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added a one-sentence Idea to holographic priniciple and added much of the references now also at self-dual higher gauge theory
expanded the list of exampl;es at holographic principle
Added to holographic principle the computations (in the Idea-section, still) that show that if higher dimensional Chern-Simons theory is holographically related to anything, then that anything must by self-dual higher gauge theory.
(It’s a bit terse, but I have to call it quits now.)
Is this any way related to the recent interest in the holomorphic version of Chern-Simons (e.g. work of Albert Schwarz) ?
By abstracts like this one there must be a close relation. But I haven’t looked into this aspect at all yet. I should.
Briefly added further example to the AdS/CFT section at holographic principle
Aren’t $n$ and $n+1$ swapped in the Idea section? I’m thinking to an $(n+1)$-dimensional topological quantum field theory where (essentially by definition) the partition function $Z(M)$ of an $(n+1)$-dimensional manifold $M$ is a state in the space of states $\mathcal{H}_{\partial M}$ associated with the $n$-dimensional boundary of $M$.
Hi Domenico,
maybe “TQFT in dimension $n+1$” sounds misleading? I have changed it to “TQFT of dimension $n+1$”. I also added more explanation.
I think it says the right thing. But if it sounds misleading to your ear, then please feel free to re-phrase it.
Hi Urs,
I’m not an expert in the holographic principle so I won’t attemp rephrasing. But in TQFTs I’d say that the holographic principle is the morphism $Z(\Sigma):\mathbf{1}\toZ(\partial\Sigma)$, which precisely identifies the partition function $Z(\Sigma)$ with an element in the space of states $Z(\partial\Sigma)$. In my interpretation of the physicists’ jargon, if we keep the $(n+1)$-dimensional manifold (with boundary) $\Sigma$ fixed, then this is the worldsheet the QFT lives on, so we are in presence of an $(n+1)$-dimensional QFT, whose partition function is $Z(\Sigma)$. Also, if we look at the $n$-dimensional manifold $\partial\Sigma$ as fixed, then this is teh worldsheet of an $n$-dimensional QFT, whose space of states is $Z(\partial\Sigma)$. So the holographic principle would be the identification of the partition function of a $(n+1)$-dimensional QFT with a state for an $n$-dimensional QFT.
In other words, I suspect that what Atiyah did in axiomatizing TQFTs has been making the holographic principle be an axiom for topological theories.
Hi Domenico,
let’s see what you write here does not sound right to me, but maybe I am misreading it:
Also, if we look at the $n$-dimensional manifold $\partial \Sigma$ as fixed, then this is teh worldsheet of an $n$-dimensional QFT, whose space of states is $Z(\partial \Sigma)$. So the holographic principle would be the identification of the partition function of a $(n+1)$-dimensional QFT with a state for an $n$-dimensional QFT.
I don’t think so, because $Z(\partial \Sigma)$ cannot be the space of states of the $n$-dim QFT, because an $n$-dim QFT does not assign spaces of states to $n$-dimensional manifolds such as $\partial \Sigma$.
But CFTs of dimension $n$ do assign vector spaces in codimension 0, namely spaces of conformal blocks_ . These are precisely vector spaces of possible correlators. Roughly, a conformal block is a function that has all the properties of the correlator of a CFT, except maybe that it cannot be globally glued to solve the sewing constraints. Conversely, constrtucting a full CFT amounts to choosing for each $n$-dim manifold a conformal block such that these choice glue to satisfy the sewing constraints.
In other words, I suspect that what Atiyah did in axiomatizing TQFTs has been making the holographic principle be an axiom for topological theories.
No, this is not true. There is some genuine and nontrivial content in the holographic principle, that goes beyond just the definition of a TQFT. In particular there is no claim (no hint) that every TQFT is holographically related to a lower dimensional theory. Those for which it is known are all of higher Chern-Simons theory type.
Which leads me over to announcing the following addition to the Idea-section of the entry…
Have added the following to the Idea-section at holographic principle
The idea that some systems in physics are governed by other systems “localized at a boundary” in this kind of way was originally suggested by the behaviour of black holes in general reltivity: their black hole entropy is proportional to their “surface”, as reflected by the generalized second law of thermodynamics. This made Gerard ’t Hooft suggest a general principle, called the holographic principle , which however remained somewhat vague.
Later two more precise classes of correspondences were identified, that are regarded now as precise examples of the general idea of the holographic principle:
Systems of Chern-Simons theory and higher dimensional Chern-Simons theory can be shown explicitly to have spaces of states that are canonically identified with correlator spaces of CFTs (conformal blocks) and self-dual higher gauge theory on their boundary.
Systems of quantum gravity in various dimensions as given by string theory on asymptocially anti de-Sitter spacetimes have been checked not in total but in a multitude of special aspects in special cases to be dual to supersymmetric CFTs on their asymptotic boundary – this is called AdS/CFT correspondence.
Below we list some systems for which something along these lines is known.
In view of this it is maybe noteworthy that one can see that also closed string field theory, which is supposed to one side of the AdS/CFT correspondence, has the form of an infinity-Chern-Simons theory (schreiber), as discussed there, for the L-infinity algebra of closed string correlators. So maybe the above two different realizations of the holographic principle are really aspects of one single mechanism for $\infty$-Chern-Simons theory.
Hi Urs,
I see I’ve written things in a confusing way.. :(
by writing QFT I meant a no better specified quantum field theory, not a topological quantum field theory. in particular a QFT in what I wrote above could be a CFT (I may be misreading here, but in the idea section of conformal field theory it seems that “conformal field theory” is an abridged name for “conformal quantum field theory”).
so in the vague terms I had in mind we have an $(n+1)$ manifold with boundary $(\Sigma,\partial\Sigma)$, some quantum field theory “A” defined on $\Sigma$, with some attached object called “partition function”, and denoted $Z_A(\Sigma)$, and some quantum field theory “B” defined on $\partial\Sigma$, with some attached “space of states” $\mathcal{H}_B(\partial\Sigma)$. and I would think that the theory “A” on $\Sigma$ and the theory “B” on $\partial\Sigma$ are holographically related if the partition function $Z_A(\Sigma)$ is naturally interpreted as a vector in $\mathcal{H}_B(\partial\Sigma)$.
concerning TQFT I didn’t mean that the holography principle is something trivial, but that TQFT implements a toy version of the holography principle (but this was just an impression/speculation of mine, which could be easily be wrong).
I think our misunderstanding here comes from what I meant by “$n$-dimensional QFT”. by this vague term I meant something involving a fixed $n$-dimensional manifold, to be called the worldsheet of the theory. that is, I did not mean “$n$-dimensional toplogical QFT”, which has a completely different (and precise) meaning: sorry for the bad choice of terms
I remain with the suspect that are partition functions on the bulk to be states on the boundary, and not vice versa, but as you see I’m too confused here, so I’ll come back to this in a few months or years when my mind will eventually be clearer on what the holography principle is :)
so in the vague terms I had in mind we have an $(n+1)$ manifold with boundary $(\Sigma,\partial\Sigma)$, some quantum field theory “A” defined on $\Sigma$, with some attached object called “partition function”, and denoted $Z_A(\Sigma)$, and some quantum field theory “B” defined on $\partial\Sigma$, with some attached “space of states” $\mathcal{H}_B(\partial\Sigma)$.
So this is where I would still disagree :-)
You, see, since $B$ is an $n$-dimensional QFT, it does not assigns an $\mathcal{H}_B$ to the $n$-dimensional $\partial \Sigma$. It assigns spaces $\mathcal{H}_B$ instead to $(n-1)$-dimensional manifolds.
and I would think that the theory “A” on $\Sigma$ and the theory “B” on $\partial\Sigma$ are holographically related if the partition function $Z_A(\Sigma)$ is naturally interpreted as a vector in $\mathcal{H}_B(\partial\Sigma)$.
So $Z_A(\Sigma)$ is always naturally interpreted as a vector in $Z_A(\partial \Sigma)$, of course.
The crucial question now is: what is this vector from the point of view of an $n$-dimensional QFT $B$. It is not a state of $B$, since $Z_A(\partial \Sigma)$ cannot be identified with a state space “$\mathcal{H}_B(\partial \Sigma)$”, since that is not provided by $B$.
So the question is: what could the space $Z_A(\partial \Sigma)$ be instead, from the point of view of $B$. And if $B$ is holographically related to $A$, then the statement is: $Z_A(\partial \Sigma)$ can be identified with a space of conformal blocks of $B$. Sometimes one says: the value of the modular functor of $B$ on $\partial \Sigma$.
Now, “conformal block” is just another term for “potential correlator” for CFTs. And “correlator” means “partition function with field insertions” if you wish. So the canonical vector given by the image
$Z(\Sigma) : 1 \to Z_A(\partial \Sigma)$in $Z_A(\partial \Sigma)$ can be indentified (if we have holography) with a correlator/partition function of the theory $B$.
I should maybe emphasite that this aspect is precisely what makes holography as in higher Chern-Simons theory and in AdS/CFT interesting and subtle: it’s not just about restricting states from higher to lower dimensions. It’s that states of a higher dimensional theory are mapped to correlators/partition functins of a lower dimensional theory.
There was once a proposal for how to formalize AdS/CFT – now sometimes called Rehren duality – but it fell short of precisely this subtlety: in that proposal a local net in higher dimensions was simply restricted to lower dimensions. But this is not what AdS/CFT is about.
Hi Urs,
it seems I’m lost with the terminology here, the following three cannot be all true :)
i) an $n$-dimensional QFT $B$ does not assigns an $\mathcal{H}_B$ to the $n$-dimensional $\partial \Sigma$; it assigns spaces $\mathcal{H}_B$ instead to $(n-1)$-dimensional manifolds.
ii) CFTs of dimension $n$ do assign vector spaces in codimension 0, namely spaces of conformal blocks
iii) a CFT is a particular kind of QFT
Hi Domenico,
they are all true. But the point is that the space in codimension 0 is not a space of states. It is a space of correlators.
I am sorry if I am saying this in a unreadable way. I don’t know, maybe we have been agreeing all along and just not understanding each other. Let’s try to sort it out.
Here is an example to keep in mind:
consider a TQFT $B$. To a cobordism $\Sigma$ it assigns not a vector in a space of states, but a linear map $Z_B(\Sigma) : Z_B(\partial_{in} \Sigma) \to Z_B(\partial_{out} \Sigma)$.
But then we may reformulate this as follows: we can say: there is a space of “topological conformal blocks” assigned to $\Sigma$, namely the vector space
$TCFBlocks(\Sigma) := Z_B(\partial_{in} \Sigma)^* \otimes Z_B(\partial_{out} \Sigma) \,.$The actual correlator $Z_B(\Sigma)$ is canonically identified with a vector in that space of topological conformal blocks
$Z_B(\Sigma) \in TCFBlocks(\Sigma) \,.$Given all spaces $TCFBlocks(\Sigma)$ for all $\Sigma$, (re)constructing the TQFT $B$ amounts to choosing such vectors in all these spaces such that they satisfy the “sewing constraints”, which is, under the above adjunction, the condition that $Z_B : Bord \to Vect$ is a functor.
So we can ask now: is there a higher dimensional TQFT $A$ such that whenver $\Sigma$ is the boundary of a $\hat \Sigma$ we have an isomorphism
$\phi : Z_A(\partial \hat \Sigma) \simeq TCFBlocks(\Sigma)$such that
$Z_B(\Sigma) = image( 1 = Z_A(\emptyset) \stackrel{Z_A(\hat \Sigma)}{\to} Z_A(\Sigma) \stackrel{\phi}{\to} TCFBlocks(\Sigma) ) \,.$If so, we say that $B$ is A holographic boundary theory of $A$.
To fill this with more substance, maybe you want to look at section 3 and section 4 of
Jens Fjelstad, Jürgen Fuchs, Ingo Runkel, Christoph Schweigert, Uniqueness of open/closed rational CFT with given algebra of open states (arXiv:hep-th/0612306)
This discusses, rigorously, the holographic construction of any rational 2d CFT from a 3d TQFT obrtained from a modular tensor category.. The conformal block assignment is in section 3.3. Picking the correlators in there is in 3.4. Then in 4 it is shown how the correlators can be picked in a sewing-consistent way by using the 3d TQFT as a “holographic dual” (that term does not appear there, though).
I’m beginning to see something, but my picture is still a bit foggy: I can look at a manifold $\Sigma$ with incoming boundary $\Sigma_{in}$ and outgoing boundary $\Sigma_{out}$ as a manifold with incoming boundary $\emptyset$ and outgoing biundary $\partial\Sigma=\Sigma_{in}^{op}\coprod \Sigma_{out}$. so we are redueced (as usual in TQFT) to $TCFBlocks(\Sigma)=Z_B(\partial \Sigma)$, and so to $Z_b(\Sigma)\in Z_B(\partial\Sigma)$. ok, I agree wuth this. but then we are assuming $\Sigma=\partial\hat{\Sigma}$ (so $\hat{\Sigma}$ will be a manifold with corners, right?), and we are asking for an isomorphism $\phi:Z_A(\Sigma)=Z_A(\partial\hat{\Sigma})\simeq Z_B(\partial\Sigma)$. then the second ondition (the one concerning $Z_B(\Sigma)$ reads as follows: the isomorphism $\phi$ identifies the element $Z_A(\hat{\Sigma})\in Z_A(\partial\hat{\Sigma})=Z_A(\Sigma)$ with the element $Z_B(\Sigma)$ in $Z_B(\partial\Sigma)$. In other words it seems we are asking for
$Z_A=Z_B\circ\partial$edit: thanks for the reference, I’ll now look into that!
Hey Domenico,
you suggest to write, for that holographic encoding of the “topological modular functor” (which is one half of the story)
$Z_A = Z_B \circ \partial$
Yes! That’s a good point. This way it does actually look like we are applying a coboundary operator after all.
so $\hat \Sigma$ will be a manifold with corners, right?
Yes, right. Or in more sophisticated language: for the full story we need to be talking about extended cobordisms.
That’s the difference between talking about the partition function or more generally about correlators. If $\Sigma$ itself has no boundary then
$Z_B(\Sigma) : 1 = Z_B(\emptyset) \to Z_B(\emptyset) = 1$is the partition function of $B$ over $\Sigma$. This $\Sigma$ can be the boundary of an ordinary $\hat \Sigma$ and so we can ask if $Z_B(\Sigma)$ can be identified with the image of $Z_A(\hat \Sigma)$.
If however $\Sigma$ itself has a boundary, say all incoming, and all of shape $X$ then
$Z_B(\Sigma) : Z_B(X)^{\otimes n} = Z_B(\partial \Sigma) \to Z_B(\emptyset) = \mathbb{C}$is a correlator or $n$-point function. And for this to be holographically related to anything, we need to allow $\hat \Sigma$ to have boundaries and corners, yes.
I’ll type out some of these things now at holographic principle. You can then go over it and see which pieces are unclear.
Okay, I have added some of the above discussion to holographic principle in a new section Idea – More details.
now that’s clear! sorry for having been pestering yesterday :)
sorry for having been pestering yesterday :)
I am glad you did!
At holographic principle I have expanded the section Of ordinary CS-theory / WZW model . In reply to this MO question.
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