The gravitational path integral in the AdS/CFT correspondence depends on the boundary submanifold of the bulk manifold

Question

On page 131 of these notes, a precise formulation of the AdS/CFT correspondence is given by the GKPW dictionary

$$Z_{\text{grav}}[\phi_{0}^{i};\partial M] = \langle \exp \left( - \frac{1}{\hbar} \sum_{i} \int d^{d}x\ \phi_{0}^{i}(x)O^{i}(x) \right) \rangle_{\text{CFT on } \partial M},$$

with

1. boundary conditions on the bulk scalars given by

$$\phi^{i}(z,x) = z^{d-\Delta}\phi_{0}^{i}(x) + \text{subleading as } z \to 0,$$ and

2. the relation of the mass of the bulk scalar to the scaling dimension of the CFT operator given by

$$m^{2} = \Delta(d-\Delta), \qquad \qquad \Delta = \frac{d}{2} + \sqrt{\frac{d^{2}}{4}+m^{2}\ell^{2}}.$$

Page 132 of the notes then goes on to mention the following:

Similar statements apply to all bulk fields, including the metric, though the boundary condition and formula for the dimension is slightly modified for fields with spin. The boundary conditions on the metric involve a choice of topology as well as the actual metric, which is why we've indicated explicitly that $Z_{\text{grav}}$ depends on the boundary manifold $\partial M$.

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I am confused by the final sentence in the above paragraph.

Why are both a choice of topology and a choice of metric needed to write down the boundary conditions on the metric?

Why does this mean that $Z_{\text{grav}}$ depends on the boundary manifold $\partial M$?

Answer 1

Looks like he's just talking doing that integral over different boundary manifolds.

• The volume form (what he wrote as $d^dx$), depends on a choice of the metric on the boundary
• The domain of the integration (which he doesn't really write explicitly, imagine it as a subscript on the integral sign or something) depends on the topology of $\partial M$.

Not the best choice of notation on Hartman's part obviously, the confusion is understandable.