# Background independence in string theory

• tom.stoer
ST but not in GR).The problem of background independence has nothing to do with the geometric nature of the theory or even the fact that it's a theory of gravity. Rather, it's an issue that arises when we try to quantize a theory that was not originally formulated in terms of fields. This is the case for GR because the dynamical variables are diffeomorphism-invariant, which makes it tricky to formulate a perturbative expansion around a background (a background being a configuration in which the metric is not dynamical). As a result, perturbative GR is not background-independent: it depends on a choice of background metric that you're expanding around. If you try to

#### tom.stoer

Can anybody explain to me what string theorists mean when they are talking about background independence in string theory?

I understand this concept in the context of general relativity and loop quantum gravity; there it means invariance with respect to active, local diffeomosphisms. In string theory the AdS/CFT conjecture is mentioned, but I do not understand, how this guarantuees background independence. Both the string and the conformal field theory are put on top of a certain background.

Thanks
Thomas

tom.stoer said:
active, local diffeomosphisms
Usually what I thought is called "active" is a transformation (mapping) was that actually changes the object leaving the coordinate system untouched, by opposition to "passive" where only coordinates change, the object being untouched. Is it how you use "active" ? Which difference does it make ?

Can a diffeomorphism be non-local ? I'm sorry, but I think this needs to be clarified !

Note that, the gravity theory in the bulk should safely be assumed to be diffeomorphism invariant in the statement of background independence of string theory via AdS/CFT. If the formulation of gravity theory is not manifestly diffeomorphism invariant, we know that in other formulations (namely GR for instance !) it is.

String theory (not just matrix theory or AdS/CfT) is by construction completely diffeomorphism invariant (see chapter 1 of Polchinski).

Using that definition is thus problematic b/c it doesn't capture a difference between theories.

diffeomosphism invariance and background independence are two different things. One should not confuse the two. A theory may be diffeomospism invariant but not background independent. As stated in the last post string theory is by construction diffeomorphism invariant; one my descibe the theory in any coordiante system.

Background independence is a different kettle of fish. What it means is that there is no a priori background geometry on which the theorey "lives". GR is background independent but a string theory that lives on a given spacetime maifold is not. Even if the effect of gravity is too add quantum effects to the geometry one is still putting a geometry in by hand at some point.

As for what Ads/CFT says about background independance I am no expert. But as far as i know all string theories that are used are not background independant. At some point string theory should probably be made background indpendent, possibly ADS/CFT helps with this. My guess is that Ads/CFT points towards a more general duality which could one day lead to a extremely elegant theory possibly based on the holographic principle.

Sorry for the confusion.

@Humanino: with "local" I mean that you are free to chose any transformation you like. In special relativity only global/ rigid transformations are allowed.

@Haelfix: the diffeomorphism invariance in string theory means (as far as I know) that the theory is diffeomorphism invariant with respect to the world sheet coordinates. But does that mean that this induces diffeomorphism invariance in the world "manifold" automatically?

"What it means is that there is no a priori background geometry on which the theorey "lives""

That is the essence of what many people mean, but its still rather vague and imprecise and there are language problems going on.

For instance, what is a 'background geometry'? Off the top of my head I can think of several inequivalent meanings.

Urs Schreiber years ago mentioned two possibilities (which you can easily generalize) that I think makes it pretty clear. You could say that a background is either (sometimes both):

1) A classical solution (read a saddle point) which we do perturbation theory around
or
2) An object or set of objects (scalars, vectors, tensors, paramaters whatever) that is non dynamical in the action of the theory, eg something that is fixed and never varied.

When string theorists refer to "a background" in string theory they are usually talking about sense (1) and you can specify whether that is referring to the worldsheet or the full space. Thus, every single string theory vacua (all 10^500 or so of them) is 'a background' in their language.

If you use (1) as your definition of a background, the problem of background independance in st is then logically equivalent to saying that perturbative string theory is perturbative and not resummable in an obvious way. Personally, I don't find that particular statement very deep or insightful, and the intended geometric and/or relational significance seems to be lost in translation, but that's fine.

Otoh if you think (2) is the more profound meaning, well then s.t. is fully background independant b/c there is absolutely nothing in the lagrangian that's fixed once the boundary conditions are specified.

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tom.stoer said:
@Haelfix: the diffeomorphism invariance in string theory means (as far as I know) that the theory is diffeomorphism invariant with respect to the world sheet coordinates. But does that mean that this induces diffeomorphism invariance in the world "manifold" automatically?

Yea... Ok that's a good question, and you really should ask a string theorist b/c my understanding of this gets really murky. Here is what I think:

Certainly, at low energies you indeed have a good notion of a manifold to begin with and hence diffeomorphism invariance is enforced. Strings are only allowed (by consistency requirements) to propagate on certain backgrounds that satisfy certain field equations (read Einsteins equations). At very strong coupling its a little harder to identify a coherent state of gravitons that induce a target metric so instead we have a bit of a quantum foam like problem (eg: different degrees of freedom become important). Its a little hard to talk about diffeomorphism invariance of the target space there of course. In some sense the (weyl*diffeo) symmetry of the 2d worldsheet remains the fundamental symmetry of the system.

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Finbar said:
GR is background independent but a string theory that lives on a given spacetime maifold is not.

This statement is based on misunderstanding. Geometry is dynamic to the exact same degree in both GR and in ST. Choosing a particular Calabi-Yau manifold on which to compactify your string theory is not akin to choosing a background: you are specifying a topology, not a geometry. In GR the topology is almost always chosen to be a simply connected manifold of dimension 4. This is not a prediction of GR, and the theory works fine on other topologies (but then it doesn't describe reality).

I am prepared to debate any claim that GR is somehow more 'background dependent' than ST, as long as the person making the claim is willing to define their terminology.

I try to be as precise as possible:

First the standard example for background dependence and why it fails:

In perturbative QG you fix a classical metric (a solution of GR) and you quantize the quantum fluctuation h with respect to g (you use g to fix the light cone structure and to determine the commutaton relation). This program fails (non-renormalizable) because of the fixed background g. This is 1) in Haelfix's terminology.

In perturbative ST you chose a target manifold with a classical metric, which has to be a solution of GR (due to a consistency condition). You do this whileyou are writing down the action of the string theory. This is a background geometry, not only a topology (!) and can never be changed by any quantum effect. Therefore this theory is background-dependent (this has nothing to do with the Calabi-Yau space you eventually chose on top; the background dependence enters already via the classical metric in the very first step). So this is again 1) in Haelfix's terminology.

I know that ST is diffeomorphism invariant and background independent on the level of the world sheet. But I do not see how this induces diffeomorphism invariance in the target manifold (or whatever this becomes after quantization).

Let's compare it with LQG: In this approach you fix a topology (T * R³) which is a global foliation of spacetime, but the R³ has no background structure and the theory is fully diffeomorphism invariant on the that level.

Where is this fully dynamical spacetime hidden in string theory? or in AdS/CFT? It's definately not there in perturbative ST!

ExactlySolved said:
This statement is based on misunderstanding. Geometry is dynamic to the exact same degree in both GR and in ST. Choosing a particular Calabi-Yau manifold on which to compactify your string theory is not akin to choosing a background: you are specifying a topology, not a geometry. In GR the topology is almost always chosen to be a simply connected manifold of dimension 4. This is not a prediction of GR, and the theory works fine on other topologies (but then it doesn't describe reality).

I am prepared to debate any claim that GR is somehow more 'background dependent' than ST, as long as the person making the claim is willing to define their terminology.

No your probably right. But what if one does define a non-dynamic geometry and fixes it as a background with a quantum correction to this then this cannot be considered background independent. I think ST is probably Background independent.

http://arxiv.org/abs/0809.3962v2

I have to understamd this article in more detail - and this will take some some time.

Nevertheless it would be nice to get a short answer right here, so I will rephrase my question as follows:

I know that ST is diffeomorphism invariant and background independent on the level of the world sheet. But I do not see how this induces diffeomorphism invariance in the target manifold ...

1) where is this fully dynamical spacetime hidden in string theory or in AdS/CFT?
2) and would does this help, as any "realistic" string theory (reproducing something similar to the standard model U(1)*SU(2)*SU(3) and certainly NOT AdS!) starts with an own background, for which similar dualities are unknown.

Its worth keeping in mind what's a physical requirement from aesthetic appeal.

You cannot have a physical theory that depends on a coordinate choice (the requirement is coordinate reparamatrization invariance). In other words if you do physics in Rieman normal coordinates on a flat spacetime, you better get the same physics for a solution than you do if you did it in polar coordinates. If you didn't, the theory would be wrong. More strongly, if you have two different backgrounds A and A' in sense 1 that I gave above, and that are in a sense close to each other in Moduli space (the space of all such backgrounds), you have a consistency requirement:

You should be able to directly calculate the observables for both backgrounds, and find that string theory is still the underlying theory behind each solution, such that the form of the equations are preserved (similar to finding that Minkowski and Schwarzschild metrics are both solutions to Einsteins field equations). Moreover, it should be possible to find a path in moduli space that interpolates between such regimes so long as they are sufficiently close.

Now assume that A and A' are NOT close in moduli space. So for instance A is described by type IIA and A' naturally lives in a portion described by type IIB. Notice that the language used is completely different, in other words you no longer have MANIFEST background independance (notice the terminology generalization here). Still by clever use of dualities and things like that you can show that indeed they are still described by string theory, otoh it would be aesthetically nice to have a formalism that doesn't rather arbitrarily change languages. Thus when string theorists talk about background independance, that's really what they are hoping for. A suitable formalism that either is nonperturbative to begin with or slightly weaker, to be able to describe the moduli space in an invariant and consistent way.

As to your question about worldsheet vs target space. The target space must obey coordinate reparamatrization invariance, and that's definitely guarenteed. Unfortunately it can be a little hard to calculate things directly. You really have a gadget (the worldsheet) that spits out answers.

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So as far as I understand you correctly the situation is the following: Your requirement is that two different backgrounds A and A' (can be two different Riemann manifolds to start with, can be two differet Calabi-Yau spaces, can even be to different types of string theory) must be connected in moduli space (as long as they are not "too far away" from each other - e.g two different superselection sectors).

OK, I can live with that as a requirement for background independence.

Two questions regarding the first possibility (two different Riemann manifolds):
is there a proof that dynamics of string theory will produce the same answers even if we start with two different backgrounds?
is there a proof that string theory can produce all those different manifolds as a low energy / semicalssical limit?

To your first question, I don't think so, at least not as a mathematically rigorous proof. The web of dualities are in some cases pretty rigorous and in others are barely past the conjecture stage. Otoh I think that most people are qualitatively happy with the state of the results there at a physics level of rigour as there is a lot of nontrivial evidence (see 2nd superstring revolution and relevant support material)... The literature is long and complex in that regard, and thoroughly outside of my field of knowledge. Slightly more troubling, some of those backgrounds are prohibitively hard to calculate in, sometimes its not even clear what the observables are (this is a general problem with quantum gravity in highly curved backgrounds) so its a little hard to exactly carry out what I said above. Still when calculation is possible and easy, people have done it and found agreement but absent a nice manifestly BI formulation its unlikely a full rigorous proof will exist any time soon (but who knows).

To your second question, I think that follows directly from the equations of motion so I think that yes that's true. In the limit where all the extra dilaton and the string coupling is small other objects and so forth are small, you will get Einsteins field equations (with matter).

Fine, thanks.

What I learned is that in addition to the well-known dualities between different types of string theories one should add moduli for the Calaby-Yau spaces plus "moduli" for the classical background geometry (I didn't know that these are called moduli as well). My feeling is then that all these different geometries add up to the so-called landscape. Selecting a vacuum means selecting a background-geometry.

Two STs (two backgrounds) are equivalent if a duality relation can be found. Two STs are inequivalent if no such duality exists. So AdS/CFT is not only one string theory with AdS(5) * S(5) but a whole class of equivalent theories, where all geometries "close" to AdS(5) * S(5) are incorporated (e.g. all geometries with the same topology, I guess). For all other classes of string theories a similar relation should hold.

For some classes the results are rather rigorous, for others there are only indirect hints. In the very end one expects that the equivalence classes are rather large, such that different backgrounds can all be related by dualities or by dynamics.

It seems that there still a long way to go.

Is this conclusion correct?

Yes, the moduli space is the space of all stringy backgrounds (eg classical solutions of s.t), so from the target space point of view that includes the relevant shape of the 11 dimensional object + all the additional fluxes, supersymmetries, Dbranes etc etc

I think I agree roughly with the rest (although caveat its subtler even than topology, since we know of topology changing transitions within st, read up on mirror symmetry) but yea in the end people really think there is just *one* single string theory (whatever that means ultimately, since st is no longer just about strings) such that all these different corners are ultimately just part of a single unifying theory. Of course, that's still conjecture at this point, and its highly nontrivial, but its backed by some pretty compeling arguments and a lot of work.

I've heard about this mirror symmetry; yes, one should include this possibility when one is talking about topology - it's topology plus something :-).

I am familiar with rather old-fashioned physics. You write down a Lagrangian, you derive the Hamiltonian or the path integral; you quantize the Hamiltonian; ...

Is there a glimpse of a Lagrangian or something similar in "background independent s.t."? How could it look like in M-theory? Just to get a feeling! Reading review articles regarding s.t. you always see the "old-fashioned" objects in perturbative s.t., so you can imagine how it works there, but you immediately lose background independence (as I just learned). As soon as it comes to M-theory or branes no Lagrangian or something similar appears. So that's why I can't understand how this should work.

Compare the s.t. landscape with solid state physics: In the latter case I can write down the QED Langrangian and say "that's it! it's all you need." Of course many people would not be happy with it and it would certainly not help if one is interested in calculating the melting temperature of ice - but in some way it's the most fundamental principle.

Is there something similar in s.t or M-theory?

Haelfix said:
To your second question, I think that follows directly from the equations of motion so I think that yes that's true. In the limit where all the extra dilaton and the string coupling is small other objects and so forth are small, you will get Einsteins field equations (with matter).

Just a short question: there is no way to get the full nonlinear field equations for the metric out of string theory, right? Considering this, can there be made a statement in how far string theory is a realistic model of gravitational interactions?

My simplistic grasp of Rovelli & co Witten etc, is they mean a local u,v gauge independent of c the background. I think they mean they want a theory independent of EFE's invariant (but whadda I, u no?)

ps, I am pun-intensive

Regarding Rozali's paper http://arxiv.org/abs/0809.3962v2: [Broken]

It's positive and negative at the same time: as far as I understand it claims that the AdS theory is background independent in the sense that all dynamically reachable metrics compatible with AdS can indeed be reached and that only the condition of "asymptotically AdS" serves as a "superselection sector". Fine.

At the same time it claims that the fully background independent theory of gravity in this AdS theory can be rewritten in terms of the CFT in flat four dimensional spacetime (this is the duality conjecture). I think this is not needed directly for the argument of background independence; it is mentioned because non-perturbative calculations seem to possible in CFT and can therefore be used to proof the background independence in AdS.

But already the string theory in AdS has a certain fixed background, namely S(5). (it's never mentioned but in reality the AdS theory is a low energy limit of a string theory living on AdS(5)*S(5), i.e. a special compactification on S(5), as far as I remember).

So my questions are:

Is this duality proven?
(if yes this would require non-perturbative calculations in both theories because you have to map the full theories, not only certain limits like weak coupling and large N)

Is there a way to establish full BI in s.t.?
(in AdS/CFT: a way to remove the fixed S(5))

Is there a way to find a BI independet formulation of your world, which requires not AdS(5) but some X(4)?

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