What is the role of combinatorics in the depths of string theory?

[atyy]

I'm not sure I understand this correctly, but http://www-library.desy.de/preparch/desy/proc/proc02-02/Proceedings/pl.6/deboer_pr.pdf seems to say that in gauge/gravity, if the gauge theory is run from IR to UV fixed point, the bulk theory goes from supergravity to string theory.

(i) Are supergravity and string theory also RG fixed points of some theory?

(ii) Would the IR fixed point of the gauge theory correspond to a UV fixed point of supergravity (asymptotic safety of supergravity)?

(iii) When the gauge theory is a CFT, then there is no IR/UV trajectory, since there is only one fixed point. It seems the CFT energy scale is a dimension of the bulk. Does the picture of the energy scale as a dimension this still hold when the gauge theory does have a IR/UV trajectory?

 

[mitchell porter]

I would really like to have a discussion about RG flow and AdS/CFT, but I need to prepare a little first. One more day and I hope I'll be ready. :-)

 

[atyy]

Ah, danke!

 

[Fra]

Also, how does AdS/CFT fit in with this? It is naively a working theory of quantum gravity, maybe not of our universe, but one would imagine that all the issues of QM apply to it.

I moved my comment here as I think it's more related to this thread.

I won't even attempt to comment on the specific Maldacena conjecture in the context of ST, since I'm neither a string theorist nor a fan of it's constructing principles.

But I think there are deep conceptual connections between theory scaling and theory dualities (in general), that are worthy of further research. The conjectured Ads/CFT correspondence is as I see it an "example of" dualities betwheen theories.

Are two theories that produce the same expectation, fully physically equivalent? I personally think the answer is both yes and now, depending on the perspective.

IMHO, there are two energy-based "theory-scaling" paths, (or there SHOULD be).
- scaling the energy/mass of the encoded theory (ie. what's behind the screen, holding the screen fixed)
- scale the observational resolution or the "screen size", holding the code of the theory fixed.

(the different between the two is apparent when you consider the theories themselves evolving and interacting)

When I read Rovelli's papers, I get a feeling of that he is mixing these two things, and I also have the distinct impression that the reason for this is that Rovelli subscribes to a certain structural realism that I don't. The observer independent laws is according to him, beyond verification and thus not subject to measurement. They are just assumed to exists. To ME this is against what I see as one of the the constructing principles of QM.

RG scales the observational resolution, not the observer itself; because "the observer" as RG is applies to the SM of particle physics is of course given: It's simply tha laboratory frame, and the scientific knowledge of human science. This does not ever scale, not even in the most extreme HEP experiments. What scales is the resolution of our probes. But my point is that consistency of reasoning here, suggest that these probes are also in fact "inside observers" and we should required the consistenct that when the THEORY is SCALED also down to the inside coding, this should give the same predictions as the effective SM as knowd from experiment. This owuld be a much more powerful constraint that currently known. But a lot of things remains to be worked out.

Two dual theories, can yield the same expectations, if defined on a common boundary (state space) where they interact. But still the way the theories are encoded is different. And it's first when you take the encoding structure seriously, that the answer as to wether they are physically equivalent is possible no.

I think they aren't equivalent, that are just in agreement. So to me the duality of theories, is a conjecture of an equilibrium.

What I seek, is an understanding that also scales the observer and the context where theories are encoded. In this sense the duality of theories that are apparently defined in totally different context, that doesn't even have the same dimensionalty is things that can happen.

This is all part of what I'm trying to understand as well. For me this touches the foundations of QM as well as GR.

/Fredrik

 

[Fra]

I possibly don't adress your questions, but to continue the associations from the other thread and what it does have to do with theory scalings and what I expect.

IMHO, there are two energy-based "theory-scaling" paths, (or there SHOULD be).
- scaling the energy/mass of the encoded theory (ie. what's behind the screen, holding the screen fixed)
- scale the observational resolution or the "screen size", holding the code of the theory fixed.

To just summarize some of me personal views, I think that a generic prediction from this scheme is that as the OBSERVER is scaled, (ie the INTERactions between smaller and smaller things in HEP) is that some kind of asymptotic freedom is BOUND to happen - like we see in QCD for example, and this corresponds to points of unification. Some forces simply cease to be distinguishable and thus effective in that range.

But the reason why it's in fact EXPECTED in this scheme is that when you SCALE the THEORY, at some point it's not possible the represent complex interactions from the inside perspective, therfor the interacting theories in the scale limit is bound to face asymptotic freedom, or alternatively disapperance the interaction.

This is a KEY point in my thinking. I'm not exactly sure how this relates to string theory. The last time I did think of this, I did make some associations about the S and T dualities, but it's not really important to me so I have not spent any efforts at all to care. I think the general topic is better understood in general terms.

I have still a very long way to go indeed to make contact to the standard model, like QCD. But it doesn't seem out of question to me that at some point down in that asymptotic freedom domain, interesting things that does included gravity and spacetime will happen.

I'm not sure if you see my point, but another way of thinking is that the interacting theories are "lobotimized" as you scale them down. At some point, they just be come unable to make inference about each other and thus interact strongly.

The whole trick is then of course to have a mathematical model of how a generic theory looks like. An of course I think of them as "inference models", which are discrete computational flows seems as generalizations of probability theory. Quantum logic is a product of this scheme. It's not an input. Well that's the vision.

/Fredrik

 

[mitchell porter]

atyy, I believe the answers to your questions are (i) no (supergravity appears because it's the low-energy limit of string theory), (ii) unsure but I think no, (iii) yes. But I still don't grasp the basic technicalities of AdS/CFT well enough to be sure of answer (ii). What I can do is outline a program of personal study which, when completed, should answer the question.

First, an overview regarding how I approach this whole subject. From a string-centric perspective, the study of gauge/gravity duality is a way to learn more about the workings of string theory (or as I prefer, M-theory, since that is the most unified picture that's available). I see this duality as an aspect of the "branes in bulk" phenomenon, i.e. the relationship between physics on the brane and physics in the bulk space: It's a study of those special cases where the brane physics (the gauge theory) completely determines the bulk physics.

This is a working philosophy I developed when trying to figure out the role of the holographic principle in M-theory. There's a lot of perplexity regarding the origin of bulk locality. From the perspective of the gauge theory, the extra, radial, AdS dimension corresponds to energy scale, so there is no apparent reason why it should mesh with the Poincare invariance of boundary space-time to produce 5D Poincare invariance in the bulk, and yet it does. My philosophy is to reverse the perspective and say that bulk physics, such as locality and Poincare invariance in the bulk, is logically prior, and that locality and Poincare invariance on the boundary are the consequences of this. That is, the higher-dimensional perspective is the more fundamental one. It goes a little against the holographic philosophy as usually understood. But it's useful psychologically as a way to put the issue aside for now, without ignoring it completely.

Now for the ingredients of the promised program of study. To get started on AdS/CFT, from a string-theory perspective, I would suggest studying three versions of the duality in order.

First up, one should think about the primordial instance of the duality, that between d=4, N=4 super-Yang-Mills theory, and the Type IIB superstring on AdS₅ x S⁵ (which should be equal to M-theory on some manifold AdS₅ x Y⁶, but I'm not sure what manifold Y⁶ is). This case matters for anyone asking "what is string theory, fundamentally?", since its connection to N=4 SYM establishes a link to the twistor string of Witten and the Grassmannian of Arkani-Hamed et al (see http://motls.blogspot.com/2011/01/twistor-minirevolution-goes-on.html" ). The relationship between all these different perspectives is still waiting to be clarified, and should be of fundamental significance.

Next, I'd suggest looking at the "pp-wave" version of this, the "BMN limit", because that's where it became possible to reconstruct the full string spectrum from the gauge theory.

Finally, for the RG flows - which haven't featured so far in this study plan, because N=4 SYM is perturbatively finite to all orders - I'd look at a worked example from http://arxiv.org/abs/hep-th/0011207" , just to maintain the conceptual connection to string theory.

With respect to the renormalization group itself, I found two papers helpful. http://arxiv.org/abs/hep-th/0212049" makes some thought-provoking points while discussing the RG in the context of gravity, e.g. (part F) that CFTs are the "backbone" defining all other QFTs.

Returning to AdS/CFT, another fundamental aspect of it is the http://arxiv.org/abs/hep-th/9805114" , which is related to the open-closed string duality, and probably to the UV/IR mixing of noncommutative QFTs, noncommutative worldvolume theories of D-brane stacks, and string field theory. But I don't even have a study plan for those topics yet.

 

[atyy]

mitchell porter, thanks for the clarifications, and especially for the Warner reference.

 

[Fra]

Maybe mitchell can answer this?

(i) Are ... string theory also RG fixed points of some theory?

How about a picture where strings are emergent, maybe from something like string bits, has anyone toyed with the idea that some form of string theory would be an intermediate fixed point? Ie. somewhere between infinity and say QCD scale?

I never heard of it, but I thought I'd throw the question out to see if you're aware of and string-related papers that have speculated about this.

The reason why I ask is beacuse this is the closest connection I see to strings, if you don't accept the continum and instead consider them to be the simplest possible continuum objects; as you lower the energy.

My own pictures is highly incomplet yet, but I picture that it would be some discrete things going on so the continuum RG would fail first of all, but maybe somewhere wthere the continuum is "born" so to speak, couldn't we picture fitting ST there as an intermediate fixed point?

/Fredrik

 

[mitchell porter]

Well, there are a lot of topics that intersect here. String bits go back to Charles Thorn 1994 if you want to look it up. They are an approach to "tensionless strings", so it's a type of limit, the zero-tension limit.

In AdS/CFT, you have a mapping between string states (on the gravity side) and a trace of a product of field operators (on the gauge side). The sequence of field operators that enters into the product can be mapped onto a spin chain. Also, you can take a physical string - a line in space - and approximate it by a sequence of piecewise linear segments; these segments are the string bits. In the 4-dimensional gauge theory (N=4 SYM) which appears in the original example of AdS/CFT duality, there are six scalar fields and they give rise to the five compact extra dimensions - the fifth dimension, which turns Minkowski space into AdS₅, comes from the 4D renormalization group, but then the other five dimensions in the string theory form a compact space, S⁵, and they come from those scalar fields. And part of the BMN correspondence involves building string bits out of the scalar fields, and then superstrings out of the string bits, chained together. I mentioned the spin chain first because I believe the two perspectives are related, and the spin chain approach is the more fundamental. The spin chain ends up being like a string worldsheet, described from the 2D CFT perspective. But I don't know the details.

Long ago there were ideas that at high temperatures, superstrings might deconfine like QCD strings, and you would end up with something else, such as a purely topological field theory. But the current understanding of high temperatures in string theory (see Hagedorn temperature or Hagedorn transition) involves quantum production of many new strings (just the same as in field theory, where high energies will lead to particle production), and eventually to statistical ensembles dominated by long tangled strings - i.e. the string will self-intersect, and at lower energies that might lead to the emission of a closed string, but here it just rearranges the tangle a little.

 

[Fra]

Thanks Mitchel for your comments! I never heard of Thorn, I'll look him up later.

I've only skimmed some old papers relating to string bits and I didn't recall them as what I had in mind. I remember that it was lined out as a technical tool to make it easier, to basically just consider parts of the string and then treat it combinatorically and then take continuum limit.

I was thinking about something that isn't string theory at all; ie a new theory non-continuum based where one has ordered distinguishable events (not embedded in a background - just interacting with other siimlar structures - ie the "background" is just the environment of other index-systems), we can index and line out as a "string", which I see as an index. And that this may rather explain from different first principles actions and emergence of strings in a low energy limit (or rater high energy by absolute menas but at an intermediate scale) so as to possibly CONNECT to current string thinking. But the same reasoning of course, these bits could by ordering of the strings equially map out higher dimensional objects and the duality would be more naturall as it could be related at pure combinatorics. I think the strings bits they corresponds to something physical.

Is there any string theorists that are working on some serious models of this? Ie that continuum strings are "approximations" or idealisations of a discrete orderded index, which is the real thing?

/Fredrik

 

[mitchell porter]

The most fundamental perspective on string theory that I have is M-theory, understood as "11-dimensional supergravity, plus 2-branes and 5-branes". I think certainly there is something deeper, but that is as deep as I go with any confidence; when I want to put something in context, I take it back to that framework. Maybe there is a combinatorial origin to M-theory, but the specific proposals I've seen are all so weak that they might all be wrong.

The known combinatorial structure with the best claim to illuminate the depths of string theory would surely be the momentum-twistor representation of perturbative N=4 super-Yang-Mills theory, where you have all these amazing identities connecting different scattering processes. Those identities may ultimately have a continuum explanation, in terms of subdivisions of integration regions in twistor space, but the simplicity of the integrals makes me wonder if they are just a highly redundant representation of something combinatorial and perhaps topological, vaguely analogous to the relationship between Grothendieck's http://en.wikipedia.org/wiki/Dessin_d%27enfant" and the corresponding Riemann surface.