# In which sense is string theory background-independent?

1. Jul 8, 2013

### tom.stoer

In another thread Ben indicated that string theory formulated as non-linear sigma model using world-sheet action is - in some sense - background independent. To discuss this I start with a generalization of the Polyakov action

$S_G[X] = \frac{1}{4\pi\alpha}\int d^2\sigma \, \sqrt{g} \, g^{ab} \, \partial_a X^\mu \, \partial_b X^\nu \, G_{\mu\nu}(X)$

Here g is the world sheet metric, X are scalar fields, and G is usually identified with the target-space metric.

So we do not have one single action S, but a class of actions SG, labelled by G. My conclusion is that G is a non-dynamical background; the string X does not back-react on G; the dynamics of X does not connect different G-sectors; G is introduced by hand, it is not subject to the dynamics of the theory.

This is what is usually called background-dependency.

What is wrong in my reasoning?

Last edited: Jul 8, 2013
2. Jul 8, 2013

### Demystifier

Nothing is wrong with your reasoning, string theory formulated in that way is background dependent.

However, such a formulation of string theory is a perturbative formulation. On the other hand, most string theorists believe that a fundamental formulation of string theory is different. First, it is almost certainly non-perturbative. Second, there are indications that it could be background-independent. Unfortunately, this non-perturbative formulation, usually identified with the mysterious M-theory, is very very far from being well understood. It is not even clear what the basic principles of M-theory are.

3. Jul 8, 2013

### tom.stoer

Fine. But I am looking forward to Ben's answer.

So Ben's claim that string theory formulated as a non-linear sigma model is background-independent is not correct?

I know that.

4. Jul 8, 2013

### tom.stoer

But the above mentioned world-sheet formulation is background-dependent.

It does not depend on the interpretation of X and G. The existence of G in the action means that the theory depends on the background G.

I fully agree, but still G is a background field.

The non-linear sigma model does assume the existence of G(X).

5. Jul 8, 2013

### Demystifier

Before answering, I would like to see more details on this claim. Where can I see them?

6. Jul 8, 2013

### tom.stoer

I agree that the world-sheet theory is coordinate invariant w.r.t. to the world-sheet. But G(X) is contained in the action SG[X], so this action depends on some background.

I do not rely on the interpretation of G(X) as target space metric. Yes, the approximation (and the interpretation) may break down, but at least on the formal level G(X) is still present.

7. Jul 8, 2013

### Ben Niehoff

As far as I can tell, Tom, you're right. I think it slipped my mind that G appeared there explicitly.

However, consistency of the 2d CFT (i.e., vanishing of anomalies) demands that, to first approximation, G must be Einstein. Which is, in the Lorentzian-signature case, a dynamical equation (in fact, the exact same dynamical equation one gets from GR). So I think the answer is subtle and may require a more fundamental definition.

8. Jul 8, 2013

### tom.stoer

No problem; at least we agree on the definition of background-dependence via G(X).

Yes, I know.

9. Jul 8, 2013

### julian

I think this might be where Smolin's comment is relevant:

"Although we sometimes use the Einstein’s equations as if they were a machine for generating solutions, within which we then study the motion of particles of fields, this way of seeing the theory is inadequate as soon as we want to ask questions about the gravitational degrees of freedom, themselves. Once we ask about the actual local dynamics of the gravitational field, we have to adopt the viewpoint which understands general relativity to be a background independent theory within which the geometry is completely dynamical, on an equal footing with the other degrees of freedom. The correct arena for this physics is not a particular spacetime, or even the linearized perturbations of a particular spacetime. It is the infinite dimensional phase space of gravitational degrees of freedom. From this viewpoint, individual spacetimes are just trajectories in the infinite dimensional phase or configuration space; they can play no more of a role in a quantization of spacetime than a particular classical orbit can play in the quantization of an electron."

String people then claim we are dealing with gravitational degrees of freedom cus we have a theory of interacting spin-2 particles. But as Penrose points out adding a finite number of gravitons doesn't change the target spacetime. Plus I dont think all general relativists completely agree with the possibility of GR being a theory of interacting gravitons starting from Minkowski spacetime.

Last edited: Jul 8, 2013
10. Jul 8, 2013

### tom.stoer

I agree with julian; I think these are the ideas most people have in mind when criticising string theory as background-dependent.

This is due to the fact that 90% of all people learning and practicing QFT are doing nothing else but perturbation theory. So they do not even understand the problem in the standard model.

But then it seems that I have to agree with
which means that asking for a background-independent formulation is the same as asking for a fundamental definition of M-theory. So as long as the latter unknown I should stop asking regarding the former ...

11. Jul 8, 2013

### Haelfix

Hi Tom, I just want to correct one thing. The worldsheet fields does backreact on the target space metric. That this happens is absolutely not obvious, and comes into play in subleties regarding the implementation of vertex operators.

I've since forgotten the exact details (picture changing operators and the like), but i'll try to find a reference when I get time.

12. Jul 8, 2013

### tom.stoer

Is the backreaction restricted to certain "superselection-sectors" in G?

13. Jul 8, 2013

### fzero

A simple way to see the backreaction is to note that perturbative interactions are treated analogous to ordinary QFT, by deforming the action by invariant operators, $V_I$, called vertex operators in this context, coupled to sources:

$$S = S_G + \sum_I c_I V_I.$$

In particular, the graviton vertex operator for the bosonic string (expanding around flat target space) is

$$V_{\mu \nu} = \frac{1}{4\pi\alpha'}\int d^2\sigma \, \sqrt{g} \, g^{ab} \, \partial_a X_\mu \, \partial_b X_\nu e^{i p\cdot X}.$$

The effect of adding this term is to shift

$$G_{\mu\nu} = \eta_{\mu \nu} \longrightarrow \eta_{\mu \nu} + c_{\mu \nu}e^{i p\cdot X} ,$$

where $c_{\mu \nu}$ is the corresponding source. For the fermionic string theories, the definition of the graviton and other vertex operators is indeed more complicated, as Haelfix alluded to by mentioning picture-changing. The complications don't really change the physical picture of backreaction, but are mainly related to self-consistency of the quantization in the presence of conformal (super)symmetry.

In geometric models, the collection of such vertex operators completely describes the size and shape parameters of the space that the theory is compactified on. This also leads to the description of topology change via deformations by appropriate vertex operators (for example, in the twisted sector of some orbifold description).

There are certainly superselection sectors, such as the ones defined by the value of the cosmological constant. In order to transition between $\Lambda > 0, =0 , < 0$ requires an infinite amount of energy and is outside the realm of perturbation theory.

14. Jul 8, 2013

### tom.stoer

Thanks to Haelfix and fzero for clarification. I remember these picture changing operators and topology changes vaguely ;-)

But if we agree that the Polyakov formulation is background dependent, then the next question is, what the indications are to believe in a background independent formulation, even so there is no explizit formulation known.

15. Jul 9, 2013

### fzero

Old evidence consists of background independent open string and closed string field theories. I haven't studied string field theory in any detail, but I am aware that the bosonic versions have been somewhat successful, but the fermionic versions suffer from various technical problems that complicate their study.

The best argument for background independence now comes from AdS/CFT. Rather than repeat myself, I found a post in an older thread that sums up the situation. It's a useful starting point and we can elaborate as necessary.

16. Jul 9, 2013

### Haelfix

Unfortunately i'm in the process of a move, and I couldn't track down the discussion about vertex operators that I had in mind (it's possibly in Ortin but i'm not sure). Anyway, to add to what Fzero said, the full massless sector of the string theory under consideration typically deforms the backgrounds in some nontrivial ways and there is a host of additional consistency requirements on what type of backgrounds you can allow (typically only backgrounds with full unbroken spacetime susy lead to actual tractable results but I am far from an expert in any of this material).
Now, I completely agree that when you analyze the calculations (at least to my untrained eye), there is an element of clumsiness and blackbox like qualities to a lot of this material (perhaps mathematical relics from the old Smatrix program where st was born) as well as certain types of questions that you can ask. The miracle is that it seems to work at all, and really hints at the extra structure that seems so compelling to theorists.

I think what I really object too, is to use the word 'background independance' as a sort of theory cudgel. Not only is a proper understanding of this concept vague and frought with mathematical subleties and difficulties in even defining what it means classically (heuristically it is quite simple, but surprisingly difficult to really nail down when analyzed in depth). But I mean more profoundly, it's not quite clear to me what such a thing really 'ought' to look like even in principle.

String theory as it currently stands is a complete maze of ideas, where to define what a background even is, is really more of a convention than anything else. I mean there are perturbative backgrounds, nonperturbative backgrounds, emergent low energy backgrounds when some coupling constant is sent to zero, 'meta' backgrounds that are intermediary steps in various calculations, backgrounds on the worldsheet, backgrounds in 11d, various arbitrary backgrounds where you only fix the asymptotic values of the fields etc etc. There is also a complete zoo of relationships between these concepts, many times linking perturbative with nonperturbative, mixing dimensionalities, and so forth and so on. At the end of the day, the real degrees of freedom are simply unknown, and the possibility exists that there might not even exist a simple description that you can write down and enummerate in a straightforward way.

More trivially, it might be that at the end of the day, the final theory might be so constraining as to simply spit out a single generalized 'background' that we call the universe (and is thus trivially background dependant). I simply don't know.

17. Jul 10, 2013

### tom.stoer

Is there a clear idea how to generalize the AdS/CFT approach to general spacetimes = superselection sectors? Is AdS/CFT strictly proven, or is this still a duality which is true for certain limits?

And in which sense is this proposal really background independent? I understand that AdS represents one superselection sector (I think it is acceptable to have such superselection sectors) with bulk d.o.f. mapped to boundary d.o.f. But the 10-dim. spacetime has an additional S5 which is rather strange b/c this seems to be rather artificial, and I do not see what happens to the S5 deformation d.o.f., not to mention topology changes.

18. Jul 12, 2013

### Ilja

Another question is if it is reasonable to look for a background-independent theory of quantum gravity.

In arXiv:0909.1408 I argue that already in a quasiclassical situation, where superpositions of different gravitational fields can be considered, a common background becomes observable.

19. Jul 13, 2013

### Haelfix

Ads/CFT is not strictly proven (moreover there are many ads/cft's under the general moniker, some are closer to being proven than others), but I think its safe to say that if it's wrong (and people are beginning to write probing papers about this very possibility) it would have to be wrong in a very subtle fashion.

The amount of evidence that has been accumulated for it to be at least approximately true, is really quite overwhelming (and has considerable empirical support).

There are numerous proposals to generalize this to different superselection sectors ds/ds, higher spin gravity, ds/cft etc But I think its fair to say that none of them have the conceptual clarity and usefulness as the original and you are looking at much more speculative type of research programs.

As far as the stringy parts of the AdS proposals. Those are really microscopic 'curled' up dimensions that are mostly important for quantum gravity questions, but yes the whole apparatus of string theory is required in order to provide a sensible UV limit, as usual.

20. Jul 15, 2013

### Ilja

What means "empirical support" for Ads/CFT?