A Slant on Warped Extra Dimensions

[Careful]

Of course; do you seriously think that string theory could possibly be inconsistent for such a reason?

My comment was more inspired by my own line of thought (that is, the insights coming from my own work which I believe to be more correct). Therefore, I was careful in mentioning it, I didn't deem it worthwhile to investigate it properly since this is the task of string theorists, not mine.

Let me point out how it roughly works, and take for the problem of unitarity the decoupling of the longitudinal modes of the graviton (it works for gauge bosons in the same way). In a consistent theory these modes must decouple, as if they wouldn't, unitarity would be immediately lost. Now, in string theory those modes correspond to BRST exact operators that indeed lead to the vanishing of the relevant amplitudes, by making use of contour deformation arguments. All what is necessary to make these arguments work is that the world-sheet theory is conformally invariant.

Hmm, I thought you were playing with D-branes which have no conformal invariance I guess and moreover, one could even drop the requirement of an anomaly free Virasoro algebra and work in the non-critical dimensions. This is called non-critical string theory as far as I remember, so how would your argument apply to that? Moreover, conformal invariance forces your background to satisfy an equation with an infinite number of terms. As far as I know, nobody has ever made sense out of this from a nonperturbative point of view and therefore it might even well be that no exact solutions exist (apart from a few trivial ones like local Minkowski). Therefore, is it really known that these CY compactifications give a unitary theory beyond say - third order perturbation theory in the string coupling constant? The devil usually is in the details. I don't know, so I ask.

This is what is meant when people say that string theory implies Lorentz invariance.

This is then an abuse of language as far as I can see...

Btw. Decoupling is a necessary condition, do you also know it is a sufficient one? You did not give a proof yet as far as I see.

 

[Careful]

Well I have seen plenty of philosophical ideas as to what the nature of GR and QM "should" be. But there _is_ indeed a difference between philosophy and phyics. Unless one can present a concrete mathematical formulation (lets not even ask for physical predictions), one still is stuck at level zero. The step from words to meaningful formulas is a trillion times more complicated, in fact that's the real problem - there was never a lack of ideas, but of ideas that actually can be shown to concretely "work".

Sure, but ad hoc formula's without good philosophical insight are equally meaningless... That is what is usually forgotten too

 

[Fra]

Well I have seen plenty of philosophical ideas as to what the nature of GR and QM "should" be. But there _is_ indeed a difference between philosophy and phyics. Unless one can present a concrete mathematical formulation (lets not even ask for physical predictions), one still is stuck at level zero.

If you are referring to MY personal ideas, then I will do my best. I need many many years from now before I am ready to present in terms of explicit predictive models. I wouldn't speak if I didn't think that I had a chance at doing better, in despite of the odds.

Other than that you are right, but I would like to add an important point. Not only are their plenty of philosophical ideas as to what physics and reality should be; there are also plenty of mathematics as to what what reality must be, that consumes a lot of research resources. I am as impressed with a random mathematical model that lacks connection to reality as I am with philosophical ramblings. One isn't better than the other, not from the point of view of physics as I see it.

But the task isn't to come up just with mathematics, the task is to find the RIGHT framework or at least some optimally rational framework, here the philosophical style analysis or the logic is a guide.

I have always had the opinion that there are a lot of brilliant technical mathematicians doing physics, but the people that in addition to the basic skills neede AND have the deeper insight and capability to see the mathematics required by nature are the rare geniouses. In that perspective, I do not see what is wrong with also analysing the way mathematical models has evolved and been constructed. IT does in neither way contradict the goal that we seek explicit mathematics for computing and simulating systems in our environment in a way that allows predictions.

/Fredrik

 

[suprised]

Hmm, I thought you were playing with D-branes which have no conformal invariance

They do have!

.. one could even drop the requirement of an anomaly free Virasoro algebra and work in the non-critical dimensions. This is called non-critical string theory as far as I remember, so how would your argument apply to that?

Those indeed break Lorentz symmetry due to the Virasoro anomaly, and are not consisered as consistent (I mean the usual non-critical strings, there are other ones not based on a world-sheet formulation).

Alas… there were attempts to invent different quantization schemes which do not know about anomalies, and it has been claimed that with these one could quantize the non-critical string consistently. AFAIK this is not taken serious by many; this is similar to certain quantization schemes in LQG which also do not see anomalies (and thus, do not even quantize the harmonic oscillator correctly…).

Moreover, conformal invariance forces your background to satisfy an equation with an infinite number of terms. As far as I know, nobody has ever made sense out of this from a nonperturbative point of view and therefore it might even well be that no exact solutions exist (apart from a few trivial ones like local Minkowski).

Well the higher expansion terms are strongly suppressed by inverse powers of the Planck scale so there is all reason to expect that they would represent small corrections when away from singularities. The situation is expected to be significantly different near singularities; ie they might not exist etc. Moreover, the convergence of the perturbation series is a hard issue, but probably a red herring.

Therefore, is it really known that these CY compactifications give a unitary theory beyond say - third order perturbation theory in the string coupling constant? The devil usually is in the details.

Indeed it is in the details. So let me spell this out. Lorentz invariance is in fact coming from conserved world-sheet Kac-Moody currents that exist on all higher genus Riemann surfaces, so naively it can't be violated. But what still could happen is that the longitudinal components don't decouple, and then there would be an anomaly which destroys unitarity. Again naively these decouple because the usual contour arguments should go through on arbitary Riemann surfaces, this means at arbitrary loop order.

But there is the devil in the details. The contour argument can fail for singular Rieman surfaces, and this is where potential anomalies could come from. This is a complicated issue and requires a careful analysis of the various degeneration limits of Rieman surfaces. AFAIK this issue is rigorously settled to one-loop order and convincingly settled to some higher orders, like 2 or 3. At higher orders there are problems as to how to rigorously define the measure on supermoduli space, I am not sure what the status is. So AFAIK there is no rigorous proof that anomalies do not appear at say, 5th loop order. On the other hand, this is pretty unlikely as anomalies usually appear at lowest order, ie 1 loop, I never have seen something else.

. Decoupling is a necessary condition, do you also know it is a sufficient one?

I guess so… the symmetries are preserved at tree level and also at the quantum level, assuming the absence of anomalies. Violation of unitarity can arise only from degeneration limits of (super-)Riemann surfaces, which is what we discussed above. AFAIK there are no hard proofs at arbitrary loop orders; I didnt follow the literature on this recently. But there has always been circumstantial evidence that this works, there are no concrete indications that it ever goes wrong; the issue is rather to prove it rigorously.

 

[Careful]

They do have!

Ok, thanks for your detailed answer . Just a simple question about your first comment which is somewhat mysterious to me: I always thought conformal invariance was somehow thightened to 2 d unless you use Weyls trick of course. So I guess the action depends upon the Weyl tensor and not the simple area ? Or do you add another field which transforms appropriately under Weyl rescalings ?

Cheers,

Careful

 

[suprised]

Just a simple question about your first comment which is somewhat mysterious to me: I always thought conformal invariance was somehow thightened to 2 d unless you use Weyls trick of course.

True… now I understand what your aim is. The issue is not to quantize the world-volume of general branes, for which indeed there is all sorts of problems. Specifically D-branes are "dual" representatives of branes that can be represented perturbatively, which means in terms of 2d CFT with boundaries. So the trick behind D-branes is that instead quantizung solitonic world-volumes, one can describe the non-perturbative closed string dynamics by perturbative open strings based on boundary CFTs, and similar computational techniques can then be applied as for closed strings.

Note that not all p-branes are equivalent to D-branes, like the NS fivebrane, and for those these methods fail.

 

[Careful]

True… now I understand what your aim is. The issue is not to quantize the world-volume of general branes, for which indeed there is all sorts of problems. Specifically D-branes are "dual" representatives of branes that can be represented perturbatively, which means in terms of 2d CFT with boundaries. So the trick behind D-branes is that instead quantizung solitonic world-volumes, one can describe the non-perturbative closed string dynamics by perturbative open strings based on boundary CFTs, and similar computational techniques can then be applied as for closed strings.

Note that not all p-branes are equivalent to D-branes, like the NS fivebrane, and for those these methods fail.

Well, as I said, I do not know enough details to make a definite statement about string theory (I studied it only intensely for like three months) ; I can only judge it from the perspective where I am standing (about the other approaches like CDT, LQG and causal set, I do know all the fine print). Actually, the reason why I would say that my approach is not so far from string is because it treats gravity in a similar dual way and insights such as the holographic principle appear to show up directly (also, my black holes do not have an horizon, so they are not perfectly black). On the other hand, the geometry and quantum theory I develop is far more general (and sophisticated) and locality on spacetime is a sacred principle for me (altough the construction is nonlocal on the tangent bundle). So, we will see what happens in the future...

Careful