An elementary view of duality?

[A. Neumaier]

Are there simple analogues of AdS/CFT duality that are understandable without any knowledge of string theory?

 

[Haelfix]

There are many types of dualities. The simplest ones that are often taught to students is the Massless Sine-Gordon and Massive Thirring model duality.

Also the Montonen–Olive duality between Electric and magnetic charges.

Perhaps you wanted something more sophisticated?

 

[A. Neumaier]

There are many types of dualities. The simplest ones that are often taught to students is the Massless Sine-Gordon and Massive Thirring model duality.

Also the Montonen–Olive duality between Electric and magnetic charges.

Perhaps you wanted something more sophisticated?

I'd like to have something that helps me to get a good idea of what is going on in AdS/CFT duality without having to work with the latter directly.

Ideally something that is just one step simpler so that the analogies to AdS/CFT [which is what I ultimately aim at] can be seen in fair detail (and not only the vague ''duality = two equivalent theories expressed in very different form''). So that working with the simpler model helps me to tackle later the real thing.

 

[Demystifier]

Yes, I would also like to first understand AdS/CFT on a simpler model. But I'm afraid such a model is not known yet.

 

[Ynaught?]

Have you considered the way in which a holographic image emerges from a hologram? This is an article in "plain English" by Juan Maldacena discussing AdS/CFT correspondence and the Holographic Principle. http://www.sns.ias.edu/~malda/sciam-maldacena-3a.pdf

 

[Haelfix]

I'd like to have something that helps me to get a good idea of what is going on in AdS/CFT duality without having to work with the latter directly.

Yea, this doesn't really exist. The closest that I know off would be the derivation of Brown and Henneaux in 2+1 dimensional gravity which is in some sense a precursor of AdS/CFT. It has the advantage that you don't need to know string theory, and motivates the holographic principle. Historically it was more of an odd calculation that struck people as interesting, but it wasn't necessarily appreciated that a duality could be hiding there. This does serve as a decent warmup though!

Brown, J. D.; Henneaux, M. (1986). "Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity". Communications in Mathematical Physics 104 (2): 207–226

 

[Fra]

So that working with the simpler model helps me to tackle later the real thing.

I'v understood from your other posts that you are working on some axiomatisations and formalisations of QM: But from which perspective is AdS/CFT the "real thing"? That sounds strange to me.

As I understand it, AdS/CFT as in the AdS5 × S5 / N=4 YM is merely a specific example of an implementation of some version of the holographic principle. But to me the "real thing" would be to arrive at some in depth, general understanding of what the physical holographic principle really is, and why. This is not not existing. The reason AdS/CFT becomes to discussed is because it's as far as I know one of the few detailed examples.

So I'm not sure to what extent all details, specific to say AdS5 × S5 / N=4 YM is relevant to the core of the new principle we seek.

I don't have any answers but my curiosity likes in trying to elude the core principles underlying statemenets like you admit is vague

"and not only the vague ''duality = two equivalent theories expressed in very different form''"

Interesting question is to understand if there is a physical basis behind the notion of "theory" such as nature beeing one-2-one with theories in some sense. I like to think of two theories as interacting in the sense that the two interacting systems; connected by an communicating interface or commong boundary, where the system encodes the theory physically, and thus the complexity of the theory is constrained by the physical information capacity, which then also constrains the action.

As to what the "real thing" is here, I like how Shannon put things although (he talked about information theory and thermodynamics)

"the essential content of ... does not dot lie in the equations; it lies in the ideas that lead to those equations". I suspect the real thing here would imply that we need a new understanding and new way of thinking of what a theory is in a inferencial theory, beyond our old realist thining of theory.

/Fredrik

 

[A. Neumaier]

I'v understood from your other posts that you are working on some axiomatisations and formalisations of QM: But from which perspective is AdS/CFT the "real thing"? That sounds strange to me.

As I understand it, AdS/CFT as in the AdS5 × S5 / N=4 YM is merely a specific example of an implementation of some version of the holographic principle. But to me the "real thing" would be to arrive at some in depth, general understanding of what the physical holographic principle really is, and why. This is not not existing. The reason AdS/CFT becomes to discussed is because it's as far as I know one of the few detailed examples.

I am interested in AdS/CFT not because of unification (which I think is a still elusive far away goal) but because it can successfully be used to model hadrons, e.g., http://arxiv.org/pdf/0802.0514 , and hence be compared with experiment.

So I want to understand something about its background.

 

[A. Neumaier]

Yea, this doesn't really exist. The closest that I know off would be the derivation of Brown and Henneaux in 2+1 dimensional gravity which is in some sense a precursor of AdS/CFT. It has the advantage that you don't need to know string theory, and motivates the holographic principle. Historically it was more of an odd calculation that struck people as interesting, but it wasn't necessarily appreciated that a duality could be hiding there. This does serve as a decent warmup though!

Brown, J. D.; Henneaux, M. (1986). "Central charges in the canonical realization of asymptotic symmetries: an example from three-dimensional gravity". Communications in Mathematical Physics 104 (2): 207–226

Thanks, I'll look into that.

 

[mitchell porter]

I think it would be conceptually helpful to also study (1) http://www.staff.science.uu.nl/~hooft101/gthpub/planar_diagram_theory.pdf" . The string worldsheet is the continuum limit of Feynman diagrams in the 1/N expansion of the boundary field theory, displaced into the AdS space in a way corresponding to energy scale. At least, that's how it looks to me, at my level of understanding.

 

[FredericGos]

I don't know if this might help you, but I just finished watching some of the latest susskind lectures on string theory. In these he explains a lot about T duality in a very simplified way.
At least I understood most of it :)

http://www.youtube.com/view_play_list?p=202191442DB1B300

 

[A. Neumaier]

Have you considered the way in which a holographic image emerges from a hologram? This is an article in "plain English" by Juan Maldacena discussing AdS/CFT correspondence and the Holographic Principle. http://www.sns.ias.edu/~malda/sciam-maldacena-3a.pdf

This seems to be about holography, not duality.

Or is duality just a name for different descriptions of equivalent theories? If that, I am somewhat surprised -- I had expected something justifying the label ''duality''. What was the reason for choosing this word?

 

[suprised]

Or is duality just a name for different descriptions of equivalent theories? If that, I am somewhat surprised -- I had expected something justifying the label ''duality''. What was the reason for choosing this word?

The word duality is used in a variety of different meanings; it can mean literally equivalent theories, in which case duality is a symmetry, but it can also mean a relationship between different formulations of a theory at different parameter values.

Eg AdC/CFT is not really a duality, but it relates different regimes of one theory (large and small N); so one may prefer to call it a correspondence.

Historically I know at least two origins of the word duality in this sense: electric-magnetic duality, and Kramers–Wannier duality. Roughly, electro-magnetic "S"-duality in N=4 susy gauge theories maps the gauge coupling to its inverse, and at unity value it becomes a symmetry of the theory. Kramers–Wannier duality maps high and low temperature phases of the Ising model into each other and becomes a symmetry of the theory at the phase transition point.

Electric-magnetic duality in itself is related to the mathematical notion of Hodge duality, so there is some justifiable reason to use this word.

 

[Physics Monkey]

I think it's fair to say that duality often refers to two descriptions of the same physical system. There is an informal requirement that these descriptions be "sufficiently different" and address "complementary" aspects of the physical system.

Let me give a few examples (I apologize in advance as I'm sure much of this well known):

The first example of duality I know of occurs in the 2d Ising model. Here the Ising model can be reformulated in terms of non-local "disorder variables". These disorder variables also form an Ising model, but the high temperature phase is exchanged with the low temperature phase. Thus the disorder variables have a finite expectation value in the disordered high temperature phase, hence their name. The critical point is a self dual point.

The 3d Ising model is dual to Z2 gauge theory. This is an interesting duality because one reformulates the original theory in terms of new degrees of freedom which are partially redundant, hence the gauge structure. Here again there is a non-trivial mapping of phases of the theory. As in the 2d case, there is a sense of complementarity with some phases better described by the gauge theory language and others better described in terms of the Ising variables.

Some other examples of note:
a) duality between the XY model and U(1) gauge theory in 3d
b) duality between spin models and Luttinger liquids in 1+1 (and bosonization)

Seiberg duality in gauge theory equates the IR fixed point of two different gauge theories. This duality is a generalization of electromagnetic duality and is a weak strong duality in the sense that the dual theory is weakly coupled if the original theory is strongly coupled. The sense of complementarity here is obvious: when one theory is hard to deal with, the other is simple to work with.

Finally, we come to holographic duality. Gauge theory is already a redundant description of a physical system. In holographic duality we trade a description in terms of gauge fields for another redundant description in terms of gravity and matter fields in a higher dimensional spacetime. This is also a weak strong duality, so the gravity description is only weakly coupled when the gauge description is strongly coupled. This had better be so since perturbative qcd is obviously not equivalent to gravity in a higher dimension. There is usually a large N limit at work which renders both theories classical in a certain sense, but it is believed that this is not essential. Everything one can calculate in the quantum field theory can also be computed in a different way in the gravity description. Or if you like, the gravity theory in asymptotically AdS defines a quantum field theory, and sometimes we have a different name for this quantum field theory.

 

[Physics Monkey]

Are there simple analogues of AdS/CFT duality that are understandable without any knowledge of string theory?

In addition to my last long post, I would also point out that one doesn't really need string theory to understand AdS/CFT.

All the basic computations can be carried out with limited reference to string theory. I give a few examples below for the dictionary at large N.

Examples:
1) Two point correlation functions in the qft are obtained from solving wave equations in the bulk. Operator dimensions are related to masses in the gravity theory. A simple limit of this procedure requires one to evaluate something like e^{- m s} with s a geodesic length to find correlation functions.

2) Expectation values of Wilson lines can be computed by studying open string world sheets that terminate on the wilson line in the boundary. Again in a classical limit one may just need to minimize the area of this worldsheet. Depending on the background geometry one may find confinement or a Coulomb phase etc.

3) Thermodynamic entropy can be computed from the areas of black holes in the bulk.

4) Entanglement entropy in the field theory appears to be related to minimal surfaces in the bulk.

5) Higher point correlation functions in the qft come from including interactions between fields in the bulk.

Another way to see the appearance of AdS in the gauge theory is to ask about the moduli space of instantons. In the conformal \mathcal{N} = 4 theory, the moduli space of an instanton on S^4 is AdS. The extra "radial" coordinate records the size of the instanton. This is one of the reasons why the radial coordinate is identified with energy scale in the field theory.

Of course, this is only the tip of the iceberg.

 

[A. Neumaier]

I think it's fair to say that duality often refers to two descriptions of the same physical system. There is an informal requirement that these descriptions be "sufficiently different" and address "complementary" aspects of the physical system.

Let me give a few examples (I apologize in advance as I'm sure much of this well known):

Maybe to the insider, but for me it is just the right level of explanation.

In addition to my last long post, I would also point out that one doesn't really need string theory to understand AdS/CFT.

All the basic computations can be carried out with limited reference to string theory. I give a few examples below for the dictionary at large N. [...]

Of course, this is only the tip of the iceberg.

Thanks. I need some time even to digest the tip...

One question: Holography apparently just adds one dimension, if I understood it right.

AdS/CFT is apparently used to get info on 3+1D QCD by embedding it into the boundary of a 4+1D (or 3+2D?) AdS.

And the Maldacena paper was about reducing 3+1D gravity to a 2+1D boundary field theory, if I got it right.

But I thought string theory is supposed to live in 10 or 11 dimensions. So how do these
items connect?

 

[suprised]

But I thought string theory is supposed to live in 10 or 11 dimensions. So how do these
items connect?

There is an extra S^5 in the game...

I don't think you can concretely access this duality, which involves quantum gravity in an essential way, without making your hands dirty with quantum gravity! Which means strings; though not for all questions you need a detailed knowledge of strings.

 

[marcus]

This discusses the correspondence in low dimension & elementary terms

http://arxiv.org/abs/0804.0632
Reconstructing AdS/CFT
Laurent Freidel
(Submitted on 4 Apr 2008)
In this note we clarify the dictionary between pure quantum gravity on the bulk in the presence of a cosmological constant and a CFT on the boundary. We show for instance that there is a general correspondence between quantum gravity "radial states'' and a pair of CFT's. Restricting to one CFT is argued to correspond to states possessing an asymptotic infinity. This point of view allows us to address the problem of reconstructing the bulk from the boundary. And in the second part of this paper we present an explicit formula which gives, from the partition function of any 2 dimensional conformal field theory, a wave functional solution to the 3-dimensional Wheeler-DeWitt equation. This establishes at the quantum level a precise dictionary between 2d CFT and pure gravity.
34 pages

Does anyone wish comment on the statement that in general the correspondence is to a pair of fields?

I'd be curious to know of some other papers which successfully address the problem of explicitly reconstructing the bulk state?

 

[JollyJoker]

Does anyone know a simple case where an extra dimension emerges? I'm thinking about something like how sound is basically just air pressure over time, with frequency giving you a third dimension that is convenient in some cases and adds odd properties like an uncertainty principle.

 

[atyy]

Does anyone know a simple case where an extra dimension emerges? I'm thinking about something like how sound is basically just air pressure over time, with frequency giving you a third dimension that is convenient in some cases and adds odd properties like an uncertainty principle.

Apart from AdS/CFT, other cases where an extra dimension emerges are:

1) 3D topological field theory is dual to a 2D CFT
http://arxiv.org/abs/0707.1889

2) a computational scheme in lattice QCD to simulate chiral fermions
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.113.4689&rep=rep1&type=pdf, http://latticeqcd.blogspot.com/2006/02/exactly-chiral-fermions.html

3) a computational scheme in condensed matter called MERA
http://pirsa.org/10110076, http://arxiv.org/abs/1106.1082

Links between 1 and AdS/CFT in the specific case of 2+1D bulk is discussed in http://arxiv.org/abs/hep-th/0403225

A link between 3 and AdS/CFT is conjectured in http://arxiv.org/abs/0905.1317

3 and 1 are tightly linked http://arxiv.org/abs/0712.0348.

 

[suprised]

Does anyone know a simple case where an extra dimension emerges? I'm thinking about something like how sound is basically just air pressure over time, with frequency giving you a third dimension that is convenient in some cases and adds odd properties like an uncertainty principle.

The dualities here under question are more complicated than just adding a formal dimension. The point is that "miraculously" a higher dimensional Lorentz invariance is generated. Another example is the strong coupling limit of 10d type IIA strings, which generates an 11 dimensional Lorentz symmetry.

 

[A. Neumaier]

I think it would be conceptually helpful to also study (1) http://www.staff.science.uu.nl/~hooft101/gthpub/planar_diagram_theory.pdf"

Isn't the 1/N limit a classical theory? So where is the dual quantum theory?

 

[mitchell porter]

Isn't the 1/N limit a classical theory? So where is the dual quantum theory?

The 1/N expansion is a way of organizing the Feynman diagrams, according to the genus of the minimal surface capable of containing them. The planar limit is where you only consider planar diagrams. But the other diagrams exist and correspond to multiloop amplitudes in the string theory (loops in the string theory = handles on the string worldsheet = genus of the worldsheet-CFT Riemann surface = genus of the summed diagrams in the gauge theory).