Rehren Duality

October 16, 2006

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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.

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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 cosheaf¹ 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 cosheaf² $\mathcal{A}_{\text{CFT}} = f_* \mathcal{A}_{\text{AdS}}$

One wants to claim³ 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.

Posted by distler at October 16, 2006 3:07 AM

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36 Comments & 5 Trackbacks

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…

Posted by: ignoramus on October 16, 2006 9:13 AM | Permalink | Reply to this

Re: Rehren Duality

Dear Jacques,
thanks for your explanations.

Still, it is probably easier to say that the free energy density in d-dimensional local theory scales like temperature^d at high temperatures, and because these powers look different at different “d”, the theories can’t be equivalent.

When you read the Rehren’s paper, their “proof” of the duality is literally equivalent to the following:

Every theory in d1 dimensions is equivalent to a theory in d2 dimensions because you can always write

phi(x,y,z)

phi_z(x,y)

which means that you change one of the coordinates into a (continuous) index of the fields, converting a higher-dimensional theory into a lower-dimensional one. From a physics viewpoint, this is of course a nonsensical operation because the continuous “index” makes the theory in lower dimensions violate the usual axioms, but they are satisfied to view it as a proof of what they call “holography” even though it is a completely childish game with symbols without any physical consequences.

Best wishes
Lubos

Posted by: Lubos Motl on October 16, 2006 1:10 PM | Permalink | Reply to this

Re: Rehren Duality

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Every theory in d1 dimensions is equivalent to a theory in d2 dimensions because you can always write $\phi(x,y,z)$ as $\phi_z(x,y)$ which means that you change one of the coordinates into a (continuous) index of the fields, converting a higher-dimensional theory into a lower-dimensional one.

Well, that’s not (literally) what he claims in his paper, but I gather that Rehren would “agree” with you. Take a Poincaré-invariant QFT on $\mathbb{R}^{1,d}$ and orthogonally project $f:\mathbb{R}^{1,d}\to \mathbb{R}^{1,d-1}$ . He would, apparently, call the resulting direct-image theory a $d$ -dimensional AQFT.

Of course, you are right that this “AQFT” would be very pathological. In particular, it would have senseless thermodynamics. Most AQFT people would, however, reject this “AQFT” because it fails to satify Buchholz-Wichmann Nuclearity (which is the AQFT way of saying that it fails to have sensible thermodynamics).

Even leaving thermodynamics aside, Rehren’s construction in AdS has other pathological features, which is what I tried to emphasize.

Posted by: Jacques Distler on October 16, 2006 1:40 PM | Permalink | PGP Sig | Reply to this

Re: Rehren Duality

Dear Jacques, you are quite right. The statement about the equivalence between theories in different dimensions is not literally written in these papers.

It turns out that Prof. Rehren wrote it to me in a private letter. After I showed the “construction” replacing the coordinates by indices and after I asked whether we are allowed to say that we have constructed a lower-dimensional theory in any useful sense (with my answer being obviously No), the answer was:

YES we are. YES it is allowed. It is a local 3D quantum field theory.
It has some properties which one may dislike - especially its continuous
mass. Your example has been considered 30 years ago (Borchers) for the
purpose of showing that there are relations between QFT’s in different
dimension. (But in this flat space case, there is of course no way back
to 5D.)

…

Some time before this correspondence, the papers looked too abstract to me and I wasn’t 100% certain what to think about them and whether there was any chance that there would be something interesting about these papers, but this answer of Prof. Rehren has answered these questions fully.

Have a nice week
Lubos

Posted by: Lubos Motl on October 16, 2006 3:02 PM | Permalink | Reply to this

Re: Rehren Duality

9905179 -

Two open regions in anti-deSitter space are called “causally disjoint” if none of their points can be connected by a time-like geodesic. The largest open region causally disjoint from a given region is called the causal complement. In a causal quantum field theory on the quotient space AdS_1,s, observables and hence algebras associated with causally disjoint regions commute with each other.

The reader should be worried about this definition, since causal independence of observables should be linked to causal connectedness by time-like curves rather than geodesics. But on anti-deSitter space, any two points can be connected by a time-like curve, so they are indeed causally connected, and the requirement that causally disconnected observables commute is empty. Yet, as our Corollary 1 shows, if the boundary theory is causal, then the associated bulk theory is indeed causal in the present (geodesic) sense. We refer also to [3] where it is shown that vacuum expectation values of commutators of observables with causally disjoint localization have to vanish whenever the vacuum state has reasonable properties (invariance and thermodynamic passivity), but without any a priori assumptions on causal commutation relations (neither in bulk nor on the boundary).

Posted by: Anonymous Coward on October 16, 2006 7:51 PM | Permalink | Reply to this

Re: Rehren Duality

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Your ability to quote, verbatim, from Rehren’s paper, is admirable. However, it contributes nothing to the present discussion.

As to reference [3], you’ll note that their result is that $A(U)$ and $A(U')$ commute if $U$ and $U'$ are causally-disconnected (in the usual sense, not in this phony-baloney ‘timelike geodesic’ sense) on the universal covering space of AdS.

Since, AdS, itself, is foliated by closed timelike curves, it’s pointless to even try to make sense of locality on AdS. The statement, on the universal covering space, is that the usual notion of locality holds.

Posted by: Jacques Distler on October 16, 2006 8:56 PM | Permalink | PGP Sig | Reply to this

Re: Rehren Duality

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Probably, I should explain that the paper of Buchholz et al is an attempt to derive (rather than postulate) what the correct notion of locality should be in AdS (or its universal covering space), from some notion of what “sensible physics” should emerge.

Unfortunately, “sensible physics in AdS” (as opposed to its universal cover) is an oxymoron. So I’m really only inclined to take seriously their ‘derivation’ for the latter case, where it yields the conventional answer.

In the case of AdS itself, my best interpretation of their result is that, if there existed a QFT with the stated property (that $A(U)$ and $A(U')$ commute for $U,U'$ causally connected, but not connected by a timelike geodesic) — and/or if pigs could fly — then such a QFT would yield “sensible” physics in AdS.

Posted by: Jacques Distler on October 16, 2006 9:34 PM | Permalink | PGP Sig | Reply to this

Re: Rehren Duality

Slightly stronger
http://xxx.arxiv.org/math-ph/0407011

Posted by: Anonymous Coward on October 17, 2006 4:43 AM | Permalink | Reply to this

Re: Rehren Duality

Please allow me a few comments on the ongoing discussion of
“Rehren duality”, which I called “algebraic holography” (AH).

1. It is true that AH, as it is published, pertains to AdS, which is
not the universal covering but the Z2 quotient of the hyperboloid.

It is, however, indicated in the “speculations” section of my paper
that one should extend AH to the covering both of AdS and of the
conformal boundary. Indeed, the paper by Buchholz and Summers
suggests that locality (in my sense) on AdS without covering excludes
scattering, in accord with the absence of scattering in CFT on
conformal Minkowski space without covering, due to Huygens’ principle.

I don’t see what could possibly go wrong with extending AH to the
coverings, using exactly the same argument (because all that matters
is the covariance of the assignment of corresponding regions.)
In this case, one would get the “usual” locality in the covering of AdS.

2. My definition of locality with respect to “absence of timelike
geodesics” was chosen at the time so that the axiom of locality is
not void. (There are indeed better reasons, see below.) But requiring
local commutativity only with respect to “absence of timelike curves”
(= requiring NOTHING), would allow QFT’s that are even more pathologic.

In fact, it would be most appropriate to “define locality locally”.
That is: for every globally hyperbolic subregion X of AdS one has:
if x and y in X cannot be connected by a timelike curve within X, then
fields at x and y should commute. These globally hyperbolic regions
are never all of AdS, but rather compact “diamonds”. This definition
amounts to the same as referring to “timelike geodesics” globally in
the case of AdS, and “timelike curves” in the case of its covering.

Also the time-slice axiom can and should be formulated “locally”,
that is, within globally hyperbolic subsets on AdS. Massive Klein-Gordon
fields are perfect examples of “sensible physics on AdS” in all senses,
except that they have no interaction.

These “local” formulations of the axioms are quite in the spirit of
general relativity: physics in any spacetime looks LOCALLY very much
like physics in Minkowski spacetime. Adopting this notion of generally
covariant QFT (made more precise by Brunetti, Fredenhagen, Verch)
seems to be a promising way out of the problem to make sense of
“sensible physics on AdS” (covered or not).

3. Lubos’ quotation from my letter is correct (only this was a comment
on a remark of his, and by no means a rephrasing of my argument).
The result of the projection R4 -> R3 (in the flat case) is a 3D QFT.
In fact, I have two options for the “projection”: If I choose the
family phi_z(t,x,y) for ALL z to be the fields of the 3D QFT, then
it will have funny thermodynamics. If I choose only phi_z(t,x,y) with
z=0 to be the fields of the 3D QFT (this is more appropriately called
“restriction” rather than projection), then it will presumably violate
the time-slice axiom.

One may discard these boundary theories for those reasons.

But (a): This is precisely what happens in Witten’s toy model of
AdS-CFT with the free scalar field in the bulk: the dual conformal
boundary field is Gaussian and has non-canonical dimension. This
field is well-known under the name “generalized free field”, invented
long ago precisely as an example which violates the time-slice axiom,
and has non-standard thermodynamics. Witten says that this model does
not capture the miraculous effects of gravity in the bulk. True.

But then (b): The non-standard-ness is on the bulk side. Not reliably
knowing any properties of the local algebras of a “string or gravity
theory on AdS” (or any other), who will claim that its restriction to
the boundary is NOT possibly a better-behaved CFT? The standard
temperature behavior of free energy in the bulk is not a theorem,
and it could be different in curved spacetime, or in theories of a
non-standard type. Isn’t the volume behavior of entropy in 5D AdS
“like 4D” according to the holographic principle? Aren’t the field
content and Hilbert space of 2D boundary CFT like those of a 1D CFT?

4. Jacques rightly insists in the CONCEPTUAL difference between
“restriction” (boundary fields are bulk fields evaluated at z=0) and
“duality” (prescribed boundary values of bulk fields are sources for
operators in the boundary theory). But in hep-th/0204123, Duetsch
and I point out that in perturbation theory on AdS, where there are
two possible choices for the bulk propagator, “restriction” (with
one choice of the propagator) gives COMPUTATIONALLY (graph by graph)
exactly the same Feynman integrals as “duality” (with the other choice).
Hence the field theories thus defined are the same.

5. AH is formulated in terms of observables localized in regions,
which may not be representable in terms of point-like fields. (Lack
of imagination is not a good argument that something makes no sense.)
It is, however, geometrically clear that point-like fields on the
boundary would correspond to the restriction of bulk fields,
discussed also by Bertola, Bros, Moschella, Schaeffer. Thus, the
observation in 4. applies to the “point-like subtheory” of the
boundary CFT obtained by AH, reconciling it with duality.

Posted by: rehren on October 18, 2006 2:35 AM | Permalink | Reply to this

Re: Rehren Duality

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Please allow me a few comments on the ongoing discussion of “Rehren duality”, which I called “algebraic holography” (AH).

Thanks for stopping by.

My definition of locality with respect to “absence of timelike geodesics” was chosen at the time so that the axiom of locality is not void.

And it would not have been void, had you been working on the covering space.

(There are indeed better reasons, see below.) But requiring local commutativity only with respect to “absence of timelike curves” (= requiring NOTHING), would allow QFT’s that are even more pathologic.

AdS (as opposed to its covering space) is foliated by closed timelike curves. So of course you are going to have trouble formulating sensible physics on it.

The solution is not to change the notion of locality (especially not to change it to something violated by any interacting field theory). The solution is to pass to the covering space.

In fact, it would be most appropriate to “define locality locally”. That is: for every globally hyperbolic subregion X of AdS one has: if x and y in X cannot be connected by a timelike curve within X, then fields at x and y should commute. These globally hyperbolic regions are never all of AdS, but rather compact “diamonds”.

It might be “appropriate”, but it’s not helpful, in that you cannot extend the time-evolution for arbitrary times (which is what hyperbolicity was supposed to buy you).

Are you also going to replace the Hamiltonian (which generates global time evolution) by some set of individual Hamiltonia, which generate time-evolution in each of these diamonds? If so, then you are certainly not doing QFT any more.

In any case, this talk about “compact causal diamonds” is just a temporary reprieve from the inescapable fact that null geodesics propagate out to the boundary and return in finite affine parameter.

So, in any theory with massless degrees of freedom (QED, anyone?), you cannot get a hyperbolic system without specifying boundary conditions.

If you restrict yourself to field theories without massless degrees of freedom (a restriction that, as far as I can tell, you did not make in your paper), then rather than having flat-out incompleteness of your Cauchy data, you find exponential sensitivity to the Cauchy data near the boundary.

This definition amounts to the same as referring to “timelike geodesics” globally in the case of AdS, and “timelike curves” in the case of its covering.

No it doesn’t. Timelike geodesics on AdS lift to timelike geodesics on the covering space (since the geodesic equation is a local one).

Also the time-slice axiom can and should be formulated “locally”, that is, within globally hyperbolic subsets on AdS. Massive Klein-Gordon fields are perfect examples of “sensible physics on AdS” in all senses, except that they have no interaction.

Massive free field theory is, indeed, the only case where signals propagate (only) along timelike geodesics.

Do you have another example of a field theory where you expect your modified notion of locality to hold?

I doubt it.

These “local” formulations of the axioms are quite in the spirit of general relativity:

But you’re not quantizing general relativity! You don’t have diffeomorphism invariance, and the Principle of Equivalence doesn’t hold.

But then (b): The non-standard-ness is on the bulk side. Not reliably knowing any properties of the local algebras of a “string or gravity theory on AdS” (or any other), who will claim that its restriction to the boundary is NOT possibly a better-behaved CFT?

I gave the argument above (and you can find it, in more detail, in the papers I linked to).

The bulk field, $h_{\mu\nu}$ , whose boundary value is the source for $T_{\mu\nu}$ of the boundary theory, is a massless spin-2 field.

On very general grounds, this means that the bulk theory is a gravitational one.

Conversely, if you insist that the bulk theory is non-gravitational (i.e., that it is some QFT in AdS, with a fixed metric), then the boundary theory (in the usual prescription for the dictionary between the bulk and the boundary theories) does not have a conserved local stress-energy tensor.

Posted by: Jacques Distler on October 18, 2006 9:00 AM | Permalink | PGP Sig | Reply to this

Restriction

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If I choose only phi_z(t,x,y) with z=0 to be the fields of the 3D QFT (this is more appropriately called “restriction” rather than projection), then it will presumably violate the time-slice axiom.

Restriction to $z=0$ (as opposed to projection) violates far more than the time-slice axiom. It violates nearly all of the Haag-Kastler axioms.

Posted by: Jacques Distler on October 18, 2006 2:28 PM | Permalink | PGP Sig | Reply to this

Re: Restriction

Dear Jacques,

1. You try once more to convince me that physics on AdS without
covering might be trivial (only free fields), which I already admitted.

On the other hand, you preferred to ignore that I said in the next
line, that algebraic holography holds as well on the coverings, and
with the “right” notion of causality.

2. It is indeed a consequence of AH, formulated in terms of local
algebras, that IF the bulk has local fields, then the CFT violates the
time-slice property, and even worse, its observables cannot all be
representable by point fields. This does not exclude the possibility
that the CFT has a subtheory of point fields.

Suppose you grant me that AH is MATHEMATICALLY correct. Suppose you
give me your favorite sensible QFT on the covered AdS (with gravity or
without). Then my construction gives a covariant and local cosheaf on
the boundary. Instead of discarding it because of lack of time-slice
property, I prefer “thin it out” by keeping only its point fields.
On the bulk side, this process amounts to throwing away phi(t,x,y,z) except phi(t,x,y,0) (i.e., restriction).

Restriction to z=0 PRESERVES ALL THE WIGHTMAN AXIOMS except time-slice. Translation into the Haag-Kastler framework just requires some
technical chin-ups.

So I can get an associated CFT of point fields. It can even have a
stress-energy tensor, generating the time evolution. This has been
explicitly demonstrated for the Klein-Gordon field (math-ph/0209035).
This stress-energy tensor is only somewhat more singular than usual
(if smeared, it is a quadratic form = well-defined matrix elements,
rather than an unbounded operator). Although technically weaker than
“time-slice”, it is perfectly acceptable for a local dynamics.

There are now two logical possibilities: either this CFT has something
to do with Maldacena, or not. In the latter case, it is another
relation between QFT in different dimensions.

By hep-th/0204123, the reinterpretation of the restricted bulk theory
as a boundary CFT (i.e., with the thinned-out local algebras) is
perturbatively indistinguishable from “duality”. This speaks for the
first possibility. Before thinning out, I just have some extra
CFT observables without representation in terms of point fields.

Why are you so offended by this picture?

3. I never claimed that there cannot be gravity (or spin 2) in the
bulk. I only maintain that it is not necessary for the argument.
Maybe gravity helps in singling out some more interesting cases
among all those covered by the general AH proposition.

4. What do you mean by this comment:

>> These “local” formulations of the axioms are quite in the spirit of
>> general relativity:

> But you’re not quantizing general relativity! You don’t have
> diffeomorphism invariance, and the Principle of Equivalence
> doesn’t hold.

Diffeomorphism covariance doesn’t require that gravity is quantized.
Neither means the presence of a spin 2 field that gravity is quantized.

Posted by: rehren on October 19, 2006 5:00 AM | Permalink | Reply to this

Re: Restriction

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You try once more to convince me that physics on AdS without covering might be trivial (only free fields), which I already admitted.

No, that’s not what I tried to convince you of, because it’s not true.

I was trying to convince you of 3 separate things and, evidently, my presentation was confusing. So let me try again.

There is no causal physics on AdS. Period. Not for free fields, not for anything. Physics in AdS is strictly periodic in global AdS time (the coordinate I called $\tau$ above). That’s as far from hyperbolicity as you can get. So let’s agree that we will speak no further about AdS itself and will, henceforth, only discuss the universal convering space. That’s what everyone else (from Hawking and Ellis to Maldacena) do. So let us follow in their footsteps.
Passing to the universal covering space, your “timelike geodesic instead of timelike curve” notion of locality is incorrect for everything except for free field theory. The usual (“timelike curve”) notion of causality is not vacuous and is the correct thing to impose.
Even after doing that, you don’t get a globally hyperbolic problem (even locally, you can’t extend your time evolution for $\Delta\tau\gt \pi/2$ ) unless you impose boundary conditions.

None of those points should be controversial.

Suppose you grant me that AH is MATHEMATICALLY correct. Suppose you give me your favorite sensible QFT on the covered AdS (with gravity or without). Then my construction gives …

What you have convinced me of is that, under your “AH” construction, either the bulk theory or the boundary theory violates the Haag-Kastler axioms (I am going to insist on the usual form of the locality axiom in the bulk; otherwise, there are no examples, except for free field theory).

Perhaps, having started with a healthy QFT on one side and obtained a sick theory on the other side of your AH ‘duality,’ you can further butcher the latter, to obtain something that does satisfy the Haag-Kastler axioms. But, then, the result is no longer dual to the theory you started with.

So I think that, at best, all you’ve managed to do is show (in mathematically rigourous fashion) that this is not the correct way to obtain an AdS/CFT duality.

I never claimed that there cannot be gravity (or spin 2) in the bulk. I only maintain that it is not necessary for the argument.

In your previous comment, you said that

Jacques rightly insists in the CONCEPTUAL difference between “restriction” (boundary fields are bulk fields evaluated at z=0) and “duality” (prescribed boundary values of bulk fields are sources for operators in the boundary theory). But in hep-th/0204123, Duetsch and I point out that in perturbation theory on AdS, where there are two possible choices for the bulk propagator, “restriction” (with one choice of the propagator) gives COMPUTATIONALLY (graph by graph) exactly the same Feynman integrals as “duality” (with the other choice). Hence the field theories thus defined are the same.

I haven’t studied hep-th/0204123 yet, but if it does achieve what you say, then it demonstrates the existence of a massless spin-2 field in the bulk whose boundary-value is either (Maldacena) the source for the stress-energy tensor of the boundary CFT, or (Rehren) is the stress-energy tensor of the boundary CFT.

Diffeomorphism covariance doesn’t require that gravity is quantized.

Without quantizing gravity, the physics of a QFT coupled to a background metric is not invariant under arbitrary diffeormorphisms; it is only invariant under isometries of the background metric.

Neither means the presence of a spin 2 field that gravity is quantized.

An interacting massless spin-2 field? Oh yes it does …

Posted by: Jacques Distler on October 19, 2006 9:12 AM | Permalink | PGP Sig | Reply to this

Re: Restriction vs. Duality

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A while ago in the above exchange some arguments about the central issue at stake here were exchanged: the claim in Karl-Henning Rehren’s work on AdS/CFT duality, which is usually understood as saying that

the restriction of bulk fields to the boundary are sources for the boundary theory

is that this can equivalently be thought of as what seems to be the orthogonal statement:

the restriction of the bulk field to the boundary are the boundary fields.

The argument for this somewhat surprising statement is from the article

M. Dütsch and K.-H. Rehren, A comment on the dual field in the AdS-CFT correspondence

whose abstract summarizes

the dual field whose source are the prescribed boundary values of a bulk field in the functional integral, and the boundary limit of the quantized bulk field are the same thing.

the statement which technically amounts to this claim:

The functional integral $Z(f)$ over fields $\phi$ on AdS space $X$ with asymptotic boundary $\partial X$ for the AdS bulk theory equals its functional Fourier transformation evaluated at $i f$ , i.e.

$Z(f) = \int D\phi\; \exp(-I(\phi)) \; \delta(\phi|_{\partial X} - f) \;\;\;\; (1.1, p. 1)$

and

$\tilde Z(f) = (\mathcal{F}Z)(if) = \int D\phi\; \exp(-I(\phi)) \; e^{\int \phi|_{\partial X} f } \;\;\;\; (1.2, p. 2)$

are equal (at least in their perturbatve expansion).

On p. 3 it is acknowledged that

This looks like a straight absurdity.

Then the argument is given for why this should still be true.

I found that an overview of the argument is reproduced in

K.-H. Rehren, QFT Lectures on AdS-CFT

where the two functional integrals in question appear as (2.4) and (2.9) on pages 11 and 12, on which the argument is sketched. Details are in the appendix.

It is noteworthy that really all the crux of “Rehren duality” is in this statement, which is a statement in quantum perturbation theory, not in AQFT.

The formulation of Algebraic Holography as a bijection between certain nets of algebras on Minkowski space and on AdS space is, by itself, really just a comparatively elementary (but nice) classical statement about the geometry of the action of $SO(2,n)$ on AdS space and its boundary. I found the best quick exposition of this idea on p.2 and 3 of the letter

K.-H. Rehren, Local quantum observables in Anti-deSitter - Conformalm QFT correspondence

which states how the restriction

$\iota : W \mapsto K$

of any spatial AdS wedge region $W$ to its asymptotic boundary Minkowski double cone $K$ respects the action of $SO(2,n)$ on bulk and boundary. This is the lemma on p. 3 of Algebraic Holography.

From that fact it is immediate that with $B$ a net of algebras in the bulk $A := \iota^* B = B \circ \iota$ is a net on the boundary (even though $A$ will not satisfy the time slice axiom if $B$ does). This is the corollary on p. 3 of Algebraic Holography.

So in summary:

1) Algebraic holography itself is a comparatively obvious and by itself unsurprising relation between nets of algebras that is directly induced by the geometry of the action of the AdS isometry group which is the conformal group on the asymptotic boundary.

2) Given that geometric/algebraic statement, using the QFT interpretation of nets of algebras, one is naturally lead to wonder what this means physically.

The only known QFT phenomenon relating the isometry action on the AdS bulk to the conformal action on its boundary is AdS/CFT. So one is naturally lead to wonder whether Algebraic Holography is the essence behind AdS/CFT.

3) While on first sight obvious, this interpretation is at least rather subtle, because from AdS/QFT one expects the relation between the boundary net $A$ and the bulk net $B$ not to come from the simple restriction $\iota$ . But (if I understand the history here correctly, quite a while after AH was proposed) there is a claim that indeed both notions coincice. This claim is the crucial aspect of the discussion. And its proof involves Feynman-diagrams, Greens functions and the like, no AQFT.

Posted by: Urs Schreiber on May 31, 2008 10:37 AM | Permalink | Reply to this

Re: Restriction vs. Duality

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If we are going to start “proving” identities, it would behoove us to state precisely what quantity we are trying to compute.

The fields, $\phi$ are all assumed to vanish at the boundary of $X=\text{AdS}_{d+1}$ . The “boundary values”, $f$ , that are the sources in the boundary CFT are given by $\lim_{y\to \partial X} \phi(y) = u^{-\Delta +d} f$ where $\Delta(\Delta-d) = m^2$ and $u(y)$ is any function with a simple zero along the boundary, so that the induced metric on the boundary ${ds}^2_{\text{bdry}}= \left(u^2 {ds}^2_X\right)|_{\partial X}$ Changing $u\to e^w u$ rescales the boundary metric by ${ds}^2_{\text{bdry}}\to e^{2w} {ds}^2_{\text{bdry}}$ . Under this transformation, $f$ transforms as a conformal tensor of weight $d-\Delta$ .

In the usual AdS/CFT correspondence, $f$ is the source for a conformal field, $\mathcal{O}$ , of conformal weight $\Delta$ (so that $\int f \mathcal{O}$ is conformally-invariant).

If I understand you correctly, you want to claim that a conformal tensor of weight $d-\Delta$ (i.e. $f$ ) and a conformal tensor of weight $\Delta$ (i.e. $\mathcal{O}$ ) are “the same thing” (at least, perturbatively).

Surely, you don’t mean that.

Posted by: Jacques Distler on May 31, 2008 11:54 AM | Permalink | PGP Sig | Reply to this

Re: Restriction vs. Duality

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If I understand you correctly, you want to claim

Sorry if that was unclear, but I didn’t claim anything in the above comment. I just went back to the K.-H. Rehren’s various articles and tried to summarize the claims he has.

The central claim he has, from which he deduces that algebraic holography describes AdS/CFT is that for the correlation functions on the boundary it does not matter whether one thinks of $f$ , the (rescaled, yes) boundary restriction of the bulk fields, as source or field, somehow.

That’s this equality between the expressions $Z(f)$ (1.1) and $\tilde Z(f)$ (1.2) in hep-th/0204123.

I am not claiming anything about the truth of this claim, but I thought it worthwhile to point out that he has this claim in his texts and that this is what makes him interpret “restriction” as the right process for understanding AdS/CFT:

it’s his lesson 6 on p. 13 of his lecture notes.

So it seems there are two questions:

1) is the claim correct that $Z(f) = \tilde Z(f)$ ?

2) $Z(f)$ corresponds to the “usual” interpretation. $\tilde Z(f)$ justifies the identification with “algebraic holography”. Does the equality in 1) then really imply that AH is about AdS/CFT?

Posted by: Urs Schreiber on May 31, 2008 1:37 PM | Permalink | Reply to this

Re: Restriction vs. Duality

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Please explain to me what the stated formula (1.2) is supposed to mean.

I really am going to insist that we impose boundary conditions such that the bulk fields vanish at the boundary. The issue is: at what rate should they vanish?

[The reason why I am going to insist on this is that we are interested in considering non-constant $f$ at the boundary. If we took $\lim_{y\to\partial X} \phi(y)= f$ instead of

(1)

\lim_{y\to\partial X} \phi(y)= u^{-\Delta+d}f

then the resulting field configuration would have infinite bulk action.]

Now, as I said, the source, and the boundary operator to which it couples have different conformal weights. With the above prescription, (1), $f$ transforms as a conformal tensor of weight $-\Delta +d$ . If (as in conventional AdS/CFT) we view this as a source, then the operator that it couples to transforms as a conformal tensor of weight $\Delta$ .

Bulk fields which satisfy the Breitenlohner-Freedman bound on $m^2$ , yield positive values of $\Delta$ . Surely, you don’t want boundary fields of arbitrarily negative conformal weight (which is what a naïve interpretation of what you have written would lead to).

So, could you please explain at what rates the bulk fields are supposed to vanish at the boundary, such that the alleged identity on the functional Fourier transform holds.

Posted by: Jacques Distler on May 31, 2008 2:24 PM | Permalink | PGP Sig | Reply to this

Re: Restriction vs. Duality

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So, could you please explain

I can try to recount what Rehren says about this.

The precise statement is apparently (1.6), which says

$Z^-(f) = \tilde Z^+(c f)$

where

- in $Z^-(f)$ the integral is over fields that vanish as $\sim z^{\Delta_-}$

- in $\tilde Z^+(f)$ the integral is over fields that vanish as $\sim z^{\Delta_+}$

- $c = 2 \Delta_+ - d$

and where

$\Delta_\pm = \frac{d}{2} \pm \frac{1}{2}\sqrt{d^2 + 4 M^2} \,,$

for $d+1$ the dimensional of AdS and for $M$ the mass of the scalar field for which Rehren does this computation.

Posted by: Urs Schreiber on May 31, 2008 3:00 PM | Permalink | Reply to this

Re: Restriction vs. Duality

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in $Z^−(f)$ the integral is over fields that vanish as $\sim {z}^{\Delta_-}$

in $\tilde{Z}^+(f)$ the integral is over fields that vanish as $\sim {z}^{\Delta_+}$

And therein lies your problem.

For $m^2\gt 0$ , $\Delta_- \lt 0$ . So it’s deceptive to speak of “fields that vanish as $\sim {z}^{\Delta_-}$ ”.

Rather, you are talking about fields that diverge at the boundary. These fields have divergent bulk action. So I have no idea what $Z^-(f)$ is supposed to mean.

Posted by: Jacques Distler on May 31, 2008 3:44 PM | Permalink | PGP Sig | Reply to this

Re: Restriction vs. Duality

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For $m^2 \gt 0$ , $\Delta_- \lt 0$ . So it’s deceptive to speak of “fields that vanish as $\sim z^{\Delta_i}$ ”.

Ah, now I get what you mean. Sorry for being slow. :-) Thanks for bearing with me!

All right, so the intregrand involved in $\tilde Z^-(f)$ does not actually exist. Hm. So that’s maybe a flaw in A comment on the dual field in the AdS-CFT correspondence.

Well, that would explain what is going on, then…

Posted by: Urs Schreiber on May 31, 2008 3:59 PM | Permalink | Reply to this

Re: Restriction vs. Duality

Thanks for bearing with me!

No problem!

I guess it is worth having the problem stated succinctly here, rather than forcing people to wade through the paper.

Posted by: Jacques Distler on May 31, 2008 5:04 PM | Permalink | PGP Sig | Reply to this

net on spacelike wedges

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Yes, that certainly helped me to see clearer. I was thinking about posting a summary somewhere.

But then I found I am still confused about some other issue. I am not sure if the earlier discussion here had come to any conclusion on this:

if you’d ask me what an AQFT on AdS space should be, I’d say it ought to be at least a net of algebras on all causal subsets on AdS (where, as we said above, causal subset should mean: within any globally hyperbolic subspace the interior of the intersection of the past of one with the future of another point).

I’d also prefer it to be “locally local” and to satisfy local time slice axioms (namely satisfy the time slice axiom in each globally hyperboloic subspace of AdS), but maybe we can worry about this after I get the following straight:

The AH construction does not yield such a net, does it? Because the construction in corollary 1 yields only a net on space-like wedge regions. Maybe I am missing some AdS-specifics here, but ordinary AQFT nets don’t assigns anything to space-like wedge regions.

Posted by: Urs Schreiber on May 31, 2008 7:10 PM | Permalink | Reply to this

Re: net on spacelike wedges

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if you’d ask me what an AQFT on AdS space should be, I’d say it ought to be at least a net of algebras on all causal subsets on AdS (where, as we said above, causal subset should mean: within any globally hyperbolic subspace the interior of the intersection of the past of one with the future of another point).

I agree that the use of spacelike wedges instead of causal subsets in AdS seems totally ad-hoc. But it’s not the only ad-hoc feature of Rehren’s attempt at defining what he means by “AQFT in AdS.”

That definition is cooked up by demanding that it be equivalent to (not dual to, as in conventional AdS/CFT) a QFT on the boundary.

Rehren tried to argue that that equivalence/duality distinction isn’t very meaningful. But, as we have just reviewed, that argument is wrong.

Indeed, I have just explained to you why, if you tried to define a field theory in the bulk whose boundary values were the correlation functions (as opposed to sources for correlation functions) of a conventional CFT on the boundary, the result would be nonsense (which, ultimately, is the conclusion Rehren comes to as well).

But it’s also clear why the construction failed, and why the conventional AdS/CFT prescription relating bulk to boundary is the correct thing to do instead.

(And, as an added bonus, if you put in the additional assumption — not typically made in AQFT, but seen as a crucial feature by many of us peasants — that the boundary theory has a local, conserved stress tensor, then you immediately see that the bulk theory is a gravitational one. QED.)

Posted by: Jacques Distler on June 1, 2008 8:44 AM | Permalink | PGP Sig | Reply to this

n-functorial holography

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After that summary of Karl-Henning Rehren’s work I might be allowed to add another one of my own thoughts:

As you know, I am claiming that the axioms of AQFT essentially follow from those of $n$ -functorial QFT. And as you may recall from my ideosyncratic remarks over at the $n$ -category café, I see evidence that in terms of $n$ -functorial QFT physicist’s holography is essentially the general abstract nonsense statement that

a transformation between two $n$ -functors (hence $n$ -dim QFTs) is itself, in components, an $(n-1)$ -functor (hence $(n-1)$ -dim QFTs)

In my work with Jens Fjelstad we are currently fleshing this out for the relation between rational 3dTFT (e.g. Chern-Simons) and 2-d CFT (e.g. WZW).

So, while I am far from claiming that the situation is completely understood (by me), there is significant nontrivial evidence that on the $n$ -FQFT side of life “holography” is an incarnation of the above abstract $\infty$ -functorial statement.

In light of these two pet observations of mine I would now like to connect this to K.-H. Rehren’s observations. From this point of view, the main obstacle is the failure of the time slice axiom for the boundary net in Algebraic holography:

in theorem 1 (p. 15) I claim that AQFT nets obtained from $n$ -functorial extended QFTs using my construction def. 9, necessarily always satisfy the time slice axiom.

Posted by: Urs Schreiber on May 31, 2008 10:58 AM | Permalink | Reply to this

Re: Restriction

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Suppose you give me your favorite sensible QFT on the covered AdS (with gravity or without).

I should point out the obvious fact that adding dynamical gravity is not some innocent little modification of the Haag-Kastler axioms.

In a gauge theory, we must demand that the local algebras of observables consist of gauge-invariant operators. We demand covariance under Poincaré (or under the anti-de Sitter group), but invariance under gauge transformations.

In gravity, diffeomorphisms are gauge transformations, and we must demand invariance under diffeomorphisms. So there are no local observables (the algebras, $A(U)$ associated to contractible open sets, $U$ , are all trivial).

Of course, that’s part of the point, from the point of view of Maldacena’s duality. From the point of view of your proposed duality, it would make the gravitational case rather … vacuous.

Posted by: Jacques Distler on October 19, 2006 3:39 PM | Permalink | PGP Sig | Reply to this

locality on general pseudo-Riemannian spacetimes

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A while ago Karl-Henning Rehren wrote about the AQFT locality axiom on general pseudo-Riemannian spacetimes

it would be most appropriate to “define locality locally”. That is: for every globally hyperbolic subregion $X$ of AdS one has: if $x$ and $y$ in $X$ cannot be connected by a timelike curve within $X$ , then fields at $x$ and $y$ should commute.

I am currently in the process of finalizing an article on how the AQFT axioms follow from the point of view of extended functorial QFT:

AQFT from $n$ -functorial QFT (pdf)

and found this idea confirmed. See what is currently section 8.3.

Posted by: Urs Schreiber on May 30, 2008 4:08 PM | Permalink | Reply to this

Re: locality on general pseudo-Riemannian spacetimes

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I’m afraid I don’t understand your section 8.3 at all.

What’s your definition of causal sets for $d\gt 2$ ?
What category structure do you impose upon them (analogous to what is done in section 3 for $d=2$ )?

One of the key features that you aregue for is the ability, in $d=2$ , to glue two causal sets along their common boundary and obtain another causal set.

I would argue that a key feature for Rehren is that (again, in $d=2$ ), the intersection of two causal sets is a causal set.

Neither property seems to hold, for any natural definition of “causal set” in $d\gt 2$ .

Posted by: Jacques Distler on May 30, 2008 5:41 PM | Permalink | PGP Sig | Reply to this

Re: locality on general pseudo-Riemannian spacetimes

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Hi Jacques,

I was just concentrating on $d=2$ for the time being.

What’s your definition of causal sets for $d \gt 2$ ?

I’d think the definition should be: inside any globally hyperbolic subspace the interior of any intersection of the future of one point with the past of another.

What category structure do you impose upon them (analogous to what is done in section $3$ for $d=2$ )?

The category structure on causal subsets should always be that coming from inclusions.

What’s less immediate to say is what the $d$ -category structure on things obtained from gluing causal subsets is in $d \gt 2$ .

If the time slice axiom is assumed to hold, then it does not matter so much whether one looks at causal subsets or at arbitrary open subsets, it seems.

I would argue that a key feature for Rehren is that (again, in $d=2$ ), the intersection of two causal sets is a causal set.

I see, I’d have to look at the deatils again. The notion of locality on arbitrary pseudo-Riemannian spacetimes is of relevance independent of that application to holography. It seems to me that this “local locality” is obviously the right concept, but it doesn’t really appear in the literature.

Currently I am wondering if maybe “Algebraic Holography” can be thought of rather in the context Chern-Simons/WZW than in AdS/CFT. But I need to have a closer look again.

Posted by: Urs Schreiber on May 30, 2008 6:29 PM | Permalink | Reply to this

Re: locality on general pseudo-Riemannian spacetimes

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I was just concentrating on $d=2$ for the time being.

I think there’s a good chance that Rehren’s AQFT approach works in $d=2$ .

What’s your definition of causal sets for $d\gt 2$ ?

I’d think the definition should be: inside any globally hyperbolic subspace the interior of any intersection of the future of one point with the past of another.

I think that’s the only reasonable definition.

What’s less immediate to say is what the $d$ -category structure on things obtained from gluing causal subsets is in $d\gt 2$ .

Sorry, I should have said “ $d$ -category structure.” For $d\gt 2$ , the above-defined causal subsets don’t glue.

If the time slice axiom is assumed to hold, then it does not matter so much whether one looks at causal subsets or at arbitrary open subsets,

That’s why I tried to phrase my discussion above in terms of (pre) cosheaves of operator algebras (which has a chance of generalizing to higher dimensions). But I don’t think Rehren was particularly happy with that.

I would argue that a key feature for Rehren is that (again, in $d=2$ ), the intersection of two causal sets is a causal set.

I see, I’d have to look at the deatils again.

Lemme know what you conclude. My recollection is that this was pretty darned important for Rehren’s program.

Posted by: Jacques Distler on May 30, 2008 7:34 PM | Permalink | PGP Sig | Reply to this

Re: Rehren Duality

Distler, I have to wonder if you always truely know what you are writing or if you don’t really know what you are writing but are giving an impression of being very knowledgable because of your frequent use of high falutin’ razzle-dazzle jargon.

AQFTs aren’t described by cosheaves. To take the definition of a cosheaf that you gave,

[ \coprod{i < j} A \left( Ui \bigcap Uj \right) \rightarrow \coprod{i} A \left( U_i \right) \rightarrow A(U) \rightarrow 0 ]

This definition does not apply for a generic AQFT. Your definition implies that the homomorphism $\coprod_{i} A \left( U_i \right) \rightarrow A(U)$ is surjective. But this need not be the case for a gauge theory. To see this, let U be the open set corresponding to the interior of a torus. Let’s also split U into $U_1$ and $U_2$ by “cutting” U into two simply connected halves and then giving both $U_1$ and $U_2$ a slight overlap because they are both open sets. The observable algebra $A(U)$ contains the gauge invariant Wilson loop observable around the interior of the torus but this operator is nowhere to be found in the image of $A(U_1) \coprod A(U_2)$, the free product of both algebras, in $A(U)$. But this is besides the point as we have to assume the surjectiveness of $\coprod_{i} A \left( U_i \right) \rightarrow A(U)$ anyway to derive the Rehren duality as we are restricting ourselves to double cones and wedges. BUT the sequence that you gave can’t be exact in general. To see this, let $U_1$ and $U_2$ be disjoint this time. This means that $A(U_1 \bigcap U_2)$ is the trivial algebra, or in your notation $0$. Exactness now means that $A(U_1) \coprod A(U_2) \rightarrow A(U_1 \bigcup U_2)$ is an isomorphism. But this is not the case. It suffices to consider the case where $U_1$ and $U_2$ are spacelike separated. Then $A(U_1 \bigcup U_2)$ is isomorphic to the ”commutative” tensor product of $A(U_1)$ and $A(U_2)$, which is a very different thing from the free product.

You claim that a nongravitational Kaluza-Klein model treated as a lower dimensional theory is “unacceptable” because it gives rise to the wrong thermodynamics; the free energy scales as the “wrong” power of temperature. But this is precisely what you’d expect with most quantum field theory with a infinite “tower” of fields. And besides, in string theory, which admittedly can’t be described by AQFT because it is a theory of quantum gravity, we have a Hagedorn temperature, which is even “worse” according to your criteria.

PS: The itex code that I just submitted works perfectly fine on http://pear.math.pitt.edu/mathzilla/itex2mmlFrag.html but not on your blog!!!!!!!!! Which is why I am submitting it under Markdown.

Posted by: Hostile Anonymous Coward on October 23, 2006 12:45 PM | Permalink | Reply to this

Re: Rehren Duality

I don’t understand the surjective arguments and I guess that it is not such a loss.

But the comments about thermodynamics are rather clear, and those written above are clearly wrong. First of all, the Hagedorn behavior is not “worse” (faster growth of states) than the behavior of (strongly coupled) quantum gravity. The number of states in the Hagedorn regime grows like exp(C.M) with the mass M. But the number of states in quantum gravity grows like exp(C’.M^2) because the black hole entropy that dominates the counting scales like M^2 in four dimensions. The difference becomes even more dramatic in higher dimensions.

At any rate, neither quantum gravity nor perturbative string theory is a local quantum field theory according to the conventional definitions (the existence of a local stress energy tensor, for example) even though perturbative weakly coupled string theory is closer to being a local theory. This is the reason why these Hagedorn comments are completely irrelevant.

It is very important in holography to distinguish which theories are local and which theories are gravitational. The bulk theories must be gravitational and they have no natural local gauge-invariant degrees of freedom. The boundary theories are local, and they do follow the right power law for free energy at high temperatures.

Jacques gave a nice and simple general argument why the bulk must be gravitational whenever the boundary theory has a conserved stress-energy tensor: spin 2 fields can’t couple differently. The only possible loophole from the conclusion about gravity in the bulk is the case when at least one of the theories is topological. But whenever there are local excitations, Jacques’ argument is robust and valid.

Posted by: Lubos Motl on October 23, 2006 3:50 PM | Permalink | Reply to this

Re: Rehren Duality

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AQFTs aren’t described by cosheaves. To take the definition of a cosheaf that you gave, $\coprod_{i \lt j} A \left( U_i \cap U_j \right) \rightarrow \coprod_{i} A \left( U_i \right) \rightarrow A(U) \rightarrow 0$ This definition does not apply for a generic AQFT. Your definition implies that the homomorphism $\coprod_{i} A \left( U_i \right) \rightarrow A(U)$ is surjective. But this need not be the case for a gauge theory.

I (personally) would be perfectly happy if the definition of AQFT involved pre-cosheaves of operator algebras instead of cosheaves. Unfortunately, this conflicts (I gather) with the desire to impose the condition that inclusions map to monomorphisms. (The dual statement — that having restrictions to map to epimorphisms, requires a sheaf, rather than just a presheaf — is probably more familiar.)

I really don’t care how this tension is resolved, as it’s pretty much irrelevant to my argument. I’ll leave it to the AQFT experts to resolve whether they wish to talk about pre-cosheaves or cosheaves.

As to the thermodynamics of AdS/CFT, Luboš has addressed most of your points. I’ll just add that the bulk string theory matches quite nicely with the boundary field theory. For instance, the Hawking-Page transition on AdS $_5\times S^5$ corresponds to the Gross-Witten phase transition of large- $N$ $SU(N)$ $\mathcal{N}=4$ super-Yang-Mills theory.

Posted by: Jacques Distler on October 23, 2006 8:35 PM | Permalink | PGP Sig | Reply to this

Re: Rehren Duality

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Regarding the question of whether a given net (= co-flabby co-presheaf) is actually a cosheaf:

one has to be a bit careful that the Haag-Kastler nets are not defined on all open subsets, but just on the “causal subsets”: those which are intersections of the future of one point with the past of another.

This means first of all that neigther the torus interior nor the union of two disjoint causal subsets – the two proposed counterexamples in the comment by the commenter who signed as “anonymous hostile coward” # qualify.

More generally it means that we need to check the co-sheaf property on covers which exist as such in the category of causal subsets: we can cover one causal subset $O$ by a bunch $O_i \subset O$ of smaller ones all sitting inside, such that $\cup_i O_i = O$ .

Whether or not a local net satisfies codescent (i.e. the co-sheaf property) with respect to such a cover is a little subtle, I suppose, if we are talking about von Neumann algebra valued nets, as we should.

I am not really sure yet about the details, but I notice that sometimes a certain extra property is found on the local net $A$ , namely that it has the split property for wedges (see reference below). This property implies that:

for $O_1$ and $O_2$ causal subsets touching “in one point”, and for $O$ the causal hull of that (smallest causal subset containing both), we have $A(O_1) \vee A(O_2) = A(O) \,,$ where $C \vee D$ is the von Neumann algebra generated by vN algebras $C$ and $D$ .

The split property also implies the time slice axiom on $A$ . In total, this seems to imply that $A$ is a cosheaf, but I’d need to check that.

This stuff about split property I am taking from Lechner: On the construction of quantum field theories with factorizing S-matrix around p. 25,, which in turn builds on Müger’s arXiv:hep-th/9705019.

Posted by: Urs Schreiber on November 13, 2008 5:19 PM | Permalink | Reply to this

Re: Rehren Duality

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one has to be a bit careful that the Haag-Kastler nets are not defined on all open subsets, but just on the “causal subsets”: those which are intersections of the future of one point with the past of another.

That’s fine in 1+1 dimensions, where the intersection of two causal sets (often called “causal diamonds” in that context) is a causal set.

That property no longer holds in higher dimensions, which means that one has to extend the definition a bit. The subject of how, exactly, to extend the definition was the subject of debate between me and my anonymous interlocutor.

I’m not sure about this “split property for wedges” business. But I have a suspicion that it, too, may be special to 1+1 dimensions.

Posted by: Jacques Distler on November 13, 2008 9:25 PM | Permalink | PGP Sig | Reply to this

nets and co-sheaves

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True. On the other hand, I suppose one could easily fix the failure of higher dimensional causal diamonds to be closed under intersection by passing to something like “convex open subsets the boundary of whose closure is piecewise lightlike”. The main point still being, I think, that all these sets are diffeomorphic to open balls.

But generally, there has been surprisingly little (as far as I can see: none(?)) discussion in the AQFT literature on cosheaf properties. Even the plain fact that a “net” is a special kind of co-presheaf (or pre-cosheaf if you like) is usually not mentioned. The only place that I know of is in writing of Bert Schroer, for instance his Lectures on AQFT and operator algebras #, where the term itself is mentioned when nets are introduced, though the sheaf-theoretic implications are not further explored.

(One also finds on the net the curious text Non-abelian quantum algebraic topology # which mentions AQFT co-presheaves and also the van Kampen theorem (which is secretly # about codescent, i.e. about cosheaves) but does not seem to connect them.)

In summary: it is not clear to me if the answer to “Should Haag-Kastler nets be taken to satisfy the co-sheaf condition?” is really “No.”

On a more speculative note: as we know, the chiral deRham complex # is a sheaf of vertex operator algebras on target space. On the other hand, vertex operator algebras are expected (people are telling me) to give rise to local nets (on parameter space, i.e on the worldvolume) by smearing vertex operators with test functions supported in causal subsets. It seems not unnatural therefore to speculate that the chiral deRham complex may be regarded as something which is a sheaf with respect to target space and a co-sheaf with respect to base space. Would seem to make sense. But I don’t know.

Posted by: Urs Schreiber on November 14, 2008 6:26 AM | Permalink | Reply to this

Re: nets and co-sheaves

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a sheaf with respect to target space and a co-sheaf with respect to base space.

Sorry, “base space” should read “parameter space” (= worldvolume):

a sheaf with respect to target space and a co-sheaf with respect to parameter space.

Posted by: Urs Schreiber on November 14, 2008 6:30 AM | Permalink | Reply to this

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October 16, 2006

Rehren Duality

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