- #1

Andrei

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I studied string theory for 2 years officially and much more on my own, chance made it that I spent the last year reading "serious" algebraic topology books and differential geometry specific for mathematicians. After gaining this insight I am very skeptical about string theory and the holographic principle... I posted these questions also on another forum and I was directed towards physicsforum...

1. I would like to understand if the t'hooft diagrams in the N->Infinity limit can be mapped back to a field theory? Is there a general proof for that? What I mean is that the whole construction adds additional degrees of freedom by extending the original manifold (lines) to bands. There must be some structure in order for the quantisation to be possible in that case. See this paper, I read it today and many things look very interesting:

http://arxiv.org/pdf/hep-th/9812012.pdf

While I agree with Witten (d'oh) it seems to me that the conditions for even having some reasons to try to apply AdS/CFT conjecture are pretty strict and rather inaccessible for experiments in "our world". Also, there are the obvious conditions for quantization one has to satisfy: symplectic manifold, choice of polarization, integrability condition (closure relation). How comes that one can assume that there exists a space (with spin structure, by the way) that satisfies all these criteria and is even remotely compatible to what exists in experiments? I am a bit afraid that people are trying out some formulation of AdS/CFT without satisfying the known criteria for the conjecture... how can they verify it correctly in that case?

2. Is t'hooft diagram "coverage" of the topological objects (sphere, torus, etc) complete? Are the diagrams in the N->infinity limit covering the whole object as pretended? How many inequivalent knots are there? Can the structure obtained in that way be classified in another way? Maybe an easier one? using some weaker equivalence, not necessary homotopy?

3. Is the map of this connection 1 to 1? it seems not...

let's see what I come with in the next days... for now, this should be it...

update: here is the question I asked on another forum... more or less the same but the general opinion was this was not well formulated:

Is there a provable map between a Feynman diagram and a 't Hooft diagram extended theory on a manifold in the N→∞ limit that produces in reverse always the QFT one is starting from?

Ok some of you did not go well with the simplicial decomposition or the simplicial complex. The idea of AdS/CFT starts with the fact that because of the two indices of the Feynman "lines" in the adjoint representation one can in principle extend them to two lines [1].

Obviously, while doing this you start obtaining simplexes [2] but if you look at the figure at page 2 you see that the retraction (aka going back to Feynman diagrams) is not unique and although the topology doesn't change you don't work day in day out in nature with only topological QFTs...

restated:

When you construct the "bands" you add complexity you don't know about. First, indeed you make Cell complex manifest and "imagine" you can cover some topological objects [3]. There can be Fibre-bundles, there can be deformations etc.

Is geometric quantisation still correct in this context?

Don't you need, in order to get a closed theory (if so, see BRST operators and the Poincaré lemma and what means δ2=0) some extra conditions?

Like, for example some sort of condition that assures you that you are covering your surface?

What if an artificial hole appears where you thought it was a sphere just because the diagrams don't "close up"? Anyone proved this? Anyone proved some integrability conditions on a sphere, torus, etc.?

Has anyone ever tried to prove that the series expansion obtained in that way (using 't Hooft diagrams) is restricted by some other criteria (like quantisation)? Why, I am not even sure that string theory is compatible with quantum mechanics...

Is this aspect not dependent on various quantisation prescriptions?

How does all this behave in the context of geometric quantisation? This is a very good question: you change line diagrams with topological objects i.e. you change the manifold. Quantisation requires you to start from a symplectic manifold and impose a relation between a Poisson structure and a commutator. In the case of string theory you just assume you can go over to the stringy side in the same way.

But what if this is not that simple? Mainly doing this transition to 't Hooft diagrams. What if the "integration" done there is not over the right manifold that assures correct quantisation?

Is the quantum polarisation maintained when doing this? Is the original group structure preserved when doing this?

When doing the quantisation does the structure needed for quantisation keep the same variables? If I remember correctly, in conformal field theories you do radial ordering instead of time ordering for very good reasons. That ordering assures a polarisation. You can also choose some Kahlerian polarisation if you want but what happens to that choice when you double and extend the Feynman lines just like that? Are you sure you are really quantising something?

[1]: (sort of linear extension, like the inverse of a deformation retraction, see page 2 in Hatcher)

[2]: (simplexes are the funny multi-sided polygons that cover a manifold, again, see Hatcher chapter 0, Cell complexes)

[3]: (a sphere S2, a torus T2, etc.)

1. I would like to understand if the t'hooft diagrams in the N->Infinity limit can be mapped back to a field theory? Is there a general proof for that? What I mean is that the whole construction adds additional degrees of freedom by extending the original manifold (lines) to bands. There must be some structure in order for the quantisation to be possible in that case. See this paper, I read it today and many things look very interesting:

http://arxiv.org/pdf/hep-th/9812012.pdf

While I agree with Witten (d'oh) it seems to me that the conditions for even having some reasons to try to apply AdS/CFT conjecture are pretty strict and rather inaccessible for experiments in "our world". Also, there are the obvious conditions for quantization one has to satisfy: symplectic manifold, choice of polarization, integrability condition (closure relation). How comes that one can assume that there exists a space (with spin structure, by the way) that satisfies all these criteria and is even remotely compatible to what exists in experiments? I am a bit afraid that people are trying out some formulation of AdS/CFT without satisfying the known criteria for the conjecture... how can they verify it correctly in that case?

2. Is t'hooft diagram "coverage" of the topological objects (sphere, torus, etc) complete? Are the diagrams in the N->infinity limit covering the whole object as pretended? How many inequivalent knots are there? Can the structure obtained in that way be classified in another way? Maybe an easier one? using some weaker equivalence, not necessary homotopy?

3. Is the map of this connection 1 to 1? it seems not...

let's see what I come with in the next days... for now, this should be it...

update: here is the question I asked on another forum... more or less the same but the general opinion was this was not well formulated:

Is there a provable map between a Feynman diagram and a 't Hooft diagram extended theory on a manifold in the N→∞ limit that produces in reverse always the QFT one is starting from?

Ok some of you did not go well with the simplicial decomposition or the simplicial complex. The idea of AdS/CFT starts with the fact that because of the two indices of the Feynman "lines" in the adjoint representation one can in principle extend them to two lines [1].

Obviously, while doing this you start obtaining simplexes [2] but if you look at the figure at page 2 you see that the retraction (aka going back to Feynman diagrams) is not unique and although the topology doesn't change you don't work day in day out in nature with only topological QFTs...

restated:

When you construct the "bands" you add complexity you don't know about. First, indeed you make Cell complex manifest and "imagine" you can cover some topological objects [3]. There can be Fibre-bundles, there can be deformations etc.

Is geometric quantisation still correct in this context?

Don't you need, in order to get a closed theory (if so, see BRST operators and the Poincaré lemma and what means δ2=0) some extra conditions?

Like, for example some sort of condition that assures you that you are covering your surface?

What if an artificial hole appears where you thought it was a sphere just because the diagrams don't "close up"? Anyone proved this? Anyone proved some integrability conditions on a sphere, torus, etc.?

Has anyone ever tried to prove that the series expansion obtained in that way (using 't Hooft diagrams) is restricted by some other criteria (like quantisation)? Why, I am not even sure that string theory is compatible with quantum mechanics...

Is this aspect not dependent on various quantisation prescriptions?

How does all this behave in the context of geometric quantisation? This is a very good question: you change line diagrams with topological objects i.e. you change the manifold. Quantisation requires you to start from a symplectic manifold and impose a relation between a Poisson structure and a commutator. In the case of string theory you just assume you can go over to the stringy side in the same way.

But what if this is not that simple? Mainly doing this transition to 't Hooft diagrams. What if the "integration" done there is not over the right manifold that assures correct quantisation?

Is the quantum polarisation maintained when doing this? Is the original group structure preserved when doing this?

When doing the quantisation does the structure needed for quantisation keep the same variables? If I remember correctly, in conformal field theories you do radial ordering instead of time ordering for very good reasons. That ordering assures a polarisation. You can also choose some Kahlerian polarisation if you want but what happens to that choice when you double and extend the Feynman lines just like that? Are you sure you are really quantising something?

[1]: (sort of linear extension, like the inverse of a deformation retraction, see page 2 in Hatcher)

[2]: (simplexes are the funny multi-sided polygons that cover a manifold, again, see Hatcher chapter 0, Cell complexes)

[3]: (a sphere S2, a torus T2, etc.)

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