General relativity from string theory

[tom.stoer]

@suprised: Can you please explain which metrics are not compatible with or do not emerge from string theory? which conditions are required? (Ricci-flatness, static / stationary; Killing vectors, ...)? what about dS, FRW, ...?

 

[nrqed]

It's done in almost all the textbooks out there, in particular GSW. Online I think David Tong has some lecture notes: (see here: http://www.damtp.cam.ac.uk/user/tong/string/seven.pdf). The Einstein Hilbert action emerges on page 168. Alternatively I believe Susskind goes over it in his lectures on youtube (this will be at the level of Zweibach)

Actually calculating the full one loop beta functions is a bit of a chore, and I have never done it, but you will get the picture.

Tong says on page 158 that inserting a factor e^V amounts to inserting a coherent state of gravitons (with V defined in his equation (7.2))
I have seen that statement many times before, of course. But I don't understand what it means. Is that in the usual sense of coherent state, i.e a state that, in the operator language, is an eigenstate of the annihilation operator? If so, how do we see that the exponential form in the path integral language corresponds to an eigenstate of the annihilation operator in the operator language?

 

[suprised]

@suprised: Can you please explain which metrics are not compatible with or do not emerge from string theory? which conditions are required? (Ricci-flatness, static / stationary; Killing vectors, ...)? what about dS, FRW, ...?

No I can't - metrics do not appear in isolation, there are other fields coupled to it, and only the whole package is consistent or not; so that's essentially a question about the swampland and there is no easy answer for that AFAIK.

 

[haushofer]

I've read that Calabi's conjecture was not motivated by GR (although Yau's interest in it was). So maybe there's a "pure" reason for this.

But the Einstein equations already pop up without compactification, right? It's only after this that one considers CY-compactification.

The first time I saw Einstein's equations popping up as a QM-consistency in string theory I was really impressed, especially in combination with the fact that the string spectrum contains gravitons. But I've never really understood how stringent this result is. Is it really "a deep physical result", or can it be understood more directly?

Perhaps a vague question, so never mind if it doesn't make sense ;)

 

[suprised]

But the Einstein equations already pop up without compactification, right? It's only after this that one considers CY-compactification.

Right.

The first time I saw Einstein's equations popping up as a QM-consistency in string theory I was really impressed, especially in combination with the fact that the string spectrum contains gravitons. But I've never really understood how stringent this result is. Is it really "a deep physical result", or can it be understood more directly?

One could vaguely say that by construction the effective action must be Lorentz invariant (the symmetry currents are conserved), so it is natural that GR pops out. But Lorentz invariance does not imply GR, eg one may contemplate about a theory with many massless higher spin fields or some other crazy theory. In fact crazy theories do pop out in certain singular limits (like tensionless strings), so that Einstein gravity emerges at all, in the limit where it is desired (small energies and curvature), seems nontrivial and I wouldn't know of a direct shortcut to prove this.

And there goes much more into that, eg the absence of anomalies which ensures conservation of symmetry currents also at the quantum level. That this actually appears to work in all detail is one of the main reasons for excitement.

 

[tom.stoer]

No I can't - metrics do not appear in isolation, there are other fields coupled to it, and only the whole package is consistent or not; so that's essentially a question about the swampland and there is no easy answer for that AFAIK.

Sounds OK, but why can one read something regarding Ricci-flatness? Is this just an oversimplification?

 

[suprised]

Ricci-flatness has to do with supersymmetry, ie this is a necessary condition for a compactification manifold for the existence of unbroken supercharges. That's written in all textbooks, eg GSW.

 

[tom.stoer]

So I got this completely wrong? I mean neither is there target space SUSY nor is the Universe Ricci-flat.

 

[Haelfix]

Tong says on page 158 that inserting a factor e^V amounts to inserting a coherent state of gravitons (with V defined in his equation (7.2))
I have seen that statement many times before, of course. But I don't understand what it means. Is that in the usual sense of coherent state, i.e a state that, in the operator language, is an eigenstate of the annihilation operator? If so, how do we see that the exponential form in the path integral language corresponds to an eigenstate of the annihilation operator in the operator language?

Hi Nrqed, that is a truly excellent question and way above the level of Tong.

(for the identical statement said slightly differently, see Polchinksi):
http://books.google.com/books?id=k4...age&q=coherent states vertex operator&f=false

I think the answer to your first question is almost but not quite. The problem is there are gauge fixing ambiguities creeping into the calculation, and you have to ensure the symmetries of string theory (Virasoro constraints) are enforced. Consequently the naive definition of a coherent state must be slightly generalized to ensure this.

But once that is done, then yes there is a sense in which you can show that what you get is an eigenstate of the annihilation operator, although the paper I am looking at is technically challenging...

See
http://arxiv.org/abs/0911.5354v2 starting on page 27 for a discussion and the calculation for eg closed strings in lightcone gauge is on page 36, although the vertex operators are more general (DDF vertex operators). Maybe one of the stringy experts here knows a simpler calculation, but I have never seen it done nor could I find it in a quick literature search. I am actually a little surprised that I couldn't find the calculation done in texts regarding nonminimal sigma models, since this is very much isomorphic.

 

[Haelfix]

So I got this completely wrong? I mean neither is there target space SUSY nor is the Universe Ricci-flat.

The only thing you are guarenteed to get in low energy string theory (at least the usual constructions, where we are not talking about non critical s.t), are Einstein's equations, in particular (minimally) classical gravitational waves. You are most assuredly not guarenteed to get any sort of cosmological constant. In fact classically I believe there are no go theorems that preclude the existence of say a positive cosmological constant, thus that would need to be generated by quantum effects (for instance the KKLT solution). Very nontrivial!

As far as cosmological solutions. That is also a very difficult question, b/c you need to ensure a way that all the extra typical stringy stuff (gravitinos, dilatons, blah blah blah) doesn't forbid universes like our own, and you therefore need mechanisms so that you don't have too much of it floating around (otherwise it could say overclose the universe or mess with big bang nucleosynthesis constraints). There have been many proposals on how to do all this, and the literature and phenomenology is vast.

 

[atyy]

Some of the classical no-go theorems that Haelfix mentions are referenced in http://arxiv.org/abs/hep-th/0106109v1 , one of which is http://arxiv.org/abs/hep-th/0007018 .

 

[Finbar]

oh come on ricci flatness is hardly deep if you didn't have it then you'd violate energy conservation.

 

[nrqed]

Hi Nrqed, that is a truly excellent question and way above the level of Tong.

(for the identical statement said slightly differently, see Polchinksi):
http://books.google.com/books?id=k4...age&q=coherent states vertex operator&f=false

I think the answer to your first question is almost but not quite. The problem is there are gauge fixing ambiguities creeping into the calculation, and you have to ensure the symmetries of string theory (Virasoro constraints) are enforced. Consequently the naive definition of a coherent state must be slightly generalized to ensure this.

But once that is done, then yes there is a sense in which you can show that what you get is an eigenstate of the annihilation operator, although the paper I am looking at is technically challenging...

See
http://arxiv.org/abs/0911.5354v2 starting on page 27 for a discussion and the calculation for eg closed strings in lightcone gauge is on page 36, although the vertex operators are more general (DDF vertex operators). Maybe one of the stringy experts here knows a simpler calculation, but I have never seen it done nor could I find it in a quick literature search. I am actually a little surprised that I couldn't find the calculation done in texts regarding nonminimal sigma models, since this is very much isomorphic.

Thank you Haelfix, I truly appreciate your explanations. I will look at the references with great interest.