Why I am REALLY disappointed about string theory

[suprised]

Oups, that turned out to be a Pandora's box. Let me put M-Theory aside, and comment on QCD.

Of course QCD is non-perturbatively defined, sorry for the imprecise way of writing. What I meant is that the degrees of freedom, in terms of which you write the QCD lagrangian, are ill-suited to describe IR physics, because they are strongly coupled there. They don't exist as asymptotic states! There is no scattering process where they would figure as incoming and outgoing states, at larges distances, so in this sense quarks and gluons are not meaningful observable quantities at low energies. Thus the usual QCD lagrangian is the "wrong" formulation to describe IR physics.(*)

One could say that the QCD lagrangrian encodes the observables of the UV theory in a direct, perturbative way, which is amenable to explicit computations, but the IR observables in such a complicated, non-perturbative way that it is practically useless for describing IR physics (I am talking about analytical, not numerical lattice computations). For describing IR physics, other variables, like meson fields, should be introduced. Similarly, for describing physics at strong coupling for a large number of colors, the good variables become type II strings on AdS5xS5 (putting Susy aside).

So I agree with Tom on most, perhaps not all points.

At any rate, the important issue we all agree on is that a QFT is more general than just perturbation theory, or Feynman diagrams, which is derived from some lagrangian containing weakly coupled degrees of freedom. While in the case of QCD a regime (namely the UV regime) with a weakly coupled lagrangian description exists, there are other theories, like certain M-branes, for which such a lagrangian formulation (apparently) does not exist, so that these are intrinsically quantum and not approximations to some classical theory. This is what one loosely refers to as "non-lagrangian theories".


(*) Edit: removed "of course, strictly speaking, it defines the theory everywhere, in the sense of spanning the complete Hilbert space".

 

[suprised]

Could you give examples of some papers that use this approach? (I'm a biologist, so it'll be all over my head anyway, so even very abstruse ones are fine.)

Any good book on CFT should do. For example "Conformal Field Theory" by Di Francesco et al. Google for conformal bootstrap and you are led to page 186 of this book where this is explained (I have probs linking it).

 

[MTd2]

Tom, look at above again. I am not suprised. You wouldn't be surprised that I agree with you. I am just trying to understand him.

 

[tom.stoer]

@suprsised & MTd2: I perfectly understand, I see you understand - and I see that we agree in many points; so Pandora's box is now closed again (but do you know what Pandora did not set free when closing the box?)

I think there is one point where I do not agree, namely that "IR observables ... in such a complicated, non-perturbative way that it is practically useless for describing IR physics (analytically, not numerically)". But that's not our point here, b/c we discuss M-theory, not QCD.

But it seems that this is a much more severe problem in M-theory than in QCD; that weakly coupled versions do not span the whole theory space (whereas in QCD the degrees of freedom which are weakly coupled in the UV are complicated but do span the whole theory space).

One question: does that really mean that M-theory cannot be defined in the whole theory space in principle? has this been ruled out? or does it only mean that it's hard to solve M-theory but that there's still hope to find an appropriate definition which allows a global definition?

 

[MTd2]

Let me see if I understand what you mean.

Between the 5 fundamental superstring theories there are dual relations. But regarding 11d sugra compactified on a circle there might be nothing relevant on the sugra side of the duality, and that this is just a coincidence.

 

[tom.stoer]

I am not sure if I understand.

Let's try it differently:

1) are there results showing that a fundamental representation (globally valid in the full theory space) is not available in principle? or are we simply not clever enough to identify / construct it?

2) if this description does not exist: are the current approaches sufficient to cover the full theory space using the known "patches" (5 * ST, 1 * SUGRA, some M-theory corners)? or is still something missing?

 

[atyy]

Any good book on CFT should do. For example "Conformal Field Theory" by Di Francesco et al. Google for conformal bootstrap and you are led to page 186 of this book where this is explained (I have probs linking it).

Let's see if I got this straight. The view is nth order correlations are all that can be measured experimentally. So we specify the correlations directly by constraining their symmetry (instead of indirectly via a Hamiltonian or Lagrangian)?

 

[tom.stoer]

It's great that this thread is still kept alive ...

... but in order not to lose focus I would liketo come back to the basic questions:

What is the most promising research direction in order to identify the unique, underlying, (pre-geomeric) structure of string theory and to identify the fundamental principle that explains how string- / M-geometry and their degrees of freedom emerge from it? Are there fundamental degrees spanning the whole theory space and allow for non-perturbative calculations?

Another bunch of questions: we have this web of dualities relating 5 * ST, 1 * 11-dim. SUGRA and 1* M-theory. As we still do not know what M-theory really IS: why can we be sure that M-theory is really the mother of all other theories just mentioned - and not "just another theory related by a certain duality"? And why do we expect that there are no other theories still to be identified? What makes M-theory special - not b/c of our expectations bjut b/c of the facts we know about it?

Do I still ask the right questions? If not, what are the appropriate ones?

 

[Haelfix]

The nonperturbative sector of string theory is something that they started to really probe in the 90s and 2ks and of course is where it starts to get difficult interpretation wise.

I believe its important to distinguish the word "M theory" in the sense of the 11 dimensional theory that when compactified on a circle yields type d=10 type IIA string theory and Mtheory in the sense of 'mother theory' which contains everything ever learned about string theory.

The former is just another limit of the stringy 'configuration' space and/or an equivalent description of the same physics of some other corner. The latter is something else entirely and the answer to the question 'what is string theory'?

I think it was thought for awhile that the 11 dimensional theory really was the master nonperturbative description of all of string theory, but for one reason or another that point of view is passe.

As to what objects 'span' the full stringy Hilbert space. Well, its highly ambigous. The naive answer would be to say the fundamental string. But then that object doesn't exist in the 11 dimensional Mtheory (only M2 and M5 branes) which is the nonperturbative limit of maximal supergravity. However Mtheory is dual to type IIA/type IIB which are derived starting from the fundamental string.

How are you supposed to interpret that? In some sense you want to mod out this 'theory space' by all the dualities and look for whatever mathematical objects or structures that remain and are irreducible. Obviously, this is technically a formidable task and so remains only speculation.

 

[tom.stoer]

Haelfix, thanks for this clarification. So we will distinguish explicitly between the M2/M5-theory and the mother-theory. Very important point!

 

[suprised]

Indeed the current way of thinking by many is that the M2/M5-theory (whose low energy limit is 11-dim sugra) is one of many limits, or "coordinate patches", of some "mother" theory, and unfortunately the word "M-Theory" has been used for both.

One of the questions is whether there are other, more "fundamental" degrees of freedom, out of which the various theories we know, emerge. Somewhat similar to QCD, where the non-abelian gauge theory with quarks plays the role of a necessary UV completion of the meson theory; at high energies, "new" degrees of freedom need to be liberated, and render the theory consistent at high energies.

As for strings, they are supposedly UV complete; there is no reason (known to me at least) why new, "more elementary" degrees of freedom would be needed when going up in energy. As we know eg. from state counting in black holes, the string spectrum seems just right to yield a consistent theory. So what we call string theory is morally close to QCD and not close to the meson theory.

This supports the picture that all there is, is the "big blob" master theory, which can be approximated at different "coordinate patches" by different perturbative theories, and there are no further, "more elementary" microscopic degrees of freedom that would be revealed by going to high energies.

All what happens, say when moving from "theory patch A" to "theory patch B", that certain degrees of freedrom, that were non-perturbative in theory A, start becoming relevant and turn into the weakly coupled degrees of freedom of theory B when the approproate duality frame is chosen to represent them.

On of the current questions of the present thread, is, I think, whether the whole theory blob can be reconstructed out of one patch, say, the M2/M5-patch, when taking the full non-perturbative quantum theory into account.

I believe the answer depends on what one means by patch - does one throw away information, or not, when restricting to a patch? Let's come back to the QCD example, please forgive me, but I think it is helpful.

The analog of the big mother blob of theories is "abstract QCD", the fully non-perturbatively defined theory, with a certain definite Hilbert space. Analogs of coordinate patches are the quark-gluon theory, which is the patch relevant in the UV; the meson theory, which is the relevant patch in the IR (and if you like, type II strings on AdS5xS5 which is the relevant patch at strong coupling at large N; and there might be more).

Let's focus in the meson theory patch. Can quarks and gluons be reconstructed from it? When taking at face value, not, I believe; the theory has less degrees of freedom than the quark-gluon model, because by definition the meson theory results from integrating out, or throwing away many degrees of freedom, and a lot of information is lost in this way. In other words, the meson model is an incomplete theory that does not faithfully represent the full QCD Hilbert space.

Conversely, going to the UV, does the quark-gluon patch represent faithfully the abstract QCD Hilbert space including all the meson fields, etc? I don't think that this is a trivial question. Somehow one believes "yes, when all non-perturbative information is taken into account", which boils down to the question how the patch is defined. But can one actually meaningfully define it such that it faithfully represents QCD at all energies? We do know that all the quantities one uses to write the UV theory become ill-defined in the IR, and naive extrapolation fails because one hits a singularity, or phase transition, at approx 1 GeV. A priori it is not clear whether one can extrapolate the quark-gluon model past this singularity or not (note again that this is not a question about the abstract QCD that is defined everywhere, but rather about whether the quark-gluon model faithfully represents it everywhere).
I would be tempted to say, not, but on the other hand, the numerical computations of lattice QCD seem to say the opposite.

Sorry again for the QCD detour, but I think this is morally close to this remark, when we replace "M2/M5-theory" by the "quark gluon theory" and "mother-theory" by "abstract QCD":

Haelfix, thanks for this clarification. So we will distinguish explicitly between the M2/M5-theory and the mother-theory. Very important point!

 

[MTd2]

What is the failure of the proposed non perturbative matrix theories?

 

[suprised]

What is the failure of the proposed non perturbative matrix theories?

It's not a failure. They are just useful in their domain of validity, as are the other approximations. In particular they mess up in lower dimensions.

 

[suprised]

Let's see if I got this straight. The view is nth order correlations are all that can be measured experimentally. So we specify the correlations directly by constraining their symmetry (instead of indirectly via a Hamiltonian or Lagrangian)?

Right. Sometimes one can solve a theory despite a lagrangian or hamiltonian is not explicitly known.

 

[MTd2]

It's not a failure. They are just useful in their domain of validity, as are the other approximations. In particular they mess up in lower dimensions.

What I meant is why do the fail to be THE m theory?

 

[mitchell porter]

suprised, the points in the string "theory space" basically differ by cosmological parameters - first they are distinguished by discrete differences like differences in topology, and then you have moduli spaces for continuously varying parameters like radius of compactification. As far as I can see, in string theory, the "definite Hilbert space" is always defined relative to a particular background or class of backgrounds. The meaning of the dualities is that the same Hilbert space sometimes admits multiple geometric interpretations. In other words, the overlapping "theory coordinate patches" exist on a space of Hilbert spaces. I doubt that we can clarify the issue without keeping that in mind.

 

[atyy]

Right. Sometimes one can solve a theory despite a lagrangian or hamiltonian is not explicitly known.

Thanks. I saw a book by Blumenhagen that said this could be done in 2D CFTs because enough symmetries are known to constrain the correlation functions completely. Is this why people like Hermann Nicolai talk about finding the symmetries of M-theory?

Also, can correlation functions be defined when there is no background geometry?

 

[suprised]

suprised, the points in the string "theory space" basically differ by cosmological parameters - first they are distinguished by discrete differences like differences in topology, and then you have moduli spaces for continuously varying parameters like radius of compactification. As far as I can see, in string theory, the "definite Hilbert space" is always defined relative to a particular background or class of backgrounds. The meaning of the dualities is that the same Hilbert space sometimes admits multiple geometric interpretations. In other words, the overlapping "theory coordinate patches" exist on a space of Hilbert spaces. I doubt that we can clarify the issue without keeping that in mind.

I fully agree. That's why one issue is to try to find a formulation that captures the physically relevant, observable "invariant" content, rather than being focused on particular geometric representation (extra dimensional geometry, brane worlds, etc) of a model. Perhaps one should only focus on the effective action.

As for connectedness of the string parameter space, this is an important and difficult point, which is not reflected by the simple picture in 10, 11 dimensions, where all theories are continuously connected. That won't be generically the case for lower dimensional theories with less supersymmetries. One of the main features of Susy theories is that they tend to have moduli spaces that allow for analytic continuation between different patches, and generically one can avoid hitting singularities or phase transitions by doing so.

It is unclear to what extent the lessons drawn from these simplified toy settings apply to real-world models, which neither have Susy nor moduli spaces. This is analogous to the relationship between N=2 Susy gauge theories, which have been solved by making use of their moduli space and the underlying quantum geometry (SW curve), and real world QCD, which doesn't have this structure.

 

[suprised]

Thanks. I saw a book by Blumenhagen that said this could be done in 2D CFTs because enough symmetries are known to constrain the correlation functions completely.

That applies only to certain theories, say minimal models or rational CFT or integrable models in 2d. In general theories don't have enough symmetries to completely fix the correlation functions.

Also, can correlation functions be defined when there is no background geometry?

Yes - in topological field theories the correlation "functions" are numbers and do not depend on any positions of operators; so they don't refer to a background geometry, rather to the topology of a background. Essentially they are zero or one.

 

[tom.stoer]

@suprised: thanks again for you post #311. The QCD detour is welcome; I was insisting on it as it always helps to compare speculative ideas to structurally similar but well-understood problems.

 

[tom.stoer]

suprised, the points in the string "theory space" basically differ by cosmological parameters - first they are distinguished by discrete differences like differences in topology, and then you have moduli spaces for continuously varying parameters like radius of compactification. As far as I can see, in string theory, the "definite Hilbert space" is always defined relative to a particular background or class of backgrounds. The meaning of the dualities is that the same Hilbert space sometimes admits multiple geometric interpretations. In other words, the overlapping "theory coordinate patches" exist on a space of Hilbert spaces. I doubt that we can clarify the issue without keeping that in mind.

Very good point.

Using M-theory in the 11-dim. M2/5 sense we can introduce coordinate patches in the usual sense; this results in the duality maps ST₁ <==> ST₂ (including M-theory and 11-dim. SUGRA) between the different theories ST_n.

Now let's come to Hilbert spaces constructed on top of certain backgrounds. Afaik these constructions are at not only restricted to a certain coordinate patch but they are restricted to a certain background. Chosing e.g. a certain CY space one gets different quantum theories defined via (CY, H_CY) where the Hilbert space H is constructed on top of the CY space and consists of string vibrations restricted to this specific CY.

So below the duality maps ST₁ <==> ST₂ we have a second web of maps, namely between(CY₁, H₁) <==> (CY₂ H₂).

How does the web of dualities act between different Hilbert spaces?

First of all there is the trivial isomorphism between any two separable Hilbert spaces, so that statement alone does not help much. One has to construct a map that does the following
- a map CY₁ <==> CY₂
- a map between states |f₁> <==> |f₂>
- a map between physically relevant operators (Hamiltonian, generators of symmetries, creation / annihilation operators, ...)
This is by no means trivial - even if one restricts to one coordinate patch.

How can one analyze the physical content of such a relation in general?

Let's come back to the QCD example to discuss my problem: Assume that both QCD and meson theory are both described ba a separable Fock space (hopefully the non-renormalizibility of the meson theory doesn't spoil my argument). We now that both theories are isomorphic mathematically. But we also know that the meson theory contains less degrees of freedom (because color has been integrated out). So something strange must happen when constructing the isomorphism. The degrees of freedom cannot simply vanish, they must show up in a different form.

I think that the map is mathematically ill-defined. The argument goes as follows: In QCD we can count physical degrees of freedom; two physical polarizations of gluons with a certain color factor; quarks with a certain color factor; that means a finite number of degrees of freedom for each point in three-space. But in the meson (meson-nucleon) theory we have an infinite number of degrees of freedom as we con to higher representations. Nothing prevents us mathematically from constructing pentaquark states, tensor glueballs etc. So the local color degrees of freedom are transformed into non-local degrees of freedom (and to be honest: nobody is able to write down an equation for this transformation in practice).

In addition we know that the meson theory must break down; this can be seen physically as we have deep inelastic scattering and QG plasma where no mesons are present. So we know that the breakdown of this isomorphism has something to do with a phase transition.

As for strings, they are supposedly UV complete; there is no reason (known to me at least) why new, "more elementary" degrees of freedom would be needed when going up in energy. As we know eg. from state counting in black holes, the string spectrum seems just right to yield a consistent theory. So what we call string theory is morally close to QCD and not close to the meson theory.

What about the Hagedorn temperature which indicates the phase transition in QCD? Is there a hint in certain string-related theories that the partition sum diverges due to an exponential growth of the state density? Is there an argument that - based on the absence of this phenomenon - one can conclude that the theory is UV complete in whole theory space?

 

[tom.stoer]

really no idea?

 

[mitchell porter]

It might be best to start with the simple case of T-duality between IIA and IIB with one compact dimension, which maps momentum states to winding states. Mirror symmetry between CYs is quite complicated, since it involves analysing the CY into a three-torus fiber over a 3D base, and then performing a separate T-duality along each of the three directions on the torus (http://arxiv.org/abs/hep-th/9606040" ).

 

[tom.stoer]

Let me again summarize some remarks and ideas from my last post.

How does the web of dualities (regarding different string theories) act between different manifolds and Hilbert spaces?

Is there a hint in certain string-related theories that the partition sum diverges due to an exponential growth of the state density? Or can one show (based on the absence of this phenomenon) that string theory is UV complete in whole theory space?

 

[Chronos]

I think any QG solution, or TOE, must be consistent with information theory. This where I perceive the big problem is with string theory. String theory will never be proven wrong, because it is not. It fails at the fundamental level because it relies on an unsupported assumption - a background.

 

[Fra]

because it relies on an unsupported assumption - a background.

IMHO, I do not have a problem with that their beeing a background per see, because there is always a background - the observer, and it's the observer that does "processing and registering". Without the context where information is encoded or processed, I see no objective meaning of information.

What I think is the problem with ST, is that the nature of the choice of background, and in particular the evolution of the background, is incomplete at best.

I also think that there is a problem with the continuum nature of the background, when it comes to the information processing analogy.

Edit: So I think the existenec of a "background" shouln't be seen as an "assumption", it should be seen as a premise, or prerequisite for beeing for formulate a question, conduct and experiment, process infomation etc. The question is rather, why the PARTICULAR background. Or rather, how does a given background evolve along with the processing itself? Unless of course you think there is a fixed processing, which I don't.

/Fredrik

 

[tom.stoer]

It fails at the fundamental level because it relies on an unsupported assumption - a background.

We had this discussion a couple of times. I think we agreed that it is a major weakness of string theory that (up now) it cannot be formulated w/o referencing background structures. But it could very well be that the restriction to certain backgrounds can be overcome. THis is what we are discussing here.

So for me this is a major weakness but not a proof of its failure.

My questions are intending to find approaches how to get rid of backgrounds or how to understand the relation between different backgrounds better. If you have two backgrounds and if you can construct an exact mapping between the backgrounds, the corresponding full string Hilbert spaces and the operators acting on them, then the backgrounds become irrelevant in principle. They may be a technical obstacle as they hide the true nature of the theory, but background dependence is not a disaster then.

 

[marcus]

Tom everybody at PF can be proud of the evenhandedness and objectivity shown in much of this thread. There's stuff here worth looking back for and re-reading. But you omitted the word "not" in a sentence where I think you meant to have it. Here's a slightly edited version:

We had this discussion a couple of times. I think we agreed that it is a major weakness of string theory that (up to now) it cannot be formulated w/o referencing background structures. But it could very well be that the restriction to certain backgrounds can be overcome. THis is what we are discussing here.

So for me this is a major weakness but not a proof of its failure.

My questions are intending to find approaches how to get rid of backgrounds or how to understand the relation between different backgrounds better...

 

[marcus]

... It fails at the fundamental level because it relies on an unsupported assumption - a background.

Originally the term "background independence" meant the theory was formulated without reference to a metric---that is, in essence, without a prior geometry specified by a distance function defined on the spacetime manifold.

Any theory relies on assumptions. For example Gen. Rel. assumes a differential manifold, a smooth continuum. Nevertheless, the theory is background independent because it does not start by assuming a metric on the manifold. No metric background---no prior geometry background.

It leads to confused and pointless conversations when people mistakenly begin to use the word "background" more broadly and vaguely. As far as I know, the precise meaning of background independence is still the same as it always has been, in this context of discussion.

I don't understand your post, Chronos, because it sounds like you consider it a flaw for a theory to rely on an unsupported assumption. That can not be right, since theories commonly do rely on unsupported assumptions, and one checks their predictions by empirical tests.

There is something especially wrong with assuming a prior 4D metric. For example it fixes the lightcone structure in an unphysical way. It prescribes a causality setup which cannot respond dynamically to events. As far as we know, Nature does not operate with a prior 4D metric. So that assumption flies in the face of reality much more seriously than making some other unsupported assumption (like representing spacetime by a differential manifold, for instance, which several background independent theories do.)

 

[marcus]

This thread has over 300 posts, some of which were exceptionally enlightening. Not mere bickering about appearances, public relations, obfuscation, complaints about damaged prestige etc. But really coming to grips honestly and frankly with pressing issues. Now it seems to be in a lull. What can we do? Should we go back and index the good parts? Should we make a list of posts where there is especially interesting dialogue?

 

[Fra]

I don't understand your post, Chronos, because it sounds like you consider it a flaw for a theory to rely on an unsupported assumption. That can not be right, since theories commonly do rely on unsupported assumptions, and one checks their predictions by empirical tests.

There is something especially wrong with assuming a prior 4D metric. For example it fixes the lightcone structure in an unphysical way.

I've understood that there is disagreement in these questions on here, but to speak for myself, it is NOT a misunderstanding on my behalf when extending the "background" to be MORE than just a metric.

The reason for this has also been discussed before, but to me it's about coherence of reasoning. If you take the information theoretic view where any expectation follows by counting and rating evidence, then I do not see why information about the metric, and the abstraction of metric is special and would need special treatment. After all these information and measurement perspectives wasn't around when GR was formulated. So although GR certainly has some deep lessions, it still remains to understand them in the more modern measurement setting. So far I see this has failed.

So while I respect that people disagree, I don't think it's due to confusion that people insist on the extended meaning of B/I. I just think that B/I as a scientific statement, necessarily needs to be understood different in a measurement theory, than in a classical theory like GR.

So if we say seek a deeper abstraction and understanding, in terms of information processing, at minimum we need to explain why it's ok to use any background assumption EXCEPT the metric? I'm not defening backgroundf metrics here, I'm saying that it seems that the better version of the principle really can not distinguisha between particular assumptions.

/Fredrik

 

[tom.stoer]

Tom everybody at PF can be proud of the evenhandedness and objectivity shown in much of this thread. There's stuff here worth looking back for and re-reading. But you omitted the word "not" in a sentence where I think you meant to have it. Here's a slightly edited version:

Thanks marcus for reading carefully. I corrected my statement accordingly.

 

[tom.stoer]

This thread has over 300 posts, some of which were exceptionally enlightening. Not mere bickering about appearances, public relations, obfuscation, complaints about damaged prestige etc. But really coming to grips honestly and frankly with pressing issues. Now it seems to be in a lull. What can we do? Should we go back and index the good parts? Should we make a list of posts where there is especially interesting dialogue?

Marcus, I am afraid I can't do more than indicate what the central problems and questions are which have been identified throughout the discussion (to be honest, I don't think that we found out something new; we only collected facts and questions well-known to the experts). I tried to do this a couple of times in order to focus the discussion; it could make sense to write such a short summary and conclude this thread instead of reiterate and spin in circles.

 

[mitchell porter]

How does the web of dualities (regarding different string theories) act between different manifolds and Hilbert spaces?

The very simplest case of T-duality is a closed string wrapped around a circle. There are two quantum numbers, a quantized momentum m describing the direction and speed of movement of the whole string, and the "winding number" n which counts the direction and the number of times the string is wrapped around the circle. The state (m,n) for a circle of radius R has the same energy as the state (n,m) for a circle of radius 1/R, because kinetic energy and winding energy have an opposite dependence on the radius. (references: http://ncatlab.org/nlab/show/T-duality#a_first_rough_look_4")

So in the change of perspective you do two things at once. You switch the interpretation of two quantum numbers, and you change a parameter in the background geometry (R goes to 1/R). You can get much more technical than this in describing T-duality (e.g. see the rest of reference 1), but I think that's the essence of it.

Is there a hint in certain string-related theories that the partition sum diverges due to an exponential growth of the state density? Or can one show (based on the absence of this phenomenon) that string theory is UV complete in whole theory space?

What happens at high temperatures is that long strings dominate - instead of a gas of many strings, you get one long tangled string. Also see the comments http://press.princeton.edu/chapters/s8456.pdf" , page 4-5, about UV-IR open-closed duality. The essence of open-closed duality is that you can get a cylindrical worldsheet in two ways - a closed string (circle) over a finite time interval (line segment), or an open string (line segment) over a periodic time interval (circle).

I don't actually know how UV completeness in string theory works, but those would be two of the ingredients.

 

[tom.stoer]

Regarding T-duality: I know this example and I think this is rather sound mathematically. Can we learn from T-duality how this may work for other dualities which have only been shown to be true for certain regimes?

Let's make an example: in a scale-invariant regime a toy and a real object can be identified, but leaving the scale-invariant regime this duality (identity) break down.

So one has to understand the general construction principle plus the proof; single examples may be helpful to identify the guiding principles, but what is missing is the global picture.

Regarding promises - which was one of my starting point: about 10 - 15 years ago there was the promise that M-theory is the mother of all string theories. But as I just learned the M2/M5 brane theory we have today is only another coordinate patch w/o the potential being the mother. I think that identifying further patches and related dualities will no longer be helpful to identify this mother theory.

So what one really needs in order to identify it is a unique construction principle which provides a means to understand all patches and dualities instead of messing around with one single duality.

That was the idea behind my question.

 

[mitchell porter]

Maybe we should talk about F-theory rather than M-theory, for a while? The other basic duality is S-duality, and (for IIB at least) F-theory explains it as a symmetry of the two extra dimensions (F-theory being 12-dimensional).

 

[tom.stoer]

Then the question is whether F-theory provides a better understanding of the true nature of these dualities or a hint for the underlying structure of string theory globally = including all versions / vacua etc.

Afaik F-theory is not able to achieve this; it is just another theory focussing on a small portion of theory space.

I do not say that it's not interesting to understand more about F-theory, but I would like to ask whether F-theory is the right strategy to identify an underlying mother-theory and to identify globally applicable fundamental degrees of freedom.

 

[suprised]

Afaik F-theory is not able to achieve this; it is just another theory focussing on a small portion of theory space.

True!

...to identify an underlying mother-theory and to identify globally applicable fundamental degrees of freedom.

You assume there are such things...

 

[mitchell porter]

I just found http://arxiv.org/abs/hep-th/9706155" in which he claims to derive F-theory from M-theory. "F–theory backgrounds are simply a subset of the possible M–theory compactifications." Add to that the central role that M-theory plays in the web of dualities, and I find myself reverting to the view that figuring out M-theory is the key after all.

For the string theories we have a reasonably straightforward picture (the sum over Riemann surfaces), but I don't know of anything like that for M-theory. Since M-theory has M5-branes, with M2-branes ending on strings internal to the M5-branes, the analogous construction ought to be a sum over six-dimensional manifolds (M5 worldvolumes) with internal string histories connected externally by three-dimensional manifolds (M2 worldvolumes). And just as in string theory, the "target space" in which the strings move gives rise to a 2D conformal field theory on the worldsheet (the Riemann surface), there should be a 6D conformal field theory in the M5 part of this construction, and a 3D conformal field theory in the M2 part, which corresponds to the geometry through which the M-branes are moving. But I've never seen anything like this. Then again, even if it is a valid and tractable construction, you couldn't make it work unless you had figured out the M-brane worldvolume theories, and that's work in progress.

http://arxiv.org/abs/0905.2720" has something about the 6D theory, though I think it's somewhat simplified from its full, physically relevant form - see the remarks on page 13 about "more elaborate constructions".

 

[MTd2]

I thought that 6D theory from witten was not related to M5 branes, but just a theory with not much relation to physics. I mean, that is a N=2 theory, right?

 

[mitchell porter]

No, it really is supposed to be the worldvolume theory of the M5-brane. There's a lot of recent work in which N=2 4D theories are derived from the 6D theory compactified on a Riemann surface (see "Gaiotto duality"). The D3-brane of IIB / F-theory is an M5 compactified on T^2, and shows up in the F-theory GUTs (http://arxiv.org/abs/1006.5459" ).

I think the best papers to read, for progress regarding M-branes, might be those by http://arxiv.org/find/hep-th/1/au:+Berman_D/0/1/0/all/0/1". 0710.1707 reviews the situation on the eve of the M2-brane "minirevolution", and can be supplemented by, e.g, Pei-Ming Ho's 0912.0445. Berman's most recent (1008.1763) even brings U-duality groups into the picture, and so begins to make contact with the school of thought which would derive M-theory from an E10 or E11 symmetry.

 

[atyy]

No, it really is supposed to be the worldvolume theory of the M5-brane. There's a lot of recent work in which N=2 4D theories are derived from the 6D theory compactified on a Riemann surface (see "Gaiotto duality"). The D3-brane of IIB / F-theory is an M5 compactified on T^2, and shows up in the F-theory GUTs (http://arxiv.org/abs/1006.5459" ).

I think the best papers to read, for progress regarding M-branes, might be those by http://arxiv.org/find/hep-th/1/au:+Berman_D/0/1/0/all/0/1". 0710.1707 reviews the situation on the eve of the M2-brane "minirevolution", and can be supplemented by, e.g, Pei-Ming Ho's 0912.0445. Berman's most recent (1008.1763) even brings U-duality groups into the picture, and so begins to make contact with the school of thought which would derive M-theory from an E10 or E11 symmetry.

Great, thanks for the recommended reading! The E10,11 stuff is really intriguing, but I've always wondered if it's applicable only in the vicinity of a spacelike singularity.

 

[MTd2]

No, it really is supposed to be the worldvolume theory of the M5-brane.

It was not really a yes/no question. Actually, I wanted to know why M5 branes should be N=2 instead of N=1 like SUGRA 11d. Why?

 

[mitchell porter]

Enhanced supersymmetry is a common phenomenon, e.g. N=8 d=4 SUGRA is N=1 d=11 SUGRA compactified on S^7. I don't have anything insightful to say about the M5-brane case but see page 9 of http://www.ggi.fi.infn.it/talks/talk1782.pdf" .

 

[MTd2]

Enhanced supersymmetry is a common phenomenon, e.g. N=8 d=4 SUGRA is N=1 d=11 SUGRA compactified on S^7.

Those are compactified theories, but if it is a source of a theory, like M5, it should be expected to have the same number of supersymmetries, right?

 

[mitchell porter]

We need to distinguish between "number of supersymmetries" and number of supersymmetry generators. See http://en.wikipedia.org/wiki/Supersymmetry#Extended_supersymmetry". The rule of thumb is that in dimension d, N=1 supersymmetry has 2^d/2 generators if d is even, 2^(d-1)/2 if d is odd (because of the size of spinors in the different dimensions). So N=1 d=11 has 32 supercharges; N=1 d=4 has just 4 supercharges; and a 32-generator susy algebra in four dimensions therefore consists of eight copies of the N=1 algebra, i.e., it's "N=8".

For the M5-brane: The 11d bulk has 32 supersymmetries; half of them are lost in the projection onto the 6d worldvolume; but that's still twice the number of supersymmetries in N=1 d=6, which by the formula above has just 8 generators. So that's where "N=2" comes from, for the M5-brane.

Those are compactified theories, but if it is a source of a theory, like M5, it should be expected to have the same number of supersymmetries, right?

That's something for which I don't have any insight - how the doubling of supersymmetry comes about, from brane to bulk, if you take an "M5-centric" view of things.

 

[suprised]

A brane is a background configuration that spontaneously breaks some of the translational symmetries because it is localized. This applies to the supersymmetries as well, and depending on the configuration, only 1/2 or 1/4 or 1/8 or 0 of the supercharges survive. Spontanous breaking means that the full symmetry is still there, albeit the broken part is non-linearly realized.

From the viewpoint of the world-volume of a brane, the broken translational symmetries manifest themselves as massless scalars, which represent the goldstone modes for those broken symmetries (and which realize the broken symmetries in a non-linear way). For the broken supersymmetries this works analogously, except that the goldstone modes are fermions; they realize the broken supersymmetries in a non-linear way.

For some info, see eg p3 in: http://arXiv.org/pdf/hep-th/0011018v1
and here: http://arXiv.org/abs/hep-th/9612080

 

[MTd2]

I am more or less aware about this issues. The point I am trying to make it is how many supersymmetries I should expect from M5 branes if that is a fundamental object.

For example, Mitchell Porter wrote:

"For the M5-brane: The 11d bulk has 32 supersymmetries; half of them are lost in the projection onto the 6d worldvolume; but that's still twice the number of supersymmetries in N=1 d=6, which by the formula above has just 8 generators. So that's where "N=2" comes from, for the M5-brane."

Would that reasoning work for finding the number of supercharges the fundamental strings? In case that is not true, why should I expect that to work with M5 branes, which are also fundamental?

 

[mitchell porter]

There is a relationship between worldsheet supersymmetry and spacetime supersymmetry in string theory too. But these relations are all rather complicated. It's like the classification of Lie algebras, there's a logically determined pattern but it's not simple. At least it's not simple for me. I am reduced to just quoting from a few papers.

"In string theory, the interplay between worldsheet symmetries and their consequences in spacetime remains largely mysterious. Certain results, however, indicate strong connections between the two. For example, it is well-known that N = 4 supersymmetry on the worldsheet implies N = 2 supersymmetry in spacetime, and likewise it has been demonstrated that N = 2 supersymmetry on the worldsheet implies N = 1 supersymmetry in spacetime." -- http://arxiv.org/abs/hep-th/9505194"

"There are a number of different branes in string theory and M-theory, most conveniently characterised by their field content when seen as a field theory on the world-volume. The simplest ones, the so-called p-branes, have a scalar multiplet on the world-volume. D-branes contain a vector multiplet, coupling to string endpoints, and the M5-brane has a self-dual tensor." -- http://arxiv.org/abs/hep-th/0105176"

And http://arxiv.org/abs/hep-th/0301005" is a highly efficient review of susy and sugra theories in 4 to 11 dimensions.

 

[MTd2]

So, probably the fundamental object of m-theory has N=2 Supersymmetry, right? And M2 has N=1 supersymmetry?