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Harlow, Ooguri: entanglement/AdS proof that there are no global symmetries in QG

Daniel Harlow wrote or co-wrote numerous interesting papers about quantum gravity, entanglement, locality, error code correction, weak gravity conjecture etc. within AdS/CFT.

John Preskill has tweeted about an interesting, soon-to-be-published result by Harlow and Hiroši Ooguri:

They seem to make some lore rigorous.

The lore says that there are no global symmetries in consistent theories of quantum gravity. In a moral sense, this implication results from the "gravity" part of "quantum gravity", not from the "quantum" one. Why? Because Einstein's theory of gravity, the general theory of relativity, makes everything "local". In particular, the existence of a gravitational field depends on what you are "locally doing".

In a small lab e.g. Einstein's elevator, the accelerated motion is indistinguishable from the gravitational field of a large object. This principle, the equivalence principle, really means that you may imagine the Universe to be a grid of many labs and the switch to an accelerating coordinate system may be done independently in each elevator (cell).

Well, I should have been more clear. What general relativity is may be described as an extension of the special relativity's symmetry principles. In special relativity, you may rotate, translate, and Lorentz boost coordinate systems. In general relativity, you may make the translations different at different points of the spacetime – which also allows the boosts or any nonlinear transformations of the coordinates, too.

So general relativity says that the rigid Poincaré group has to be extended to the infinite-dimensional diffeomorphism group. It's not hard to believe that morally speaking, general relativity also "should" promote all other global symmetries to their local counterparts. Why? Look at the Kaluza-Klein theory. That's a theory in which the \(U(1)\) or \(G\) gauge group becomes geometrically realized as an isometry group of the compactification manifold – e.g. an extra (fifth) circular dimension of the spacetime. But general relativity allows the translations in this periodic coordinate \(x^5\to x^5 + \varepsilon\) to be spacetime-dependent, i.e. \(\varepsilon=\varepsilon(x^0,x^1,x^2,x^3)\), so this isometry which could be a global symmetry in a non-gravitational theory must become a local, gauge symmetry, too. (I hope that you're OK with my counting, 0,1,2,3,5, which makes things clearer LOL.)

This argument is just an analogy, although a persuasive one. Symmetries don't have to be geometrized in the Kaluza-Klein way, you could object. So does the conclusion hold in general? Do all global symmetries have to be promoted to local ones? You may surely find classically consistent theories of gravity with global symmetries.

Within some particular superselection sectors or descriptions of some vacua of quantum gravity, this lore may be proven, however. As the Preskill's photograph of Ooguri's lecture indicates, Ooguri and Harlow are going to argue that they have a proof that a global symmetry within the AdS space would ruin the locality in the AdS space. At the linguistic level, it sounds plausible: globalists (such as George Soros and the people whom he has bought) fight against the localists. ;-) However, this "understanding" is fake – it is mostly a coincidence that "global symmetries" are incompatible with "locality" in this sense. In spite of the randomness of this "simple to understand result", it seems to be right and they have a proof.

So all the global symmetries that we observe in the boundary CFT must still exist in the AdS space in some sense. But the right sense is that they are and they have to be extended to a local, gauge group. Moreover, Ooguri and Harlow will hopefully offer us a clear proof that physical states transforming in every finite-dimensional irreducible representation of the group \(G\) have to exist in the bulk theory.

This is a statement we would often discuss in the context of the weak gravity conjecture. Once you determine that the charges must lie in a lattice \(\Gamma\), for example, is it guaranteed that for every lattice site, there exists a physical object/state that has exactly this value of the charges? This statement is a "strengthening" of the Dirac quantization rule but this strengthening can't be proven too easily – the usual proof of the Dirac quantization rule isn't enough, for example. The minimum magnetic monopoles could be \(N\) times more charged than the "theoretically minimal" ones, after all.

But in string theory, they always seem to have the minimum charges. The representations of \(G\) are a non-Abelian generalization of this lattice discussion. So they should prove that everything that seems to be allowed must be allowed, indeed.

There are many relationships to the weak gravity conjecture. Harlow already wrote a related paper two years ago. First, the weak gravity conjecture is a stronger version of the statement that there are no global symmetries. Why? It's because the weak gravity conjecture bans very small values of the coupling \(g\) – and the small values of \(g\) would allow a gauge symmetry to "mimic" the physical behavior of a global symmetry arbitrarily accurately, which should be forbidden as well. When you ban murders, i.e. stabbing of another person that is at least 1 inch deep, you should better discourage the 0.99 inch stabbings as well, shouldn't you? ;-)

Also, I think that their proofs could have amusingly simple and group-theoretical consequences for some particular vacua, e.g. for the pure AdS three-dimensional gravity which is dual to the monster group CFT if the radius is minimal. I've discussed my proposal that Schur's orthogonality relations have something to do with the ER-EPR correspondence. The monstrous AdS/CFT vacuum obviously uses all 194 irreps of the monster group but I believe that they're acting on each other in some interesting ways and some nontrivial identities will be proven to hold for the monster group that follow from physics in \(AdS_3\). I believe that the monstrous pure gravity vacuum is the only quantum gravity vacuum where the gauge group is purely finite and discrete (massless states are absent etc.) and this could mean some special identities obeyed by this finite group that no other group obeys.

The black hole microstates in this \(AdS_3\) come in all 194 irreps of the monster group. But because the monster group is a local gauge group, as Harlow and Ooguri also seem to prove, "cosmic strings" should be allowed whose monodromy around them is equal to any element of the monster group. In three-dimensional spacetimes, "cosmic strings" (codimension two objects in the spacetime) should be interpreted as zero-dimensional particles. So shouldn't they be the same objects as the microstates in 194 irreps?

If it's so, the 194-dimensional space may have two natural bases which are parameterized either as conjugacy classes of the monster group (by the monodromy); or as the irreps. The characters should give you the matrix elements to switch from one basis to another. But I don't know what to do with the "other" degrees of freedom that aren't purely group-theoretical.

You could be surprised that I am saying that these states have monodromies around them – but the black hole microstates are mutually local in the CFT. Well, that doesn't mean that they have to be mutually local in the bulk. I believe that a natural "Dirac string" or the branch cut may be drawn as a line between the bulk object (black hole) and a point on the AdS boundary.

I expect the number of target readers who have made it to this point to be comparable to a dozen so it's possible that I won't proofread this text.

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