# Boundary conditions for bulk partition function in AdS/CFT

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.

• 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. Commented Jun 5, 2022 at 20:43
• @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. Commented Jun 6, 2022 at 11:55

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.