Celestial spheres, CCFTs and holography...?

Question

I was reading some papers* on celestial holography which aims to apply the holographic principle to a flat spacetime, making a duality between a celestial sphere and a celestial conformal field theory (CCFT).

I did a question to Andrew Strominger, namely:

"Since in your paper you say that celestial amplitudes can be used to study UV complete theories, and therefore, to study possible consistent theories of quantum gravity, does it mean that we can use them to study every possible UV completion theory? All possible GUTs and all possible theories of quantum gravity? Could we use it to study every possible theory of everything like string theory, M-theory, loop quantum gravity, asymptotically safe quantum gravity...?".

His answer was basically a "yes"

This is consistent with Laurent Freidel and collaborators' works on corner symmetries and local holography (https://arxiv.org/abs/2212.12469, https://arxiv.org/abs/2302.12799 & https://arxiv.org/abs/2407.11132) which are related to celestial holography, where I asked Luca Ciambelli, another physicist working on these models, if all possible models of quantum gravity could be applied to their models, and he again basically answered that yes, they could.

Then, this makes me wonder: is it possible that different CCFTs corresponding to these different fundamental theories of quantum gravity could exist in different celestial spheres? Or something similar to this? Or is it forbidden?

*the papers:

https://arxiv.org/abs/2212.12469

https://arxiv.org/abs/2012.04208

https://arxiv.org/abs/2309.03932

Answer 1

The question is imprecise. We must track the definitions to understand what is going on. An asymptotically flat spacetime is one spacetime that has one asymptotic boundary containing the null surfaces ${\cal I}^\pm\simeq \mathbb{R}\times S^2$. This is a boundary condition singling out a class of Lorentzian manifolds. The celestial sphere is this $S^2$ factor appearing in ${\cal I}^\pm$. At this point, one could say that there are two celestial spheres (one in the past and another in the future), but there is actually a matching relation between these to make the scattering problem well-defined that picks out one single sphere. So, given an asymptotically flat spacetime, there is one celestial sphere.

I don't see what you mean then when you say whether different theories of gravity live on different celestial spheres. The conjecture is that if you have some quantum theory of gravity that describes the dynamics of this asymptotically flat spacetime, there is going to be some two-dimensional Celestial CFT on the celestial sphere. So, different quantum theories of gravity would correspond to different theories on the celestial sphere, not to different celestial spheres.

But even then, this is all a conjecture. The difference between Celestial Holography and AdS/CFT is that while for the latter, there are plenty of examples of the duality available, where you can construct both bulk and boundary theories from some fundamental superstring theory, the only example of Celestial Holography so far is restricted to the self-dual sector of gravity in $(2,2)$ signature.

So, given some concrete theory of gravity, determining whether a corresponding Celestial CFT exists is an open problem. And just to close, I'd like to comment on the notion of existence here. One could say that a theory is defined by prescribing all possible correlators. As such, given a quantum theory of gravity, you know all celestial amplitudes and, hence, the correlators of the dual theory. So, in this sense, the theory always exists. I mean existence in a different sense: is there some theory that can be intrinsically constructed in two dimensions and that has useful properties to study the bulk that reproduces that particular quantum theory of gravity? That is not understood completely. Notice that, in AdS/CFT, the fact that the dual theory is a CFT is central to being able to use the boundary as a definition of the bulk quantum gravity theory. It is not clear at this point whether Celestial CFT is a CFT or a different kind of theory that resembles a CFT in some aspects, differing in others.