Holographic renormalization and reconstruction of space-time. Kostas Skenderis Southampton Theory Astrophysics and Gravity research centre

Transcription

1 Holographic renormalization and reconstruction of space-time Southampton Theory Astrophysics and Gravity research centre STAG CH RESEARCH ER C TE CENTER Holographic Renormalization and Entanglement Paris, 26 January 2015

2 Introduction According to holography, gravity in (d + 1) dimensions is related to QFT in d dimensions. A rough picture of how the bulk is reconstructed by QFT data emerged already in the early days of AdS/CFT. The AdS radial direction is related to the renormalization scale of the dual QFT and RG transformations are related to bulk field equations. In this talk I will summarise progress in making this more precise.

3 References S. de Haro, S. Solodukhin, KS, Holographic reconstruction of space-time and renormalization in AdS/CFT hep-th/ M. Bianchi, D. Freedman, KS, Holographic renormalization, hep-th/ KS, Lecture notes on holographic renormalization, hep-th/ I. Papadimitriou, KS, AdS/CFT correspondence and geometry, hep-th/ KS, B. van Rees, Real-time gauge/gravity duality: prescription, renormalization and examples, KS, Ariana Christodoulou, to appear

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6 QFT To define a theory we typically give the Lagrangian. Depending on context we may be interested in: vacuum correlators h0 O 1 (x 1 ) O(x n ) 0i correlators in a non-trivial states, h O 1 (x 1 ) O(x n ) i For example, i may be a state that spontaneously breaks some of the symmetries, a thermal state, a non-equilibrium state, etc. The correlators may be time-ordered, Wightman functions, advanced, retarded...

7 QFT Given a theory, astate and a set of operators there is a unique answer for their correlators. The state can be alternatively described by its one-point functions, ho i i. Any dual formulation should have the same properties. How does it work in holography?

8 Holography Within the gravity approximation: The dual QFT is represented by: An (asymptotically) AdS solution of (d + 1)-dimensional gravity coupled to matter. Operators are dual to bulk fields and the state is encoded in subleading terms in bulk solutions.

9 Euclidean vs If we are interested in time-independent states (vacuum state, thermal state) we may Wick rotate to. In Lorenzian signature there are different types of correlators (time-ordered, Wightman, retarded, etc.). In Euclidean QFT there is a unique type of correlators. So to simplify matters let s initially focus on. We will return to full fledged Lorentzian discussion afterwards.

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11 Bulk scalar field Bulk scalar fields of mass m 2 = ( d) are dual to operators O of dimension. The solution of the bulk field equation has an asymptotic expansion of the form [de Haro, Solodukhin, KS (2000)]: (x, r) =r d (0) + + log r 2 (2 d) + r (2 d) + (0) is the source for the dual operator and all blue terms are locally determined by it. (2 d) is related with the expectation value of O, ho i (2 d)

12 QFT vs gravity We argued earlier that the Lagrangian and the state uniquely specify the correlators. The counterpart of this statement on the gravity side is that ( (0), (2 d)) uniquely specify a bulk solution. One way to see this is to use a radial Hamiltonian formalism, where the radial direction plays the role of time. The renormalised radial canonical momentum is = ho i (2 d) By a standard Hamiltonian argument, specifying the conjugate pair ( (0), ) uniquely picks a solution. [Papadimitriou, KS (2004)]

13 Holographic reconstruction I Given QFT data ( (0), ho i) there is a unique solution (x, r) of the bulk field equations.

14 QFT vs gravity In QFT the vacuum structure is a dynamical question: in general, one cannot tune the value of ho i. The counterpart of this statement in gravity is that regularity in the interior selects ho i. A a generic pair ( (0), ) leads to a singular solution.

15 Fefferman-Graham radial coordinate and dilatations The metric and scalar field in Fefferman-Graham gauge read ds 2 = dr2 + 1 r 2 (g (0)ij(x)+O(r 2 ))dx i dx j r 2 = r d ( (0) + O(r 2 )) From here one gets that the QFT dilatation operator Z D = d d x 2g (0)ij (d ) (0) g (0)ij (0) is realised in the bulk as a radial derivative D (1 + O(r2 )) RG transformation are linked with the radial coordinate.

16 Holographic RG flows Holographic RG-flows are described by asymptotically AdS spacetimes of the form with A( )! as!1. ds 2 = d 2 + e 2A( ) dx i dx i = ( ) If =' 0 e (d ) as!1then the flow describes the QFT with Lagrangian L = L CFT + ' 0 O If =' 0 e as!1then we have the CFT in a state that spontaneously breaks conformal invariance, ho i ' 0

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18 New issues On the gravity side: In the bulk equations of motion do not have a unique solution given boundary conditions. One needs in addition initial conditions. There are non-trivial normalisable modes that vanish at the boundary. On the QFT side: Given a Lagrangian and operators there is more than one type of correlator one may wish to compute. One may wish to compute ttime-ordered, anti-time ordered, Wightman products, etc. on non-trivial states.

19 QFT On the QFT side one can summarise the new data needed by providing a contour in the complex time plane specifying where operators are inserted in this contour.

20 Examples t T C0 C1 T C3 C2 a b c a Vacuum-to-vacuum contour: computes correlators h0 T ( ) 0i b In-in contour: computes correlators h i c Thermal contour: computes thermal correlators

21 Examples Wightman in-in 2-point function, h0 O(x)O(y) 0i t t1 0 t2 Thermal Wightman 2-point function. t t1 0 t2

22 Real-time gauge/gravity duality [van Rees, KS (2008)] The holographic prescription is to use "piece-wise" holography: Real segments are associated with Lorentzian solutions, Imaginary segments are associated with Euclidean solutions, Solutions are matched at the corners.

23 Examples a E L E b E L L E c E L L

24 Comments In the extended space-time associated to a given contour, boundary conditions and regularity in the interior uniquely fix the bulk solution. The Euclidean caps can also be thought of as Hartle-Hawking wave functions.

25 Holographic reconstruction II: Lorentzian case Given QFT data ( (0), ho i) and a contour in the complex time plane there is a unique solution (x, r) of the bulk field equations.

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27 Summary We have just argued that QFT data uniquely reconstruct the bulk. This is an existence theorem. There is no explicit formula for the bulk fields, except near the boundary. QFT data can detect non-trivial topology behind black hole horizons, as for example in the 3d wormhole solutions [van Rees, KS (2009)].

28 The extrapolation dictionary An a priori different dictionary was suggested in the early days of AdS/CFT [Banks, Douglas, Horowitz, Martinec (1998)]. Boundary correlation functions can be obtained as a limit of bulk correlation functions: ho (x 1 ) O (x n )i QF T =lim r!0 r n h (x 1,r) (x n,r)i Bulk This prescription is equivalent to the standard one for scalars in a fixed background [Harlow, Stanford (2011)]. It also agrees with the real-time prescription for non-equilibrium scalar 2-point functions [Keranen, Kleinert (2014)].

29 The extrapolate dictionary suggests the following map between boundary and bulk operators: Z (x, r) = d d x 0 K(x 0 r, x)o (x 0 ) K is called the smearing function. Bulk correlation functions are then related to boundary correlation functions by h (x 1,r 1 ) (x n,r n )i Bulk = Z d d x d d x 0 nk(x 0 1 r 1,x 1 )...K(x 0 n r n,x n )ho (x 0 1) O (x 0 n)i QF T [Banks, Douglas, Horowitz, Martinec (1998)],... [Hamilton, Kabat, Lifschytz, Lowe (2006)]...

30 Remarks The idea is that the smearing function depends only on the operator, so once it is determine by, for instance, looking at 1-point functions it can be used for other correlators. The bulk field so constructed should be local (bulk operators at space like separation should commute). This type of bulk reconstruction has been used in order to study the information loss paradox/firewall controversy. In the reminder of this talk we will discuss a number of checks one can perform on this map using the standard dictionary.

31 The case of global AdS [Christodoulou, KS, to appear] We will now derive the smearing function for linear scalar fields in global AdS 3 using the real-time gauge/gravity duality. We are looking at local bulk excitations so (x, r) r as r! 0. By the standard dictionary such solutions should correspond to considering the theory in a non-trivial states i with As we shall see, the state i is h O i 6=0 i = O 0i

32 The time contour and the bulk space-time In a CFT all states are obtained by acting with an operator on 0i. Holographically this means that we need to turn on source on the boundary of EAdS. The time contour and the corresponding bulk space-time are:

33 Obtaining the solution The Lorentzian solution is a sum of normalisable modes L(t, r, )= X n,k (c + nk e i!+ nk t+ik + c nk e i! nk t+ik )g(! nk, k,r) where g(! nk, k,r) r as r! 0. In the future cap we consider a solution with a delta function source: + E = rd + (0) (, )+..., + (0) (, )= ( 0) ( 0) and similar for the past Euclidean cap. The matching conditions imply: c + nk (0) (!+ nk,k), c nk + (0) (! nk,k)

34 The solution across the matching surface Euclidean manifold Exponentially decaying source mode Lorentzian Manifold Matching surface Oscillatory excitation mode Amplitude of a single mode as a function of (Euclidean and then Lorentzian) time for fixed r,.

35 Smearing function The smearing function can now be derived using similar steps to those in [Hamilton, Kabat, Lifschytz, Lowe (2006)] One defines the boundary field as O(x) =lim r!0 r L(r, x) After some algebra this leads to Z L(x, r) = d d x 0 K(x 0 r, x)o (x 0 ) The main added value here is that the smearing function is automatically convergent due to i s that originate from the Euclidean caps. In Hamilton et al these insertions were added by hand.

36 Beyond linear fields This boundary-to-bulk map cannot hold in this form in complete generality. If the QFT contains operators with dimensions, O 1,O 2,O 3, with 3 = Then [KS, Taylor (2006)], ho 3 i = 3 + c where c 12 is a constant. This suggests Z Z 3(x, r) = d d x 0 K(x 0 r, x)o 3 (x 0 )+ d d x 0 K 1 (x 0 r, x)o 1 (x 0 )O 2 (x 0 ) Locality also suggest non-linear terms should be added to the map [Kabat, Lifschytz, Lowe (2011).

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38 I discussed how QFT data reconstruct classical fields in the bulk, both in Euclidean and. The map works in full generality. It is however implicit in the sense that one knows explicitly only the near boundary behaviour of the solution. I reviewed the holographic reconstruction of bulk operators. More work is needed to understand whether this construction works in general.