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ER=EPR as Schur orthogonality relations

The AdS/CFT correspondence relates a murky, effective description with quantum theory in the bulk – in AdS – to a well-defined, microscopic, non-gravitational theory on the boundary – CFT. I think that most people would agree that at least at present, the CFT side is the "more well-defined one", and the relationship therefore helps us to understand what quantum gravity (in this case in AdS) actually is.

I would like to have a more universal definition of quantum gravity that works for any superselection sector, whether the boundary behavior of the spacetime is flat Minkowski, AdS, or otherwise. What is the relationship between the low-energy field and some "detailed microscopic theory" in the most general case?

Witten's monstrous model of pure gravity in \(AdS_3\) has been one of my favorite toy models that I have employed to check and refine various tools that I proposed for quantum gravity in general. Just to recall, the AdS/CFT dual should describe pure gravity in a 3-dimensional space. In \(D=3\), the Ricci-tensor \(R_{\mu\nu}\) and the Riemann tensor \(R_{\kappa\lambda\mu\nu}\) both have six components. So the Ricci-flatness, i.e. Einstein's vacuum equations, imply the Riemann flatness. The vacuum must be flat. However, sources may create a deficit angle.




In the \(AdS_3\) space, this statement is deformed by the extra cosmological constant, and the BTZ black hole becomes the most important black hole solution of classical general relativity in \(AdS_3\). The simplest theory with no extra fields aside from the metric tensor must still allow black hole microstates by consistency.




A decade ago, Witten conjectured that there exists a cool AdS/CFT pair dual to the pure gravity in \(AdS_3\). Davide Gaiotto later showed that the conjectured duality only holds for the minimum radius, i.e. the \(N=1\) case. So that's the only case of interest. The dual CFT is a \(CFT_2\) with \(c=24\) which is pretty much the same CFT that was used to clarify the monstrous moonshine. It has the discrete monster group symmetry.

So one of the simplest realizations of general relativity – pure gravity in \(AdS_3\) – seems to secretly carry the most impressive sporadic finite group that relates its microstates. The CFT with the monster group symmetry may be constructed in analogy with the bosonic construction of the heterotic string. But one doesn't use a 16-dimensional even self-dual lattice. Instead, one has to pick the 24-dimensional even self-dual lattice. There are some 24 inequivalent ones.

We must pick one of them, the Leech lattice, which is the unique 24-dimensional even self-dual lattice that only has sites with the length \(\ell^2=0\), \(\ell^2=4\), \(\ell^2=6\), and so on. There are no sites with \(\ell^2=2\) at all – which is linked to the fact that the dual AdS theory contains no massless fields aside from the metric tensor – which has no allowed vacuum waves, due to the Ricci-Riemann equivalence that I previously mentioned.

Great. So this theory only has the unit operator with \(\ell^2=0\), at the origin of the lattice, and then various operators at \(\ell^2=4\) and other operators with similar dimensions that transform as\[

{\bf 196,883}\oplus {\bf 1}

\] OK, all these objects must obviously be understood as black hole microstates – there's nothing else in the theory. Their density increases quasi-exponentially with the mass, as you know from CFTs, and they transform as representations of the monster group \(M\). The GR intuition should be basically right qualitatively but you must be ready to embrace the fact that the corrections to some quantities may be of order 100%.

Now, the monster group has \(K=194\) conjugacy classes. If you know the basic representation theory of finite groups, you're familiar with the amazing statement\[

K = R

\] saying that the number of conjugacy classes is equal to the number of irreducible representations of the group, too. The dimensions of these 194 irreps may be found e.g. on this Subwiki page. The smallest ones have dimensions \(1\) and \(196,833\), of course, while the largest one has the dimension\[

258823477531055064045234375.

\] Many of the large irreps are rather close in size to this one. \(146\) of these irreps are real, \(48\) of them are complex, coming as \(24\) pairs of mutually complex conjugate irreps. (These 24 pairs contain all the DNA chromosome information for chimps, which are included in the monster, and the 146 non-paired chromosomes are those of a unicorn doll, but I don't want to overwhelm you with advanced monster biology.) Another fact that you remember from the basic representation theory is\[

|G| = \sum_{i=1}^{194} d_i^2.

\] The number of elements of the monster group, about \(8\times 10^{53}\), is the sum of the squared dimensions of all the irreps. This numerical fact, along with \(K=R\) I mentioned before, may be understood as trivial consequences of stronger Schur orthogonality relations – which hold nicely for finite as well as compact Lie groups.

Wikipedia tells us: The space of complex-valued class functions of a finite group G has a natural inner product:\[

\left \langle \alpha, \beta\right \rangle := \frac{1}{ \left | G \right | }\sum_{g \in G} \alpha(g) \overline{\beta(g)}

\] Just to be sure, a class function is a function mapping the group to the complex numbers that is constant all over each conjugacy class, i.e. one that obeys\[

\forall g,h\in M:\quad \alpha(hgh^{-1}) = \alpha(g).

\] Can you find some really apt physical realization for these things? I think you can. Consider an Einstein-Rosen bridge – a non-traversable wormhole as discussed in the ER-EPR correspondence – and twist the throat by an element \(g\in M\). So instead of connecting the two black holes, in our case the \(AdS_3\) black holes, using the most trivial entanglement\[

\ket\psi = \sum_{i} \ket i \otimes \ket{i'}

\] where the sum goes over some interval of masses or something like that, you replace \(\ket{i'}\) above by the \(g\)-transformed element \(g\ket{i'}\) for some \(g\in M\). If there were a single wormhole microstate for every \(g\), they would form a Hilbert space and the \(\left \langle \alpha, \beta\right \rangle \) inner product could simply be the inner product on their Hilbert space. I think it's appropriate that we demand "class functions". Why? Because I believe that all symmetries in a theory of quantum gravity (including discrete symmetries) are gauge symmetries. So the physical states must be invariant under all these symmetries. In particular\[

\forall g\in M: \quad g \ket\psi = \ket\psi

\] for all localized objects that may be isolated. Now, a wormhole with a twist given by \(h\in M\) may also be transformed by the action of \(g\), but if you do so, I believe that the twist \(h\) gets conjugated. So the action of \(g\) on the \(h\)-twisted wormhole is the \(ghg^{-1}\)-twisted wormhole. Does it make sense so far? So the condition that the wave functions on the space of the twisted wormholes are class functions is just a mathematical translation of the gauge invariance for objects with some monodromy – such as cosmic strings or wormholes.

If you continue reading the Wikipedia article, you will also learn that under the very same "sum over \(M\)" inner product above, the characters are orthogonal to each other:\[

\left \langle \chi_i, \chi_j \right \rangle = \begin{cases} 0 & \mbox{ if } i \ne j, \\ 1 & \mbox{ if } i = j. \end{cases}

\] Here, \(\chi_i\) where \(i=1,2,\dots,194\) is the character i.e. the trace over the \(i\)-th irrep framed as a function of \(g\)\[

\chi_i(g) = {\rm Tr}_i (g).

\] The orthogonality also works in the opposite direction. If you sum over all \(194\) irreps (instead of summing over elements of the group),\[

\sum_{\chi_i} \chi_i(g) \overline{\chi_i(h)} = \begin{cases} \left | C_M(g) \right |, & \mbox{ if } g, h \mbox{ are conjugate } \\ 0 & \mbox{ otherwise.}\end{cases}

\] you may determine whether \(g,h\) belong to the same conjugacy class or not. The normalization \(|C_M(g)|\) is the number of elements in the centralizer of \(g\) within \(M\). The centralizer of \(g\) is the subgroup of all \(h\in M\) that commute with \(g\), i.e. \(gh=hg\).

Many of these identities are deeply suggestive of the ER-EPR correspondence. In the Schur orthogonality relations, we're basically switching between two bases – one given by elements \(g\in M\) and their conjugacy classes on one side; and one given by components of irreps and these irreps (and their characters \(\chi_i\)) on the other side. Within the ER-EPR correspondence, I propose to identify the former – the elements with the monster group – with the ER "bulk" description of operators within the wormhole; while the latter – the representations – should be identified with the EPR side of ER-EPR, i.e. with the representations of the monster group and their entanglement.

So I propose to study a "metaduality":

\(ER=EPR\) is dual to the Schur orthogonality changes of bases.
Something like that should work, I think, but one should get much further. In the construction above, one only discussed the exact symmetry of the vacuum – which are completely unbroken – namely the monster group. This group should be considered a toy model for all gauge and global symmetries (and isometries of compactification etc.) that one encountered in a generic string/M-theoretical vacuum. (I hope that it's not too terrifying for you to call a monster your toy. As a kid, you should have played with monsters, too.)



A monster and a baby monster apparently eat tomatoes and kiwis, respectively.

But I would like to get much more from similar considerations than just some new duality that allows you to pick two different bases in a Hilbert space. I would like to generalize these constructions from the "twist of a wormhole" to all conceivable localized – and then local – operators you may think of, including the low-energy quantum fields. Those aren't generators of exact symmetries but if their energy is low, they may be close to it – think of all low-energy fields as some counterparts of the Nambu-Goldstone bosons (with the same idea that explains why these bosons are massless or light). So all these operators (especially when located in the new region of the spacetime, inside the wormhole) reshuffle some nearby microstates of the two black holes.

If there were some low-energy (and even not so low-energy) quantum fields inside the wormhole, they should be able to act on the Hilbert space of the two black holes – basically the tensor square of the state of all black hole microstates – in a certain way that generalizes the Schur orthogonality relations above. At the end, you should be able to see that the two black holes can't be an "exact" representation of an algebra of low-energy quantum fields. Instead, you should collide with some limits or restrictions of the Raju-Papadodimas type: the number of insertions can't be arbitrarily high etc.

Above, the idea was to start with the known spectrum of black hole microstates, use two such black holes, and study interesting operators acting on that space. In this way, we should get access to the "new region of the spacetime", namely inside the wormhole. The wormhole spacetime has a classically new, non-trivial topology. So the general lesson is that you could construct more complicated spacetimes and states with them from simpler ones.

It's possible that some complex enough spacetime, e.g. one we inhabit, could be constructed from many simpler ones, perhaps even \(AdS_3\) spacetimes, by similar Schur-like transitions to completely new observables etc. The monster group example shows that quantum gravity demands huge and almost absolute constraints on the spectrum of the "pieces" – the two individual black holes' microstates – that you may entangle and where you may study the interesting operators that may be embedded. In fact, the pieces' microstates are constrained by the new operators in between and vice versa.

There should exist a well-posed definition of this problem and all perturbative string theory vacua – associated with a conformal field theory on the world sheet – should be a subclass of solutions to this problem.

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