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Do all theories of quantum gravity* have a duality with some CFT? Can all theories of quantum gravity be approximated by some EFT? Can CFT and/or EFT be equivalent to all possible theories of physics?

*including: String theory, M-theory, supergravity, loop quantum gravity, causal sets, causal dynamical triangulation, twistor theory...etc (https://en.wikipedia.org/wiki/Quantum_gravity#Other_theories)

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To call something "quantum gravity" at all, it needs to be a UV complete theory which has Einstein gravity as its effective theory at low energies. So for the questions "can quantum gravity always be approximated by an EFT" and "can quantum gravity always be equivalent to an EFT", the answers are "yes" and "no" respectively. Both by definition. Note that the basic consistency check of reducing to GR at low energies is already something that loop quantum gravity has trouble with.

Exact equivalence to a CFT is another story. There is currently much debate over flat and dS backgrounds. But for quantum gravity around a background that is asymptotically AdS, it is widely believed that some CFT dual is inevitable.

The modern approach to AdS/CFT which starts with QFT in AdS (example reference) is enough to show, just from geometry, that the boundary limits of local operators in the QFT will have conformally covariant correlators. It is then quite believable that making gravity dynamical in the bulk will give this system of correlators a stress tensor and turn it into a CFT. So we can say that being able to describe asymptotically $AdS_{d + 1}$ quantum gravity theories by a $CFT_d$ is another consistency check. String theory passes this check and moreover provides enough microscopic information to suggest which CFT you will get in many cases.

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    $\begingroup$ +1, nice answer. A super nitpicky question: does "quantum gravity" necessarily need to be UV complete? For example, suppose there was some effective field theory that (a) reproduced GR below the Planck scale, (b) was consistent for energies above the Planck scale but below some huge cutoff $\Lambda$, (c) broke down above $\Lambda$ because of some non-gravitational physics. Would we not call that quantum gravity? I don't have an example in mind and I'm fine with the answer "I'm presenting the standard story for simplicity", I'm just wondering if "necessary" is slightly too strong a word. $\endgroup$
    – Andrew
    Commented Sep 3 at 17:40
  • $\begingroup$ Good point. Problems at the Planck scale are the main reasons why people don't want to refer to the "perturbative quantum gravity" done today as just "quantum gravity". A theory solving these, even if it breaks down at some higher scale, would certainly be qualitatively new. Presumably it would be found by imposing the constraint that some UV completion exists. But the physics above $\Lambda$ in that case could still be as mysterious as trans-Planckian physics is to us now. $\endgroup$ Commented Sep 3 at 18:04
  • $\begingroup$ @ConnorBehan Even in the case described by Andrew that theory would be approximated to an EFT (namely GR) right? Also, do all LQG models have troubles reproducing GR (physics.stackexchange.com/questions/545314/…)? Could there be quantum gravity theories having other possible EFT approximations that wouldn't necessarily be GR? $\endgroup$
    – vengaq
    Commented Sep 6 at 9:58
  • $\begingroup$ I am not knowledgeable about LQG. But the low energy EFT of any quantum gravity theory should be one that has spin-2 gauge interactions. Whether this is referred to as GR depends on the community. Sometimes the number of dimensions is not 4, sometimes the metric field has superpartners and sometimes it is topological like Kodaira-Spencer theory. $\endgroup$ Commented Sep 6 at 16:44

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