar properties of the CFT under consideration.

“The resolution of this puzzle lies in the peculiar properties of the CFT under consideration [7].”

• cites [7] [7] O. Aharony, S.S. Gubser, J. Maldacena, H. Ooguri and Y. Oz, Phys. Rep. 323 , 183 (2000) [arXiv:hep-th/9905111]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The two contributions in (20) are dual to each other under interchange of β and L.

“We see that these contributions are dual to each other under interchange of β and L .”

The characteristic times for thermalisation of an instantaneous perturbation are determined by the imaginary part of the poles of D_R(ω,k) in the complex ω-plane.

“Eqn. (6) then shows that the characteristic times for the thermalisation of an instantaneous perturbation is determined by the imaginary part of the poles of D R ( ω, k ) in the complex ω -plane.”

Consequently, O still sees a finite volume.

“Consequently, O still sees a finite volume.”

This is a sharp transition in the limit k = ∞, which is the case when the supergravity description is valid.

“This is a sharp transition in the limit k = ∞ , which is the case when the supergravity description is valid.”

For a generic CFT, summing over images does not produce the correct finite volume correlator.

“Indeed, for a generic CFT, summing over images does not produce the correct finite volume correlator.”

The exponential decay of the free fermion correlation function in the limit of large L is consistent with its periodicity with period L.

“It is important to observe that the exponential decay of the function in the limit of large L is consistent with periodicity of the real time correlation function (16) with period L .”

It was shown in [11] that there is a 1-1 correspondence between the poles of the retarded finite temperature Green's function and the quasinormal modes for fields of spin s = h - hbar in the BTZ background.

“However, it was shown in [11] that there is a 1-1 correspondence between the poles of the retarded finite temperature Green's function and the quasinormal modes for fields of spin s = h -¯ h in the BTZ background.”

• cites [11] [11] D. Birmingham, I. Sachs and S.N. Solodukhin, Phys. Rev. Lett. 88 , 151301 (2002) [arXiv:hep-th/0112055]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

There is a correspondence between (quasi)normal modes in the gravitational background and the poles of the retarded Green's functions in the conformal field theory.

“In addition, there is a correspondence between (quasi)normal modes in the gravitational background and the poles of the retarded Green's functions in the conformal field theory [8].”

• cites [8] [8] U. H. Danielsson, E. Keski-Vakkuri and M. Kruczenski, Nucl. Phys. B 563 (1999) 279 [arXiv:hep-th/9905227]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The behaviour in the intermediate regime, when both L and T are finite, cannot be derived entirely from conformal symmetry.

“However, the behaviour in the intermediate regime, when both L and T are finite, cannot be derived entirely from conformal symmetry.”

On the supergravity side, the justification to sum over images comes from the fact that the correlator solves a Green's function equation.

“On the supergravity side, the justification to sum over images comes simply from the fact that the correlator solves a Green's function equation.”

The operator O does not distinguish between simply wound and multiply wound strings.

“The reason for this lies in the fact that the operator O does not distinguish between simply wound and multiply wound strings.”

• AssumptionRests on an assumption: The operator O does not distinguish between simply wound and multiply wound strings.

The supersymmetric conformal field theory dual to string theory on AdS3 describes the low energy excitations of a large number of D1- and D5-branes.

“This theory describes the low energy excitations of a large number of D1- and D5-branes [4, 7, 15].”

• cites [4] [4] J. M. Maldacena, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [arXiv:hep-th/9711200]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [7] [7] O. Aharony, S.S. Gubser, J. Maldacena, H. Ooguri and Y. Oz, Phys. Rep. 323 , 183 (2000) [arXiv:hep-th/9905111]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [15] [15] J. Maldacena and A. Strominger, JHEP 9812 , 005 (1998) [arXiv:hep-th/9804085]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The correlation function of the dual operators contains the sum of two contributions, given by (20).

“The correlation function of the dual operators then contains the sum of two contributions [14] as”

• cites [14] [14] J. Maldacena, arXiv:hep-th/0106112; and private communication. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

Using AdS/CFT, the correlation functions on the torus were computed for the case of a scalar field (h = hbar).

“Using the AdS/CFT correspondence, the correlation functions on the torus were computed in [24, 25] for the case of a scalar field ( h = ¯ h ).”

• cites [24] [24] E. Keski-Vakkuri, Phys. Rev. D 59 (1999) 104001 [arXiv:hep-th/9808037]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

• cites [25] [25] L. Chekhov, arXiv:hep-th/9811146; L. Chekhov, Mod. Phys. Lett. A 14 , 2157 (1999) [arXiv:hep-th/9903209]; -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The bulk perturbation decays via these quasinormal modes in precisely the same way as the linear response of the conformal field theory given by (6).

“Indeed, one finds that the bulk perturbation decays via these quasinormal modes in precisely the same way as the linear response of the conformal field theory given by (6).”

The limit of large L is meaningful only for times much smaller than L.

“Thus, the limit of large L is meaningful only for times much smaller than L .”

For understanding the Hawking-Page phase transition, it suffices to consider the BTZ black hole and thermal AdS space.

“For the purpose of understanding the Hawking-Page phase transition, it suffices to consider the BTZ black hole and thermal AdS space, corresponding to anti-de Sitter space filled with thermal radiation.”

The considerations so far are quite general, and rely only on the conformal properties of the operators driving the perturbation.

“The considerations so far are quite general, and rely only on the conformal properties of the operators driving the perturbation.”

For finite k, the correlator must ultimately exhibit Poincaré recurrence.

“However, for finite k the correlator must ultimately exhibit Poincar´ e recurrence.”

Typically, one expects the correlation function for a system with finite entropy to be a quasiperiodic function, with incommensurate frequencies.

“Typically, one expects the correlation function for a system with finite entropy to be a quasiperiodic function, with incommensurate frequencies.”

This explains why we found complex poles corresponding to quasinormal modes of the black hole.

“This explains why we found complex poles corresponding to quasinormal modes of the black hole.”

These recurrences will occur on a timescale of the order t ∼ e^S.

“These recurrences will occur on a timescale of the order t ∼ e S .”

This Hawking-Page transition is related to linear response theory in the dual CFT via quasinormal and normal modes in the two backgrounds.

“This Hawking-Page transition is then related to linear response theory in the dual CFT via quasinormal and normal modes in the two backgrounds.”

The remaining members of the SL(2,Z) family of solutions with torus boundary will be parametrically of the same order as the thermal AdS contribution.

“However, they will be parametrically of the same order as the thermal AdS contribution.”

The Hawking-Page transition is a transition between oscillatory behaviour at low temperature and exponentially decaying behaviour at high temperature.

“The Hawking-Page transition is thus a transition between oscillatory behaviour at low temperature and exponentially decaying behaviour at high temperature.”

The correspondence between normalisable modes and the poles of D_R(ω,k) in the low temperature phase is valid for any s = h - hbar.

“We thus expect that the correspondence between normalisable modes and the poles of D R ( ω, k ) in the low temperature phase is, in fact, valid for any s = h -¯ h .”

On the AdS side, the set of frequencies (15) is identical to the set of normalisable modes.

“On the AdS side, the set of frequencies (15) is identical to the set of normalisable modes [27]: these modes are regular at the origin of AdS 3 and are normalisable at infinity.”

• cites [27] [27] V. Balasubramanian, P. Kraus and A.E. Lawrence, Phys. Rev. D 59 , 046003 (1999) [arXiv:hep-th/9805171]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

As shown in [28], higher spin s equations of motion in AdS3 can be reduced to that of massive scalar fields for any s.

“However, as shown in [28], higher spin s equations of motion in AdS 3 can be reduced to that of massive scalar fields for any s .”

• cites [28] [28] R. R. Metsaev, Nucl. Phys. B 563 (1999) 295 [arXiv:hep-th/9906217]. -Not matched

  Couldn’t be matched to a paper in Crossref, so it can’t be checked.

The oscillating behaviour of the bulk perturbation is mirrored by the oscillating behaviour of the linear response.

“As a result, the oscillating behaviour of the bulk perturbation is mirrored by the oscillating behaviour of the linear response.”

As a consequence, the system will necessarily show Poincaré recurrence.

“As a consequence, the system will necessarily show Poincar´ e recurrence.”

Therefore, the linear response will exhibit oscillatory behaviour.

“Therefore, the linear response will exhibit oscillatory behaviour.”

The total correlator will still include the decaying BTZ part with complex frequencies.

“The total correlator will still include the decaying BTZ part with complex frequencies.”

In this way, one can obtain quantitative, non-perturbative information about thermal Green's functions also in higher dimensions.

“In this way, one can obtain quantitative, non-perturbative, information about thermal Green's functions also in higher dimensions.”

The energy spectrum for a system with finite entropy S is discrete, with level spacing of the order e^{-S}.

“As pointed out in [16]-[19], the energy spectrum for a system with finite entropy S is discrete, with level spacing of the order e -S .”

• cites [16]-[19]Not checked yet

  Not checked in the free run.

For a system with discrete spectrum, the retarded propagator has simple poles on the real axis.

“Indeed, as we saw in section 2, for a system with discrete spectrum, the retarded propagator has simple poles on the real axis.”