# In which sense is string theory background-independent?

## Main Question or Discussion Point

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:

Related Beyond the Standard Model News on Phys.org
Demystifier
Gold Member
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.

Nothing is wrong with your reasoning, string theory formulated in that way is background dependent.
Fine. But I am looking forward to Ben's answer.

However, such a formulation of string theory is a perturbative formulation.
So Ben's claim that string theory formulated as a non-linear sigma model is background-independent is not correct?

... 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.
I know that.

... I just explained how the worldsheet theory is, in fact, background-independent. (And yes, for real; not just "for arbitrary backgrounds").
But the above mentioned world-sheet formulation is background-dependent.

I think some of the confusion comes from the fact that introductory textbooks will usually draw pictures of strings propagating in spacetime, and will refer to the 10 scalars as "spacetime coordinates", talk about things like "induced metrics", etc. But these are all words that apply only in the low-energy limit.
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.

In reality the worldsheet theory contains a bunch of scalar fields that interact in some conformal field theory. The theory is fully quantum, and only in the classical limit can these fields even be viewed as smooth functions, let alone coordinates on a manifold. In the full theory, they are a quantum mess.
I fully agree, but still G is a background field.

String theory doesn't even assume the existence of these basic structures (smoothness, topological, etc.)
The non-linear sigma model does assume the existence of G(X).

Demystifier
Gold Member
So Ben's claim that string theory formulated as a non-linear sigma model is background-independent is not correct?
Before answering, I would like to see more details on this claim. Where can I see them?

The worldsheet theory is 2-dimensional coordinate-invariant, so I hope you would agree from classical differential geometry (and GR) that one can discuss the 2-d theory without making any reference to a putative 10-dimensional ambient space.
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.

But what's more, is that these 10 scalars only have an interpretation as geometry in the low-energy approximation. At higher energy, the very notion of a background geometry breaks down.
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.

Ben Niehoff
Gold Member
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.

As far as I can tell, Tom, you're right. I think it slipped my mind that G appeared there explicitly.
No problem; at least we agree on the definition of background-dependence via G(X).

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).
Yes, I know.

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

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.
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
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.
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 ...

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.

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

fzero
Homework Helper
Gold Member
Is the backreaction restricted to certain "superselection-sectors" in G?
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.

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.

fzero
Homework Helper
Gold Member
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.
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.

As you say, perturbative string theory (let's call it PST for short), in particular its action, depends on a chosen background geometry. So it is not background independent in the same way that GR is. However, in PST infinitesimal variations in the background metric arise by adding local operators to the string action. The coefficients of some of these local operators are connected directly to the matter distribution. Large variations are smoothly generated from these infinitesimal variations by considering, for example, coherent state operators. 2d conformal invariance actually enforces that all the geometries generated in PST satisfy Einstein's equation. In fact, PST also connects backgrounds with different topology through local operators. There is no analogue of this in GR.

In a certain sense, PST turns background-independence upside down. All backgrounds are smoothly connected in the way described above. Any particular point in the solution space can be seen to satisfy a condition that can be expressed in a background-independent manner. However, in order to actually see this, we must choose a particular splitting of fields into "background" and "excitation." As you allude to, this choice has consequences for the operator algebra. Only specific backgrounds (flat space and a few others) turn out to clean enough to compute with.

I'm not sure that I would choose to call this state of affairs "non-manifest background independence" myself. It's clear that PST does (at least in principle) everything that GR does as far as producing a geometry from a specified matter distribution. Since it also describes topology change on essentially the same footing, it actually does more than GR in this respect. The background independent Einstein equation even comes out in a roundabout manner, but I would not necessarily insist on using the connectedness of backgrounds to call PST background independent itself.

Now, AdS backgrounds actually demonstrate background independence in a clearer way. These form a superselection sector of spacetime solutions in whatever (consistent) theory you want to consider. We can't generate an AdS space from a closed or flat geometry by using a finite amount of energy. So we should really fix the asymptotics of the solutions that we will admit (in the same way that we fix the topology in GR) and then study what metric geometries are allowed in the bulk. This is a very mild degree of background fixing compared to what is done in PST, but here it is actually forced upon us by the physics rather than the formalism.

Then we would argue that string theory on spaces which are asymptotically AdS is in fact background independent. The reason is that we have a CFT description of the states and their dynamics and all bulk geometry is completely emergent from CFT degrees of freedom. As explained before, we need to fix the asymptotics because the AdS spaces are not smoothly connected to the other superselection sectors like flat or closed geometries. The asymptotic geometry is the only part of the geometry that directly appears in the CFT.

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.

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.

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.

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.

What means "empirical support" for Ads/CFT?

Demystifier