Why do people say string theory is non-manifestly background-indepndent?

[julian]

Background-independence is the requirement that the theory be formulated based only on a bare differentiable manifold but not on any prior geometry. General relativity is the first example of such a theory. This is a radical shift as all theories before General relativity had part of their formulation a pre-existing geometry, e.g. Maxwell's equations are based on Minkowski spacetime.

All perturbative string theories are based on a prior background geometry (by the very definition of perturbative). So perturbative string theory is not background independent. But some people then claim that perturbative string theory is just non-manifestly background independent, as if background independence is being gauge fixed...but string theories on distinct background geometries are obviously physically distinct situations, so how can they be related to each other by a gauge transformation?

 

[julian]

Perturbative string theory on different background spacetimes are different. For example, QFT axiomatically assumes causality with respect to the background metric, and this makes it way into the operator algebra - so perturbative string theory based on different geometries are physically distinct.

 

[fzero]

Background-independence is the requirement that the theory be formulated based only on a bare differentiable manifold but not on any prior geometry. General relativity is the first example of such a theory. This is a radical shift as all theories before General relativity had part of their formulation a pre-existing geometry, e.g. Maxwell's equations are based on Minkowski spacetime.

All perturbative string theories are based on a prior background geometry (by the very definition of perturbative). So perturbative string theory is not background independent. But some people then claim that perturbative string theory is just non-manifestly background independent,

It would probably help the discussion if you'd cite a particular source for this statement. I'm familiar with the general ideology, but many are probably not.

as if background independence is being gauge fixed...but string theories on distinct background geometries are obviously physically distinct situations, so how can they be related to each other by a gauge transformation?

First off, distinct background geometries are not necessarily related to each other by a gauge transformation, even in general relativity. In GR, no diffeomorphisms connect solutions of different topology. Smoothly varying the matter distribution leads to smooth changes in the metric geometry.

Perturbative string theory on different background spacetimes are different. For example, QFT axiomatically assumes causality with respect to the background metric, and this makes it way into the operator algebra - so perturbative string theory based on different geometries are physically distinct.

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.

 

[julian]

Thanks for your interesting response. If PST is non-manifestly background independent it is subtle.

Have to log off now but will come back to you tomorrow.

Can I just mention on the issue of whether GR can be derived from string theory. Like you say, a necessary condition for PST to be consistent is that the two dimensional world sheet QFT that defines the theory be conformally invariant, to leading order this condition is equivalent to Einstein's equations. However, in PST, you are lead to having to impose spacetime super symmetry, but this requires the spacetime admit a time-like Killing vector field, that is, the spacetime is stationary. So PST only works on stationary spacetimes which are of measure zero in the space of solutions of Einstein's equations.

 

[fzero]

Can I just mention on the issue of whether GR can be derived from string theory. Like you say, a necessary condition for PST to be consistent is that the two dimensional world sheet QFT that defines the theory be conformally invariant, to leading order this condition is equivalent to Einstein's equations. However, in PST, you are lead to having to impose spacetime super symmetry, but this requires the spacetime admit a time-like Killing vector field, that is, the spacetime is stationary. So PST only works on stationary spacetimes which are of measure zero in the space of solutions of Einstein's equations.

In a supersymmetric theory, only very special types of spacetimes will be stable solutions. Whether or not these are of measure zero is not for me to say (the presence of so-called "flat directions" in SUSY vacua would suggest that the SUSY solutions are not). Nonsupersymmetric spacetimes can appear as metastable objects (if these are of measure zero, as expected, it slightly improves the landscape problem), which is the case in string theory. This obviously leads to various complications, but there are still a few things that one can say.

First is the question of how Einstein's equation comes out. This we understand in the sense of an effective field theory. The terms involving the Einstein tensor arise at tree-level. In general, all possible curvature invariants would be generated by quantum effects, but where SUSY holds, some types of corrections will be forbidden. Where we can concretely formulate PST, we can in principle use it to compute the coefficients of the terms in the EFT.

Now, it's clear that non-supersymmetric and even non-stationary spacetimes are connected to SUSY points in the "solution space." We can turn on non-SUSY and even time-dependent deformations of a SUSY background. Where we will run into trouble is if we then try to perform the different split into background and excitation at the new point. We will be quickly aware of the instability of the background we have chosen and this will compromise our efforts to make computations. However, no one would believe that the EFT description breaks down here, just the predictability of the coefficents appearing therein.

There has been a limited amount of success in studying certain time-dependent backgrounds somewhat outside of the geometric, perturbative string paradigm. These include backgrounds in noncritical string theory as well as dS/CFT ideas.

 

[julian]

Can I just mention on the issue of whether GR can be derived from string theory. Like you say, a necessary condition for PST to be consistent is that the two dimensional world sheet QFT that defines the theory be conformally invariant, to leading order this condition is equivalent to Einstein's equations. However, in PST, you are lead to having to impose spacetime super symmetry, but this requires the spacetime admit a time-like Killing vector field, that is, the spacetime is stationary. So PST only works on stationary spacetimes which are of measure zero in the space of solutions of Einstein's equations.

As I'm no expert in string theory, you obviously know a lot more, I'd better give a reference: one source I used was from Smolin's book "The trouble with physics" p184-186. I think I used another reference but I can't find it at the moment.

 

[julian]

The other reference is Smolin's paper "The case for background independence" http://fr.arxiv.org/pdf/hep-th/0507235

"Some string theorists have also claimed that string theory does not need a background independent formulation, because the fact that string perturbation theory is, in principle, defined on many different backgrounds is sufficient. This assertion rests on exaggeration and misunderstanding. First, string perturbation theory is so far only defined on stationary backgrounds that have timelike killing fields. But this is a measure zero of solutions to the Einstein equations. It is, however, difficult to believe that a consistent string perturbation theory can be defined on generic solutions to the Einstein equations because, in the absence of timelike killing fields, one cannot have spacetime supersymmetry, without which the spectrum will generally contain a tachyon21."

He then goes on to say:

"More generally, this assertion misses completely the key point that general relativity is itself a background independent theory. 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."

I think he is saying that there is a difference between doing physics on curved spacetime, even if the spacetime is a solution of Einstein's equations, which I think is what PST is, and doing general relativity proper.

 

[fzero]

The other reference is Smolin's paper "The case for background independence" http://fr.arxiv.org/pdf/hep-th/0507235

"Some string theorists have also claimed that string theory does not need a background independent formulation, because the fact that string perturbation theory is, in principle, defined on many different backgrounds is sufficient. This assertion rests on exaggeration and misunderstanding. First, string perturbation theory is so far only defined on stationary backgrounds that have timelike killing fields. But this is a measure zero of solutions to the Einstein equations. It is, however, difficult to believe that a consistent string perturbation theory can be defined on generic solutions to the Einstein equations because, in the absence of timelike killing fields, one cannot have spacetime supersymmetry, without which the spectrum will generally contain a tachyon21."

This is an issue for any supersymmetric theory, not just string theory. Nonsupersymmetric solutions are at best metastable. Typically such theories will have a tachyon in them. A prototypical example is the usual potential that illustrates the Higgs mechanism. At the symmetry preserving local maximum, the scalar field is a tachyon. In the true vacuum, the scalar is no longer tachyonic.

He then goes on to say:

"More generally, this assertion misses completely the key point that general relativity is itself a background independent theory. 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."

I think he is saying that there is a difference between doing physics on curved spacetime, even if the spacetime is a solution of Einstein's equations, which I think is what PST is, and doing general relativity proper.

String theorists were actively aware by the early 90s that perturbation theory would not suffice to properly analyze the string vacuum structure. By this, I mean the community at large, not just the deepest thinkers. Few people (except for Witten and a few others) were actively worrying about background independence. Instead people were thinking about quantitative problems like strong coupling and nonperturbative contributions to low-energy effective potentials.

The discovery of the dualities of the mid 90s and their culmination into AdS/CFT was a huge, but by no means complete, step towards understanding nonperturbative phenomena. As I said earlier, in AdS/CFT some important elements of background independence are actually explict, though perhaps not as completely as Smolin would like. In the other cases, without a detailed nonperturbative definition, there are a host of other problems at least as important as background (in)dependence. I realize that Smolin's main point is that perhaps BI is somehow the key to solving all of these other issues. I can't really argue with that, but I would also be happy with a background dependent theory (string or nonstring) that predicted realistic phenomenology.

In any case, the fact that PST has serious shortcomings was known well more than a decade before 2005. Like any perturbation theory, it has regions of usefulness, while being completely inadequate to address other important phenomena. Where we have our most useful definition of string theory, we also have background independence to the appropriate degree (within the superselection sector).

 

[julian]

This is an issue for any supersymmetric theory, not just string theory. Nonsupersymmetric solutions are at best metastable. Typically such theories will have a tachyon in them. A prototypical example is the usual potential that illustrates the Higgs mechanism. At the symmetry preserving local maximum, the scalar field is a tachyon. In the true vacuum, the scalar is no longer tachyonic.

String theorists were actively aware by the early 90s that perturbation theory would not suffice to properly analyze the string vacuum structure. By this, I mean the community at large, not just the deepest thinkers. Few people (except for Witten and a few others) were actively worrying about background independence. Instead people were thinking about quantitative problems like strong coupling and nonperturbative contributions to low-energy effective potentials.

The discovery of the dualities of the mid 90s and their culmination into AdS/CFT was a huge, but by no means complete, step towards understanding nonperturbative phenomena. As I said earlier, in AdS/CFT some important elements of background independence are actually explict, though perhaps not as completely as Smolin would like. In the other cases, without a detailed nonperturbative definition, there are a host of other problems at least as important as background (in)dependence. I realize that Smolin's main point is that perhaps BI is somehow the key to solving all of these other issues. I can't really argue with that, but I would also be happy with a background dependent theory (string or nonstring) that predicted realistic phenomenology.

In any case, the fact that PST has serious shortcomings was known well more than a decade before 2005. Like any perturbation theory, it has regions of usefulness, while being completely inadequate to address other important phenomena. Where we have our most useful definition of string theory, we also have background independence to the appropriate degree (within the superselection sector).

Thanks for your informed response.

As I understand it for the string, the world sheet super symmetry is established by adding fermionic coordinates to the two-dimensional world-sheet coordinates, spacetime supersymmetry is acheived by adding fermionic coordinates to the `bosonic' coordinates x^\mu. Spacetime supersymmetry can be understood to come about from world-sheet supersymmetry together with the imposition of the GSO condition (the GSO projection is the truncation of the spectrum that eliminates the tachyon). It's the stronger condition of spacetime supersymmetry that imposes the need for timelike Killing vector field. Is supergravity from the outset based on spacetime supersymmetry? I wonder would the solution space freed up by symmetry breaking at lower energies?

Would be interested in comments made by Witten and other "deep thinkers" on background independent string theory. You mention Smolin indeed thinking BI is key to resolving problems in ST. Didn't Green in his pop book "Fabric f the universe" say that he thinks that a BI formulation of ST would bring about a 3rd revolution in string theory?

On the shortcomings of PST, scattering amplitudes break down at the Plank level - not good. (do you know the status on whether SUSY PT is Borel summable? A need for essential non-perturbative corrections?) You mention the most useful definition of string theory...I only really know about LQG, I've read that Rovelli tries to summarize:

"For instance, attempts are made to describe the bulk quantum geometry of spacetime by using the ADS-CFT conjecture, thus trying to describe what we do not know (quantum gravity) in terms of conceptual tools that we control (flat-space quantum field theory on the boundary). Analogously, the string theory calculations of black hole entropy exploit the relation between the strong-coupling genuinely-gravitational regime of interest, and the weak-coupling regime where conventional flat-space tools can be used, and states can be counted. Again, string cosmology often addresses the highly non-Minkowskian geometry of early cosmology by an hypothesis, that sounds bizarre to relativists: an overall larger Minkowski space where everything happens."

I think BI string theory would be extremely interesting, maybe this decade it will be formulated and consequences established...

 

[fzero]

Thanks for your informed response.

As I understand it for the string, the world sheet super symmetry is established by adding fermionic coordinates to the two-dimensional world-sheet coordinates, spacetime supersymmetry is acheived by adding fermionic coordinates to the `bosonic' coordinates x^\mu.

Worldsheet SUSY comes about by adding the appropriate number of 2d fermions to the 2d scalar fields. Adding fermionic coordinates to the worldsheet is a part of the superspace formalism, which isn't strictly needed for SUSY, but can be very convenient in practice.

Spacetime supersymmetry can be understood to come about from world-sheet supersymmetry together with the imposition of the GSO condition (the GSO projection is the truncation of the spectrum that eliminates the tachyon).

Essentially, though there are GSO projections that result in no spacetime fermions at all, and so no spacetime SUSY. These are the Type 0 strings.

It's the stronger condition of spacetime supersymmetry that imposes the need for timelike Killing vector field. Is supergravity from the outset based on spacetime supersymmetry? I wonder would the solution space freed up by symmetry breaking at lower energies?

Classical supergravity looks to me to be as background independent as GR. The bosonic sector is essentially the same as GR (with some particular matter sector). Adding the fermion sector can be reduced to a topological restriction on spacetime, but there is no way around that. The supersymmetric transformation is local, so it does not require any globally-defined Killing spinor or vector.

Now when we consider a specific background, we're led to ask if SUSY is preserved. This will only be the case if there is a globally-defined Killing spinor (to generate the global SUSY transformation). Furthermore, the SUGRA Lagrangian leads to a potential energy function. Given a specific background, this leads to a potential on the space of shape and size parameters of the background (called moduli). The SUSY points are the zero-energy minima, while the other points break SUSY spontaneously and are unstable, or at best metastable "solutions."

Would be interested in comments made by Witten and other "deep thinkers" on background independent string theory. You mention Smolin indeed thinking BI is key to resolving problems in ST. Didn't Green in his pop book "Fabric f the universe" say that he thinks that a BI formulation of ST would bring about a 3rd revolution in string theory?

I'm not familiar with Greene's popular books, but I would not argue against that possibility. Some specific papers by Witten are these, though I think they are very short on philosophy and much more focused on details. I mentioned them mainly to show that BI was not completely ignored.

On the shortcomings of PST, scattering amplitudes break down at the Plank level - not good. (do you know the status on whether SUSY PT is Borel summable? A need for essential non-perturbative corrections?)

I'm only familiar with an old paper by Gross and Periwal that argues that it is not Borel summable. This is a very qualitative argument, since complete two-loop computations had not been done at the time (and higher-loops remain out of reach to this day).

You mention the most useful definition of string theory...I only really know about LQG, I've read that Rovelli tries to summarize:

"For instance, attempts are made to describe the bulk quantum geometry of spacetime by using the ADS-CFT conjecture, thus trying to describe what we do not know (quantum gravity) in terms of conceptual tools that we control (flat-space quantum field theory on the boundary). Analogously, the string theory calculations of black hole entropy exploit the relation between the strong-coupling genuinely-gravitational regime of interest, and the weak-coupling regime where conventional flat-space tools can be used, and states can be counted. Again, string cosmology often addresses the highly non-Minkowskian geometry of early cosmology by an hypothesis, that sounds bizarre to relativists: an overall larger Minkowski space where everything happens."

I think BI string theory would be extremely interesting, maybe this decade it will be formulated and consequences established...

I wouldn't call Rovelli's summary too inaccurate, but a one-sentence summary shouldn't be expected to satisfy anyone's curiosity.