### Rehren Duality

Recently, there have been some quite lively discussions of Sean’s review of Lee Smolin’s book and Clifford’s synopsis of Lee’s radio appearance with Jeff Harvey. One of the things that one discovers about such discussions is that the same issues keep cropping up over and over, and one gets the sense of very little progress being made, as the participants don’t seem to have assimilated the lessons of previous discussions.

So, I found myself bashing my head against the keyboard when I saw the following comment

Bert Schroer, in the paper here says, beginning on page 18:

A profound mathematical theorem reveals that there is even a unique correspondence between Local Quantum Physics {QFT both Lagrangian and non-Lagrangian} models in n+1 AdS spacetime with a n-dimensional conformal invariant Local Quantum Physics model?..I have tried all possibilities of what Maldacena could have meant and none of them seem to be consistent with the above structural theorem.

As I understand it, most string theorists regard Maldacena?s AdS/CFT as a done deal. Will they continue to believe this in the face of a rigorous proof of the contrary by a non-string theorist?

The commenter, here, is referring to Rehren Duality, a proposal, in the context of AQFT, that a conformal field theory on the boundary of AdS is isomorphic to an ordinary, (non-gravitational) QFT in the bulk of AdS. If *true*, this would, presumably, be incompatible with the AdS/CFT, which posits that the theory in the bulk is a gravitational one.

Since it’s tiresome to have to explain what’s wrong with this proposal over and over again, every time the subject comes up anew, I decided to grit my teeth, and try to write a post so that, in future, one can simply get away with linking back here.

I’m not not going to attempt to give an introduction to AQFT. That would be boring as hell, but – fortunately – it’s mostly unnecessary. All you need to know is the the object of study is a cosheaf^{1} of operator algebras. For each open set, $U$, we associate an algebra, $A(U)$, which roughly, corresponds to “the algebra of observables supported on $U$”. Of course, not just any old cosheaf of operator algebras will do. One needs to build in the usual properties of locality, Poincaré invariance (if we are working on Minkowski space), *etc*. You can amuse yourself by trying to figure out how to phrase QFT in the language of cosheaves of operator algebras, or you could look at the papers, or you can simply trust that it can be done.

One thing that we *will* need to borrow from the Haag-Kastler Axioms is the Locality Axiom.

If $U$ and $U'$ are causally-disconnected (that is, if there does not exist an everywhere timelike path in $X$ connecting some point in $U$ with a point in $U'$), then $A(U)$ and $A(U')$ mutually-commute (as subalgebras of $A(X)$).

Anyway, the metric on AdS_{$d+1$} can be written in “global coordinates” as
${\mathrm{d}s}^2 = \frac{R^2}{\cos^2\rho}\left(-{\mathrm{d}\tau}^2 +{\mathrm{d}\rho}^2 +\sin^2\rho\, {\mathrm{d}\Omega}^2 \right)$
where ${\mathrm{d}\Omega}^2$ is the round metric on the unit $(d-1)$-sphere. The boundary of AdS is located at $\rho=\pi/2$. For AdS itself, $\tau$ is periodically identified $\tau\sim \tau+2\pi$ and the boundary has topology $S^1\times S^{d-1}$. String theorists usually work on the universal cover where $\tau\in \mathbb{R}$, and the boundary has topology $B=\mathbb{R}\times S^{d-1}$. Rehren works on AdS itself. For most of what I will say, it won’t matter.

The boundary metric, ${\mathrm{d}s}_{\text{b}}^2 = -{\mathrm{d}\tau}^2+ {\mathrm{d}\Omega}^2$ is conformally flat.

Rehren’s prescription for establishing an isomorphism of the observables of the bulk and boundary theories is a little obscure. Here’s my best attempt at rendering it into English.

- Choose a null geodesic, $L$, of the boundary metric.
- The complement, $B\setminus L\simeq \mathbb{R}^{1,d-1}$, and the subgroup of $SO(2,d)$ which preserves it can be identified $d$-dimensional Poincaré group ($\times$ dilatations).
- For any point, $p$, in the interior of AdS, there is a unique future-directed null geodesic, $\gamma_p$, which starts at some point on $L$, and passes through $p$.
- Let $f(p)$ be the subsequent intersection of $\gamma_p$ with $B$.
- $f$ thus defines a “stereographic” projection of AdS space onto its boundary, $B$. Moreover, it maps an open dense subset of AdS onto $B\setminus L\simeq \mathbb{R}^{1,d-1}$.

Rehren’s prescription for constructing an isomorphism between an “AQFT” in AdS and a conformal AQFT on the boundary is now simple to state. From the cosheaf of algebras of local observables, $\mathcal{A}_{\text{AdS}}$ on AdS, construct the direct image cosheaf^{2}
$\mathcal{A}_{\text{CFT}} = f_* \mathcal{A}_{\text{AdS}}$

One wants to claim^{3} that the resulting theory is a conformal QFT on $B$.

Timelike geodesics in AdS get mapped to time-like curves on the boundary. However, not all timelike *curves* do. As an extreme case, on AdS itself (as opposed to the universal cover) *every* pair of points can be connected by a timelike curve. That’s just one of the screwy consequences of the fact that AdS is foliated by closed timelike curves. But, even on the universal covering space, it’s a well-known consequence of the focussing of timelike geodesics in AdS that not every pair of points, which are causally connected, can be connected by a timelike *geodesic*. (This is unlike the case of Minkowski space.)

In any case, it is timelike geodesics that, Rehren finds, map to the right thing on the boundary. And so Rehren, quite arbitrarily, decides to change the nature of the Locality Axiom, replacing “timelike path” with “timelike geodesic”. This is an incredibly strong restriction on a putative “QFT” in AdS: operators which are timelike-separated (but not connected by a geodesic) are required to commute. I don’t think there are any examples of interacting QFTs that satisfy it.

The Locality Axiom is, of course, not the only axiom that needs to be savaged, in order to establish the correspondence Rehren is after. Since neither AdS, nor its universal cover have a Cauchy surface, it’s impossible to implement the Time Slice Axiom (which states that, for $U$ the neighbourhood of a Cauchy surface, $A(U)\simeq A(X)$).

The bottom line is that the “AQFT,” that Rehren would like to construct as the “dual” of a CFT on the boundary, bears little resemblance to a bona fide quantum field theory.

To obtain a bona fide field theory on AdS, one first needs to pass to the universal covering space (to eliminate the closed timelike curves) and then impose *boundary conditions*, to obtain a well-defined Cauchy problem.

Of course, once one imposes boundary conditions, then the physics depends on those boundary conditions. And it’s that dependence, that leads, inexorably, to Maldacena’s version of AdS/CFT.

Well-known arguments lead to the conclusion that the partition function (as a function of the boundary-value data) satisfies the requisite properties to be the *generating functional* of a (conformal) QFT on the boundary. Among the local operators in any local, Poincaré-invariant quantum field theory is a conserved stress tensor, $T_{\mu\nu}$. The source for $T_{\mu\nu}$ is the boundary value of a massless spin-2 field of the bulk theory. And this leads, inexorably, to the conclusion that the bulk theory is a gravitational one.

One can’t, strictly speaking, say that Rehren’s proposal is **wrong**. He is perfectly free to *define* what he means by “AQFT in AdS space.” Having made a suitably cockeyed definition, he can then go on to assert that the result is isomorphic to a CFT on the boundary.

This is, *in no way*, incompatible with Maldacena’s conjecture which isn’t an isomorphism between the bulk and boundary theories, but rather a duality: the boundary values of bulk fields are *sources* for operators in the boundary theory. Needless to say, the bulk theory that one obtains in Maldacena’s case is a much more interesting object to study.

^{1} Since someone will surely asked, I will pause to define a “cosheaf.”

Recall the definition of a sheaf. Let $Top(X)$ be the category of open sets on $X$, with inclusions as morphisms. A **presheaf**, $\mathcal{A}$, on X is a contravariant functor from $Top(X)$ to some other category, $C$ (of sets, of abelian groups, of algebras, …). A **sheaf** is a presheaf that satisfies an additional requirement. Let $\{U_i\}$ be a collection of open sets on $X$ and $U=\cup_i U_i$. A presheaf,$\mathcal{A}$, is a sheaf iff, for any such collection $\{U_i\}$, the sequence
$0 \to A(U)\to \underset{i}{\textstyle{\prod}} A(U_i) \stackrel{\text{diff}}{\to} \underset{i\lt j}{\textstyle{\prod}} A(U_i\cap U_j)$
is exact.

A **pre-cosheaf** is a *covariant* functor from $Top(X)$ to a category, $C$. A **cosheaf** is pre-cosheaf, $\mathcal{A}$, for which
$\coprod_{i\lt j} A(U_i\cap U_j) \to \coprod_i A(U_i) \to A(U) \to 0$
is exact. So cosheaves are “dual” to sheaves.

In AQFT, all the cosheaves are such that inclusions map to monomorphisms in $C$. This is dual to the notion of a flabby sheaf.

^{2} If $\mathcal{A}$ is the cosheaf on $X$ which assigns
$V \mapsto A(V)$
for each open sent $V\subset X$, and if $f: X\to Y$, then $f_* \mathcal{A}$ is the cosheaf which assigns
$U \mapsto A(f^{-1} (U))$
for each open set $U\subset Y$.

^{3} Moreover, one would like a map going the other way, presumably using the inverse-image cosheaf. And one would like the composition of the two to be an isomorphism. I won’t attempt to see why that’s true; I’ll just grant that Rehren has gotten it to work out for the particular case of interest, where $C$ is the category of unital $C^*$ algebras.

## Re: Rehren Duality

Thank you very much for this public service(timelike geodesics, timelike curves, and Locality Principle). It is important that such topics be dicussed in a dispassionate way(as in this case) so outsiders can draw their own conclusions.

No expert on ADS/CFT myself, I read the classic review on the subject and find the conjecture to be extremely plausible. While rigorous arguments have their place, in physics it is well known that rigorous formulations often arrive much later than successful use(classic example: Dirac delta function and distributions).

In this case, it seems to me that Rehren’s duality may also be correct, but he is talking about something other than Maldacena’s duality…