What is the dual asymptotic spacetime of a CFT on a particular flat manifold?

Question

According to AdS/CFT correspondence, the dual theory of a boundary CFT on flat spacetime is defined on an asymptotically AdS spacetime. The nature of the bulk spacetime depends on the topology of the boundary spacetime. For example, it is well known that if the boundary CFT base space has the topology of a non-compact globally flat spacetime i.e. $\mathbb{R}^{d-1,1}$ then the bulk spacetime is asymptotically Poincare AdS, but if the boundary CFT is defined on an Einstein universe (which still has the flat space metric, but different topology $\mathbb{R} \times S^{d-1}$), the bulk spacetime is asymptotically global AdS instead.

Restricting to flat spacetime on the boundary side *, but with topologies other than the two described above, what do the bulk spacetimes look like? What is the most general analysis done in this direction? In other words, is there a recipe to find the bulk spacetime topology, given the manifold on which the boundary theory is defined? For example, what is the bulk spacetime corresponding to CFT on a cone with a flat metric on it or a flat torus? For simplicity, one can look at ground states on the CFT side, so that the dual spacetime is always "empty" i.e. you might simply drop the word "asymptotic" in the above para.

$*$ The reason, why I am restricting to flat spacetime for the boundary theory is because theories on curved spacetimes don't enjoy conformal symmetries in general, so that the bulk dual spacetime might not be even asymptotically AdS in that case. I would be interested in this as well, but I guess the answer is not known, because essentially I would be asking to extend AdS/CFT to ???/QFT.

Answer 1

Any asymptotically AdS spacetime admits a Fefferman-Graham expansion of the form $$ ds^2 = L^2 \frac{dz^2}{z^2} + \frac{1}{z^2} [ \gamma_{ij} + O(z^2) ] dx^i dx^j . $$ The boundary metric is $\gamma_{ij}$ and it is arbitrary. The boundary topology is also arbitrary. Given a metric and topology of boundary manifold $M$ on which you put your CFT, the bulk geometry can be deduced by solving the bulk Einstein's equations with $\gamma_{ij}$ as a boundary condition. You will also need additional data such as the expectation value of the stress tensor $\langle T_{ij} \rangle$ which depends on the state of the CFT and the one-point functions of any other operators that are dual to bulk fields.

Answer 2

Arguably the most well known method to construct bulk spacetime metric from CFT data using Fefferman Graham expansion (as mentioned in Prahar's answer) was pioneered by Haro Solodukhin and Skenderis. In principle we can use that, but even for well known systems like AdS Schwarzschild geometries, one usually instead guesses the bulk geometry and verifies the conjectures by some non-trivial tests. This is explained at some depth in this answer. That answer touches upon some important caveats which I emphasize now: there can be no or more than one bulk geometry corresponding to a given particular boundary data! In case there is no bulk geometry, the CFT is most probably not holographic (or falls outside the code subspace). In case there is more than one bulk geometry for some given boundary data, one has to systematically sum over all topologies. One famous example is that for a thermal state in CFT, the bulk geometry can be either thermal AdS (at low temperatures) or AdS black hole (at high temperatures). The phase transition between these two phases called Hawking-Page transition is topological in this sense. This is reviewed briefly in that answer and also in detail in the excellent lecture notes by Hartman.

Having said that, reconstructing the bulk metric is one of the major goals of a program in holography called "Bulk reconstruction". There are plenty of results in this program. For a recent brief overview: see these TASI notes by Harlow. Reconstruction of the metric is fairly recent and not covered there, which is why I am going to say a few words about two of my favourites. I will like to highlight these two not because, simply because I know a bit about these two only (having worked with some of the authors directly), but let me warn you that there are plenty of other techniques available in the literature.

1. Intersecting Modular Hamiltonians

Using two intersecting Ryu Takayanagi surfaces in the bulk (whose area gives entanglement entropy associated to the subregion of the boundary it is anchored to), one can localize a bulk field $\Phi$ at a point in the bulk by demanding that the bulk field operator commutes with Modular Hamiltonians associated with both the subregions. These relations can be directly solved to yield $\Phi$ in terms of $H_1$ or $H_2$ as described in Kabat and Lifschytz. Using the expressions for bulk fields (derived completely from CFT data) one can find that the two point correlators exactly match with a "well-known" expression for SUGRA bulk propagator at the leading order of geodesic distance, in a certain WKB type limit i.e. for "heavy" scalars. From that, the geodesic distance and finally the metric is extracted. This was done in a paper by Roy and Sarkar. The main limitation is that this has been so far to my knowledge, done only for static geometries.

2. Light cone cuts

From the location of certain spacelike slices obtained from the intersection of a bulk point's light cone with the timelike boundary called light cone cuts, we can get the bulk metric. The location of the cuts in turn are obtained from the boundary CFT correlators' nature of light-like separated point's singularities. Here the limitation is the construction works only within the causal wedge. Some extensions and discussions of other methods to construct the metric given CFT data is explored in section 1 of Bao, Cao, Fischetti and Keeler, where for disk topologies the author is able to find bulk metric merely from entanglement entropy!

Answer 3

In "Connectedness Of The Boundary In The AdS/CFT Correspondence" by Witten and Yau, they prove that the bulk manifold $M$ must have $H_{n}(M;Z)=0$ if the boundary metric has positive scalar curvature. This is some information about the topology of the bulk spacetime from the CFT data and is not much well known. I don't know any results on information about the topology of the bulk spacetime, or any progress on Witten's work, so I am keeping this partial answer Community wiki so that someone who knows about this can add similar results.