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what does string theory have to say about the trace anomaly in the expectation value of the stress energy tensor of massless quantum fields on a curved background and its interpretation as the emergence of an extra scalar degree of freedom in the semiclassical limit of GR + QFT? Is it something that can be proved? that emerges naturally? Or that can be shown to be false?

The question has actually two quite distinct parts: 1) the presence of the anomaly and its form and 2) the possibility of interpreting it as the emergence of an extra degree of freedom at the classical level with its own EOM and a specific stress energy tensor satisfying the constraint Trace (T_mu_nu) = some function of the curvature.

Thanks a lot

NOTE ADDED As a warm-up I will reduce the scope of the question.

String theory and the triangle anomaly in Q.E.D.?

Are there examples where, starting from "first principles" in string theory, one compactifies to 3+1 dimensions and reproduce the physical effects of the so called/chiral triangle anomaly. When I say physical effects, I mean I don't care how it shows up, as long as it reproduces the final result in some approximation. Than we can think of the fact that there are different kind of anomalies etc.

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The way you've phrased this, it sounds a lot like a homework question. What prompted you to ask it? –  David Z Apr 22 '11 at 0:26
    
I just recently got interested in understanding the spherical collapse in semiclassical gravity, but I din't have time yet to explore the literature in string theory and in QFT in curved spacetime. This seemed to me something the string theory could address in some generic way, and I wanted just to see if someone out there have some useful info on the topic. For me at the moment it's not clear yet if the question is well posed and if it obvious or too complex. Thanks –  Curious George Apr 22 '11 at 5:29
    
OK, but consider this: when a question includes a command (like "in the answer please consider") it comes across as rude, either because it's a homework question where the poster couldn't even be bothered to reword (much less attempt) the question, or because it sounds like giving an order rather than asking for help. I'd suggest rewording that a bit. –  David Z Apr 23 '11 at 6:33
    
I see your point, take into account that Curious George is a monkey. I reworded a bit. –  Curious George Apr 24 '11 at 5:47
    
Much better :-) –  David Z Apr 24 '11 at 7:04
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The quantum conformal anomaly is not present in string theory if the dimension of space time is fixed to 26 for bosonic string theory and 10 for superstring theory. In string theory conformal symmetry in the world sheet is necessary for the theory to be consistent, so the trace of the energy momentum tensor vanishes. If not we don't have string theory at all.

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My question is more: you have you 10 dimensional theory, fine. Now compactify it to 4 dimension (3+1) anyway you want. I guess you should reproduce some approximate version of QFT+GR with massive and massless particles. Now, is the trace of the stress energy tensor of the massless particle equal to 0 in this limit or proportional to some function of the curvature? Or if you want, is there a limit where string theory reproduce QFT+GR, and if so, is this limit equivalent to the old fashioned QFT in curved spacetime studied since the 80' where massless field have a trace anomaly? –  Curious George Jun 25 '11 at 5:52
    
It is known that the process of compactification preserves the virasoro algebra. That is why compactification is allowed in the first place. If you were to destroy the quantum symmetry and reintroduce the conformal anomaly to reproduce known physics then you have achieved nothing. That is why I said you remove the anomaly, because is gone for good. I recommend you to move on from the ill-defined one-loop quantum gravity, it is a relic from the past. If you do so, the conformal anomaly with all the mass energy tensor renormalization ambiguities will also be gone. –  Ernesto Ulloa Jun 28 '11 at 13:07
    
I think you are being a little extreme. But I'll wait until I get more familiar with the subject before wrongly labeling your answer as "string propaganda". –  Curious George Jul 12 '11 at 21:42
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What happened to my comments? I downvoted and explained that this answer is confusing the world-sheet physics with spacetime physics. –  Ron Maimon Aug 28 '12 at 8:52
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