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In AdS/CFT, we are told that the bulk and boundary functions are equal:

$$ \tag{1}Z_{bulk}[J]= Z_{CFT}[J], $$

where on the left hand side of the equality, $J$ is interpreted as a boundary condition at spacelike infinity for the bulk path integral (i.e. it fixes the values of the bulk dynamical fields on the cylindrical surface of the "tin can" diagram of AdS), and on the right hand side, $J$ is interpreted as a source for the CFT. Eq. (1) is known as the GKPW dinctionary, after Gubser, Klebanov, Polyakov and Witten.

But on both sides of the equality, further boundary conditions are required. To make my point clear, let's restrict the domain of the path integral to a finite time interval, i.e. a finite segment of the tin can. Like a real tin can, in addition to the cylindrical surface, this segment has boundaries on its top and bottom. So for $Z_{bulk}$ to make sense, we must specify the incoming and outgoing bulk states on the top and bottom of the tin can (respectively). Similarly, on the CFT side, we must also specify in and out states at the initial and final times.

My first guess is that we use vacuum in/out states on both sides of the duality. But although this would seem to make sense on the CFT side, where there is a unique vacuum, things aren't so easy on the AdS side: the Hamiltonian density $\mathcal{H}$ vanishes on the physical state space, so the Hamiltonian $H_{bulk}$ consists entirely of the ADM energy acting at the boundary. Hence the ground space of $H_{bulk}$ is highly degenerate, and there is no obvious way to choose a preferred bulk vacuum state.

So my question is: What boundary conditions ensure that Eq. (1) is true? An ideal answer would give a general way to construct the appropriate boundary state given an arbitrary bulk state, and vice versa. However, I'd also be satisfied with a single example of a pair of bulk/boundary states which make (1) true.

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  • $\begingroup$ Path integrals are made to be cut. It doesn't make sense to ask what the initial and final states are because that's up to you. $\endgroup$ Jun 5, 2022 at 20:43
  • $\begingroup$ @ConnorBehan Good point. But once we've chosen bulk in/out states, this must somehow fix the boundary in/out states that appear in Eq. (1). And vice versa. So really what I meant to ask is how we map between bulk and boundary states in such a way that Eq. (1) holds. I've edited my question to make this clearer. $\endgroup$ Jun 6, 2022 at 11:55

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You should interpret “partition function” in this dictionary as the CFT path integral on a closed manifold, without any boundaries. The sources $J$ include the metric of this manifold, which might be Euclidean, or might contain Euclidean and Lorentzian pieces joined together.

Now states come in because parts of this path integral might be interpreted as preparing initial and final states. The simplest example is the vacuum, which is prepared by a path integral over semi-infinite Euclidean time. Another nice example (particularly relevant in CFT due to the state-operator correspondence) is the path integral over a hemisphere with an operator inserted at the pole (a delta function source, if you like). This produces a state on the equatorial sphere. By gluing together two such hemispheres you get a sphere with operator insertions on both poles, and the path integral then computes the overlap of the two states.

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  • $\begingroup$ So, for example if we prepare the vacuum state on the CFT side, what state should we use on the AdS side? Is there a unique AdS vacuum? In fact I don't know anything about what the AdS quantum gravity Hilbert space looks like, so a reference on this would help. $\endgroup$ Jun 21, 2022 at 14:40
  • $\begingroup$ Yes, the vacuum state of the CFT on a spatial sphere is the vacuum state of empty global AdS space. The CFT and bulk theory are different descriptions of the same thing: they have the same states and same Hamiltonian, so in particular the same unique vacuum. It is unique, being the only state invariant under all the conformal symmetries. $\endgroup$ Jun 23, 2022 at 2:58
  • $\begingroup$ One introductory reference is arxiv.org/abs/1608.04948 $\endgroup$ Jun 23, 2022 at 3:01
  • $\begingroup$ A priori, the bulk and boundary Hilbert spaces are different: the former is the space of wavefunctionals (of 3-geometries on a bulk Cauchy slice, plus maybe other fields) satisfying the Hamiltonian constraint, whereas the latter is a space of wavefunctionals of field data on the boundary. So I'm interpreting your answer as saying that AdS/CFT picks out an isomorphism between these spaces (an isomorphism which maybe seems so natural that eventually we just call the spaces "the same"). I'd like to know the precise form of this isomorphism, but I can't find that in the reference you provided. $\endgroup$ Jun 23, 2022 at 12:17
  • $\begingroup$ Yes, your interpretation is correct. I would also like to know the precise form of the isomorphism! This is certainly not known. The problem is that we don't even have a precise definition of the bulk Hilbert space in the language you are using (except in perturbation theory around AdS, for example). $\endgroup$ Jun 25, 2022 at 0:18

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