In recent years I’ve found there’s no point to trying to have an intelligible argument about “string theory”, simply because the term no longer has any well-defined meaning. At the KITP next spring, there will be a program devoted to What is String Theory?, with a website that tells us that “the precise nature of its organizational principle remains obscure.” As far as I can tell though, the problem is not one of insufficient precision, but not knowing even the general nature of such an organizational principle.

What one hears when one asks about this these days is that the field has moved on to focusing on the one part of this that is understood: the “AdS/CFT conjecture.” I’ve gotten the same answer when asking about the meaning of the “ER=EPR conjecture”, and recently the claim seems to be that the black hole information paradox is resolved, again, somehow using the “AdS/CFT conjecture.” Today I noticed this twitter thread from Jonathan Oppenheim raising questions about the “AdS/CFT conjecture” and the discussion there reminded me that I don’t understand what the people involved mean by those words. What exactly (physicist meaning of “exactly”, not mathematician meaning) is the “AdS/CFT conjecture”?

To be clear, I have tried to follow this subject since its beginnings, and at one point was pretty well aware of the exact known statements relating type IIB superstring theory on five-dim AdS space times a five-sphere with M units of flux to N=4 U(M) SYM. While this provided an impressive realization of the old dream of relating a large M QFT to a weakly coupled string theory, it bothered me that there was no meaning to the duality in the sense that no one knew how to define the strongly coupled string theory. This problem seemed to get dealt with by turning the conjecture into a definition of string theory in this background, but it was always unclear how that was supposed to work.

So, my question isn’t about that, but about the much more general use of the term to refer to all sorts of gravity/CFT relationships in various dimensions. There are hundreds if not thousands of theorists actively working on this these days, and my question is aimed at them: what exactly do you mean when you say “the AdS/CFT conjecture”?

**Update**: The ongoing discussion between Jonathan Oppenheim, Geoff Pennington and Andreas Karch about this on Twitter is very interesting, indicates that it isn’t so clear exactly what “the AdS/CFT conjecture” is. For me and presumably many others, would be great to have a source for an authoritative discussion of what is known about this topic. The Twitter format is very much not optimal for discussions like this.

I actually think of AdS/CFT as being a bit like the orbit method in representation theory. Each is really a collection of results rather than a top-down theory. In both cases, the results are statements that you can do some calculations in two quite different looking setups and somewhat mysteriously arrive at the same answer. In AdS/CFT, one of the important results is the Ryu-Takayanagi formula. This formula says an entanglement entropy in certain CFTs is equal to the area of an extremal surface in an AdS spacetime. This reminds me of the orbit method result that relates dimensions of irreducible representations to volumes of coadjoint orbits. In both examples, something from quantum mechanics is related to something geometrical. Of course, AdS/CFT is much more limited than the orbit method in the sense that it only applies to certain “large N” CFTs, while the orbit method is supposed to be about all of representation theory. At any rate, I would call “AdS/CFT” the AdS/CFT correspondence and use it to refer to this collection of correspondences.

Bob P,

In the case of the orbit method (or of the Langlands program that Geoff Pennington made an analogy to on Twitter), a top-down explanation of why they work may be missing, but a minute of googling will get you fairly precise technical definitions of what “orbit method” or “Langlands program” means. You’ll easily find the details of how specific co-adjoint orbits are supposed to match with specific Lie group representations or how specific modular forms are supposed to match with specific Galois representations. My question “what is the AdS/CFT conjecture” was not intended to be tendentious or get people arguing, but, seriously, to try and just understand exactly what people working on this mean when they use the term.

The proper experts will hopefully correct me if I’m wrong, but my amateur answer is that the old school version of AdS/CFT was that (i) the partition function of the bulk AdS theory (e.g. type IIB string theory in AdS5xS5) is the same as the partition function of the boundary CFT (e.g. N = 4 supersymmetric Yang–Mills theory ). And (ii) the Hilbert spaces of both theories are the same. This would give us a correspondence between bulk and boundary observables and correlation functions. I think there is a lot of evidence for (i).

The status of (ii) is very unclear to me. It would say that there is a 1-1 map between bulk states and boundary states. But https://arxiv.org/abs/1411.7041 gives an example where it can’t be true in its original form because it would lead to a violation of the no-cloning theorem. So in order to preserve some sense of (ii), they postulate that the bulk is encoded in the boundary as an error correcting code. In other words, each physical state in the bulk, needs many states of the boundary in order for it to be encoded in such a way that a violation of the no-cloning theorem is avoided. So (ii) might be true in the sense that there is a correspondence between physical states of the bulk, and “code states” of the boundary. I don’t know that there is much independent evidence that the CFT is an error correcting code, but it certainly has to be if some notion of (ii) is to be preserved.

But even this is not enough, because old black holes can have “too many states”. And so a more recent claim is that various bulk states which we think of as being orthogonal, are not really orthogonal. See for example https://arxiv.org/abs/2207.06536. So, I would say that (ii) is not regarded as true anymore, but it may be true in some sense (although I feel this is speculative).

There are other additional parts of the correspondence which are more speculative but are generally acknowledged as such (e.g. complexity = something). And there are elements which are claimed to be “established beyond reasonable doubt”, and yet I see no evidence for (and if anything, evidence against). Chief among them is the claim that because the CFT undergoes unitary evolution, the bulk quantum gravity theory must also be unitary, and therefore, black hole’s preserve information. I see no reason for (i) and (ii) to imply this, and I see no evidence for this part of the AdS/CFT conjecture. On the contrary, it forces researchers to embrace all manner of perversions, from violations of the equivalence principle, to post-selection, to the absence of free will and local degrees of freedom. I suppose this was the cause of my Sunday morning rage-tweeting. If only I’d just gone for a walk….

Hey Jonathan,

so we moved on to the long-text format. I think in your rant, I mean argument, you are getting yourself/us into a lot of trouble because you have a very old-fashioned notion of what states are in a theory of gravity. Diffeomorphisms are a gauge redundancy in a theory of gravity. Every statement that makes reference to local fields, local states, maybe “counts” how many states there are inside a black hole, is not really about physical states. Gauge invariant states in AdS are associated only with the boundary. It’s almost true by tautology that the states of a theory of gravity in AdS are those of a CFT. The only questions is really “which CFT”. I don’t think anyone has come up with a better definition of what we would even mean by the states of quantum gravity in AdS other than boundary observables. This is the AdS analog of the statement that the only observables in a theory in asymptotically flat quantum gravity are S-matrix elements. Any local question in Minkowski space is gauge redundant.

You (and the papers you cite) are asking interesting questions about how local physics in AdS can arise as an approximate notion from the dual CFT perspective. There are lots of puzzles there. But I really don’t see how this calls AdS/CFT into question. None of the things you call physical states in the bulk are physical.

And now I go hide in a hole …..

This is, by the way, also one of the main shortcomings of these tensor network models of AdS/CFT that are so popular these days. They capture many interesting aspects of fields on AdS space, except for the most important one: if we do gravity on AdS, none of the local states in the bulk is physical. They are all gauge redundant. Surely there is a lot one can learn from these toy models, but one shouldn’t take them too seriously. It’s a tall order having a toy model of quantum gravity that assumes gravity is not important.

(and now I should REALLY go hide in a hole ….)

I would have said that the AdS/CFT correspondence is that any quantum gravitational theory in an anti-de-Sitter space of dimension D+1 is described by a non-gravitational quantum theory in dimension D. Fundamentally, this is because a gravitational theory is believed to not have gauge invariant local observables; the true gauge -invariant observables must therefore live on the boundary, The isometries of AdS ensure that this boundary theory is conformal.

I do think that this idea has not changed; the original correspondence between type IIB in AdS5 times S5, and N=4 SYM was only the best worked out case.

The more general correspondence leaves the nature of the boundary theory somewhat vague, and is therefore quite plausibly true. The more specific one becomes about the boundary theory, the more tests are possible. There are a few very specific duality conjectures, including the original one; I don’t think the evidence for them is overwhelming.

Pace Jonathan Oppenheim, I think that if the boundary theory is unitary and the bulk theory is not, then this means that the AdS/CFT correspondence is manifestly false. But YMMV.

@Andreas, I think it is a bit too strong to say that gauge invariant states are associated only with the boundary. There are plenty of examples in AdS/CFT (e.g. the Lunin-Mathur and Lin-Lunin-Maldacena constructions) where a BPS spectrum is matched on both sides of the duality with exquisite precision. Specific BPS states in the CFT are mapped to specific BPS bulk geometries. Just because there is gauge symmetry (on both sides) doesn’t mean that one is unable to describe physical states in both bulk and boundary descriptions, and say what they look like deep in the middle of AdS. Of course, when one gets away from situations where calculations on both sides of the duality are under control, one is unable to construct the map. That doesn’t mean there isn’t a bulk theory of quantum gravity, or that one must somehow refer all its physical properties to the conformal boundary of spacetime.

@Jonathan, it seems to me that much of your complaint stems from the fact that we don’t understand much about the duality map as a detailed isomorphism in complicated dynamical situations, and that certainly includes those where black holes are involved. Even then, there are examples where one has confidence that the map is still valid. I am thinking specifically of AdS3/CFT2, where there is an infinite dimensional conformal symmetry whose representation theory highly constrains both sides. There is a moduli space of theories on both sides that includes a weak-coupling domain for the CFT where the Hilbert space is well-understood. The conformal symmetry continues to govern the asymptotic density of states through the Cardy formula over the entire moduli space, and it agrees with gravitational thermodynamics in the regime where gravity is weakly coupled. So either there is some non-unitary quantum theory of gravity which lives nowhere on this moduli space and whose thermodynamics has nothing to do with an underlying quantum statistical mechanics and its associated state-counting, or quantum gravity is indeed unitary and the thermodynamics does have such an underlying interpretation. In the latter case, it would be unreasonable to suppose there is no duality map, since the CFT is explicitly such a quantum system with exactly the same thermodynamics, with a precise match of BPS protected quantities, etc.

Many of your other quibbles with the correspondence have to do with more recent work much of which is on rather less sound a footing. I am not surprised if you are confused by it, since in some cases the authors themselves appear to be confused about how AdS/CFT works. If some of these ideas turn out to be wrong, it will not invalidate AdS/CFT but will rather tell us that certain ideas about the properties of the duality map are incorrect. I have not seen ANY evidence to date that suggests that the duality itself is in question.

@Peter, the most established part of the conjecture has to do with maps of protected quantities where (super)symmetry allows calculations on both sides of the duality that can be compared. In addition to those mentioned above, there are now quite sophisticated evaluations of the gravitational path integral that match the asymptotic series of BPS partition sums computed in the CFT. The semiclassical expansion of the gravitational path integral (with the sum over saddles generating subleading terms in the asymptotic expansion) takes one beyond the leading order gravitational thermodynamics to a fuller understanding of the correspondence. Here one is not asking gravity to describe microstates in the Hilbert space directly, that is likely the wrong thing to do (just as the terms in the asymptotic expansion of an elliptic function obtained by modular transform are not associated with individual terms in its original q-expansion); rather, gravity seems to know how to count states in aggregate (including subleading terms in the expansion) without resolving the microscopic details.

These calculations are not grand statements of principle, but rather detailed computations in specific examples where one can work out a match between the same quantity in AdS and CFT. Many checks of this sort have been done, amassing a great deal of evidence for the conjecture, but what the full scope of the duality is remains unclear. The most solid part of the correspondence is a list of explicit examples for which there is a wealth of supporting evidence. This success has led to a reasonable belief that any other example one can motivate as a limit of string theory will hold as well.

Where people seem to be running into trouble is in the arena of trying to abstract some grand principle that underlies and unifies (and perhaps extends) all the particular examples, in the process abandoning much of the string theory which makes the details work in the examples we know. One such is ER=EPR, but there are others (to paraphrase Groucho Marx). Many focus on toy models which may or may not contain the essential physics. Many have given up on understanding the bulk dynamics in favor of making inferences/conjectures about what the bulk physics must be like by analyzing the CFT, which ends up processing bulk physics through some poorly understood nonlocal map in a strongly coupled field theory; there seems to be rather less effort in the opposite direction (doing string theory in AdS), likely because the calculations are rather more difficult.

You can’t hide in a hole Andreas, the boundary sees everything! 😉

I think even you have to admit that this proposal that you’re advocating along with Suvrat Raju and others, is very controversial within the string theory community. My impression is that it is a small minority of people who believe it, while the majority believe what you call the “old fashioned notion”. But that could just be that I tend to hang out with a different crowd of string theorists…. Namely, those who believe that GR makes sense in our universe without a boundary, and “the value of the field where my finger is pointing at” is a diffeomorphism invariant statement.

At any rate, that’s besides the point, because the one thing that is obvious here, is that string theorists have extremely different opinions about what counts as a state of the bulk, and yet they all seem to agree that the AdS/CFT conjecture is obviously true. In the same way as all string theorists believe the black hole information paradox has been solved, but completely disagree on the solution.

I find this tendency frustrating because I learn a lot from my string theory colleagues, but it is often made difficult by overly bold claims. And it is this sort of tendency which led to the wormhole-on-a-chip fiasco, and which distorts our view of which research directions are likely to be fruitful.

What I describe in my comments is much more basic than my work with Suvrat. The fact that there are no local gauge invariant observables in a theory of quantum gravity other than those defined at the boundary has been hammered away by people like Witten from the earliest days of AdS/CFT. Please don’t mix this up with somewhat more conjectural statements about black holes. Local states in the bulk aren’t physical. I don’t think you’ll find many people who disagree with this assertion. An interesting question is how to reproduce approximately local quantum field theory observables from the CFT.

PS: for the record, I am referring here to Witten’s talk at the Strings 98 conference in Santa Barbara where he pointed out that the fact that there are no local observables in a theory of quantum gravity basically tells you that just by asking what you mean by gravity in AdS, the only observables we know how to define are correlation functions at the boundary. At this level, AdS/CFT is really a tautology. The fact that you find this even controversial is somewhat puzzling to me.

Thanks Andreas. Some genuine questions:

1) Is there agreement that it’s a tautology? What then do you make of https://arxiv.org/abs/2207.06536 which claims that there is not an isomorphism between local bulk and boundary states.

2) When you say that the only observables we know how to define are correlation functions at the boundary, would you agree that there might be others that you don’t know how to define (e.g. relating to observables inside a black hole)?

3) There is a community of researchers who study relational observables in GR. E.g. “the value of the field where Alice’s finger is pointing,” is a diff invariant statement. I’ve always regarded this program as moreally equivalent to string theorists relating their observables to the boundary (just replace Alice with the boundary). Would you agree?

4) I of course agree that there are diff invariant observables that you can relate to the boundary, e.g. the value of the field along a geodesic some proper time from the boundary. And via some extrapolate dictionary, reconstruct regions in the bulk from the boundary, But don’t you think that https://arxiv.org/abs/1411.7041 successfully argues that you can get a potential violation of the no-cloning theorem, because naively, it appears you can reconstruct the same bulk observable from different parts of the boundary?

Just a note that due to the way moderation of comments works, Emil Martinec’s very helpful comment just now was posted, even though it is earlier in the time sequence.

Thanks Jonathan. I think your points 1, 2, and 4 are all about the question how apparently semi-classic physics in the bulk can arise within a large N CFT. There are open questions there. But this is a higher order question: if there are no local observables in a theory of quantum gravity, how come local QFT works so well to describe the experiences of a low energy bulk creature? We happily do experiments in labs and don’t worry about this not being strictly defined an observable. But I don’t think many would argue that this program contradicts the assertion that there are no local gauge invariant observables.

Sometime it feels like rephrasing AdS/CFT in quantum information language, despite all its new insights, has made people forget some very basic lessons learned long ago.

As to your 3, the relational observables, they are defined as integrals over all of space and as such only depend on boundary data. They are boundary observables that have the power to give you an approximate sense of local physics in the bulk. That’s exactly the sort of thing I am talking about. This program accepts that there are no gauge invariant local observables and tries to do interesting physics with it.

Thanks Andreas. When I hear people invoke AdS/CFT, it is often to claim that the CFT is a quantum theory of gravity. And so for me, the question of how (or whether) semi-classical physics arises in the bulk, is the central question. In order for the statement that the CFT is quantum gravity to be a testable and meaningful statement, it must reproduce semi-classical gravity (or something which could reasonably be said to be the correct low-energy or classical limit). Otherwise, it is not a quantum theory of gravity, but a quantum theory of something else. That doesn’t mean it can’t teach us a lot about quantum gravity, but it wouldn’t be quantum gravity.

Since you call them second order questions, maybe they are impossibly to answer right now, and that’s fine. But I feel like people act like it’s settled, which is what I take issue with.

Regarding relational observables – I think we mostly agree here. I suppose I would say they can be defined in a space-time without a boundary and so I don’t need a boundary to have gauge invariant observables which are the moral equivalent of local ones.

@Emil, thanks so much for such a detailed answer, much appreciated. Indeed, it is in these more complicated situations where I feel people’s apparent certainty is less supported by evidence. Although these are complicated situations, they are important ones — the black hole information paradox & black hole thermodynamics seem to be the key questions for quantum gravity.

In classical gravity, observables need to be diff invariant, but in quantum gravity, an observable is (expected to be) defined through an integral over all bulk metrics with some fixed boundary value. This is much more restrictive than the classical case. For example, “the value of the field along a geodesic some proper time from the boundary” cannot be an observable in QG; when the metric changes, the geodesics themselves change. It is for this reason that only boundary-to-boundary correlations are believed to be gauge invariant. I didn’t think this was controversial, but things may have changed.

And as everybody has said, this doesn’t mean that it is not interesting to talk about local bulk physics; we just have to be cognizant that this must be in some sense approximate.

@Jonathan, just to be clear: Among the complicated situations that ARE well-understood are BPS black holes with macroscopic horizon area, where there is a robust CFT state counting that agrees with the AdS gravitational path integral.

It is useful to divide the black hole information problem into two aspects: the information storage problem, and the information recovery problem. The fact that the BPS spectrum has been precisely matched tells us that AdS/CFT knows how to solve the information storage problem for this class of extremal black holes, in that on the CFT side of the duality one is explicitly counting the microstates, and the AdS gravity computation reproduces this counting at the level of an asymptotic expansion for large charge/mass.

The more complicated information retrieval problem where we throw something into such an extremal black hole and ask what comes back, and how it comes back, and which of our assumptions that lead to the information paradox is breaking down in AdS quantum gravity, is currently out of reach. Not because AdS/CFT is wrong, but because we currently lack the tools to ascertain what is going on on the AdS gravity side of the duality, let alone calculate correlators in a strongly-coupled CFT, or produce an explicit map between the two. But again, this is a statement about our poor understanding of the duality map, not about its existence.

I think most people assume that because we have all these precise tests of the things we can calculate, that the duality holds not only for symmetry-protected quantities but in general. Again, that it is unreasonable that it holds only for some class of states (which includes extremal black holes) but not others. And then the duality tells you that the information retrieval problem is solved in principle by the unitarity of the CFT under the assumption that the duality is an exact (if inexplicit) map.

I think this is where most people in the subject are at the moment — at some point you have done enough checks of the duality, assume it is true and ask what are the consequences. Much as people no longer question the truth of the Riemann hypothesis, and instead work out its implications. Which is not to say that it is unimportant to prove either conjecture; rather that until that day arrives there are useful consequences to explore assuming their respective validity.

@AR, good luck describing what is going on today say at the LHC using only the asymptotic data from the inflaton fluctuations in the far past which eventually assembled into it and its beams, and wherever those beams (and the atoms in the LHC) end up in the asymptotic future.

If (as some people were briefly fantasizing before the LHC turned on) the fundamental Planck scale had been sufficiently low for the LHC to have made small black holes, I suspect we would be trying to use a bulk quantum theory of gravity to calculate the scattering amplitudes for this process in a small spacetime region around the collision rather than worrying about where the LHC is hiding in the primordial inflaton wavefunction.

Hi Jonathan,

I’m just entering the discussion so maybe I’ve missed a lot of the background. But just reading your first few comments. 1411.7041 doesn’t say anything to the effect that the Hilbert space cannot be the same. The point is simply that approximately local operators in the bulk can be written in different ways on the boundary. More precisely, we have multiple operators \phi_{i} which have the property that within a low-energy set of states, the different \phi_{i} act in approximately the same way. This is quite well understood and this phenomenon occurs even about empty AdS. Where is the violation of no-cloning here?

The issue about the growing volume in the black-hole interior that leads to a seeming puzzle with the entropy is a more interesting problem. Mathur pointed it out several years back. But one popular proposed resolution is simply that states that *appear* to be independent in the bulk are not really independent. One kinematic observation is that this can happen quite easily in high-dimensional Hilbert spaces. See https://arxiv.org/pdf/2010.03575.pdf

In a Hilbert space of dimension e^{S}, with S >> 1, it is possible to find e^{S’} vectors (|\Psi_{i}>) where S’ >> S which have the property that their mutual inner product || << 1 for all i,j.

There are many things we don't understand about the duality and about black holes. But I agree with Andreas. I don't think it is accurate to suggest that there are somehow well-known problems with the original formulation of AdS/CFT in terms of the equivalence of Hilbert spaces.

Thanks @Emil. Here I have to disagree. The fact that string theory gets the entropy right for a class of symmetry protected extremal black holes, does not imply that AdS/CFT implies that black hole evaporation has to be unitary. I can predict the entropy of a box of gas from it’s microscopic or macroscopic properties, and that holds just as well in a unitary theory as well as an open system. I think it’s perfectly reasonable to believe that black-hole evaporation is unitary for various other reasons. But that’s very different from claiming that AdS/CFT tells us that black-hole evaporation is unitary (which is what is often claimed). In fact, I would say that AdS/CFT is either silent on this, or appears to lead to the opposite conclusion.

I agree that it is reasonable to assume the conjecture is correct and explore the consequences. I think that’s a solid research direction to take. We do this all the time in quantum information theory and computer science, but we’re always careful to append some line such as “assuming the polynomial hierarchy doesn’t collapse.” Even though in that example, we are virtually certain that it doesn’t. This level of care is what I often find missing, and I do think it leads to genuine confusion, even amongst experts, as to what is solid and what is less solid.

In many cases, I feel that the conclusions that are drawn assuming the AdS/CFT conjecture would cause me to question the assumptions, at least in the particular setting under consideration. But I never see that happening, and so am struck by people’s confidence.

Jonathan, I’m not sure what you have in mind. Among the BPS protected quantities are the operators that introduce and absorb (super)gravitons from the AdS conformal boundary, that can again be matched on both sides of the duality. For instance, such operators acting on the CFT vacuum/global AdS yield protected single-particle states. So we can begin with the AdS vacuum and send in some gravitons with enough energy (order $n^2$) that in the bulk description would make a black hole (this could even be two very high energy gravitons that in semiclassical gravity look like colliding shockwaves). On the CFT side, one has a strongly coupled field theory in which such excitations rapidly thermalize. On the gravity side, one has black hole formation. Now, I suppose one could try to claim that the same operator that makes a graviton excitation on the AdS vacuum does not do the same thing when acting on the single-particle excited state generated by the first graviton operator, but that seems rather unreasonable — the interpretation of the asymptotics of the operators (eg single-trace operators in N=4 U(n) SYM being dual to supergraviton creation/annihilation operators) is independent of the state they are acting on. The intermediate state is a microstate in the thermal ensemble of the CFT, and a black hole in the gravity theory, with energy of order $n^2$. The density of intermediate states is the same on both sides.* One can then de-excite this state long after thermalization by removing low-energy supergravitons near the boundary in the AdS description, dual to the same local single-trace operators in the gauge theory. The gauge theory description of this process is explicitly unitary. It has a dual AdS description in terms of incoming and outgoing supergravitons.

So as long as we stay outside the black hole and compare the two descriptions of the evolution, there seems to be a complete parallel. Of course, I’m not giving you the exact bulk description everywhere in the bulk because we don’t have one at the moment. But the description of what’s going on as seen from far away is built from parts of the duality that ARE well-understood.

Could you be a bit more explicit about what you find lacking in this line of reasoning? Do you think there are “hidden variables” that don’t participate in the gauge theory description, but are there on the gravity side, that somehow invalidate the duality? The matching of the non-extremal entropy makes this unlikely. I’m puzzled as to why you say that AdS/CFT is silent or even negative on the issue of unitarity in black hole evaporation.

* There is a well-known factor of 3/4 between weak and strong coupling entropies in the gauge theory that is understood to be due to interaction corrections in the strongly coupled gauge theory. If this bothers you, we can work in the AdS3/CFT2 examples where the thermodynamics doesn’t change between weak and strong coupling in the CFT2, due to the additional power of conformal symmetry in 2d.

@Emil, since we are talking about extremal or near-extremal black holes, and they don’t really evaporate, to what extent can we even pose the question here, much less answer it? And even if we could pose the question, we can’t check the other side of the duality, so is this not a question of faith?

In terms of what I find lacking, I would say that any attempt to reconstruct the bulk when we have a late-time evaporating black hole results in conclusions that would prompt me to reconsider the premise. The conclusions being either a break-down of the equivalence principle at low energies for large black holes, or the requirement for post-selection (which already signals a breakdown of unitarity) etc. There is no need to introduce hidden variables to consider a break-down of unitarity.

The need to consider island corrections to the entropy (because AdS/CFT seems to over-count the entropy of radiation), or the need to consider non-isometric bulk to boundary maps as in https://arxiv.org/abs/2207.06536 (and I see now Suvrat mentions https://arxiv.org/abs/2010.03575) seem to me to be symptoms of that. Which is not to say that it can’t all be made to hold together, but only to say that at the moment, it doesn’t. In my view we don’t have a clue what is going on, and shouldn’t claim otherwise.

@Suvrat, apologies, your comment was moderated so only appears now. Regarding no-cloning and error correction, this is the content of 1411.7041, and I mean no more than what is contained in that paper. My naive quantum information theorist’s view of an error-correcting code is that one needs to encode the logical states in a larger physical Hilbert space. Is this still compatable with the Hilbert space’s being equal?

Regarding the more important question of proposals to deal with the over-counting of states (as in 2207.06536 or 2010.03575): It is certainly possible that bulk states which we think of as being orthogonal are not actually orthogonal. But this seems unlikely to me, and one is introducing yet another radical concept to fix a problem with the original proposal. We also appear to require that states with high complexity do not have semi-classical bulk duals. These concepts may be required to save AdS/CFT in this regime, and I applaud them being explored. But if one keeps on having to introduce new mechanisms, then the project starts to look less elegant and less likely to be true. Which is not to say that it can’t be made to work, only that if we need to keep introducing new mechanisms along our path, we might want to at least question whether we’re heading in the right direction.

@Jonathan, a black hole in AdS can be made to evaporate by extracting energy at the conformal boundary by hand — there is a finite probability to find Hawking radiation at any radius, and we can collect it near the boundary and spirit it away. We are free to decide when and where we want to introduce the operators that accomplish this. On both sides of the duality, this corresponds to acting with field operators at the conformal boundary by a well-understood limiting procedure, which on the CFT side amounts to using local operators to extract energy from it. So as I said, one can make a black hole form by sending in quanta from the conformal boundary, let the system thermalize, wait as long as you want, and then extract energy at the conformal boundary. So yes, we can pose the question.

I have been trying to argue that all of this is much more than a matter of faith, rather it is a reasonable hypothesis supported by mountains of evidence. And that evidence points to the existence of a unitary bulk theory of quantum gravity. Of course there is more to be understood about black hole interiors, where/how does the Hawking calculation go wrong, etc. The information problem isn’t solved (claims to the contrary in the popular press notwithstanding).

The utility of any duality is that it provides an alternate route for calculation; one accepts the hypothesis and then uses whichever side of the duality is simpler to analyze for the question at hand. Otherwise, what is the point? To do every calculation twice because we might find the N+1st check of the duality invalidates it? So perhaps the more germaine issue might be why, if one is interested in answering questions about physics inside the horizon, one wants to attack that problem by using the CFT side of the duality, where it is filtered though all sorts of ill-understood, nonlocal properties of the duality, and moreover in a strongly-coupled field theory. I would have thought using the side of the duality where quasi-locality in the bulk doesn’t need to be “reconstructed” would be more appropriate.

You mentioned recent proposals about islands, non-isometric codes, etc. They are a work in progress, the sort of toy model that I was referring to as perhaps having discarded essential ingredients. There is no “need” to consider any of them; and they may very well be wrong-headed. They are examples of what I was calling attempts to formulate grand principles — models and interpretations layered over the basic facts of AdS/CFT. And they are not guaranteed to be correct. In some cases, they represent a radical departure from what had seemed to be settled about how AdS/CFT works. And if indeed they turn out to be red herrings, the basic facts of AdS/CFT will still stand.

I will add one more option to the menu, which obviates all the discussion about islands, old black holes, bags of gold, overcounting/non-orthogonality of states etc. We have several examples of non-geometric phases of string theory, where gravity is not the most important actor and its effects are overwhelmed by the plethora of other degrees of freedom in string theory. Examples include the high-temperature Hagedorn phase, Landau-Ginsburg phases of string compactifications, and the formation of “giant gravitons”. In AdS/CFT, gravity is a set of collective modes, order N^0 among N^2 degrees of freedom. In geometric phases, most of the N^2 degrees of freedom are confined/not activated/in their ground state. In AdS/CFT, the formation of a black hole involves on the CFT side a deconfinement transition. It might then be that on the AdS side, the horizon is a phase boundary, and there is no geometric interior (the analytic continuation of the vacuum geometry past the horizon presumes that the N^2 degrees of freedom are still confined there).

Much of the confusion at the moment stems from trying to hold onto a geometrical picture of the black hole interior, and presuming that the only place we need strings and branes is well inside the horizon near the singularity where tidal forces are large. We then get painted into a corner where all of a sudden we are invoking effects over much larger distances than the horizon scale in order to rescue the theory (like saying that the black hole interior at late times is non-locally encoded in the early radiation arbitrarily far away). So the lack of a geometrical interior is a possibility which bypasses many of the things you find objectionable, though there is plenty to object to about this alternative as well. Anyway, best to shut up and figure out a way to calculate, if we can.

@Emil Martinez: If there is no “need” to consider any of these unconventional proposals about islands, non-isometric codes, etc., why are physicists considering them, and claiming in their papers that they solve certain problems that arise in AdS-CFT? I have too much respect for the authors of these papers to believe that they are introducing these seeming contrived ideas solely to lengthen their publication lists.

So I can only conclude that AdS-CFT is a conjecture with many unanswered questions about how it can be applicable to our universe.

Daniel Harlow and Daniel Jafferis claim that Jackiw-Teitelboim gravity is a quantum gravity in 1+1D AdS space which has no CFT dual:

https://arxiv.org/abs/1804.01081

Thanks to all who have provided some useful explanations about AdS/CFT. I feel I’m still though lacking what I’m looking for, a specific general conjecture about which two theories are in duality. More specifically:

I see the claim for N=4 SYM in d=4 and its supposed relation to a string theory/supergravity low energy limit, compactified on S^5. I also can see how different compactifications of this in different dimensions will give different pairs of conjecturally dual theories. But:

1. In dimensions above 3/2 does the conjecture go beyond this and how do I know what the paired dual theories are?

2. In dimension 2/1 and 3/2, are the dual pairs supposed to be related by compactification to the higher dimensional story?

3. In 3/2 d, writing gravity in CS form, there’s a clear analog of the usual Chern-Simons story relating 3d gauge theory to 2d CFT on the boundary. What is the relation of this to AdS3/CFT2?

4. What exactly is the current best statement of how AdS/CFT relates 2d JT gravity to 1d SYK (presumably addressing the issue in the Harlow/Jafferis paper linked above)?

I believe the “best” (in the sense of being as close to the rigor of mathematical physics as it is possible in the present state of our understanding of QFT and string theory) work along the lines of proving the AdS3/CFT2 correspondence is currently due to Rastelli’s and Gaberdiel’s school. More precisely, this school is trying to show that type IIB string theory on AdS3 × S3 × M4 with p units of NS flux contains an integrable subsector, isomorphic to the minimal (p, 1) bosonic string. Some relevant papers:

https://arxiv.org/abs/hep-th/0507037

https://arxiv.org/abs/1011.2986

https://arxiv.org/abs/1207.6697

For some recent work on the original form of the Maldacena’s proposal, using the pure spinor formalism, see:

https://arxiv.org/abs/1903.08264

@Peter:

1. Dual pairs can be constructed by taking low-energy limits of brane configurations in string theory. The basic idea is that the low energy limit of the open string theory on the branes is a gauge theory coupled to matter. On the gravity side, the low-energy limit is the near-source geometry of the branes. The brane construction determines the field content and then one analyzes the field theory RG to determine whether it is (i) scale invariant or (ii) has an interesting IR fixed point. At large N this is dual to (i) a self-similar geometry, or (ii) to a non-scale invariant geometry with an AdS limit as one scales it toward the brane source. If the field theory happens to be scale-invariant from the start, then the geometry will have the AdS form.

1a. For instance, the IR limit of N D2-branes is maximally supersymmetric gauge theory in d=3. This theory is not conformal, but has a nontrivial IR fixed point that is dual to M-theory on AdS4xS7. It was subsequently realized that this theory and many other AdS4/CFT3 dual pairs can be realized as Chern-Simons gauge theories coupled to matter, going by the name of the ABJM construction and various elaborations of it.

1b. The IR limit of fivebranes yields a nontrivial 6d SCFT with (2,0) superconformal symmetry, dual to AdS7xS4. Webs of intersecting fivebranes yield 5d SCFT’s dual to a variety of examples of AdS6/CFT5. So there are examples in all dimensions where there is field-theoretic superconformal symmetry.

2. Compactification of the CFT does not usually give lower-dimensional correspondences, rather one typically intersects more branes. For instance, maximally susy AdS3/CFT2 comes from intersecting two stacks of three-branes, rather than by compactifying a single stack of three-branes. The 2d SCFT lives at the intersection. There is now a rich zoology of such constructions with less supersymmetry.

3. First let’s review how things work in 2d JT gravity, which can be written as a BF topological gauge theory for SL(2,R). Because it has no bulk dynamics, the dynamics can be pushed onto a set of edge modes, whose action is the Schwarzian (the natural action of coadjoint orbits of Diff(S^1)/SL(2,R). This story lifts to 3d Chern-Simons gravity, whose only dynamics is again that of boundary diffeomorphisms, governed by a 2d version of the Schwarzian theory originally studied by Alekseev-Shatashvili in 1988 and derived in this context by Cotler-Jensen 1808.03263. Note that this is not the CFT of the duality, rather it is simply a generating function for stress tensor Ward identities and does not realize the BTZ black hole density of states.

@anonymice, the version of this statement for the JT gravity/Schwarzian theory is the content of the paper you referenced. JT gravity and the Schwarzian theory have almost no states and so do not realize (n)AdS2 BH microstates; but because gravity knows about BH thermodynamics, it can be used to compute the BH density of states (as usual, the thermal free energy is the gravitational action).

4. The Schwarzian theory can be derived as the dynamics of a set of collective modes of the SYK model via a standard set of standard manipulations, but SYK has more content than just these collective modes — for instance it actually has 2^N states while the Schwarzian has just a few. The Schwarzian theory is equivalent to JT gravity — the latter, being topological, can be reduced to a set of edge modes. Then one could ask whether there is some sort of 2d theory that is exactly dual to the 1d SYK. It is unlikely that there is a quasi-local bulk theory exactly dual to the entire SYK model, in contrast to the examples derived from string theory where it is believed that string theory in the bulk and the large N CFT are exactly dual. At the very least, such a theory would be non-local at the AdS scale.

One might say (and has) that gravity a thermodynamic collective field theory, while string theory realizes the underlying statistical mechanics. AdS/CFT makes this more than a slogan, by relating string theory in AdS to a standard quantum system with a standard quantum statistical mechanics. The astonishing fact about gravity is that it knows about the BH equation of state without knowing much about the microstates; whereas in conventional systems, one has to have the microphysics in hand first, and use it to derive the equation of state in order to then start doing thermodynamics.

Well, either there is a proof, or it might be that the duality doesn’t actually hold in all conceivable cases. It is thus a conditional calculation. Or else you are making a definition that the result of what is happening on one side is literally the thing that happens on the other, and no one has checked that this is consistent with the definition not using the duality.

It’s not about doing calculations twice, but actually proving conjectures. Is there some solid statement of some conjecture here that is proven, with the hypotheses of some proof of at least a wide class of special cases spelled out explicitly? Otherwise I fail to see how a conjectured mechanism that has only been calculated in a bunch of cases, but not proven for all the cases people care about, can be considered anything other than an open problem. Experimental mathematics is full of examples of patterns that fail to continue, and famously some of them continue for ranges that make the volume of the universe in Planck volumes look approximately zero.

@Emil, as far as I understand the process you’re describing, you are inserting operators into the action, and thus changing the physical content of the theory during evolution? This doesn’t seem to have much to do with black-hole evaporation, and I’d be loath to draw any conclusions about that. But I’d be happy to come away from this discussion with some reading homework if you have suggestions. It sounds like a check of the partition function, which I am not challenging.

I think evaporating black holes is more than just another N+1 check. And this is made apparent by the fact that when one tries to reconstruct the bulk using the standard rules one gets nonsense, unless one adopts the various proposals (island formula, non-isometric codes) that have recently been put forward. Maybe I’m biased, being part of It from Qubit, but I am with @Peter here — these proposals are being considered because we are being forced to — otherwise there are paradoxes. And whether these new proposals can be made consistent is very much a great challenge and an open question.

@Jonathan, no, I am not changing the action or the Hamiltonian of the CFT. The suggestion is to compute correlators in the same theory, with the insertion of a few operators at early times that make black holes as intermediate states, and then at late time insert many operators that gradually collect the Hawking radiation. The procedure for introducing/extracting such quanta goes back to the classic papers of Gubser-Klebanov-Polyakov hep-th/9802109 and Witten hep-th/9802150. That a small number of such insertions can make a black hole is exemplified by calculations in AdS3 of two energetic particles making a BTZ black hole, see for instance gr-qc/9809087. The same operators, looking at much lower energy modes, can be used at late times to collect the radiation.

A closely related procedure is to couple the CFT to a bath (which indeed involves a modification of the CFT Hamiltonian by a term that couples the same local CFT operators discussed above to creation operators for bath excitations) and thereby transfer energy from the CFT to the bath. If we turn on such a coupling adiabatically at very late times, it does the same thing as I was describing above — remove Hawking radiation from the CFT.

As to your last comment, I think we’re largely in agreement that there is a problem somewhere; we just disagree on whether the takeaway is that AdS/CFT duality is wrong or whether people are making incorrect inferences about the duality map.